Configuration Interaction in Quantum Chemistry
Jun-ya HASEGAWA
Fukui Institute for Fundamental Chemistry
Kyoto 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.
Hˆ  ri , rA    ri   E   ri  for fixed rA 
ri : Coordinates for electrons
rA : Coordinates for nucleus
• Electronic Hamiltonian operator (non-relativistic)
Hˆ  Tˆ  Vˆen  Vˆee  Vnn
elec
elec
nuc
Z Z
1 2 elec nuc Z A
1
  i  

 A B
i 2
i
A ri  rA
i  j ri  r j
A B rA  rB
• Potential energy
– E = E  rA  parametrically depends on rA 
• Wave function
– The most important issue in electronic structure theory
–   ri  parametrically depends on rA 
5
Many-electronwave function
• Orbital approximation: product of one-electron orbitals
  , ri ,
, rj ,
    r   r 
1
1
2
2
i  ri   j  r j 
• The Pauli anti-symmetry principle
Pˆi  j   , ri ,
, rj ,
   
, rj ,
, ri ,

Pˆi  j : Permutation operator
• Slater determinant
 SD  r1 , r2 ,

1  r1  1  r2 
1 2  r1  2  r2 
N!
 N  r1   N  r2 
1  rN 
2  rN 
 N  rN 
 Aˆ 1  r1  i  ri   N  rN  
Aˆ : Anti - symmetrizer
– Anti-symmetrized orbital products
– One-electron orbitals are the basic variables in MO theory
6
One-electron orbitals
• Linear combination of atom-centered Gaussian functions.
AO
i    r Cr ,i
r
Cr ,i : MO coefficient, the variable in MO theory
 r : Contracted atom - centered Gaussian functions
 r  ri , rA , lx , l y , lz    g  ri , rA , lx , l y , lz ,  d ,r

g : Primitive Gaussian function
d ,r : Contraction coefficient (pre - defined)
• Primitive Gaussian function

g  ri , rA , lx , l y , lz ,     xi  x A  x  yi  y A  y  zi  z A  z exp a ri  rA
l
l
l
2

a : Exponent of Gaussian function (pre - defined)
7
Variational determination of the MO coefficients
• Energy functional
E   Hˆ    hi    J i , j  K i , j 
elec
elec
i
i j
hi : One - electron integrals, J i , j : Coulomb integral, K i , j : Exchange integral
hi  i Tˆ  Vˆen i
J i , j  i j i j   i*  r1  *j  r2  r1  r2 i  r1  j  r2  dr1dr2
1
K i , j  i j  ji   i*  r1  *j  r2  r1  r2  j  r1 i  r2  dr1dr2
1
• Lagrange multiplier method

L  E    i , j i  j   i , j

i, j
 i , j : Multiplier, Real symmetric,  i , j =  j ,i , when i  are real function.
Constratint : Orthonormalization of i  , i  j   i , j
8
Hartree-Fock equation
• Variation of MO coefficients
L
  r Tˆ  Vˆen i    r j i j   r j  ji   k ,i   r i
Cr ,k
j
2
 j  r2 
• Hartree-Fock equation
r1  r2
f r ,s Cs ,i  S r ,s Cs ,i i ,k
   c.c.  0
1
 r  r1   s  r1 
f r ,s   r Tˆ  Vˆen  s    r j  s j   r j  j  s
 r  r1  j  r1 
j
Sr ,s   r  s
r
j
 s  r2  j  r2 
s
r1  r2
1
• A unitary transformation that diagonalizes the multiplier matrix
T
 mcan m,l  U mi
 i ,kU k ,l
Crcan
,i   Cr ,mU m ,i
m
i ,k
• Canonical Hartree-Fock equation
can can
f r ,s Cscan
,i  S r , s Cs ,i  i
→Eigenvalue equation
Eigenvalue: Multiplier (orbital energy)
Eigenvector: MO coefficients
9
Restricted Hartree-Fock (RHF) equation
• Spin in MO theory: (a)spin orbital formulation → spatial orbital
rep. )
(b) Restricted
i  i  
i  i  
i  i  
(c) Unrestricted
i  i  
fr ,sCs,i Hartree-Fock
 Sr ,sCs ,i i
• Restricted
(RHF)
equation for a closed shell (CS)
Nocc
system f r ,s   r Tˆ  Vˆen  s   2  r j  s j   r j  j  s
j
RHF
RHF
 Hˆ , Sˆ 2   0
Sˆ 2  CS
 0  0  1  CS


• RHF wf
an eigenfunction
ofspin
a proper relation
RHF
RHF
ˆ , Sˆ operators:

Sˆz is
0


H

0
CS
CS
z

10
Electron correlations
− Introduction to Configuration Interaction −
11
Definition of “electron correlations” in Quantum Chemistry
• Electron correlations defined as a difference from Full-CI energy
E Corr  E Full CI  E HF
E HF : Energy of a single determinant (independent particle)
E Full CI : Full - CI energy (exact limit) for a set of one - electron basis functions
Restricted HF
• Two classes of electron correlations
Dynamical correlations
Static correlation
is dominant.
– Lack of Coulomb hole
Static (non-dynamical) correlations
– Bond dissociation, Excited states
– Near degeneracy
No explicit separation between dynamical
and static correlations.
Numerically Exact
Dynamical correlation
is dominant.
Fig. Potetntial energy curves of H2 molecule.
6-31G** basis set. [Szabo, Ostlund, “Modern
Quantum Chemistry: Introduction to
Advanced Electronic Structure Theory”,
Dover]
Dynamical correlations: lack of Coulomb hole
• Slater det. : Products of one-electron function
ˆ   r   r    r   r  
 SD  
i
i
i
j
 1 1 1 2

→Independent particle model
• Possibility of finding two electrons at r1 , r2 
case
1   r   s    r    s 
 SD  r1 , r2  
P  r1 , r2    
i
1
1
i
1
: H2–like molecule
1
2 i  r2   s2  i  r2    s2 
SD
 r1 , r2 
2
ds1ds2
 i  r1  i  r2 
2
2
i  i  
i  i  
No correlation between r1 and r2 : P  r1 , r2  is a product of one - electron density.
– At r1 = r2 , P  r1 , r2   0  Lack of Coulomb hole
–
Introducing dynamical correlations via configuration
interaction
• Interacting a doubly excited configuration
  r1 , r2   C1 Aˆ i  r1   s1 i  r2    s2   C2 Aˆ a  r1   s1 a  r2    s2 
• Some particular sets of C1 and C2 decrease P  r1 ,r2  .
P  r1 , r2   C1i  r1 i  r2   C2a  r1 a  r2 
2
–
At r2  r1

lim P  r1 , r2   C1 i  r1   C2 a  r1 
r2 r1
2

2 2
C1C2  0
• Chemical intuition: Changing the orbital picture
 p  i  xa

q  i  xa
x    C2 C1 
12

→   r1 , r2   C1 Aˆ  p  r1   s1 q  r2    s2    Aˆ  p  r1    s1 q  r2   s2  




2
－
Left-right correlation
 
•     in olefin compounds

2
2
 p  i  xa 
-x
=
q  i  xa 
+x
=
x    C2 C1 
12

• Avoiding electron repulsion by introducing      configuration
2
2
No correlations
included
=
-
Configuration
interaction
15
Angular correlation
• One-step higher angular momentum  2s    2 px 
2
•
 p  i  xa 
-x
=
q  i  xa 
+x
=
2
x    C2 C1 
12
Avoiding electron repulsion by introducing  2s    2 px  configuration
2
=
2
-
Configuration
interaction
No correlations
included
16
Static correlations: improper electronic structure
• 2-electron system in a dissociating homonuclear diatomic molecule
a   A   B
i   A   B
A
B
• Changing orbital picture into a local basis:   A ,  B 
  r1 , r2   Aˆ   A  r1    B  r1   s1  A  r2    B  r2    s2  
 Aˆ   A  r1   s1   A  r2    s2    Aˆ   A  r1   s1   B  r2    s2  
Ionic configuration: 2 e on A
Covalent config.: 2 e at each A and B
 Aˆ   B  r1   s1   A  r2    s2    Aˆ   B  r1   s1   B  r2    s2  
Covalent config.: 2 e at each A and B Ionic configuration: 2 e on B
– Each configuration has a fixed weight of 25 %.
– No independent variable that determines the weight for each configuration
when the bond-length stretches.
Introducing static correlations via configuration interaction
• Interacting a doubly excited configuration
 CI  C1  C2  ia,,ia

  C1  C2  Aˆ   A  r1   s1   B  r2    s2    Aˆ   B  r1   s1   A  r2    s2  
 B  r2 
 A  r1 
A
B

 A  r2 
 B  r1 
A
B

  C1  C2  Aˆ   A  r1   s1   A  r2    s2    Aˆ   B  r1   s1   B  r2    s2  
 A  r1   A  r2 
A
– Some particular  C1 , C2 
configurations.
B
 A  r1   A  r2 
A
B
change the weights of covalent and ionic

Configuration Interaction (CI)
and
Coupled-Cluster (CC)
wave functions
19
Some notations
• Notations
c
b
a
– Occupied orbital indices: i, j, k, ….
– Unoccupied orbital indices: a, b, c, …..
†
– Creation operator: aˆ a
Annihilation operator:aˆi
• Spin-averaged excitation operator
1 †
Sˆia 
aˆa aˆi  aˆa† aˆi 

2
c
b
a
i
j
k
+
i
j
k
c
b
a
i
j
k
≡
c
b
a
i
j
k
– Spin-adapted operator (singlet)
• Reference configuration: Hartree-Fock determinant
0  0
• Excited configuration
ia  Sˆia 0 , ia,,jb  Sˆia, ,jb 0  Sˆia Sˆ bj 0 , ia,,jb,,kc  Sˆia Sˆ bj Sˆkc 0
2
– Correct spin multiplicity (Eigenfunction of Sˆ and Sˆ z
operators)
20
Configuration Interaction (CI) wave function: a general form
• CI expansion: Linear combination of excited configurations
CI  CHF  HF   Cia ia 
i ,a
or
CI  C0 0   Cia
a
i
i ,a
C
 C
a ,b
i, j
ia,,jb 
i , j ,a ,b
a ,b a ,b
i, j i, j
i , j ,a ,b
 C
 C
a ,b ,c
i , j ,k
i , j ,k ,a ,b ,c

  CK  K
ia,,jb,,kc 
a ,b ,c a ,b ,c
i , j ,k i , j ,k
K

  CK K
i , j ,k ,a ,b ,c
K
c
b
a
c
b
a
c
b
a
c
b
a
i
j
k
i
j
k
i
j
k
i
j
k
∙∙∙∙
CI Singles (CIS)
CI Singles and Doubles (CISD)
CI Singles, Doubles, and Triples (CISDT)
Full configuration interaction (Full CI)
–  HF ,ia  ,ia,,jb  ,ia,,jb,,kc  , K  : Excited configurations
CHF ,Cia  ,Cia, ,jb  ,Cia, ,jb,k,c  ,CK  : Coefficients
– Full-CI gives exact solutions within the basis sets used.
21
Variational determination of the wave function coefficients
• CI energy functional
E  CI Hˆ CI   CI I Hˆ J CJ
I ,J
• Lagrange multiplier method
– Constraint: Normalization condition
CI CI  1
L  CI Hˆ CI    CI CI  1



ˆ
  C I H J C J     CI I J C J  1
I ,J
 I ,J


I
• Variation of Lagrangian
L
  CI I Hˆ K     CI I K  (c.c.)  0
CK
I
I ,J
• Eigenvalue equation

I
K Hˆ I CI  E  K I CI
I
 E
22
Availability of CI method
– Difficulty in applying large systems
Percentage (%)
• 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)
Full-CI
• Energy is not size-extensive
CISD
HO
HO
HO
HO
except for CIS and Full-CI
R ~ large



2
2
2
H2O
• Full-CI: number of configurations rapidly
increases with the size of the system.
– kα + kβ electrons in nα + nβ orbitals
determinants
→
n Ck
n Ck
– Porphyrin: nα = nβ =384 , kα =kβ =152
→ ~10221 determinants
2
H2O
H2O
H2O
Number of water molecules
Fig. Correlation energy per water molecule as
a percentage of the Full-CI correlation energy
(%) . The cc-pVDZ basis sets were used.
23
Coupled-Cluster (CC) wave function
• CI wf: a linear expansion
CI  C0 0   Cia
i ,a
a
i

C
a ,b a ,b
i, j i, j
i , j ,a ,b
• CC wf: an exponential expansion


Cia, ,jb,k,c
a ,b ,c
i , j ,k
i , j ,k ,a ,b ,c

CC  exp   Cia Sˆia   Cia, ,jb Sˆia, ,jb   Cia, ,jb,k,c Sˆia, ,jb,k,c 
i , j ,k ,a ,b ,c
 i ,a (CCS) i , j ,a ,b
CC Singles
 0
CI Singles and Doubles (CCSD)
CC Singles, Doubles, and Triples (CCSDT)
  Cia Sˆia HF

  CK K
K

0

∙∙∙∙
Single excitations
i ,a

1
a ,b ˆ a ,b
a b ˆ a ˆb 
Double excitations
   Ci , j Si , j   Ci C j Si S j  0
2! i ,a
 i , j ,a ,b



2
1
   Cia, ,jb,k,c Sˆia, ,jb,k,c   Cia C bj ,,kc Sˆia Sˆ bj ,,kc   Cia C bj Ckc Sˆia Sˆ bj Sˆkc  0
 i , j ,k

2! i , j ,k
3! i , j ,k
a ,b , c
a ,b ,c
Tripleexcitations
 a ,b , c
Non-linear terms
24
Linear terms =CI
Why exponential?
• Size-extensive
No interaction
– Non interacting two molecules A and B
Hˆ A exp Sˆ A 0 A  EA exp Sˆ A 0 A
Hˆ B
 
exp  Sˆ  0
B
B
 EB
 
exp  Sˆ  0
– Super-molecular calculation
 Hˆ
A
 

B
Hˆ A E A
Far away
ETot  EA  EB
B
   
 exp  Sˆ  Hˆ exp  Sˆ  0 0
  E  E  exp  Sˆ  Sˆ  0 0
 Hˆ B exp Sˆ A  SˆB 0 A0 B  Hˆ A exp Sˆ A exp SˆB 0 A0 B
A
A
↔ CI case
Hˆ A  Hˆ B

  Sˆ
A

Hˆ B EB
B
B
B
A

  Sˆ A , SˆB   0
A B
B
A B

 SˆB 0 A0B   E A  EB  Sˆ A  SˆB 0 A0B
• A part of higher-order excitations described effectively by products
of lower-order excitations.
– Dynamical correlations is two body and short range.
Solving CC equations
• Schrödinger eq. with the CC w.f.


ˆ
H  E exp   Cia Sˆia   Cia, ,jb Sˆia, ,jb   Cia, ,jb,k,c Sˆia, ,jb,k,c 
i , j ,a ,b
i , j , k , a ,b ,c
 i ,a


 0 0

• CC energy: Project on HF determinant

E  0 Hˆ exp   Cia Sˆia   Cia, ,jb Sˆia, ,jb   Cia, ,jb,k,c Sˆia, ,jb,k,c 
i , j , a ,b
i , j ,k ,a ,b ,c
 i ,a

0

• Coefficients: Project on excited configurations (CCSD case)

a†
a ˆa
a ,b ˆ a ,b 
ˆ
ˆ
0 Si H  E exp   Ci Si   Ci , j Si , j  0  0
i , j ,a ,b
 i ,a


a ,b †
a ˆa
a ,b ˆ a ,b 
ˆ
ˆ
0 Si , j H  E exp   Ci Si   Ci , j Si , j  0  0
i , j ,a ,b
 i ,a

– 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.




26
Hierarchy in CI and CC methods and numerical performance
– 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.
Error from Full-CI (hartree)
• Rapid convergence in the
CC energy to Full-CI
energy when the excitation
order increases.
Excitation order in wf.
SD
SDT
SDTQ SDTQ5 SDTQ56
CI法
~kcal/mol
“Chemical accuracy”
CC法
Fig. Error from Full-CI energy. H2O molecule with cc-pVDZ
basis sets.[1]
Table. Error from Full-CI energy. H2O at
equilibrium structure (Rref) and OH bonds
elongated twice (2Rref)). cc-pVDZ sets
were used.[1]
– Conventional CC method is
for molecules in
structure.
[1]“Molecularequilibrium
Electronic Structure
Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.
27
cc-pVDZ
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-pVTZ
cc-pVQZ
HF
MP2
CCSD
CCSD(T)
CISD
[1]“Molecular Electronic Structure
Theory”, Helgaker, Jorgensen, Olsen,
Wiley, 2000.
Error/pm=0.01Å
Error/pm=0.01Å
Error/pm=0.01Å
28
Statistics: Atomization energy
Error from experimental value in kJ/mol (200 kJ/mol=48.0 kcal/mol)
•
•
Normal distribution
F2, H2, HF, H2O, HOF, H2O2, HNC, NH3, etc (total 20 molecules)
[1]“Molecular Electronic Structure Theory”, Helgaker, Jorgensen, Olsen, Wiley, 2000.
29
Statistics: reaction enthalpy
•
Normal
distribution
•
CO+H2→CH2O
HNC→HCN
H2O+F2→HOF+HF
N2+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.
30
Multi-Configurational
Self-Consistent Field method
31
Beyond single-configuration description
• Single-configuration description
– Applicable to molecules in the ground state at near equilibrium structure
Hartree-Fock method
• Multi-configuration description
– Bond-dissociation, excited state, ….
– Quasi-degeneracy
→ Linear combination of configurations
to describe STATIC correlations
A
A
B
+
A
B
B
• Multi-Configuration Self-Configuration Field (MCSCF) w.f.
MCSCF 
Config .

i
i Ci ,
i  Aˆ 12
 N 
elec
– Ci  : CI coefficients, i  : MO coefficients  Optimized
– Complete Active Space SCF (CASSCF) method
CI part = Full-CI: all possible electronic configurations are involved.
32
MCSCF method: a second-order optimizaton
• Trial MCSCF wave function is parameterized by
 pq ,Ci 
MCSCF  exp  ˆ 
0  Pˆ C
1  C Pˆ C
0 : Reference CI state
Pˆ  1  0 0 : Projector
†
– Orbital rotation: unitary transformation exp  ˆ  , ˆ  ˆ
ˆ    pq Eˆ pq  Eˆ qp
Eˆ pq  a †p aq +a †p aq
p q


– CI correction vector
C   i Ci
i
• MCSCF energy expanded up to second-order
Calc. E  & E
1
2
1
(2)  κ 
κ
C
E

  
2
C
E trial
E trial
(1)
( 2)  κ 
 0,
 0  E E    0
 pq
Ci
C
– At convergence(κ  0, C  0), E(1)  0
ˆ ˆ 0  0 : Generalized Brillouin theorem
F  F  0 : MCSCF condition, i PH
E trial  κ , C   E (0)   κ C  E(1) 
pq
qp
33
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
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.
34
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)


CASPT 2  1   Ct ,u ,v , x Eˆt ,u Eˆ v , x  MCSCF
 t ,u ,v , x

– Coefficients are determined by the 1st order eq.
– Energy is corrected at the 2nd order eq. ← MP2 for MCSCF
• MRCC (Multi-Reference Coupled-Cluster)


I
MRCC  exp   CK  SˆK  I CI
 K

– One of the most accurate treatment for the electron correlations.
35
Theory for Excited States
36
Excited states: definition
• Excited states as Eigenstates
Hˆ  I  EI  I
I  1, 2,
• Mathematical conditions for excited states
– Orthogonality
 J  I   J ,I
– Hamiltonian orthogonality
 J Hˆ  I  EI  J , I
• CI is a method for excited states
– CI eigenequation
Hˆ k Ck , I  EI k Ck , I I  1, 2,
– Hamiltonian matrix is diagonalized.
CJT,l H l ,k Ck , I   J Hˆ  I  EI  J , I
Hamiltonian orthogonality
– Eigenvector is orthogonal each other
C JT,l l k Ck , I  J I  J , I
Orthogonality
37
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
0  Aˆ 12
HF    exp  ˆ  0 ,
N 
– Unitary transformation for the orbital rotation
exp  ˆ  ,
ˆ †  ˆ

ˆ    pq Eˆ pq  Eˆ qp
p q

Eˆ pq  a †p aq +a †p aq
– HF energy expanded up to the second order
E trial  E    κ TE   1 2  κ TE κ ,
0
1
2
– Stationary condition
E trial
 0  E(1)  E( 2 )κ  0
 pq
1
E p,q = 0  Eˆ pq  Eˆ qp , Hˆ  0
At convergence

κ = 0, E(1) = 0
38
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
1
E p,q = 0  Eˆ pq  Eˆ qp , Hˆ  0  0 
0 Eˆia Hˆ 0  0
– CIS wave function
CIS   Eˆ 0 C a
ai
i
a ,i
– Hamiltonian orthogonality & orthogonality
ˆ ˆ 0 C a  0,
0 Hˆ CIS   0 HE
ai
i
a ,i
0 CIS   0 Eˆ ai 0 Cia  0
a ,i
→ 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
ˆ ˆ Eˆ 0  4  ia | jb   2  ib | aj   0
0 HE
bj ai
39
Excited states for Coupled-Cluster (CC) ground state [1]
• CC wave function (or symmetry-adapted cluster (SAC) w. f.)


CC  exp   CI SˆI  HF
 I

Excitation operators and coefficients:
ˆ  C a Sˆ a 
C
S
 I I  i i
I
i ,a

Cia, ,jb Sˆia, ,jb 
i , j ,a ,b
• CC w.f. into Schrödinger eq.
CC Hˆ  E CC  0
• Differentiate the CC Schrödinger eq.

CC Hˆ  E CC  CC Hˆ  E SˆK CC   c.c.  0
CK


• Generalized Brillouin theorem (GBT) → Structure of excited-state w.
f. CC Hˆ  E Sˆ CC  0


I
[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
ˆ ˆ CC ,
PS
I

Pˆ  1  CC CC

CC Hˆ  E SˆI CC  0
GBT from CC equation
– Orthogonality
CC Pˆ SˆI CC  0
– Hamiltonian orthogonality
ˆ ˆ CC  CC Hˆ  E Sˆ CC  0
CC Hˆ PS
I
I
ˆ ˆ CC satidfies the conditions for excited - state w.f.
→ PS


I
• SAC-CI wave function
ˆ ˆ CC d
SAC  CI   PS
K
K
K
[1]H. Nakatsuji, Chem. Phys. Lett., 59(2), 362-364 (1978); 67(2,3), 329-333 (1979); 334-342 (1979).
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)
Wave function
HF determinant
Up to Doubles
Electron correlations
No
Yes
Yes
Size-extensivity
Yes
No
Yes
Single excitations
Singles and doubles
Singles, doubles,
effective higher
excitations
Ground state
0


Sˆ 0  Sˆ S  Sˆ D  0
CCSD level


exp Sˆ S  Sˆ D  0
Excited state
Wave function
Sˆ S  0
 Sˆ
0

 Sˆ S  Sˆ D  0
 Sˆ
S

 Sˆ D  CC
Electron correlations
No
Not enough.
Near Full-CI result.
Size-extensivity
Yes
No
Yes (Numerically)
Qualitative description
for singly excited
states
No. Excitation energy
is overestimated
Quantitative
description for singly
excited states
O(N6)
O(N6)
Applicable targets
Number of operation ((N: O(N4)
# of basis function)
Hierarchical view of CI-related methods
EQ: Equilibrium IP: Independent Particle model
GS: Ground states Corr: Correlated model
EX: Excited states
Dynamical
correlations
CC
Corr
CC level
Full-CI
MRCC
SAC-CI
MP2
Perturbation 2nd order
CIS(D), CC2
CASPT2
Hartree-Fock
IP
GS EQ
CIS
EX
Excited states
Uncorrelated
Non-EQ
Applicability
to structures
MCSCF
Static
correlations
45
Practical aspect in CI-related methods
Maximum number of active orbitals
Fragment based approximated methods (divide & conquer, FMO, etc.) were excluded.
Nact: Number of active orbitals , MxEX: The maximum order of excitation
Nact
CCSD, SAC-CISD(MxEX in linear terms)
~1000
CCSDTQ (MxEX in linear terms)
~100
RASSCF
RASPT2[1]
32
15
Challenge: Speed up
2
4
Challenge
CASSCF, CASPT2[1]
10
16
Maximum number of excitations
MxEX
[1] P.-Å. Malmqvist, K. Pierloot, A. R. M. Shahi, C. J. Cramer, and L. Gagliardi, JCP 128, 204109 (2008).
46
End
47
Some important conditions for an electronic wave function
• The Pauli anti-symmetry principle
Pˆi  j   , ri ,
, rj ,
   
, rj ,
, ri ,

Pˆi  j : Permutation operator
• Size-extensivity
Hˆ
Tot

Frag
I Hˆ I
(non - interacting limit, Hˆ I  J = 0)  E
• In some CI wave functions, E
Tot
• Cusp conditions
  
1
lim 

  rij  0 

rij 0  r 
 ij ave 2

Frag
I EI
Tot

Frag
I EI
E
Coordinates
• Spin-symmetry adapted (for the non-relativistic Hamiltonian op.)
Sˆ 2   S  S  1 
Sˆz   M 
 Hˆ , Sˆ 2   0


 Hˆ , Sˆz   0


48