Non-interacting electrons

One can utilize some very simple cases to illustrate this approach. Suppose one considers a solution for non-interacting electrons i.e. in equation A1.3.1 the last temi in the Hamiltonian is ignored. In diis limit, it is... 

Perhaps the simplest description of a condensed matter system is to imagine non-interacting electrons contained within a box of volume, Q. The Scln-ddinger equation for this system is similar to equation Al.3.9 with the potential set to zero ... 

Kolm and Sham [25] decompose G[p] into the kinetic energy of an analogous set of non-interacting electrons with the same density p(r) as the interacting system. 

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... 

Thus the interacting multi-electron system can be simulated by the noninteracting electrons under the influence of the effective potential l eff(r)- Kohn and Sham [51] took advantage of the fact that the case of non-interacting electrons allows an exact computation of the particle density and kinetic energy as... 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

Kohn and Sham provided a further contribution to make the DFT approach useful for practical calculations, by introducing the concept of fictitious non-interacting electrons with the same density as the true interacting electrons [8]. Non-interacting electrons are described by orthonormal single-particle wavefunctions y/i (r) and their density is given by ... 

The calculation of the induced electron density may be done in the context of the Kohn-Sham approach to density functional theory, because the response of a KS system to a change in the one particle effective potential (r) corresponds to that of a system of non-interacting electrons. 

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+). 

For disordered systems, then, a quite different form of metal-insulator transition occurs—the Anderson transition. In these systems a range of energies exists in which the electron states are localized, and if at zero temperature the Fermi energy lies in this range then the material will not conduct, even though the density of states is not zero. The Anderson transition can be discussed in terms of non-interacting electrons, though in real systems electron-electron interaction plays an important part. 

R is the distance between atoms. In a theory of non-interacting electrons a metal-insulator... 

Metal-insulator transitions in both crystalline and non-crystalline materials are often associated with the existence of magnetic moments. Moments on atoms in a solid are of course an effect of correlation, that is of interaction between electrons, and their full discussion is deferred until Chapter 3. But even within the approximation of non-interacting electrons in crystalline solids, metal-insulator transitions can occur. These will now be discussed. 

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. 

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