Perturbation theory, Moeller-Plesset

The exact FCI (frill configuration interaction) solution of the PPP or Hubbard model is possible for molecules with up to about 16 atoms in the pi system. Any of the standard methods for performing approximate ab initio calculations, such as limited configuration interaction, Moeller-Plesset perturbation theory, or coupled cluster theory, may be applied to these models as well. All are expected to be very accurate at low order when U is small, but all will have to be pushed to higher order as U increases. 

Three other approaches towards the problem of incorporating electron correlation should be mentioned. The first is Moeller-Plesset perturbation theory, a method first introduced in 1934 by Moeller and Plesset [35]. Suppose that a perturbed Hamiltonian is defined by... 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

The data are obtained from large basis set calculations (see Table 7.6) which include electron correlation (second-order Moeller-Plesset perturbation theory) and which are corrected for basis set superposition error (BSSE), (Boys and Bernardi 1970). While the results for the stronger hydrogen bonds are qualitatively the same with or without the inclusion of correlation, the weaker interactions require the better description of the wave function. 

In another study of the polarizability and hyperpolarizability of the Si atom Maroulis and Pouchan6 used the finite field method with correlation effects estimated through Moeller-Plesset perturbation theory. Correlation effects are found to be small. 

The most commonly used method is the Hartree-Fock calculation in which interactions between electrons are treated as the interaction of one electron within an average field of the remaining electrons. Electron interactions are, of course, much more specific than this, and include Pauli repulsions as well as electrostatic ones. Electron correlation can be addressed by various methods, but among the most commonly used are configuration interaction and Moeller-Plesset perturbation theory. 

Moeller-Plesset Perturbation Theory. After having computed the quasi one-electron orbitals q>j and quasi one electron energies e,- one can apply (following Rice and Handy113,114) the MP/2 expression of the second order correlation correction of the quasi total energy for given , a), t, Est and 0. In this case the Moeller-Plesset perturbation will be... 

P. Pulay and S. Saebo, Theor. Chim. Acta, 69, 357 (1986). Orbital-invariant Formulation and Second-Order Gradient Evaluation in Moeller-Plesset Perturbation Theory. 

As first step second order Moeller-Plesset perturbation theory (MP2) /20/ has been applied for different chains with small unit cells (see point 3). The MP2 correction to the Hartree-Fock energy can be written with the help of (15) in the simple form, ... 

A Moeller-Plesset Cl correction to v / is based on perturbation theory, by which the Hamiltonian is expressed as a Hartree-Fock Hamiltonian perturbed by a small perturbation operator P through a minimization constant X... 

MBPT(2) stands for second-order many-body perturbation theory, which is also known by the Hamiltonian partitioning scheme it employs, Moeller-Plesset (see references 68 and 69). 

In Moeller-Plesset theory, the mixing in of excited states is treated as a series of perturbations with designations MPn (usually MP2, MP3, MP4), where n designates the point at which the series is truncated. Moeller-Plesset theory is less laborious than Cl, and thus has displaced the latter method in most ab initio calculations, where the computational labor is already high. 

We can apply for the diagonal elements of the self-energy matrix, X(ft>,) in the Moeller-Plesset (MP) many body perturbation theory (MBPT) in the second order (MP2) approximation... 

WFT treatment such as Moeller-Plesset second-order perturbation theory (MP2) or coupled-cluster with single and double excitations (CCSD(T)) with correlation energy [51]. This approximation works fairly well for large interfragment distances but is obscured for shorter bonds by exchange and intrafragment correlation effects. 

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