Solutions of the Schrödinger equation for the ground. helium by finite element method
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1 EGEE 50 Spng 007 Semeste Pape. Instcto: D. Elswoth Soltons of the Schödnge eqaton fo the gond. Intodcton helm b fnte element method b Jaha Go Mlt-bod Colomb poblems ae tadtonal challengng poblems []. The fale of theo to descbe pecsel the sstem stmlated man mathematcans and phscsts to devote themselves n sng vaos methods to obtan the eneges and othe epectaton vales. Few-electon sstems lke helm ae tpcal models. Thee ae two electons ot of the ncles of the helm atom; and each of them has thee feedoms (wthot consdeng ts spn. So the sstem s descbed b a s-dmensonal Schödnge eqaton (fo fed-ncles poblem. One sal appoach n qantm mechancs s mean-feld method []. Each electon s consdeed ndependentl to be n a cental electc feld fomed b the ncles and othe electons. The ognal poblem s tansfeed nto a sstem of nonlnea patal dffeental eqatons of low-dmenson then solve t teatvel. The othe sal appoach named vaatonal method [3-4] seaches the stats fncton to fnd the mnmm eneg vale whch has been poved to be close to the pecse vale. A common feate of these calclatons s a hghe eo n the non-hamltonan epectaton vales than n the eneg an ndcaton that the appomate wave fnctons ae less accate than one mght ognall assme fom the well-conveged eneges [5]. Ths behavo povdes the motvaton fo applng a new method to the poblem. The fnte element method (FEM s a nmecal algothm that ses local ntepolaton methods to solve second-ode dffeental eqatons descbng bonda-vale poblems. A local ntepolaton scheme shold be speo to global vaatonal appoaches n eldng an accate wave fncton becase of the ease b whch local mpovements to the appomate wave fncton can be ntodced n the FEM. In pncple theefoe t s epected that ths algothm ma povde new nsghts nto the stcte of molecles and atoms [5]. The man woks of the fnte element method appled to atomc and molecla poblems appeaed n 970 s n one- and two-dmensonal cases [6-7] whee smplct and effcenc of FEM was shown. The fst wok of FEM appled to
2 thee-dmensonal case was done b evn and Shete [5]. The calclated the helm n the gond state and the s-dmensonal Schödnge eqaton was tansfeed nto thee-dmensonal sstems of eqatons goosl [8]. Woks abot thee-dmensonal FEM appled to thee-bod poblems appeaed heeafte [9-]. The all obtaned ve pecse eslts.. Govenng Eqatons In Catesan coodnates the spn-ndependent nonelatvstc Schödnge eqaton fo the two electons n the helm atom s [5] : E H whee H s called Hamltonan opeato stands fo the wave fncton and opeato calclates the knetc eneg of electon and electon (all the epessons ae n the atomc nts. Also we have: sn sn (sn ( / ( In sphecal coodnates the Hamltonan of the sstem can be wtten as [] : H Whee s the sqae of the angla momentm opeato of the th electon: B the law of chan fo the dffeental calclatons: sn sn sn sn Smlal we have and :
3 sn sn sn sn sn sn Snce: ( sn Ths: sn sn ( Theefoe the Hamltonan can be tansfeed to an opeato wth thee vaables: sn sn H whee can be epessed b and... Bonda condtons In ode to pefom FEM the nfnte volme of coodnate space spanned b and was made fnte b tncaton.e. and wee each lmted to the doman c 0. The wave fncton was set eqal to eo fo 0 and c. Smlal 0 and 0 when 0 and.. Fomlaton COMSO Mltphscs descbes the coeffcent fom of egenvale PDE as: d a c ( In ode to match the eqaton fom the Hamltonan shold be fomlated as: ctg H Then the followng coeffcent vales ae sed:
4 c ctg a 3. Solton The solton povdes a nmbe of the lowest egenvales. The gond state eneg s solved as: E = hatee = -74. ev close to the epemental vale eV. Fg. shows the calclaton eslts plottng wth the vale of the wave fncton whch s ntepeted as the pobablt denst. Theefoe the ed aea shold stand fo the most possble dstbton of electons when at the gond state eneg. Accodng to Fg. when E = hatee one electon shold be ve close to the helm ncle wth anothe electon fa fom t. One thng s left to be pled that snce echangng the postons of the two electons shold not change the sstem the fge shold have been smmetc. (Actall I am not able to well ndestand ths fge bt the egenvale s close to the epemental eslt. Fg. The slce scheme of the wave fncton wth the lowest egenvale.
5 4. Valdaton In chemst an atomc obt s a egon n whch an electon ma be fond. One specfc egon coesponds to one egenvale whch s the vale of one eneg level. Snce the atomc obts and eneg levels of hdogen atom have been well docmented than an othe atoms the same model and solton pocess wll be appled to hdogen atom to valdate the coecton of o eslts fo helm. 4.. Eneg levels Accodng to Boh model the eneg levels of hdogen atom can be solve as: E n E n n whee E s the lowest eneg level abot -0.5 hatee v. -3.6eV. The followng 0 table compaes the eneg vales got fom Femab and the vales calclated fom Boh model. Eneg Eneg vale based on Boh Eneg vale calclated b Eo level model (hatee Femab (hatee n= % n= % Moeove as mentoned above the eneg egenvale of helm atom s calclated as hatee close to the epemental eslt hatee. 4.. Atomc obts Smla to the eneg levels the atomc obts of hdogen atom ae also well known. The followng fges valdate o calclatons b compang the sosfaces of wave fnctons plotted b Femab and the known atomc obts of hdogen. s s p p
6 p 5. Conclson Based on the dscssons above t can be conclded that:. Schödnge eqaton can be smplfed b deceasng some vaables makng t an eqaton wth fewe dmensons.. FEMlab s a good tool when tng to fnd ot the egenvales of eneg of some thee-dmensonal sstems (e.g. hdogen and helm atoms. Howeve t can t deal wth a complcated man-bod Schödnge eqaton. Refeences [] W. Zheng. Yng P. Dng Nmecal soltons of the Schödnge eqaton fo the gond lthm b the fnte element method Appled Mathematcs and Comptaton 53 ( [] Ch. Foese-Fsche The Hatee-Fock Method fo Atoms Wle-Intescence New Yok 977. [3] C.A. Colson B. O ea R.B. Mallon Hückel Theo fo Oganc Chemsts Academc Pess ondon 978. [4] A. Aska and M. Demalp the Jonal of Chemcal Phscs 60 ( [5] F.S. evn J. Shete Fnte-element solton of the Schödnge eqaton fo the helm gond state Phscal Revew A 3 ( [6] A. Aska Fnte element method fo bond state calclatons n qantm mechancs the Jonal of Chemcal Phscs 6 ( [7] M. Fdman Y. Rosenfeld A. Rabnovtch R. Thebege Fnte element method fo solvng the two-dmensonal Schödnge eqaton Jonal of Comptatonal Phscs 6 ( [8] W. Zheng Applcaton of the Hlleaas-Bet tansfomaton to the Schödnge eqatons of helm lke atoms Chnese Jonal of Atomc and Molecla Phscs 3 (00 9.
7 [9] M. Ban W. Schwee H. Heold Fnte-element calclatons fo the S states of helm Phscal Revew A 48 ( [0] A. Scn A 3-dmensonal fnte elements pocede fo qantm mechancal applcatons Compte Phscs Commncatons 86 ( [] J. Ackemann Fnte-element epectaton vales fo coelated two-electon wave fnctons Phscal Revew A 5 ( [] C. Bottche Nmecal solton of the Petekop-Rabk Temkn-Hahn model of electon mpact onaton Jonal of Phscs B 4 ( [3] J. Echle I.V. Hetel N. Stoltefoht The Phscs of Electonc and Atomc Collsons Elseve New Yok 984 [4] W.K. Fod and F.S. evn Channel-coplng theo of molecla stcte. Fnte-element method solton fo H + Phscal Revew A 9 ( [5] P.G. Calet The Fnte Element Method fo Ellptc Poblems Noth-Holland Amstedam New Yok Ofod 978. [6] K. Bathe E. Wlson Nmecal Methods n Fnte Element Analss Pentce-Hall New Jese 973. [7] C. Felppa Intodcton of Fnte Element Method Unvest of Coloado 00. [8] Wekpeda (005 Fnte Element Method [9] Femab Use s Gde Comsol. [0] Femab Modelng Gde Comsol.
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