Chapter 10 Electronic Correlation (相关)Methods Hartree-Fork 方法 用平均相互作用描述电子-电子相互作用 HF ~99% of the total energy ~1% very important for chemical phenomena Electronic Correlation Energy = Difference between HF and the lowest possible energy in a given basis set
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Chapter 10 Electronic Correlation (相关)Methods
Hartree-Fork 方法 用平均相互作用描述电子-电子相互作用 HF ~99% of the total energy ~1% very important for chemical phenomena Electronic Correlation Energy = Difference between HF and the lowest possible energy in a given basis set
Coulomb 相关: largest contribution! Starting point : HF wave function
∑=
Φ+Φ=Ψ1
HF0i
iiaa
10.1 Excited Slater Determinants
N电子M基函数体系: 占据轨道:N/2 个,空轨道M-N/2
Replacing occupied MOs with virtual MOs ⇒ General Excited Slater Determinants 1: singly 2: doubly 3: triply 4: quardruply …
Accuracy • chemical accuracy ~1kcal/mol only for small systems! • relative energies constant errors! the core orbitals and the valence orbitals frozen core approximation
(0) (1) (0) * (0) (0) (1)| |m n m m m nE Eψ ψ ψ ψ< > = < >
(0) (0) (1) (0) (0) (1) (1) (0) (1)| | | ' |m m n n m n n mn m nE E E Hψ ψ ψ ψ δ ψ ψ− = −
(0) (0) (0) (1) (1) (0) (0)( ) | | ' |m n m n n mn m nE E E Hψ ψ δ ψ ψ− = −
m = n *(1) (0) (0) (0) (0)| ' | 'n n m nE H H dψ ψ ψ ψ τ= = ∫
*(0) (1) (0) (0) (0)'n n n n n nE E E E H dψ ψ τ= + = + ∫
Example:
214
(0) 20
xe
ααψ
π
−⎛ ⎞= ⎜ ⎟⎝ ⎠
212
3 4( )xe cx dx dxααπ
∞ −
−∞
⎛ ⎞= +⎜ ⎟⎝ ⎠ ∫
2
34dα
=
(1) (0) (0)| ' |n n nE Hψ ψ=
0 3 4'H H H cx dx∧∧ ∧
= − = +
E0(1) = ψ0
(0) | (cx3 + dx4 ) |ψ0(0)
2
4 2 2
364dhv mπ
=
(0) (0) (0) (1) (1) (0) (0)( ) | | ' |m n m n n mn m nE E E Hψ ψ δ ψ ψ− = −
(0) (1)|m m na ψ ψ=where
m n≠
First order wavefunction
(1) (0)n m m
maψ ψ=∑
(0) (0) (0) (1) (0) (0)( ) | | ' |m n m n m nE E Hψ ψ ψ ψ− = −
(0) (0) (0) (0)( ) | ' |m n m m nE E a Hψ ψ− = −
(0) (0)
(0) (0)
| ' |m nm
n m
Ha
E Eψ ψ
=−
m n≠
(0) (0)(1) (0)
(0) (0)
| ' |m nn m
m n n m
HE E
ψ ψψ ψ
≠
=−∑
1λ = '(0) (0)
(0) (0)mn
n n mm n n m
HE E
ψ ψ ψ≠
≈ +−∑
(0) (2) (0) (2) (2) (0) (1) (1) (1)'n n n n n n n nH E E E Hψ ψ ψ ψ ψ− = + −
(0)*mψ
(0) (0) (2) (0) (0) (2) (2) (1) (0) (1)
(0) (1)
| | |
| ' |
m m n n m n n mn n m n
m n
E E E E
H
ψ ψ ψ ψ δ ψ ψ
ψ ψ
− = +
−
(0) (1) 2 (2) ( )k kn n n n nψ ψ λψ λ ψ λ ψ= + + + + +L L
Second-order W. F. Energy Correction
(2) (0) (1)| ' |n n nE Hψ ψ=
(0) (0)| ' |n m mm n
H aψ ψ≠
= ∑
m n=
(0) (0)' (0) (0)
(0) (0)'
| ' || ' |m n
n mm n m
H HH H
E Eψ
ψ=−∑
2(0) (0)'
(0) (0)'
| ' |m n
m n m
H H
E E
ψ=
−∑
2'(0) '
(0) (0)mn
n n nnm n n m
HE E H
E E≠
= + +−∑
Møller-Plesset Perturbation Theory (MPPT) 多体微扰
H0:可以求解的,H’ 小项
Ψ=Ψ WH
!
!
+Ψ+Ψ+Ψ+Ψ=Ψ
=
33
22
11
00
33
22
11
00
λλλλ
λλλλ ++++ WWWWW
H =H0 +λH'
H0Φi = EiΦi, i = 0,1, 2!∞
[ ]∑∑< < −−+
−=
occ
ji
vir
ba baji
abjibajiEεεεε
φφφφφφφφ2
)MP2(
H
H0 = Fii∑
MP2: ~M5 method ~80~90% 相关能, 是一个经济的方法 MP4: ~M6
~95~98%相关能 MP5: ~M8
MP6: ~M9
MP方法的能量可能低于真实能量,大小一致
10.4 Coupled Cluster Methods (CC)耦合簇方法
0Φ=Ψ TCC e
∑∞
=
=++++=0k
k32T TTT21T1e
!1
61
k!
N321 TTTTT ++++= !
∑∑
∑∑
< <
Φ=Φ
Φ=Φ
occ
ji
vir
ba
abji
abij
occ
i
vir
a
ai
ai
t
t
0
0
2
1
T
T
eT = 1+T1 + T2 +12T12!
"#
$
%&+ T3 +T2T1 +
16T13!
"#
$
%&
+ T4 +T3T1 +12T22 +12T2T1
2 +124T14!
"#
$
%&+!
如果取T=T1+T2,包括 , 等 CCSD:大小一致性
22T 4
1T
CISD,MP2,MP3,MP4,CCSD,CCSD(T) CISD: variational, but not size consistent MP and CC: not variational, but size consistent. CISD and MP: in principle non-iterative but CISD usually is so large that it has to be done iteratively CC: iterative
CCSD(T)MP4CCSD~MP4(SDQ)CISDMP2HF <<<<<<
HF: (Minimal basis set) give results which are worse than AM1 and PM3,but 100 times computational cost. (medium and large basis set) does not give absolute results
CCSD(T) with sufficiently large basis set is able to meet the goal of an accuracy of ~1kcal/mol.
The use of CI methods has been declining in recent years
Electron motions are correlated, which arises from the electron-electron repulsion operator
It would there seem natural that the interelectronic distance would be a necessary variable.
R12 Method
ΨR12 =ΦHF + aijabΦijab
ijab∑ + bijrijΦHF
ij∑
R12-HF, R12-CI, R12-MP, R12-CC.
Overwhelming problem
ΦHF H rijΦHF = ΦHF h rijΦHF + ΦHF g rijΦHF
rijΦHF H rijΦHF = rijΦHF h rijΦHF + rijΦHF g rijΦHF
Three-/four-center integrals
φi (1)φ j (2)φk (3)r12r13
φi ' (1)φ j ' (2)φk ' (3)
φi (1)φ j (2)φk (3)r12r23r13
φi ' (1)φ j ' (2)φk ' (3)
φi (1)φ j (2)φk (3)φl (4)r12r23r14
φi ' (1)φ j ' (2)φk ' (3)φl ' (4)
Solution: Resolution of the identity (RI)
F12 method
Extension of R12 method
10.7 Excited States
A. Different symmetry from the ground state (easy)
A HF wave function may be obtained by a proper specification of the occupied orbitals, and improved by adding electron correlation by for example CI, MP, or CC methods.