Hartree-Fock approximation perturbed energy

Configuration interaction (Cl) is a systematic procedure for going beyond the Hartree-Fock approximation. A different systematic approach for finding the correlation energy is perturbation theory... 

Another distinguishing aspect of MO methods is the extent to which they deal with electron correlation. The Hartree-Fock approximation does not deal with correlation between individual electrons, and the results are expected to be in error because of this, giving energies above the exact energy. MO methods that include electron correlation have been developed. The calculations are usually done using MoUer-Plesset perturbation theoiy and are designated MP calculations." ... 

Practical calculations require approximations in the self-energy operator. Perturbative improvements to Hartree-Fock, canonical orbital energies can be generated efficiently by neglecting off-diagonal matrix elements of the selfenergy operator in this basis. Such diagonal, or quasiparticle, approximations simplify the Dyson equation to the form... 

This equation is the main result of the present considerations. In order to define the two-particle self energy (w) and for establishing the connection to the familiar form of Dyson s equation we adopt a perturbation theoretical view where a convenient single-particle description (e. g. the Hartree-Fock approximation) defines the zeroth order. We will see later that the coupling blocks and vanish in a single-particle approximation. Consequently the extended Green s function is the proper resolvent of the zeroth order primary block which can be understood as an operator in the physical two-particle space ... 

Since the total electronic energy from the Hartree-Fock approximation is the starting point in the perturbation calculation, we see from (5.22) that it is convenient to consider the operator... 

Since the zeroth-order equations are solved only in the Hartree-Fock approximation, the perturbation corrections account not only for intermolec-ular interactions, but for the intramolecular correlation energy as well. These two effects cannot be separated in the Lowdin basis set, but one may subtract the contributions of those [ij kl] integrals which result in local correlation. 

For Inter Molecular Perturbation Theory (IMPT) see Hayes, I. C. Stone, A. J. An intermolecular perturbation theory for the region of moderate overlap, Mol. Phys. 1984, 53, 83-105 papers of this kind, however, contain a large amount of theoretical and mathematical detail and are not transparent to the uninitiated. For Symmetry-Adapted Perturbation Theory (SAPT) see e.g. Bukowski, R. Szalewicz, K. Chabalovski, C. F. Ab initio interaction potentials for simulations of dinitramine solutions in supercritical carbon dioxide with cosolvents, J. Phys. Chem. 1999, A103, 7322-7340, and references therein. The Morokuma decomposition scheme is described in Kitaura, K. Morokuma, K. A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation, Int. J. Quantum Chem. 1976,10, 325-340. 

In all cases where the question concerning the relative stabilities of equidistant versus bond alternating structures arises (polyyne [20,21, polyacetylene 22-27, polymethineimine 28,29 ) the latter are more stable within the framework of the restricted Hartree Fock approximation. For polyyne and polyacetylene this issue is in accord with the well known concept of a Peierls distortion jsoj. The occurence of Hartree Fock instabilities (see e.g. refs. 31,32 ) in the case of the equidistant, metallic structures of polyyne (cumulene) and all-trans polyacetylene points, however, to the need for improved methods going beyond the independent particle model. First efforts in this direction 27 show that at the level of second order Moller-Plesset perturbation theory the alternant configuration of polyacetylene is still preferred energetically although as expected the energy difference to the equidistant structures becomes smaller. 

A Perturbation Theory is developed for treating a system of n electrons in which the Hartree-Fock solution appears as the zero-order approximation. It is shown by this development that the first order correction for the energy and the charge density of the system is zero. The expression for the second order correction for the energy greatly simplifies because of the special property of the zero order solution. It is pointed out that the development of the higher order approximation involves only calculations based on a definite one-body problem. 

The difference between the Hartree-Fock energy and the exact solution of the Schrodinger equation (Figure 60), the so-called correlation energy, can be calculated approximately within the Hartree-Fock theory by the configuration interaction method (Cl) or by a perturbation theoretical approach (Mpller-Plesset perturbation calculation wth order, MPn). Within a Cl calculation the wave function is composed of a linear combination of different Slater determinants. Excited-state Slater determinants are then generated by exciting electrons from the filled SCF orbitals to the virtual ones ... 

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