1950年中期，MO方法Huckel规则得到实验验证1952年Fukui发表了前线分子轨道理论光谱，化学反应等应用的成功，计算方法的发展（EHMO，半经验方法）1965年Woodward和Hoffmann发表轨道对称守恒原理对化学反应的研究取得巨大的成功VB方法停滞不前，没有给出新东西，无法应用于较大分子体系
1950 年中期， MO方法 Huckel规则得到实验验证 1952年Fukui发表了前线分子轨道理论 光谱，化学反应等应用的成功， 计算方法的发展（EHMO, 半经验方法） 1965年Woodward和Hoffmann发表轨道对称守恒原理 对化学反应的研究取得巨大的成功 VB方法停滞不前，没有给出新东西，无法应用于较大分子体系
DensityfunctionaltheoryTheHohenberg-KohnTheoremmDensity Is EverythingKohn played the leading role in the development of densityfunctionaltheory,whichmadeitpossibletocalculate quantummechanicalelectronicstructurebyeguationsinvolvingtheelectronicdensity (ratherthanthemany-bodywavefunction).A=-↓ZV?+Zv(c)+21externalfieldi<iriZv(r)=-Once v(r,) and N are specified, wave function is determined.There exists a one-to-one correspondence between the electrondensity of a system and the energy
Density functional theory The Hohenberg-Kohn Theoremm Density Is Everything Kohn played the leading role in the development of density functional theory, which made it possible to calculate quantum mechanical electronic structure by equations involving the electronic density (rather than the many-body wavefunction). There exists a one-to-one correspondence between the electron density of a system and the energy i i i j ij i i r Z v r r H r ( ) 1 ( ) 2 1 2 external field Once v(ri ) and N are specified, wave function is determined
ProofbycontradictionIf the theorem is wrong, one p(r) must correspond to least two external potential, v(r)and v'(r), so there must be two Hamiltonian system:H= T+ V+ VeeH'=T+V+Vee()The operator T= T", Ve.= V'eesowehaveH=H'+V-V(2) H4=EH'Y'-E'Y'For system 1: p(r) = /F(r)P2: p(r) = /4(r)2
Proof by contradiction If the theorem is wrong, one (r) must correspond to least two external potential, v(r) and v’(r), so there must be two Hamiltonian system: H = T + V + Vee H’ = T’ + V’ + V’ ee (1)The operator T = T’ , Vee = V’ee so we have H = H’ + V - V’ (2) H=E H’’=E’’ For system 1: (r) = |(r)|2 2: (r) = |’(r)|2
(3) According to variation principle:If the exact ground state Y(r) is found, the energy isE=<YHY>E'=<P'H">If the ground state Y(r) is not fully optimized, then E< <Y|H/4>
(3) According to variation principle: If the exact ground state (r) is found, the energy is E = <|H|> E’ = <’|H’|’> If the ground state (r) is not fully optimized, then E < <|H|>
E=<H><<H">=<4H'+V- V4>=<H">+<4IV- V'甲'>=E"+<y'IV- V'{4">Because:(P"V-V")= [dr'(r)(V-V)P"(r)= [dr(V-V)P"(r)Y"(r)= [dr(V-V")p(r)(*)So we have:E<E'+[dr(V-V')p(r)
E = <|H|> < <’|H|’> = <’| H’ + V - V’ |’> = <’| H’ |’> + <’| V - V’ |’> = E’ + <’| V - V’ |’> ( ') ( ) ( ') '( ) '( ) ' ' ' '( )( ') '( ) d r V V r d r V V r r V V d r r V V r Because: So we have: E E' dr(V V')(r) (*)