Integral approximation, semiempirical molecular orbital theory

By ab initio we refer to quantum chemical methods in which all the integrals of the theory, be it variational or perturbative, are exactly evaluated. The level of theory then refers to the type of theory employed. Common levels of theory would include Hartree-Fock, or molecular orbital theory, configuration interaction (Cl) theory, perturbation theory (PT), coupled-cluster theory (CC, or coupled-perturbed many-electron theory, CPMET), etc. - We will use the word model to designate approximations to the Hamiltonian. For example, the zero differential overlap models can be applied at any level of theory. The distinction between semiempirical and ab initio quantum chemistry is often not clean. Basis sets, for example, are empirical in nature, as are effective core potentials. The search for basis set parameters is not usually considered to render a model empirical, whereas the search for parameters in effective core potentials is so considered. 

Another way of performing calculations using the cluster model is the use of a hybrid method. It is a theoretical method, which uses different approaches for different parts of the molecular system. The ONIOM method is one of the hybrid methods developed quite recently to facilitate accurate ab initio calculations of large chemical species. The ONIOM method (n-layered integrated molecular orbital and molecular mechanics approach) [29] is a multi-level extrapolation method, in which the studied molecular system is divided into two or more parts or layers. The most important part of the system from the chemical point of view (the inner part, IP) is treated at a high" level of theory (the HL method - a high level of ab initio molecular orbital method) and the rest of the system is described by a computationally less demanding method (the LL method - the lowest ab initio approximation or even semiempirical or molecular mechanic approximations) [30]. 

This is a semiempirical all-valence electron quantum mechanical method, apart from the Tr-approximation and the neglect of overlap integrals, as those of Hiickel molecular orbital (HMO) theory. The method reproduces, relatively well, the shapes and the order of the energy levels of molecular orbitals. To consider the overlapping, it is possible to describe the net destabilization caused by the interaction of the two doubly occupied orbitals, the effect of which is not reproduced by HMO theory. 

Semiempirical approaches to quantum chemistry are thus characterized by the use of empirical parameters in a quantum mechanical framework. In this sense, many current methods contain semiempirical features. For example, some high-level at initio treatments of thermochemistry employ empirical corrections for high-order correlation effects, and several advanced density functionals include a substantial number of empirical parameters that are fitted against experimental data. We shall not cover such approaches here, but follow the conventional classification by considering only semiempirical methods that are based on molecular orbital (MO) theory and make use of integral approximations and parameters already at the MO level. 

A qualitative semiempirical valence-electron-only quantum mechanical method which uses the same approximations, apart from TT-approximation and neglect of overlap integrals, as those of the Hiickel molecular orbital (HMO) theory. 

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