Perturbation theory multireference

As an alternative for the variational methods, one can also apply multireference perturbation theory (MRPT) to calculate magnetic interactions. In principle, this type of calculations makes it possible to treat larger systems with more unpaired electrons. Among the many different implementations, two schemes are especially popular in the field of magnetic interactions CASPT2 [13] and NEVPT2 [14, 15], which will be shortly overviewed here. 

The standard Mpller-Plesset perturbation theory uses a single determinant reference wave function and defines the zeroth-order Hamiltonian as the sum of the Fock operators 

By Koopmans theorem, the eigenvalues of the Fock operator applied on occupied orbitals are proportional to ionization potentials and the eigenvalues corresponding to unoccupied orbitals are related to the electron affinities. CASPT2 extends the applicability to multireference cases by defining an effective one-electron Fock-type operator as zeroth order Hamiltonian in the following way 

The structure of the matrix is schematically presented in Fig. 4.11. The inactive-virtual block of the matrix is given by 

In this case, the first and third term are again zero for the same reason as exposed above. However, the operators in the second term annihilate an electron in the bra and the ket, which result in N — 1 H A - 1), the energy of the ionized system. 

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E. Rosta and P. R. Surjan, Two-body zeroth order Hamiltonians in multireference perturbation theory the APSG reference state. J. Chem. Phys. 116(3), 878-890 (2002). 

An appealing feature of multireference perturbation theory is that it can correct for some deficiencies associated with an incomplete active space. For instance, the relative energies... 

One area in which Cl methods appear to be thoroughly superior to perturbation theory is in the treatment of multireference problems problems with substantial nondynamical correlation effects. Even UHF-based single reference perturbation theory methods may not cope with some such situations, and multireference perturbation theory, despite many efforts over the years, still appears to be far from developing a general flexible approach that is competitive with MRCI. Transition-metal chemistry, in particular, is a graveyard for UHF-based MP methods. 

One way of achieving size-consistency for a dissociation process is to use an MCSCF wave function as the reference. Unfortunately, as noted above, there are as yet no general multireference perturbation theory or multireference coupled-cluster treatments that can be applied to such an MCSCF reference function. For rather few electrons, as we shall see, the MRCI approach performs acceptably. 

The exclusion of a given correlation treatment from this survey should not necessarily be taken as a condemnation. Constraints of tiipe and space preclude the inclusion of all treatments in current use I have simply attempted to cover as many as possible while concentrating on those with which I am most familiar. I stand by my assertion that there is no generally useful multireference perturbation theory or coupled-cluster... 

EXCITED STATES IN METAL OXIDES BY CONFIGURATION INTERACTION AND MULTIREFERENCE PERTURBATION THEORY... 

Excited states in metal oxides by configuration interaction and multireference perturbation theory 227... 

P. M. Kozlowski, M. Dupuis, and E. R. Davidson,/. Am. Chem. Soc., 117, 774 (1995). The Cope Rearrangement Revisited with Multireference Perturbation-Theory. 

RELATIVISTIC MULTIREFERENCE PERTURBATION THEORY COMPLETE ACTIVE-SPACE SECOND-ORDER PERTURBATION THEORY (CASPT2) WITH THE FOUR-COMPONENT DIRAC HAMILTONIAN... 

Keywords Relativity, Four-Component, Electron Correlation, Multireference Perturbation Theory,... 

H.J. Werner, Third-order multireference perturbation theory— The CASPT3 method. Mol Phys. 89 (2) (1996) 645-661. 

P. Celani, H.J. Werner, Multireference perturbation theory for large restricted and selected active... 

Multireference Formalisms. - Whilst the generalization of MPn theory1111 -and, in particular, because of its efficiency, MP2 theory -is obviously an important requirement if many-body perturbation theory is to be applied to bond breaking processes, radicals, excited states and the like where a multireference formalism is mandated, a robust theory that be applied routinely to a wide range of problems has been elusive for over 25 years (see, for example, the discussion of the problems associated with multireference perturbation theory in my monograph53 Electron correlation in molecules published in 1984). 

When MCSCF wavefunctions are used as the reference, the most commonly used methods for recovering the electron correlation are multireference configuration interaction (MRCI) (15) and multireference perturbation theory (MRPT)... 

Maranzana, A., Ghigo, G., Tonachini, G., Diradical and Peroxirane Pathways in the [ji2 + Ji2] Cycloaddition Reactions of 1 Ag Dioxygen with Ethene, Methyl Vinyl Ether, and Butadiene a Density Functional and Multireference Perturbation Theory Study, J. Am. Chem. Soc. 2000, 122, 1414 1423. 

In multireference perturbation theory, defining a proper zero-order Hamiltonian is anything but straightforward. The reference wavefunction, in general, is not an eigenfunction of the zero-order Hamiltonian. A second complication arises as interactions between the FOIS functions and zero-order wavefunction through the zero-order Hamiltonian cannot be excluded. Therefore, projection techniques are commonly employed. In NEVPT2, the zero-order Hamiltonian takes the form... 

In many multireference perturbation theories, the energy spectrum computed with the approximated Hamiltonian is ill conditioned. Functions outside the reference space become artificially degenerate with reference wavefunction. The phenomenon is known in the literature as intruder state, and it results in unphysical energy corrections and spurious bumps along a potential energy surface. Several schemes, such as ad hoc level shift parameter, have been proposed [97-100], yet... 

MULTIREFERENCE PERTURBATION THEORY AND VALENCE BOND DESCRIPTION OF ELECTRONIC STRUCTURES OF MOLECULES... 

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