Configuration Interaction in Quantum Chemistry

Jun-ya HASEGAWAFukui Institute for Fundamental

ChemistryKyoto University

1

Prof. M. Kotani (1906-1993)

2

Contents

• Molecular Orbital (MO) Theory• Electron Correlations• Configuration Interaction (CI) & Coupled-Cluster

(CC) methods• Multi-Configuration Self-Consistent Field (MCSCF)

method• Theory for Excited States

• Applications to photo-functional proteins

3

Molecular orbital theory

4

Electronic Schrödinger equation

• Electronic Schrödinger eq. w/ Born-Oppenheimer approx.

• Electronic Hamiltonian operator (non-relativistic)

• Potential energy–

• Wave function– The most important issue in electronic structure theory–

2

ˆ ˆ ˆ ˆ

1 1

2

e n e e n n

elec elec nuc elec nucA A B

ii i A i j A Bi A A Bi j

H T V V V

Z Z Z

r r r rr r

ˆ ,i A i i AH E r r r r r for fixed

ir : Coordinates for electrons

Ar : Coordinates for nucleus

E E A A= r parametrically depends on r

i Ar parametrically depends on r5

Many-electronwave function

• Orbital approximation: product of one-electron orbitals

• The Pauli anti-symmetry principle

• Slater determinant

– Anti-symmetrized orbital products– One-electron orbitals are the basic variables in MO theory

ˆ , , , , , , , ,i j i j j iP r r r r

i jP : Permutation operator

6

1 1 2 2, , , ,i j i i j j r r r r r r

1 1 1 2 1

2 1 2 2 21 2

1 2

1 1

1, ,

!

ˆ ˆ

N

NSD

N N N N

i i N N

N

A A

r r r

r r rr r

r r r

r r r

: Anti -symmetrizer

One-electron orbitals

• Linear combination of atom-centered Gaussian functions.

• Primitive Gaussian function

,

,

,

, , , , , , , , ,

r i

r

r i A x y z i A x y z r

r

C

l l l g l l l d

g

d

r r r r

: MO coefficient, the variable in MO theory

: Contracted atom-centered Gaussian functions

: Primitive Gaussian function

: Contrac tion coefficient (pre-defined)

7

,

AO

i r r ir

C

2, , , , , expx y zl l l

i A x y z i A i A i A i Ag l l l x x y y z z a r r r r

a : Exponent of Gaussian function (pre-defined)

Variational determination of the MO coefficients

• Energy functional

• Lagrange multiplier method

8

, ,,

, , ,

,,

i j i j i ji j

i j i j j i

i j i j

L E

i

i

: Multiplier, Real symmetric, = , when are real function.

Constratint : Orthonormalization of

, ,

, ,

1* *, 1 2 1 2 1

ˆ

ˆ ˆ

elec elec

i i j i ji i j

i i j i j

i i e n i

i j i j i j i j i

E H h J K

h J

h T V

J

r r r r r

:One-electron integrals, : Coulomb integral, K : Exchange integral

2 1 2

1* *, 1 2 1 2 1 2 1 2

j

i j i j j i i j j i

d d

K d d

r r r

r r r r r r r r

Hartree-Fock equation

• Variation of MO coefficients

• Hartree-Fock equation

• A unitary transformation that diagonalizes the multiplier matrix

• Canonical Hartree-Fock equation　　　　　　　　　　　　　　　　

9

,,

ˆ ˆ . . 0r e n i r j i j r j j i k i r ijr k

LT V c c

C

, , , , ,

,

,

ˆ ˆ

r s s i r s s i i k

r s r e n s r j s j r j j sj

r s r s

f C S C

f T V

S

, , ,,

can Tm m l mi i k k l

i k

U U

, , , ,can can can

r s s i r s s i if C S C

, , ,canr i r m m i

m

C C U

→Eigenvalue equation　　　 Eigenvalue: Multiplier (orbital energy) Eigenvector: MO coefficients

2

2j r

1 1r s r r

1

1 2

r r

1

1 2

r r

1 1r j r r 2 2s j r rj

s

r

Restricted Hartree-Fock (RHF) equation

• Spin in MO theory: (a)spin orbital formulation → spatial orbital rep. )(b) Restricted

(c) Unrestricted

• Restricted Hartree-Fock (RHF) equation for a closed shell (CS) system

• RHF wf is an eigenfunction of spin operators: a proper relation　　　　　　　　　　　　　　　　

10

i i

i i

i i

i i

, , , ,r s s i r s s i if C S C

2 2ˆ ˆˆ0 0 1 , 0RHF RHFCS CSS H S

ˆ ˆˆ0 , 0RHF RHFz CS CS zS H S

,ˆ ˆ 2

occN

r s r e n s r j s j r j j sj

f T V

Electron correlations

− Introduction to Configuration Interaction −

11

• Electron correlations defined as a difference from Full-CI energy

• Two classes of electron correlationsDynamical correlations– Lack of Coulomb hole

Static (non-dynamical) correlations– Bond dissociation, Excited states– Near degeneracyNo explicit separation between dynamical

and static correlations.　　　　　　　　　　　　　　　　

Definition of “electron correlations” in Quantum Chemistry

Corr Full CI HF

HF

Full CI

E E E

E

E

: Energy of a single determinant (independent particle)

: Full -CI energy (exact limit) for a set of one-electron basis functions

Restricted HF

Numerically Exact

Fig. Potetntial energy curves of H2 molecule. 6-31G** basis set. [Szabo, Ostlund, “Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory”, Dover]

Static correlation is dominant.

Dynamical correlation is dominant.

• Slater det. : Products of one-electron function

→Independent particle model

• Possibility of finding two electrons at : H2–like molecule case

– –

Dynamical correlations: lack of Coulomb hole

1 1 1 2ˆSD

i i i j r r r r

2

2

1 2 1 2 2

2

2

1

1

, ,SD

i i

P ds ds

r

r r

r

r r

1 1 1 11 2

2 2 2 2

1,

2i iSD

i i

s s

s s

r r

r rr r

1 2,r r

i i

i i

1 2 1 2,Pr r r rNo correlation between and : is a product of one-electron density.

1 2 1 2,P r r r rAt = , 0 Lack of Coulomb hole

• Interacting a doubly excited configuration

•

–

• Chemical intuition: Changing the orbital picture

→

Introducing dynamical correlations via configuration interaction

1 2 1 2 .C C PSome particular sets of and decrease r ,r

2 1r rAt

1 2 1 1 1 2 2 2 1 1 2 2ˆ ˆ, a ai iC A s s C A s s r r r r r r

2

1 2 1 1 2 2 1 2, i i a aP C C r r r r r r

2 1

22 2

1 2 1 1 2 1lim , i aP C C

r r

r r r r

p i ax q i ax 1 2

2 1x C C

1 2 0C C

11 2 1 1 2 2 1 1 2 2

ˆ ˆ,2 p q p q

CA s s A s s r r r r r r

－

15

Left-right correlation

• 　　 in olefin compounds

•

1 2

2 1x C C

Configuration interaction

22

22 Avoiding electron repulsion by introducing configuration

p i ax - x =

q i ax + x =

-=

No correlationsincluded

16

Angular correlation

• One-step higher angular momentum 　　　

•

1 2

2 1x C C

222 2 xs p

p i ax - x =

q i ax + x =

222 2 xs pAvoiding electron repulsion by introducing configuration

Configuration interaction

-=

No correlationsincluded

• 2-electron system in a dissociating homonuclear diatomic molecule

• Changing orbital picture into a local basis:

– Each configuration has a fixed weight of 25 %.– No independent variable that determines the weight for each

configuration when the bond-length stretches.

Static correlations: improper electronic structure

i A B a A B

A B

,A B

1 2 1 1 1 2 2 2

1 1 2 2 1 1 2 2

1 1 2 2 1 1 2 2

ˆ,

ˆ ˆ

ˆ ˆ

A B A B

A A B

B A

A

B B

A s s

A s s A s s

A s s A s s

r r r r r r

r r r r

r r r r

Ionic configuration: 2 e on A

Ionic configuration: 2 e on B

Covalent config.: 2 e at each A and B

Covalent config.: 2 e at each A and B

• Interacting a doubly excited configuration

– Some particular change the weights of covalent and ionic configurations.

Introducing static correlations via configuration interaction

,1 2 ,

1 1 2 2 1 1 2 2

1 1 2 2 1

1 2

1 2 21 2

ˆ ˆ

ˆ ˆ

A B B

CI a a

A

i

A B B

i

A

C C

A s s A s s

A s s A

C

C s

C

sC

r r r r

r r r r

1 2C C,

A B

1A r 2B r

A B

1B r 2A r

A B

1A r 2A r

A B

1A r 2A r

Configuration Interaction (CI) and

Coupled-Cluster (CC) wave functions

19

Some notations

• Notations– Occupied orbital indices: i, j, k, ….– Unoccupied orbital indices: a, b, c, …..– Creation operator: Annihilation operator:

• Spin-averaged excitation operator

– Spin-adapted operator (singlet)

• Reference configuration: Hartree-Fock determinant

• Excited configuration

– Correct spin multiplicity (Eigenfunction of operators)

20

†ˆaa ˆia

† †1ˆ ˆ ˆ ˆ ˆ2

ai a i a iS a a a a

0 0

abc

ijk

abc

ijk

abc

ijk

+ ≡abc

ijk

, , , ,, , , ,

ˆ ˆ ˆ ˆ ˆ ˆ ˆ0 , 0 0 , 0a a a b a b a b a b c a b ci i i j i j i j i j k i j kS S S S S S S

2ˆ ˆzS S and

21

Configuration Interaction (CI) wave function: a general form

• CI expansion: Linear combination of excited configurations

–

– Full-CI gives exact solutions within the basis sets used.

, , , , , ,, , , , , ,

, , , , , , , , ,

CI a a a b a b a b c a b cHF HF i i i j i j i j k i j k K K

i a i j a b i j k a b c K

C C C C C

abc

ijk

abc

ijk

abc

ijk

abc

ijk

CI Singles (CIS)

CI Singles and Doubles (CISD)

CI Singles, Doubles, and Triples (CISDT)

Full configuration interaction (Full CI)

∙∙∙∙

, , ,, , ,, , , ,a a b a b c

HF i i j i j k KC C C C C : Coefficients

, , ,, , ,, , , ,a a b a b c

HF i i j i j k K : Excited configurations

, , , , , ,0 , , , , , ,

, , , , , , , , ,

0 a a a b a b a b c a b ci i i j i j i j k i j k K

i a i j a b i j k a b c K

CI C C C C C K or

22

Variational determination of the wave function coefficients

• CI energy functional

• Lagrange multiplier method– Constraint: Normalization condition

• Variation of Lagrangian

• Eigenvalue equation

,

ˆ ˆI J

I J

E CI H CI C I H J C

,

ˆ ( . .) 0I II I JK

LC I H K C I K c c

C

ˆI I

I I

K H I C E K I C E

, ,

ˆ 1

ˆ 1I J I JI J I J

L CI H CI CI CI

C I H J C C I J C

1CI CI

23

Availability of CI method

• A straightforward approach to the correlation problem starting from MO theory

• Not only for the ground state but for the excited states

• Accuracy is systematically improved by increasing the excitation order up to Full-CI (exact solution)

• Energy is not size-extensive except for CIS and Full-CI– Difficulty in applying large systems

• Full-CI: number of configurations rapidlyincreases with the size of the system.– kα + kβ electrons in nα + nβ orbitals

→

– Porphyrin: nα = nβ =384 , kα =kβ =152

→ ~10221 determinants

Fig. Correlation energy per water molecule as a percentage of the Full-CI correlation energy (%) . The cc-pVDZ basis sets were used.

Number of water molecules

Perc

enta

ge (

%)

H2O

H2O

H2O

H2OH2O

H2OH2O

H2O

R ~ large

n k n kC C

determinants

CISD

Full-CI

24

Coupled-Cluster (CC) wave function

• CI wf: a linear expansion

• CC wf: an exponential expansion, , , , , ,

, , , , , ,, , , , , , , , ,

,

, ,, ,

, , , ,

, , , ,, , , , ,

, ,, ,

ˆ ˆ ˆexp 0

0

ˆ

1ˆ ˆ ˆ 02!

2ˆ2!

a a a b a b a b c a b ci i i j i j i j k i j k

i a i j a b i j k a b c

a ai i

i a

a b a b a b a bi j i j i j i j

i j a b i a

a b c a b c ai j k i j k i j

i j ka b c

CC C S C S C S

C S HF

C S C C S S

C S C C

, ,,

, , , ,, , , ,

1ˆ ˆ ˆ ˆ ˆ 03!

b c a b c a b c a b ck i j k i j k i j k

i j k i j ka b c a b c

S S C C C S S S

Single excitations

Double excitations

Triple excitations

, , , , , ,0 , , , , , ,

, , , , , , , , ,

0 a a a b a b a b c a b ci i i j i j i j k i j k K

i a i j a b i j k a b c K

CI C C C C C K

CC Singles (CCS)CI Singles and Doubles (CCSD)

CC Singles, Doubles, and Triples (CCSDT)∙∙∙∙

Linear terms =CI Non-linear terms

25

Why exponential?

• Size-extensive– Non interacting two molecules A and B

– Super-molecular calculation

↔ CI case

• A part of higher-order excitations described effectively by products of lower-order excitations.– Dynamical correlations is two body and short range.

ˆ ˆˆ ˆ exp 0 0 ˆˆ exp 0

ˆ

ˆ ˆe

exp

ˆexp 0

ˆˆ e

xp

p 00x

00

B B

B B B

A B A B A A A A

A A

AB B

B

A B A

H H S S

E

S

H S

S

H

S

S

S

E

ˆ ˆˆ exp 0 exp 0A A A A A AH S E S

ˆ ˆˆ exp 0 exp 0B B B B B BH S E SFar away

No interaction

ˆBHˆ

AH AEBE

ATot BEE E

ˆ ˆ ˆ ˆˆ ˆ 0 0 0 0A B A B A B A B A B A BH H S S E E S S

ˆ ˆ , 0A BS S

26

Solving CC equations

• Schrödinger eq. with the CC w.f.

• CC energy: Project on HF determinant

• Coefficients: Project on excited configurations (CCSD case)

– Non-linear equations. – Number of variable is the same as CI method.– Number of operation count in CCSD is O(N6), similar to CI

method.

, , , , , ,, , , , , ,

, , , , , , , , ,

ˆ ˆ ˆˆ exp 0 0a a a b a b a b c a b ci i i j i j i j k i j k

i a i j a b i j k a b c

H E C S C S C S

, , , , , ,, , , , , ,

, , , , , , , , ,

ˆ ˆ ˆˆ0 exp 0a a a b a b a b c a b ci i i j i j i j k i j k

i a i j a b i j k a b c

E H C S C S C S

† , ,, ,

, , , ,

ˆ ˆˆ0 exp 0 0ˆ a a a b a bi i i j i j

i a i ai

b

a

j

H E C S C SS

, ,, ,

, †

, , ,,

,

ˆ ˆˆ0 expˆ 0 0a a a ba b a bi i i j i j

i a i j a bi j H E C SS C S

27

Hierarchy in CI and CC methods and numerical performance

• Rapid convergence in the CC energy to Full-CI energy when the excitation order increases.– Higher-order effect was

included via the non-linear terms.

• In a non-equilibrium structure, the convergence becomes worse than that in the equilibrium structure.– Conventional CC method is

for molecules in equilibrium structure.

SD SDT SDTQ SDTQ5 SDTQ56

Excitation order in wf.

Err

or

from

Full-

CI

(hart

ree)

CI 法

CC 法

Fig. Error from Full-CI energy. H2O molecule with cc-pVDZ basis sets.[1]

~kcal/mol“Chemical accuracy”

Table. Error from Full-CI energy. H2O at equilibrium structure (Rref) and OH bonds elongated twice (2Rref)). cc-pVDZ sets were used.[1]

[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.

28

Statistics: Bond length

• Comparison with the experimental data (normal distribution [1])

• H2, HF, H2O, HOF, H2O2, HNC, NH3, … (30 molecules)

• “CCSD(T)” : Perturbative Triple correction to CCSD energy

cc-pVDZ cc-pVTZ cc-pVQZ

HF

MP2

CCSD

CCSD(T)

CISD

Error/pm=0.01Å Error/pm=0.01Å Error/pm=0.01Å

[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.

29

Statistics: Atomization energy

• Normal distribution• F2, H2, HF, H2O, HOF, H2O2, HNC, NH3, etc (total 20 molecules)

Error from experimental value in kJ/mol (200 kJ/mol=48.0 kcal/mol)

[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.

30

Statistics: reaction enthalpy

• Normal distribution

• CO+H2→CH2O

HNC→HCN

H2O+F2→HOF+HFN2+3H2→2NH3

etc. (20 reactions)

• Increasing accuracy in both theory and basis functions, calculated data approach to the experimental values.

Error from experimental data in kJ/mol (80 kJ/mol=19.0 kcal/mol)

[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.

Multi-Configurational Self-Consistent Field method

31

• Single-configuration description– Applicable to molecules in the ground state at near

equilibrium structureHartree-Fock method

• Multi-configuration description– Bond-dissociation, excited state, ….– Quasi-degeneracy → Linear combination of configurations

to describe STATIC correlations

• Multi-Configuration Self-Configuration Field (MCSCF) w.f.

– – Complete Active Space SCF (CASSCF) method

CI part = Full-CI: all possible electronic configurations are involved.

Beyond single-configuration description

32

A B

A B

A B

+

.

1 2ˆ,

elec

Config

ii NMCSCF i iC A

i iC : CI coefficients, : MO coefficients Optimized

• Trial MCSCF wave function is parameterized by

– Orbital rotation: unitary transformation

– CI correction vector

• MCSCF energy expanded up to second-order

–

MCSCF method: a second-order optimizaton

33

ˆ0

ˆexpˆ1

PMCSCF

P

C

C C

†ˆ ˆ ˆexp ,

† †ˆ ˆ ˆˆ + pq qp pq p q p qp q

pq E E E a a a a

iii CC

0 : Reference CI state

, ipq C

(0) 21,

2trialE E

(1) ( ) κκ C κ C E κ C E

C

ˆ 1 0 0P : Projector

20, 0 trial trial

pq i

E E

C

(1) ( ) κE E 0

C, (1)κ 0 C 0 E 0At convergence ( ),

ˆ ˆ0 0 0pq qpF F i PH : MCSCF condition, : Generalized Brillouin theorem

21E ECalc. &

MCSCF applications to potential energy surfaces

• CI guarantees qualitative description whole potential surfaces– From equilibrium structure to bond-dissociation limit– From ground state to excited states

34

Soboloewski, A. L. and Domcke, W. “Efficient Excited-State Deactivation in Organic Chromophores and Biologically Relevant Molecules: Role of Electron and Proton Transfer Processes”, In “Conical Intersections”, pp. 51-82, Eds. Domcke, Yarkony, Koppel, Singapore, World Scientific, 2011.

Dynamical correlations on top of MCSCF w.f.

• MCSCF handles only static correlations.– CAS-CI active space is at most 14 elec. in 14 orb. → For main configurations. → Lack of dynamical correlations.

• CASPT2 (2nd-order Perturbation Theory for CASSCF)

– Coefficients are determined by the 1st order eq. – Energy is corrected at the 2nd order eq. ← MP2 for MCSCF

• MRCC (Multi-Reference Coupled-Cluster)

– One of the most accurate treatment for the electron correlations. 35

, , , , ,, , ,

ˆ ˆ2 1 t u v x t u v xt u v x

CASPT C E E MCSCF

ˆexp IK K I

K

MRCC C S I C

Theory for Excited States

36

Excited states: definition

• Excited states as Eigenstates

• Mathematical conditions for excited states– Orthogonality

– Hamiltonian orthogonality

• CI is a method for excited states– CI eigenequation

– Hamiltonian matrix is diagonalized.

– Eigenvector is orthogonal each other

37

ˆ 1,2,I I IH E I

,ˆ

J I I J IH E

,J I J I

, ,ˆ 1,2,k I I k IH k C E k C I

, , , ,ˆ T

J l l k k I J I I J IC H C H E

, , , TJ l k I J IC l k C J I

Hamiltonian orthogonality

Orthogonality

Excited states for the Hartree-Fock (HF) ground state

• From the HF stationary condition to Brillouin theorem– Parameterized Hartree-Fock state as a trial state

– Unitary transformation for the orbital rotation

– HF energy expanded up to the second order

– Stationary condition

38

1 2ˆˆexp 0 , 0 NHF A

†ˆ ˆ ˆexp ,

† †ˆ ˆ ˆˆ + pq pq qp pq p q p qp q

E E E a a a a

0 2 1,

ˆ ˆ ˆ1 2 , = 0 , 0 trialp q pq qpE E E E E H

1T Tκ E κ E κ

20 , trial

pq

E

(1) ( ) (1)E E κ 0 κ = 0 E = 0At convergence

Excited states for the Hartree-Fock (HF) ground state

• CI Singles is an excited-state w. f. for HF ground state– Brillouin theorem: Single excitation is Hamiltonian

orthogonal to HF state

– CIS wave function

– Hamiltonian orthogonality & orthogonality

→ CIS satisfies the correct relationship with the HF ground state

• CI Singles and Doubles (CISD) does not provide a proper excited-state for HF ground state 39

1,

ˆ ˆ ˆ ˆ ˆ= 0 , 0 0 0 0 0p q pq qp iaE E E H E H

,

ˆ 0 aai i

a i

CIS E C

, ,

ˆ ˆ ˆ ˆ0 0 0 0, 0 0 0 0a aai i ai i

a i a i

H CIS HE C CIS E C

ˆ ˆ ˆ0 0 4 | 2 | 0bj aiHE E ia jb ib aj

Excited states for Coupled-Cluster (CC) ground state [1]

• CC wave function (or symmetry-adapted cluster (SAC) w. f.)

• CC w.f. into Schrödinger eq.

• Differentiate the CC Schrödinger eq.

• Generalized Brillouin theorem (GBT) → Structure of excited-state w. f.

ˆexp I II

CC C S HF

, ,, ,

, , , ,

ˆ ˆ ˆa a a b a bI I i i i j i j

I i a i j a b

C S C S C S Excitation operators and coefficients:

ˆ 0CC H E CC

ˆˆ 0ICC H E S CC

ˆˆ ˆ . . 0KK

CC H E CC CC H E S CC c cC

[1]H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978); 67(2,3), 329-333 (1979); 334-342 (1979).

Symmetry-adapted cluster-Configuration Interaction (SAC-CI)[1]

• A basis function for excited states

– Orthogonality

– Hamiltonian orthogonality

→

• SAC-CI wave function

ˆˆC 0CCC ISH E

[1]H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978); 67(2,3), 329-333 (1979); 334-342 (1979).

ˆˆˆ , 1 CC CC C CI PPS GBT from CC equation

CˆC 0ˆC CISP

ˆˆ ˆCC CC CCˆˆ CC 0IIH H E SPS ˆˆ CCIPS satidfies the conditions for excited-state w.f.

ˆˆ CCK KK

SAC CI PS d

SAC-CI(SD-R)compared with Full-CI

Accurate solution at Single and Double approximation→Applicable to molecules

Summary

43

CIS, CISD, SAC-CI (SD-R) are compared

HF/CIS CISD SAC/SAC-CI (SD-R)

Ground state

Wave function HF determinant Up to Doubles CCSD level

Electron correlations

No Yes Yes

Size-extensivity Yes No Yes

Excited state

Wave function Single excitations Singles and doubles Singles, doubles, effective higher excitations

Electron correlations

No Not enough. Near Full-CI result.

Size-extensivity Yes No Yes (Numerically)

Applicable targets Qualitative description for singly excited states

No. Excitation energy is overestimated

Quantitative description for singly excited states

Number of operation ((N: # of basis function)

O(N4) O(N6) O(N6)

00

ˆ ˆ ˆS DS S S 0 0ˆ ˆexp S DS S

0ˆ SS 0

0ˆ ˆ ˆS DS S S ˆ ˆS D

CCS S

Hierarchical view of CI-related methods

45

Dynamical correlations

Non-EQ

Excited states

Applicabilityto structuresEQ

EQ: EquilibriumGS: Ground statesEX: Excited states

GS

EX

Corr

IPHartree-Fock

MP2

CC

CIS

CIS(D), CC2

SAC-CIFull-CIMRCC

CASPT2

MCSCF

Perturbation 2nd order

CC level

Uncorrelated

IP: Independent Particle modelCorr: Correlated model

Static correlations

Practical aspect in CI-related methods

46[1] P.-Å. Malmqvist, K. Pierloot, A. R. M. Shahi, C. J. Cramer, and L. Gagliardi, JCP 128, 204109 (2008).

Fragment based approximated methods (divide & conquer, FMO, etc.) were excluded.Nact: Number of active orbitals , MxEX: The maximum order of excitation

Nact

MxEX

CCSD, SAC-CISD(MxEX in linear terms)

2 4

~1000

CASSCF, CASPT2[1]

16

15

32

10

~100CCSDTQ (MxEX in linear terms)

RASSCFRASPT2[1]

Maximum number of excitations

Maxim

um

num

ber

of

act

ive o

rbit

als

ChallengeChallenge: Speed up

End

47

Some important conditions for an electronic wave function

• The Pauli anti-symmetry principle

• Size-extensivity

•

• Cusp conditions

• Spin-symmetry adapted (for the non-relativistic Hamiltonian op.)

ˆ ˆ ˆFrag Frag

Tot TotI I J I

I I

H H H E E (non-interacting limit, =0)

ˆ , , , , , , , ,i j i j j iP r r r r

i jP : Permutation operator

0

1lim 0

2ijij

rij ave

rr

2 2ˆ ˆˆ1 , 0S S S H S

48

FragTot

II

E EIn some CI wave functions,

Coordinates

E

ˆ ˆˆ , 0z zS M H S