Development of Barostats for Finite Systems Born-Oppenheimer Molecular Dynamics Simulations

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1 J. Me. Che. Soc. 0, 56(3), Develoent rticle of Barostats for Finite Systes Born-Oenheier Molecular Dynaics Siulations 0, Sociedad Quíica de Méico 79 ISSN X Develoent of Barostats for Finite Systes Born-Oenheier Molecular Dynaics Siulations Gariel Ulises Gaoa, * Patrizia Calainici and ndreas M. Köster Deartaento de Quíica, CINVESTV, venida Instituto Politécnico Nacional 508,.P Méico D.F , Méico. calain@cinvestav., akoster@cinvestav. Present ddress: Deartent of Physics, Virginia Coonwealth University, Richond, Virginia 384. gugaoaart@ vcu.edu Received Noveer 9, 0; acceted ril 0, 0 stract. new ethod for ressure control in first-rincile olecular dynaics siulations for finite systes is resented. The etended Lagrangian ethodology is alied to generate the equations of otion and the syste s volue is otained y a urely geoetrical rocedure, which is ineensive in ters of coutational cost. The ileentation of all discussed algoriths was carried out in the rogra demonk where a roust achinery for auiliary density functional theory calculations eists. The here descried ethodology etend our effort on roerty calculations eyond the olyatoic ideal gas aroiation on the asis of first-rincile electronic structure calculations. Key words: Molecular Dynaics, Finite Systes, Barostats. Resuen. Se resenta un nuevo étodo ara controlar la resión en siulaciones de dináica olecular de rieros rinciios. La etodología del Lagrangiano etendido es alicada ara la generación de las ecuaciones de oviiento y el voluen del sistea se otiene de un rocediiento uraente geoétrico, el cual es econóico en térinos de costo coutacional. La ileentación de los algoritos discutidos se llevó a cao en el rograa demonk, donde eiste una aquinaria rousta ara cálculos de teoría de funcionales de la densidad auiliar. La etodología aquí descrita alía nuestro esfuerzo en cálculos de roiedades ás allá de la aroiación del gas ideal con ase en cálculos de estructura electrónica de rieros rinciios. Palaras clave: Dináica olecular, sisteas finitos, arostatos. Introduction Molecular Dynaics (MD) siulations are used to study the natural tie evolution of a syste of N articles in a volue V. In such siulations the total energy E is a constant of otion. The integration of the (classical) equations of otion for such a syste leads, in the liit of infinite saling, to a trajectory which as onto a icrocanonical (NVE) ensele of icrostates. lication of ressure to a given syste ay lead to electronic structure changes which will also odify the roerties of the syste. Many eeriental easureents are ade under constant teerature and ressure conditions, and thus, siulations in the isotheral-isoaric (NPT) ensele can e ost directly related to eeriental data. lthough therodynaic results can e transfored etween enseles, this is only reliale in the liit of infinite syste size (the therodynaic liit). In the case of olecules it ay, therefore, e desirale to erfor siulations in a secific ensele. NPT siulations of ulk aterials have een erfored since the eighties, ainly due to the develoent of etended Lagrangians [-6]. There, the volue of the syste is well defined and identical to that of the siulation cell. The volue and the eternal ressure, as its conjugate variale, are included in the Lagrangian as a PV ter. n artificial kinetic energy corresonding to the cell fluctuations is also introduced and the cell is treated as a dynaical variale, allowing the olecular dynaics trajectory to sale the NPT ensele. These techniques have een alied to different systes and ecellent agreeents etween eerient and theoretical siulations were found [7-9]. On the other hand, ressure control for finite systes is uch less elored. Recent eerients of finite systes under ressure, such as clusters, nano-crystals, roteins and iological systes [0-8], have stiulated the interest in finite syste arostats. There are two ain aroaches to control the ressure in MD siulations of finite systes. The first one is ased on the therodynaic descrition of ressure as the result of (linear) oentu echange etween a article and its environent. Here the environent can consist of different tyes of other articles or a container. In this treatent, a olecule is iersed into a classical reulsive fluid, with classical interactions etween the olecule and the fluid. The araeters are chosen such that there is a clear volue artition etween the two systes, and no fluid article is allowed to e inside the siulated olecule. The whole syste, liquid and olecule, is siulated at constant volue conditions. The fluid ehaves as a ressure reservoir, equilirating its ressure with the olecule s internal ressure. This aroach is closed to eerient, where a finite syste is in contact with a gas or a fluid. The ressure then deends on the fluid density and teerature. Fro a ractical oint of view, a certain drawack of the reservoir ethod is the necessity to use a relatively large nuer of fluid articles in order to avoid crystallization [9-30]. The second aroach is ased on the etended Lagrangian foralis where the ressure is taken into account y including a PV ter in the descrition of the syste. The Lagrangian for a syste under eternal ressure P et then reads: L M E ( R ) P V ( R ) () ot et

2 80 J. Me. Che. Soc. 0, 56(3) Gariel Ulises Gaoa et al. Where the ters on the right side reresent the kinetic energy, otential energy and echanical work on the syste, resectively. In this aroach there is no need for an auiliary liquid to eert ressure on the olecule. However, due to the lack of eriodic oundary conditions, the volue of the syste (V ) is not iosed y the siulation cell ut has to e calculated as a function of the atoic coordinates (R). For this reason, considerale ehasis has een laced uon accurate calculations of the volue of olecules within the last years. Sun et al. [3] defines the total volue of the syste as a su of individual atoic volues. Landau [3] estiates the volue ased on the average inter-article distance. Cococcioni et al. [33] have introduced a definition ased on the quantu volue enclosed y a charge isosurface, and Calvo and Doye [34] have roosed a volue definition y finding the inial olyhedron enclosing the finite syste. Other definitions are ased on an isotroic aroiation to the finite syste volue, e.g. y relacing the syste y a shere of a given radius, which is calculated fro a coination of atoic coordinates [35]. Our work contriutes to the second aroach, i.e. the etended Lagrangian foralis. fter a rief introduction of Born-Oenheier olecular dynaics eloying auiliary density functional theory (DFT) [36] in the net section we introduce a sile, yet eact, volue definition for finite systes in Section 3. The coutational details are given in Section 4. The results of our validation calculations are discussed in Section 5. Concluding rearks are drawn in the last section. Born-Oenheier Molecular Dynaics with DFT In a full descrition of the energetics and dynaics of a syste, all articles, which in our case are electrons and nuclei, should e treated quantu echanically. However, for cheical studies, hysical and ractical considerations otivate calculations within the Born-Oenheier (BO) aroiation [37]. This aroiation introduces a searation of the tie scales for the nuclear and electronic otions. It is at this level of aroiation that the fundaental cheical concet of a olecular structure aears in the descrition of atter. To further silify the equations of otion for the eleentary articles we aly in our ileentation the classical equations of otion for the roagation of the nuclei. Therefore, a BOMD ste as discussed in this article consists of solving the static electronic structure role, i.e. to solve the stationary Kohn-Sha equations [38], and the roagation of the nuclei via classical olecular dynaics. The resulting BOMD ethod is defined y: ^ H KS y i (r) = e i y i (r) () M Ä = - E ot (3) Where H ^ KS, y i (r) and e i are the Kohn-Sha Hailtonian, oritals and their eigenvalues resectively, M stands for the ass of the ato, Ä denotes its acceleration and E ot is the otential energy function of the syste. In the aove descried BOMD ste, the solution of the Kohn-Sha equations () reresents the coutational ottleneck. In the fraework of a localized asis set aroach this task requires the calculation of 4-center electron reulsion integrals (ERIs). This calculation scales forally with the 4 th ower of the size of the asis set, therefore, the size of treatale systes is severely liited y this ste. The use of the variational fitting of the Coulo otential aroiation fro Dunla [39], reduces the scaling of the ERI calculation y one order ecause only 3-center integrals have to e calculated. For this urose a second function set is introduced, the so called auiliary function set. In the rogra demonk [40] the density calculated fro this auiliary function set is also used for the calculation of the echange-correlation otential. This aroach is naed auiliary density functional theory (DFT) in the literature [36]. Here the olecular oritals of a syste are reresented y a linear coination of Gaussian tye oritals and the auiliary functions are reresented y riitive Herite Gaussian functions which share the sae eonents within a set. Whereas the Kohn-Sha density scales quadratically with the nuer of asis functions, the auiliary density scales only linearly with the nuer of auiliary functions. This reresents a considerale reduction in the grid work. Equations of otion (3) can e integrated with the velocity Verlet algorith [4, 4], which yields a good coroise etween accuracy and coutational cost [43]. This erforance oens then the ossiility of couting olecular roerties as a function of tie with first-rincile ethods u to the nanosecond tie scale [44]. Volue Definition for Finite Systes In localized atoic orital electronic structure ethods a natural volue definition is otained y the radial cutoffs of asis functions used in the analytical and nuerical olecular integral calculations. Based on these cutoffs a sherical volue can e assigned to each ato. The radii of these atoic sheres are defined y the ost diffuse asis functions at the atos. Thus, the atoic volues deend fro the asis set of the ato and the asis function cutoff value, t V. Once the atoic volues are calculated the olecular volue is otained y the suerosition of the atoic sheres. Insired y the GEPOL rogra [45] we ileented a geoetrical rocedure for the olecular volue calculation. The algorith can e suarized y the following stes:. Each ato, centered in, is surrounded y a shere with radius R calculated according to the (radial) asis function cutoff value, t V.. Each shere is then divided into 60 sherical triangles of equal area. This is equivalent to the rojection of a entakisdodecahedron (Figure, left) on each shere.

3 Develoent of Barostats for Finite Systes Born-Oenheier Molecular Dynaics Siulations 8. Now the tessellation is increased. Each initial triangular tesserae is divided into four triangles (Figure, right). Let and e the coordinates of two vertees of the initial triangle. Then c is the coordinate corresonding to the idoint along the chord (straight line assing y and ) calculated as: c. This calculation is reeated for all sides of the 60 triangles. Please note that new c coordinates are not lying on the surface of the shere (see Figure, left). 3. Calculation of the coordinates of the vertees of the new triangles on the shere as deicted in Figure. These coordinates corresond to the idoint along the arc on the shere surface and are given y: c. Here R c is the distance fro the origin of the shere to c. By reeating this calculation for all c one otains a set of 40 sherical triangles er shere. 4. Stes 3 and 4 are reeated until the division rocess has yielded 3840 sherical triangles er shere (ato). 5. Calculation of the center coordinates { n } for each sherical triangle. This rocedure requires the calculation of the triangle s arycenter (see Figure 3, left), given y: R R c i j k. where i, j and k stands for the vertees after colete tessellation. The coordinates of the triangle center on the sherical surface (see Figure 3, right) are then the coonents of the vector: n 3 R R (4) (5) (6) (7) Fig.. Scheatic reresentation for the tesserae division rocedure. The left side shows the difference etween coordinates c (idoint etween and along the straight line joining the) and (idoint etween and along the arc joining the, on the surface of the shere). The right side shows a new sherical triangle created y the tessellation, here and are the new vertees. Fig. 3. Barycenter (left) of the new triangle after tessellation. This oint lies in the center of the lane fored y the coordinates i, j and k, while the center of the sherical triangle n (right) is the rojection of the arycenter on the surface of the shere. 6. Eliination of all triangles contained within the olecular surface, such that the reaining sherical triangles now for the enveloe surface. The distance etween the center of a triangle and the center of another shere to which the triangle does not elong is calculated as - n (see Figure 4, left). If this distance is less or equal to the radius of this shere, i.e. - n R, the triangle is discarded. This ste is reeated with all sheres to which the triangle does not elong to. If the triangle is not discarded in this loo it elongs to the enveloe surface. This rocess is reeated for all triangles foring the sherical surfaces. 7. Calculation of the olecular volue y suing all solid volues sanned y the triangular surfaces and the origin of each shere as deicted in Figure 4 (right). The olecular volue is given y: Fig.. Scheatic reresentation of a entakisdodecahedron rojected on each atoic shere (left) and tessellation rocedure (right). The urle triangle on the right is the original tesserae y the rojection of the entakisdodecahedron. The lue triangle divides the tesserae into four (lanar) triangles and, thus, increases the tessellation. N V v i R a i ( ) (8) 3 i N. Where a i () is the area of the i th triangle elonging to the shere of ato given y: i

4 8 J. Me. Che. Soc. 0, 56(3) Gariel Ulises Gaoa et al. t n t n t n k B T0 (7) Fig. 4. Left: Scheatic reresentation of the triangle eliination rocedure. The oint n is a center of a triangle that elongs to the ato B. Because - n < R, the volue of ato already contains the triangle n, so it has to e eliinated. Right: Triangular solid volue corresonding to the ato. a i ( ) 4 R Here n t denotes the nuer of triangles (3840) on each atoic shere. In this way, the volue for olecules is calculated y a urely geoetrical rocedure, which in ters of coutational cost is not eensive and has een alied already to ioacroolecules [45]. Once the olecular volue is defined the equations of otion for an isotheral-isoaric ensele [46,47] can e forulated. Here we give these equations for a syste of N articles couled to a Nosé-Hoover chain therostat [48-50] with n links and a arostat: F M N 3V ( P P ) int et M t i t t i t i f n t 3 3 N f M (9) (0) () () (3) (4) t ( N f ) k B T0 t (5) k B t t i T0 t (6) i t i Here, and t, reresent the ositions of the atos and the corresonding variales for the arostat and therostat, resectively. The associated oenta are denoted y, and t. The dot on to of variales indicate first derivative with resect to tie. N f reresents the nuer of degrees of freedo in the real syste, V is the calculated volue, T 0 is the teerature of the ath, P et is the eternal (alied) ressure and P int is the instantaneous (internal) ressure of the syste, calculated as: P int 3V M F (8) The therostat ass araeters, ti, are related to the characteristic frequency of the otion of the articles, w, y: N k T t f B 0 / (9) k T / for i =,, n (0) t i B 0 The arostat ass araeter,, is defined as: ( N ) k T / f 3 B 0 () Here w reresents the couling frequency etween the arostat, the articles and the therostat. The equations of otion (0)-(7) are integrated with the velocity Verlet algorith, which requires the inversion of a (3N + n + ) atri at least ties every tie ste. Coutational Details The erforance of the descried ethodology was tested with BOMD siulations for a water olecule running 50,000 stes with a ste size of 0. fs. The local density aroiation eloying the echange functional fro Dirac [5] in coination with the correlation functional roosed y Vosko, Wilk and Nusair [5] was used. The target teerature was 500 K in all cases controlled y a Nosé-Hoover chain therostat with 3 chain links with a couling frequency of 500 c -. The target ressure was 0 at using the here descried ethodology. ll calculations were erfored in the fraework of DFT with the doule-ζ valence olarization asis set and auiliary function set [53]. Results and Discussion In the first set of validation calculations we varied the asis function cutoff value, t V, used for the calculation of the olecular volue. In these calculations t V was set to 0-0 a.u., 0-0 a.u. and 0-30 a.u. and the arostat couling frequency was fied to 500 c -. Figure 5 deicts the average teera-

5 Develoent of Barostats for Finite Systes Born-Oenheier Molecular Dynaics Siulations 83 Fig. 5. Profiles of the average teerature (left) and average ressure (right) of the descried water olecule BOMD siulation with the etended syste arostat. Target teerature is 500 K and target ressure is 0 at in all cases. Barostat frequency, w, is 500 c - ; t V = 0-0 (to), t V = 0-0 (iddle) and t V = 0-30 (otto). ture and ressure rofiles of these runs. s this figure shows the asis function cutoff value affects critically the BOMD ressure. For the here aied ressure of 0 at a t V of 0-0 a.u. yields est results. The other cutoff values fail to equilirate to the target ressure of 0 at. However, the ressure stailizes also in these runs. With t V = 0-0 a.u. the equilirated ressure stailizes around 50 at whereas it stailizes around at with a cutoff value of 0-30 a.u. as Figure 5 shows. This indicates that the asis set cutoff value ust e chosen according to the target ressure. This is articularly critical for sall ressures elow 00 at. Figure 5 also shows that the ressure equiliration is usually faster than the teerature equiliration. Therefore, it is recoended to first adjust the asis set cutoff values in NPT BOMD siulations with the here roosed ethodology. Once this cutoff value is chosen it usually will work successfully for a large range of different teeratures. While teerature is controlled y a chain of therostats, where each chain link acts as a article uffering kinetic variales, the ressure control is driven y only one direct interacting quasi-article, aking teerature equiliration slower than ressure equiliration. In the here discussed eales all three calculations reach the target teerature in aou0 s indeendent of the equilirated ressure. In a second set of validation calculations the asis function cutoff value, t V, was fied to 0-0 a.u. and the arostat couling frequency was varied (500 c -, 500 c - and 3000 c - ). In Figure 6 the corresonding average teerature and ressure rofiles are deicted. s can e seen fro this figure the target

6 84 J. Me. Che. Soc. 0, 56(3) Gariel Ulises Gaoa et al. Fig. 6. Profiles of the average teerature (left) and average ressure (right) of the descried water olecule BOMD siulation with the etended syste arostat. Target teerature is 500 K, target ressure is 0 at and t V = 0-0 a.u. in all cases. Barostat frequency, w = 500 c - (to), w = 500 c - (iddle) and w = 3000 c - (otto). ressure of 0 at is otained in all three runs indeendently of the arostat couling frequency. Pressure equiliration is siilar, however, est for the run with w = 3000 c -. The arostat frequency ainly affects teerature equiliration. In our case here the low frequency run (weak arostat couling) slightly accelerates teerature equiliration. Our validations shows that ressure control in finite systes can e realized with the here roosed araeter free volue definition. However, the descried etended syste arostat is sensile to the choice of the asis set cutoff value, t V. Because of the relative short tie for ressure equiliration, a suitale t V value can e found in advance to the NPT siulation. Once this value is deterined NPT and NVT siulations are very siilar. In articular, the here roosed volue definition does not deteriorate the coutational erforance of our BOMD ileentation. The therostat couling to the olecule is siilar as in NVT siulations where no arostat is used. The here descried ethodology allows NPT siulations of finite systes as required for olecular hase diagra calculations. Figure 7 deicts the ehavior of the instantaneous and average ressure of the erfored BOMD siulation. The ressure fluctuations are usually uch larger than the total energy fluctuations in a NVE siulation. This is eected ecause the

7 Develoent of Barostats for Finite Systes Born-Oenheier Molecular Dynaics Siulations 85 Fig. 7. Profiles of the average (lack) and the instantaneous (red) ressures of the descried H O BOMD siulation with the etended syste arostat. ressure is related to the roduct of the ositions and the osition derivatives of the otential energy function, i.e. the forces. This roduct, F, changes faster with than does the internal energy, hence the greater fluctuation in the ressure. Conclusions In this contriution we descrie a new ethod for ressure control in first-rincile olecular dynaics siulations for finite systes ased on the etended Lagrangian ethodology. The volue is defined via the asis function cutoff value, t V, of the ato centered asis set. It is calculated highly efficient y an analytical geoetrical rocedure. Our validation calculations show that the ain adjustale araeter for ressure control is the asis function cutoff value. Because ressure equiliration is rather fast this value can e easily adjusted in short tie test runs. Once the asis function cutoff value is adjusted the arostat ehaves very stale in NPT siulations. With this ileentation in demonk first-rincile calculations of finite systes hase diagras ecoe ossile. cknowledgeents G.U. Gaoa gratefully acknowledges a CONCyT Ph.D. fellowshi (003). P. Calainici and.m. Köster acknowledge funding fro ICyTDF (PICCO-0-47), CONCyT (607-U, 3076) and CIM References. ndersen, H. C. J. Che. Phys. 980, 7, Parrinello, M.; Rahan,. Phys. Rev. Lett. 980, 45, Parrinello, M.; Rahan,. J. l. Phys. 98, 5, Nosé, S.; Klein, M. L. Mol. Phys. 983, 50, Tuckeran, M. E.; Liu, Y.; Ciccotti, G.; Martyna, G. J. J. Che. Phys. 00, 5, Baltazar, S. E.; Roero,. H.; Rodríguez-Lóez, J. L.; Terrones, H.; Martoňák, R. Co. Mat. Sci. 006, 37, Bernasconi, M.; Chiarotti, G. L.; Focher, P.; Parrinello, M.; Tosatti, E. Phys. Rev. Lett. 997, 78, Serra, S.; Chiarotti, G.; Scandolo, S.; Tosatti, E. Phys. Rev. Lett. 998, 80, Laio,.; Bernard, S.; Chiarotti, G. L.; Scandolo, S.; Tosatti, E. Science 000, 87, Nellis, W. J. J. Phys.: Condens. Matter 004, 6, S93-S98.. Silva, J. L.; Foguel, D.; Suarez, M.; Goes,. M. O.; Oliveira,. J. Phys.: Condens. Matter 004, 6, S99-S944.. shcroft, N. W. J. Phys.: Condens. Matter 004, 6, S945-S ngilella, G. G. N. J. Phys.: Condens. Matter 004, 6, S953- S Tsuji, K.; Hattori, T.; Mori, T.; Kinoshita, T.; Narushia, T.; Funaori, N. J. Phys.: Condens. Matter 004, 6, S989-S Hattori, T.; Kinoshita, T.; Narushia T.; Tsuji, K. J. Phys.: Condens. Matter 004, 6, S997-S Utsui, W.; Okada, T.; Taniguchi, T.; Funakoshi, K.; Kikegawa, T.; Haaya, N.; Shioura, O. J. Phys.: Condens. Matter 004, 6, S07-S Dessat, R.; Hel, L.; Merach,. E. J. Phys.: Condens. Matter 004, 6, S07-S Sikka, S. K. J. Phys.: Condens. Matter 004, 6, S033-S Seller, L.; Fidy, J.; Hereans, K. J. Phys.: Condens. Matter 004, 6, S053-S Torrent, J.; lvarez-martinez, M. T.; Heitz, F.; Liautard, J.; Balny, C.; Lange, R. J. Phys.: Condens. Matter 004, 6, S059-S066.. Kiełczewska, K.; Czerniewicz, M.; Michalak, J.; Brandt, W. J. Phys.: Condens. Matter 004, 6, S067-S070.. Struzhkin, V. V.; Heley, R. J.; Mao, H. J. Phys.: Condens. Matter 004, 6, S07-S Huel, H.; van Uden, N. W..; Fau, D..; Dunstan, D. J. J. Phys.: Condens. Matter 004, 6, S8-S Hara, K.; Baden, N.; Kajioto, O. J. Phys.: Condens. Matter 004, 6, S07-S4. 5. Balny, C. J. Phys.: Condens. Matter 004, 6, S45-S Marchal, S.; Lange, R.; Tortora, P.; Balny, C. J. Phys.: Condens. Matter 004, 6, S7-S Huertz, H.; Ee, H. J. Phys.: Condens. Matter 004, 6, S83-S Deianets, L. N.; Ivanov-Schitz,. K. J. Phys.: Condens. Matter 004, 6, S33-S Martoňák, R.; Molteni, C.; Parrinello, M. Phys. Rev. Lett. 000, 84, Martoňák, R.; Molteni, C.; Parrinello, M. Co. Mater. Sci. 00, 0, Sun, D. Y.; Gong, X. G. J. Phys.: Condens. Matter 00, 4, L487-L Landau,. I. J. Che. Phys. 00, 7, Cococcioni, M.; Mauri, F.; Ceder, G.; Marzari, N. Phys. Rev. Lett. 005, 94, Calvo, F.; Doye, J. P. K. Phys. Rev. B 004, 69, Cheng, H. P.; Li, X.; Whetten, R. L.; Berry, R. S. Phys. Rev. 99, 46, Köster,. M.; Reveles, J. U.; del Cao, J. M. J. Che. Phys. 004,, Born, M.; Oenheier, J. R. nn. Physik 97, 84, Kohn, W.; Sha, J. Phys. Rev. 965, 37, Dunla, B. I.; Connolly, J. W. D.; Sain, J. R. J. Che. Phys. 979, 7, Köster,.M.; Geudtner, G.; Calainici, P.; Casida, M. E.; Doin-

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