Quasidegenerate perturbation theory

In this section, the spin-orbit interaction is treated in the Breit-Pauli [13,24—26] approximation and incoi porated into the Hamiltonian using quasidegenerate perturbation theory [27]. This approach, which is described in [8], is commonly used in nuclear dynamics and is adequate for molecules containing only atoms with atomic numbers no larger than that of Kr. 

Nakano H (1993) Quasidegenerate perturbation theory with multiconfigurational self-consistent-field reference functions. J Chem Phys 99 7983... 

Nakano H (1993) MCSCF reference quasidegenerate perturbation theory with EpsteinNesbet partitioning. Chem Phys Lett 207 372-378... 

A Study of the Ground State of Manganese Dimer Using Quasidegenerate Perturbation Theory. 

H. Nakano, J. Chem. Phys. 99, 7983 (1993). Quasidegenerate Perturbation Theory with... 

M. R. Hoffmann, Quasidegenerate perturbation theory using effective Harrriltonians, in Modem Electronic Structure Theory, Part II, Volume 2 of Advanced Series in Physical Chemistry, World Scientific, 1995, p. 1166. 

Thus, indeed it may be viewed as a kind of localized function, although the localization condition (1-56) cannot easily be interpreted. Other localization schemes are also possible99- 106. For instance, one could ask that F can be obtained from 0 by the action of the Bloch operator of the quasidegenerate perturbation theory. In this... 

Effective Hamiltonians play a key role in quasidegenerate perturbation theory. If an eigenvalue is degenerate or near-degenerate, the standard perturbation cannot be applied, because zero or small energy denominators would arise. However the first block diagonalization can often be done by perturbation theory, the final diagonalization of Hg/f must be done nonperturbatively. 

We continue this section by pointing out that calculations exist that combine (PT) and MC methods. For example, methods such as complete active space, second order perturbation theory (CASPT2) and multi-configuration quasidegenerate perturbation theory MCODPT use a MC wave function as the reference. PT, generally to second order, is used to estimate the contribution from excited states that arise from excitation outside the original active space of the MC calculation. 

Shavitt and L. T. Redmon, Quasidegenerate Perturbation Theories. A Canonical van Vleck Formalism and Its Relationship to Other Approaches. J. Chem. Phys. 73, 5711-5717 (1980). 

One should simply mention briefly the application of the effective Hamiltonian approaches which use them as technical tools to solve numerically complex problems. The uses of partitioning techniques and of quasidegenerate perturbation theory are especially frequent in solving the configuration-interaction (Cl) problem in molecular physics. 

However, it becomes a little more subtle when one wants to describe a collection of states of a quantum system that are close in energy, or when states with a marked multiconflgurational character have to be described. The reference space is now spanned by several Slater determinants that define a collection of electronic states. Most approaches first diagonalize the reference space and then introduce the effect of the external determinants with perturbation theory. In contrast, quasidegenerate perturbation theory (QDPT), first addresses the external determinants for all the matrix elements among the reference determinants and then diagonalizes the reference space to obtain the energies and wave functions of the states of interest. 

Model core potential (MCP) methods replace core orbitals by a potential just as in ECP. On the other hand, MCP valence orbitals preserve the nodal structure of valence orbitals, unlike ECP valence orbitals. The expectation values of (r ) for the valence orbitals show that the results of MCP are closer to those calculated with all-electron orbitals when comparing MCP, ECP, and the all electron case. Comparisons between MCP and an all electron basis utilizing the full Breit-Pauli spin-orbit Hamiltonian based on multiconfigura-tional quasidegenerate perturbation theory (MCQDPT) calculations show good agreement between the two methods for hydrides of P, As, and Sb. The MCP based spin-orbit calculation appears to be a promising technique, but systematic studies of many different molecular systems are still needed to assess its characteristics and accuracy. 

Below, we discuss specific examples of the one-step and two-step approaches. Although most of these have been applied with pseudopotentials or model potentials, they are by no means dependent on this, and they may also be adapted to all-electron calculations. The discussion is not exhaustive, and among the topics not covered below are the approaches pioneered by Yarkony (1992) (based on quasidegenerate perturbation theory) and Agren et al. (1996) (based on response theory), both of which have found... 

Excited states may be treated using the CIS, TDDFT, multi-configurational quasidegenerate perturbation theory (MCQDPT) (Nakano 1993a, b), and equation-of-motion coupled-cluster (EOM-CC) methodologies. 

Nakano, H. (1993a). MCSCF reference quasidegenerate perturbation theory with Epstein-Nesbet partitioning. The Journal of Chemical Physies, 99, 7983-7992. 

The second theoretical calculation is by Tatewaki et al. [22] They used four-component relativistic multiconfigurational quasidegenerate perturbation theory to take account of correlation effects between the active core [Ce + 4s. ..5p )F (2s 2p )] and three valence electrons in (4/), (5d), (6s), (6p), and intra-valence correlation effects. For the energy levels of the two excited states they obtained 0.144 eV and 0.351 eV. They did not reach conclusions about the higher excited states because of the limitations of the theory they used. A further study of the CeF excited states using larger RASCI and the/-shell Omega decomposition method (which is described below) is under way. 

H. Moriyama, Y. Watanabe, H. Nakano, and H. Tatewaki, Electronic structure of LaF+ and LaF from frozen-core four-component relativistic multiconfigurational quasidegenerate perturbation Theory, J. Phys. Chem. A112, 2683-2692 (2008). 

Zaitsevskii A, JP Mahieu. Multi-partitioning quasidegenerate perturbation theory. A new approach to multireference MpUer-Plesset perturbation theory. Chem Phys Lett. 1995 233 597. J Finley, P-A Malmqvist, BO Roos, Serrano-Andres L. The multi-state CASPT2 method. Chem Phys Lett. 1998 288 299. 

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