Perturbation method three-orbital systems

Second-order Moller-Plesset perturbation theory (MP2) is the computationally least expensive and most popular ab initio electron correlation method [4,15]. Except for transition metal compounds, MP2 equilibrium geometries are of comparable accuracy to DFT. However, MP2 captures long-range correlation effects (like dispersion) which are lacking in present-day density functionals. The computational cost of MP2 calculations is dominated by the integral transformation from the atomic orbital (AO) to the molecular orbital (MO) basis which scales as 0(N5) with the system size. This four-index transformation can be avoided by introduction of the RI integral approximation which requires just the transformation of three-index quantities and reduces the prefactor without significant loss in accuracy [36,37]. This makes RI-MP2 the most efficient alternative for small- to medium-sized molecular systems for which DFT fails. 

Abstract The periodic orbits play an important role in the study of the stability of a dynamical system. The methods of study of the stability of a periodic orbit are presented both in the general case and for Hamiltonian systems. The Poincare map on a surface of section is presented as a powerful tool in the study of a dynamical system, especially for two or three degrees of freedom. Special attention is given to nearly integrable dynamical systems, because our solar system and the extra solar planetary systems are considered as perturbed Keplerian systems. The continuation of the families of periodic orbits from the unperturbed, integrable, system to the perturbed, nearly integrable system, is studied. 

This inclnsion of electron correlation can be accomplished in several ways. One method has been to nse Mpller-Plesset (MP) pertnrbation theory. This theorized that the electron correlation was a perturbation of the wavefnnction, so the MP perturbation theory conld be applied to the HF wavefnnction to inclnde the electron correlation. As more perturbations are made to the system, more electron correlation is inclnded (these methods are denoted MP2, MP3, and MP4). Another method is to calculate the energy of the system when electrons are moved into vacant orbitals. These methods move electrons either one at a time (single), two at a time (double, such as the QCISD method), or three at a time (triple, such as the QCISDT method). These methods calculate energy values more accurately but at greater computational cost. 

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