Other methods to consider electron correlation:
Coupled-Cluster and Perturbation Theory
Péter G. Szalay
Eötvös Loránd University
Institute of Chemistry
H-1518 Budapest, P.O.Box 32, Hungary
[email protected]
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Inclusion of the electron correlation
Eötvös Loránd University, Institute of Chemistry
1
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Inclusion of the electron correlation
• Perturbation Theory (PT) - use HF as start
• Configuration Interaction (CI) - expand the wave function on several
determinants
• Coupled Cluster (CC) - exponential expansion of the wave function
• (Density Functional Theory - DFT)
Eötvös Loránd University, Institute of Chemistry
1
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Inclusion of the electron correlation
• Perturbation Theory (PT) - use HF as start
• Configuration Interaction (CI) - expand the wave function on several
determinants
• Coupled Cluster (CC) - exponential expansion of the wave function
• (Density Functional Theory - DFT)
Eötvös Loránd University, Institute of Chemistry
1
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Inclusion of the electron correlation
• Perturbation Theory (PT) - use HF as start
• Configuration Interaction (CI) - expand the wave function on several
determinants
• Coupled Cluster (CC) - exponential expansion of the wave function
• (Density Functional Theory - DFT)
Eötvös Loránd University, Institute of Chemistry
1
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Inclusion of the electron correlation
• Perturbation Theory (PT) - use HF as start
• Configuration Interaction (CI) - expand the wave function on several
determinants
• Coupled Cluster (CC) - exponential expansion of the wave function
• (Density Functional Theory - DFT)
Eötvös Loránd University, Institute of Chemistry
1
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Perturbation theory (PT)
Usual Rayleigh-Schrödinger Perturbation Theory with
Ĥ0 =
X
fˆ(i)
i
i.e. sum of the one-electron Fock-operators (Møller-Plesset partitioning)
1st order: Hartre-Fock method
2nd order: MP2 or MBPT(2) method
3rd order: MP3 of MBPT(3) method
etc.
Eötvös Loránd University, Institute of Chemistry
2
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Perturbation theory (PT)
MP2: cheap way to include electron correlation
MP3: usually not any better than MP2
MP4: often very good but expensive
Main problems:
• series may not converge
• HF must be a good starting point
• gets very expensive
Eötvös Loránd University, Institute of Chemistry
3
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Perturbation theory (PT)
If the HF is not a good starting point, use a CAS wave function instead →
CASPT2 and CASPT3 methods
Disadvantages:
• definition of the reference space is not straightforward
• not variational → gradient is expensive (no public program yet!)
• with the increasing size of the CAS gets expensive
• states are not orthogonal → transition moments can be calculated by
some tricks
Eötvös Loránd University, Institute of Chemistry
4
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Wave function:
Eötvös Loránd University, Institute of Chemistry
ΨCC = eT Φ0
5
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
ΨCC = eT Φ0
Wave function:
T = T1 + T2 + ...
Tn is an excitation operator:
1
T n Φ0 =
n!
Eötvös Loránd University, Institute of Chemistry
X
abc..
tabc..
ijk.. Φijk..
abc...ijk...
5
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
ΨCC = eT Φ0
Wave function:
T = T1 + T2 + ...
Tn is an excitation operator:
1
T n Φ0 =
n!
X
abc..
tabc..
ijk.. Φijk..
abc...ijk...
Truncated versions:
• CCSD (T = T1 + T2)
• CCSD(T) (T = T1 + T2 + approximate T3)
Eötvös Loránd University, Institute of Chemistry
5
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
ΨCC = eT Φ0
Wave function:
T = T1 + T2 + ...
Tn is an excitation operator:
1
T n Φ0 =
n!
X
abc..
tabc..
ijk.. Φijk..
abc...ijk...
Truncated versions:
• CCSD (T = T1 + T2)
• CCSD(T) (T = T1 + T2 + approximate T3)
Very popular and very accurate for ground states!
Eötvös Loránd University, Institute of Chemistry
5
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Advantages of CCSD:
• size-extensive since “multiplicatively separable”:
eTA eTB
= eTA+TB
where TA and TB are the cluster operators for system A and B
respectively.
• higher excitations are present with respect to CISD:
1 2
1 2
e = 1 + T1 + T2 + ... + T1 + T1T2 + T2 ...
2
2
T
i.e. also quadruple excitations are present which where used before to
correct CISD!!
Eötvös Loránd University, Institute of Chemistry
6
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Advantages of CCSD:
• size-extensive since “multiplicatively separable”:
eTA eTB
= eTA+TB
where TA and TB are the cluster operators for system A and B
respectively.
• higher excitations are present with respect to CISD:
1 2
1 2
e = 1 + T1 + T2 + ... + T1 + T1T2 + T2 ...
2
2
T
i.e. also quadruple excitations are present which where used before to
correct CISD!!
Eötvös Loránd University, Institute of Chemistry
6
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Why do we not use CC instead of CI?
Eötvös Loránd University, Institute of Chemistry
7
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Why do we not use CC instead of CI?
Disadvantages of the CC methods:
• not variational → gradient calculations cost twice as much
• not an eigenvalue problem → only for ground state
(But: excited states with symmetry different from ground state)
• multireference extension is not straightforward
• spin-contamination for open shell systems ← because of the products
Eötvös Loránd University, Institute of Chemistry
7
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Why do we not use CC instead of CI?
Disadvantages of the CC methods:
• not variational → gradient calculations cost twice as much
• not an eigenvalue problem → only for ground state
(But: excited states with symmetry different from ground state)
• multireference extension is not straightforward
• spin-contamination for open shell systems ← because of the products
Eötvös Loránd University, Institute of Chemistry
7
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Why do we not use CC instead of CI?
Disadvantages of the CC methods:
• not variational → gradient calculations cost twice as much
• not an eigenvalue problem → only for ground state
(But: excited states with symmetry different from ground state)
• multireference extension is not straightforward
• spin-contamination for open shell systems ← because of the products
Eötvös Loránd University, Institute of Chemistry
7
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
The Coupled-Cluster method
Why do we not use CC instead of CI?
Disadvantages of the CC methods:
• not variational → gradient calculations cost twice as much
• not an eigenvalue problem → only for ground state
(But: excited states with symmetry different from ground state)
• multireference extension is not straightforward
• spin-contamination for open shell systems ← because of the products
The second problem can be solved (see later) while the other ones not →
use MR-AQCC if multireference method is needed!!
Eötvös Loránd University, Institute of Chemistry
7
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Excited states in CC theory
CC equations are not eigenvalue equations → only the ground state can be
calculated.
But again Linear Response Theory (LRT) or equivalently Equation of Motion
(EOM) formalism can be used.
A simple representation of the EOM-CC method:
Consider CI:
Ĥ(φ0 +
X
ciφi) = E(φ0 +
i
X
ciφi)
i
Ĥ(1 + Ĉ)φ0 = E(1 + Ĉ)φ0
We know that CC is a very good wave function for the ground state
→ replace φ0 by the CC wave function:
Ĥ(1 + Ĉ)eT̂ φ0 = E(1 + Ĉ)eT̂ φ0
Eötvös Loránd University, Institute of Chemistry
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P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Excited states in CC theory
Since T̂ and Ĉ commute, we can write:
ĤeT̂ (1 + Ĉ)φ0 = EeT̂ (1 + Ĉ)φ0
Now we multiply from the left by e−T̂ :
e−T̂ ĤeT̂ (1 + Ĉ)φ0 = E(1 + Ĉ)φ0
Introduce the notation:
H̄
= e−T̂ ĤN eT̂
and our equation now becomes:
H̄(1 + Ĉ)φ0 = E(1 + Ĉ)φ0
Excitation energies can be obtained by diagonalizing H̄!!!!!
Eötvös Loránd University, Institute of Chemistry
9
P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Excited states in CC theory
Disadvantages of EOM-CCSD:
• only states dominated by single excitations
• if the CCSD is not adequate to describe the ground state, the reults are
unreliable → problem potential energy surfaces!!
• not fully variational (because of T̂ ) → extra costs to obtain gradients
Advantages of EOM-CCSD
• very accurate if the above requiremts are fulfilled (0.1-0.2 eV)
• straightforward to use, no definition of multireference space are neede
Eötvös Loránd University, Institute of Chemistry
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P.G. Szalay: CC and PT methods
Rio de Janeiro, Nov. 27 - Dec. 2, 2005
Vertical excitation energies (in eV) of
trans-1,3-butadiene
Method/State
EOM-CCSDa
EOM-CCSD(T̃ )a
CASPT2b
CASPT2c
MR-SDCId
MR-AQCCe
experiment
11Bu
6.42
6.13
6.23
6.06
6.70
6.20
5.92f
21Ag
7.23
6.76
6.27
6.27
6.78
6.55
a J.D. Watts, R. Gwaltney and R.J. Bartlett, J. Chem. Phys., 105, 6979 (1996).
b L. Serrano-Andres, M. Merchan, I. Nebot-Gil, R. Lindh and B.O. Roos, J. Chem. Phys., 98, 3151 (1993).
c B. Ostojic and W. Domcke, Chem. Phys., 269, 1 (2001).
d P.G. Szalay, A. Karpfen and H. Lischka, Chem. Phys., 130, 219 (1989).
e M. Dallos and H. Lischka, Theor. Chem. Acc., 112, 56 (2004).
f O.A. Mosher, W.M. Flicker and A. Kuppermann, J. Chem. Phys., 59, 6502 (1973).
Eötvös Loránd University, Institute of Chemistry
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