# Multi-state Multireference Rayleigh–Schrödinger Perturbation Theory for Mixed Electronic States: Second and Third Order

## Abstract

**:**

**1998**, 288, 299–306].

## 1. Introduction

- The choices of orbitals and zero-order Hamiltonian try to mimic the Møller–Plesset procedures that have been found to be very effective in single-reference perturbation expansions.
- The zero-order Hamiltonians need not be diagonal.
- A non-Hermitian effective Hamiltonian is generated, based on the Bloch equation.
- The method is formulated in configuration space, allowing flexibility in the choice of the reference space (including incomplete active spaces), and enabling easy implementation in a CI program.
- Uncontracted configuration state functions are used as a basis for the perturbation expansions.
- Procedures for both second and third order in the energy are included.

## 2. Rayleigh–Schrödinger perturbation theory for arbitrary zero-order functions

**E**

^{(0)}defined so that ${E}_{\alpha \alpha}^{(0)}\equiv {E}_{\alpha}^{(0)}$ and ${E}_{i\alpha}^{(0)}=0(i\ne \alpha )$. We then have

## 3. Multireference perturbation treatment with a single MCSCF zero-order wave function

#### A. The zero-order functions

_{1}, Θ

_{2}, …, Θ

_{m}is used as ${\Phi}_{\alpha}^{(0)}$:

_{i}(i > m).

**E**

^{(0)}matrix is diagonal. Therefore

_{i}(including ${\Phi}_{i}^{(0)}$ ≡ Θ

_{i}for i > m). We also have

_{ij}.

#### B. The zero-order Hamiltonian

**E**

^{(0)}is defined as

**E**

^{(0)}in which one or both indices refer to orthogonal-complement functions ${\Phi}_{\beta}^{(0)}$ (β ≠ α, β ≤ m) do not enter into the calculation of the first-order wave function and the second- and third-order energy, and need not be specified in the present treatment (except that ${E}_{\alpha \beta}^{(0)}$ = 0 for β ≠ α). The principal difference between ${\widehat{H}}_{0}$ and $\widehat{F}$ is the omission from ${\widehat{H}}_{0}$ of any contributions from $\langle {\Phi}_{i}^{(0)}\left|\widehat{F}\right|{\Phi}_{\alpha}^{(0)}\rangle $ elements with i > m; assuming ${\Phi}_{\alpha}^{(0)}$ is an optimized MCSCF wave function that satisfies a generalized Brillouin theorem [19], it should be possible to choose the Fock operator so that all or most such matrix elements vanish.

_{p}(${\Phi}_{i}^{(0)}$) is the occupation number of the p-th orbital in ${\Phi}_{i}^{(0)}$, so that

**f**is diagonal, so is the

**E**

^{(0)}matrix, because

#### C. Choice of the generalized Fock operator

_{pq}elements involving virtual orbitals, since density matrix elements involving at least one virtual orbital vanish. Other choices of the generalized Fock operator and of ${\widehat{H}}_{0}$ have been discussed, e.g., by Hirao [24], Kozlowski and Davidson [25], Zaitsevskii and Malrieu [26], Dyall [20], Andersson [27], and Roos et al. [18(a)], and are often aimed at providing a better balance between the treatments of closed- and open-shell states, thus improving the calculation of excitation energies.

**f**as diagonal as possible. But even if

**f**is not diagonal, the matrix

**E**

^{(0)}is very sparse, because $\widehat{F}$ is a one-electron operator and can have nonvanishing matrix elements only between configuration state functions differing by at most one orbital. This property makes the solution of the linear system (12) relatively easy. A limited use of two-body terms in ${\widehat{H}}_{0}$, as in Dyall’s Hamiltonian [20], does not greatly increase the difficulty of this procedure.

#### D. Evaluation of the perturbation series in a CI program

**H**is defined in terms of the original configuration state functions Θ

_{j}, with ${\Phi}_{j}^{(0)}$ ≡ Θ

_{j}for j > m), we obtain the $\Delta {\mathbf{c}}_{\alpha}^{(1)}$ vector of first-order coefficients by solving the linear equations (12) or, in the diagonal case, from Eq. (14). The computational effort of this step is quite small, because the nonzero part of ${\mathbf{c}}_{\alpha}^{(0)}$ is very short. The first-order wave function is then

## 4. The treatment of mixed electronic states

**C**

^{(0)}of l columns, in which at most the first m rows (corresponding to the reference configurations) are nonzero.

**E**

^{(0)}taken to be diagonal, so that we have

**E**

^{(0)}involving orthogonal-complement functions ${\Phi}_{\beta}^{(0)}$, l < β ≤ m, do not enter into the calculation of the first-order wave function and the second- and third-order energies.

**H**

^{EFF}=

**H**

_{0}+

**W**=

**H**

_{0}+

**W**

^{(1)}+

**W**

^{(2)}+ …,

**E**

^{(0)},

**CH**

^{EFF}=

**HC**,

**C**, which is a representation of the wave operator, has the order-by-order expansion

**C**=

**C**

^{(0)}+

**ΔC**

^{(1)}+

**ΔC**

^{(2)}+ …

**H**(unlike

**H**

_{0}in Eq. (38)) is defined in terms of the individual configuration state functions Θ

_{i}. Equation (41) is a representation of the Bloch equation [9] in the configuration-state-function basis. Note that the first m rows of

**ΔC**

^{(1)}are zero because of intermediate normalization and the noninteracting nature of the functions ${\Phi}_{\alpha}^{(0)}$ (α ≤ m).

**C**

^{(1)}=

**C**

^{(0)}+

**ΔC**

^{(1)}constructed from the first-order vectors ${\mathbf{c}}_{\alpha}^{(1)}={\mathbf{c}}_{\alpha}^{(0)}+\Delta {\mathbf{c}}_{\alpha}^{(1)}$ (α = 1, 2, …, l) calculated for each of the zero-order functions by Eq. (12) or (14), analogously to the single-state calculations described in the previous section. The second-order non-Hermitian shift operator is then obtained as

**W**

^{(3)}cannot be fully obtained from the first-order wave functions. Instead it is computed from the second-order wave functions,

**E**

^{(0)}to have no off-diagonal elements coupling the higher excitations with the single and double excitations, we can decouple the equations for the single- and double-excitation coefficients from the others and solve for these from the linear equations system

**H**

^{(3)}to obtain the third-order energies . While a transformation matrix

**X**

^{(2)}is also obtained, and can be used to obtain the transformed coefficients matrix $\tilde{\mathbf{C}}(2)$ for the relevant part of the second-order wave functions ${\Psi}_{\alpha}^{(2)}$, it does not provide the complete second-order wave function, because of the omission of the higher excitations and the orthogonal complement functions. Once the truncated second-order wave function has been obtained, the construction of the third-order effective Hamiltonian matrix and the diagonalization require little computational effort, and thus the total effort is still about l times the effort in the single-state case.

**E**

^{(0)}and

**V**, and to be less analogous to the Møller–Plesset partitioning of single-reference perturbation theory. Nevertheless, the

**E**

^{(0)}matrix should remain very sparse, because it is the matrix representation of a one-electron operator (the state-averaged Fock operator). Another potential problem is that it may be very difficult to choose a model space that does not vary discontinuously over a potential energy surface.

## 5. Discussion

## References

- Kahn, L. R.; Hay, P. J.; Shavitt, I. J. Chem. Phys.
**1974**, 61, 3530–3546. - Meyer, W. Methods of Electronic Structure Theory; Schaefer, H. F., III, Ed.; Plenum: New York, 1977; Chapter 11; pp. 413–446. [Google Scholar]
- Siegbahn, P. E. M. Int. J. Quantum Chem.
**1980**, 18, 1229–1242. - Werner, H.-J.; Reinsch, E.-A. J. Chem. Phys.
**1982**, 76, 3144–3156. - Werner, H.-J.; Knowles, P. J. J. Chem. Phys.
**1988**, 89, 5803–5814. - Andersson, K.; Malmqvist, P.-Å.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. J. Phys. Chem.
**1990**, 94, 5483–5488. [PubMed]Andersson, K.; Malmqvist, P.-Å.; Roos, B. O. J. Chem. Phys.**1992**, 96, 1218–1226. - Werner, H. Mol. Phys.
**1996**, 89, 645–661. - Knowles, P. J.; Werner, H.-J. Theor. Chim. Acta
**1992**, 84, 95–103. - Bloch, C. Nucl. Phys.
**1958**, 6, 329–347. Bloch, C.; Horowitz, J. ibid.**1958**, 8, 91–105. - Brandow, B. H. Rev. Mod. Phys.
**1967**, 39, 771–828. Brandow, B. H. Adv. Quantum Chem.**1977**, 10, 187–249. - Primas, H. Rev. Mod. Phys.
**1963**, 35, 710–712. - Docken, K. K.; Hinze, J. J. Chem. Phys.
**1972**, 57, 4928–4936. - Ruedenberg, K.; Cheung, L. M.; Elbert, S. T. Int. J. Quantum Chem.
**1979**, 16, 1069–1101. - Daudey, J.-P.; Malrieu, J.-P. Current Aspects of Quantum Chemistry 1981; Carbo, R., Ed.; Elsevier: Amsterdam, 1982; pp. 35–64. [Google Scholar] Spiegelman, F.; Malrieu, J.-P. J. Phys. B
**1984**, 17, 1235–1257, 1259–1279. - Sheppard, M. G.; Schneider, B. I.; Martin, R. L. J. Chem. Phys.
**1983**, 79, 1364–1368. - Lisini, A.; Decleva, P. Chem. Phys.
**1992**, 168, 1–13. Lisini, A.; Decleva, P. J. Phys. B**1994**, 27, 1325–1339. - Nakano, H. Chem Phys. Lett.
**1993**, 207, 372–378. Nakano, H. J. Chem. Phys.**1993**, 99, 7983–7992. Nakano, H.; Nakatani, J.; Hirao, K. J. Chem. Phys.**2001**, 114, 1133–1141. - Roos, B. O.; Andersson, K.; Fülscher, M. P.; Malmqvist, P.-Å.; Serrano-Andrés, L. Adv. Chem. Phys.
**1996**, 93, 219–331. Finley, J.; Malmqvist, P.-Å.; Roos, B. O.; Serrano-Andrés, L. Chem. Phys. Lett.**1998**, 288, 299–306. - Levy, B.; Berthier, G. Int. J. Quantum Chem.
**1968**, 2, 307–319, (erratum ibid.**1969**, 3, 247). - Dyall, K. G. J. Chem. Phys.
**1995**, 102, 4909–4918. - Mahapatra, U. S.; Datta, B.; Mukherjee, D. Chem. Phys. Lett.
**1999**, 299, 42–50. - Angeli, C.; Cimiraglia, R.; Evangelisti, S.; Leininger, T.; Malrieu, J.-P. J. Chem. Phys.
**2001**, 114, 10252–10264. - Hinze, J. J. Chem. Phys.
**1973**, 59, 6424–6432. - Hirao, K. Chem. Phys. Lett.
**1992**, 196, 397–403. - Kozlowski, P. M.; Davidson, E. R.; Kozlowski, P. M.; Davidson, E. R. J. Chem. Phys.
**1994**, 100, 3672–3682. Kozlowski, P. M.; Davidson, E. R. Chem. Phys. Lett.**1994**, 222, 615–620, 226, 440–446. - Zaitsevskii, A.; Malrieu, J.-P. Chem. Phys. Lett.
**1995**, 233, 597–604. - Andersson, K. Theor. Chim. Acta
**1995**, 91, 31–46. - Bartlett, R. J. (1981). Ann. Rev. Phys. Chem.
**1981**, 32, 359–401. - Sheppard, M. G. J. Chem. Phys.
**1984**, 80, 1225–1229. - Meissner, L.; Jankowski, K. Int. J. Quantum Chem.
**1989**, 36, 705–726. - Pople, J. A.; Binkley, J. S.; Seeger, R. Int. J. Quantum Chem., Quantum Chem. Symp.
**1976**, 10, 1–19. - Lepetit, M. B.; Pélissier, M.; Malrieu, J. P. J. Chem. Phys.
**1988**, 89, 998–1008. - Gill, P. M. W.; Pople, J. A.; Radom, L.; Nobes, R. H. J. Chem. Phys.
**1988**, 89, 7307–7314. - Del Bene, J. E.; Shavitt, I. Int. J. Quantum Chem. Quantum Chem. Symp.
**1989**, 23, 445–452. Del Bene, J. E.; Stahlberg, E. A.; Shavitt, I. ibid.**1990**, 24, 455–466. Del Bene, J. E.; Shavitt, I. J. Mol. Struct. (THEOCHEM)**1991**, 234, 499–508. - Löwdin, P. O. J. Math. Phys.
**1962**, 3, 969–982. - Huron, B.; Malrieu, J. P.; Rancurel, P. J. Chem. Phys.
**1973**, 58, 5745–5759. - Lindgren, I. J. Phys. B
**1974**, 7, 2441–2470. - Kaldor, U. J. Chem. Phys.
**1975**, 63, 2199–2205. Hose, G.; Kaldor, U. J. Phys. B**1979**, 12, 3827–3855. Hose, G. J. Chem. Phys.**1986**, 84, 4505–4518. Hose, G. Many-Body Methods in Quantum Chemistry; Kaldor, U., Ed.; Springer: Berlin, 1989; pp. 43–64. [Google Scholar] - Hegarty, D.; Robb, M. A. Molec. Phys.
**1979**, 37, 1455–1468. Baker, H.; Hegarty, D.; Robb, M. A. ibid.**1980**, 41, 653–668. McDouall, J. J. W.; Peasley, K.; Robb, M. A. Chem. Phys. Lett.**1988**, 148, 183–189. - Sun, H.; Sheppard, M. G.; Freed, K. F. J. Chem. Phys.
**1981**, 74, 6842–6848. Freed, K.F. Many-Body Methods in Quantum Chemistry; Kaldor, U., Ed.; Springer: Berlin, 1989; pp. 1–21. [Google Scholar] - Malrieu, J.-P.; Durand, P.; Daudey, J. P. J. Phys. A
**1985**, 18, 809–826. - Wolinski, K.; Sellers, H. L.; Pulay, P. Chem. Phys. Lett.
**1987**, 140, 225–231. Wolinski, K.; Pulay, P. J. Chem. Phys.**1989**, 90, 3647–3659. - Kucharski, S. A.; Bartlett, R. J. Int. J. Quantum Chem., Quantum Chem. Symp.
**1988**, 22, 383–405. Meissner, L.; Bartlett, R. J. J. Chem. Phys.**1989**, 91, 4800–4808. - Cave, R. J.; Davidson, E. R. J. Chem. Phys.
**1988**, 88, 5770–5778, 89, 6798–6814. Murray, C.; Racine, S. C.; Davidson, E. R. Int. J. Quantum Chem.**1992**, 42, 273–285. - Jeziorski, B.; Moszinski, R.; Rybak, S.; Salewicz, K. Many-Body Methods in Quantum Chemistry; Kaldor, U., Ed.; Springer: Berlin, 1989; pp. 65–94. [Google Scholar]
- Murphy, R. B.; Messmer, R. P. Chem. Phys. Lett.
**1991**, 183, 443–448. Murphy, R. B.; Messmer, R. P. J. Chem. Phys.**1992**, 97, 4170–4184. - Hoffmann, M. R. Chem. Phys. Res.
**1991**, 2, 27–32. Hoffmann, M. R. Chem. Phys. Lett.**1992**, 195, 127–134. - Hirao, K. Chem. Phys. Lett.
**1992**, 190, 374–380. Hirao, K. ibid.**1993**, 201, 59–66. - Angeli, C.; Cimiraglia, R.; Persico, M.; Toniolo, A. Theor. Chem. Acc.
**1997**, 98, 57–63. Angeli, C.; Persico, M. ibid.**1997**, 98, 117–128. - Celani, P.; Werner, H.-J. J. Chem. Phys.
**2000**, 112, 5546–5557. - Grimme, S.; Waletzke, M. Phys. Chem. Chem. Phys.
**2000**, 2, 2075–2081. - Angeli, C.; Cimiraglia, R.; Malrieu, J.-P. Chem. Phys. Lett.
**2001**, 350, 297–305. - Heully, J. L.; Malrieu, J. P.; Zaitsevskii, A. J. Chem. Phys.
**1996**, 105, 6887–6891. - Mahapatra, U. S.; Datta, B.; Mukherjee, D. Mol. Phys.
**1998**, 94, 157–171. - Shavitt, I.; Stahlberg, E. A. (1991). State-specific multireference perturbation theory with Møller–Plesset-like partitioning: second and third order. In presented at the Symposium on Current Methods and Applications in Quantum Chemistry, Youngstown State University, Youngstown, Ohio, September 1991. presented at the 32nd Sanibel Symposium, St. Augustine, Florida, March 1992.
- Stahlberg, E. A. Application of Multireference Based Correlation Methods to the Study of Weak Bonding Interactions. Ph.D. Dissertation, Ohio State University, Columbus, Ohio, 1991. [Google Scholar]
- Shepard, R.; Shavitt, I.; Pitzer, R. M.; Comeau, D. C.; Pepper, M.; Lischka, H.; Szalay, P.; Ahlrichs, R.; Brown, F. B.; Zhao, J.-G. Int. J. Quantum Chem. Quantum Chem. Symp.
**1988**, 22, 149–165. Lischka, H.; Shepard, R.; Pitzer, R. M.; Shavitt, I.; Dallos, M.; Müller, T.; Szalay, P. G.; Seth, M.; Kedziora, G. S.; Yabushita, S.; Zhang, Z. Phys. Chem. Chem. Phys.**2001**, 3, 664–673, For information on the latest version and its availability see the columbus home page at http://www.itc.univie.ac.at/~hans/Columbus/columbus.html. - Lischka, H.; Dallos, M.; Shavitt, I. Unpublished.
- Durand, P.; Malrieu, J.-P. Adv. Chem. Phys.
**1987**, 67, 321–412. Malrieu, J.-P.; Heully, J.-L.; Zaitsevskii, A. J. Phys. A**1985**, 18, 809–826. - Nakano, H.; Yamanishi, M.; Hirao, K. Trends Chem. Phys.
**1997**, 6, 167–214. Hirao, K.; Nakayama, K.; Nakajima, T.; Nakano, H. Computational Chemistry; Lesczinski, J., Ed.; World-Scientific: Singapore, 1999; pp. 227–270. [Google Scholar]

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**MDPI and ACS Style**

Shavitt, I.
Multi-state Multireference Rayleigh–Schrödinger Perturbation Theory for Mixed Electronic States: Second and Third Order. *Int. J. Mol. Sci.* **2002**, *3*, 639-655.
https://doi.org/10.3390/i3060639

**AMA Style**

Shavitt I.
Multi-state Multireference Rayleigh–Schrödinger Perturbation Theory for Mixed Electronic States: Second and Third Order. *International Journal of Molecular Sciences*. 2002; 3(6):639-655.
https://doi.org/10.3390/i3060639

**Chicago/Turabian Style**

Shavitt, Isaiah.
2002. "Multi-state Multireference Rayleigh–Schrödinger Perturbation Theory for Mixed Electronic States: Second and Third Order" *International Journal of Molecular Sciences* 3, no. 6: 639-655.
https://doi.org/10.3390/i3060639