MR-MBPT

Just as single reference Cl can be extended to MRCI, it is also possible to use perturbation methods with a multi-detenninant reference wave function. Formulating MR-MBPT methods, however, is not straightforward. The main problem here is similar to that of ROMP methods, the choice of the unperturbed Hamilton operator. Several different choices are possible, which will give different answers when the tlieory is carried out only to low order. Nevertheless, there are now several different implementations of MP2 type expansions based on a CASSCF reference, denoted CASMP2 or CASPT2. Experience of their performance is still somewhat limited. 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

Third-order MR MBPT calculation taken from [72]. 

MR-MBPT methods, however, is not straightforward. The main problem here is similar With the coupled cluster wave function (4.46) the Schrodinger equation becomes... 

The need for the inclusion of higher-order effects increases with the degree of quasidegeneracy of the state considered. For this reason, much effort has been devoted to the formulation of the so-called MR MBPT [28-30]. Here, however, a number of ambiguities arises, which often limits the development of practical algorithms (c/, e.g. attempts to extend the so-called CAS-PT2 method, which is based on the complete active space self-consistent field (CAS SCF) reference, to higher than the second order). In fact, we shall see that the same problem manifests itself, even when extending the standard SR CC theory to the MR case. 

We should mention here that the historically oldest MR many-body theories using 7/gff have been the MR perturbation theories. Successful implementation of all the traditional multi-reference many-body perturbation theories (MR-MBPT), which were developed within the H ff framework [6], was mainly confined to calculation of energy differences of spectroscopic interest, and not to the smdy of PES. 

MR-MBPT multi-reference many-body perturbation theory MR-MPPT multi-reference Mpller-Plesset perturbation theory CCSD single-reference coupled cluster with single and double replacements... 

The MC-QDPT(2) theory starts with fixed coeflicients for the model space functions, unlike in our SS-MRPT theories. It is also not fully extensive. We also present the second order SR-MBPT and the effective hamiltoni in based MR-MBPT results, since we want to investigate to what extent the SR-MBPT results behave poorly at the quasi-degenerate regions, and the MR-MBPT results sense the presence of intruders. Hq in both SR-MBPT and MR-MBPT is taken as in the standard EN partitioning, which is structurally closer to our choice for Hq the two SS-MRPT methods. 

Apart from specifying the reference (P) space, the only variability in this and all MR-MBPT methods lies in the choice of orbitals, orbital energies, and the definition of the zeroth order Hamiltonian Hq since the perturbation approximation is completely determined by these choices. The zeroth order Hamiltonian (i.e., the partitioning of the exact Hamiltonian into Ho and V), may, in principle, be specified at our disposal, but, in practice, this choice strongly affects the perturbative convergence. Generally, the zeroth order Hamiltonian is prescribed as a sum of one-electron operators. 

In some points of the previous analysis we have used formulations based on the Hartree-Fock (HF) expression of the quantum problem, mainly for simplicity of exposition. As a matter of fact, there are no formal reasons to limit continuum solvent approach to the HF level. Actually, PCM solvation procedures have been extended to MCSCF (Aguilar et al., 1993b), Cl (Persico and Tomasi, 1984), MBPT (Olivares del Valle et al., 1991, 1993), CASSCF, MR-SDCI (Aguilar et al., 1993b), DFT (Fortunelli and Tomasi, 1994) levels of the quantum description. The other continuum solvation methods have, at least in principle, the same flexibility in the definition of the quantum theory level to be used in computations. 

It is well known that for CS and HS OS systems the widely employed second order MBPT [often referred to as the M0ller-Plesset PT (MP2) when a RHF or UHF operator is used as the unperturbed Hamiltonian (14)] provides computationally the cheapest, yet reasonably reliable, results. Surprisingly enough, no such method exists for the low spin OSS state mentioned above, as far as we know. This is likely due to the fact that the zero order wave function describing these states involves two Slater determinants and thus cannot be handled by the conventional MBPT relying on the spin orbital formalism. At the same time, in the MR theory... 

Polarizable Continuum Model (PCM) This method was developed by Tomasi s group in 1981 and many applications have been proposed [2]. The most distinctive feature of this method is to be able to treat a molecular shaped cavity. Applications not only to Hartree-Fock methods, but to UHF, MCSCF, MBPT, CASSCF, MR-SDCI and DFT etc. have been reported. They also proposed a extension to nonequilibrium solvation problems. The basic concept of their method is that the reaction potential may be described in terms of an apparent charge distribution on the cavity boundary s surface. The charge distributions a and the potential from them can be evaluated as... 

In Table I we compare the full Cl correlation energies with those obtained from single-reference type treatments which aim for size consistency, such as the Davidson corrected CI(SD) (and the CI(SD) itself, of course), the MBPT(2) and MBPT(4), ° the CCSD, the symmetry-adapted cluster methods S AC-A and SAC-B and the CPF methods. (We have not included MR-CI(SD)... 

The presented quasiparticle framework offers a simple way to develop the MR versions of the single-refer-ence-based correlation methods, such as the Cl or the CC approaches [37], To introduce the MR variant of the MBPT [6], first the Hamiltonian should be separated into a zeroth-oder and a perturbation part, H = + V. Following the... 

BWPT = Brillouin-Wigner perturbation theory EN = Epstein-Nesbet FCI = full Cl MBPT = many-body perturbation theory MRS PT = multireference state perturbation theory PT = perturbation theory RSPT = Rayleigh-SchrSdinger perturbation theory SRS PT = single-reference state perturbation theory. 

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