chapter 10 electronic correlation (相关)methodsweiwu.xmvb.org/ppt_cc/cc10.pdf · chapter 10...
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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
电子相关
相同轨道的两个电子(自旋相反)~20kcal/mol 不同轨道的两个电子(自旋相反,相同)~1kcal/mol
Coulomb 相关:不同自旋电子
Fermi 相关: 相同自旋电子
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
Three main methods
Configuration Interaction (CI)
Many Body Perturbation Theory (MBPT)
Coupled Clusters (CC)
Notation: level/basis level2/basis2//level 1/basis 1
10.2 Configuration Interaction (组态相互作用) Method
i0i
iTT
TDD
DSS
SSCF0CI Φ=Φ+Φ+Φ+Φ=Ψ ∑∑∑∑=
aaaaa !
]1[ −ΨΨ−ΨΨ= CICICICI H λL
ΨCI H ΨCI = Φi H Φ jj=0∑
i=0∑ = a0
2Ei +i=0∑ aiaj Φi H Φ j
j≠i∑
i=0∑
ΨCI ΨCI = aiaj Φi Φ jj=0∑
i=0∑ = ai
2 Φi Φii=0∑ = ai
2
i=0∑
0)(
0)(
022
=ΦΦ+−
=ΦΦ+−ΦΦ
=−ΦΦ=∂
∂
∑
∑
∑
≠
≠
jij
ijii
jij
ijjii
ijj
iji
aEa
aa
aaaL
H
HH
H
λ
λ
λ
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
−
−
−
!
!
!
!
!!!!!!!!!!!!!!!!!
0
00
1
0
0
11110
00100
jjjj
jE
j
a
aa
EHH
HHHHHEH
aHaaIH
EE=
=− 0)(本征值问题
CI Matrix Elements
jiijH ΦΦ= H
0=ijH If Фi与Фj 具有不同的自旋,<α|β>=0
abjibajiabij
ajjij
jajiaiai
φφφφφφφφ
φφφφφφφφφφ
−=ΦΦ
−+=ΦΦ ∑
H
hH
0
0 (
aiajjij
jajiai φφφφφφφφφφφφ Fh =−+∑(
将MO表示为AO的线性组合
∑∑∑∑
∑∑
=
=
M M
lkji
M M
lkji
M M
jiji
cccc
cc
α βδγβαδγβα
γ δ
α ββαβα
χχχχφφφφ
χχφφ hh
∑∑ ∑∑=δ
δγβαδα γ
γβ
βα χχχχφφφφ lkjilkji cccc (((
~M8
~M5
Size of the CI matrix
H2O with 6-31G(d) 10 电子,38 自旋轨道
个电子激发到空轨道nnn
=⎟⎟⎠
⎞⎜⎜⎝
⎛⋅⎟⎟⎠
⎞⎜⎜⎝
⎛ 2810
总数 )!1038(!10!38281010
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛⋅⎟⎟⎠
⎞⎜⎜⎝
⎛∑n nn
减少个数:自旋考虑, 对称性考虑 7536400
Full CI (完全CI) 包括所有的组态(行列式) 包括所有电子相关 N电子,M基函数 组态个数:
!12
!2
12
!2
)!1(!
⎟⎠
⎞⎜⎝
⎛ +−⎟⎠
⎞⎜⎝
⎛ −⎟⎠
⎞⎜⎝
⎛ +⎟⎠
⎞⎜⎝
⎛
+=
NMNMNNMM CSFs ofNumber
对H2O 6-31G ~30×106 (N=10,M=19) 6-311G(2d,2p) ~ 106×109 (N=10, M=41)
对H2O Full CI 应用:for DZP 24基函数 451681246 行列式 实验值相比:~0.2 a.u (~125kcal/mol) Full CI: 没有实际应用意义,标志性!
Truncated CI methods
CIS: 包含所有单重激发态行列式
00 =ΦΦ aiF Brillouins 定理
基态: ECIS=EHF
CID: 包含所有双重激发
CISD:包含所有单双重激发 ~M6
CISDT:S,D,T ~M8
CISDTQ: S,D,T,Q ~M10 ECISDTQ ≈ Efull
最普遍 ,CISD ~80~90% 相关能
Direct CI Methods
由于组态太多(~106,CISD,小分子),无法直接对角化 使用迭代方法
0)( =− aIH E给初始值a(1,0,0…),计算 ,改变a,… Davidsion算法
aIH )( E−
Size Consistency and Size Extensivity
CISD
Infinity: H2+H2 H2…H2
cdkl
abij
cdkl
abij
ck
ai
ΦΦ
ΦΦΦΦΦΦ 00 ,,
E(H2+H2)≠2E(H2)
Size inconsitency (大小不一致)
n n nH Eψ ψ∧
=
perturbation
2 20 2
2
12 2
dH kxm dx
∧
= − +h
2 22 3 4
2
12 2
dH kx cx dxm dx
∧
= − + + +h
' 0H H H∧ ∧∧
= −0 'H H H
∧ ∧∧
= +
0 0 (0) (0)0 nH Eψ ψ
∧
=slightly different
0 'H H Hλ∧ ∧∧
= +
10.3 Many-body Perturbation Theory (MBPT) 多体微扰
( , )n n qψ ψ λ=
( )n nE E λ=
ψn =ψn(0) +λψn
(1) +λ 2ψn(2) +!+λ kψn
(k ) +!
0( ')n n n nH H H Eψ λ ψ ψ∧∧ ∧
= + =
0 (0) (0) (0)n n nH Eψ ψ
∧
=
En = En(0) +λEn
(1) +λ 2En(2) +!+λ kEn
(k ) +!
1= ψn(0) |ψn
(0) +λ ψn(0) |ψn
(1) +λ 2 ψn(0) |ψn
(2) +!
Suppose
ψn(0) |ψn
(1) = 0 , ψn(0) |ψn
(2) = 0 , ! etc.
(0) (0)| 1n nψ ψ =
(0) | 1n nψ ψ = intermediate normalization
H 0∧
ψn(0) +λ(H '
∧
ψn(0) +H 0
∧
ψn(1) )+λ 2 (H (0)
∧
ψn(2) +H '
∧
ψn(1) )+!
= En(0)ψn
(0) + λ(En(1)ψn
(0) + En(0)ψn
(1) )+λ 2 (En(2)ψn
(0) +
En(1)ψn
(1) + En(0)ψn
(0) )+!0 (0) (0) (0)
n n nH Eψ ψ∧
=
ψm(0) |H 0
∧
|ψn(1) − En
(0) ψm(0) |ψn
(1) = En(1) ψm
(0) |ψn(0) − ψm
(0) |H '∧
|ψn(0)
(0) 0 (1) (1) (0) (0) (1)' n n n n n nH H E Eψ ψ ψ ψ∧∧
+ = +
0 (1) (0) (1) (1) (0) ' (0)n n n n n nH E E Hψ ψ ψ ψ
∧ ∧
− = −
(0) 0 (1) (1) 0 (0) *ˆ ˆ| | | |m n n mH Hψ ψ ψ ψ< >=< > =
(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
Three levels HF MP2 CCSD(T) 5000 bs 800 bs 300~400bs Note: Bs: basis functions
10.5 Multi-reference Based Method
Complete Active Space self-consistent Field (CASSCF)
Restricted Active Space self-consistent Field (RASSCF)
Multi-reference Configuration Interaction (MRCI) 多参考组态相互作用
CI方法的激发行列式是基于HF方法(单个行列式) MRCI方法是采用多行列式(MASCF方法)作为参考态 MRCI方法可以得到很好的计算结果
Seniority Number Based Configuration Interaction
Seniority number is defined as the number of unpaired particles in a determinant.
Multi-reference Perturbation Theory, MRPT2 (CASPT2)
Multi-reference Coupled Cluster, MRCC
10.6 Methods Involving Interelectronic Distances
1r1 − r2
=1r12
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.
B. Same symmetry as the ground state (difficult)
CI method MCSCF (state-averaged MCSCF), CASPT2