量子化学教学课件苏州大学第八章密度泛函理论简介课件

?量子化学?樊建芬Chapter8IntroductionofDensityFunctionTheory第八章密度泛函理论简介1从20世纪60年代密度泛函理论〔DFT〕提出以来，并在局域密度近似下导出著名的Kohn-Sham方程以来，DFT已逐渐成为量子化学计算领域的强有力的工具。在量子化学计算领域，据INSPEC数据记录显示:①直至80年代末，分子轨道HF方法一直占主导地位；②90年代中期，DFT和HF方法的论文并驾齐驱；之后，DFT的工作按指数级数增加。2最近几年，密度泛函方法得到了广泛的应用，通常可以得到比HF方法更好的计算结果，但只需中等程度的价格，对于中型或大型分子体系，其计算本钱远低于MP2法。

1998年，Kohn因DFT的开创性工作与Pople共同获得诺贝尔化学奖。从中可见DFT在化学领域中的地位。3电子动能算符核－电子吸引能电子－电子排斥能电子相关交换能+++[]

=E

注：①电子相关能：主要包括自旋━轨道偶合作用能、自旋━自旋偶合作用能、轨道━轨道偶合作用能。②电子交换能，电子具有全同粒子特性，满足保里原理，波函数必须具有反对称性，它与非对称波函数的能量差为交换能。分子体系的Schrödinger方程234两种状态的波函数不同，能量也有所差异。例：Li+的某一激发态1s12s1,假设电子排布状态为1s2s为非对称波函数为反对称波函数5但是它在描述化学键时有些缺乏，而且没有相关能校正时不宜处理热化学问题。HF理论能够在分子尺寸提供较准确的交换能处理，同时也是大体系的实际计算工具。HF超自洽场的校正，如MP多级微扰、组态相互作用CI也只能处理中小分子，对大体系是不实用的。6Densityfunctionaltheory(DFT)attemptstocalculateandotherground-statemolecularpropertiesfromtheground-stateelectrondensity

Proof:TheelectronicHamiltonianisexternalpotential11目录9InDFT,iscalledtheexternalpotentialactingonelectroni,sinceitisproducedbychargesexternaltothesystemofelectrons.Oncetheexternalpotentialandthenumberofelectronsnarespecified,theelectronicwavefunctionsandallowedenergiesofthemoleculearedeterminedasthesolutionsoftheelectronicSchrödingerequation.

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

emphasizesthedependenceofontheexternalpotential,whichdiffersfordifferentmolecules.

12Here,However,thefunctionalsareunknown.

13isalsowrittenasHere,Thefunctionalisindependentoftheexternalpotential.

目录

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

16LetususethewavefunctionasatrialvariationfunctionforthemoleculewithHamiltonian.Accordingtothevariationtheorem

17SincethelefthandsideofthisinequalitycanberewrittenasHohenbergandKohnprovedtheirtheoremsonlyfornondegenerategroundstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

Onegets

目录

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)目录268.4CommentsontheDFTmethods(1)TheKSequationsaresolvedinaself-consistentfashion,liketheHFequations.(2)ThecomputationtimerequiredforaDFTcal