Electron interacting terms

Hamiltonians equivalent to (1) have been used by many authors for the consideration of a wide variety of problems which relate to the interaction of electrons or excitons with the locaJ environment in solids [22-25]. The model with a Hamiltonian containing the terms describing the interaction between excitons or electrons also allows for the use of NDCPA. For example, the Hamiltonian (1) in which the electron-electron interaction terms axe taken into account becomes equivalent to the Hamiltonians (for instance, of Holstein type) of some theories of superconductivity [26-28]. 

Imagine a model hydrogen molecule with non-interacting electrons, such that their Coulomb repulsion is zero. Each electron in our model still has kinetic energy and is still attracted to both nuclei, but the electron motions are completely independent of each other because the electron-electron interaction term is zero. We would, therefore, expect that the electronic wavefunction for the pair of electrons would be a product of the wavefunctions for two independent electrons in H2+ (Figure 4.1), which I will write X(rO and F(r2). Thus X(ri) and T(r2) are molecular orbitals which describe independently the two electrons in our non-interacting electron model. 

The main difficulty in solving the Schrodinger equation (Eq. II. 1) for a many-electron system comes from the two-electron interaction terms... 

The individual terms, e , in Eq. (2.27) are termed one-electron orbital energies and correspond to the ionization potential (-e ) of an electron in MO i r assuming that no reorganization of the core nuclei, or the other (2N — 1) electrons, takes place during ionization. The total energy of the system [Eq. (2.27)] is clearly not the sum of the one-electron energies because electron-electron interaction terms are included twice. 

The Electronic Interaction Term. A question raised earlier was is the magnitude of J sufficiently small to justify the weak-interaction assumption This is a question which can be answered, at least for mixed-valence complexes, by interpretation of the oscillator strength of the intervalence band. In Table III, some data (24, 25) for the intervalence band of the... 

The magnitude of the electronic interaction term, J, is also critical in determining the degree of electronic non-adia-baticity [such as Newton has been discussing] that may be present. The electronic transmission coefficient, K, can be expressed as (21) ... 

This is referred to as the Neglect of Diatomic Differential Overlap or NDDO approximation. It reduces the number of electron-electron interaction terms from 0(N ) in the Roothaan-Hall equations to 0(N ), where N is the total number of basis functions. 

Because of the electron interaction terms, the matrix elements in this approximation are themselves a function of the coefficients of the atomic orbitals. Initially, therefore, a reasonable guess is made as to the electronic distribution in the molecule and the calculation carried out until... 

With R > RC the Coulomb repulsion of the electrons when they are on the same ion is the dominant electron-interaction term, so that the lowest states correspond to an exact number of electrons on each ion rather than to running waves. Although the mutual repulsion of electrons on the same ion prevents the permanent occupation of... 

P and S. Note that states with different values of /. or. S have different energies, partly because of the electron-electron interaction term in the electronic Hamiltonian. A further symmetry classification that should be mentioned is the parity of an atomic state which depends on the behaviour of the total wave function under space-fixed inversion. This is either even (g) or odd (u) and is determined by, summed over all the electrons in the atom. 

Here the 7 s are core-valence electron interaction terms, of which more will be said later, and nj is the number of valence electrons /. The first term gives the energy shift produced by chemical effects on the parent atom and the second refers to charge changes on the other lattice sites associated with the flow of charge... 

In order to take into account the two-electron interaction terms we define a supermatrix notation that we illustrate for The components of a supervector t are the matrix elements t ij in dictionary order. Supervectors are transformed by supermatrices. We need the supermatrix V, defined by... 

The site term H- consists of the chemical-potential term He the electron-electron interaction term, the phononic term and the electron-phonon... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

Note that here and later on r denotes the single-particle coordinate whereas R is still used as abbreviation for all nuclear positions as in Eq. (1). The potential (5) consists, on one hand, of an external potential V(r,R), which in our case is time-dependent owing to the atomic motion R( ). On the other hand, there are electron-electron interaction terms, namely the Hartree and the exchange-correlation term, which depend both via the density p on the functions tpj. The exchange-correlation potential VIC is defined within the so-called adiabatic local density approximation [25] which is the natural extension of the lda from stationary dpt. It is assumed to give reliable results for problems where the time scale of the external potential (in our case typical collision times) is larger than the electronic time scale. 

The exchange-correlation potential coming from the non-local part of the electron-electron interaction term is ... 

These considerations are, of course, really bound up with the Hiickel theory s failure to deal with the electron-interaction terms which arise— l/rfJ for each pair of electrons, i and j. In the Hiickel scheme, an attempt is made to average such terms in order to obtain the effective, one-electron Hamiltonian, effective ( 2.2). It is during this averaging process that the singlet-triplet distinctions which we have been describing in this subsection are lost. 

The derivational hierarchy of Figure 1 does not explicitly indicate any standard MO-theoretic model, though this can be rectified through the use of more elaborate many-body models, of the general PPP-Hubbard type [30,31 ], all defined on the same space as for H o. and still most simply dependent solely on the system graph G. Of these the first is the Hubbard model which is the sum of H o and a second electron-electron interaction term... 

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