量子化學教學課件蘇州大學第八章密度泛函理論簡介課件

?量子化學?樊建芬Chapter8IntroductionofDensityFunctionTheory第八章密度泛函理論簡介1從20世紀60年代密度泛函理論〔DFT〕提出以來，并在局域密度近似下導出著名的Kohn-Sham方程以來，DFT已逐漸成為量子化學計算領域的強有力的工具。在量子化學計算領域，據(jù)INSPEC數(shù)據(jù)記錄顯示:①直至80年代末，分子軌道HF方法一直占主導地位；②90年代中期，DFT和HF方法的論文并駕齊驅；之后，DFT的工作按指數(shù)級數(shù)增加。2最近幾年，密度泛函方法得到了廣泛的應用，通?？梢缘玫奖菻F方法更好的計算結果，但只需中等程度的價格，對于中型或大型分子體系，其計算本錢遠低于MP2法。

1998年，Kohn因DFT的開創(chuàng)性工作與Pople共同獲得諾貝爾化學獎。從中可見DFT在化學領域中的地位。3電子動能算符核－電子吸引能電子－電子排斥能電子相關交換能+++[]

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注：①電子相關能：主要包括自旋━軌道偶合作用能、自旋━自旋偶合作用能、軌道━軌道偶合作用能。②電子交換能，電子具有全同粒子特性，滿足保里原理，波函數(shù)必須具有反對稱性，它與非對稱波函數(shù)的能量差為交換能。分子體系的Schr?dinger方程234兩種狀態(tài)的波函數(shù)不同，能量也有所差異。例：Li+的某一激發(fā)態(tài)1s12s1,假設電子排布狀態(tài)為1s2s為非對稱波函數(shù)為反對稱波函數(shù)5但是它在描述化學鍵時有些缺乏，而且沒有相關能校正時不宜處理熱化學問題。HF理論能夠在分子尺寸提供較準確的交換能處理，同時也是大體系的實際計算工具。HF超自洽場的校正，如MP多級微擾、組態(tài)相互作用CI也只能處理中小分子，對大體系是不實用的。6Densityfunctionaltheory(DFT)attemptstocalculateandotherground-statemolecularpropertiesfromtheground-stateelectrondensity

Proof:TheelectronicHamiltonianisexternalpotential11目錄9InDFT,iscalledtheexternalpotentialactingonelectroni,sinceitisproducedbychargesexternaltothesystemofelectrons.Oncetheexternalpotentialandthenumberofelectronsnarespecified,theelectronicwavefunctionsandallowedenergiesofthemoleculearedeterminedasthesolutionsoftheelectronicSchr?dingerequation.

10(1)theexternalpotential(exceptforanarbitraryadditiveconstant);Itcanbeprovedthattheground-stateelectronprobabilitydensitydetermines:(2)thenumberofelectrons.911“Forsystemswithanondegenerategroundstate,theground-stateelectronprobabilitydensitydeterminestheground-statewavefunctionandenergy,andotherproperties”。

emphasizesthedependenceofontheexternalpotential,whichdiffersfordifferentmolecules.

12Here,However,thefunctionalsareunknown.

13isalsowrittenasHere,Thefunctionalisindependentoftheexternalpotential.

目錄

148.2TheHohenberg-kohnvariationaltheoremthefollowinginequalityholds:whereisthetrueground–stateenergy.”“Foreverytrialdensityfunctionthatsatisfiesandforall,目錄15Letsatisfythatand.BytheHohenberg-Kohntheorem,determinestheexternalpotentialandthisinturndeterminesthewavefunctionthatcorrespondstothedensity.Proof:

16LetususethewavefunctionasatrialvariationfunctionforthemoleculewithHamiltonian.Accordingtothevariationtheorem

17SincethelefthandsideofthisinequalitycanberewrittenasHohenbergandKohnprovedtheirtheoremsonlyfornondegenerategroundstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

Onegets

目錄

188.3TheKohn-ShammethodIfweknowtheground-stateelectrondensity,theHohenberg-Kohntheoremtellsusthatitispossibleinprincipletocalculatealltheground-statemolecularpropertiesfrom,withouthavingtofindthemolecularwavefunction.419

Theirmethodiscapable,inprinciple,ofyieldingexactresults,butbecausetheequationsoftheKohn-Sham(KS)methodcontainanunknownfunctionalthatmustbeapproximated,theKSformationofDFTyieldapproximate

results.1965,KohnandShamdevisedapracticalmethodforfindingandforfindingfrom.[Phys.Rev.,140,A1133(1965)].

20.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingelectronsthateachexperiencethesameexternalpotentialthatmakestheground-stateelectronprobabilitydensityofthereferencesystemequaltotheexactofthemoleculeweareinterestedin:

21

Sincetheelectronsdonotinteractwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

22TheKSoperatoristhesameastheHFoperatorexceptthattheexchangeoperatorsintheHFoperatorarereplacedby,whichhandlestheeffectsofbothexchangeandelectroncorrelation.

23VariousapproximatefunctionalsareusedinmolecularDFTcalculations.Thefunctionaliswrittenasthesumofanexchange-energyfunctionalandacorrelation-energyfunctional:4

24Amongvariousapproximations,gradient-correctedexchangeandcorrelationenergyfunctionalsarethemostaccurate.Commonlyusedand.

25Lee-Yang-Parr(LYP)functionalP86(thePerdew1986correlationfunctional)PW91(PerdewandWang’s1991functional)PW86(PerdewandWang’s1986functional)B88(Becke’s1988functional)目錄268.4CommentsontheDFTmethods(1)TheKSequationsaresolvedinaself-consistentfashion,liketheHFequations.(2)ThecomputationtimerequiredforaDFTcal