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一种简化的Kohn-Sham方程推导方法
Title:ASimplifiedDerivationApproachfortheKohn-ShamEquations
Abstract:
TheKohn-Shamequationsplayacrucialroleinthefieldofdensityfunctionaltheory,enablingaccuratepredictionsoftheelectronicpropertiesofmoleculesandsolids.However,thederivationoftheseequationsinvolvescomplexmathematicalrelationshipsandintricacies,makingitchallengingfornovicestocomprehend.Inthispaper,wepresentasimplifiedderivationapproachfortheKohn-Shamequations,aimingtomaketheunderlyingconceptsandderivationsmoreaccessibletoawideraudience.Wefocusonpresentingastep-by-stepexplanationofthekeyprinciplesandfundamentalassumptionsinvolvedintheKohn-Shamformulation,whilestreamliningthemathematicalformalismtoenhanceclarityandunderstanding.Byemployingthissimplifiedapproach,wehopetoprovideausefulresourceforresearchersandstudentslookingtogainadeeperunderstandingoftheKohn-Shamequations.
1.Introduction
Densityfunctionaltheory(DFT)hasrevolutionizedthefieldofcomputationalchemistry,providinganefficientwaytopredictelectronicpropertiesofatoms,molecules,andsolids.CentraltothesuccessofDFTistheKohn-Shamformulation,whichmapsthecomplicatedmany-bodyproblemofinteractingelectronsontoanequivalentsingle-electronproblem.However,thederivationoftheKohn-Shamequationscanbeintimidatingduetotheinvolvedmathematicalformalismanditsinherentcomplexity.Inthispaper,wepresentasimplifiedapproachtoderivetheKohn-Shamequations,breakingdowntheprocessintointuitivestepsandexplanations.
2.Background
Toappreciatethesimplifiedderivation,webeginbyprovidingabriefoverviewofthefoundationalprinciplesofDFTandthenecessityoftheKohn-Shamformulation.Weintroducetheconceptofelectrondensityanditsroleindeterminingelectronicproperties,alongwiththeconceptofexchange-correlationfunctional.UnderstandingtheseconceptsestablishesthemotivationforthedevelopmentoftheKohn-Shamequations.
3.SimplifiedDerivation
WeproceedwiththesimplifiedderivationoftheKohn-Shamequations,focusingonthefollowingkeysteps:
3.1Non-interactingReferenceSystem
Weintroducetheconceptofthenon-int