configuration interaction in quantum chemistry jun-ya hasegawa fukui institute for fundamental...
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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