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Configuration Interaction in Quantum Chemistry. Jun-ya HASEGAWA Fukui Institute for Fundamental Chemistry Kyoto University. Prof. M. Kotani (1906-1993). Contents. Molecular Orbital (MO) Theory Electron Correlations Configuration Interaction (CI) & Coupled-Cluster (CC) methods - PowerPoint PPT Presentation
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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 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
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
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
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 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
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 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
[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→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)
[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) 21,2
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
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