METHODS TOWARD LARGE-SCALE AND HIGHLY- ACCURATE QUANTUM MECHANICS SIMULATIONS

Transcription

1 METHODS TOWARD LARGE-SCALE AND HIGHLY- ACCURATE QUANTUM MECHANICS SIMULATIONS OF MATERIALS CHEN HUANG A DISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS ADVISER: EMILY A. CARTER SEPTEMBER 011

2 Copyight by Chen Huang 011. All ights eseved.

3 Abstact Computational simulation has become an impotant tool in mateial science fo both undestanding expeiment esults and pedicting unknown mateial popeties. In this thesis we focus on two basic poblems: how to pefom fast quantum mechanics simulations fo lage scale featues in mateials and how to apply highlyaccuate quantum mechanics methods to mateials while keeping the computational cost affodable. Fo the fist poblem we wok on the obital-fee density functional theoy (OF- DFT). Unlike any othe obital-based methods in OF-DFT the total enegy is fomulated solely based on electon density which makes OF-DFT a pomising linea-scaling method fo mateial simulations. The most challenging pat in OF-DFT is to constuct a good appoximation to the ohn-sham kinetic enegy density functional (EDF). A second challenge is to accuately descibe the electon-nuclea inteaction. In this thesis we intoduce a new EDF fo semiconductos based on the low-q limit of the esponse function in semiconductos. We also discuss attempts to fomulate EDFs fo teating tansition metals and phase changes in semiconductos. We also descibe an efficient means of constucting accuate local electon-ion pseudopotentials. To tackle the second poblem we wok on the density-based embedding theoy in which the mateial is divided into a local egion of inteest (a cluste) and the est of the mateial (the envionment). The cluste is teated with highly-accuate quantum mechanics methods and the envionment is eplaced with an embedding potential. We fist intoduce a constaint to emove the non-uniqueness in conventional embedding potentials and show how one can eliminate use of appoximate EDF potentials that wee employed in pevious density-based embedding theoies. Then we popose a unified potential-functional embedding theoy which couples the cluste and its envionment in a seamless and self- iii

4 consistent way. Studies on lage systems in which detailed electonic stuctues of local egion ae equied ae made possible with this novel embedding theoy fo example the excitation and polaization of molecules on sufaces which is elated to photocatalysis. Lastly we intoduce an efficient diect minimization method fo calculating an optimized effective potential (OEP). An OEP is equied fo solving the S equations when obital-dependent exchange-coelation functionals (ODXCFs) ae employed. Compaed with othe widely-used electon-density-based XCFs ODXCFs have many stiking advantages such as being fee of self-inteaction eo and epoducing the discontinuity of XC potential duing chage tansfe. Howeve the taditional integal equations fo calculating an OEP ae cumbesome to solve. With ou efficient diect minimization method it is now possible to apply S-DFT-ODXCFs calculations to many impotant poblems such as spin canting in molecules local toque on spins in spin devices electic esponse in molecules etc. iv

5 Wok pesented in this thesis has been published o submitted as follows: C. Huang and E. A. Cate Optimized effective potential: diect minimization made simple submitted to Physical Review Lettes. C. Huang and E. A. Cate Potential-functional embedding theoy fo molecules and mateials submitted to The Jounal of Chemical Physics. C. Huang M. Pavone and E. A. Cate "Quantum mechanical embedding theoy based on a unique embedding potential" The Jounal of Chemical Physics (011). C. Huang and E. A. Cate "Nonlocal obital-fee kinetic enegy density functional fo semiconductos" Physical Review B (010). (Edito s suggestion) C. Huang and E. A. Cate "Tansfeable local pseudopotentials fo magnesium aluminum and silicon" Physical Chemisty Chemical Physics (008). Confeences: Invited speake inetic enegy density functionals fo and lage scale applications of obital-fee density functional theoy the Intenational Chemical Congess of Pacific Basin Societies Hawaii USA (010) Poste Nonlocal kinetic enegy density functional fo semiconductos the Intenational Wokshop on "Fonties in Density Functional Theoy" Montauk NY USA (009) v

6 Table of Contents Abstact... iii Acknowledgements... ix Chapte I Intoduction... 1 Chapte II Tansfeable bulk-deived local pseudopotentials Intoduction Inveting the S equations Building a BLPS Calculation details Results....6 Conclusions Acknowledgements Chapte III Nonlocal obital-fee kinetic enegy density functional fo semiconductos Intoduction inetic enegy density functional fom Numeical implementations Results and discussions Local pseudopotentials fo Ga In P As and Sb Conclusion Acknowledgements Chapte IV Extending OF-DFT fo tansition metals and semiconductos Intoduction OF-DFT fo tansition metals Towads a univesal EDF fo semiconductos Conclusions Acknowledgement Chapte V Quantum mechanical embedding theoy based on a unique embedding potential Intoduction vi

7 5. Theoy Numeical implementation Results and discussions Self-consistent embedding scheme Genealization to spin-polaized quantum systems Connection to patition density functional theoy Implementation in the Pojecto Augmented Wave (PAW) method Conclusions Acknowledgements Chapte VI Potential-functional embedding theoy fo molecules and mateials Intoduction Theoy Numeical Details Results and Discussion Conclusions Acknowledgements... 1 Chapte VII Optimized effective potential: diect minimization made simple Intoduction Theoy Numeical methods Results and discussions Extension to the cuent-spin-density-functional theoy Conclusion Acknowledgements... 7 Chapte VIII Conclusions and Outlook Conclusions Outlook Appendix A Inveting the ohn-sham equations: codes and use manual Appendix B PROFESS code fo pefoming Huang-Cate EDF calculations vii

8 Appendix C OF-DFT fo semiconductos using a λ-field EDF: code and use manual Appendix D OF-DFT towad tansition metals Appendix E Density-based embedding potential builde and use with quantum chemisty: codes and use manual Appendix F Potential-based embedding theoy: code and how to use Appendix G Diect minimization of OEP: code and use manual Bibliogaphy viii

9 Acknowledgements I am tuly gateful to my adviso Pofesso Emily Ann Cate fo leading me into this computational mateials field. I eally appeciate he kindness fo having me in he goup when I finally ealized that I was inteested in computational physics at such a late stage the winte of my thid yea at Pinceton. He insight pesistence and stong cuiosity in science influenced me vey much in my fou-yea life in the goup. I eally appeciate he advice suppot and encouagement wheneve I was facing difficulties in my eseach. I am also vey gateful to Pofesso David Huse fo being my co-adviso. It is a geat pleasue fo me to discuss my eseach and pogess with him duing these yeas. I am thankful to him fo much advice on my academic caee. I am also vey thankful to my pevious colleagues D. Geg Ho D. Vincent Lignèes and D. Linda Hung fo many valuable discussions on OF-DFT. I benefited a lot fom thei temendous help in my OF-DFT pojects. Without thei contibution to the PROFESS code the OF-DFT wok pesented hee would not have been possible. Discussions with Ilgyou Shin on vaious OF-DFT eseach topics ae also vey intiguing. I also want to thank D. Michele Pavone D. Tsz S. Chwee Peilin Liao D. Saha Shaifzadeh David isiloff Ting Tan and D. Patick Huang fo teaching me quantum chemisty and many othe things that I knew nothing about befoe joining the goup. I am thankful to D. Youqi e fo many helpful discussions on vaious aspects of physics especially electonic tanspot. I want to thank Andew Ritzmann fo teaching me how to geneate eye-catching plots with the Tecplot softwae. I want to thank Pof. Ashwin Ramasubamaniam and Junchao Xia fo being such good officemates. I leaned mechanics fom Ashwin. Many discussions on a wide ange of science with Ashwin wee vey inteesting and memoable. I also benefited ix

10 a lot fom the discussions with Junchao ove these basic concepts in physics fom statistical mechanics to quantum mechanics. I also want to thank D. Floian Libisch fo his citical eading and comments on the dafts of the potential-functional embedding wok and the optimized effective potential wok and the appendices in this thesis. Of couse it was my geat pleasue to wok with all of you in the Cate goup. They ae D. Ivan Milas D. Donald Johnson D. Nicholas Mosey D. isten Maino D. Maytal Tooke D. Doon Naveh D. John eith D. Ana Munoz Nima Alidoust Dalal. anan Leah Isseoff Victo Oyeyemi Jin Cheng D. Philip Casey and Sevatsan Mualidhaan. I am vey lucky to have been involved in such a geat goup! I want to eseve my special thanks to Pof. Robet H. Austin Pof. Nai Phuan Ong Pof. Heschel A. Rabitz and Pof. Antoine ahn. Thank you fo giving me the oppotunity to ty out expeimental wok. I am pofoundly gateful to my dea wife Wenjing Qiu. Without he love it would have been impossible fo me to devote myself completely into my studies. She always encouaged and suppoted me at these most difficult moments duing these yeas. Finally I am pofoundly gateful to my fathe Ting Huang and my mothe LiLi ong. They did thei best to aise me and povided me with a good education. They taught me to be a peson of diligence. I could not have accomplished this thesis wok without thei encouagement all though these yeas. Anytime no matte what happened to my eseach caee o life they ae always thee being my suppot. x

11 Chapte I Intoduction Compute simulation is an inceasingly impotant tool fo pedicting mateial popeties poviding the undelying ationale fo mateials behavio as well as helping to intepet measuements. 1 The most geneal and accuate methods available fo simulating mateials ae those based on quantum mechanics. To apply quantum mechanics to mateial simulations nowadays we ae facing seveal geat challenges. Among those challenges one is how to appoximately solve the manybody Schödinge equation in an efficient way in ode to pefom lage-scale mateials simulations. Such simulations ae equied fo studying those featues and phenomena that happen at the length scale of nanometes o lage such as dislocation movement in solids potein folding dynamics etc. At pesent the most common quantum mechanics method fo studying mateials is ohn-sham density functional theoy (S-DFT) which is based on the Hohenbeg and ohn theoems. 3 S-DFT implementations geneally exhibit a good balance between accuacy and computational cost. Howeve fo lage scale mateial simulations e.g. thousands of atoms o moe the computational cost of conventional S-DFT becomes pohibitive. This is patially due to S-DFT s use of one-electon wavefunctions (ohn-sham obitals) which leads to 3 N degees of feedom with N the numbe of electons. Anothe challenge is how to study a local egion of inteest in a mateial with highly-accuate quantum mechanics methods while still taking into consideation the effect of the extended condensed matte envionment. Accuate local electonic stuctue is of geat impotance fo undestanding pocesses in mateials. Fo example an accuate desciption of the electonic stuctue at the intefaces between cabon monoxide (CO) molecule and tansition metal sufaces ae citical 1

12 fo undestanding of CO edox catalysis yet standad implementations of S-DFT cannot even be counted on to obtain the ight binding site fo CO on a metal suface. Unfotunately many cases exist whee conventional S-DFT fail to povide the accuacy equied: van de Waals (vdw) inteactions stong coelation excited states chage tansfe open-shell multiplets adsoption of molecules on metal sufaces etc. Fo such phenomena sophisticated coelated wavefunction (CW) theoy fo example the configuation inteaction method geneally povides supeio accuacy. Howeve CW methods ae too computationally expensive to be applied to moe than ~50 atoms even when fast educed scaling algoithms ae used. Theefoe we need a seamless scheme in which the egion of inteest is solved with highly-accuate quantum mechanics methods and the est of the mateial is solved with low-level quantum mechanics methods in ode to keep the computational cost affodable. In this thesis I have advanced obital-fee density functional theoy (OF-DFT) 4 5 which is a pomising method fo pefoming fast lage-scale mateials simulations. In OF-DFT the total enegy functional is fomulated only based on the electon density instead of the S obitals. The key in OF-DFT is to eplace the oiginal S kinetic enegy with a kinetic enegy density functional (EDF) which solely depends on electon density. Consequently OF-DFT educes the degees of feedom in the computation fom 3 N to 3. Howeve pevious wok has demonstated that OF-DFT is only capable of S-DFT-level accuacy fo nealy-fee-electon-like main goup metals with only maginal pogess made in teating othe types of mateials. In this thesis I have poposed a novel EDF that is designed to teat semiconductos to some extent. To tackle the second challenge mentioned above I have developed an advanced density-based embedding theoy 6 to seamlessly embed accuate electonic stuctue calculations of the local egion of inteest inside mateials. In densitybased embedding theoy the whole system is patitioned into a cluste (the egion

13 of inteest) and its envionment (the est of the mateial). The idea of density-based embedding theoy is to eplace the envionment with an embedding potential. Theefoe highly-accuate quantum mechanics methods can be applied to the cluste in the pesence of the embedding potential. The density-based embedding theoy was poposed moe than two decades ago. Howeve the theoy still suffeed fom seveal difficulties including how to geneate accuate and unique embedding potentials fom fist pinciples and how to pefom tuly self-consistent embedding calculations. To solve the fist difficulty I have emoved this non-uniqueness by constaining the embedding potential to be the same fo both the cluste and the envionment effectively endeing the embedding potential to be an inteaction potential between subsystems. I also intoduced an optimized-effective potential (OEP) method 7 by efomulating the pocedue of solving fo the embedding potential into an optimization poblem which geatly inceases the numeical efficiency of the embedding theoy and eliminates the use of appoximate EDF potentials that had been used in all pevious density-based embedding theoies. Futhemoe it is the fist time that the density-based embedding theoy is extended to magnetic mateials. To solve the second difficulty I intoduced a potential-functional embedding theoy that allows fully self-consistent embedding calculations to be ealized in which each subsystem in pinciple can be teated with diffeent quantum mechanics methods as needed. Lastly I focus on how to pefom efficient S-DFT calculations employing obital-dependent exchange-coelation functionals (ODXCFs). To wok with ODXCFs in the past we have to solve these numeically challenging OEP integal equations. In this thesis I showed how to pefom OEP calculations efficiently with a novel diect minimization method. With this efficient diect minimization eliable S-DFT calculations employing ODXCFs fo many exciting applications such as spin canting in magnetic molecules and spin-dynamics in spin devices become possible. This thesis is oganized as follows. 3

14 In Chapte II we build tansfeable local pseudopotentials (LPSs) fo the electon-ion inteaction by inveting the S equations fo bulk valence electon densities deived fom nonlocal PS (NLPS) S-DFT calculations to obtain an atomcenteed LPS. We build LPSs fo the elements Mg Al and Si and then test them in S-DFT calculations of static bulk popeties fo seveal Mg Al and Si bulk stuctues as well as β -Al3Mg. Ou Mg Al and Si LPSs poduce coect gound state popeties and phase odeings. These LPSs ae then tested in S-DFT calculations of suface enegies fo seveal low-index Mg and Al sufaces point defect popeties in hexagonal-close-packed (hcp) Mg face-centeed cubic (fcc) Al and cubic diamond Si and stacking fault enegies in fcc Al. All of these LPS esults agee quantitatively with the esults fom NLPSs with eos less than o equal to 40 mev pe atom. Finally we pefom OF-DFT calculations fo vaious Mg and Al stuctues. The OF- DFT esults geneally agee well with the coesponding S-DFT esults. The main pat of this chapte is published as C. Huang and E. A. Cate "Tansfeable local pseudopotentials fo magnesium aluminum and silicon" Physical Chemisty and Chemical Physics (008). In Chapte III we popose a nonlocal EDF fo semiconductos based on the expected asymptotic behavio at the low-q limit of its susceptibility function. The EDF s kenel depends on both the electon density and the educed density gadient with an intenal paamete fomally elated to the mateial s static dielectic constant. We detemine the accuacy of the EDF within OF-DFT by applying it to a vaiety of common semiconductos. With only two adjustable paametes the EDF epoduces quite well the S-DFT pedictions fo bulk moduli equilibium volumes and equilibium enegies. The two paametes in ou EDF ae sensitive pimaily to changes in the local cystal stuctue such as atomic coodination numbe and exhibit good tansfeability between diffeent tetahedally-bonded phases. This local cystal stuctue dependence is ationalized by consideing Thomas-Femi dielectic sceening theoy. The main pat of this chapte is published as C. Huang and E. A. Cate "Nonlocal obital-fee kinetic 4

15 enegy density functional fo semiconductos" Physical Review B (010) (Edito s suggestion). In Chapte IV we extend OF-DFT to teat tansition metals and phase changes in semiconductos. Fo tansition metals fist we discuss how to build a physical local pseudopotential fo tansition metals. Then we show how to apply a EDF to tansition metals whose d electons ae faily localized. To achieve this we decompose the valence electon density into two pats: localized and delocalized pats. Afte the decomposition we apply the Wang-Govind-Cate 1999 EDF fo the delocalized pat. The localized pat is teated with a linea combination of the Thomas-Femi EDF and von Weizsäcke (vw) EDF. Using this technique basic bulk popeties fo seveal silve phases can be epoduced. To extend OF-DFT to coectly teat shea-elated defomations in semiconductos and thei phase changes I discuss how to update on-the-fly the EDF λ paamete intoduced fo the nonlocal EDF in Chapte III as a function of local coodination numbe. By intoducing bond-bending enegies I demonstate that OF-DFT is able to give basic bulk popeties of seveal silicon phases including shea-elated elastic constants of cubic diamond (CD) silicon and vacancy and self-intestitial point defect fomation enegies in CD silicon in a easonable ageement with the S-DFT. In Chapte V we emove the non-uniqueness of the embedding potential that exists in most pevious quantum mechanical embedding schemes by letting the envionment and embedded egion shae a common embedding (inteaction) potential. To efficiently solve fo the embedding potential we intoduce an OEP method. This embedding potential is then used to descibe the envionment while a coelated wavefunction (CW) teatment of the embedded egion is employed. We fist demonstate the accuacy of this new embedded CW (ECW) method by calculating the van de Waals binding enegy cuve between a hydogen molecule and a hydogen chain. We then examine the pototypical adsoption of CO on a metal suface hee the Cu(111) suface. In addition to obtaining coect site odeing (top site most stable) and binding enegies within this theoy the ECW 5

16 exhibits damatic changes in the p-chaacte of the CO 4σ and 5σ obitals upon adsoption that agee vey well with X-ay emission specta poviding futhe validation of the theoy. Finally we genealize ou embedding theoy to spinpolaized quantum systems and discuss the connection between ou theoy and patition density functional theoy. The main pat of this chapte is published as C. Huang M. Pavone and E. A. Cate "Quantum mechanical embedding theoy based on a unique embedding potential" The Jounal of Chemical Physics (011). In Chapte VI we intoduce a potential-functional embedding theoy by efomulating ou ecently poposed density-based embedding theoy in tems of functionals of the embedding potential. This potential-functional based theoy completes the dual poblem in the context of embedding theoy fo which densityfunctional embedding theoy has existed fo two decades. With this potentialfunctional fomalism it is staightfowad to solve fo the unique embedding potential shaed by all subsystems. We conside chage tansfe between subsystems and discuss how to teat factional numbes of electons in subsystems. We show that one is able to employ diffeent enegy functionals fo diffeent subsystems in ode to teat diffeent egions with theoies of diffeent levels of accuacy if desied. The embedding potential is solved fo by diectly minimizing the total enegy functional and we discuss how to efficiently calculate the gadient of the total enegy functional with espect to the embedding potential. Foces ae also deived theeby making it possible to optimize stuctues and account fo nuclea dynamics. We also extend the theoy to spin-polaized cases. Numeical examples of the theoy ae given fo some homo- and heteo-nuclea diatomic molecules and a moe complicated test of a six-hydogen-atom chain. We also test ou theoy in a peiodic bulk envionment with calculations of basic popeties of bulk NaCl by teating each atom as a subsystem. Finally we demonstate the theoy fo molecula adsoption on insulating and metallic sufaces. A publication based on this wok has been submitted to The Jounal of Chemical Physics. 6

17 In Chapte VII we intoduce an efficient diect minimization method fo calculating the optimized effective potential (OEP). To avoid solving eithe the difficult-to-solve optimized effective potential (OEP) integal equation o the singula diffeential equation of obital shifts we intoduce an efficient diect minimization scheme fo pefoming OEP calculations within ohn-sham density functional theoy (S-DFT). Diect minimization within S-DFT is a geneal and pomising appoach howeve in pactice its application has been stongly hindeed by the damatically inceasing computational cost as systems become lage. In contast to the peviously poposed diect minimization scheme based on petubation theoy ou method is fomulated based on efficient finite diffeences and only equies the obitals involved in the constuction of the S exchangecoelation functionals. We demonstate ou scheme by pefoming exact-exchange OEP fo sodium clustes in which only occupied S obitals ae needed to obtain the OEP. Ou efficient diect minimization scheme should aid futue development of obital-dependent density functionals and ende OEP to be a pactical choice fo vaious applications. A publication based on this wok has been submitted to Physical Review Lettes. In Chapte VIII I daw conclusions and give an outlook to futue eseach based on the esults and methods developed in this thesis. In Appendix A I document the details of the code fo inveting the S equations fo a given electon density and use manual fo poducing local pseudopotentials. In Appendix B I explain how the PROFESS CVS was updated to include the Huang- Cate 010 EDF and the spin-polaized OF-DFT (the latte witten by Junchao Xia). In Appendix C I document the details of my code fo using the Huang-Cate 010 EDF fo semiconductos and povide the coesponding use manual. In Appendix D I illustate how to pefom OF-DFT towad tansition metals. In Appendix E I give the details of the code fo geneating the embedding potential fo density-based embedding theoy and povide the use manual. In Appendix F I document the details of the code fo pefoming potential-functional embedding theoy 7

18 calculations and its coesponding use manual. In Appendix G I give the details of the code fo pefoming efficient diect minimization fo the OEP poblem and povide its use manual. 8

19 Chapte II Tansfeable bulk-deived local pseudopotentials.1 Intoduction Among vaious fist-pinciples quantum mechanics methods ohn Sham density functional theoy (S-DFT) which is based on the Hohenbeg ohn (H) theoems 3 stikes a good balance between accuacy and computational cost. Unfotunately the cost of S-DFT typically scales oughly cubically with system size although methods that scale linealy fo nonmetals ae now available. 8 The situation is wose fo metals since the conventional cubic scaling can be accompanied by a lage pefacto due to the need fo dense Billouin zone sampling. Consequently it is impactical at pesent to use S-DFT to simulate metals with thousands of atoms. On the othe hand such lage samples can be studied eadily with obital-fee density functional theoy (OF-DFT) 5 which is also based on the H theoems but equies much less computational time with a cost that can be made to scale linealy with system size. OF-DFT simulations of thousands of sodium atoms wee demonstated aleady a decade ago. 9 Although OF-DFT is impessively efficient computationally it is difficult to obtain the accuacy of S-DFT within OF-DFT. The lack of obitals in OF-DFT endes two tems in the total enegy quite challenging to epesent accuately: the kinetic enegy and the ion-electon potential enegy (by ion we mean the nucleus plus coe electons which ae typically epesented togethe as a pseudopotential acting on the valence electons). S-DFT intoduces obitals which povides the 9

20 means to evaluate the electon kinetic enegy in tems of the exact kinetic enegy of non-inteacting electons (the coection to this kinetic enegy fo inteacting electons is subsumed into the exchange coelation functional). By contast OF- DFT expesses the kinetic enegy only in tems of the electon density in a kinetic enegy density functional (EDF). Recent advances in EDF theoy include the development of nonlocal EDFs that epoduce the linea esponse of a unifom electon gas Fo example OF-DFT with the Wang Govind Cate (WGC) EDF 13 epoduces S kinetic enegies of simple main goup metals vey well. Consequently popeties of bulk aluminum magnesium and an Al Mg alloy can be pedicted by OF-DFT almost as accuately as by S-DFT Howeve fo othe types of mateials an accuate EDF emains to be developed limiting the scope of what OF-DFT can cuently be used to study. Besides the need fo an accuate EDF in OF-DFT accuate and tansfeable electon-ion local pseudopotentials (LPSs) ae also equied to pefom a meaningful OF-DFT calculation. Unlike S-DFT in which nom-conseving 17 and ulta-soft nonlocal pseudopotentials 18 (NLPSs) and the elated nonlocal pojecto augmented-wave method 19 have been widely used cuently only LPSs can be used in OF-DFT as no S obitals exist in OF-DFT onto which diffeent angula momentum-dependent potentials can be pojected. Many schemes have been poposed fo LPS constuction. One appoach is to design an analytic fom fo the LPS 0-5 fo which the paametes ae optimized to fit to expeimental data o some othe constaints. Anothe stategy is a diect numeical method. Hee the valence electon density is fist geneated fom a S-DFT calculation using a NLPS 6-8 o is geneated by defining the valence density pofile manually. 9 A fist attempt in this diection inveted the obital-fee H equation diectly employing a EDF and the S-NLPS valence electon density to deive a LPS. A elated OF method to build a LPS fo lithium involved embedding a lithium ion into an electon gas. 31 The accuacy of the latte two methods is limited by the quality of the EDF used. An altenative pocedue is to invet the S-DFT equations instead of the H 10

21 equation so as to utilize the exact non-inteacting kinetic enegy opeato. A local S effective potential is then deived that is able to epoduce the valence electon density. Then this potential is unsceened by subtacting off the electon electon Coulomb epulsion and exchange coelation potentials to obtain the ionic extenal potential. If the valence electon density is geneated fom a single atom then the ionic potential is aleady the LPS fo that atom. 7 9 If the density is geneated in a bulk cystalline envionment then the bulk cystal s stuctue facto is needed to futhe convet the ionic extenal potential into an atom-centeed ionic potential which is the LPS of that atom. The method pesented in this chapte is of the second type descibed above and is based on the pevious wok of Zhou et al. 8 They efeed to this kind of LPS as a bulk-deived local pseudopotential (BLPS) and demonstated fo silicon that the Si BLPS was moe tansfeable than the Si LPS built fom the valence electon density of a Si atom. Howeve Zhou et al. s Si BLPS still has two emaining deficiencies: (1) it gives an incoect enegy odeing fo seveal Si bulk stuctues elative to pedictions fom a NLPS (the latte is pesumed to be moe accuate) and () it equies a lage kinetic enegy cutoff to convege the planewave basis set. To addess the above two poblems we outline two impovements on the method of Zhou et al. both of which involve how the tails of BLPS ae handled in eal and ecipocal spaces. The new Si BLPS obtained with ou impoved scheme is able to give the coect enegy odeing fo all Si bulk stuctues studied in the wok of Zhou et al. and is a much softe LPS allowing convegence of the planewave basis set at much smalle kinetic enegy cutoffs. The same appoach is also applied to build Mg and Al BLPSs. This chapte is oganized as follows. Fist we outline how the S equations ae inveted a key step in building BLPSs. Then we pesent a detailed pocedue fo building a BLPS with numeical details given aftewads. The esulting Mg Al and Si BLPSs ae then pesented and analyzed. The pefomance of ou BLPSs is fist compaed within S-DFT to esults fom NLPSs. Last we compae esults of OF-DFT 11

22 calculations using ou BLPSs and the WGC EDF to S-DFT fo vaious Mg and Al systems. Inveting the S equations Ou fist task is to seek a local extenal potential that can epoduce a given electon density. Accoding to the fist H theoem 3 thee exists a unique local extenal potential associated with a given v-epesentable electon density. To obtain this local extenal potential we fist invet the S equations to extact a local S effective potential able to epoduce the taget electon density. Many ways have been poposed to invet the S equations. One is an iteative method 3 which was used by Zhou et al. in building Si 8 and Ag 6 BLPSs. Hee we use a diffeent technique: the diect optimization method developed by Wu and Yang. 7 A bief intoduction is given below. In the Wu Yang method a functional is defined as W 3 ( ) ( )] T [ V ] d [ ( ) ( )] V ( ) (.1) [ Veff 0 S eff 0 eff whee 0 ( ) is the taget electon density. With a tial local S effective potential () we solve the S equations to get the total S kinetic enegy T S and the V eff electon density ( ). Then we inset these T S ( ) back into Eq. (.1) to evaluate the functional W [ V ( ) 0 ( )]. The optimal Veff ( ) will maximize the functional eff W[ V eff ( ) 0( )] which in tun will be the Veff () that epoduces the taget electon density 0 ( ). Specifically we expand V eff () in a basis set gt () as V eff ( ) bt gt ( ). t The gadient of W with espect to b t is 1

23 W b t 3 d ( ) ( )] g ( ). [ 0 t We have implemented this diect optimization method in the ABINIT code. 33 In the code () is expanded in a plane wave basis set and the coefficients ae then V eff optimized to maximize the functional W using a conjugate gadient optimization ~ code. We note that the Fouie component of V eff () at q=0 denoted as V eff ( q 0) can be set to any value which just shifts the entie () by a constant. Because the ~ shifted V eff () yields the same electon density we manually set V eff ( q 0) to zeo duing the optimization..3 Building a BLPS V eff The fist step in building a BLPS is to geneate the taget bulk electon densities used duing inveting the S equations with the Wu Yang method. We select the face-centeed cubic (fcc) body-centeed cubic (bcc) simple cubic (sc) and diamond (dia) stuctues in ode to geneate a wide ange of bulk electon densities. These fou paticula stuctues span a ange of coodination numbes fom 1 fo fcc down to diamond s 4 with sc and bcc exhibiting intemediate coodination numbes of 6 and 8 espectively. By geneating densities ove this ange of coodination we anticipate that ou BLPSs should be able to wok well in both close-packed and moe open bulk envionments. The taget bulk electon densities ae obtained by solving the S-DFT equations using NLPSs with bulk cystal stuctues in which both cell vectos and ion positions have been optimized. Numeical details of these calculations can be found in the next section. We then invet the S equations using these taget bulk electon densities to obtain the local S effective potentials () fo these bulk stuctues using the Wu Yang method descibed above. Since we need to solve the S equations again fo evey tial local S effective potential in the diect optimization method the settings of k-point meshes Femi Diac smeaing 13 V eff

24 widths and planewave kinetic enegy cutoff values used ae the same as those used in geneating the taget bulk electon densities (see Table.1). Afte inveting the S equations we unsceen the local S effective potential by subtacting off the electon electon Coulomb epulsion and exchange coelation potentials to extact out the bulk ionic potential V bulk ( ) V eff J[ ] E xc[ ] ( ) ( ) ( ). We then Fouie tansfom V bulk () to obtain the atom-centeed ionic potential ~ V atom ( q) in ecipocal space by dividing by the stuctue facto S(q) of each bulk cystal ~ V atom ~ bulk V ( q) ( q) (.) S( q) ~ whee q q To avoid the singulaity of V atom ( q) at q 0 we wok with the non- ~ Coulomb pat of V atom ( q) ~ V nc ~ atom 4Z ( q) V ( q) (.3) q whee Z is the pseudochage (nuclea chage minus coe electon chage). Thus Z is fo Mg 3 fo Al and 4 fo Si..4 Calculation details Peiodic S-DFT calculations to obtain the bulk densities fo BLPS constuction static bulk popeties fo Mg Al and Si suface enegies fo Mg and Al stacking fault enegies fo fcc Al and defect fomation enegies in hcp Mg and fcc Al ae pefomed using the ABINIT planewave DFT code. 33 Peiodic OF-DFT calculations ae done 14

25 with the FORTRAN90-based code PROFESS (Pinceton Obital-Fee Electonic Stuctue Softwae) 34 developed in ou goup. The OF-DFT code also uses a planewave basis set. In all S-DFT and OF-DFT calculations we use the local density appoximation (LDA) fo electon exchange and coelation deived fom the quantum Monte Calo esults of Cepeley and Alde 35 as paameteized by Pedew and Zunge. 36 Calculations with the ABINIT code ae done using Toullie Matins (TM) NLPSs 37 geneated with the FHI98PP code. 38 We use the default coe cutoff adii in the FHI98PP code fo Mg and Si. The Mg TM-NLPS has coe cutoff adii of and.476 boh fo s- p- d- and f- channels espectively. The Si TM-NLPS has coe cutoff adii of and.0 boh fo s- p- and d- angula momentum channels espectively the same settings as in the wok of Zhou et al. 8 Fo the Al TM-NLPS we set the coe cutoff adii to be.1 boh fo all angula momentum channels (the default coe cutoff adii ae and.14 boh fo s- p- d- and f- angula momentum channels espectively). Use of the default coe cutoff adii fo the Al TM-NLPS poduced negligible changes in the Al BLPS s softness and quality justifying ou simple choice of.1 boh fo all channels. Fo all Mg Al and Si TM-NLPSs the d-angula momentum channel is used as the local pseudopotential to constuct the leinman Bylande fom 39 of the NLPSs. In all cases the NLPSs ae unsceened with LDA exchange coelation. In OF-DFT calculations we use Madden and cowokes LPS fo Mg 30 Goodwin and cowokes LPS fo Al 4 Zhou and cowokes Si BLPS 8 and ou new Mg Al and Si BLPSs fom this wok. Some S-DFT calculations ae pefomed with the CASTEP code 40 with which we use its default ultasoft NLPSs. All numeical details ae summaized in Table.1 including planewave kinetic enegy cutoffs used in both S and OF-DFT. Fo consistency we used the highest planewave cutoff equied fo convegence of any pseudopotential employed fo a given system in both S and OF calculations. We use pimitive cells thoughout this wok: one atom in fcc bcc and sc stuctues; two atoms in diamond body-centeed tetagonal 5 (bct5) hexagonal-close-packed (hcp) and -tin stuctues; fou atoms 15

26 in the hexagonal diamond (hd) stuctue; and eight atoms in the complex bcc (cbcc) stuctue as used by Zhou et al. 8 We use Femi Diac smeaing with a width of 0.1 ev to smooth out the Femi suface fo Mg Al and all metallic phases of Si (cbcc -tin bct5 sc bcc and fcc Si). No smeaing is used fo the Si diamond and hd phases both of which ae semiconductos. All k-point meshes in the Billouin zone ae geneated using the Monkhost Pack method. 41 In all OF-DFT calculations the WGC EDF 13 is used with the paametes: γ =.7 α= ( 5 5) / 6 and β= ( 5 5) / 6. Those paametes ae optimum fo bulk Al. 13 Pevious wok demonstated that these paametes also wok well fo Mg and β - Al3Mg. 15 To ende evaluation of the WGC EDF linea scaling we compute it appoximately by Taylo expanding the WGC EDF kenel aound the elevant aveage bulk electon density. In the case of Al and Mg sufaces the kenel of the WGC EDF is Taylo expanded aound the aveage electon density of bulk fcc Al o hcp Mg at thei espective equilibium volumes. All calculations of bulk cystal popeties stat by obtaining optimized stuctues by elaxing cell vectos and ion positions. Thesholds below which the stess tenso 7 elements and foces on ions ae consideed minimized ae 5 10 Hatee/boh 3 and hatee/boh espectively. Once the equilibium cell vectos ae detemined the atios between cell vectos ae kept fixed duing changes in the bulk volume used to calculate bulk moduli equilibium volumes and enegies. We compess and expand the unit cell isotopically fom 0.95V0 to 1.05V0 whee V0 is the equilibium volume. Then the enegy vesus volume cuve is fitted to Munaghan s equation of state 4 which yields the bulk modulus. 16

27 Table.1 Numeical details in this wok. Ecut (in ev) and Econv (in mev) efe to both S-DFT and OF-DFT calculations. k-point meshes Femi suface smeaing and Ek (in mev) ae only fo S-DFT. Ecut is the kinetic enegy cutoff used to tuncate the planewave basis set. Esmea is the smeaing width used fo the Femi Diac smeaing. Econv and Ek ae the convegence of the total enegy pe atom with espect to Ecut and k-point meshes espectively. / means not applied. Systems Ecut Econv Esmea k-point Ek sc bcc and fcc Al and Mg 800 < <1 diamond and hcp Al and Mg 800 < <1 β -Al3Mg 600 < <1 cbcc β-tin hcp and bct5 Si 1000 < <1 sc bcc and fcc Si 1000 < <1 diamond and hd Si 1000 <0.5 / <1 vacancy fomation enegy (fcc Al) 800 < < vacancy migation enegy (fcc Al) 600 < <5 vacancy fomation enegy (hcp Mg) 800 < < vacancy migation enegy (hcp Mg) 600 < < Al fcc(111) sufaces 800 < <1 Al fcc(100) and fcc(11) sufaces 800 < <1 Mg hcp(0001) suface 800 < <1 Mg bcc(100) and bcc(110) sufaces 800 < <1 stacking fault enegies in fcc Al 480 < <0.5 vacancy fomation and migation 760 <0.1 / <1.3 enegies in diamond Si with LPSs vacancy fomation and migation enegies in diamond Si with NLPS 500 <0.3 / <1.3 17

28 Point defects in Al Mg and Si ae set up as follows. A vacancy in bulk fcc Al is constucted by putting eight Al fcc cubic unit cells togethe in a fashion to fom a supecell with 3 Al atoms. One Al atom is then emoved fom one of the supecell s fou cones to ceate a vacancy. The vacancy fomation enegy calculated with this 3-site cell is conveged to within 0.04 ev with espect to cell size. 43 To constuct a vacancy in hcp Mg 18 Mg hcp pimitive cells ae put togethe in a 3 3 fashion and then one Mg atom is emoved fom one of the supecell s cones. This 35-atom supecell yields a easonable vacancy fomation enegy. 44 To constuct the vacancy and intestitial defects in a Si diamond stuctue we use the same supecell as in the wok of Zhou et al.: 8 a cubic Si supecell with 64 lattice sites constucted by putting 8 cubic unit cells togethe in a fashion. One vacancy is ceated by emoving the atom at the cente of the supecell. The intestitial defect is ceated by inseting an exta atom at the tetahedal site. Both the supecell and ion positions ae fixed duing the calculations. Due to the pesence of dangling bonds spin-polaized S-DFT is employed to calculate defect fomation enegies using the CASTEP code. The lattice vectos of each supecell ae fixed to the equilibium bulk lattice vectos but the ion positions ae fully elaxed in the pesence of these vacancies. We use these same supecells to calculate activation enegies fo vacancy migation between neaest neighbo sites. The initial and final states fo migation ae obtained by fully elaxing the ions with the supecell lattice vectos fixed. The vacancy fomation enegy ( E ) is calculated as in Gillan s wok 45 vf N 1 N 1 E vf E N 11 E( N0 ) N N and the intestitial fomation enegy ( E ) is calculated as if 18

29 E if N 1 N 1 E N 11 E( N0 ) N N whee E( n m ) is the total enegy fo the cell of volume with n atom and m defects. In S-DFT the vacancy migation enegy is calculated using the linea/quadatic synchonous tansit method 46 in the CASTEP code with a maximum foce theshold of 0.05 ev/å. In OF-DFT the vacancy migation enegy is calculated using the climbing-image nudged elastic band (CINEB) method implemented in PROFESS code with a maximum foce theshold of 0.01 evå -1. We model the Al fcc(110) (100) and (111) sufaces with seven five and five layes of Al atoms espectively and with vacuum laye thicknesses of 1.6 Å 11.9 Å and 13.7 Å espectively. These models have been tested to give easonable Al suface enegies. 13 Mg hcp(0001) bcc(001) and bcc(110) sufaces ae modeled with nine seven and five layes of Mg atoms espectively and vacuums of 1.4 Å 14.8 Å and 14.8 Å espectively. Mg and Al suface unit cells contain only one atom in the lateal diection with lateal lattice constants taken fom the equilibium bulk stuctue but with all ions elaxed. Suface enegies ae defined as E slab NE 0 /(A) whee E slab is the total enegy of slab E 0 is the total enegy pe atom of bulk hcp Mg o fcc Al at its equilibium volume N is the numbe of atoms in slab and A is the lateal aea of each slab. The facto of in the denominato is due to the ceation of two sufaces upon foming the slab. The phase tansition pessue in Si is calculated using the common tangent ule de dv de dv phase 1 phase P tansition. 19

30 To calculate stacking fault enegies in bulk fcc Al we use the same setup (Fig..1) as in Benstein and Tadmo s wok. 49 The layes in the illustation maked by aows tanslate lateally togethe. Afte each small tanslation step we pefom calculations by elaxing ion positions only in the z-diection with lattice vectos in the xy plane fixed to the bulk fcc Al equilibium values but the lattice vecto in the z- diection is elaxed. Unit cells contain only one atom in the lateal diection with 0 [setup (a)] and [setup (b)] atoms in the z-diection. 0

31 Figue.1 Illustation (afte Benstein and Tadmo 49 ) of the atomic laye configuation setup used to compute vaious stacking fault enegies in fcc Al. Setup (a) and setup (b) ae used to calculate the stacking fault enegy changes fom d =0 to d = 1.0 and fom d = 1.0 to d =.0 in Fig..4 espectively. A B C denote the atomic layes in fcc Al. We use the pimitive unit cell of β -Al3Mg which contains thee Al atoms and one Mg atom to model the bulk alloy. The alloy fomation enegy pe atom is defined as E Al 3 3Mg E Al EMg E f 4 1

32 whee E 3 Al Mg is the total enegy of the pimitive cell of the β -Al3Mg alloy at its equilibium volume. at thei equilibium volumes..5 Results.5.1 Bulk-deived LPS E Mg and E Al ae the total enegy pe atom of hcp Mg and fcc Al ~ The non-coulomb BLPS V nc ~ ( q) and the complete BLPS V atom ( q) of Mg Al and Si ae displayed in Fig... It is evident that fo each element the data points fom diffeent bulk stuctues almost constitute a common cuve. This finding confims ou basic assumption undelying this wok: thee exists a common LPS suitable fo descibing a wide vaiety of stuctual envionments as illustated hee by fcc bcc sc and dia stuctues. ~ With just the data points in Fig.. we still cannot complete V nc ( q) because of ~ the lack of the data point at q=0. As explained above V nc ( q 0) is undetemined at ~ this stage. So we teat V nc ( q 0) as a fitting paamete. With a tial value fo ~ V nc ~ ( q 0) we can complete V nc ( q) by smoothing these data points in Fig.. using the least-squaes method and then applying piecewise cubic Hemite intepolation to the smoothed data points. 50 Subsequently we add back the Coulomb pat back to ~ V nc ~ ( q) to obtain a tial BLPS in the q-space V BLPS ( q). ~ All Mg Al and Si data fo V BLPS ( q) exhibit a small oscillating tail along the q-axis as shown in the insets of Fig... Mg data exhibits the lagest amplitude oscillations. Those oscillating tails in q-space can cause oscillating tails in eal space when we ~ Fouie tansfom V BLPS ( q) as

33 V BLPS 0 1 ~ BLPS qsin( q) ( ) V ( q) dq. ~ To solve this poblem we simply multiply ou V BLPS ( q) with a cutoff function f (q). We use the same cutoff function as in the wok of Goodwin et al. 4 f 6 ( q) exp ( q / qc ) ~ whee we set q C to be the thid zeo of the V BLPS ( q). We find that inceasing q C i.e. ~ including moe of the tail in the V BLPS ( q) does not impove the quality of the BLPSs. In pactice qc is set to and 6.14 boh fo Mg Al and Si espectively. We have also exploed using a step function as the cutoff function. The change in the BLPS quality due to diffeent cutoff functions is negligible but use of the cutoff function geatly educes the kinetic enegy cutoff equied to convege the plane wave basis. In addition to optimizing the ecipocal space asymptotic behavio we also need to detemine the eal space Coulomb tail of the BLPS. Zhou et al. 8 enfoced a Coulomb tail onto V BLPS () beyond a cetain point C. They agued that the smalle was the moe tansfeable the final V BLPS () C should be as is commonly found fo NLPSs. In actuality V BLPS () should have its own intinsic Coulomb tail but it ~ cannot be accuately ecoveed afte Fouie tansfoming V BLPS ( q) into eal space because of the death of data at low enough q-vectos. A systematic means to detemine the adial cutoff beyond which we ecove the intinsic Coulomb tail in eal space is needed. ~ We now descibe a detailed method to detemine C and V nc ( q 0) the only two fitting paametes in ou method. We take aluminum as an example hee; the 3

34 method is the same fo Mg and Si. Fist we set C to a lage value; in pactice C is set ~ to 1 boh fo all thee elements. In the second step we fix C and tune V nc ( q 0) to see if we can obtain a V BLPS () that yields bulk moduli and equilibium volumes in good ageement with the coesponding Al TM-NLPS esults fo all bulk stuctues used fo building the Al BLPS i.e. the Al fcc bcc sc and diamond stuctues. We also aim to epoduce with this V BLPS () the enegy diffeences between these bulk ~ stuctues calculated with the Al TM-NLPS. If we cannot find a V nc ( q 0) to make a V BLPS () that satisfies the above equiements we conclude that the cuent C is too lage and we decease C by 0.5 boh. With this new C we epeat the second ~ step until we find a combination of C and V nc ( q 0) that poduces a good V BLPS (). This V BLPS () is ou final BLPS with a Coulomb tail stating at C. The C of the final Mg Al and Si BLPSs (shown in Fig..3) ae and 10.5 boh espectively. When building the BLPS we only equie that the final BLPS epoduces bulk popeties of a subset of stuctues that agee well with esults using the TM-NLPS. Below we test the tansfeability of the BLPS in calculations of othe bulk stuctues suface enegies defect enegies and the Al3Mg alloy fomation enegy. None of the ~ esults fom these latte tests ae used to detemine C and V nc ( q 0)..5. Compaing the BLPS with othe PSs in S-DFT In this section we will fist compae S-DFT pedictions of vaious bulk defect and inteface popeties of Mg Al and Al3Mg using the new Mg and Al BLPSs vesus using TM-NLPSs in ode to evaluate the accuacy and tansfeability of these Mg and Al BLPSs. We also compae the quality of ou BLPSs to Madden and cowokes Mg LPS and Goodwin and cowokes Al LPS. The quality of ou new Si BLPS is tested aftewad. 4

35 Static bulk popeties As shown in Table. static bulk popeties calculated using ou Mg and Al BLPSs ae in excellent ageement with those calculated using TM-NLPSs except fo the bulk modulus of sc Mg and the equilibium volume of dia Mg whee the deviation is moe significant. The TM-NLPS enegy diffeence between any two bulk stuctues ae epoduced by ou BLPSs with an eo of about 40 mev pe atom which is within the eo of othe appoximations made in conventional S-DFT. This small 40 mev pe atom eo again confims that the data points fom diffeent bulk stuctues (Fig..) can be intepolated well with smooth cuves. The tansfeability of ou BLPSs can be evaluated by examining the stuctues not used to fit the BLPSs. Fo example the hcp stuctue is not used to build eithe Mg o Al BLPS. Ou BLPSs ae able to accuately epoduce the small enegy diffeence between the hcp and fcc stuctues. Like Zhou et al. 8 we believe that the oigin of good tansfeability exhibited by ou BLPSs is because they ae constucted fom bulk athe than atomic densities. BLPSs ae built fom bulk envionments whee the valence electons ae polaized upon bonding to neighboing atoms and expeience a change in potential compaed to the isolated atom. Ou BLPSs explicitly captue changes in potential necessay fo good tansfeability. Oveall ou BLPSs fo Mg and Al pefom bette than Madden and cowokes Mg LPS and Goodwin et al. s Al LPS at least fo the popeties investigated hee. This enhanced pefomance is not supising since Madden and cowokes Mg LPS makes use of an obital-fee EDF in its constuction which is a souce of eo and Goodwin et al. s Al LPS is an empiical one fitted only to the expeimental lattice paametes of fcc Al and zincblende AlAs. 5

36 ~ Figue. The main gaphs display V nc ( q) [Eq. (.3)]: the non-coulombic pat of the final (a) Mg (b) Al and (c) Si BLPSs in ecipocal space (solid lines). The data points ae discete values of atom-centeed local pseudopotentials geneated fom diffeent bulk stuctues (see legends fo details). Data points in the insets ae the complete atom-centeed local pseudopotentials [Eq. (.)] containing the Coulombic pat. Fo each of the thee elements the data points fom diffeent bulk stuctues almost constitute a common cuve. 6

37 Suface and vacancies Since ou BLPSs ae built completely fom data fom pefect bulk cystals calculations of the enegies of low-index Mg and Al sufaces as well as vacancy enegetics futhe challenge the tansfeability of the BLPSs. An atom in a suface laye bonds to the atoms undeneath but has no bonds on the vacuum side. So a suface epesents a special envionment intemediate between the bulk cystal and an isolated atom. The pesence of a vacancy epesents anothe case of undecoodination elative to a pefect cystal. As shown in Table.3 the Al BLPS and TM-NLPS pedicted suface enegies ae the same to within about 0 mj/m coesponding to a BLPS eo of about 8 mev pe atom well within the expected eo of S-DFT-LDA. This analysis consevatively ascibes all the eo to the desciption of the atoms at the suface which is a easonable assumption given that the bulk cystal is so well descibed by the Al BLPS. Of couse if we assumed the eo was distibuted ove the non-suface atoms also the eo on a pe atom basis would be even smalle. Similaly the Mg TM-NLPS suface enegies fo thee Mg sufaces ae epoduced to within about 10 mj/m with ou Mg BLPS coesponding to a ~7 mev pe atom eo in the BLPS by the same analysis. 7

38 Table. (Uppe table) S-DFT-LDA esults fo bulk moduli (B0 in GPa) bulk equilibium volumes (V0 in Å 3 ) and equilibium total enegies (Emin in ev pe atom) calculated using a TM-NLPS Mg BLPS and Madden and cowokes Mg LPS. (Lowe table) Compae the same quantities using a TM-NLPS Al BLPS and Goodwin et al. s Al LPS. The equilibium total enegies of hcp Mg and fcc Al stuctues ae given while fo all othe stuctues the enegy diffeences elative to the gound state stuctues ae shown. The TM-NLPS data should be viewed as the benchmak. Ou BLPSs pefom bette oveall than pevious LPSs Mg hcp fcc bcc sc dia B 0 TM-NLPS BLPS Madden LPS V 0 TM-NLPS BLPS Madden LPS E min TM-NLPS BLPS Madden LPS Al fcc hcp bcc sc dia B 0 TM-NLPS BLPS Goodwin LPS V 0 TM-NLPS BLPS Goodwin LPS E min TM-NLPS BLPS Goodwin LPS

39 Figue.3 (a) Mg (b) Al and (c) Si BLPSs (solid lines) in eal space. Madden and cowokes Mg LPS 30 Goodwin and cowokes Al LPS 4 Zhou and cowokes Si BLPS (all in dashed lines) 8 ae shown togethe with ou Mg Al and Si BLPSs espectively. We see that ou new Si BLPS is softe than Zhou and cowokes Si BLPS wheeas ou Mg and Al BLPSs ae hade than those deived empiically (Goodwin and cowokes 4 ) o via OF-DFT invesion (Madden and cowokes 30 ). The Coulombic tails of the Mg Al and Si BLPSs stat at and 10.5 boh espectively (see insets). 9

40 Table.3 (Uppe table) S-DFT-LDA esults fo the suface enegies (in mj/m ) of Mg hcp(0001) bcc(110) and bcc(001) sufaces. (Lowe table) Suface enegies fo fcc Al (110) (100) and (111) sufaces. The TM-NLPS esults should be viewed as the benchmak. Ou Mg and Al BLPSs yield significantly moe accuate suface enegies than Madden and cowokes Mg LPS and Goodwin et al. s Al LPS. Mg hcp(0001) bcc(110) bcc(001) Mg TM-BLPS Mg BLPS Madden LPS Al fcc(111) fcc(100) fcc(110) Al TM-NLPS Al BLPS Goodwin LPS We find similaly good ageement between the TM-NLPS and BLPS pedictions fo Mg and Al vacancy enegetic (Table.4). Vacancy fomation enegies agee to within ~0 mev pe vacancy fo Mg and ~4 mev pe vacancy fo Al while vacancy migation enegies diffe by no moe than ~11 mev pe vacancy fo Mg and ~6 mev pe vacancy fo Al. All these deviations between the TM-NLPS and ou BLPS ae within the expected accuacy of the numeical and physical appoximations made in S-DFT-LDA. By contast Madden and cowokes Mg LPS 30 and Goodwin et al. s Al LPS 4 yield significantly lage discepancies fo the same quantities. 30

41 Table.4 S-DFT-LDA esults fo vacancy fomation (Evf in ev) and migation enegies (Eva in ev) in hcp Mg (uppe table) and fcc Al (lowe table) calculated using Mg and Al BLPSs as well as Madden and cowokes Mg LPS and Goodwin et al. s Al LPS. As benchmaks (labeled as NLPS in the table) we use Mg and Al s TM-NLPSs to calculate the Evf wheeas Eva is calculated with Mg and Al s ultasoft NLPSs in CASTEP. 40 Ou Mg and Al BLPS esults diffe less than 30 mev pe defect fom the benchmaks. Mg Evf Eva Expeiment 0.81 a 0.58 b 0.79 c 0.90 d NLPS BLPS Madden LPS Al Evf Eva Expeiment e e NLPS BLPS Goodwin LPS a Fom ef 51. b Fom ef. 5. c Fom ef. 53. d Fom ef. 54. e Fom ef. 55 Stacking fault enegies in fcc Al We futhe test ou Al BLPS by calculating vaious stacking fault enegies in fcc Al. As shown in Table.5 ou Al BLPS epoduces the TM-NLPS twinning enegy vey well. Othe stacking fault enegies pedicted by the Al BLPS ae all about 50 mj/m smalle than those fom the TM-NLPS as seen moe clealy in Fig..4. Assuming the Al BLPS accuately descibes the Al atoms away fom the stacking 31

42 fault inteface and ascibing the entie eo of ~50 mj/m to the less accuate desciption of the Al atoms at the stacking fault inteface then the Al BLPS eo is only 10 mev pe atom which is again within the eo of othe appoximations made in S-DFT-LDA. The loweing of ~50 mj/m in stacking fault enegies is consistent with the level of eo aleady epoted in Table. whee TM-NLPS enegy diffeences between diffeent Al stuctues ae epoduced by the Al BLPS with an eo of ~40 mev pe atom. The Al BLPS appeas at fist glance to yield bette esults fo suface enegies because the suface enegies ae about 10 times lage than the stacking fault enegies. In fact fo both suface and stacking fault enegy calculations the accuacy of ou Mg and Al BLPSs is consistently bette than 10 mev pe atom. By contast pevious LPSs yield consistently lowe and less accuate suface and stacking fault enegies. Table.5 S-DFT-LDA esults fo vaious stacking fault enegies in fcc Al (in mj/m ) (twinning enegy γt unstable stacking fault enegy γus intinsic stacking fault enegy γisf unstable twinning enegy γut extinsic stacking fault enegy γesf) calculated using ou Al BLPS. The benchmak values ae calculated using the Al TM- NLPS. We see that Goodwin et al. s Al LPS gives much lowe stacking fault enegies γt γus γisf γut γesf Al TM-NLPS Al BLPS Goodwin LPS

43 Figue.4 The stacking fault enegy (E in mj/m ) as a function of the factional displacement d. d is the tanslation (shown in Fig..1) along the [11] diection 1 a with ten steps ( a is the lattice constant of fcc Al at equilibium). The cuves 10 6 fom d=0 to d=1 and fom d=1 to d= ae calculated with the setup (a) and (b) in Fig..1 espectively. The small discontinuity at d=1.0 is due to the slight diffeence between setup (a) and (b). Only twin boundaies exist at d=0.0. The unstable stacking fault is fomed at d~0.5 the intinsic stacking fault is fomed at d~1.0 the unstable twinning fault is fomed at d~1.5 and the extinsic stacking fault is fomed at d~.0. β -Al3Mg alloy Thus fa we have tested ou Mg and Al BLPSs fo cases which contain only Mg o Al atoms. We next test them in an alloy envionment: the Al3Mg alloy in its β phase. 33

44 Ou BLPSs do vey well fo this mixed system (Table.6). The TM-NLPS esults fo the bulk modulus and equilibium volume ae epoduced accuately by Mg and Al BLPSs a clea impovement ove pevious LPSs. Most encouagingly the small alloy fomation enegy calculated with ou BLPSs diffes by only 0.4 mev pe unit cell fom the TM-NLPS pediction. Calculation of the alloy fomation enegy is the toughest test yet and we see that the combination of Madden and cowokes Mg LPS and Goodwin et al. s Al LPS gives much smalle alloy fomation enegy. Table.6 S-DFT-LDA esults fo the bulk modulus (B0 in GPa) equilibium volume (V0 in Å 3 ) and the alloy fomation enegy ( Ef in mev) of the β -Al3Mg alloy using Mg and Al TM-NLPSs (as the benchmak) Mg and Al BLPSs as well as Madden and cowokes Mg LPS and Goodwin et al. s Al LPS. The TM-NLPS esults fo all popeties including the small alloy fomation enegy ae accuately epoduced by ou BLPSs B0 V0 Ef TM-NLPS BLPS Madden & Goodwin New Si BLPS In this wok we make two key impovements on the pevious wok of Zhou et al.: 8 (1) a cutoff function is used to emove the oscillating tail of the BLPS in q-space and () the BLPS s Coulomb tail is systematically detemined. The fist impovement makes ou BLPSs softe i.e. a smalle kinetic enegy cutoff conveges the plane wave basis set. The second impovement ecoves the intinsic Coulomb tail of the BLPS which helps peseve the tansfeability of the BLPS. With 34

45 these two efinements we e-build the Si BLPS with othe settings kept the same as in Zhou et al. s wok. As shown in Table.7 although Zhou et al. s Si BLPS poduces accuate enegies and volumes fo the common metallic phases (hcp bcc and fcc) it gives incoect enegy odeings between β-tin and bct5 Si and between bcc and fcc Si compaed to TM-NLPS pedictions. Ou new Si BLPS pedicts the coect enegy odeing fo all nine Si stuctues consideed albeit with some loss of accuacy fo the highest enegy metallic phases. Ou softe Si BLPS poduces significantly impoved bulk moduli but slightly wose equilibium volumes fo all phases. We find a mixed outcome fo ou new Si BLPS compaed to Zhou et al. s BLPS when compaing phase tansition pessues and defect fomation enegies to those pedicted by the TM-NPLS. Table.8 eveals that ou new Si BLPS pefoms bette fo the diamond to β-tin phase tansition pessue but wose fo the diamond to bct5 phase tansition pessue compaed to Zhou et al. s Si BLPS. Table.9 shows that ou Si BLPS poduces smalle defect fomation enegies than Zhou et al. s Si BLPS but the diffeence between self-intestitial and vacancy fomation enegies is slightly bette epoduced by this new Si BLPS. 35

46 Table.7 TM-NLPS (labeled NLPS) and ou new Si BLPS (labeled BLPS) ae used in these S-DFT-LDA calculations of bulk moduli (B0 in GPa) equilibium volumes (V0 in A 3 ) and total enegies (Emin in ev) fo nine phases of Si. The esults of Zhou et al. s Si BLPS (labeled Zhou s) ae quoted fom Ref. 8. The equilibium total enegy of the Si diamond (dia) stuctue is given. Fo othe Si stuctues the enegy diffeences between them and Si diamond stuctue ae shown. Ou new Si BLPS gives a coect enegy odeing fo all the Si stuctues while Zhou et al. s Si BLPS gives incoect enegy odeings between β-tin and bct5 and between bcc and fcc. B 0 dia hd cbcc β-tin bct5 sc hcp bcc fcc NLPS BLPS Zhou s V 0 NLPS BLPS Zhou s E min NLPS BLPS Zhou s

47 Table.8 Tansition pessues (in GPa) calculated with S-DFT-LDA using the Si TM-NLPS and ou new Si BLPS. Results fom Zhou et al. s Si BLPS ae also listed diamond β-tin diamond bct5 Expeiment 1.5 a TM-NLPS BLPS (this wok) Zhou s Si BLPS a Fom ef. 56 expeiment Table.9 Self-intestitial (Es) and vacancy (Ev) fomation enegies in cubic diamond Si calculated with spin-polaized S-DFT-LSDA using the default Si ultasoft NLPS in CASTEP 40 (as the benchmak) and ou new Si BLPS. Results fom Zhou et al. s Si BLPS 8 ae newly calculated in this wok. All quantities ae in the units of ev/defect. Es Ev Es - Ev Othes 3.76 a 3.6 b NLPS BLPS (this wok) Zhou s Si BLPS a Fom ef. 57 spin-esticted S-LDA with a lage supecell and a elaxed stuctue. b Fom ef. 58 and 59 - expeiment estimate. 37

48 .5.3 OF-DFT tests Given the status of cuent EDFs OF-DFT is especially suitable fo studying main goup metals and thei alloys which ae nealy-fee-electon-like systems. Fo bulk Al Mg and Al3Mg it has been shown that the S-DFT pedictions of many key popeties can be epoduced vey well by OF-DFT when the WGC EDF is employed At pesent the accuacy of OF-DFT when applied to these nealy-fee-electon-like metals is mainly hindeed by the quality of available LPSs. We now compae the pefomance of OF-DFT to S-DFT using ou new Mg and Al BLPSs and the WGC EDF fo all the Mg and Al cases studied above. Static bulk popeties Vaious static bulk popeties pedicted by S-DFT-BLPS theoy ae epoduced well by OF-DFT-BLPS theoy (Table.10). It is clea that OF-DFT woks bette fo Mg s and Al s fcc and bcc stuctues than fo thei cubic diamond and sc stuctues since the fome ae moe close-packed which makes the electon density distibuted moe evenly close to a nealy-fee-electon gas. Since the WGC EDF is based on the latte it makes sense that the close-packed stuctues ae descibed moe accuately. Sufaces vacancies and stacking faults OF-DFT with the WGC EDF does an oveall excellent job at descibing defects and intefaces in Mg and Al. The eos ae all less than 0.1 ev pe atom compaed to S-DFT with the same BLPS and often much smalle. Fo example OF-DFT pedicts the coect enegy odeing fo diffeent sufaces of both Mg and Al (Table.11). OF-DFT Al suface enegies diffe by ~40 mj/m fom the coesponding S-DFT- BLPS quantities. If we again assume that all the eo is due to suface Al atoms then this 40 mj/m deviation indicates an eo of ~96 mev pe atom. Similaly OF-DFT epoduces the enegy fo suface Mg atoms with an eo of ~67 mev pe atom. As 38

49 befoe these ae uppe bounds to the pe atom eo. Secondly OF-DFT epoduces the S-DFT-BLPS vacancy fomation enegies to within ~0.1 ev fo Al and ~0.07 ev fo Mg (Table.1). Diffeences between OF-DFT and S-DFT fo vacancy migation enegies ae < 0.03 ev pe vacancy fo both Mg and Al. Table.10 Simila to Table. OF-DFT-LDA static bulk popeties calculated with the WGC EDF and the BLPS. B0 is in GPa. V0 is in Å 3. Emin is in ev. S-DFT-LDA esults using the BLPS ae epeated fom Table. fo ease of compaison. Mg hcp fcc bcc sc dia B0 OF S V0 OF S Emin OF S Al fcc hcp bcc sc dia B0 OF S V0 OF S Emin OF S

50 Table.11 Simila to Table.3 OF-DFT-LDA suface enegies (mj/m ) calculated with the WGC EDF. S-DFT-LDA esults fom Table.3 ae quoted in paentheses fo compaison Mg hcp(0001) bcc(110) bcc(001) OF (S) 730 (631) 755 (674) 875 (808) Al fcc(111) fcc(100) fcc(110) OF (S) 1143 (1010) 1343 (1104) 1360 (11) Table.1 Simila to Table.4 OF-DFT-LDA vacancy enegetics (ev) calculated with the WGC EDF. The coesponding S-DFT-LDA esults fom Table.4 ae quoted in paentheses fo compaison Mg Evf Eva OF (S) 0.96 (0.8) (0.419) Al Evf Eva OF (S) (0.784) 0.69 (0.638) 40

51 Table.13 Simila to Table.5 OF-DFT-LDA stacking fault enegies (mj/m ) fo fcc Al calculated with the WGC EDF and the BLPS. The coesponding S-DFT-LDA quantities calculated with the BLPS ae shown undeneath fo compaison γt γus γisf γut γesf OF S The S-DFT esults (using BLPS) fo unstable stacking fault twinning and unstable twinning enegies ae well epoduced by OF-DFT using the WGC EDF and BLPS (Table.13). On the othe hand the intinsic (isf) and extinsic stacking fault (esf) enegies fom OF-DFT ae smalle by mj/m than pedicted by S- DFT. To undestand the oigin of the lowe isf and esf enegies in OF-DFT note that the fomation of eithe an isf o esf in fcc Al can be thought of as the eplacement of one fcc plane by one hcp plane. Table.10 aleady indicates that the S-DFT-BLPS enegy diffeences between fcc and hcp Al is about 0 mev pe atom highe than fom OF-DFT. As discussed above this 0 mev pe atom can lowe the isf o esf enegies by about 45 mj/m. So the smalle isf and esf enegies fom OF-DFT ae lagely due to the fact that OF-DFT is still unable to accuately captue the small enegy diffeence between fcc and hcp Al. Consequently as shown in Fig..4 the baies fom d = 1 to d = 0 and fom d = to d = 1 ae both 0 mev pe atom lowe in OF-DFT than those calculated with S-DFT. This small change of baie height is on the ode of othe expected eos in DFT-LDA. β -Al3Mg alloy OF-DFT pefoms vey well fo this alloy as shown in Table.14 independent of which set of LPSs ae used. Howeve ou new Mg and Al BLPSs impove the OF-DFT pediction fo the alloy fomation enegy substantially when compaed with esults 41

52 obtained using pevious LPSs. Moeove Caling and Cate epoted 15 that this small alloy fomation enegy cannot be coectly epoduced by EDFs with density-independent esponse kenels. The WGC EDF which is a EDF with a density-dependent kenel was the only one to poduce this alloy fomation enegy with the coect sign. 15 Table.14 Bulk modulus B0 (in GPa) unit cell s equilibium volume V0 (in Å 3 ) equilibium total enegy E0 (ev pe unit cell) and alloy fomation enegy ΔEf (mev pe unit cell) of β -Al3Mg ae calculated using OF-DFT-LDA with the WGC EDF. The S-DFT-LDA esults ae shown in paentheses fo compaison. With ou BLPSs the OF-DFT-LDA esults fo the small alloy fomation enegy ae much impoved. M & G LPSs efes to the esults calculated with Madden and cowokes and Goodwin et al. s LPSs OF-DFT B0 V0 E0 ΔEf Mg & Al BLPSs 66.8 (67.5) (67.131) ( ) -5.6 (-8.6) M & G LPSs 59.5 (58.0) (70.008) ( ) -1.7 (-.7).6 Conclusions In this wok we made two impotant impovements to Zhou et al. s 8 appoach to building BLPSs. We intoduced a potential cutoff in Fouie space and systematically detemined the cutoff adius beyond which BLPS s Coulomb tail is ecoveed in eal space. Consequently ou new BLPSs ae moe efficient to use (smalle plane-wave basis expansion equied) and moe accuate in many 4

53 instances. We built Mg Al and Si BLPSs and tested thei tansfeability and accuacy by applying them in S-DFT calculations of static bulk popeties of seveal Mg Al and Si bulk stuctues defect enegetics in hcp Mg fcc Al and diamond Si suface enegies fo low-index Mg and Al sufaces and stacking fault enegies in fcc Al. Compaison of S-DFT-BLPS and S-DFT-NLPS esults demonstated the excellent tansfeability and accuacy of ou BLPSs. In these tests the eo due to the Mg and Al BLPSs is always less than 40 mev pe atom and in most cases is only ~10 mev pe atom. These BLPSs ae accuate enough in S-DFT that in addition to thei use in OF-DFT they could find use in lage scale S-DFT calculations whee calculation of NLPS tems can become pohibitively expensive. We also tested the quality of the WGC EDF in combination with ou new BLPSs by pefoming OF-DFT calculations fo many of the same popeties mentioned above. We demonstated yet again that OF-DFT pefoms as well as S-DFT in systems which ae close to behaving in a fee-electon-like manne. The method outlined hee is eady fo use in building BLPSs fo othe elements fo futue use in OF-DFT. Moeove with ou impoved BLPSs and the WGC EDF OF-DFT is now a pactical and tustwothy tool fo the lage-scale simulation of main goup metals and thei alloys..7 Acknowledgements Funding fo this wok is povided by the National Science Foundation and the Depatment of Enegy. The autho would like to thank D. Geg Ho D. Vincent Lignèes and D. Linda Hung fo thei temendous contibution to the PROFESS code. 43

54 Chapte III Nonlocal obital-fee kinetic enegy density functional fo semiconductos 3.1 Intoduction Computational mateial science becomes vey impotant fo undestand and pedicting mateial popeties. The most common quantum mechanics method fo studying mateials is ohn-sham density functional theoy (S-DFT) which is based on the Hohenbeg and ohn theoems. 3 By employing local density appoximations conventional implementation of S-DFT exhibits a good balance between accuacy and computational cost. Howeve fo systems with thousands of atoms o moe S-DFT becomes pohibitive due to the exponential incease of computational cost geneally popotional to the cube of the numbe of atoms in the mateials. Obital-fee density functional theoy (OF-DFT) 5 60 is a possible altenative to S-DFT fo lage scale mateials simulation. In OF-DFT the total enegy functional depends only on the electon density instead of the S obitals. Theefoe OF-DFT educes the degees of feedom in the computation fom 44 3 N to 3 which geatly simplifies the fomalism. A ecent implementation 61 has made the computational cost of all pats of the OF-DFT calculation scale linealy with espect to system size fo all sizes i.e. thee is no cossove between cubic and linea scaling as in linea scaling S-DFT. 8 The linea scaling coupled with paallelization via domain decomposition now allows an unpecedented numbe of atoms (~10 6 ) to be teated explicitly with quantum mechanics. Such a fast fist pinciples method is tantalizing

55 to use to study mateials phenomena at the mesoscale peviously uneachable with quantum mechanics methods. Howeve pevious wok has demonstated that OF-DFT is only capable of S-DFT-level accuacy fo nealy-feeelecton-like main goup metals with only maginal pogess made in teating othe types of mateials. 6 6 The key element detemining the accuacy of OF-DFT is the expession used to evaluate the electon kinetic enegy in tems of the electon density namely the kinetic enegy density functional (EDF). In S-DFT obitals ae used to evaluate the usual quantum mechanical expectation values of the Laplacian giving ise to accuate values of the exact non-inteacting electon E. A fa geate challenge is posed when evaluating the E solely fom the electon density since an analogous exact expession fo the EDF is unknown except in cetain idealized limits. Poposals of new EDFs have occued ove many decades. These EDFs geneally can be gouped into two classes: (1) local/semilocal EDFs and () nonlocal EDFs Local/semilocal EDFs ae constucted based on the local electon density and its density gadient. Recently a meta-gga (genealized gadient appoximation) EDF 65 was also poposed which adds the Laplacian of the electon density into the EDF. A detailed suvey on local/semilocal EDFs was given ecently by Gacía-Aldea and Alvaellos 66 and an olde eview was povided by Thakka. 67 Nonlocal EDFs typically have nonlocal kenels which elate any two points in space. Most commonly nonlocal EDFs have been deived fom linea esponse theoy fo the petubed unifom electon gas; i.e. they ae based on the Lindhad esponse function 68 since its fom is known analytically exactly in momentum space. Howeve the Lindhad esponse function at most can be expected to epesent popely nealy-fee-electon-like metals with nealy unifom densities (hence the success mentioned above in applying OF-DFT to main goup metals). Consequently the Lindhad-based EDFs cannot be expected to teat semiconductos well since thei linea esponse behavio is fa diffeent. In contast to the unifom electon gas no exact analytic fom exists fo the linea esponse 45

56 function of semiconductos othe than some models The pupose of the pesent wok is to examine the basic physics that should be incopoated into a EDF fo semiconductos and then to build and test such a EDF. This chapte is stuctued as follows. Fist we discuss the asymptotic behavio of a semiconducto s susceptibility function ~ ( q ) at the q 0 limit whee q is the electon momentum and then we popose a new EDF based on this behavio. Then we test the new EDF on a vaiety of binay semiconductos as well as diffeent phases of and defects in silicon. The tansfeability of the two paametes in ou EDF is also analyzed both fomally and numeically. We also popose ecommended values fo the two paametes of the EDF that wok easonably well acoss a boad ange of tetahedally bonded semiconductos. 3. inetic enegy density functional fom In S-DFT the total enegy functional is patitioned as E[ ] TS [ ] V[ ] (3.1) whee T [ S ] is the S EDF of the non-inteacting electon gas and V [] contains the Hatee electon-electon epulsion enegy the electon exchange-coelation enegy and the extenal potential enegy (usually the latte is just the ion-electon attaction enegy whee the ionic potentials consist of eithe bae nuclea potentials o pseudopotentials accounting fo attaction of the valence electons to nuclei sceened by the coe electons). Within linea esponse theoy the S susceptibility function is defined by ( ) S S 3 ( ' ) v ( ') d ' (3.) 46

57 S whee is simply a function of '. ( ) is the change in electon density induced by a petubation of the S effective potential v S () which includes the electostatic (i.e. electon-electon plus ion-electon) pat v ele () and the exchange-coelation pat v XC () : v S ele XC ( ) v ( ) v ( ). (3.3) Eq. 3.3 can be ewitten as S ele XC v ( ) v ( ) v ( ) 1 F F F ( ') ( ') ( ') ~ S ( q) (3.4) whee F is the Fouie tansfom opeato. The geneal behavio of ~ S ( q) at q 0 limit has been shown by Pick et al. 7 to be ~ S ( q 0) a (metal) ~ S ( q 0) bq (insulato) whee a and b ae positive numbes. Hee we give anothe poof fo the behavio of an insulato s susceptibility function (the second elation above) within the local density appoximation (LDA ) o the GGA 73 fo exchange-coelation. Unde an extenal petubing potential the solid undegoes a polaization P(q ) (in Fouie space) accoding to standad electostatics expessed as P( q) ~ 1 4 ( q) 1 E( q) (3.5) 47

58 with ~ ( q ) being the dielectic function and with a change in electic field E(q ) given by ele E( q) iqv ( q). (3.6) The polaization P(q ) is elated to the induced density change as iq P( q) ( q). (3.7) By combining Eqs and 3.7 we obtain ( q) 1 ~ ele ( q) 1 v ( q). (3.8) q 4 Reaanging and substituting Eq. 3.8 into Eq.3.4 we elate ~ S ( q) to ~ ( q ) as 1 ~ S ( q) q 4 ˆ v ( XC ) ~ F ( ( q) 1) ( '). (3.9) Up to this point only the linea esponse appoximation has been made. We now invoke the LDA/GGA fo exchange-coelation. Unde the latte XC v is a functional only of the density (LDA) and possibly density gadients (GGA) which in tun means that the second tem in Eq. 3.9 is meely a polynomial in q. ~ ( q 0) is just the macoscopic static dielectic constant which is a finite numbe geate than 1 fo semiconductos. Consequently the second tem in Eq. 3.9 cannot cancel out the 1/ q singulaity fom the fist tem as q 0 with the LDA/GGA ~ S ( q) behaves as ~ q as q 0.. Thus fo semiconductos teated Ou simple poof above elies on use of the most common exchange-coelation functionals namely the LDA/GGA and theefoe does not include the nonlocal esponse function pat of the exchange-coelation functional. Howeve ou 48

59 conclusion was poved in geneal by Pick et al. 7 Since ou pupose hee is to deive a EDF fo use with the LDA/GGA fo exchange-coelation anyway it is sufficient fo us to pove the asymptotic behavio of ~ S ( q) within the LDA/GGA. Note also that ou poof above only woks fo semiconductos and not fo metallic systems whee ~ ( q 0). (The denominato of the fist tem in Eq. 3.9 is then indeteminate as q 0.) Howeve we expect that ~ S ( q) fo metals should appoach a finite value as q 0 based on the known linea esponse function of the unifom gas namely the Lindhad esponse function. The contasting behavio of ~ S ( q) as q 0 fo metals and semiconductos is a key featue that distinguishes thei susceptibility functions. Fom the easily-deived elationship between ~ S ( q) and T [ S ] namely ˆ TS [ ] ~ F S ( q) ( ') ( ) (3.10) and the asymptotic behavio of ~ S ( q) above we obtain the condition that T [ S ] needs to satisfy as q 0 : ˆ T ( ) S F ~ ( ') ( ) q 1 ~ ( q 0) 1 (3.11) which in tun implies the following asymptotic behavio in eal space as ' : T S ( ) ( ') ( ) 1 ~ ' ~ ( q 0) 1. (3.1) To ou knowledge the latte condition has not been accounted fo in any EDF to date. 49

60 Hee we popose to explicitly impose the asymptotic behavio given in Eq. 3.1 by genealizing a pevious EDF fom due to Wang and Tete. 11 Othe EDF foms could be genealized as well but in the pesent wok we conside only this fom. Wang and Tete patitioned the EDF as T [] T T T (3.13) S TF vw NL whee T TF / 3 3 CTF d with C 3 / 3 TF is the local Thomas-Femi (TF) EDF T vw 1 d 8 3 is the semilocal von Weizsäcke EDF 76 and the nonlocal pat is geneally witten as T NL C ( ) d. (3.14) 3 3 ( ') ' ( ') d ' Hee ( ') is an effective Femi wavevecto that may depend on densities at and ' is a dimensionless kenel C C TF 83 and and ae two exponents that satisfy 8/ 3 to ensue the coect dimensionality. The kenel depends on the distance between and ' scaled by ( '). Clealy in the Wang-Tete fom the nonlocal pat is a function of '. We genealize the kenel to make it a function of both ' and 1/ ' in ode to explicitly include the 1/ ' asymptotic behavio of semiconductos given in Eq Ou geneal kenel thus takes the fom [ ] ( ') [ ] ' Y G. (3.15) ' whee G [] and Y [] ae unknown functionals. Hee we conside a specific kenel 50

61 ( ') ( ') ' (3.16) with whee k F ( ) ( ') 1 ( ') k ( ) 1 F (3.17) 8 / 3 ' ( ) ( ) mixing between the 1/ 3 (3 ( )) is the Femi wavevecto and the paamete contols the have a kenel that depends only on ' tem and the 1/ ' tem in the kenel. If 0 we ' and the density ( ). ( ) ( ' ) in Eq is intoduced to emove the singulaity when ' 0 and the 1/ ( ) facto peseves coect dimensionality. 8 / 3 By intoducing the educed density gadient s( ) we appoximate the agument of the quadatic tem in Eq by ( ) ( ') ( ) ' (3.18) leading Eqs and 3.17 to simplify to ( ') k F ( ) 1 s( ) ' ( ) s( ). 4 / 3 ( ) Justification of the appoximation made in Eq is given late by consideing Thomas-Femi dielectic sceening theoy. Ou total EDF in this wok finally becomes 51

62 T [] T T T (3.19) S TF vw NL with T NL C 3 3 k ( ) 1 s( ) ' ( ') d ' 8/ 3 ( ) F d. (3.0) We obseve that the coefficient of s( ) is implicitly linked to the static dielectic constant by Eq. 3.1 afte we do the Taylo expansion of the kenel in Eq. 3.0 with espect to ( ) s( ) '. This implication is discussed in Section 3.4. k F To specify the kenel ( ') we enfoce the exact linea esponse of a unifom electon gas onto ou EDF as one limit we wish to satisfy ˆ TS [ ] 1 F ( ) ( ') ~ (3.1) Lindhad ( q) ( ) * ~ Lindhad k F kf 1 ( ) ln 4 1. F( ) Hee * is the unifom electon gas density and q /( kf ). Imposition of Eq. 3.1 leads to a fist ode odinay diffeential equation fo the kenel 5 ~ ( )' (5 3 ) ~ ( ) F ( ) (3.) whee ~ ( ) is the Fouie tansfom of the kenel ( ' ) and aises fom the functional deivatives of the density exponents in Eq The paamete does not appea in the kenel equation which indicates that should not have any effect as the system appoaches a unifom electon gas. We integate Eq. 3. fom 5 to zeo with the Runge-utta method implemented in the RSUITE code 77 in ode to evaluate the kenel.

63 Although it is an impotant limit to etain an inconsistency aises by focing ou EDF to epoduce the Lindhad esponse function in Eq Analysis of Eq. 3. eveals that ~ ( ) is zeo at q 0 since F ( ) 1 at q 0. Consequently ou kenel does not exhibit the coect behavio fo semiconductos as q 0. Ou kenel uns into this poblem patially because we selected the Wang-Tete EDF fom. In the latte T NL has to be zeo fo a unifom electon gas in ode to ecove the coect 3 3 (Thomas-Femi) limit which makes ' d d' 0 and theefoe ~ ( 0) is zeo (since ~ (0) ~ ( ) d 3 ). Even if the Wang-Tete fom is a good appoximation fo nealy-fee electon gases (simple metals) it is likely not the optimal stating point to constuct EDFs fo semiconductos. Howeve hee we still use the Wang- Tete EDF fom and exploe whethe the new EDF can be applied successfully to both semiconductos and metals. Thus ou poposed kenel (Eqs and 3.0) is admittedly compomised by using the Wang-Tete EDF fom. Imposition of the Lindhad esponse behavio itself may also be a limitation on accuacy but unfotunately no moe sophisticated analytical fom fo the esponse function is available to use fo EDF kenel constuction. A EDF fom fully coect fo both the nealy-fee electon gas and semiconductos emains unknown. 3.3 Numeical implementations Computation details This new EDF is implemented in the PROFESS code 34 which is a plane-wavebased OF-DFT code that imposes peiodic bounday conditions. S-DFT benchmak calculations ae pefomed with the ABINIT code. 33 All calculations use the LDA fo electon exchange-coelation Vaious phases of silicon ae studied in this wok. The stuctues and unit cells used fo cubic diamond (CD) hexagonal diamond (HD) complex body-centeed 53

64 cubic (cbcc) -tin body-centeed tetagonal 5 (bct5) simple cubic (sc) hexagonal close-packed (hcp) body-centeed cubic (bcc) and face-centeed cubic (fcc) stuctues ae all as given in ou pevious wok. 8 We also examine the (cubic) zincblende (ZB) and (hexagonal) wutzite (WZ) stuctues of the III-V semiconductos GaP GaAs GaSb InP InAs and InSb. In S-DFT calculations with nonlocal pseudopotentials (NLPSs) we use Toullie-Matins (TM) 37 NLPSs geneated in the FHI98 code 38 using the default cutoff adii. Fo all local pseudopotential S-DFT and all OF-DFT calculations we employ bulk-deived local pseudopotentials (BLPSs) Fo silicon we use the one peviously epoted and tested in Ref. 78 wheeas fo othe binay semiconductos we use new BLPSs built with the same method. 78 In all S-DFT and OF-DFT calculations the numbe of plane waves (i.e. the kinetic enegy cutoff) is inceased until the total enegy is conveged to within 1 mev pe cell (fo defect fomation enegy calculations the total enegy is conveged to 5 mev/cell). k -point meshes fo the S-DFT calculations ae geneated with the Monkhost-Pack method. 41 The kinetic enegy cutoff the numbe of k -points and the numbe of atoms in each peiodic cell ae listed in Table 3.1. In S-DFT calculations Femi-Diac smeaing is used (smeaing width of 0.1 ev) fo metallic phases with no smeaing fo semiconductos. In OF-DFT the kinetic enegy cutoff used is 1600 ev fo all stuctues which conveges the total enegies to within 1 mev/cell. 54

65 Table 3.1 inetic enegy cutoffs ( E in ev) and k -point meshes used fo vaious cut S-DFT calculations in this wok. The numbes of atoms pe unit cell ae given in paentheses next to each phase (sc is simple cubic ZB is zincblende WZ is wutzite CD is cubic diamond HD is hexagonal diamond and cbcc is complex bcc). E cut k-point fcc(1) hcp() bcc(1) and sc(1) silicon tin() and bct5() silicon ZB() and WZ(4) III-V semiconductos CD() HD(4) and cbcc(8) silicon point defects in CD Si (63: vacancy 65: self-intestitial) A vaiety of popeties wee calculated to test the new EDF including equilibium volumes bulk moduli phase enegy diffeences and defect fomation enegies. The S-DFT equilibium stuctues wee detemined by elaxing each stuctue with a foce theshold of hatee/boh and a stess theshold of 7 10 hatee/boh 3. The OF-DFT equilibium stuctues wee obtained by elaxing the intenal (atomic) coodinates to within a foce theshold of 10 hatee/boh while the lattice paametes including the c/a atio wee optimized manually (as opposed to minimizing the stess tenso). We have not yet deived the stess tenso expession fo this new EDF hence the manual optimization of the lattice vectos. OF-DFT pedictions of equilibium volumes and bulk moduli wee then calculated by expanding and compessing the OF-DFT equilibium unit cell stuctue by up to 5% to obtain eight enegy-vesus-volume points which ae then fit to Munaghan s equation of state. 4 Phase enegy diffeences ae simply the diffeences in total enegy (pe atom o fomula unit) between diffeent phases at 4 55

66 thei equilibium volumes. The phase tansition pessues wee calculated using the common tangent ule: de dv phase-1 de dv phase- P tans. A vacancy in CD Si was modeled by putting eight cubic unit cells togethe in a fashion and then emoving one Si atom fom the cone. With this supecell a self-intestitial defect was constucted by inseting an exta Si atom at a tetahedal intestitial site. Fo the vacancy and self-intestitial defects spinesticted DFT was used and the Si atoms wee not stuctually elaxed duing these point defect benchmak calculations. We ae not attempting to model the actual physical defect accuately which would be bette descibed by spin-polaized DFT and stuctual elaxation; we ae only inteested in testing the EDF in compaison to S-DFT within a given S-DFT model hence the use of spin-esticted DFT and unelaxed stuctues fo simplicity. Point defect enegies ae calculated accoding to Gillan s expession 45 N 1 N 1 E defect E N 11 E( N0 ) N N whee E ( N m ) is the total enegy fo a cell of volume with N atoms and m defects. The sign is fo the self-intestitial defect and the sign is fo the vacancy defect Efficient evaluation of nonlocal EDF In ode to evaluate T NL[] and its potential T ] / with a plane-wave basis unde peiodic bounday conditions in a linea-scaling way we need to efficiently calculate two types of integals NL[ 56

67 P( ) d ( ) ' f ( ') ' 3 Q( ) d. ( ') ' f ( ') ' 3 They ae moe complicated than the standad convolution: Y( ) d ' f ( ') ' 3 whose kenel only depends on the elative distance between and ' and can be calculated efficiently with Fast Fouie tansfoms (FFTs) as ~ ~ ~ Y ( q) ( q) f ( q) ~ ~ ~ whee Y ( q ) ( q) and f ( q ) ae the Fouie tansfoms of Y and f. If an FFT can be used the computational cost is (quasi-) linea scaling i.e. O( N ln( N)). Howeve the kenels in the integals shown fo P( ) and Q( ) depend on infomation at eithe o ' espectively and theefoe the integals cannot be evaluated with FFTs. To make the evaluation of P( ) linea scaling we use an intepolation technique. 79 We fist evaluate P( ) fo selected values of { i } with i 1 / i. Fo each i the evaluation of P( ) educes to a standad convolution and then an FFT can be applied. Afte that we intepolate P( ) ove the space fo the actual distibution of (). The accuacy of this intepolation technique is contolled by the atio. The smalle is the moe accuate ou intepolation will be. In ou cases we find that 1. is enough to convege total enegy to bette than 1 mev pe atom. Using this intepolation technique the computational cost fo evaluating P( ) becomes quasi-linea scaling O( mn ln( N)) whee m is the size of the set } and is detemined by the diffeence between the maximum and minimum of (). N is 57 { i

68 the numbe of planewaves in the OF-DFT calculation. With 1. typical values of m ae aound 100 fo semiconducting CD Si due to lage electon density fluctuations in eal space; fo metallic fcc Al m is aound 0. To make the integal fo Q( ) linea scaling we note that ou kenel oscillates ove long distance. Theefoe we have to fist Fouie tansfom Q( ) obtaining ~ 1 Q( G) f ( ') e ig ' ~ 3 ( G ( ~ ')) d' (3.3) with ~ ig ( ') 3 ( G ( )) ( ') ' e d. We find that ~ does not oscillate in ecipocal space theefoe it can be efficiently splined at the beginning of the computation. We use the cubic Hemite spline which only equies the value and the fist deivative of ~. The spline is expessed as ~ ( G ( ' )) h 00 ~ ( t ) i i h n1 i1 10 ( ' ) ( ' ) ( t ) h i i ~ i i h 01 i1 ~ ( t ) i i1 h 11 ( t ) h i i ~ i 1 (3.4) whee we have n nodes { i } () is the Heaviside function and h 00 h 01 h 11 h10 ae the standad Hemite basis functions: 3 h ( t) t 3t 1 00 h 3 ( t) t t t 10 h 3 01( t) t 3t 58

69 with hi h 3 11 ( t) t t i 1 i t i ( ( ') i )/ hi and ~ ~ i is shot fo ( i ' ). Afte inseting Eq. (3.4) into Eq. (3.3) and moving the summation in font of the integal we ae able to make use of an FFT to do the standad convolution fist and then do the summation. With this spline technique the computational cost of evaluating Q( ) becomes ~ O( mn ln( N)) m is the numbe of nodes used in the spline which is again detemined by the distibution of () in eal space. Again N is the numbe of planewaves. In this way we have made the evaluation of two unconventional convolutions defined in P( ) and Q( ) both almost linea scaling which makes the evaluation of ou new EDF and its potential vey efficient albeit with an exta pefacto of m. Howeve that pefacto can be eliminated by implementing these two numeical techniques in paallel. 59

70 3.4 Results and discussions Bulk popeties To test the quality of this new EDF we focus on the bulk modulus the equilibium volume and the equilibium total enegy of vaious semiconductos. If the new EDF is a good model of the S kinetic enegy functional T S we should at least be able to epoduce the above thee popeties fo each semiconducto by adjusting the only two paametes in ou EDF: and. As a point of efeence coesponding values of the exponents and used in the Wang-Tete EDF ae = = 5/6 11 in Peot s vesion of the Wang-Tete EDF = = 1 63 in Smagiassi s and Madden s vesion of the Wang-Tete EDF = =1/ 61 and in the WGC98 vesion of the Wang-Tete EDF = ( 5 5)/ In this wok + = 8/3 which is diffeent fom the + =5/3 in the Wang-Tete and WGC98 EDFs because ou kenel as defined in Eq. 3.0 is dimensionless wheeas the Wang-Tete and WGC98 EDF kenels have dimensions of the electon density. Table 3. lists the optimal and fo the gound state phase of each semiconducto which wee fitted to the S-DFT equilibium enegies and volumes of each semiconducto gound state. The bulk moduli wee not pat of the fit and theefoe epesent a veification test of the EDF. These bulk popeties ae mainly contolled by the paamete that scales the educed density gadient wheeas the paamete has a much smalle effect. We adjusted only to efine the final OF- DFT equilibium total enegy. Table 3. shows that with optimal paametes ou EDF yields vey good bulk popeties fo all these semiconductos. To ou knowledge this is the fist EDF model able to epoduce S-DFT bulk moduli equilibium enegies and equilibium volumes well fo this lage set of semiconductos with only two paametes. 60

71 Table 3. Optimal nonlocal EDF paametes and fitted to epoduce S- DFT/BLPS equilibium volumes ( V 0 in Å 3 ) and total enegies ( E 0 in ev) pe unit cell fo CD Si and vaious ZB semiconductos. OF-DFT/BLPS and S-DFT/BLPS (in paentheses) pedictions of bulk moduli (B in GPa) as well as best fits to V 0 and E0 ae also given. B V0 E 0 ( 10 ) Si 97 (98) (39.56) (-19.58) AlP 91 (90) (40.637) (-40.18) AlAs 76 (80) ( ) (-3.908) AlSb 61 (60) (56.607) ( ) GaP 87 (80) (37.646) ( ) GaAs 81 (75) (40.634) ( ) GaSb 58 (56) (5.488) ( ) InP 66 (73) (46.040) (-35.7) InAs 63 (65) (49.13) (-8.537) InSb 49 (50) (6.908) (-0.387)

72 Figue 3.1 and Figue 3. futhe illustate the close coespondence between S and OF DFT pedictions via total enegy vesus isotopic volume cuves fo CD and -tin silicon as well as ZB GaAs. Optimal and wee used fo CD silicon ( =0.01 and =0.65) and fo ZB GaAs ( =0.013 and =0.783) poducing tuly excellent ageement. By contast the OF-DFT total enegy vesus volume cuve fo -tin silicon deviates fom the S-DFT cuve due to use of a non-optimal. In pinciple if one wanted to model solely -tin silicon within OF-DFT it would be best to optimize as well as to obtain highe accuacy in the EDF. Figue 3.1 Obital-fee (OF) DFT and ohn-sham (S) DFT total enegy vesus volume cuves fo cubic diamond (CD) and -tin silicon. Fo CD silicon the optimal EDF paametes ae used ( =0.01 and =0.65). Fo -tin silicon the optimal = is used with =0.65 (optimal fo CD silicon). 6

73 Figue 3. OF-DFT and S-DFT total enegy vesus volume cuves fo zincblende (ZB) GaAs. The OF-DFT cuve is calculated with optimal =0.013 and = Next we examine biefly the sensitivity of bulk popeties to the choice of. We focus on CD silicon with fixed to 0.65 (its optimal value) and vay ±40% aound its optimal value fom to which is the ange of optimal values found acoss all semiconductos examined thus fa. Figue 3.3 eveals that the bulk modulus is the stongest function of vaying ±17% aound the optimal value of while the equilibium volume and the total enegy pe atom change moe modestly with [±4% and ±0.04% (<0.1 ev) espectively]. As inceases the bulk modulus inceases while the total enegy and equilibium volume tend to decease. As mentioned ealie the stength of the educed density gadient tem in the nonlocal EDF kenel is detemined by ; we see that subtle coections to the physics ae povided by this tem given that the magnitude of is quite small. 63

74 Figue 3.3 Vaiation in CD silicon bulk modulus equilibium volume pe atom and total enegy pe atom with diffeent. The physics contained in ou new EDF offes a significant impovement ove an ealie attempt to develop a EDF fo covalent mateials. 6 In that case a nonlocal EDF based on unifom-gas linea esponse theoy was poposed again with only two tunable paametes: (1) the aveage valence density * used in a Taylo expansion of the EDF (used to achieve algoithmic linea scaling) and () the exponent defining the two-body Femi wavevecto in the density-dependent kenel of the WGC99 EDF. These two paametes wee optimized but the best pedicted bulk modulus of CD silicon was in eo by 34%. By contast ou bulk modulus of CD silicon lies within % of the S-DFT esult. Moeove although this ealie EDF was able to obtain CD Si as the gound state fo the fist time in an OF-DFT calculation the thee basic stuctual popeties (bulk modulus equilibium volume and equilibium enegy) and the equation of state (enegy vesus volume cuve) of 64

75 CD silicon could not be simultaneously epoduced well by that EDF in contast to the EDF poposed hee Electon density To evaluate the pefomance of the new EDF with espect to epoducing electon densities we compute the self-consistent valence electon density fo CD Si and ZB GaAs using OF-DFT with ou nonlocal EDF whee we set eithe to zeo o a nonzeo value to see how affects the density distibution. The latte case consides the inhomogeneity of the electon distibution while the fome case educes ou new EDF to a WGC99-like EDF that should wok well fo metallic phases. (The WGC99 EDF is double-density dependent wheeas ou new EDF is single-density dependent; WGC99 gives quantitative accuacy fo nealy-feeelecton-like metals.) We then compae the esulting electon densities with benchmak S-DFT electon densities. Figue 3.1 and Figue 3. eveal that ou EDF with nonzeo poduces a density close to the S-DFT density in the coe and lowe density egions (both ae highly inhomogeneous egions); howeve the density in the bond egion (left side of plots) is wose than the 0 case. 65

76 Figue 3.4 The electon density of CD silicon along the [111] diection. Black solid line: S-DFT. Red dashed line: OF-DFT with =0.0. Blue dotted line: OF-DFT with = 0.01 and =0.65. Vetical axis is electon density in 1/boh 3 ; hoizontal axis epesents the gid. Figue 3.5 The electon density of ZB GaAs along the [111] diection. Black solid line: S-DFT. Red dashed line: OF-DFT with λ=0.0. Blue dotted line: OF-DFT with λ=0.013 and β= Vetical axis is electon density in 1/boh 3 ; hoizontal axis epesents the gid. 66

77 3.4.3 Validity of the semilocal appoximation One key assumption made is the intoduction of the educed density gadient in Eq which appoximates the nonlocal pat of the kenel with a semilocal tem. The question aises as to whethe it is physically justified to make this eplacement. We now show that this eplacement appeas valid based on Resta s Thomas-Femi dielectic sceening theoy 80 and ou numeical tests. Let us take CD silicon as an example. Conside the sceening length fo each silicon ion in the cystal. If the sceening length is long ove seveal neaestneighbo distances then it is invalid to make the appoximation given in Eq Howeve accoding to Resta s theoy 80 the sceening length in CD silicon is oughly equal to the neaest-neighbo distance. Resta extended the Thomas-Femi dielectic sceening theoy to semiconductos; peviously the theoy had been used exclusively fo metals. With the sceening length denoted as R and the ionic chage of each silicon ion as Z the sceened electostatic potential of each single silicon ion beyond a cetain distance R is modeled as Z V ( ) R whee is the static dielectic constant in CD silicon and is the distance fom the silicon ion. The unsceened electostatic potential inside R is assumed to obey the Thomas-Femi theoy and is detemined fom the Thomas-Femi equation with the geneal solution of Z q q V ( ) ( e e ) A R 67

78 whee and A ae paametes detemined by the bounday conditions of V () at 0 and R. Hee k / q with 4 F k F 1/ 3 ( 3 ). Resta finally obtains an equation fo R as sinh( qr ) / qr. (3.5) This equation fo R is easily solved using silicon s aveage valence density and its static dielectic constant as input. The sceening length R fo CD silicon obtained fom this equation is vey close to the neaest-neighbo distance which indicates that only neaest-neighbo silicon atoms paticipate in sceening. Simila esults wee found fo CD gemanium and cabon by Resta. 80 Thus based on this Thomas-Femi dielectic sceening theoy we ague that the nonlocality of the EDF is weak outside fist neaest neighbos and theefoe it is valid to make the appoximation given in Eq This featue was also exploited implicitly in Cotona s wok 81 in which he successfully calculated vaious bulk popeties of many semiconductos using his embedding theoy. In Cotona s appoach he divides the bulk into atoms and solves S-DFT equations fo each atom with an atom-centeed Gaussian basis with the atoms consideed to be embedded in the bulk. He numeically showed that fo CD Si and many othe semiconductos the embedding potential due to the kinetic enegy inteactions between atoms is appoximated easonably well with the Thomas-Femi EDF (a local EDF) and his quite good esults povide suppot fo ou agument: the nonlocality of the EDF fom one atom to anothe in solids with band gaps is weak beyond fist neaest neighbos. 68

79 3.4.4 Tansfeability and cystal stuctue dependence of λ and β Given that the sceening length R deived fom Eq. 3.5 is shot-anged fo semiconductos then we may infe that the local bonding envionment in the cystal contols most of the dielectic sceening. We theefoe would expect the paametes λ and β to be tansfeable between solids of simila local bonding motifs. We acknowledge that this hypothesis ignoes the fact that the dielectic function is a global popety. To test this hypothesis we investigate the tansfeability of λ and β optimized fo CD Si fo pedicting popeties of HD and cbcc Si all of which ae tetahedallybonded (Table 3.3). Likewise we test paamete tansfeability fo the tetahedally-bonded ZB and WZ stuctues of vaious binay semiconductos (Table 3.4). In othe wods λ and β wee fist fit to epoduce the equilibium volume and enegy of CD Si and then these paametes wee used to calculate the bulk popeties of HD and cbcc Si. Fo binay semiconductos and wee fist fit to the ZB stuctues and then wee applied to WZ stuctues. Table 3.3 and Table 3.4 clealy show that and ae tansfeable between CD HD and cbcc silicon as well as between the ZB and WZ stuctues fo each binay semiconducto in tems of pedicting the small S-DFT enegy diffeences between vaious phases (to within 5 mev). The tansfeability of and is also evident in the epoduction of S-DFT bulk moduli (aside fom the 15% deviation fo WZ GaP) and equilibium volumes fo HD and cbcc silicon as well as fo the WZ stuctues. 69

80 Table 3.3 Bulk popeties of silicon in its CD HD and cbcc phases as pedicted by S-DFT/BLPS (in paentheses) and OF-DFT/BLPS using the new nonlocal EDF with 110 and (paametes optimized fo CD silicon only). Bulk moduli (B) ae in GPa and equilibium volumes pe atom ( V 0 ) ae in Å 3. The equilibium total enegy pe atom ( E 0 ) fo CD silicon and the enegies of othe stuctues elative to the CD phase ae in ev. Si stuctues B V 0 E 0 CD 97 (98) (19.781) ( ) HD 98 (99) (19.64) (0.015) cbcc 105 (10) (17.517) (0.1) 70

81 Table 3.4 S-DFT/BLPS (in paentheses) and OF-DFT/BLPS bulk moduli ( B in GPa) and equilibium volumes pe fomula unit ( V 0 in Å 3 ) fo WZ stuctues of vaious binay semiconductos. Enegy diffeences pe fomula unit ( E E in mev) between WZ and ZB stuctues ae also listed. c/a atios optimized in OF-DFT ae compaed to S-DFT atios (in paentheses). OF-DFT/BLPS esults ae calculated using the and listed in Table 3. (fitted fo each ZB semiconducto). WZ WZ WZ ZB B WZ V 0 WZ WZ EZB E c/a AlP 9 (90) (40.608) (9) 1.64 (1.64) AlAs 79 (80) (43.61) 5 (11) 1.66 (1.65) AlSb 59 (58) (56.548) 16 (13) 1.65 (1.65) GaP 76 (88) (37.65) 6 (18) 1.65 (1.65) GaAs 76 (76) (40.611) 5 (19) 1.65 (1.65) GaSb 59 (57) 5.86 (5.397) 15 (16) 1.64 (1.65) InP 73 (73) (46.037) 19 (3) 1.64 (1.64) InAs 66 (65) (49.19) 3 (7) 1.65 (1.64) InSb 48 (50) (6.884) 17 (11) 1.64 (1.65) Although we have just agued and then numeical demonstated good tansfeability as long as the local bonding envionment is simila once the coodination numbe changes the optimal is no longe tansfeable. To illustate this we conside the tend in optimal fo vaious phases of Si with fixed (since as mentioned ealie is found to be much moe impotant than in detemining bulk popeties). We optimize fo each Si phase to yield the best equilibium enegy and volume with fixed to 0.65 which is optimal fo CD silicon. Table 3.5 eveals the geneal tend that the optimal becomes smalle fo stuctues with highe coodination numbes (moe metallic). This tend fo makes complete 71

82 sense when one consides the elationship between and static dielectic constant pointed out in Section 3.. Theefoe 0 coesponds to the infinite static dielectic constant case i.e. a metal wheeas finite lambda coesponds to a finite static dielectic constant i.e. a semiconducto. If we let 0 ou EDF becomes physically simila to the Wang-Tete 11 and WGC99 13 EDFs in which the fome s kenel has no density dependence while the latte s kenel has a double-density dependence. Ou EDF s kenel has single density dependence. Like the Wang-Tete and WGC99 EDFs that descibe nealy-fee-electon-like metals well ou new EDF with 0 gives a good desciption fo metallic stuctues of Si aside fom pedicting the hcp phase to be less stable than it should be (howeve the enegy diffeences ae vey small and cetainly within the typical uncetainty of S-DFT). With these optimal the equilibium volumes ae all faily well epoduced by ou EDF except fo the hypothetical bct5 phase (14% deviation). OF-DFT bulk moduli of the -tin though the fcc stuctues ae all unifomly shifted downwads by 0-30 GPa fom the S-DFT pedictions suggesting that if these metallic phases ae of inteest optimization of the paamete may be citical. Ealie OF-DFT studies of these Si phases using a e-paameteization of the WGC99 EDF by Zhou et al. 6 had touble epoducing the small enegy diffeence between CD and HD silicon wheeas ou new EDF captues this small enegy diffeence quite well. The notable cystal stuctue dependence of exhibited hee suggests that a potential futue avenue of eseach could be to paameteize as a function of coodination numbe. 7

83 Table 3.5 Bulk moduli B 0 (in GPa) equilibium volumes ( V 0 ) (in Å 3 /atom) and equilibium total enegies ( E 0 ) (in ev/atom) fo vaious silicon phases (abbeviations defined in the text) calculated using S-DFT/BLPS (in paentheses) and OF-DFT/BLPS with the optimal (value listed should be multiplied by 10 ) fo each stuctue and fixed at 0.65 (optimal fo CD Si). c.n. stands fo coodination numbe. Si λ B0 V0 E0 c.n. CD (98) (19.781) ( ) 4 HD (99) (19.64) (0.015) 4 CBCC (10) (17.517) (0.1) 4 -tin (99) (14.660) (0.168) 6 bct (96) (16.905) (0.15) 5 sc (11) (15.484) 0.6 (0.9) 6 hcp (91) (14.157) (0.340) 1 bcc (98) (14.60) (0.351) 8 fcc (83) (14.37) (0.381) 1 We also tested ou EDF s tansfeability by calculating the tansition pessue fo Si tansfoming fom the CD to the -tin stuctue. When the EDF paametes optimized fo CD Si ( 110 and ) ae used fo both phases the pedicted tansition pessue is -.3 GPa (!) compaed to S-DFT tansition pessues of 5.4 (using the BLPS) and 7.4 GPa (using the NLPS). This unphysical OF- DFT esult is undoubtedly due to the non-optimal fo the -tin stuctue which is metallic and has a lage coodination numbe than the CD stuctue. As discussed ealie and shown in Figue 3.3 when is too lage it poduces too low a total enegy. In this case the too-lage value of ovely stabilizes the -tin stuctue so 73

84 its enegy is below the CD stuctue of silicon! As a esult the tansition pessue becomes negative. If instead the optimal smalle is used fo -tin silicon (see Figue 3.1 whee fo both phases) the OF-DFT tansition pessue is pedicted to be a physically easonable 6.4 GPa quite close to the S- DFT/BLPS value of 5.4 GPa. coodination numbe changes seems clea. Fom these esults the need to let vay as the In ealie wok Zhou et al. 6 obtained an OF-DFT/BLPS CD to -tin tansition pessue fo silicon of 1.0 GPa compaed to a S-DFT/BLPS tansition pessue of 10. GPa using a diffeent BLPS and the e-paameteized WGC99 EDF. Given that the equations of state obtained fo these two phases in that wok exhibited lage eos the good ageement of the tansition pessues was likely fotuitous. Using and fitted to pefect CD silicon we futhe tested tansfeability by calculating the vacancy and self-intestitial defect fomation enegies in CD silicon. The vacancy fomation enegy is only off by 0.35 ev: S-DFT/BLPS pedicts 3.04 ev while OF-DFT/BLPS yields.69 ev. Howeve the self-intestitial fomation enegy fom OF-DFT with ou EDF is again the wong sign (S: 3.9 ev vesus OF: ev) which again is likely due to an impope used fo the inseted Si atom at the intestitial position. The intestitial Si atom and its neighbos now have highe coodination numbes which would be bette descibed by a smalle value of (see tend in Table 3.5). As mentioned above fo -tin we find that too lage a ovestabilizes close-packed stuctues. Theefoe the inappopiately lage used fo the intestitial Si atom poduces too low an enegy which in tun esults in the negative self-intestitial fomation enegy. Again this atifact could vey well disappea if we wee to paameteize based on the local envionment instead of using a constant thoughout the cell. 74

85 3.4.5 Recommended EDF paametes fo tetahedallybonded semiconductos Although the inteelationship of and the static dielectic constant fomally pecludes a single optimum value of fo all semiconductos fo pactical calculations it would be pefeable to have one set of paametes to use fo any semiconducting mateial. Consequently we tested the tansfeability of the aveage values of and whee we take a simple aveage of the optimal values shown in Table 3.. These aveage paamete values ( = and =0.7143) ae used to define the EDF. As Table 3.6 shows S bulk moduli ae epoduced to within 5-10% fo most semiconductos (except fo GaP which is off by ~0%) and S equilibium volumes ae epoduced to within 4%. The total enegy is moe sensitive to changes in and with maximum eo of about 1% (eos of <.3 ev). The total enegy eo incued using OF-DFT is faily unifom fo all the semiconductos as evidenced by examining the enegy diffeences between CD silicon and othe ZB binay semiconductos (Fig. 3.6). The S enegy odeing tends i.e. elative stability among these semiconductos ae well epoduced by OF-DFT with this single set of and. Thus we conclude that these aveaged values of and will be a good fist choice fo modeling most tetahedally-bonded semiconductos. 75

86 Figue 3.6 OF-DFT and S-DFT elative enegy diffeences between CD silicon and ZB semiconductos (pe pimitive cell). OF-DFT esults using an aveaged = and = closely match the enegy odeing fom S-DFT. See Table

87 Table 3.6 Bulk moduli (B) equilibium volumes (V0) and equilibium total enegies (E0) pe unit cell fo CD silicon and ZB semiconductos calculated by OF-DFT with = and = (aveaged fom Table 3.). S-DFT values ae given in paentheses. B (GPa) V0 (Å 3 ) E0 (ev) Silicon 100 (98) (39.56) (-19.58) AlP 89 (90) (40.637) (-40.18) AlAs 76 (80) (43.616) (-3.908) AlSb 61 (60) (56.607) ( ) GaP 94 (80) (37.646) ( ) GaAs 78 (75) (40.634) ( ) GaSb 6 (56) (5.488) ( ) InP 68 (73) (46.040) (-35.7) InAs 61 (65) (49.13) (-8.537) InSb 49 (50) (6.908) (-0.387) 3.5 Local pseudopotentials fo Ga In P As and Sb Fo all local pseudopotential S-DFT and OF-DFT calculations in this wok we employ bulk-deived local pseudopotentials 78 (BLPSs) and the local density appoximation (LDA) Fo silicon we use the BLPS peviously epoted and tested in Ref. 78 wheeas fo othe binay semiconductos we use new BLPSs built with the same method. 78 The new BLPSs fo Ga In P As and Sb ae shown in Figs

88 Figue 3.7 Gallium BLPS in eal space. Coulombic tail is enfoced beyond 3.5 boh. Figue 3.8 Indium BLPS in eal space. Coulombic tail is enfoced beyond 4.0 boh. 78

89 Figue 3.9 Phosphous BLPS in eal space. Coulombic tail is enfoced beyond 3.5 boh. Figue 3.10 Asenic BLPS in eal space. Coulombic tail is enfoced beyond 4.0 boh. 79

90 Figue 3.11 Antimony BLPS in eal space. Coulombic tail is enfoced beyond 4.0 boh. The two paametes equied to build these BLPSs i.e. (1) the value of the non- Coulomb pat of the Fouie-tansfomed BLPS at q 0 and () the position in eal space beyond which a Coulomb tail is enfoced ae chosen to epoduce the S-DFT nonlocal pseudopotential (S-DFT/NLPS) enegy odeing fo face-centeed cubic (fcc) simple cubic (sc) body-centeed cubic (bcc) and cubic diamond (CD) phases of these elements. These stuctues ae chosen because they span a wide ange of coodination numbe envionments in a solid. In building the BLPSs Toullie- Matins 37 nonlocal pseudopotentials ae geneated with the FHI98 code 38 using default cutoff adii. Femi-Diac smeaing with a smeaing width of 0.1 ev is used all though the pocess of building of BLPSs and the planewave basis kinetic enegy cutoff is 000 ev fo all cases to obtain accuate taget S-DFT/NLPS electon densities used in the BLPS constuction scheme. The k -point mesh used duing BLPS constuction is fo fcc sc and bcc and 1 11 fo CD. The 80

91 numbe of atoms pe unit cell used is as given in Table 3.1 of the main text fo elemental Si. The BLPSs ae then tested on the known gound states of these elements i.e. - gallium (Ga) 8 -asenic (As) 83 and -antimony (Sb) 84 body-centeed-tetagonal (bct) indium (In) 85 and A17 phosphous (P) 86 and thei popeties ae compaed to othe phases. In these tests planewave basis kinetic enegy cutoffs of 800 ev and k -point meshes ae used to convege the total enegy pe cell to within 1 mev. No Femi smeaing is used fo insulatos. Fo metallic solids a Femi-Diac smeaing width of 0.1 ev is used. The numbe of atoms used pe unit cell fo the gound state stuctues ae as follows: two atoms each fo -As and -Sb and eight atoms each fo α-ga and A17 P. The S-LDA-NLPS enegy odeings equilibium volumes and bulk moduli of diffeent phases ae mostly qualitatively if not quantitatively epoduced with these new BLPSs as shown in Tables Only two enegy odeings ae inveted fo two elements the bcc vesus CD phases of phosphous and asenic but the enegy diffeences between these two phases in both cases ae quite small ( 5 mev) cetainly within the uncetainty of S-LDA oveall. The phosphous enegy diffeences between phases show the lagest eos but at least the coect gound state is obtained. -gallium is not pedicted to be the gound state stuctue with eithe the NLPS o the BLPS which might be due to the lack of a nonlinea coe coection in the Ga NLPS o due to the LDA desciption of exchange-coelation. The equilibium volumes of all phases of all elements ae quite well epoduced by the BLPSs; eos in bulk moduli in some cases ae significantly lage. 81

92 Table 3.7 Compaison of NLPS and BLPS S-LDA bulk popeties of the fcc hcp bcc sc and CD phases of gallium. The bulk modulus (B) is in GPa the equilibium volume pe atom (V0) is in Å 3 and the equilibium total enegy (E0) fo -Ga is in ev/atom. The total enegies (ev/atom) of othe Ga stuctues ae listed elative to -Ga s equilibum total enegy. -Ga is the expeimental gound state at low tempeatue. 8 The same convention and units fo B V0 and E0 ae used in all subsequent tables. Gallium fcc hcp bcc sc CD B V0 E0 NLPS BLPS NLPS BLPS NLPS BLPS Table 3.8 Compaison of NLPS and BLPS S-LDA bulk popeties of the bodycenteed-tetagonal (bct) fcc bcc sc and CD phases of indium. bct is the expeimental gound state stuctue. 85 Indium bct fcc bcc sc CD NLPS B BLPS NLPS V0 E0 BLPS NLPS BLPS

93 Table 3.9 Compaison of NLPS and BLPS S-LDA bulk popeties of the A17 sc CD bcc and fcc phases of phosphous. The A17 stuctue i.e. black phosphous is the gound stuctue unde ambient conditions. 86 Phosphous A17 sc CD bcc fcc B NLPS BLPS V0 NLPS BLPS E0 NLPS BLPS Table 3.10 Compaison of NLPS and BLPS S-LDA bulk popeties of the sc bcc CD and fcc phases of asenic. -As is the expeimental gound state at low tempeatue. 83 Asenic sc bcc CD fcc B NLPS BLPS V0 NLPS BLPS E0 NLPS BLPS

94 Table 3.11 Compaison of NLPS and BLPS S-LDA bulk popeties of the sc bcc fcc and CD phases of antimony. -Sb is the expeimental gound state at low tempeatue. 84 Antimony sc bcc fcc CD B NLPS BLPS V0 NLPS BLPS E0 NLPS BLPS Table 3.1 povides a tansfeability test of the bulk popeties poduced by these BLPSs fo zincblende (ZB) binay semiconductos which we see agee quite well with the NLPS pedictions (except fo the bulk modulus of InP). The numbe of atoms used pe unit cell is as given in Table 3.1 of the main text. The absolute total enegies ae included fo completeness but thee is no eason that diffeent PSs should give the same absolute total enegies so deviations in the last column ae not meaningful. As we ae inteested pimaily in using these BLPSs fo studying such binay compounds athe than the pue elements these esults ae encouaging. 84

95 Table 3.1 S-LDA-BLPS pedictions of the bulk popeties fo zincblende binay semiconductos: bulk modulus (B) equilibium volume (V0) and equilibium total enegy (E0) pe fomula unit. NLPS esults ae in paentheses fo compaison. B (GPa) V0 (Å 3 ) E0 (ev) AlP 90 (89) (39.577) ( ) AlAs 80 (75) (43.708) (-3.169) AlSb 60 (57) (55.917) ( ) GaP 80 (90) (37.575) ( ) GaAs 75 (79) (4.169) ( ) GaSb 56 (60) (5.855) ( ) InP 73 (88) (44.001) ( ) InAs 65 (73) (48.794) (-31.45) InSb 50 (55) (60.890) ( ) In addition to veifying electonic stuctual popeties we tested the electonic stuctue tansfeability of these BLPSs by compaing pedicted S-LDA/NLPS band gaps (S eigenvalue gaps) with S-LDA/BLPS band gaps fo the ZB and wutzite (WZ) stuctues of each binay semiconducto as well as fo CD and hexagonal diamond (HD) silicon (Table 3.13). Again the numbe of atoms used pe unit cell is as given in Table 3.1 of the main text. Geneally ou BLPSs give compaable band gaps fo most semiconductos except fo HD Si (zeo band gap) and ZB and WZ InAs and InSb (too lage gaps). Moe difficult is the coect pediction of the natue of the gaps namely whethe they ae diect o indiect. Howeve since the main pupose of this wok is to test ou new kinetic enegy density functional on semiconductos as long as ou BLPSs pedict these mateials to be semiconductos within S-DFT (which they do except fo HD silicon) we should be on solid gound as thee is no band stuctue within OF-DFT anyway. 85

96 Table 3.13 Compaison of NLPS and BLPS S-LDA eigenvalue band gaps (in ev) of vaious semiconductos (ZB: zincblende WZ: wutzite). (I) indicates an indiect band gap and (D) a diect band gap. Hexagonal diamond (HD) silicon has zeo band gap using the silicon BLPS. silicon AlP AlAs AlSb GaP GaAs GaSb InP InAs InSb NLPS BLPS CD 0.46 (I) 0.9 (I) HD 0.8 (I) zeo ZB 1.41 (I) 1.16 (I) WZ 1.79 (I) 1.4 (I) ZB 1.18 (I) 1.17 (I) WZ 1.51 (I) 1.33 (D) ZB 1.1 (I) 0.74 (I) WZ 1.01 (D) 0.66 (I) ZB 1.38 (I) 1.08 (I) WZ 1.36 (I) 1.10 (D) ZB 0.90 (D) 1.13 (I) WZ 0.83 (D) 0.99 (D) ZB 0.44 (D) 0.47 (I) WZ 0.19 (D) 0.4 (I) ZB 1.51 (I) 1.3 (I) WZ 1.61 (D) 1.43 (D) ZB 0.45 (D) 1.6 (I) WZ 0.57 (D) 1.55 (D) ZB 0.0 (D) 1.06 (I) WZ 0.4 (D) 1.04 (I) 86

97 3.6 Conclusion In this wok we discussed the ~ q asymptotic behavio at the 0 q limit of the susceptibility function in semiconductos. We pointed out that any EDF designed fo semiconductos theefoe should behave as 1/ ' as '. Based on this equiement we poposed a geneal EDF fom whose kenel explicitly contains 1/ ' and that has only two adjustable paametes. One of the paametes ( ) was shown to be elated to the static dielectic constant. We tested ou EDF on popeties of vaious binay semiconductos and a vaiety of phases of silicon. Since each semiconducto has a diffeent dielectic function a univesal value fo cannot exist. Howeve the two paametes and in ou EDF can be adjusted to simultaneously epoduce thee bulk popeties of each semiconducto: S-DFT bulk moduli equilibium volumes and equilibium enegies. The paametes and in ou EDF ae obseved to depend on coodination numbe; ou numeical esults can be explained by appealing to Resta s Thomas- Femi dielectic sceening theoy which demonstates that in semiconductos the effective sceening length is essentially a bond length. Consequently the EDF paametes ae quite tansfeable within phases possessing the same local coodination numbe. We also detemined a tend in the optimal fo diffeent silicon phases with lage pefeed fo small coodination numbes and smalle fo lage coodination numbes which can be undestood based on the tend in dielectic constants fo semiconducting vesus metallic phases. As moe sevee tests we calculated the tansition pessue fo the phase tansition fom CD Si to -tin Si as well as point defect fomation enegies in CD Si. Fom these latte calculations a coodination-numbe-dependent paameteization of instead of a constant thoughout the cystal appeas to be equied. Howeve fo all tetahedal-bonded semiconductos consideed in this wok a single pai of and in ou new EDF used within OF-DFT is able to epoduce quite well S-DFT pedictions of basic bulk popeties and elative enegy odeings among vaious mateial phases. We 87

98 believe this wok povides a new diection fo futue developments of EDFs fo semiconductos. 3.7 Acknowledgements We ae gateful to the National Science Foundation fo financial suppot of this wok. We thank the helpful discussion with Pofesso E.. U. Goss on the ohn- Sham esponse function. 88

99 Chapte IV Extending OF-DFT fo tansition metals and semiconductos 4.1 Intoduction In obital-fee density functional theoy (OF-DFT) one key pat is the ohn- Sham (S) kinetic enegy density functional (EDF). How to constuct a univesal EDF fo inhomogeneous mateials still emains challenging. EDFs based on the linea esponse function of the nealy-fee electon gas have been shown to yield vaious bulk popeties in good ageement with S-DFT fo most main goup metals In ode to apply OF-DFT a wide ange of mateials it is theefoe desiable to extend OF-DFT to both tansition metals and semiconductos. The difficulty in teating tansition metals using OF-DFT is that the localized electons in the outmost d angula momentum channel cannot be teated popely with the EDFs based on the nealy-fee electon gas. In this chapte an altenative way to teat tansition metals in OF-DFT is demonstated. Fist the valence electon density of tansition metals (including the outmost s and d electons) is decomposed into a delocalized and a localized pat. Each pat is then teated with a physically appopiate EDF as outlined in moe detail below. Chapte III pesented some effots fo constucting a nonlocal EDF fo semiconductos by enfocing a coect asymptotic behavio of the esponse function in the EDF at the q=0 point in Fouie space. 4 This nonlocal EDF still has two paametes with one (λ) that appeas stongly dependent on the local cystal stuctue. An intuitive stategy is then to let the value of λ vay in space and be 89

100 detemined by the local cystal stuctue via assessment of atomic coodination numbes. Although this new nonlocal EDF with a λ field (λ()) is able to pedict the bulk modulus total enegy and equilibium volume of cubic diamond (CD) silicon as well as othe silicon phases easonably well it gives shea-elated moduli of CD silicon that diffe significantly fom benchmak S-DFT esults. These shea moduli ae mainly contolled by bond-bending enegies that ae extemely difficult to descibe with EDFs based solely on electon density. To emedy these eos an explicit bond-bending enegy tem was added in an ad hoc way to the OF-DFT total enegy functional. This modified OF-DFT total enegy functional is able to yield vaious basic bulk popeties including shea moduli fo CD silicon and othe silicon phases that ae in easonable ageement with S-DFT benchmaks. 4. OF-DFT fo tansition metals 4..1 Local pseudopotentials One difficulty in applying OF-DFT to tansition metals is the equiement of high quality local pseudopotentials (LPSs). Fo tansition metals it is convenient to wok with nonlocal pseudopotentials (NLPSs) in which we constuct diffeent pseudopotentials V lm () fo the s p and d angula momentum channels espectively. The s p and d nonlocal pojectos i.e. the spheical hamonics Y lm ae attached to the coesponding V lm () to complete the NLPS17 V ( ) NLPS Ylm Vlm ( ) Ylm. lm Hence fo a NLPS the V lm () only act on the S obitals with the same angula momentum. Howeve in OF-DFT we cannot use NLPSs due to the lack of S obitals. We have to use LPSs. To build LPSs fo tansition metals we need to follow some basic 90

101 ules. Accoding to the Hellmann-Feynman theoem the lowest enegy eigenfunction in a spheically symmetic potential should have s angula momentum. This can be shown by teating the Hamiltonian Ĥ as a function of the angula momentum l theefoe l 0 gives the lowest enegy due to the fact that H ˆ [ l]/ l 0. On the othe hand usually the outemost s and d electons ae consideed as the valence electons. Fo example in the Ag atom the valence electons contains ten 4d electons and one 5s electon with the 4d obital having a lowe eigenvalue than the 5s obital. Consequently in ode to build a physical LPS fo Ag we have to include Ag s 4s and 4p electons as the additional valence electons. The impotance of 4s and 4p electons in building the LPS was not consideed in the pevious wok by Zhou et al. 6 In all calculations in this chapte the local density appoximation 36 (LDA) is used fo exchange-coelation (XC). The bulk-deived LPS (BLPS) fo Ag is built as follows. Fist Ag s 4s and 4p electon densities ae geneated by pefoming a self-consistent S-DFT-LDA calculation fo a single Ag atom using the FHI98 pogam. 38 Then bulk S-DFT-LDA electon densities (consisting of Ag s 4d and 5s electons) fo thee bulk phases (see below) ae geneated using the ABINIT pogam 33 using an NLPS that acts on Ag s 4d and 5s electons. This Ag NLPS is also built with the FHI98 pogam using its default settings. Then the oute coe electons geneated ealie ae placed at each atom in the unit cell of each stuctue and the two sets of electon densities ae supeimposed to constuct the total electon density which theefoe contains Ag s 4s 4p 4d and 5s electons. We then invet S equation fo each of these total electon densities to solve fo the coesponding S effective potential. The pocedue fo inveting the S equation has been discussed in detail in Chapte II. This pocedue is pefomed fo thee bulk stuctues: face-centeed-cubic (FCC) body-centeed cubic (BCC) and simple cubic (SC). The plane wave basis kinetic enegy cutoff used in the invesion pocedue is 4000 ev. The k-point meshes 41 ae and 8 88 fo FCC BCC and SC pimitive cells 91

102 espectively. In all calculations Femi-Diac smeaing with a smeaing width of 0.1 ev is used. With the S effective potential in hand the potocols in Chapte II wee followed to calculate the atom-centeed potentials. The data fom FCC BCC and SC ae plotted in Fig In the uppe plot of Fig. 4.1 the V nc (q) is shown in which we have emoved the Coulomb pat fom the oiginal data V (q) as V nc 4Z ( q) V ( q) q whee Z is equal to 19 fo Ag with ou Ag BLPS. In the lowe plot of Fig. 4.1 we see a smooth but deep BLPS fo Ag Fouie-tansfomed to eal space. The deepe well is due to the inclusion of Ag s 4s and 4p electons. Table 4.1 displays the esults of testing ou Ag BLPS by calculating within S- DFT-LDA basic bulk popeties fo five Ag phases; since the HCP and CD phases wee not used to geneate data to constuct the BLPS data fo these two phases offe a test of tansfeability of the BLPS. The esults ae then compaed to Ag NLPS S-DFT benchmaks. inetic enegy cutoffs of 3000 and 1500 ev with a k-point mesh of ae used to convege the total enegy to within 10 mev. Fo all stuctues pimitive unit cells ae used. Femi-Diac smeaing with a smeaing width of 0.1 ev is used thoughout the calculations. We see thee is good ageement between popeties pedicted by the BLPS and the NLPS fo all phases. We also calculated low-index suface enegies of FCC Ag with ou BLPS (Table 4.) using a kinetic enegy cutoff of 3000 ev and a k-point mesh fo all suface S-DFT calculations. The FCC(100) FCC(110) and FCC(111) sufaces ae modeled with five seven and five Ag FCC layes espectively with 10 Å vacuum in the suface nomal diection to isolate peiodic images of the suface slab in the supecell. We see in Table 4. that the suface enegies fom ou Ag BLPS ae consistently lowe than the NLPS by about 180 mj/m o 90 mev pe top-laye 9

103 atom (if all the 180 mj/m mismatch is attibuted to the eo fom the top-laye atom). This is consistent with the slightly smalle enegy diffeence between CD and FCC bulk Ag using the BLPS compaed to the NLPS (see Table 4.1) since the coodination numbe deceases both at the suface and going fom FCC to CD. Figue 4.1 Ag BLPS in Fouie space (uppe plot) and in eal space (lowe plot). See text fo moe details. 93

104 Table 4.1 Compaison between the Ag BLPS and NLPS using S-DFT-LDA fo basic bulk popeties of five phases of Ag: bulk moduli (B in GPa) deivatives of the bulk modulus with espect to pessue (db/dp dimensionless) equilibium volumes (V0 in boh 3 ) equilibium enegies (E0 in hatees Ha) and the enegy diffeences (ΔE in mha) between the gound state FCC phase and the othe HCP BCC SC and CD phases. NLPS esults ae in paentheses as benchmaks. Recall that the total enegies E0 cannot be compaed acoss diffeent PSs. Ag FCC HCP BCC SC CD B 134 (133) 133 (131) 131 (131) 103 (101) 50 (49) db/dp 5.6 (5.6) 5.6 (5.1) 5.6 (5.1) 5.6 (5.6) 4.7 (5.0) V ( ) ( ) (110.87) (17.476) ( ) E (-36.33) (-36.3) (-36.3) (-36.18) ( ) ΔE 0.0 (0.0) 0.06 (0.08) 1.0 (1.3) 10.6 (14.5) 30.4 (38.0) Table 4. Compaison between the BLPS and NLPS S-DFT-LDA pedictions of suface enegies (mj/m ) fo FCC(111) (100) and (110) Ag sufaces. Ag fcc(111) fcc(100) fcc(110) BLPS NLPS eo

105 4.. Decomposition of valence electon density Because of the localized d electons in tansition metals it is not staightfowad to apply those EDFs developed fo the nealy-fee electon gas diectly to tansition metals. To tackle such poblems we patition the valence electon density into two pats: the delocalized electon density and the localized density. In pactice the patitioning is not unique. Again let s take Ag atom as an example. In Fig. 4. we demonstate one choice of the decomposition. The taget valence electons include Ag s 4s 4p 4d and 5s electons. The delocalized electon density del (ed cuve) is constucted in such a way that it matches the valence electon density valence (blue line) beyond a pedefined coe adius C : valence ( C ) del. a ( C ) This coe adius C is chosen to be. boh. Inside the coe adius the delocalized density a goes to zeo smoothly appoaching the nuclei. The deivative d a / d is constucted to be zeo at =0 to educe the numbe of plane waves needed in OF- DFT calculation fo expanding del. In Fig 4. the integal of del in the space is set to be 4 which is found to be optimal fo yielding good esults fo diffeent Ag phases tested late. The localized electon density local (geen cuve in Fig. 4.) is obtained by subtacting the delocalized electon density del fom the valence density valence. Unde this decomposition the total OF-DFT electonic enegy is modified to E tot [ tot ] ( TS [ tot ] TS [ del ]) TS [ del ] EXC [ tot ] J[ tot ] Vext ( ) tot ( ) d 3 whee tot del local. Duing the minimization of E tot only the delocalized electons density del is elaxed while the localized electon density local is fixed all the time. To evaluate the kinetic enegy we employed local EDFs 95

106 at [ ] bt [ ] to evaluate the fist two tems in paentheses. Because of the TF vw localization of electons within C the electon density inside C is high. Fo such elatively slowly-vaying electon density on the scale of the local Femi wavelength we expect that a local EDF will be a good choice. 87 The EDF tem T S [ del ] outside the paentheses can be evaluated with any EDFs that ae constucted based on the nealy-fee electon gas. We use WGC99 EDF 13 hee. Figue 4. Decomposition of the Ag atom valence electon density (blue) into the delocalized pat (ed) and the localized pat (geen). The valence electon density (blue) contains the 4s 4p 4d and 5s electons. 96

107 In Table 4.3 we show some basic bulk popeties calculated with S-DFT-LDA (the benchmak) and OF-DFT using this decomposition technique. In both calculations the Ag BLPS built in the pevious section is used. The paametes used fo at [ ] bt [ ] ae a=0.84 and b=0.18 which ae adjusted to give the best OF- TF vw DFT modulus and volume fo FCC Ag. In Table 4.3 we find good ageement between the S-DFT benchmak and the OF-DFT esults fo bulk moduli equilibium volumes and the deivative of bulk moduli with espect to the extenal pessue fo all the Ag phases consideed hee. Howeve the OF-DFT enegy diffeences between Ag phases ae almost doubled compaed to the S-DFT esults. Despite the deviations just mentioned both the new BLPS and this new EDF ansatz ae significant impovements ove ealie wok. Zhou et al. s ealie Ag BLPS 6 will not bind Ag s cubic diamond stuctue even using S-DFT-LDA that tends to ovebind. This is likely due to the exclusion of Ag s 4s and 4p electons in Zhou et al. s constuction of Ag s BLPS which in tun geneated an unphysical Ag BLPS. Moving to OF-DFT calculations Zhou et al. found unbound enegy-vesusvolume cuves fo the gound state Ag FCC phase with eithe 0.4vW TF (vw : von Weizsäcke 76 EDF and TF : Thomas-Femi EDF) o the Wang-Tete 11 EDF. By contast by decomposing the valence electon density as we have descibed we apply physically appopiate EDFs to diffeent densities and ae able to get easonable popeties fo Ag FCC BCC HCP SC and CD stuctues fo the fist time. 97

108 Table 4.3 Compaison of basic bulk popeties of Ag phases pedicted by S-DFT- LDA (the benchmak in paentheses) and OF-DFT-LDA. See the caption of Table 4.1 fo notation and units. The Ag BLPS is used in all S and OF calculations. In the semilocal EDF at bt used to evaluate the localized electon density contibution TF vw to the kinetic enegy we set a=0.84 and b=0.18 which wee adjusted to yield a good bulk modulus and equilibium volume fo FCC Ag. The delocalized electon density contibution to the kinetic enegy is evaluated using the WGC99 nonlocal EDF. Ag FCC BCC SC CD B 141 (134) 141 (131) 104 (103) 46 (50) db/dp 3.7 (5.6) 4.0 (5.6) 5.65 (5.59) 3.6 (4.7) V ( ) ( ) (16.771) ( ) E ( ) ( ) ( ) ( ) ΔE 0.0 (0.0) 1. (1.3) 5.8 (14.5) 68.0 (38.0) 4.3 Towads a univesal EDF fo semiconductos Coodination numbe dependent λ Recall the nonlocal EDF poposed in Chapte III 3 3 s ' ] ( ) d ' T[ ] T [ ] T [ ] ( ) [ k ( ) 1 d. (4.1) TF VW F 98

109 Since the paamete λ is elated to the local cystal stuctue (see Table 3.5) hee we aim adjust its value on-the-fly based on the local coodination numbes of atoms. Ou goal is to constuct a spatial vaiable ( ) to eplace the scala λ in Eq. (4.1). We fist calculate the coodination numbe ( CN ) of each atom in the bulk accoding to the expession poposed by Gimme: 88 N CN A atom 1 f ( AB 1 exp[ k ( k ( R R ) / 1)] ) (4.) B A 1 A.cov whee A and B ae indices fo atoms and k 1 and k ae paametes to be fixed late. R is the covalent adius of atom A. Fo silicon R. 19 boh. 89 AB is the A.cov B.cov AB Si. cov distance between atoms A and B and f ) is a cutoff function to emove the ( AB contibutions fom atoms that ae fa fom atom A 1 f ( AB ) exp[ / ] 1 AB cut whee = 3.78 boh. Then we intepolate CN to a spatial vaiable cn( ) using A wa( RA ) cn( ) CN w ( R ) Aatoms j jatoms j whee w R atom A is a Gaussian weight-function on atom A and R A A A is the position of w ( ) A exp R t A. In the above t is adjusted to 0.5 boh which yields good esults fo the tests shown late. To obtain good enegy odeings between diffeent silicon phases we choose k 1 =16 k =4/3 and cut =19 boh in Eq. (4.). 99

110 Once the spatial vaiable cn( ) is built we map it to ( ). The mapping is shown in Fig This mapping is optimal fo vaious silicon phases tested in this wok. Figue 4.3 Mapping between the cn( ) and the spatial vaiable ( ). Fo FCC with coodination numbe 1 is mapped to zeo the value appopiate fo a metal. Fo CD with coodination numbe is 4 is mapped to 0.01 which is the optimal value fo CD silicon in Chapte III Bond-bending enegy Since it is difficult to teat bond-bending enegies in OF-DFT we explicitly add a bond-bending enegy to the OF-DFT total enegy functional. Fo atom A the bondbending enegy E bb A is defined as 100

111 E C ) f ( d ) f ( d ) bb A 0 ( A I J 0 A I A J I J whee A I J is the angle IAJ defined by atom I A and J. is chosen to be which is the equilibium angle between two adjacent bonds in the tetahedallybonded CD silicon. f ( d A I ) is again the cutoff function based on the distance d A I between atom A and I and is defined as exp ( d ) / 1 d I 0 A 1 with d 0 =4.0 boh and = d 0 / 0. The coefficient C is chosen to be to yield a good selfintestitial defect fomation enegy fo CD silicon (see Table 0 4.6) Numeical details In both OF-DFT and S-DFT calculations the LDA XC is used. The HC08 BLPS fo silicon 78 is used. Fo all silicon phases we use pimitive unit cells. Fo all the benchmak S-DFT calculations a plane wave basis kinetic enegy cutoff of 800 ev is used. Fo silicon metallic phases a k-point mesh is used while fo silicon semiconducto phases a k-point mesh of is used. These paametes convege the total enegies to within 0.1 mha pe atom. A Femi-Diac smeaing with smeaing width of 0.1 ev is used in all calculations. In all OF-DFT calculations a kinetic enegy cutoff of 1600 ev is used. To calculate the Huang-Cate 010 EDF we use the intepolation technique discussed in Chapte III with the atio between adjacent bins being 1.5 (this conveges the total enegy to within 1 mha pe atom). In the Huang-Cate EDF β=0.63 is used which is the optimal value fo poducing easonable enegy odeings among diffeent silicon phases when the coodination-numbe-dependent λ is used (and no bondbending coection is applied). 101

112 Results and discussion We fist show esults on seveal shea-elated elastic constants ij C. We give a geneal definition fo ij C which is elated to the enegy change of the cystal unde a small defomation away fom its equilibium stuctue i j j i ij e e C V E. Above V is the equilibium volume and { i e } ae small stains unde which the cell is defomed which ae given below. Fo the FCC stuctue the pimitive cell vectos ae 0 / / / 0 / / / a a a a a a a a a whee a is the lattice constant. Unde a small stain (a 3x3 matix) the lattice vectos change accoding to ) ( ' ' ' I a a a a a a. The stain matix is defined as / / / / / / e e e e e e e e e. Fo the ti-axial shea modulus C44 the stain is ). 000 ( e theefoe 44 3 / C V E.

113 The othohombic shea modulus C is due to the stain e ( (1 ) 1000 ) : E / V 6C'. The shea modulus C11 can also be obtained unde the defomation e (00000 ). Elastic constants C11 and C1 can also be obtained with the help of bulk modulus B C 11 3B 4C' 3 C 1 3B C'. 3 In Fig. 4.4 we show the inaccuacy of the OF-DFT in pedicting S-DFT sheaelated moduli without a bond-bending coection and then show how the bond bending coection can impove these popeties. The OF-DFT esults ae based on the nonlocal EDF discussed in Chapte III with the paamete β=0.63 fo both constant λ=0.01 and the coodination-numbe-dependent λ. Fig. 4.4 demonstates that without a bond-bending coection the OF-DFT C11 is significantly smalle (smalle cuvatue) than the S-DFT C11. Fo C OF-DFT without the bond bending tem even gives one esult with the wong sign (negative) and one esult that is positive but nea zeo. Theefoe OF-DFT with eithe coodination-numbedependent λ o a constant λ cannot captue these shea moduli coectly. This failue is because local bond-bending is not descibed popely by ou EDF. By adding bond-bending enegies explicitly we ae able to impove C44 as shown in Fig. 4.4 and Table 4.4. Howeve C11 and C ae ovecoected while C1 is geatly undeestimated. Thus even the addition of this bond bending tem does not povide unifomly impoved esults. In Table 4.5 we show the OF-DFT esults fo some basic bulk popeties fo diffeent silicon phases. OF-DFT esults ae obtained with and without bondbending coections. As expected the coodination-numbe-dependent λ vesion of the HC10 EDF is sufficient to pedict coect enegy odeings fo most of the 103

114 silicon phases. Without bond-bending coections (data in the squae backets) the enegy odeing among diffeent silicon phases is not bad compaed with S-DFT except fo the pediction of too low a total enegy fo the bct5 phase and too high total enegies fo the cbcc and β-tin phases. With the bond-bending coection the popeties of the cbcc stuctue get wose. Table 4.6 displays pedicted vacancy and intestitial defect fomation enegies in CD silicon compaing SDFT benchmaks OFDFT using coodination-numbedependent in the HC10 EDF with and without bond bending enegy coections. β=0.63 is used in these calculations. These defect fomation enegies ae calculated with the same potocol descibed in Chapte III. The self-intestitial defect fomation enegy without bond-bending coections is fa too small; adding the bond bending enegy coection geatly impoves the desciption of this defect due to the additional bonds fomed. By contast the vacancy fomation enegy is unchanged (in both defect calculations the cystal is not allowed to elax) upon adding the bond bending tem since no new bonds ae fomed and the lattice is fozen. 104

115 Figue 4.4 Elastic constants of cubic diamond silicon calculated using OF-DFT with both constant λ=0.01 and coodination-numbe-dependent λ (with and without a bond-bending enegy coection) compaed against the S-DFT benchmaks. β=0.63 is used in all OFDFT calculations. Without the bond-bending coection OF- DFT gives smalle C11 and C44 than S-DFT pedictions and the OF-DFT C is wong in sign o fa too small. 105

116 Table 4.4 Compaison of S-DFT OF-DFT and expeimental bulk moduli (B) C11 C1=(3B-C11)/ C44 and C =(3C11-3B)/4 fo cubic diamond silicon (all in GPa). The OF-DFT esults employing coodination-numbe-dependent λ with and without the bond-bending enegy ae labeled as OF-DFT-BB and OF-DFT-NBB espectively. The OF-DFT esults with a constant λ=0.01 is labeled as OF-DFT-CL. β=0.63 in all OF-DFT calculations. B C11 C1 C44 C S-DFT OF-DFT-BB OF-DFT-NBB OF-DFT-CL Expeiment

117 Table 4.5 Compaison of bulk moduli equilibium volumes and equilibium enegies between OF-DFT and S-DFT (in paentheses) popeties fo diffeent silicon phases. Fo OF-DFT esults obtained using the coodination-numbedependent λ and bond-bending enegy coections ae outside paentheses. The OFDFT esults with coodination-numbe-dependent λ and no bond-bending enegies coections ae in the squae backets. Fo CD silicon E0 (in ev) is the absolute total enegy. Fo othe phases we show the enegy diffeences elative to CD. See text fo details. Si B (GPa) V0 (Å 3 ) E0 (ev) CD 105 [105] (98) HD 10 [10] (99) cbcc 86 [16] (10) β-tin 6 [6] (99) bct5 81 [88] (96) sc 64 [64] (81) hcp 75 [75] (91) bcc 85 [85] (98) fcc 69 [69] (83) [0.6453] (19.781) [0.57] (19.64) 1.39 [0.374] (17.517) [15.774] (14.660) 0.09 [0.759] (16.905) [17.657] (15.484) [14.600] (14.157) [14.880] (14.60) [14.43] (14.37) [ ] ( ) 0.01 [0.01] (0.015) 0.60 [0.167] (0.1) 0.30 [0.30] (0.168) [0.104] (0.15) 0.30 [0.31] (0.9) [0.368] (0.340) [0.367] (0.351) 0.37 [0.381] (0.381) 107

118 Table 4.6 Compaison of vacancy fomation enegies Evf (in ev) and self-intestitial defect fomation enegies Eisv (in ev) in CD silicon as pedicted by S-DFT and OF- DFT using coodination-numbe-dependent λ with and without bond-bending coections. Evf Eisv S-DF OF-DFT (with bond-bending) OF-DFT (no bond-bending) Conclusions In this chapte we show some effots to extend OF-DFT to teating tansition metals and diffeent phases and mechanical popeties of semiconductos as one step futhe fo futue simulation of heteogeneous mateials using OF-DFT. Fo tansition metals the basic pinciples in building a high-quality BLPS wee discussed fist. To apply OF-DFT to tansition metals a pactical scheme is poposed in which the total valence electon density is patitioned into delocalized and localized pats. Fo each pat a physically well founded EDF is applied. Using this scheme OF-DFT pedicts some basic bulk popeties fo seveal Ag phases in easonable ageement with S-DFT. Howeve the OF-DFT enegy diffeences fo diffeent Ag phases (FCC SC BCC and CD) ae doubled compaed to the S-DFT esults. Despite these shotcomings this mixed semilocal nonlocal EDF desciption povides qualitatively easonable behavio fo the fist time in contast to an ealie attempt that used eithe semilocal o nonlocal EDFs but not both. To genealize ou teatment of semiconductos within OF-DFT a coodinationnumbe-dependent λ was intoduced into the HC10 EDF since we had shown ealie that the optimal value of λ is diffeent fo each phase of Si and is esponsible 108

119 fo obtaining the coect enegy odeing of most silicon phases. To obtain good shea-elated moduli we explicitly added bond-bending enegies to the OF-DFT total enegy functional. With these two impovements OF-DFT is able to give a easonable desciption fo the bulk popeties fo diffeent silicon phases point defect fomation enegies as well as the shea-elated moduli fo cubic diamond silicon. 4.5 Acknowledgement I am vey gateful to Junchao Xia fo discussions and his enomous contibution to the nonlocal EDF poject. 109

120 Chapte V Quantum mechanical embedding theoy based on a unique embedding potential 5.1 Intoduction Obtaining detailed and accuate local electonic stuctue is of geat impotance fo undestanding physical popeties of mateials. Howeve sophisticated coelated wavefunction (CW) methods ae too computationally expensive to be applied to moe than ~50 atoms even when fast educed scaling algoithms ae used On the othe hand ohn-sham density functional theoy (S-DFT) exhibits a balance between accuacy and computational cost. Unfotunately many cases exist whee standad implementations of DFT fail to povide the accuacy equied: long ange van de Waals (vdw) inteactions stong coelation between d electons in tansition metals excited states in molecules chage tansfe openshell multiplets adsoption of molecules on metal sufaces etc. Fo such phenomena CW theoy geneally povides supeio accuacy. Much effot has been expended tying to extend CW methods to condensed matte. Second ode petubation theoy 95 has been developed fo cystals and the method of incements 96 obtains impoved total enegies fo insulatos semiconductos and metals by pefoming local CW calculations in the solids. An altenative stategy that eveals local electonic stuctue is to use embedding methods. Hee the system is patitioned into a local egion of inteest often denoted as a cluste and its envionment. The cluste is teated with CW methods while the inteaction of the cluste with its envionment is teated in a vaiety of 110

121 ways. Fo embedding in ionic solids the inteaction can be well descibed by eplacing the envionment with point chages o moe sophisticated shell-models. 97 Semiconductos can be teated with classical embedding stategies as well in paticula via quantum mechanics/molecula mechanics (QM/MM) methods. 98 Fo othe mateials quantum embedding potentials ae pefeable and fall into thee classes: simple bond satuation obital-based and density-based. In addition to QM/MM caving a cluste out of a solid and simply satuating the dangling bonds with hydogens o pseudohydogens often povides a easonable desciption of covalently bonded cystals. 99 Fo metallic mateials othe quantum embedding appoaches ae used. In obital-patitioning-based embedding embedding potentials ae often fomulated with the help of the unpetubed system s Geen s function 100 o ae explicitly constucted (using exchange Coulomb and othe potential tems) fom localized molecula obitals. 101 Fo density-patitioning-based embedding appoximate kinetic enegy density functionals (EDFs) in an obitalfee (OF) DFT embedding potential fomalism have been employed In this wok we focus on the density-based embedding which was fist poposed by Cotona 81 fo fast S-DFT calculations and was late extended by Cate and cowokes to embed CW calculations in an envionment teated with S-DFT. A ecent eview of diffeent embedding stategies was given by Huang and Cate. 109 Not supisingly the accuacy of density-based embedding theoies depends heavily on the accuacy of the tems that ente the embedding potential. In paticula the choice of EDF is citical. Fo example Wesołowski et al. 110 examined how local and semi-local EDFs affected vdw inteactions between benzene and othe molecules and found that the vdw binding enegy can diffe by up to 50% and the equilibium distance can change by 10% with diffeent choices of EDFs. As anothe example Tail and Bid 111 embedded aluminum atoms in bulk aluminum and showed that nonlocal EDFs wee necessay to obtain accuate total enegies. To educe the inaccuacy of OF-DFT-based embedding potentials ecently seveal authos obtained embedding potentials by an invesion pocedue applied 111

122 diectly to the embedded egion s density which needs to be pedetemined. These pedefined embedded egion densities ae typically obtained via pojection methods such as fom Mulliken populations and theefoe ae not guaanteed to be v- epesentable. Ronceo et al. 115 avoided using such a pedefined density; howeve the embedding potential they poposed is not guaanteed to be unique. To ovecome the above-mentioned difficulties in this wok we intoduce an embedding potential that is unique by letting the envionment and the embedded egion shae a common embedding (inteaction) potential. A closely elated scheme was poposed by Cohen and cowokes in thei patition density functional theoy (PDFT) Ou theoy diffes fom theis in two espects. Fist ou method is fomulated based on an extension of an efficient optimized effective potential (OEP) technique. 7 Second we constain the electon numbe in the embedded egion to be an intege which allows ou theoy to be used with advanced CW methods. We show that this S-DFT-deived embedding potential povides an accuate desciption of the envionment s inteaction with the embedded egion fo two quite diffeent test cases. In this way we eliminate the appoximate EDF-based embedding potential entiely to povide a moe accuate embedding theoy fo futue calculations whee S-DFT is insufficient to descibe the system of inteest. Then we extend ou theoy to spin-polaized quantum systems. At the end we compae ou theoy moe completely to PDFT. 5. Theoy 5..1 The uniqueness of embedding potential In density-based embedding theoies typically the total system is patitioned into a cluste and its envionment with n tot n clu and n env the electon densities fo the total system cluste and envionment espectively ( n enegy functional can be decomposed as 11 tot n clu n env ). The total

123 E tot[ ntot clu clu env env int clu env ] E [ n ] E [ n ] E [ n n ] with the cluste-envionment inteaction enegy E int simply defined as E int E [ n ] ( E [ n ] E [ n ]). tot tot clu clu env env Then one can deive an embedding potential fo the cluste as V clu emb E n n ]/ n int[ clu env clu. Likewise the embedding potential fo the envionment due to the pesence of the cluste can be deived as V env emb E n n ]/ n int[ clu env env. As these expessions indicate the embedding potential is theefoe dependent on how the total system is patitioned. In pinciple as long as the total system can be patitioned into two v-epesentable systems this patitioning is valid. Howeve usually a quantum system can be patitioned in moe than one way. A simple example is a unifom electon gas which can be patitioned into any two unifom electon gases both of which ae always v-epesentable theefoe the density patitioning fo a unifom electon gas is not unique. To emove the non-uniqueness of density patitioning and meanwhile to make the embedding potential unique we fix the numbe of electons in each subsystem clu env and intoduce a constaint on the embedding potential i.e. V V. Physically one can think of this choice as epesenting the inteaction potential between the two subsystems. This constaint makes the cluste and the envionment shae a common embedding potential which we now show to be unique (o with a constant shift). (The existence of this embedding potential will be discussed in detail late in the connection to patition density functional theoy section.) emb emb Ou poof fo the embedding potential s uniqueness (assuming its existence) follows the oiginal fist Hohenbeg and ohn theoem. 3 Suppose that thee ae two diffeent embedding potentials V 1 and V and thei coesponding electon densities fo cluste (A) and envionment (B) ae { n A1 n B1 } and { n A n B } such that 113

124 n A 1 nb 1 ntot and na nb ntot espectively. We fix the numbe of electons in the cluste and the envionment and assume that both systems A and B ae not degeneate. One can easily show that E [ na 1 ] V n d E [ n ] V1n Ad 3 1 A1 A A 3 A E [ nb 1 ] V n d E [ n ] V1n Bd 3 1 B1 B B 3 B. Summing up these two inequalities we obtain E A[ na 1 B B1 A A B B ] E [ n ] E [ n ] E [ n ]. In the same way (exchanging the indices 1 and ) we obtain a contadictoy esult: E A[ na B B A A1 B B1 ] E [ n ] E [ n ] E [ n ]. Theefoe via poof by contadiction we have shown that this constained density patitioning and embedding potential (if it exists) ae unique. 5.. Solving fo the Embedding Potential: Optimized Effective Potential Method To solve fo the embedding potential density ( n ef V emb we fist obtain the total electon ) fo the entie system with e.g. S-DFT. The extenal potential (usually just the sum of nuclea potentials o pseudopotentials) is gouped accoding to the patitioning of the cluste and the envionment: V total ext ( ) V i ext iatoms in cluste ( ) V ( ) j ext jatoms in envionment 114

125 whee total V ext is the total extenal potential and associated with atom i. Then we seach fo a i V ext is the extenal potential V emb that makes n A and n B satisfy n ef n n (as befoe the cluste is labeled A and envionment is labeled B). To A B efficiently solve fo this V emb we extend the optimized effective potential (OEP) method 7 fom a single quantum system to multiple quantum systems. An extended Wu-Yang functional can be defined as W[ V emb ] E [ n ] Vemb ( ) A B n A B n ef d 3. (5.1) V is solved fo by maximizing the functional V ] emb W i.e. max min W. [ emb V n n Duing the OEP pocess the electon numbes in subsystems A and B ae fixed. The gadient of W is simply emb A B W V ( ) n emb A B A n B A B n V n ef emb W n ( ') 3 d' ( n n ( ') V V emb ( ) emb ( ') 3 d' ( na nb nef ) ( ) A n B n ef ). To show that W s stationay point i.e. n ef n A n is the only maximum we B need to show that W is concave globally. Hee we give a poof based only on the convexity of the enegy functional. We stat with the second functional deivative of W W[ V V ( ) V emb emb emb ] ( ') A B n ( ) n n ( '') V emb ( '') d'' ( ') 3. (5.) We define a modified total enegy functional fo the cluste 115

126 3 QA [ na[ Vemb ] Vemb ] EA[ na] Vemb nad with a simila expession fo the envionment. Fo a givenv Theefoe emb n A minimizes n A is a functional of V emb. Fixing V emb we apply a small vaiation on n A : QA E[ na] n A( ') 3 Vemb ( ') d' A[ na] n ( '') n ( '') n ( '') A Vemb whee A is the chemical potential a functional of above we obtain A A A Q A. n. Applying / V emb ( ) to Q n ] n ( '') n ( ( ) ( ) n ( ) n ( '') 3 A A A d A ( ) ( ) [ n 3 A[ na] n A d A ) Vemb A n A( ) V V emb emb. (5.3) By defining H A QA[ n ( '' ) n ( '') n A A A ] ( ) Vemb A[ na] n ( ) A Eq. (5.3) becomes: n A ) n ( '') ( 3 n A d H A( '' ) A Vemb ( ). (5.4) ( ) Substituting Eq. (5.4) into Eq. (5.) yields W[ V V ( ) V emb emb emb ] ( ') A B n V emb ( ) H ( ) n ( '' ) V emb ( '') d ( ') 3 d'' 3 The second vaiation of W is then 116

127 W A B A B A B A B n Q n n V ( ) n 0 emb ( ) H ( ) V ( ) emb Q ( ) n ( ) H ( '' ) n ( '') n ( '' ) V 3 3 ( '') d d'' Vemb n emb ( '') d ( '') V ( ') 3 d'' 3 emb 3 ( ') d d'' 3 d' 3 3 d. (5.5) In the above integal the tem involving the chemical potential is zeo: 3 n ( '') d'' 0 [ n ] 3 3 n ( ) n ( '') d d'' [ n ] n ( ) since the numbe of electons in system is fixed. In Eq. (5.5) W is always negative and theefoe we have poven that W is globally concave. The concavity of W is intinsically connected to the convexity of the modified total enegy functional Q at its minimum. Thus we can obtain a unique embedding potential fom invesion of the S-DFT equations that connects the embedded egion and its envionment without invoking any appoximations beyond S-DFT. In the following we demonstate the accuacy of ou new embedding theoy fo two vey diffeent systems: (1) the vdw inteaction between a hydogen molecule and a ten-hydogen-atom chain (H/H10) which was used as a test system in an ealie wok 11 and () chemisoption of CO on the Cu(111) suface which is a wellknown example whee S-DFT fails to give the coect binding enegies and binding site and which was also studied peviously using a EDF-based embedding potential

128 5.3 Numeical implementation Computational Details Peiodic S-DFT calculations ae pefomed using the ABINIT code. 33 The local density appoximation (LDA) 36 fo electon exchange and coelation is used in all S-DFT calculations. Ou embedding method is implemented within a modified vesion of ABINIT. To solve fo the embedding potential of Eq. (5.1) two sepaate S-DFT calculations ae pefomed simultaneously one fo the cluste and anothe fo the envionment. Fo any given tial embedding potential these two sepaate S-DFT calculations etun two total enegies and two electon densities which ae calculated with the embedding potential teated as an additional extenal potential. Then the functional W V ] is calculated accoding to Eq. (5.1). We employ a quasi- [ emb Newton method 1 to maximize W V ]. In all examples shown in this wok the [ emb numbes of electons in cluste and envionment ae set to keep the cluste and envionment neutal. The embedding potential is egulaized 13 with the coefficient 5 of the egulaized enegy functional set to 1 10 fo the H/H10 test and to 110 fo the CO/Cu(111) case. 4 Fo the H/H10 case one hydogen molecule is placed pependicula to and above a hydogen chain (Fig. 5.1). The cluste consists of the hydogen molecule and two hydogen atoms unde it that ae pat of the hydogen chain with the est of atoms defined to be the envionment. The distances between all neighboing hydogen atoms in the chain and in the hydogen molecule ae fixed at 1.4 boh while the distance (d) between the cente-of-mass of the hydogen molecule and the hydogen chain is vaied. Fo consistency we employ the same Hatee-Fockdeived nom-conseving pseudopotential fo hydogen 14 in both the peiodic S- DFT and the CW calculations. In S-DFT calculations a kinetic enegy cutoff of 000 ev fo the plane wave basis is used. The cc-pvqz basis set 15 fo hydogen is utilized 118

129 in the quantum chemisty calculations caied out in the GAMESS 16 and MOLCAS 17 codes. In GAMESS the coupled cluste singles and doubles with petubative tiples (CCSD(T)) method 18 is used fo H/H10 while othe CCSD(T) and MRSDCI 19 calculations ae done in MOLCAS. Embedded MRSDCI and CCSD(T) calculations ae pefomed with ou embedding code MOLCAS-embed which is based on the MOLCAS code. In the complete active space self-consistent field (CASSCF) calculations 18 the CAS contains 1 electons in 1 obitals fo H/H10. Fo the H/H cluste the CAS has fou electons in fou obitals. In the MRSDCI calculations (also fo the CO/Cu wok below) the dominant efeences ae chosen to be those fo which the absolute values of thei coefficients ae We do not impose symmety in any calculations pefomed in the MOLCAS and GAMESS codes. Fo CO/Cu(111) the Cu fcc(111) peiodic slab model is fou layes thick with a 3 3 peiodic cell in the suface plane. Since the Cu(111) suface elaxes vey little fom the bulk stuctue 11 the Cu slab is not elaxed. The distance between CO and the Cu suface is optimized within S-DFT-LDA. In peiodic S-DFT LDA nomconseving pseudopotentials fo Cu O and C ae built with the FHI98PP code 38 and a plane wave basis kinetic enegy cutoff (Ecut) of 1500 ev is employed. We also tested a 000 ev Ecut; the esulting S-DFT-LDA binding enegies did not change moe than 0.0 ev. All peiodic S-DFT calculations on CO/Cu ae done using Femi- Diac smeaing with a smeaing width of 0.1 ev. A Monkhost-Pack 41 k-point mesh of 5 51 is used as it was found to be a conveged mesh size in ealie wok. 11 Two embedded clustes ae consideed. One has fou Cu atoms (Cu4) and the othe has 10 (Cu10). Fo CO adsoption at the top-site the Cu4 cluste contains fou Cu atoms all within the suface laye with one cental Cu and its thee neaest neighbos. The Cu10 cluste contains fou thee and thee Cu atoms on the suface second and thid layes espectively denoted as (4 3 3) whee the fou suface atoms ae aanged as in the Cu4 cluste and the subsuface atoms ae in bulk positions in the layes beneath. Fo the hcp hollow site within the same notation the Cu4 cluste is (3 1 0) and the Cu10 cluste is (3 4 3). Stevens-Basch-auss 119

130 effective coe potentials (ECPs) and thei coesponding basis sets 130 fo oxygen and cabon ae used in MOLCAS. Fo top-site adsoption the fou Cu atoms in the top laye ae teated with a Hay-Wadt Cu ECP 131 which eplaces 18 coe electons (Cu- ECP1) leaving 11 electons teated explicitly. Fo hcp-site adsoption thee Cu atoms on the suface nea CO and one Cu atom unde CO on the second laye ae teated with Cu-ECP1. Fo all othe Cu atoms since they ae not expected to play an active ole in metal-co bonding we use a Cu ECP 13 that additionally eplaces the 3d electons leaving only the 4s electon on each Cu to be teated explicitly. In all CASSCF calculations the active space consists of CO s and * as well as one occupied and one unoccupied obital fom the metal cluste nea the Femi level; theefoe the active space has 1 electons distibuted among 10 obitals. Ou CASSCF and MRSDCI calculations yield open-shell singlet gound states fo all the embedded CO/Cu clustes studied hee. Lastly in the embeddingpotential-geneating pocess the envionments ae constucted by simply emoving these clustes fom the slabs. The embedding potentials geneated with ou modified ABINIT code fo CO/Cu(111) theefoe have the same peiodicity as these slabs. Because ou MOLAS-based embedding code does not use peiodic bounday conditions the obtained embedding potentials ae padded i.e. epeated in a 3 3 finite eplication of the oiginal peiodic slab unit cell in the suface plane to mimic the peiodic suface envionment with the embedded cluste placed in the cental unit cell befoe the embedded CW calculations ae pefomed. No such padding is necessay in the diection nomal to the suface whee vacuum is pesent. The padding allows ou CO/Cu cluste densities to be fully contained within the embedding potential Evaluating the Total Enegy To calculate the total enegy in ou embedding theoy we decompose the total enegy as usual as : 10

131 E tot E E CW cluste DFT env [ DFT [ env ] Eint CW cluste ] E DFT DFT cluste [ [ DFT env DFT cluste ] DFT cluste ] E DFT cluste [ DFT cluste ]. (5.6) In this decomposition the second set of tems in paentheses seves as the CW CW coection to the total enegy. In pinciple these tems E [ ] and DFT DFT E [ ] should be calculated by fist pefoming embedded CW and S-DFT cluste cluste calculations in the pesence of the embedding potential and then subtacting the 3 double-counted tem V emb ( ) ( ) d fom these enegies (since this tem is DFT DFT DFT aleady pesent in E [ ] ). We discuss thee diffeent schemes fo calculating E tot : int env cluste cluste cluste (1) We denote the fist scheme Emb-NO. The quantity contained in the fist set of paentheses in Eq. (5.6) is obtained by pefoming a self-consistent S-DFT fo the whole system. To evaluate the tems in the second set of paentheses in Eq. (5.6) an embedded CW calculation on the cluste is caied out and then the embedded CW s natual obitals (NOs) ae supplied to a non-self-consistent embedded S-DFT calculation to obtain the coesponding embedded S-DFT enegy. Then the quantity in the second set of paentheses in Eq. (5.6) is simply appoximated by subtacting this embedded S-DFT enegy fom the embedded CW enegy. Note that we do not subtact the double-counted tems in this Emb-NO method because the two double-counted tems in the embedded CW and S-DFT calculations will exactly cancel. This is because the NO s electon density used in the S-DFT is just DFT the CW s electon density. This method is equivalent to eplacing the cluste with DFT DFT in the E [ ] tem in the second set of paentheses of Eq. (5.6). CW cluste cluste cluste Theefoe since the subtaction of the DFT cluste egion density s contibution to the total DFT enegy is essential to the pope evaluation of Eq. (5.6) this method CW will be accuate as long as cluste does not diffe much fom DFT cluste. This scheme fo 11

132 calculating the coection tem is identical to ealie wok in ou goup albeit with a diffeent definition fo the embedding potential. () The second scheme we efe to as Emb-NV. We again apply S-DFT to the entie system to obtain the quantity in the fist set of paentheses in Eq. (5.6). Howeve hee we calculate the second tem in the second set of paentheses selfconsistently namely we use the self-consistent S-DFT enegy (calculated with ou MOLCAS-embed code so as to use the same basis as in the CW calculation) fo the cluste in the pesence of the embedding potential. This time we explicitly subtact 3 the double-counted tems V emb ( ) ( ) d fom the self-consistent embedded CW CW CW DFT DFT and S-DFT enegies to obtain E [ ] and E [ ] since now the two cluste cluste embedded cluste densities ae not equivalent and theefoe the double-counted inteaction tem will not cancel as in scheme (1) above. cluste cluste (3) The thid scheme we name Emb-V. As usual we fist calculate the enegy of the total system within self-consistent S-DFT. Howeve hee we update the DFT DFT DFT inteaction enegy E [ ] in ecognition of the fact that the cluste int env cluste DFT DFT DFT density is updated afte embedded CW calculations. To access E [ ] in pinciple we need the total enegy functional which unfotunately is unknown. DFT DFT CW Instead we update E [ ] with a fist ode appoximation (vey simila int env cluste to a concept in pseudopotential theoy): int env cluste E DFT int E E [ DFT int DFT int DFT env [ [ DFT env DFT env CW cluste ] DFT cluste DFT cluste DFT Eint ] ] Vemb ( ) DFT cluste CW DFT 3 d cluste CW cluste cluste DFT cluste d 3. (5.7) Combining Eq. (5.7) with Eq. (5.6) we obtain 1

133 E DFT DFT DFT DFT DFT DFT DFT tot Eenv [ env ] Eint [ env cluste ] Ecluste[ cluste ] CW CW CW 3 DFT DFT DFT 3 E [ ] V ( ) d E [ ] V ( ) d cluste cluste. (5.8) emb cluste cluste cluste emb cluste CW CW Hee we have gouped the double-counted tems with E [ ] and DFT DFT E [ ]. Conveniently the second and thid sets of paentheses in Eq. (5.8) ae cluste cluste just the enegies fom self-consistent embedded CW and S-DFT calculations in the pesence of the embedding potential enegies we have aleady calculated ealie. Eq. (5.8) gives a much simple way to evaluate the total enegy and also takes into account the fist ode coection to the inteaction enegy. cluste cluste We compae the accuacy of the above thee schemes in the H/H10 test. Fo the CO/Cu(111) test hee we only apply the Emb-NO method as used in ou ealie wok 11 which does not equie a self-consistent unesticted S-DFT calculation (as would be needed due to the open-shell natue of the clustes and which is difficult to convege due to the high symmeties of the clustes). We also compae ou theoy to the ONIOM method 133 in which CW E cluste and DFT E cluste in Eq. (5.6) ae simply calculated self-consistently without the pesence of the embedding potential i.e. fo bae clustes. Because of difficulty conveging unesticted S-DFT fo these open-shell high symmety Cu clustes esticted S-DFT is used fo DFT E cluste within ONIOM the same as was done in ou pevious wok. 11 In all thee schemes fo evaluating the total enegy the NOs ae deived fom embedded MRSDCI wavefunctions. To evaluate the S-DFT enegy with NOs we set the occupation numbes to be one (fo open-shell) o two (fo closed-shell) fo stongly occupied NOs. Fo weakly occupied NOs occupation numbes ae set to zeo. Fo open-shell (closed-shell) clustes unesticted (esticted) S-DFT is used. 13

134 5.4 Results and discussions vdw inteaction: H/H10 We compae the binding enegy cuves pedicted by diffeent electonic stuctue methods in Fig CCSD(T) calculations on the entie H/H10 system povide the benchmak (black cicles) since CCSD(T) povides a supeio desciption of vdw inteactions compaed to MRSDCI by avoiding the latte s sizeextensivity eo. CCSD(T) fo H/H10 pedicts an equilibium distance of 5. boh and a well depth of 17 mev. Next we conside vaious appoximations to the full CCSD(T) cuve. Fist simply neglecting the envionment entiely by pefoming CCSD(T) only on the H/H cluste poduces almost no binding at all (gey squaes in Fig. 5.1) which implies that the est of the hydogen chain also paticipates in the vdw inteactions. Fom this esult we conclude that the envionment must be teated eithe explicitly o via embedding. Second peiodic S-DFT-LDA (oange diamonds) on the entie H/H10 system (placed in a box with at least 9 Å vacuum between peiodic images) gives the expected ove-binding: too shot an equilibium distance (4.4 boh) and too lage a binding enegy (57 mev). 14

135 Figue 5.1 Binding enegy cuves (enegies ae shifted to match at lage distance) fo the vdw inteaction between a hydogen molecule and a hydogen chain (inset). The inset displays the embedded cluste atoms (oange) while the envionment atoms ae shown in tuquoise. CCSD(T) on the entie H/H10 system povides the benchmak fo compaison (black cicles). SDFT-LDA on the entie system (oange diamonds) geatly ovebinds as expected. CCSD(T) only on the bae H/H cluste (gey squaes) exhibits essentially no binding at all indicating the impotance of the envionment atoms. Coecting the S-DFT-LDA enegies using the ONIOM method (ed squaes) yields a much impoved though slightly too weak binding enegy cuve. Thee embedded MRSDCI cuves ae shown. When 15 cluste E DFT is calculated using MRSDCI natual obitals ( Emb-NO method) the system is ovebound. When the double-counting is subtacted ( Emb-NV method) the system is undebound. When we incopoate a fist ode coection to the cluste-envionment inteaction enegy ( Emb-V method) we find a cuve that essentially coincides with the benchmak.

136 To assess the quality of ou embedding method we focus on the egion whee vdw dominates: d>5.0 boh. The ONIOM pediction whee the S-DFT-LDA binding enegy cuve fo the entie H/H10 system is coected by the diffeences in enegies between CCSD(T) and S-DFT-LDA on the bae H/H cluste (ed squaes) stays consistently highe than the benchmak. We also pefomed MRSDCI on the bae H/H cluste and it gives almost the same esults as CCSD(T) since the size extensivity eo is minute fo such a small system. Consequently we conclude it is safe to use MRSDCI fo the embedded H/H cluste. In Fig. 5.1 the only cuve that coincides with the benchmak is the embedded MRSDCI cuve (ed tiangles) whee the total enegy is evaluated by the Emb-V scheme descibed above (in which the inteaction enegy is coected to fist ode). An embedded CCSD(T) cuve (using the Emb-V method) is almost on top of the embedded MRSDCI cuve and theefoe is not shown. Fo total enegies calculated with the Emb-NV scheme (subtacting the double-counted tem) we obtain a binding enegy cuve (geen tiangles) that is vey close to the ONIOM esult which indicates that the coection DFT DFT CW to the inteaction enegy tem i.e. E [ ] in Eq. (5.7) is impotant to int include fo highest accuacy. Neglect of this tem is likely why the embedding total enegy cuve stays highe than the CCSD(T) benchmak cuve in Ronceo et al. s wok. 115 Total enegies calculated by the Emb-NO scheme (evaluating the DFT cluste enegy using the MRSDCI NOs to define the density and not subtacting the double-counted tems since they should cancel) yield a binding enegy (blue tiangles) of ~4 mev with an equilibium distance of ~4.9 boh somewhat away fom the benchmak but still much bette than the S-DFT LDA esult it is meant to coect. This latte test shows that the eplacement of the S-DFT electon density DFT DFT with the CW electon density in E [ ] in Eq. (5.6) intoduces some eo; howeve the eo is still quite small. cluste env cluste cluste In Fig. 5. (a) we show the embedded cluste and envionment S-DFT-LDA densities along the H10 chain as well as the total electon density fom a S-DFT- LDA calculation on the whole H/H10 system. The cluste and envionment densities 16

137 ae outputs fom ou embedding-potential-geneating code afte the embedding potential is conveged. Along this H10 chain we also plot the embedding potential obtained fo this H/H cluste embedded in the emainde of the H10 chain. Its accuacy can be estimated fom the maximum density diffeence between the taget total density and the sum of cluste and envionment densities. The maximum 4 density diffeence is less than 5 10 boh -3. In fact the sum of the cluste and envionment densities should not be pecisely equal to the total S-DFT density because of ou estiction that the cluste etains an intege numbe of electons. Despite the estiction the density diffeence is vey small. The cluste density decays quickly into the envionment which means that the cluste egion is well localized. Moeove away fom the inteface the fluctuation of the embedding potential becomes weak. To connect the cluste and envionment the two lage wells seen in the embedding potential pull electons fom neighboing hydogen atoms while the cental peak in the potential pushes electons away. In Figs. 5. (b) (c) and (d) we show contou plots of the embedding potential fo diffeent cuts though the system. It is clea that the embedding potential is mainly concentated nea the H/H cluste. Its main featues ae that the embedding potential is positive inside the cluste (best seen in Fig. 5. (d)) and negative elsewhee (best seen Figs. 5. (a)-(c)) to push electon density out to inteact with the envionment. Of couse as mentioned ealie negative potential egions nea the two embedded nuclei that ae pat of the hydogen chain seve to etain the bond between those two atoms (Figs. 5. (a)-(c)) 17

138 Figue 5. (a) (Uppe panel) Electon densities fo the cluste (black dashes) the envionment (ed dashes) and the benchmak electon density (to be matched solid blue line) along the H10 chain. Each peak coesponds to an atom. (Lowe panel) Embedding potential along the H10 chain. Embedding potential contous ae also shown fo planes pependicula to (b) the y-axis passing though the H10 chain (c) the z-axis passing though the H10 chain and (d) the x-axis passing though the two H atoms (two black dots) above the H10 chain. The cuve in (a) is plotted along the dashed lines in (b) and (c). The coodinate system is defined in the inset of Figue

139 5.4. Metallic envionment: CO/Cu(111) Fig. 5.3 (a) shows the contou plot fo the density diffeence between the total electon density of CO/Cu(111) fom the peiodic S-DFT-LDA slab calculation and the sum of the cluste and envionment s S-DFT-LDA electon densities deived fom ou embedding-potential-geneating code. The cluste is CO/Cu4 with the CO sitting above the top-site (see Fig. 5.3 (d)). The embedding potential is deemed accuate by obseving that the maximum density diffeence in the valence egion is 3 vey small (< 1 10 boh -3 ). The density diffeences nea the nucleus ae slightly 3 highe (~ 10 boh -3 ) but this should not affect the embedding potential in the valence egion. Fig. 5.3 (b) and (c) display contou plots of the embedding potential on planes passing though the suface Cu atoms and pependicula to the suface. Like the vdw case the embedding potential is quite localized aound the cluste both in the suface plane and in the diection nomal to the suface. We expect this to be tue in geneal since we have defined this numeical embedding potential as the inteaction that connects the embedded egion to its envionment and so it is not supising that it is localized at the inteface between the two egions. 19

140 Figue 5.3 Plots fo the CO/Cu4 cluste embedded in a Cu(111) peiodic slab. The cluste contains fou Cu atoms with the CO above the top site (see (d)). (a) Contou plot fo the density diffeence between the total density fom a peiodic S-DFT- LDA slab calculation and the sum of the embedded cluste and envionment s electon densities; the plane is though the suface Cu atoms. The high quality of the embedding potential is implied by the small density diffeence in the valence egion 3 (< 1 10 boh -3 ). (b) Contou plot fo the embedding potential though the suface Cu atoms. (c) Contou plot fo the embedding potential pependicula to the suface though the dashed line in (b). The dashed lines in (a) (b) and (c) pass though the same Cu atoms. (d) The CO/Cu4 cluste geomety with these fou Cu atoms maked in (a) and (b). (a) (b) and (c) ae epeated peiodically in the lateal diection fo the pupose of illustation. The embedding potentials ae actually padded in a 3 3 fashion in the diection paallel to the suface in these embedding calculations which is explained in the numeical implementation section. All plots ae in atomic units. 130

141 Table 5.1 displays the binding enegies fo CO adsobed on the top and hcp hollow sites at vaious levels of theoy. As has been obseved by seveal othes peiodic S-DFT-LDA slab calculations ovebind and give the wong stability tend: top-site adsoption is pedicted to be less favoable than hollow-site adsoption. This conflicts with expeiments in which adsoption at the hollow site is neve obseved at low CO coveages wheeas top-site adsoption is always obseved. By contast the embedded MRSDCI calculations give positive (stable) binding enegies fo top site adsoption. The hcp-site adsoption is pedicted to be unstable with ou embedding method fo both Cu4 and Cu10 clustes and the binding enegies convege faste with the cluste size than the ONIOM method. ONIOM also gives a small positive binding enegy fo CO adsoption at the hollow site of Cu4 cluste. Compaed to the pevious EDF-based embedding wok 11 ou top-site adsoption enegy conveges moe slowly. Fo the hcp-site adsoption both methods exhibit fast convegence fo the (negative) binding enegies with espect to cluste size. In this case the self-consistently optimized EDF-containing embedding potential of Ref. 11 woks vey well; it may be that binging a selfconsistency to ou new embedding potential will be key to obtaining the most accuate esults. Late we outline how this could be done in futue. 131

142 Table 5.1 Binding enegies (ev) fo CO adsobed on top and thee-fold hollow sites of Cu fcc(111). Peiodic S-DFT-LDA slab calculations (labeled S-DFT-SLAB) ae calculated within the ABINIT code. Othe calculations ae pefomed fo clustes of fou and ten Cu atoms espectively with the MOLCAS-embed code whee we supply ou new embedding potential on a unifom gid in eal space. Pevious EDF-based embedding esults 11 ae also cited fo compaison. top site hcp site S-DFT-SLAB Cu4 Cu10 Cu4 Cu10 ONIOM with MRSDCI Embedded MRSDCI Embedded MRSDCI (this wok) Expeiment a not bound a Expeimental data ae fom Refs To semi-quantitatively validate ou embedded MRSDCI pedictions we analyze changes in the contibutions of oxygen s and cabon s p-obitals in CO s 4 and 5 obitals upon adsoption and compae with X-ay emission specta (XES) (Fig. 5.4). The Cu-CO bond is pimaily a dative bond fom the CO to the Cu; we find no evidence of -back-bonding in ou MRSDCI NOs. We only need to focus on the obitals since they compise the main components of CO s 4 and 5 obitals (z being the diection of the suface nomal). Ou analysis is based on the coefficients of the pz atomic obital basis functions in the NOs deived fom the embedded MRSDCI calculations on the CO/Cu10 cluste with CO adsobed at the top site. The changes in the cabon and oxygen 13 pz pz contibutions to the CO obitals ae summaized in Table 5.. A stiking finding in XES is that cabon's p-obital signal

143 nea the CO 4 enegy level almost disappeas upon adsoption (Fig. 5.4). 141 The calculated coefficient of cabon pz confims this XES finding vey well: we pedict a decease fom 0.3 (fee CO molecule) to 0.09 (adsobed CO) in the 4 obital. In the adsobed CO the main contibution to the 4 NO fom cabon is its s obital. Since XES only pobes the element s p-obitals due to dipole selection ules cabon s s obital does not contibute to XES specta. Fo CO s 5 obital cabon s coefficient changes much less fom 0.47 (fee CO) to 0.37 (adsobed CO) which is consistent with the mostly unchanged cabon signal denoted as 5 ~ in Fig Tuning to oxygen we find that the coefficient of O s pz pz in CO s 4 dops fom 0.66 (fee CO) to 0.45 (adsobed CO). Fo the 5 -type obitals the coefficient of O s p z inceases fom 0.15 (fee CO) to 0.47 (adsobed CO). These pedictions ae entiely consistent with the decease of oxygen's signal aound the 4 enegy level and the lage incease aound the 5 enegy level found in XES (Fig. 5.4). Fo the * obitals of -symmety thee is no noticeable filling of CO obital in the XES specta (Fig. 5.4). This is consistent with ou calculations whee we find that the * chage back-donation fom the metal s electons to CO s is negligible. 133

144 Table 5. Cabon and oxygen pz contibutions in CO 4 and 5 obitals pio to and upon adsoption fo CO adsobed at the top site of an embedded Cu10 cluste. Coefficients of atomic obital basis functions ae extacted fom the natual obitals deived fom embedded MRSDCI calculations. Down-aows and up-aows indicate the damatic change in coefficients upon adsoption. 4 5 fee CO adsobed fee CO adsobed C O Figue 5.4 Expeimental X-ay emission specta fo CO adsobed on Cu(100) and Ni(100) sufaces togethe with the specta fo a gaseous CO molecule (Ref. 141). Repinted with pemission fom J. Chem. Phys (000). Copyight 000 Ameican Institute of Physics. 134

145 5.5 Self-consistent embedding scheme The examples given above do not self-consistently update the embedding potential. We biefly descibe how to pefom a self-consistent embedding calculation. Once we obtain the embedding potential fo the cluste egion we can pefom a CW calculation on the cluste in the pesence of that embedding potential. The CW method povides an updated cluste electon density via the natual obitals of the many-body wavefunction. We can add this updated cluste electon density to the envionment s electon density to obtain an updated total electon density as done in ealie wok. 11 Then we can apply the above OEP method to solve fo an updated embedding potential associated with this updated total electon density. With this updated embedding potential we can pefom anothe CW calculation on the cluste to complete an iteation of the self-consistent embedding scheme. This is simila in spiit to Huang and Cate s 006 appoach 108 except that we have eliminated the appoximate EDF fom the embedding potential. This simple self-consistent embedding scheme may be necessay when the inteactions between cluste and envionment ae not well descibed by standad DFT alone. Howeve I need to point out that above self-consistency is only patial even if the total electon density is updated by eplacing the cluste s density with CW density. We notice that the cluste is always solved with S-DFT (using appoximated XC functionals). A tue self-consistent embedding theoy in which cluste is always calculated with highe level method such as CW theoy will be pesented in Chapte VI. 135

146 5.6 Genealization to spin-polaized quantum systems Fomalism We now genealize ou embedding theoy to spin-polaized quantum systems (fo both non-collinea spin and collinea spin cases). Fo mateials with magnetization the inteaction between the cluste and the envionment is not meely a simple scala embedding potential but should contain spin-dependent embedding potentials. Following Bath and Hedin s wok 14 the genealized Hamiltonian ( Ĥ ) fo a spin-polaized system with a genealized spin-dependent extenal potential w ( ) and the Coulomb inteaction v( ' ) can be witten unde second quantization as ˆ pˆ H ( ) ( ) m 1 ( ) ( ' ) v( ' ) ( ) 3 ( ) w ( ) ( ) d ( ' ) d 3 d' 3 (5.9) whee α and β ae spin indices pˆ is the momentum opeato m is the electon mass ( ) and ( ) ae ceation and annihilation field opeatos fo electon of spin α. The Hamiltonian ( Ĥ k ) fo each subsystem k can be witten in a simila way. The density matix ( ) is obtained as ( ) ( ) ( (5.10) ) whee is the many-body wavefunction. The expectation value of Ĥ is simply

147 E T V w ( ) ( ) d 3 (5.11) whee T and V ae the expectation values of the kinetic and potential enegies. To patition a spin-polaized quantum system we seek a common spin-dependent embedding potential w emb ( ) shaed by each subsystem such that the sum of the density matix ( ) of each subsystem (labeled k) matches the total density k matix of the entie quantum system i.e.. total k k subsystems The embedded Hamiltonian fo the subsystem k is defined as Hˆ ˆ d k emb H k k ( ) w emb ( ) k ( ) 3. (5.1) Let us patition the entie system into two subsystems with the electon numbe in each subsystem fixed. Assume that we have two diffeent patitionings which lead to ; } and '; '} such that { 1 { 1 1 1' '. (5.13) total Accoding to von Bath and Hedin 14 these two diffeent patitionings yield two diffeent many-body wavefunctions { 1; } and { 1'; '} with two diffeent spindependent embedding potentials w emb ( ) and w emb '( ). Fo subsystem 1 we have Hˆ ' Hˆ ' (5.14) 1 1 emb emb 1 which leads to E1 ( w emb ( ) 1'( ) d 3 w emb ) 1( ) d E1' 3 (5.15) 137

148 whee E 1 Ĥ 1 and E ' ' ˆ 1 H1 '. Fo subsystem Hˆ ' Hˆ ' (5.16) emb emb which leads to E 3 w emb ( ) ( ) d E ' w emb ( ) '( d ) 3. (5.17) Sum up Eqs. (5.15) and (5.17) with the identity in Eq. (5.13) we immediately get E1 E E1' E'. (5.18) With a simila pocedue we get a contadictoy esult E E E. (5.19) 1 ' E' 1 Via poof by contadiction we have shown that thee is only one unique way to patition the total system s density matix once electon numbes in subsystems ae fixed. Note that thee is a unique density matix patitioning (if it exists) but thee is no unique spin-dependent embedding potential w emb ( ) fo this patitioning because the mapping fom spin-dependent potential to spin density is many-toone. 14 Howeve the embedding potential w emb ( ) can still be calculated with the OEP method since we just need to find one potential fom many solutions as discussed by Heaton-Bugess et al. 143 Detailed examples fo this spin-dependent embedding theoy will be discussed elsewhee. The OEP fo obtaining an embedding potential w emb ( ) fo a spin-polaized quantum system having only two subsystems can be genealized to 138

149 W[ w emb E [ 1 1 ( )] ] E [ ] w emb ( )( 1 total ) d 3. (5.0) Fo the simple collinea spin case the above OEP method is just W[ V V emb emb ( ) Bemb ( )] E1[ 1 1 ] E 3 ( )( ) d B 1 total emb 1 [ ] ( )( m m m total ) d with and denotes the spin up and spin down and m and m. total is the total electon density and mtotal total total the total magnetization. Hee V emb () and B emb () ae Lagange multiplies that constain () and (). The above expession total m total povides an explicit epesentation of Eq. (5.0) fo the collinea spin case in which w ( ) Vemb ( ) Bemb ( ) and w ( ) V ( ) B ( ) emb emb Numeical examples fo spin-polaized embedding Now we show one example the C dime fo this spin-polaized embedding scheme. We pefom collinea spin-polaized calculations and the LDA 36 is used fo exchange-coelation. The two C atoms ae sepaated by 1.7 Å at equilibium as pedicted by optimizing the bond length using spin-polaized S-DFT-LDA. Each C atom is consideed as one subsystem. The C NLPS is constucted using the FHI98 code 38 with its default settings fo coe adii. The taget total and m total ae obtained by pefoming spin-polaized S-DFT-LDA on the entie C dime. In the fist iteation V emb () and B emb () ae initialized to zeo with a maximum diffeence 3 fo ) and m m m ) of and ( 1 total ( 1 total convegence the maximum diffeences ae educed to attibute these final small esiduals to numeical eos espectively. At and We

150 In Fig. 5.5 (a) we compae the subsystem electon densities (ed and blue cicles) with the taget total (black line). We find that subsystem electon densities ae localized aound each atom in this spin-polaized embedding case. This localization is also found in Fig 5.5 (b) whee we compae the subsystem magnetization m 1 and m with the taget m total. C dime is anti-feomagnetic which can be seen moe clealy with the subsystem magnetization m 1 and m in Fig 5.5 (b). The embedding potential V emb () [Fig. 5.6 (a)] and the embedding magnetic field B emb () [Fig. 5.6 (b)] povide futhe insight. V emb () is symmetic with espect to the two C atoms. Simila to pevious examples V emb () is negative between the two C atoms to attact electons to fom the bond and is epulsive aound each of the C atoms. B emb () is anti-symmetic with espect to the mio plane between the two C atoms due to the anti-feomagnetic natue of the C dime. 140

151 Figue 5.5 (a) Compaison of the subsystem electon densities of each C atom and the total electon density calculated by pefoming spin-polaized S-DFT-LDA on the C dime. (b) Compaison of the coesponding spin-density diffeences of each C subsystem and the total spin density diffeence. 141

152 Figue 5.6 (a) Embedding potential V emb () and (b) embedding magnetic field B emb () fo the C dime. Each C atom is teated as a subsystem. The two C atoms ae sepaated by 1.7 Å the equilibium bond length pedicted by spin-polaized S- DFT-LDA. Embedding potentials ae essentially zeo away fom the C dime. Coodinates ae in Å. 14

153 5.7 Connection to patition density functional theoy Duing the pepaation of this manuscipt we became awae that ou method is closely elated to the patition density functional theoy (PDFT) poposed by Cohen and cowokes They poposed a simila means to divide the system into seveal inteacting subsystems and fomulated the poblem in a way close to ou Eq. (5.1). Since each subsystem is teated with ensemble DFT in PDFT 116 the numbe of electons in each subsystem is allowed to be factional and they do not fix the numbe of electons in the subsystems though the sum of the electon numbes in subsystems must equal the total electon numbe. To show the uniqueness of thei eactivity potential (we call it embedding potential hee) they defined a functional n V ] E [ 116 S k emb E [ d S 3 n V ] E [ n ] V ( ) n ( ) k emb k k k emb k k (5.1) whee n is the electon density of subsystem k and E n ] is the enegy density k k [ k functional fo a non-embedded subsystem k. Fo a given embedding potential V emb () a set of { nk () } is detemined by minimizing E n } V ]. Via poof of S [{ k emb contadiction Cohen and Wassemann showed 116 that thee is a unique embedding potential V emb () fo E n } V ] such that the sum of electon densities fom all S[{ k emb subsystems matches the total electon density calculated fo the entie system. Note that Eq. (5.1) is just the sum of each subsystem s enegy functional i.e. 3 Ek[ nk ] Vemb ( ) nk ( ) d which is the conestone in ou poof given ealie fo the uniqueness of ou embedding potential. Theefoe the only diffeence between PDFT and ou wok is that we have an additional constaint: the total numbe of 143

154 electons in each subsystem should be fixed to an intege. This constaint is useful in pactice since most CW methods equie an intege numbe of electons in thei calculations. This additional constaint was shown not to alte the uniqueness of the density/density matix patitioning and the embedding potential (fo the non-spinpolaized case). Howeve it might cause poblems fo the existence of the solution because we ae now seaching fo the solution in the space of all the v-epesentable gound-state densities while PDFT seaches fo the solution in a much lage space containing all ensemble v-epesentable gound-state densities. 116 Since we fix the numbe of electons in each subsystem in pactice we need to assign a easonable numbe of electons fo each subsystem which can be done by pefoming some chage analysis such as Bade analysis 144 afte finishing the S- DFT calculation fo the total system. The existence of the embedding potential can also be judged numeically based on the diffeence between the total S-DFT electon density calculated fo the whole system and the sum of the cluste and envionment electon densities obtained fom the embedding-potential-geneating softwae. In this wok we have numeically demonstated the existence of embedding potentials fo both the H/H10 and CO/Cu cases. 5.8 Implementation in the Pojecto Augmented Wave (PAW) method Embedding theoy fomulated with the PAW method In this section we extend ou embedding theoy to wok with the pojecto augmented-wave (PAW) DFT method. 19 PAW-DFT was intoduced by Blöchl as an efficient technique fo pefoming all-electon (AE) fozen coe S-DFT calculations using planewave basis sets. Its fomalism is much moe complicated than the conventional NLPS The basic idea of PAW-DFT is that the AE wavefunction 144

155 nk is ecoveed though a pojection involving the pseudo (PS) wavefunction ~ nk via ~ ~ ~ nk nk i i p i ~ i nk. (5.) Hee n k ae indices fo bands and k-points in the Billouin zone fo cystals but these could be simply an obital index in a finite system. The i and ~ i ae AE and PS atomic patial waves centeed on each atom which ae equal to each othe outside a coe adius C. Theefoe the pojection in Eq. (5.) is only defined within the coe adius (PAW sphee) which leaves the intestitial egion unaffected. p~ i is the pojecto function that is dual to the PS patial waves ~ i : ~ p i ~ j. ij One of the key ideas in PAW-DFT is that both p~ i and ~ i ae not nomalized to one (just as in the ultasoft PS fomalism 18 ). Theefoe they can be constucted to be vey soft i.e. expanded with only a small numbe of plane-waves. By contast the AE patial wave i is evaluated on a adial mesh within the coe adius. Technically thee is one adial mesh (inside the PAW sphee) on each atom and one unifom mesh in the space. The patial waves and pojecto functions ae decomposed into adial and angula pats n ( ) ili ( ~ ~ n ( ) ili i ) Sl (ˆ) im ( ) S (ˆ ) i i lim and ~ ~ pn ( ) ili p ( ) S (ˆ) i i lim. (5.3) i 145

156 Above n l } is a pai label fo the enegy level index and angula momentum { i i index fo a single atom. ˆ l () is the eal spheical hamonic fo the angula S i m i momentum indices l m } and is defined as { i i 1 m Yl m ( 1) Ylm fo m 0 S lm ( ˆ) Yl 0 fo m 0. i m Y lm ( 1) Yl m fo m 0 By intoducing the pojection in Eq. (5.) the total electon density tot () of the entie system is also decomposed into thee pats tot ~ 1 ( ) ( ) ( ) ~ 1 ( ) (5.4) with the definitions ~ ~ ~ ( ) nk ' nk nk 1 ( ) i j i ' j i j ~ 1 ~ ( ) i j i ~ ' j. i j ~ ~ In the above we have defined the density matix ~ p ~ p i j nk nk i j nk. In the PAW fomalism we minimize the total enegy with espect to the soft PS wavefunction ~ nk. Skipping the technical details of PAW-DFT we diectly discuss how to implement the OEP fo solving fo embedding potential within PAW. Inteested eades ae suggested to consult Refs

157 The OEP in the PAW-DFT fomalism becomes W E V V emb emb [ A ] E ( ) ( ) A B [ ] B ~ 3 ~ ~ A B ef d ~ ~ ~ A B ef A B ef d 3 (5.5) whee EA[ A] and E B [ B ] ae simply the total enegy functionals of isolated systems A and B. Note that we have aleady used the decomposition of the total electon density fom Eq. (5.4). To evaluate the on-site tems in the last line in Eq. (5.5) we need to expand V emb () ove the ( l m) angula momenta which ae centeed at each nucleus ( R is the coodinate fo nucleus I) V emb I RI Slm ( ) lm ( ) V. (5.6) lm emb R I is the spheical angle of the vecto RI. We employ the identity between R I plane-waves spheical Bessel functions and eal spheical hamonics 146 e iq 4 l i l0 ml l j ( q) S l lm ( qˆ) S lm (ˆ). (5.7) By expanding () in a planewave basis which can be achieved easily using Fast- V emb Fouie tansfomation (FFT) in pactice and with the help of Eq. (5.7) we can expess V emb () in tems of eal spheical hamonics S ) : V emb ( ) q v~ e q iq R I e iqr I 147 lm ( R I l iqr ~ l I 4 e v q i jl q RI S lm ( qˆ) S lm ( ). (5.8) RI q l 0 ml l iqr 4 ~ l I e v qi jl q RI S lm ( qˆ) S lm ( ) R I l 0 ml q

158 Compaing Eq. (5.6) and Eq. (5.8) we have the expession fo the moments lm Vemb RI fo the atom at R I V lm emb R ) S ( qˆ). (5.9) iqr l I R 4 e v~ i j ( q I lm With Eq. (5.9) in hand we can evaluate V R q q l emb I I lm by pefoming a summation ove q-space with the Fouie coefficients of () special Bessel functions and eal spheical hamonics. V emb Now the on-site tems (the last line) in Eq. (5.5) can be evaluated as V emb R I R I ( ) ( ) d lm lm sin dddvemb R lm lm 1 V R ( ) d C 0 1 A emb 3 I lm I S lm 1 ( ) R l' m' ( ) Sl' m' ( ) I RI. (5.30) l' m' Next we deive the ohn-sham equation fo the subsystems. The embedded Hamiltonian Ĥ A fo subsystem A is Hˆ A ~ W ~ A A V E A[ A ] ~ V emb ( ) A emb ( ) ~ ~ ~ A B ef A B ef d 3. (5.31) The fist tem on the ight-hand side in Eq. (5.31) is the S Hamiltonian the non-embedded system A 146 PAW Ĥ A fo Hˆ PAW A 1 ~ V ( ) ~ D ~ eff pi Dˆ 1 1 ij D ~ ij ij p j i j 148

159 whee Dˆ ij ~ D 1 ij and D 1 ij ae the conventional PAW quantities defined as 146 Dˆ ij lm ~ V eff ( ) Qˆ lm ij ( ) d 3 D 1 ij i 1 V 1 eff j ~ 1 ~ 1 ~ 1 ~ ~ 1 lm D ij i Veff j Veff ( ) Qˆ ij ( ) d lm 3. And ~ V eff V [ ~ ˆ ~ ] V [ ~ ˆ ~ ] H Zc XC C V 1 eff 1 1 V [ ] V [ ] H Zc XC C ~ 1 1 eff H Zc XC V ~ 1 V [ ˆ ~ ] V [ ~ ˆ ~ ]. C The eason fo intoducing Zc ~ Zc c and ~ c above is to make the computation of Hatee and exchange-coelation enegies efficient. Zc is the sum of Z C the point chage density of the nuclea Z and the fozen coe chage density C. ~ Zc and ~ C ae the soft pseudized chage densities that ae equal to Zc and C espectively outside a pe-defined coe adius. ˆ Q lm ( ) ae the 50 moments of the caefully-chosen compensation chage ˆ which is placed on each atomic site to wok with soft chage densities ~ and ij 1 ~ to epoduce the coect multipole moment of the tue all-electon chage density. Fo a moe detailed definition of ˆ Q lm ( ) please see Ref ij The thid tem in Eq. (5.31) is evaluated with the help of Eq. (5.3) and Eq. (5.6) 149

160 150 j i j i ext ij j i j i l n l n lm R I lm emb lm l n l n j i j i m l l n m l l n R lm lm R I lm emb j i j i j i emb j i j i j i j i A ij emb A ij j i A A ij A emb A ij ef B A emb A p p D p p d R V g p p d d d S S S R V p p d V p p d V d V d V j j i i I c j j i i j j j j i i i i I I 1 * 0 * 3 * 3 * ~ ~ ~ ~ ) ( ) ( ~ ~ sin ) ( ) ( ) ( ~ ~ ) ( ) ( ) ( ~ ~ ) ( ) ( ) ( ~ ) ( ) ( ) ( ) ( ) ( ) ( ~ (5.3) whee d R V g D j j I C i i j j i i l n lm R l n I lm emb lm l n n l ext ij ) ( ) ( 0 * 1 and d d S S S g j j i i j j i i m l lm m l lm l n l n sin (ˆ) (ˆ) (ˆ) 4 1 ae the eal Gaunt coefficients. 146 Similaly we also have ij j i ext ij ef B A emb A p p D V ~ ~ ~ ) ~ ~ ~ ( ~ with lm R l n n l I lm emb lm l n l n ext ij I c j j i i j j i i d R V g D 0 * 1 ) ( ~ ) ( ~ ~. In summay once we obtain the embedded subsystem Hamiltonian (Eq. 5.31) S-DFT-PAW is pefomed fo each subsystem. With the etuned electon density

161 and vaious enegy tems the OEP functional W defined in Eq. (5.5) can be evaluated Smooth PAW functions One numeical issue in using the PAW functions 19 i.e. AE patial waves i PS patial waves ~ i and pojectos p~ i is that we ecove the AE wavefunctions i though Eq. (5.) which have cusps at the nuclei. These cusps cannot be expanded efficiently using plane-wave basis set. To ovecome this numeical poblem we need to pseudize the AE patial waves i to be smooth at the nuclei. This smoothing technique has been used by esse et al. in thei implementation 147 of the ultasoft-pseudopotentials. 18 To demonstate ou smoothed PAW functions we show two elements hee: H and Cu. Fo both elements thei PAW functions ae constucted with the AtomPAW code. 148 Then the AE patial waves within a cetain adius ae pseudized using the Toullie-Matin scheme 37 built in the AtomPAW code. Figs. 5.5 (a) and (b) show the two AE patial waves (solid black cuve) and the pseudized AE patial waves (ed dashes) fo the s angula momentum channel of the H atom whee the adius within which smoothing is pefomed is chosen to be 0.7 boh fo both s channels. Fig. 5.5 (c) shows the final embedding potential fo the H molecule with the embedding theoy fomulated using the PAW method discussed above. We see a deep potential well between the two H atoms in ode to attact electons to fom covalent bond. As we can see a lage penalty coefficient 1 10 potential. 4 poduces a smoothe embedding 151

162 Figue 5.5 (a) and (b) ae the two PAW functions both fo the s angula momentum channel. One is calculated at the efeence enegy of the eigenvalue of the H atom s only electon and anothe is calculated at the efeence enegy of 0.0 Hatee. In (a) and (b) we compae the oiginal AE (black cuve) and ou smoothed AE (eddashed) patial waves. (c) The conveged embedding potential (each H atom is consideed as one subsystem). We use two diffeent penalty function coefficients: (solid black) and the two H atoms is 1.4 boh (ed dashed) espectively. The distance between 15

163 Figue 5.6 The AE patial waves (black cuve) and the smoothed AE (ed-dashed) patial waves fo the s (a) p (b) and d (c) angula momentum channels fo Cu atom. The final embedding potential fo the Cu dime is shown in (d) fo two diffeent penalty coefficients: (ed dashed line) and espectively. The distance between the two Cu atoms is 5.8 boh (solid black line) The AE (solid black cuves) and pseudized AE (ed dashes) patial waves ae displayed in Fig. 5.6(a-c) fo the s p and d angula momentum channels of the Cu atom. The conveged embedding potential fo the Cu dime with each Cu atom as a subsystem is shown in Fig 5.6(d). The embedding potential nea the coe egion fluctuates consideably (ed dashes) with a smalle penalty function coefficient The embedding potential becomes much smoothe with a lage penalty 153

164 function coefficient (solid black cuve). Howeve away fom the atoms and in the bonding egion the embedding potentials ae simila fo these two penalty function coefficients. 5.9 Conclusions We poposed an accuate ab initio embedding theoy with a unique density/density matix patitioning fo both non-spin-polaized and spin-polaized quantum systems in which the electon numbes in cluste and envionment ae fixed. Fo the non-spin-polaized case this theoy leads to a unique embedding potential. The uniqueness is achieved by letting the cluste and the envionment shae a common embedding potential as shown fo both non-spin-polaized and spin-polaized cases. The embedding potential is solved fo numeically exactly and eschews use of any appoximate density functionals fo exchange-coelation o kinetic enegy. As a esult the embedding potential in pinciple should be moe accuate than any density-based embedding potential poposed peviously. By intoducing the OEP method we demonstate a numeically simple efficient and obust way to solve fo embedding potentials. In ou OEP method two S-DFT calculations ae pefomed fo the cluste and the envionment simultaneously which is moe efficient than pevious schemes in which eithe the cluste o the envionment is fozen with the othe being optimized duing each iteation i.e. the so-called feeze-and-thaw scheme. 149 Moeove since the OEP method is based on diect optimization using analytic gadients it is usually much easie to solve than an extended Lagangian dynamics appoach In the embedding famewok we also give a detailed discussion on how to compute total enegies. Afte applying a fist ode coection to the inteaction enegy tem in calculating the vdw binding enegy cuve fo the H/H10 system ou embedding method yields a binding enegy cuve that essentially coincides with the CCSD(T) benchmak cuve in the vdw-dominant egion close than the ONIOM 154

165 method o Ronceo et al. s embedding esults. 115 Employing this new embedding theoy we evisited the adsoption of CO on the Cu(111) suface and showed that the embedded MRSDCI calculations yield the pope site stability and binding enegy tends compaed to expeiment. Moeove the embedded MRSDCI wavefunctions povide an explanation fo the tends obseved in XES specta specifically the changes we obseve in the CO s 4 and 5 embedded MRSDCI natual obitals. 141 We also genealized ou embedding theoy to self-consistent embedding potentials and to spin-polaized quantum systems whee the latte expands the scope of the theoy to mateials with magnetization. We show how to implement this embedding theoy within the PAW famewok which makes it much efficient to teat fist and second ow elements as well as some tansition metals. We also compaed ou theoy with PDFT and point out that the only fomal diffeence between PDFT and ou theoy is an additional constaint in ou theoy: fixed intege numbes of electons in subsystems. With this additional constaint we seach fo the embedding potential in the space of all v-epesentable densities athe than the lage space of ensemble v-epesentable densities used in PDFT. We demonstated how to justify the existence of embedding potentials in pactice and employed an efficient OEP appoach fo pactical ealization of the theoy. The ab initio embedding theoy pesented in this wok is based on a unique density/density matix patitioning and delives exact embedding potentials with no appoximate functionals fo systems that can be decomposed into v-epesentable subsystems shaing a common embedding potential. Fo othe systems that can be appoximately decomposed into v-epesentable subsystems shaing a common embedding potential ou theoy delives embedding potentials that ae systematically close to the exact embedding potential defined in PDFT Lastly because we insist on intege numbes of electons in subsystems ou theoy can immediately be used with coelated wavefunction calculations opening the way to 155

166 pefom accuate embedded coelated wavefunction calculations on a wide vaiety of mateials Acknowledgements This wok is collaboated with D. Michele Pavone. I thank D. Qin Wu D. Saha Shaifzadeh D. Tsz S. Chwee and Peilin Liao fo insightful discussion. I also thank Pofesso Adam Wassemann fo intiguing discussions on the PDFT. We ae gateful fo financial suppot fom the U.S. National Science Foundation. 156

167 Chapte VI Potential-functional embedding theoy fo molecules and mateials 6.1 Intoduction Accuate and detailed electonic popeties and enegies ae a peequisite fo pope undestanding and eliable pediction of mateial popeties on a vaiety of length scales. Theoies of vaying levels of accuacy exist in the liteatue. Coelated-wavefunction (CW) theoies include in thei canon fomally exact means fo solving the many-body Schödinge equation (full configuation inteaction (CI)). Howeve the computational cost of CW theoies tends to scale damatically with espect to the system s size. Even though (quasi-) linea-scaling CW methods have been intoduced in the last 10 yeas it is still not feasible to apply these highly accuate methods to systems that contain hundeds of atoms. On the othe hand ohn-sham density functional theoy (S-DFT) which is based on the Hohenbeg-ohn 3 (H) DFT geatly simplifies solving the many-body Schödinge equation by intoducing the fictitious S non-inteacting electons and gouping most of the many-body infomation into the so-called exchange-coelation functionals (XCs). In pinciple S-DFT is exact and its accuacy can be impoved by using moe advanced XC functionals such as obital-dependent XC functionals. Howeve these latte functionals again equie consideable computational time. Unfotunately the most common implementations of S-DFT cannot adequately addess many poblems in mateial and life sciences such as van de Waals (vdw) inteactions between polyme stands stong coelations in tansition metal and ae eath ionic compounds adsoption of some molecules on metal sufaces singlet 157

168 diadicals etc. Fo these poblems CW methods geneally povide supeio pedictions. The nea-sightedness pinciple endes valid the application of a theoy of high accuacy to a local egion of inteest while teating the est of the quantum system with a moe appoximate theoy that equies much less computational effot. One example 6 11 fom suface science is to use a CI desciption fo an adsobate and neaby suface atoms (a cluste) with the est of the suface (the envionment) teated e.g. within S-DFT. An embedding theoy is needed to descibe the inteaction between the cluste and its envionment. Diffeent embedding schemes exist fo diffeent scenaios. In ionic solids usually the envionment is modeled sufficiently accuately with an extended point chage aay. If the polaization of the envionment is also impotant a moe sophisticated classical shell-model can be used. Anothe appoach is the ab initio model potential method 160 which delives ab initio embedding potentials that have been used to study local geometies excitation specta etc. fo defects in ionic solids. In covalently-bonded solids and in lage molecules the covalent bonds ae usually cut and satuated with pseudo-hydogen atoms. 99 Fo metallic systems density-based embedding is usually pefeable since the delocalization of electons cannot be handled by embedding theoies designed fo ionic o covalent solids. Densitybased embedding is based on the patitioning of the total electon density into subsystems as fist poposed by Cotona 81 as a choice fo fast S-DFT calculations. It was then extended by Cate and co-wokes fo embedded CW calculations. In pinciple density-based embedding which is based on DFT is fomally exact. A ecent eview of embedding schemes is available. 109 Ultimately a successful embedding theoy should attempt to satisfy a numbe of equiements. (1) It should be able to teat the long-ange electostatic inteactions between the cluste and its envionment in a self-consistent fashion which means that the cluste s and envionment s electon distibutions adjust to each othe until they each an equilibium state. This will be impotant fo cases in which the cluste 158

169 and envionment ae easily polaized. () The embedding theoy should be able to accuately descibe the shot-ange inteaction at the bounday between the cluste and its envionment. In density-based embedding theoy these shot-ange tems lagely involve patitioning of kinetic enegy and XC tems (3) The theoy should be able to equilibate the chemical potential between the cluste and its envionment by allowing chage tansfe between the two. Fo density-based embedding we have an additional equiement: (4) since multiple ways exist to patition the total electon density into a cluste and its envionment 6 the embedding theoy should have a unique patitioning of the total electon density to make the theoy tactable. The fouth equiement has been analyzed in detail in the patition-density functional theoy (PDFT) and ou ecent density-based embedding theoy. 6 In these theoies the non-uniqueness of density patitioning is emoved by applying a constaint that the embedding potential is the same fo all subsystems. Chage edistibution among subsystems (the thid point above) was also discussed in PDFT whee factional numbes of electons in subsystems ae teated using ensemble DFT. 161 Then the question becomes how to find this unique embedding potential in pactice. In PDFT Elliott et al. fomulated the embedding potential (see Eq. (11) in Ref. 119 ) in the same way as in Cotona s oiginal fomalism (see Eq. (14) in Ref 81). Both fomulated the embedding potential in tems of density functionals which makes diffeent subsystems subject to diffeent embedding potentials if the subsystem electon densities ae not caefully chosen. Even though Elliott et al wee able to impose the constaint that all subsystems shae a common embedding potential it was not staightfowad to satisfy this constaint with thei densityfunctional based fomalism. Hee we show that the key to imposing this constaint in a seamless way is to efomulate the total enegy functional in tems of embedding potential functionals instead of electon density functionals. In this famewok the embedding potential is used diectly as the only woking vaiable and the total system s gound state is obtained by diectly minimizing the total enegy with espect to the embedding potential. Duing the minimization the constaint is satisfied automatically and staightfowadly. As we show below all fou equiements enumeated above can be fulfilled simultaneously with ou potential-functional fomalism fo the embedding theoy. This potential-functional 159

170 fomalism is actually the dual poblem to the density-functional fomalism of the embedding theoy In conventional S-DFT this dual poblem i.e. the equivalence between density-based and potential-based total enegy functionals was aleady ecognized some time ago. 16 This pape is oganized as follows. Fist we intoduce the potential-functional fomalism fo the embedding theoy and pove that one has the feedom to choose any well-defined enegy functional fo descibing subsystems. We then discuss use of eithe appoximate kinetic enegy density functionals (EDFs) o optimized effective method (OEP) methods fo evaluating the S kinetic enegy component of the inteaction enegy between subsystems. We then pesent how to efficiently calculate the gadient of the total enegy with espect to the embedding potential and subsystem electon numbes. We also deive foce expessions so that geomety optimizations and dynamics will be possible. We then extend ou theoy to spin-polaized systems. To demonstate ou theoy we test it on seveal diatomic molecules an H 6 chain and bulk NaCl. Finally we show two examples involving intefaces: the inteaction between H O molecules and the MgO(100) suface; and the inteaction between O molecules and an Al(111) suface. 6. Theoy 6..1 Potential-functional fomalism fo embedding theoy In evey embedding theoy the total system is divided into two (o moe) subsystems. A common way is to patition the total extenal ionic potential V ext total ( ) fom all atoms into seveal subsystem potentials V ext ( ) (whee indexes the subsystems) V ext total( ) Vext ( ). (6.1) An embedding potential seves as an additional extenal potential to each subsystem to eplace the inteaction between that subsystem and the est of the 160

171 total system. If we apply the constaint that all subsystems shae one common embedding potential u( ) and also fix the subsystem electon numbes N } we have shown 6 that this embedding potential u( ) is unique. Once u( ) and N } ae known the electon densities { } in the subsystems can be detemined as well as the total electon density tot. Since tot uniquely detemines the gound state of the total system due to the fist H theoem 3 thee is a one-to-one mapping between u( ) and the total system with fixed N }. Consequently the poblem of finding the tot that minimizes the total enegy can be tansfomed to the poblem { k of finding u( ) and N } that minimize the total enegy { { { min E tot tot [ ] min E [ [ u{ N }]] (6.) tot u{ N } tot tot which is the key idea in this wok. The fomal expession fo the total enegy functional E in tems of u( ) and N } is tot { E tot E 1 Z Z [ u N ] Eint[ u{ N }] R R I J ' I J '. (6.3) I J ' The fist tem on the ight-hand side is a sum ove each subsystem s total enegy functional in the pesence of an additional u( ) fo a given N }. The ion-ion Coulomb inteaction enegies between diffeent subsystems ae gouped in the last tem with the coodinates and Z the full nuclea chage (fo all-electon RI I calculations) o the valence chage (fo pseudopotential calculations) of nucleus I in subsystem. E [ u{ N }] is the inteaction enegy that will be discussed late. Note int that Eq. (6.) is equivalent to min min Etot[ tot[ u{ N }] u { N } 161 { which means that N } is the minimize fo E [ u{ N }] fo a given u( ). Thoughout this wok we tot[ tot assume no degeneacies exist i.e. thee is only one unique { N } fo a given u( ). {

172 This assumption is often satisfied in eal physical systems due to chage equilibation that poduces an optimal set of { N } fo a given u( ). Consequently { N } is an implicit functional of u( ) which in tun endes E tot a functional solely of u( ). In this wok we demonstate how to detemine the embedding potential u() and the subsystem electon numbes N }. { 6.. Definition of the subsystem enegy Fomally a subsystem s enegy functional is defined as E [ u N d R I ] E [ u N R I ] 3 0 ( ) u( ) (6.4) whee E [ u N R ] 0 I is the total enegy functional fo the bae system without embedding and whee the embedding potential u( ) seves as an extenal potential. Then we ae fee to evaluate E with e.g. CW methods o othe advanced quantum mechanics methods by teating u( ) as an extenal potential. In this wok fo simplicity we use a ohn-sham-like definition fo the subsystem enegy E 0 E 0 iocc ( ) V f i ext i ( ) d 3 1 i J[ ] E Z R I Z R xc J I J I J [ ] (6.5) whee f is the occupation numbe fo the i -th obital in subsystem and i f i N. i iocc. i ae the ohn-sham obitals in subsystem with i In contast to the usual S scheme both N and f ae allowed to be factional. i J[ ] is the Hatee enegy of the electon density in subsystem. Hee we use 16

173 the local-density-appoximation (LDA) 36 fo E xc [ ] but of couse one could also use moe advanced XC functionals such as obital-dependent XC functionals (a ecent eview of obital-based XC functionals is given in Ref. 156 ). The last two tems in Eq. (6.5) ae the extenal potential enegy and the ion-ion enegy within subsystem. Ou subsystem enegy definition in Eq. (6.5) is diffeent fom PDFT whee ensemble DFT is used to teat factional electon numbes in subsystems. In ensemble DFT the enegy fo a subsystem with a factional numbe N of electons is defined as 161 E [ N ] E [ N] (1 ) E [ N 1] k whee 0 1 and E [N] is the enegy fo an intege N electons. One can ague that ensemble-dft 161 which is fomulated fo open systems is physically coect fo descibing a factional numbe of electons. Howeve we can also employ othe well-defined convenient methods fo subsystems because in the context of embedding theoy only the total electon density tot entes the total enegy functional E tot (Eq. (6.)). Consequently one has the feedom to choose a diffeent theoy fo each subsystem if it is desiable to do so. Fo example if it was advantageous one could use ensemble-dft fo some subsystems and use ou definition (Eq. (6.5)) o CW methods fo othe subsystems. In the latte case the electon numbes in those subsystems ae equied to be integes. To veify whethe a given definition fo the subsystem total enegy E is welldefined we at least need to show the existence of E / which equies a oneto-one mapping between E and. If thee ae an intege numbe of electons in subsystem then this mapping is one-to-one due to the fist H theoem. 3 We now show that thee is also a one-to-one mapping between E as defined in Eq. (6.5) 163

174 and fo the case of a factional numbe of electons. Let us assume the opposite that fo a given u( ) and N the subsystem has two diffeent densities 1 and with coesponding S effective potentials eff 1 V and V eff. Thei coesponding obitals ae { i 1 } and { i } with f i i1 1 f i i iocc iocc and whee i is the index fo these obitals. The occupation numbes { f } ae the same fo both cases: the lowe eigenstates have occupation i numbes of two (in the spin-esticted fomalism) and the highest occupied eigenstate has an occupation numbe less than two so that all the occupation numbes sum to N. Since { i 1 } is the only minimize (assuming no degeneacy) of the Hamiltonian H 1 Veff 1 u we have f i i1 Veff 1 u i1 f i i Veff 1 u i iocc iocc. (6.6) To simplify notation we define. E q f i i q Veff q i q iocc Reaanging tems we have E1 1ud E ( Veff 1 Veff ) kd ud (6.7) Similaly we have E ud E ( V V d ud. (6.8) 1 eff eff 1) k

175 By summing Eq. (6.8) and Eq. (6.7) we each 0 0 which is obviously false showing that ou oiginal assumption was false. Thus via poof by contadiction we have shown that fo a given u( ) and N the mapping between the gound state of subsystem and its density is a one-to-one mapping even fo a factional numbe of electons. Hence the subsystem enegy defined in Eq. (6.5) is a functional of fo a given u( ) and N and E / is well defined Definition of the Inteaction enegy The inteaction enegy should be added to enegy functional is then E int in Eq. (6.3) is defined to include eveything that 1 E and I J ' Z R I I Z R J ' J ' in ode to estoe the total E tot. Using the definition of the subsystem enegy in Eq. (6.4) E int Eint Eint0 u( ) totd 3 (6.9) whee E int0 J int TS [ tot ] ] ( ) [ TS [ ] Exc[ ' V ext ' tot ( ) d ] 3 E xc [ ]. (6.10) Hee T S [ ] f i i i iocc and T ] is the S kinetic enegy fo the total density tot. J int is defined as S [ tot J int ( ) ( ') 3 3 d d'. (6.11) ' 165

176 It is easy to see that by inseting Eq. (6.4) Eq. (6.9) and Eq. (6.10) into Eq. (6.3) the total enegy fo the entie system is ecoveed E tot T S 1 Z I Z J [ tot] Exc[ tot] J[ tot] Vext ( ) 3 tot( ) d (6.1) R R I J I I J which only depends on the total electon density tot and the position of ions as expected. To evaluate E int XC functionals in Eq. (6.10) can be appoximated with the LDA 161 o othe advanced XC functionals. 156 To evaluate the EDFs in Eq. (6.10) we can eithe calculate the exact kinetic enegy and its potential fo a given tot by employing the optimized effective potential 7 (OEP) method o use appoximate EDFs. In this wok we use both the OEP method fo the exact EDF and some appoximate EDFs such as Thomas-Femi (TF) von Weizsäcke (vw) 76 and Huang-Cate 4 (HC10) EDFs. Fo the HC10 EDF we set its two paametes to λ=0 and β=0.65 which is a simple nonlocal EDF with a single-density dependent kenel appopiate fo nealy-fee-electon-like mateials. The TF vw and HC10 (λ=0) EDFs allows us to test this potential-functional embedding theoy with epesentative local semilocal and nonlocal EDFs. In futue wok we plan to examine othe EDFs such as the HC10 (λ=0.01) that ae appopiate fo covalently bonded mateials. In what follows calculations with these appoximate EDFs ae labeled as emb-oep emb-tf emb-vw and emb-hc10. If we choose to use othe methods than S-DFT fo some subsystems e.g. CW o quantum Monte Calo fo subsystem A then the inteaction enegy of Eq. (6.10) becomes 166

177 E int0 J int TS [ tot ] ] ( ) [ S A T [ ] T ' V S ext ' [ A ] E ( ) d 3 xc [ ] tot A E xc [ ] E XC [ A ]. (6.13) Hee we explicitly sepaate out the S kinetic enegy T [ S A] and XC functional E [ ] XC A associated with the density A in subsystem A whee this density can be calculated fom the many-body wavefunction. T [ S A] can then be calculated by inveting the S equations using the taget density A i.e. the OEP pocess 7 and E [ ] XC A can then be calculated using whateve appoximate XC functional is used fo the othe subsystems. This is diectly analogous to how the embedding potential is fomulated in ealie embedded CW schemes Diect minimization of the total enegy Even though fomally potential u( ) in pactice E tot is a functional solely depending on the embedding E tot is minimized in two sepaate steps. Fist fo a given u() we minimize E with espect to { N } with the constaint of conseving the total electon numbe N N tot. Second with fixed { N } we minimize E tot with espect to u( ). We epeat these two steps until u( ) and N } ae both conveged. A flow chat (Fig. 6.1) is given to illustate the pocedue. { We biefly discuss why this two-step pocedue is valid fo seaching fo the global minimum of we each the condition E tot. By minimizing E tot with espect to u( ) fo a fixed { N k } Etot u( ) { } N 167 0

178 which is just the condition equied fo u( ) at the global minimum of E tot. Theefoe the second step above is valid fo seaching the global minimum with espect to u( ). Now let s focus on { N k } u() we obtain the equation:. By minimizing E with espect to N } fo a given { L N u E N u 0 (6.14) ( whee the Lagangian is defined as L E N N ). tot of In actuality the equation that { N } is equied to satisfy at the global minimum E with fixed u( ) is tot L N tot u E N u E N int u 0 (6.15) whee the Lagangian fo the total system is defined as L E N N ). tot tot ( tot Howeve the second tem in the middle of Eq. (6.15) is actually zeo as we now explain. Since the embedding potential u( ) Eint0 is defined as u( ) ( ) at the global minimum of E tot then E int N u in Eq. (6.15) is zeo due to E N int 3 Eint0 u d tot ( ) d ( ) N u u u Eint0 ( ) 3 ( ) u d 0 ( ) N u u (6.16)

179 Theefoe Eq. (6.15) becomes L N tot u E N u 0. Consequently Eq. (6.14) that { N } is equied to satisfy at each optimization step is equivalent to Eq. (6.15) equied fo { N k } at the global minimum of E tot with fixed u(). Hence ou two-step pocedue fo seaching fo the global minimum of E tot is valid. Technical details to solve fo u( ) and N } ae discussed in section 6.6 and 6.7. { k 6..5 Uniqueness of the embedding potential Following Cohen and Wasseman s appoach 116 we now show that the solution fo the embedding potential u( ) is unique up to a constant unde some assumptions independent of any specific definitions fo E and E int discussed above. Let us once again assume the opposite namely that we in fact have two diffeent embedding potentials u 1 and u with coesponding electon densities and 1 minimum of and whee 1 tot. As discussed above at the global E tot one can solve fo { N } by simply minimizing E with a fixed u(). Assuming thee ae no degeneacies fo the given ( ) minimize fo E E at the global minimum of tot E u then 1. We then have [ 1 u1] E[ u1] E[ u] tot( )( u1 u) d is the only 1 3. (6.17) Similaly we have 169

180 ) )( ( ] [ ] [ ] [ d u u u E u E u E tot. (6.18) Summing Eq. (6.17) and Eq. (6.18) we obtain u E u E u E u E ] [ ] [ ] [ ] [ (6.19) which is clealy false. Thus via poof by contadiction we have poven that the solution fo ) ( u is unique up to a constant shift Minimization of the total enegy with espect to the embedding potential To efficiently minimize tot E with espect to ) ( u fo a fixed } { N we adapt the potential-based minimization technique of Gonze. 163 The gadient is fomally } { int } { } { ) ( ] [ ) ( ] [ ) ( N N N tot u u E u u E u E. (6.0) The fist tem on ight-hand side of Eq. (6.0) is ) ( ) ( ' ) ( ') ( ) ( ' ) ( ') ( ') ( ] [ ) ( ] [ 3 3 } { d u d u u E u u E u N. (6.1) Hee is the chemical potential fo subsystem due to the constaint that the electon numbe in subsystem is fixed. The integal in Eq. (6.1) is zeo due to the consevation of the electon numbe unde a small change in ) ( u. Accoding to Gonze s appoach 163 we have the identity

181 ( ') E ( ). (6.) u( ) u( ) u( ') u( ') Eint Fo the tem u( ) in Eq. (6.0) by the chain ule we have u( ) Eint Eint ( ') 3 ( ') u( ) d'. (6.3) Inseting Eq. (6.) into Eq. (6.3) we can use a fist-ode finite-diffeence expession to calculate the functional deivative Eint u( ) 1 h Eint [ u]( ) d' [ u]( ') u( ') u h Eint [ u]( ) u( ) 3 ( ) [ u]( ). (6.4) Hee h is a small step used in the finite diffeence expession fo each subsystem. Eq. (6.4) can be calculated by pefoming anothe calculation fo each subsystem with a slightly diffeent new embedding potential u new Eint u h. [ u] u As we appoach the minimum we find that a second ode finite diffeence expession is necessay to obtain an accuate gadient Eint u( ) 1 h u h Eint [ u]( ) u ( ) u h Eint [ u]( ) u ( ). (6.5) With the gadients (Eq. (6.0)) in hand we can pefom outine optimization methods such as conjugate gadient o quasi-newton methods to minimize 171 E tot.

182 6..7 Optimizing subsystem electon numbes We give a geneal pocedue to minimize E with espect to N } fo a given embedding potential u (). In ode to satisfy the constaint N Ntot whee N tot is the numbe of electons in the entie system we employ the vecto x x } { { instead of the oiginal { N } with x N becomes a function of { x } and x Ntot. The Lagangian then L E [ x ] x Ntot. The -th element of the steepest decent diection d is theefoe d L x E x x. In the above fo a given x the Lagange multiplie is taken to be de[ x x Ntot dx ] which will be exact at the minimum. Fist we detemine the othogonal pojection of the vecto d (pependicula to x ) d d d x x x whee d x is the inne-poduct between vectos d and x. Then d is nomalized to be of length N tot 17

183 d N d N tot d d. The new x is then obtained by mixing the old x with the diection d N x new xcos d sin N in which the mixing paamete is tuned fom 0 to to minimize E [ ( )]. By using the vaiable { x } instead of the oiginal vaiable { N } we always have x new Ntot fo any. And x N pefom a line seach fo the optimal we need to compute x guaantees that N is always positive. To d E [ x d E [ x x ( )] new new ] ( x E [ x x new sin d new N ] dx new d cos ). Using the expessions above we can efficiently optimize the electon numbes in subsystems. In pactice we minimize the total enegy with espect to u( ) fo a few steps with a fixed { N } then we minimize E [ ] with espect to N } with u( ) x being fixed as shown in Fig To acceleate the convegence of electon numbes { N } simple mixing o Andeson mixing 164 which ae often used fo electondensity and potential mixing 165 in conventional S-DFT calculations can be used. { 173

184 6..8 Woking with nonlocal pseudopotentials and entopy Eq. (6.3) is exact fo local potentials such as the -Z/ potential fo an ion with positive chage Z. To wok with nonlocal pseudopotentials (NLPSs) we need to modify the subsystem enegy functional by adding the nonlocal tems. Woking with the leinman-bylande (B) NLPSs 4 the additional nonlocal electon-ion tems ae centeed at nuclei. Fotunately these tems ae shot-anged. Additionally if a smeaing technique is used in S-DFT fo efficiently sampling of the Billouin zone we must include the associated entopy tems. We add two additional tems to Eq. (6.4) to obtain E E }] TS (6.6) 3 0 ud ENLPS [{ i whee E }] is due to the use of B-NLPSs and is defined as 4 NLPS [{ i E NLPS i V V R lm R lm R lm R lm i [{ i}]. (6.7) iocc R l m V R lm R lm R lm Hee as in conventional B-NLPSs the ae the atomic pseudowavefunctions associated with the NLPS and ae the lm angula momentum channels of the V R lm NLPS fom which the local component has been subtacted i.e. V V. R lm lm loc Now we discuss the ange of the R summation in Eq. (6.7). If appoximate EDFs ae used to evaluate the inteaction enegy E int0 the sum ove R in Eq. (6.7) will be ove all the nuclei in the total system athe than ove just a single subsystem. This is because the wavefunctions in subsystem inteact with the B- NLPS pojectos fom othe subsystems and theefoe we have to include such inteactions eithe in the subsystem enegy functional (Eq. (6.5)) o in the 174

185 inteaction enegy E int0 (Eq. (6.10)). Because diect functional deivatives Eint 0 / must be pefomed in the evaluation of Eq. (6.3) E int0 needs to be an explicit functional of electon density. Since the B-NLPS enegy functional [ ]}] ( ] ae the S obitals associated with the total electon ENLPS [{ j tot j [ tot density tot (the supeposition of the subsystem electon densities)) is not an explicit functional of electon densities we cannot include ENLPS [{ j[ tot]}] in E int0. Thus to conside the inteaction between the NLPSs in one subsystem and the wavefunctions fom othe subsystems we let R un ove all atoms in the system in Eq. (6.7). Howeve the situation changes if the OEP technique is used to evaluate E int0 diectly. Via the OEP scheme we have access to the functional deivative T S E tot NLPS tot as a bundle. Consequently we can include ENLPS [{ j[ tot]}] in the inteaction enegy E int0 and we only need to include the B-NLPS pojectos fo atoms belonging to that subsystem in Eq. (6.7) i.e. R uns ove the atoms only within each subsystem in Eq. (6.7). This in tun geatly simplifies the fomalism with NLPSs and the deivation of foces late. Howeve thoughout this wok to compae esults using both EDFs and OEP we choose the case that R in Eq. (6.7) uns ove all the atoms in the total system. Because of the shot-ange natue of the nonlocal tems of the pseudopotential one could safely adopt a distance cutoff in futue calculations involving EDFs to limit the numbe of NLPS pojectos included fom neighboing subsystems. When smeaing is applied in S-DFT we need to conside an additional enegy aising fom the electonic entopy of Femions with the Femi-Diac distibution 166 S in subsystem which is due to the statistics S kb f ln f (1 f )ln(1 f ) N ( ) d 175

186 whee N ( ) is the density of states in subsystem k B is the Boltzmann constant and f is the Femi-Diac distibution of occupation numbes ove the noninteacting S enegy levels 1 k T 1 e. f / B Now let us focus on the inteaction enegy. The E int0 in Eq. (6.10) also needs to include ENLPS and the enegies due to the entopy new E int0 Eint0 ENLPS [{ j[ tot ]}] ENLPS [{ i}] TStot TS (6.8) whee j loops ove all occupied S obitals. S tot is the entopy associated with the total system. Without the OEP ] E [ ]}] and S tot cannot be j [ tot NLPS [{ j tot evaluated and so we have to set the tems in the fist paentheses and zeo i.e. new Eint 0 Eint0 TS TStot to. Fo consistency we also emove the subsystem entopy-elated as well. Fo closed-shell molecules o mateials with a modeate o lage band gap the entopy of the total system S tot should be close to zeo with a small smeaing tempeatue anyway. In the non-oep (the appoximate EDF) case the inteactions between the NLPSs in one subsystem and the wavefunctions fom othe subsystems ae appoximately teated via Eq. (6.7) by letting R un ove all atoms in the system as discussed above. With the OEP ] E [ ]}] j [ tot NLPS [{ j tot and S tot can be calculated fo a given tot theefoe Eq. (6.8) can be fully evaluated and the R in Eq. (6.7) uns ove only the atoms in that subsystem. 176

187 Figue 6.1 Flow chat fo total enegy minimization with espect to the embedding potential u( ) and subsystem electon densities N }. Afte initializing u( ) we solve fo the subsystem electon numbes { N } by minimizing E [ u N ] with u() { fixed (step ). In step 3 the gadient is calculated accoding to Eq. (6.0). With the gadient we employ a quasi-newton method to minimize E tot [u] (step 4). Once E tot is minimized we update N } with the new u( ) (step 5). If E tot and N } ae { both conveged the code exits othewise we go back to step 3. { 177

188 6..9 Fist-Ode Coection used in the Wu-Yang OEP In some we employ the Wu-Yang OEP 7 to evaluate the exact S kinetic enegy and its potential ( T S / ( ) ) fo a given electon density 0 (supeposition of subsystem electon densities). Upon including NLPSs and entopy tems ou modified Wu-Yang OEP is defined as W[ V E eff NLPS ] i [ ] E occ TS i [ ] i V eff ( )( ( ) ( )) d 0 3 V eff ( ) d 3 (6.9) whee the S obitals { i } ae solved fo each tial V eff () with ( ) iocc i. is the penalty function coefficient. Theefoe W is a functional of (). The index i loops ove all occupied obitals. By maximizing W V ] we obtain the () which epoduces the taget electon density 0 ( ). The tems in paentheses ae as [ eff defined in the oiginal Wu-Yang OEP. Hee the NLPS enegy V eff V eff E NLPS is due to the use of NLPSs. The enegy E TS TS is due to the electonic entopy discussed tot peviously. Due to the incompleteness of any finite basis set used in expanding () a penalty function 13 (the last tem in paentheses) is added to enfoce the V eff smoothness of V eff (). We find that the use of a penalty function and othe numeical inaccuacies cause the conveged solution opt afte maximizing Eq. (6.9) to be slightly diffeent fom the taget electon density 0. Theefoe the kinetic enegy calculated diectly using iocc i i is not accuate. We show hee that we can estimate the exact kinetic enegy and its potential though a fist-ode coection. If NLPSs and 178

189 smeaing ae used in S-DFT calculations the inteaction enegy (Eq. 6.8) also must include the E ] and E ]. Theefoe what we ae inteested in is the NLPS [ 0 TS[ 0 sum of the E ] E ] and the kinetic enegy all togethe. The exact sum of NLPS [ 0 TS[ 0 these thee tems estimated with a fist-ode coection is T [ ] E S 0 T [ ] E S T [ ] E S T [ ] E S W[ opt opt opt opt ] NLPS [ ] E 0 NLPS NLPS NLPS V [ [ [ eff opt opt opt TS ( ) [ ] ] E ] E ] E d 0 TS TS TS 3 [ [ [ opt opt opt ] ] ] TS ENLPS E ( ) ( V V eff eff [ [ opt opt ]( ] [ opt TS opt ) d 0 opt ( ])( opt opt ) d ) d 3 3. (6.30) In the above the integal with the chemical potential is zeo at convegence due to the fact 3 ]N N d 0 3 [ opt]( 0 opt) d [ opt tot tot. Theefoe afte emoving the penalty function W[ opt ] is just the fist-ode coected T S[ opt] ENLPS [ opt] ETS[ opt] fo opt 0. Likewise the fist ode coection to the potential of the sum of these thee tems is given by T S V ENLPS E ( ) eff [ opt TS ] [ opt 0 ] TS ENLPS E ( ') ( ) TS opt 3 d' 0 opt. (6.31) In this wok we apply the coection in Eq. (6.30) to TS [ opt] ENLPS [ opt] ETS[ opt] which geatly impoves the numeical stability of the OEP esults. No coection fo the potential is used to avoid the calculation of 179

190 the esponse function in the last integal in Eq. (6.31). We find that the method is stable without applying the coection to the potential Foces Foces ae equied fo modeling stuctual elaxation o dynamics. We show that it is staightfowad to deive foces in this potential-functional embedding theoy. Hee we assume that u( ) is expanded on a unifom gid (i.e. a plane-wave basis) independent of the nuclea coodinates. In the following we conside the case in which the OEP is used fo the inteaction enegy tem. Theefoe each subsystem enegy functional only contains the B- NLPSs of the atoms belonging to this subsystem. This geatly simplifies ou foce deivation below. Foces ae deived as de F E tot tot u( ) [ u[ R ] N [{ R }] R I { N }{ RI } dr IA u( ) d R IA 3 I I E ] N tot u{ RI } N R IA E R tot IA { N } u (6.3) whee RI A minimize of ae the coodinates fo the atom I in subsystem A. Since u( ) E tot we have is the Etot u ( ) { }{ } N R I 0. Because { N } is also the minimize of E tot with the constaint of the consevation of total electon numbe we have 180

191 181 0 } { A I A I A I R u tot R N R N R N N E I whee is the global chemical potential in the total system. Then only the last tem in Eq. (6.3) emains. With the help of Eq. (6.3) we have A J J A I J A I A I N u A I N u A I A I J J I J I A I N u A I N u A I A N u A I I tot R R Z Z R R E R E R R Z Z R R E R E R R u E F A A ' ' ' } { int ' ' ' } { int } { 1 }] { [. (6.33) To obtain the fist tem on the last line above we have used the fact that only subsystem A is affected by the change of A I R fo fixed ) ( u and } { N. If subsystems ae solved with a basis set that depends on } { I R e.g. a Gaussian basis then the subsystem electon densities I explicitly depend on } { I R. In this case we need to conside Pulay foces. 167 In Eq. (6.8) NLPS E is an implicit functional of } { I theefoe int E is a functional of } { I I R } { N and ) ( u. With () u and } { N fixed the second tem in Eq. (6.33) becomes } { 3 int0 3 int0 } { 3 int0 } { int ) ( ' ') ( ') ( A N u A I tot A I A I A A N u A I tot N u A I d u R R E d R E R ud E R E. (6.34)

192 In Eq. (6.34) we have sepaated the dependence of E int0 on RI A into two pats: (1) the fist tem in the last line in Eq. (6.34) is due to the change of E int0 though the change of A with espect to RI A. Note that with fixed u( ) and { N } the change of RI A only affects the density in subsystem A. () The middle tem in the last line in Eq. (6.34) is due to the explicit coodinate dependence of E int0 on RI A and the last tem accounts fo the implicit dependence of the total density on RI A We now show that the fist tem in Eq. (6.34) finally cancels the last tem in Eq. (6.34). This is due to Eint0 A( ') d' ( ') R IA A( ') u( ') d' R 3 3 tot 3 A IA u( ') R IA d'. The middle tem in the last line in Eq. (6.34) is E R int0 IA A dv E [ R ] E [ R ] ext A 3 NLPS tot I NLPS A A d dria RIA R IA tot I A whee V would be the local pat of the nonlocal pseudopotential of atom A. Loc A Finally the total foce contains fou pats: F F. 0 F1 FNLPS F F 0 is the fist tem in Eq. (6.33) F 0 E R A IA u N A 18

193 which is just the conventional foces in subsystem A calculated in the pesence of the embedding potential with fixed the electon densities fom the othe subsystems N A. F 1 is the foce between the nucleus I and dv F1 d ext A dria A 3. F NLPS is the foce due to the coodinate dependence of the NLPS pojectos (centeed at nuclei) E NLPS [ tot R I ] E NLPS [ A R I ] FNLPS R R IA tot IA A. Finally F is the electostatic foce between nucleus I and othe nuclei outside subsystem A F R IA J ' A Z R IA IA Z J R ' J ' Spin-polaized systems To extend ou theoy to magnetic systems we modify the total enegy expession in Eq. (6.3) in two ways to teat spin-polaized systems. One simple appoach assumes that even if subsystems now contain diffeent spin densities the inteactions between subsystems can still be descibed by one common scala potential u( ). Then the total enegy is still the same as in Eq. (6.3) howeve the inteaction enegy tem (Eq. (6.10)) is modified to teat spins. Fo the collinea spin case we have 183

194 Eint0 TS [ ] T S [ ] T S [ ] T S [ ] tot tot Exc[ ] E tot xc[ ] E tot xc[ ] E xc[ ] (6.35) J int ( ) V ext ( ) d 3 whee the total electon density is and subsystem electon tot tot tot densities ae. Moe igoously magnetic subsystems should inteact not only via a common scala embedding potential u( ) but also via an additional embedding magnetic field B() as discussed in Ref. 6. In this case the subsystem enegy functional becomes E. (6.36) 3 3 E 0 ( ) u( ) d [ ] B( ) d and the inteaction enegy tem E E int becomes (6.37) 3 3 int Eint0 tot ( ) u( ) d [ ] B( ) d tot tot whee E int 0 is defined as in Eq. (6.35). It was shown in Ref. 6 that the patitioning of the spin-density (collinea case) and spin-density matix (non-collinea case) is still unique fo a common embedding potential u( ) and embedding magnetic field B() with fixed subsystem electon numbes. If subsystem electon numbes ae allow to vay the pevious poof fo the non-spin-polaized case can be adapted to show the uniqueness of the patitioning of the spin-density o the spin-density matix. Theefoe minimization of the total enegy functional still yields a unique solution fo the spin-density o spin-density matix patitioning. We leave implementation of a spin-polaized potential-functional embedding theoy fo futue wok. 184

195 6.3 Numeical Details A FORTRAN90 code (named UOPT ) was witten fo pefoming this potentialfunctional embedding theoy. As mentioned above a flow chat of the code is given in Fig In UOPT a quasi-newton optimization code 1 is used to minimize the total enegy with espect to the embedding potential u( ). Duing the calculation UOPT calls a modified ABINIT 33 (named mod-abinit ) code with a tial u( ) to pefom standad S-DFT calculations. This u( ) seves as an additional extenal potential. Then mod-abinit etuns with a new gound state enegy and electon density. If emb-oep is pefomed we have anothe modified ABINIT code (named inv-abinit ) to find the coesponding S kinetic enegy and othe quantities given an electon density by pefoming the OEP algoithm discussed in both the Theoy section and in Ref. 7. All S-DFT calculations in this wok ae done with the ABINIT code. 33 We note that fo now ou implementation can only teat nonmagnetic systems so all calculations below ae spin-esticted calculations. The LDA 36 XC functional is used fo diatomic molecule tests (H P AlP LiH NaH and H) and the H6 chain test. The genealized-gadient-appoximation (GGA) 73 XC functional is used fo bulk NaCl HO/MgO(001) and O/Al(111) tests. In the diatomic molecule tests the equilibium distances (elaxed using S-DFT-LDA) between the two atoms in H P AlP LiH NaH and H ae 0.7 Å.0 Å. Å 1.8 Å 1.8 Å and. Å espectively. Fo the H6 setup the distance between the neaest H atoms ae 0.7 Å and the chain is linea. In diatomic molecule tests a cubic box with sides of 10 Å is used. In the H6 chain test the box is 17 Å 10 Å 10 Å to make sue the image inteaction is vey small. We used both local and nonlocal pseudopotentials in a vaiety of tests of the theoy. In the diatomic molecule and H6 chain tests we use local pseudopotentials (LPSs) in ode to test the kinetic enegy tem in the inteaction enegy 185 E int and avoid NLPS tems in Eq. (6.8). To build an LPS fo H we fist build its NLPS with

196 the FHI98 code. 38 The LPS of H is then taken to be just the s angula momentum channel. We build LPSs fo Li Na and using the same pocedue. Fo P and Al we use bulk-deived LPSs 4 78 that we had aleady constucted fo othe puposes. Fo the NaCl HO/MgO(001) and O/Al(111) tests NLPSs ae built with the FHI98 code. To achieve faste convegence of the total enegy with espect to the numbe of plane waves (i.e. a small kinetic enegy cutoff fo the plane wave basis Ecut) we build soft NLPSs fo oxygen and hydogen. Fo oxygen the NLPS cutoff adii ae and.0 boh fo the s p and d angula momentum channels. Fo hydogen the NLPS cutoff adius is 1.3 boh fo all s p and d angula momentum channels. We tested these soft NLPSs in a geomety elaxation of the wate molecule using S- DFT-GGA. The angle fo H-O-H is degees and the H-O bond is Å in good ageement with the taget S-DFT-GGA esults degees and 0.96 Å obtained with NLPSs with the default cutoff adii in the FHI98 code. 38 In H the H6 chain LiH NaH and H tests an Ecut of 100 ev is employed to convege the total enegy to within 10 mev. Fo P and AlP diatomic molecules the Ecut is 800 ev since ou Al and P BLPSs ae set to zeo at high fequency in Fouie space. 78 In the bulk NaCl test a cubic unit cell of fou Na and fou Cl atoms is employed. An Ecut of 1000 ev and a Monkhost-Pack 41 k-point mesh of ae used to convege the S-DFT total enegy to within 10 mev. To calculate the bulk modulus in the NaCl test Munaghan s equation of state 4 is used. Fo both HO/MgO(001) and O/Al(111) tests an Ecut of 100 ev conveges the total enegy to within 10 mev thanks to ou soft H and O NLPSs. A thee-laye thick MgO(001) slab with a 11 peiodic cell in the suface plane containing two Mg and two O atoms is used with one wate molecule added to one side of the slab (wate coveage of 0.5 ML). A vacuum laye 1 Å thick is added above both the MgO and Al slabs to limit inteactions with peiodic images. The dipole inteaction enegy between peiodic images in the suface nomal diection is small ~ 0.05 ev in the MgO case. The oxygen atom in the wate molecule is fixed to be diectly above one suface Mg atom and the distance between them is vaied. Fo 186

197 each Mg-wate O distance only the H atoms ae elaxed using ABINIT. These geometies ae supplied to the following calculations and no futhe stuctue elaxations ae pefomed. Fo O/Al(111) the Al suface is modeled with a fou-laye thick slab with one atom pe laye in the suface unit cell with the O molecule placed pependicula to the suface and diectly above one suface Al atom. This is a hypothetical top-site configuation at 1 ML O coveage. Again the O molecule is added on only one side of the slab and the dipole inteaction between peiodic cell images is smalle than 0.01 ev. Ou pupose hee is to simply test the potential-functional embedding theoy by compaing embedding esults with a benchmak (calculated by pefoming S-DFT on the entie O/Al(111)). Thus fa we have only implemented a non-spin-polaized vesion of the potential-functional embedding theoy and so this O/Al(111) test is pefomed with esticted S-DFT using Femi-Diac smeaing. Hence fa fom the suface O is in a singlet athe than tiplet state; thus this test is not mimicking the actual adsoption of O on Al(111) which would involve the gound tiplet state but athe just seves as a test of a diffeent type of inteaction within ou embedding theoy. Duing the calculations the O-O bond length is fixed to be 1. Å (the equilibium bond-length fom S-DFT-LDA) and the distance between O and the Al(111) suface is vaied to obtain a potential enegy cuve. In all bulk and suface tests heein Femi-Diac smeaing with a smeaing width of 0.1 ev is used to efficiently sample the Billouin zone. Fo HO/MgO(001) and O/Al(111) the k-point meshes ae and espectively which conveges the total enegy to within 10 mev. 187

198 6.4 Results and Discussion Diatomic molecules We demonstate ou theoy fo seveal diatomic molecules (H P AlP LiH NaH and H) to conside both covalent and ionic bonding as well as single and multiple bonds. In each test case each atom is a subsystem. We stat by consideing the simplest case H. In Fig. 6. the uppe plot compaes the emb-oep emb-vw emb- TF and emb-hc10 electon densities with the benchmak which is a S-DFT calculation on the entie H molecule. We see a pefect match between emb-oep and emb-vw densities with the benchmak. Since H only contains two electons the vw EDF is exact in this case. With emb-tf and emb-hc10 we see a deficiency of electon density between the two H atoms which indicates that TF and HC10 EDFs do not descibe this covalent bond well. This disadvantage of the TF EDF is manifest fo the othe diatomic molecules as well. Since TF theoy is known not to bind atoms popely it is not supising that the TF EDF is inadequate. Howeve the HC10 EDF (with λ=0) incopoates linea esponse behavio which povides a supeio desciption to emb-tf. In the lowe plot of Fig. 6. we show a contou plot of the embedding potential fom emb-oep calculations. The embedding potential is zeo fa away fom the H molecule and is negative between H atoms to attact electons to fom bond. P with its 10 valence electons and tiple bond povides a toughe test. In Fig. 6.3 the uppe plot again compaes the electon densities calculated with emb-oep emb-tf and emb-hc10 against the benchmak obtained by pefoming esticted S-DFT on the closed shell P diatomic. In esticted S-DFT each S obital is occupied equally by spin up and spin down electons. Because only the total electon density entes the total enegy functional ou nonmagnetic teatment of each P atom (subsystem) is acceptable in systems that ae oveall nonmagnetic. This is confimed numeically fo this P case and all the following tests heein; we leave 188

199 spin-polaized tests fo futue wok. The emb-oep (ed cicles) density matches the benchmak well except at the middle peak whee the emb-oep density is slightly too high. This small mismatch is pobably due to numeical inaccuacy in the OEP pocedue. The emb-tf again gives too low an electon density in the bonding egion (geen cuve) which indicates too weak covalent bonding. The emb-hc10 esult (blue cuve) ovebinds slightly with the density too high in the bonding egion. The two ed solid cuves ae the subsystem electon densities fo the two P atoms espectively which ae localized aound each P atom. The lowe plot in Fig. 6.3 shows an embedding potential contou plot calculated with emb-oep. The ed egion is epulsive which pushes electon densities to the negative blue egion to fom the covalent bond. The above two homonuclea diatomic cases have no chage tansfe between atoms due to symmety. We now conside a pola covalent bond in the AlP diatomic. As in the P case the AlP molecule is teated as nonmagnetic within esticted S- DFT. In Fig. 6.4 the uppe plot again shows that the emb-oep density (ed cicles) matches the benchmak (S-DFT calculations on the AlP diatomic solid black cuve) vey well. This time both emb-tf and emb-hc10 geneate much too high electon density peaks aound P atom. The middle plot of Fig. 6.4 shows the electon densities associated with each atom when emb-oep is used. Again the subsystem electon densities ae mainly localized at each atom and decays fast towad its neighbo. The contou plot in the lowe plot of Fig. 6.4 shows a negative embedding potential (blue egion) at the bond and a epulsive embedding potential (ed and yellow egions) nea atoms. This shows that the chaacte of the bond of AlP is not puely ionic but a pola-covalent bond. In Table 6.1 we give the total enegies fo H P and AlP calculated with both emb-oep and the benchmak. Emb-OEP enegies ae consistently highe than the benchmaks but only by at most 8 mev (fo AlP). To see how the embedding theoy woks fo moe ionic systems we conside next thee ionic diatomic metal hydide molecules LiH NaH and H whee in each case only the two valence electons ae teated explicitly. In the uppe plots of Fig. 189

200 6.5 Fig. 6.6 and Fig. 6.7 the emb-oep densities (open-cicles) match the S-DFT benchmaks on the diatomics (black cuves) vey well. The contou plots of these embedding potentials (lowe plots in Fig. 6.5 Fig. 6.6 and Fig. 6.7) show a clea tend of epulsive egions (ed aeas) moving successively towads the H atom fom Li to which shows that loses moe electons to the H atom as expected fom ionization enegy tends. In Table 6.1 we compae the total enegies between emb- OEP and the benchmak whee the enegies diffeences ae less than mev fo all thee cases. 190

201 Figue 6. (Uppe plot) Compaison of electon densities along the H molecula axis: S-DFT density (benchmak) is the black cuve. The emb-oep emb-vw emb- HC10 and emb-tf densities ae shown with black cosses black open-squaes blue dash-dots and pink dashes espectively. The electon density of each H atom fom the emb-oep calculations is shown by the ed solid cuve. (Lowe plot) Contou plot fo the embedding potential (in Hatee boh 3 ) in a plane containing the H molecule with the coodinates in Å. 191

202 Figue 6.3 (Uppe plot) Compaison of electon densities along the P molecula axis: S-DFT density (benchmak) is the black cuve. The emb-oep emb-hc10 and emb-tf esults ae given by the ed cicles blue cuve and geen cuve espectively. Electon densities fo each P atom fom the emb-oep calculations ae given by ed solid cuve. (Lowe plot) Contou plot fo the embedding potential (in Hatee boh 3 ) in a plane containing the P molecule with the coodinates in Å. 19

203 Figue 6.4 (Uppe plot) Compaison of electon densities along AlP bond axis: S- DFT density (benchmak) is the black cuve. The emb- OEP emb-hc10 and emb-tf esults ae shown by ed cicles blue dashes and geen dash-dots espectively. (Middle plot) Electon densities of Al (ed dash-dots) and P (ed dashes) atoms fom emb-oep calculations. (Lowe plot) Contou plot fo the embedding potential (in Hatee boh 3 ) in a plane containing the AlP molecule with the coodinates in Å. 193

204 Figue 6.5 (Uppe plot) Compaison of electon densities along the LiH bond axis: S-DFT density (benchmak) is given by the black cuve and the emb-oep density is shown in open cicles. The inset shows the details at the peak of the density. Electon densities of H and Li atoms fom emb-oep calculations ae shown by the ed and blue cuves. (Lowe plot) Contou plot of the embedding potential (in Hatee boh 3 ) in a plane containing the diatomic molecule with the coodinates in Å. The same convention is used in Fig. 6.6 and Fig

205 Figue 6.6 (Uppe plot) Compaison of electon densities along the NaH bond axis (Lowe plot) Contou plot of the embedding potential (in Hatee boh 3 ) with the coodinates in Å. See Fig. 6.5 s caption fo details. 195

206 Figue 6.7 (Uppe plot) Compaison of electon densities along the H bond axis. (Lowe plot) Contou plot of the embedding potential (in Hatee boh 3 ) with the coodinates in Å. See Fig. 6.5 s caption fo details. 196

207 Table 6.1 Compaison of the total enegies (ev) between the S-DFT benchmak and the emb-oep theoy fo H P AlP LiH NaH and H diatomic molecules. S-DFT benchmak Emb-OEP theoy H P AlP LiH NaH H H6 chain We also pefomed emb-oep on an H6 chain as fist step towad teating manyatom systems and compae the esults against the benchmak obtained by pefoming S-DFT on the whole H6 chain (this example is simila to one used to demonstate patition DFT in Ref. 10). Each H atom is a subsystem. Fig. 6.8(a) displays a contou plot of the final embedding potential on a plane containing the H6 chain. Negative egions of the embedding potential (geen) ae along the chain to attact electons to fom the bonds between H atoms. Suounding the chain is a epulsive embedding potential (ed and yellow egions) which futhe pushes electon density onto the chain. Fig. 6.8(b) shows a pefect match between the sum of electon densities fom all subsystems (open cicles) and the benchmak (black solid cuve). Dashed cuves show the electon densities associated with each H atom (subsystem). Again these densities ae well localized at each atom. To futhe quantify how well ou emb-oep epoduces the benchmak Fig. 6.8(c) displays a contou plot of the mismatch between the sum of all subsystem electon densities and the benchmak. A mismatch of ~ a.u. is found aound some H atoms. 197

208 Howeve the absolute value of electon density in these egions is ~ 1 10 a.u. so the elative mismatch is quite small. The total enegies of the benchmak and the emb- OEP diffe by only 19 mev a vey small deviation. We conclude that ou emb-oep epoduces the benchmak vey well. Figue 6.8 (a) Contou plot of the embedding potential (Hatee boh 3 ) fom emb- OEP theoy in a plane containing the H6 chain with the coodinates in Å. The six hydogen atoms ae maked with black dots. (b) Compaison of the electon density along the H6 chain fo the S-DFT benchmak (black solid cuve) and the emb-oep density sum (open cicles). Subsystem electon densities ae shown with dashes fo each the six hydogen atoms. (c) Contou plot of the electon density diffeence between the S-DFT benchmak and the emb-oep scheme. (coodinates in Å) 198