Self-consistent field , evaluation

In this paper a method [11], which allows for an a priori BSSE removal at the SCF level, is for the first time applied to interaction densities studies. This computational protocol which has been called SCF-MI (Self-Consistent Field for Molecular Interactions) to highlight its relationship to the standard Roothaan equations and its special usefulness in the evaluation of molecular interactions, has recently been successfully used [11-13] for evaluating Eint in a number of intermolecular complexes. Comparison of standard SCF interaction densities with those obtained from the SCF-MI approach should shed light on the effects of BSSE removal. Such effects may then be compared with those deriving from the introduction of Coulomb correlation corrections. To this aim, we adopt a variational perturbative valence bond (VB) approach that uses orbitals derived from the SCF-MI step and thus maintains a BSSE-free picture. Finally, no bias should be introduced in our study by the particular approach chosen to analyze the observed charge density rearrangements. Therefore, not a model but a theory which is firmly rooted in Quantum Mechanics, applied directly to the electron density p and giving quantitative answers, is to be adopted. Bader s Quantum Theory of Atoms in Molecules (QTAM) [14, 15] meets nicely all these requirements. Such a theory has also been recently applied to molecular crystals as a valid tool to rationalize and quantitatively detect crystal field effects on the molecular densities [16-18]. 

The obvious disadvantage of this simple LG model is the necessity to cut off the infinite expansion (26) at some order, while no rigorous justification of doing that can be found. In addition, evaluation of the vertex function for all possible zero combinations of the reciprocal wave vectors becomes very awkward for low symmetries. Instead of evaluating the partition function in the saddle point, the minimization of the free energy can be done within the self-consistent field theory (SCFT) [38 -1]. Using the integral representation of the delta functionals, the total partition function, Z [Eq. (22)], can be written as... 

Although the MEG model is essentially ionic in nature, it may also be used to evaluate interactions in partially covalent compounds by appropriate choices of the wave functions representing interacting species. This has been exemplified by Tossell (1985) in a comparative study in which MEG treatment was coupled with an ab initio Self Consistent Field-Molecular Orbital procedure. In this way, Tossell (1985) evaluated the interaction of C03 with Mg in magnesite (MgC03). Representing the ion by a 4-31G wave function, holding its... 

The Section on More Quantitive Aspects of Electronic Structure Calculations introduces many of the computational chemistry methods that are used to quantitatively evaluate molecular orbital and configuration mixing amplitudes. The Hartree-Fock self-consistent field (SCF), configuration interaction (Cl), multiconfigurational SCF (MCSCF), many-body and Mpller-Plesset perturbation theories,... 

Hiickel and extended Huckel methods are termed semi-empirical because they rely on experimental data for the quantification of parameters. There are other semi-empirical methods, such as CNDO, MINDO, INDO, in which experimental data are still used, but more care is taken in evaluating the Htj. These methods are self-consistent field procedures based on 3 SCF. They are discussed in various works on molecular orbital theory.4... 

In many cases more refined molecular orbital models give a better agreement between theory and experiment. Self-consistent field- 80 81> or CNDO-82> calculations as well as other ab initio calculations 83>84) were performed and the results of several different approaches for phosphorous compounds were critically evaluated by M. Pelavin 77). For the nitrogen compounds two different linear relationships, one for cations and for neutral molecules, the other for anoins, were observed 82), a phenomenon which might be explained with a Madelung contribution. 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

The initial Hiickel calculations can be employed to obtain preliminary values for the electron densities and bond orders, from which the self-consistent field matrix elements can be evaluated by introduction of the chosen core potentials and electron repulsion integrals.11 Table I lists the ionization potentials, electron affinities and nuclear charges employed in the present calculations. 

Julg et al.24 employed an approximation based on the self consistent field molecular orbital method to evaluate the average energy per atom for various structures. They calculate that whereas the normal b.c.c. structure is more stable for clusters containing more than 106 atoms, smaller clusters prefer to take up pentagonal symmetry. However, these authors make an important point, namely, that the calculated energies for different structures are very similar. Interconversion of different structures will be facile, and external factors such as the method of deposition, level of impurities, support effects, etc., may cause the less stable structure to grow. For example, impurities on... 

In principle, this kind of scheme may be carried out for any molecule, with any number of electrons and any number of atomic orbitals y in the LCAO basis set. The practical calculation, however, involves the tedious evaluation of a large number of integrals, a number which increases so rapidly with the number of electrons that, for large molecules, complete self-consistent field calculations are not really feasible on a large scale. 

This review is necessarily selective, and is divided into several sections. Initially we give an overview of the availability of supercomputers in the U.K., and summarise the experience gained in the implementation of various quantum chemistry packages. Optimization of Quantum Chemistry codes on the CRAY-1 is considered, with integral evaluation, self-consistent-field and integral transformation phases of quantum chemical studies being considered together with some aspects of the correlation problem. 

Ab initio MO computer programmes use the quantum-chemical Hartree-Fock self-consistent-field procedure in Roothaan s LCAO-MO formalism188 and apply Gaussian-type basis functions instead of Slater-type atomic functions. To correct for the deficiencies of Gaussian functions, which are, for s-electrons, curved at the nucleus and fall off too fast with exp( —ar2), at least three different Gaussian functions are needed to approximate one atomic Slater s-function, which has a cusp at the nucleus and falls off with exp(— r). But the evaluation of two-electron repulsion integrals between atomic functions located at one to four different centres is mathematically much simpler for Gaussian functions than for Slater functions. 

Any molecular calculation starts with the evaluation of integrals over the basis functions. This is usually, but not necessarily, followed by a self-consistent field calculation and a transformation from integrals over atomic basis functions to integrals over molecular orbitals. Full details of these particular phases of calculation are well documented elsewhere and we do not consider them further here. 

In order to proceed we first need to know the atomic basis functions from which we can construct the symmetry orbitals and how to evaluate the one-centre and multi-centre integrals of H and S. Finally one has to find an efficient and accurate way to describe the molecular potential in the self-consistent-field calculations. 

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