Other methods to consider electron correlation: Coupled-Cluster and Perturbation Theory
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1 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
2 Inclusion of the electron correlation Eötvös Loránd University, Institute of Chemistry 1
3 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
4 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
5 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
6 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
7 Perturbation theory (PT) Usual Rayleigh-Schrödinger Perturbation Theory with Ĥ 0 = i ˆf(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
8 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
9 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
10 The Coupled-Cluster method Wave function: Ψ CC = e T Φ 0 Eötvös Loránd University, Institute of Chemistry 5
11 The Coupled-Cluster method Wave function: Ψ CC = e T Φ 0 T n is an excitation operator: T = T 1 + T T n Φ 0 = 1 n! abc...ijk... t abc.. ijk..φ abc.. ijk.. Eötvös Loránd University, Institute of Chemistry 5
12 The Coupled-Cluster method Wave function: Ψ CC = e T Φ 0 T n is an excitation operator: T = T 1 + T Truncated versions: T n Φ 0 = 1 n! abc...ijk... t abc.. ijk..φ abc.. ijk.. CCSD (T = T 1 + T 2 ) CCSD(T) (T = T 1 + T 2 + approximate T 3 ) Eötvös Loránd University, Institute of Chemistry 5
13 The Coupled-Cluster method Wave function: Ψ CC = e T Φ 0 T n is an excitation operator: T = T 1 + T Truncated versions: T n Φ 0 = 1 n! abc...ijk... t abc.. ijk..φ abc.. ijk.. CCSD (T = T 1 + T 2 ) CCSD(T) (T = T 1 + T 2 + approximate T 3 ) Very popular and very accurate for ground states! Eötvös Loránd University, Institute of Chemistry 5
14 The Coupled-Cluster method Advantages of CCSD: size-extensive since multiplicatively separable : e T A e T B = e T A+T B where T A and T B are the cluster operators for system A and B respectively. higher excitations are present with respect to CISD: e T = 1 + T 1 + T T T 1 T T i.e. also quadruple excitations are present which where used before to correct CISD!! Eötvös Loránd University, Institute of Chemistry 6
15 The Coupled-Cluster method Advantages of CCSD: size-extensive since multiplicatively separable : e T A e T B = e T A+T B where T A and T B are the cluster operators for system A and B respectively. higher excitations are present with respect to CISD: e T = 1 + T 1 + T T T 1 T T i.e. also quadruple excitations are present which where used before to correct CISD!! Eötvös Loránd University, Institute of Chemistry 6
16 The Coupled-Cluster method Why do we not use CC instead of CI? Eötvös Loránd University, Institute of Chemistry 7
17 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
18 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
19 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
20 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
21 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(φ 0 + i c i φ i ) = E(φ 0 + i c i φ i ) Ĥ(1 + Ĉ)φ 0 = E(1 + Ĉ)φ 0 We know that CC is a very good wave function for the ground state replace φ 0 by the CC wave function: Ĥ(1 + Ĉ)e ˆT φ 0 = E(1 + Ĉ)e ˆT φ 0 Eötvös Loránd University, Institute of Chemistry 8
22 Excited states in CC theory Since ˆT and Ĉ commute, we can write: Ĥe ˆT (1 + Ĉ)φ 0 = Ee ˆT (1 + Ĉ)φ 0 Now we multiply from the left by e ˆT : e ˆT Ĥe ˆT (1 + Ĉ)φ 0 = E(1 + Ĉ)φ 0 Introduce the notation: H = e ˆT Ĥ N e ˆT and our equation now becomes: H(1 + Ĉ)φ 0 = E(1 + Ĉ)φ 0 Excitation energies can be obtained by diagonalizing H!!!!! Eötvös Loránd University, Institute of Chemistry 9
23 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 ˆT ) extra costs to obtain gradients Advantages of EOM-CCSD very accurate if the above requiremts are fulfilled ( ev) straightforward to use, no definition of multireference space are neede Eötvös Loránd University, Institute of Chemistry 10
24 Vertical excitation energies (in ev) of trans-1,3-butadiene Method/State 1 1 B u 2 1 A g EOM-CCSD a EOM-CCSD( T ) a CASPT2 b CASPT2 c MR-SDCI d MR-AQCC e experiment 5.92 f 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 11
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