Chapter 10 Electronic Correlation (相关)Methodsweiwu.xmvb.org/ppt_CC/cc10.pdf · Chapter 10...

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∧

=

(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

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