Hartree-Fock method perturbation theory based

In the section that follows this introduction, the fundamentals of the quantum mechanics of molecules are presented first that is, the localized side of Fig. 1.1 is examined, basing the discussion on that of Levine (1983), a standard quantum-chemistry text. Details of the calculation of molecular wave functions using the standard Hartree-Fock methods are then discussed, drawing upon Schaefer (1972), Szabo and Ostlund (1989), and Hehre et al. (1986), particularly in the discussion of the agreement between calculated versus experimental properties as a function of the size of the expansion basis set. Improvements on the Hartree-Fock wave function using configuration-interaction (Cl) or many-body perturbation theory (MBPT), evaluation of properties from Hartree-Fock wave functions, and approximate Hartree-Fock methods are then discussed. 

It is an attractive feature of ab initio wave function theory that there is a clear hierarchy of methods leading from Hartree-Fock to the exact solution of the Schrodinger equation. Post-Hartree-Fock methods can be divided into three main categories [88]. The first is based on (Mqller-Plesset) perturbation theory [89] and referred to as MPn where n is the order of the perturbation. MPn is excellent when Hartree-Fock already is giving a reasonable description, as is often the case for complexes involving Ad and 5d elements. Otherwise, it fails or might only converge slowly with the order n. MP2 can be used for medium size systems of 100-200 atoms. 

SM calculations are broadly based on either the (i) Hartree-Fock method (ii) Post-Hartree-Fock methods like the Mpller-Plesset level of theory (MP), configuration interaction (Cl), complete active space self-consistent field (CASSCF), coupled cluster singles and doubles (CCSD) or (iii) methods based on DFT [24-27]. Since the inclusion of electron correlation is vital to obtain an accurate description of nearly all the calculated properties, it is desirable that SM calculations are carried out at either the second-order Mpller-Plesset (MP2) or the coupled cluster with single, double, and perturbative triple substitutions (CCSD(T)) levels using basis sets composed of both diffuse and polarization functions. 

This is not the place for a full overview of the wave function based post Hartree-Fock methods currently applied for the calculation of intermolecular interactions and in particular molecule/surface interactions. Table 3 contains a brief characterization of the most widely applied schemes. The two most popular methods are MP2 (second order Moller-Plesset perturbation theory), because it covers large part of electronic correlation at comparably low ex-... 

However, first it should be stressed that within density-functional theory, in its most widespread formulation, it provides a method for calculating static groimd-state properties of a given atomic, molecular, crystalline, liquid,. .. system. This is the field where density-fimctional theory has found most applications and where the experience has shown that the results in by far the most cases are accurate and useftd. Typically, structural, energetical, and vibrational properties are as accurate as those obtained with Hartree-Fock methods augmented with perturbational inclusion of correlation effects at the MP2 level although there are exceptions (both for the case that the density-functional calculations are inaccurate and for the case that the Hartree-Fock-based calculations fail). In this report we have not attempted to review all the studies supporting this conclusion. Therefore, it is veiy important at this point to emphasize the importance and accuracy of such studies. 

Sometimes, as we know, the method fails and then the perturbation theory based on the Hartree-Fock starting point is a risky business, since the perturbation is very large. 

Although a wide variety of theoretical methods is available to study weak noncovalent interactions such as hydrogen bonding or dispersion forces between molecules (and/or atoms), this chapter focuses on size consistent electronic structure techniques likely to be employed by researchers new to the field of computational chemistry. Not stuprisingly, the list of popular electronic structure techniques includes the self-consistent field (SCF) Hartree-Fock method as well as popular implementations of density functional theory (DFT). However, correlated wave function theory (WFT) methods are often required to obtain accmate structures and energetics for weakly bound clusters, and the most useful of these WFT techniques tend to be based on many-body perturbation theory (MBPT) (specifically, Moller-Plesset perturbation theory), quadratic configuration interaction (QCI) theory, and coupled-cluster (CC) theory. 

QCISD(T) = quadratic Cl including single, double, and triple excitations, UCCD(ST) = coupled-cluster doubles method based on the unrestricted Hartree-Fock method and corrected for single and triple replacements, MC SCF = multiconfiguration SCF, MRD Cl = multireference singles and doubles Cl, MBPT= many-body perturbation theory, SD Cl = singles and doubles Cl. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

Recently, quantum chemical computational techniques, such as density functional theory (DFT), have been used to study the electrode interface. Other methods ab initio methods based on Hartree-Fock (HF) theory,65 such as Mollcr-PIcsset perturbation theory,66,67 have also been used. However, DFT is much more computationally efficient than HF methods and sufficiently accurate for many applications. Use of highly accurate configuration interaction (Cl) and coupled cluster (CC) methods is prohibited by their immense computational requirements.68 Advances in computing capabilities and the availability of commercial software packages have resulted in widespread application of DFT to catalysis. 

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