The Non-Interacting Electron Model

In order to investigate whether the wavefunction can indeed be written in this way, we use the separation of variables technique and so write a wavefunction of the form 

We then substitute this wavefunction into the electronic Schrodinger equation, and study the consequences. Do the substitution yourself, divide either side by 

Each term on the left-hand side separately involves the coordinates of one of the two electrons, and as the sum has to be a constant for all values of the coordinates of these two electrons, the terms must individually be constants which I can write 

You are probably used to this idea from descriptive chemistry, where we build up the configurations for many-electron atoms in terms of atomic wavefunctions, and where we would write an electronic configuration for Ne as 

The orbital model would be exact were the electron repulsion terms negligible or equal to a constant. Even if they were negligible, we would have to solve an electronic Schrodinger equation appropriate to CioHs in order to make progress with the solution of the electronic Schrodinger equation for naphthalene. Every molecular problem would be different. 

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Figure 1. Evolution of the radial electron density for argon from each of the three limits described in Table 1. The limits (top row) are those given by the Thomas-Fermi approximation, the non-interacting electron model, and the large-dimension limit. The abscissae are Hn-ear in y/r, where r is given in units of (IS/JV) for the first column, (18/Z) Co for the second, and (D/Zy a<, for the third. All curves except the Thomas-Fermi limit were obtained by adding Slater distributions whose exponents were determined from the subhamiltonian minima described in Sec. 4.

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. 

Our treatment so far has dealt with non-interacting electrons, yet we know for sure that electrons do interact with each other. Dirac (1930b) studied the effects of exchange interactions on the Thomas-Fermi model, and he soon discovered that this effect could be modelled by adding an extra term... 

Resultant energy curves in H2 and H2. u, Burrau s curve for H2 . b, Curve for H2 for non-interacting electrons, c, Approximate curve for H2 with interacting electrons. The small circle in the crook of curve b, represents the equilibrium position and energy on Hutchisson s classical crossed-orbit model of H2. Units same as figure 1 (note different scales of ordinates for Ha and H2+). 

Confining ourselves to the model of non-interacting electrons, the transition is second-order, in the sense that there is no discontinuity in the value of n, the number of carriers. The band-gap should vary as a—a0U where Oq is the value of the lattice parameter a at the transition the number n of carriers should vary as a0—al3/2 and the energy as a0—a 5/2. The conductivity of a perfect crystal at zero temperature should change from zero to infinity, but if a finite mean free path is introduced there will be no discontinuity in the conductivity. This conclusion is changed, however, when electron-electron interaction is taken into account as in Chapter 4. 

In the case of conventional (i.e. non-superconductive) metals, the inter pretation of the results of experimental studies on electron tunneling through the "metal-insulator metal junction becomes possible within the framework of the simplest model of non-interacting electrons. Here, the interaction between electrons and between electrons and phonons both inside the barrier and in the bulk of metals are neglected. 

Anderson and his coworker carried out a series quantum chemistry studies of oxygen reduction reactions.52-57 Anderson and Abu first studied reversible potential and activation energies for uncatalyzed oxygen reduction to water and the reverse oxidation reaction using the MP2/6-31G method. The electrode was modeled by a non-interacting electron donor molecule with a chosen ionization potential (IP). The primary assumption is that when the reactant reaches a point on the reaction path where its electron affinity (EA) matched the donor IP, an electron transfer is initialized. The donor s IP or reactant s EA was related to the electrode potential by,... 

In this paper, the multiphonon relaxation of a local vibrational mode and the non-radiative electronic transitions in molecular systems and in solids are considered using this non-perturbative theory. Results of model calculations are presented. According to the obtained results, the rate of these processes exhibits a critical behavior it sharply increases near specific (critical) value(s) of the interaction. Also the usual increase of the non-radiative transition rate with temperature is reversed at certain value of the non-diagonal interaction and temperature. For a weak interaction, the results coincide with those of the perturbation theory. 

The simplest and most used model possible deals with non-interacting electrons in a potential box having a constant interior potential and plane-bounding surfaces of potential walls of finite height. Here the potential is only a function of distance along the surface normal and is of the form shown in Fig. 2. Because in this model the ionic charge is smeared out uniformly within each unit cell, its properties, and specifically its cohes-... 

Time-dependent density functional theory (TDDFT) as a complete formalism [7] is a more recent development, although the historical roots date back to the time-dependent Thomas-Fermi model proposed by Bloch [8] as early as 1933. The first and rather successful steps towards a time-dependent Kohn-Sham (TDKS) scheme were taken by Peuckert [9] and by Zangwill and Soven [10]. These authors treated the linear density response of rare-gas atoms to a time-dependent external potential as the response of non-interacting electrons to an effective time-dependent potential. In analogy to stationary KS theory, this effective potential was assumed to contain an exchange-correlation (xc) part, r,c(r, t), in addition to the time-dependent external and Hartree terms ... 

We demonstrate some of the ideas for a vTS x /T8 Hubbard lattice containing 18 electrons, with periodic boundary conditions. This model has been extensively studied [11]. The Hubbard parameter was set to U/t = 4. The ground-state of the non-interacting system, being diagonalised by plane-waves, is a closed shell consisting of l- -4- -4 doubly occupied levels. There are 2M = 36 one particle orbtials, leading to Ndet = 9075135300 determinants for an... 

The current control at the one-by-one electron accuracy level is feasible in mesoscopic devices due to quantum interference. Though the electric charge is quantized in units of e, the current is not quantized, but behaves as a continuous fluid according to the jellium electron model of metals. The prediction of the current quantization dates back to 1983 when D. Thouless [Thouless 1983] found a direct current induced by slowly-traveling periodic potential in a ID gas model of non-interacting electrons. The adiabatic current is the charge... 

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