量子化学教学苏州大学密度泛函理论简介省公开课一等奖全国示范课微课金奖

《量子化学》樊建芬Chapter8IntroductionofDensityFunctionTheory第八章密度泛函理论介绍11/29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Kohn-Sham方程以来，DFT已逐步成为量子化学计算领域强有力工具。在量子化学计算领域，据INSPEC数据统计显示: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方法论文并驾齐驱；之后，DFT工作按指数级数增加。22/29最近几年，密度泛函方法得到了广泛应用，通常能够得到比HF方法更加好计算结果，但只需中等程度价格，对于中型或大型分子体系，其计算成本远低于MP2法。

1998年，Kohn因DFT开创性工作与Pople共同取得诺贝尔化学奖。从中可见DFT在化学领域中地位。33/29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：主要包含自旋━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，电子含有全同粒子特征，满足保里原理，波函数必须含有反对称性，它与非对称波函数能量差为交换能。分子体系Schrödinger方程2344/29两种状态波函数不一样，能量也有所差异。例：Li+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55/29不过它在描述化学键时有些不足，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交换能处理，同时也是大致系实际计算工具。HF超自洽场校正，如MP多级微扰、组态相互作用CI也只能处理中小分子，对大致系是不实用。66/298.1TheHohenberg-KohnTheorem8.2TheHohenberg-kohnvariationaltheorem8.3TheKohn-Shammethod8.4CommentsontheDFTmethods77/298.1TheHohenberg-KohnTheoremIn1964,HohenbergandKohnprovedthat

“Formoleculeswithanondegenerategroundstate,theground-statemolecularenergy,wavefunctionandallothermolecularelectronicpropertiesareuniquelydeterminedbytheground-stateelectronprobabilitydensitynamely,.”Phys.Rev.136,13864(1964)88/29Densityfunctionaltheory(DFT)attemptstocalculateandotherground-statemolecularpropertiesfromtheground-stateelectrondensity

Proof:TheelectronicHamiltonianisexternalpotential11目录99/29InDFT,iscalledtheexternalpotentialactingonelectroni,sinceitisproducedbychargesexternaltothesystemofelectrons.Oncetheexternalpotentialandthenumberofelectronsnarespecified,theelectronicwavefunctionsandallowedenergiesofthemoleculearedeterminedasthesolutionsoftheelectronicSchrödingerequation.

1010/29(1)theexternalpotential(exceptforanarbitraryadditiveconstant);Itcanbeprovedthattheground-stateelectronprobabilitydensitydetermines:(2)thenumberofelectrons.91111/29“Forsystemswithanondegenerategroundstate,theground-stateelectronprobabilitydensitydeterminestheground-statewavefunctionandenergy,andotherproperties”。

emphasizesthedependenceofontheexternalpotential,whichdiffersfordifferentmolecules.

1212/29Here,However,thefunctionalsareunknown.

1313/29isalsowrittenasHere,Thefunctionalisindependentoftheexternalpotential.

目录

1414/298.2TheHohenberg-kohnvariationaltheoremthefollowinginequalityholds:whereisthetrueground–stateenergy.”“Foreverytrialdensityfunctionthatsatisfiesandforall,目录1515/29Letsatisfythatand.BytheHohenberg-Kohntheorem,determinestheexternalpotentialandthisinturndeterminesthewavefunctionthatcorrespondstothedensity.Proof:

1616/29LetususethewavefunctionasatrialvariationfunctionforthemoleculewithHamiltonian.Accordingtothevariationtheorem

1717/29SincethelefthandsideofthisinequalitycanberewrittenasHohenbergandKohnprovedtheirtheoremsonlyfornondegenerategroundstates.Subsequently,Levyprovedthetheoremsfordegenerategroundstates.

Onegets

目录

1818/298.3TheKohn-ShammethodIfweknowtheground-stateelectrondensity,theHohenberg-Kohntheoremtellsusthatitispossibleinprincipletocalculatealltheground-statemolecularpropertiesfrom,withouthavingtofindthemolecularwavefunction.41919/29

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

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

2020/29.KohnandShamconsideredafictitiousreferencesystemsofnnoninteractingelectronsthateachexperiencethesameexternalpotentialthatmakestheground-stateelectronprobabilitydensityofthereferencesystemequaltotheexactofthemoleculeweareinterestedin:

2121/29

Sincetheelectronsdonotinteractwithoneanotherinthereferencesystem,theHamiltonianofthereferencesystemiswhereistheone-electronKohn-ShamHamiltonian.

2222/29TheKSoperatoristhesameastheHFoperatorexceptthattheexchangeoperatorsintheHFoperatorarereplacedby,whichhandlestheeffectsofbothexchangeandelectroncorrelation.

2323/29VariousapproximatefunctionalsareusedinmolecularDFTcalculations.Thefunctionaliswrittenasthesumofanexchange-energyfunctionalandacorrelation-energyfunctional:4

2424/29Amongvariousapproximations,gradient-correctedexchangeandcorrelationenergyfunctionalsarethemostaccurate.Commonlyusedand.

2525/29Lee-Yang-Parr(LYP)functionalP86(thePerdew1986correlationfunctional)PW91(PerdewandWang’s1991functional)PW86(PerdewandWang’s1986functional)B88(Becke’s1988functional)目录2626/298.4CommentsontheDFTmethods(1)TheKSequationsaresolvedinaself-consistentfashion,liketheHFequat