configuration interaction in quantum chemistry

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 12

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 electronsAr : 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 rr r r

r 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 li 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

L T V c cC

, , , , ,

,

,

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

11 2

r r

11 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

1,2

i iSD

i i

s ss 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 22 1x C C

1 2 0C C

11 2 1 1 2 2 1 1 2 2

ˆ ˆ,2 p q p qC A s s A s s r r r r r r

－

15

Left-right correlation• 　　 in olefin compounds

•

1 22 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 22 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 cHF 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


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

L C I H K C I K c cC

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

0ˆ

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


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 AE BE

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


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


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 SDTQ56Excitation order in wf.

Erro

r fro

m Fu

ll-CI

(h

artre

e)

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


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)


30

Statistics: reaction enthalpy

• Normal distribution

• CO+H2→CH2OHNC→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)


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) 21,2

trialE E

(1) ( ) κκ C κ C E κ C EC

ˆ 1 0 0P : Projector

20, 0 trial trial

pq i

E EC

(1) ( ) κE E 0C

, (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

34Soboloewski, 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 other37

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 comparedHF/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 YesExcited 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-CI Full-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

~100 CCSDTQ (MxEX in linear terms)

RASSCFRASPT2[1]

Maximum number of excitations

Max

imum

num

ber o

f act

ive

orbi

tals

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 02ij

ijrij 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

configuration interaction in quantum chemistry

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