Electronic structure methods independent-particle models

If we are interested in describing the electron distribution in detail, there is no substitute for quantum mechanics. Electrons are very light particles and they cannot be described correctly even qualitatively by classical mechanics. We will in this chapter and in Chapter 4 concentrate on solving the time-independent Schrodinger equation, which in shorthand operator form is given in eq. (3.1). 

If solutions are generated without reference to experimental data, the methods are usually called ab initio (latin from the beginning ), in contrast to semi-empirical models, which are described in Section 3.10. 

Introduction to Computational Chemistry, Second Edition. Frank Jensen. 

The HF model is a kind of branching point, where either additional approximations can be invoked, leading to semi-empirical methods, or it can be improved by adding additional determinants, thereby generating models that can be made to converge towards the exact solution of the electronic Schrddinger equation.  

HF theory only accounts for the average electron-electron interactions, and consequently neglects the correlation between electrons. Methods that include electron correlation require a multi-determinant wave function, since HF is the best singledeterminant wave function. Multi-determinant methods are computationally much more involved than the HF model, but can generate results that systematically approach the exact solution of the Schrodinger equation. These methods are described in Chapter 4. 

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Electronic Structure Methods Independent-Particle Models... 

ELECTRONIC STRUCTURE METHODS INDEPENDENT-PARTICLE MODELS... 

The most widely used qualitative model for the explanation of the shapes of molecules is the Valence Shell Electron Pair Repulsion (VSEPR) model of Gillespie and Nyholm (25). The orbital correlation diagrams of Walsh (26) are also used for simple systems for which the qualitative form of the MOs may be deduced from symmetry considerations. Attempts have been made to prove that these two approaches are equivalent (27). But this is impossible since Walsh s Rules refer explicitly to (and only have meaning within) the MO model while the VSEPR method does not refer to (is not confined by) any explicitly-stated model of molecular electronic structure. Thus, any proof that the two approaches are equivalent can only prove, at best, that the two are equivalent at the MO level i.e. that Walsh s Rules are contained in the VSEPR model. Of course, the transformation to localised orbitals of an MO determinant provides a convenient picture of VSEPR rules but the VSEPR method itself depends not on the independent-particle model but on the possibility of separating the total electronic structure of a molecule into more or less autonomous electron pairs which interact as separate entities (28). The localised MO description is merely the simplest such separation the general case is our Eq. (6)... 

Since his appointment at the University of Waterloo, Paldus has fully devoted himself to theoretical and methodological aspects of atomic and molecular electronic structure, while keeping in close contact with actual applications of these methods in computational quantum chemistry. His contributions include the examination of stability conditions and symmetry breaking in the independent particle models,109 many-body perturbation theory and Green s function approaches to the many-electron correlation problem,110 the development of graphical methods for the time-independent many-fermion problem,111 and the development of various algebraic approaches and an exploration of convergence properties of perturbative methods. His most important... 

Most chemists picture the electronic structure of atoms or molecules by invoking orbitals. The orbital concept has its basis in Hartree-Fock theory, which determines the best wavefunction I ) under the approximation that each electron experiences only the average field of the other electrons. This is also called the one-electron, or independent particle model. While the Hartree-Fock method gives very useful results in many situations, it is not always quantitatively or even qualitatively correct. When this approximation fails, it becomes necessary to include the effects of electron correlation one must model the instantaneous electron-electron repulsions present in the molecular Hamiltonian. 

One of the main reasons for the good results obtained with the Hartree-Fock SCF method in electronic structure calculations for atoms and molecules is that the electrons keep away from each other due to the Pauli exclusion principle. This reduces the correlation between them, and provides a basis for the validity of the independent-particle model. The question arises as to the mechanisms that account for the validity of the SCF approximation in the vibrational case, which are obviously quite unrelated to the Pauli principle. 

It was the hope that by the introduction of localized molecular orbitals (LMOs) one can come closer to chemical intuition, to understand the transferability and it will also lead to a convenient study of the electron correlation. The localization of electron density in many atomic system was dealt mainly by the method of the independent particle model. Most of studies refer to closed shell systems, although open shell structures were also investigated. 

Of course, all electrons are the same and any theory of molecular electronic structure which claims to have any validity must reflect this elementary fact whether it uses a single-determinant model or something more advanced. But in the single-determinetnt ( independent particle ) model, while each electron is the same in the sense of charge, mass and laws of motion etc., each electron has its own, different distribution. If we are to have a method of calculating the distributions of the electrons in such a model (whether it be a real distribution as in HF theory or a fictitious one as in KS theory) we are trapped between the two conflicting requirements ... 

As in the Hartree-Fock molecular orbital theory, which is based on the independent particle model, the above Hartree product method also lacks enough correlation among the orbitals, and thereby the resultant accuracy is limited. To overcome the drawback, one can take account of the interaction among possible configurations (or the Hartree products) as in the configuration interaction method and multiconfiguration SCF methods in electronic structure theory. The multiconfigulational time-dependent Hartree... 

Unfortunately, the determination of exact solutions of the SchrOdinger equation is intractable for almost all systems of practical interest. On the other hand, independent particle models are not sufficiently accurate for most studies of molecular structure. In particular, the Hartree-Fock model, which is the best independent particle model in the variational sense, does not support sufficient accuracy for many applications. Some account of electron correlation effects has to be included in the theoretical apparatus which underpins practical computational methods. Although the energy associated with electron correlation is a small fraction of the total energy of an atom or molecule, it is of the same order as most energies of chemical interest. However, such theories may not be true many-body theories. They may contain terms which scale non-linearly with electron number and are therefore unphysical and should be discarded. Any theory which contains such unphysical terms is not acceptable as a true many-body method. Either the theory is abandoned or corrections, such as that of Davidson [7] which is used in limited configuration interaction studies, are made in an attempt to restore linear scaling. 

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