The Independent-Electron Approximation

In previous chapters we have dealt with the motion of a single particle in various potential fields. When we deal with more than one particle, new problems arise and new techniques are needed. Some of these are discussed in this chapter. 

In constructing the hamiltonian operator for a many electron atom, we shall assume a fixed nucleus and ignore the minor error introduced by using electron mass rather than reduced mass. There will be a kinetic energy operator for each electron and potential terms for the various electrostatic attractions and repulsions in the system. Assuming n electrons and an atomic number of Z, the hamiltonian operator is (in atomic units) 

The numbers in parentheses on the left-hand side of Eq. (5-1) symbolize the spatial coordinates of each of the n electrons. Thus, 1 stands for xi, i,zi, or n, 6i, (j i, etc. We shall use this notation frequently throughout this book. Since we are not here concerned with the quantum-mechanical description of the translational motion of the atom, there is no kinetic energy operator for the nucleus in Eq. (5-1). The index i refers to the electrons, so we see that Eq. (5-1) provides us with the desired kinetic energy operator for each electron, a nuclear electronic attraction term for each electron, and an interelectronic repulsion term for each distinct electron pair. (The summation indices guarantee that l/ri2 and l/r2i will not both appear in H. This prevents counting the same physical interaction twice. The indices also prevent nonphysical self-repulsion terms, such as l/r22, from occurring.) Frequently used alternative notations for the double summation in Eq. (5-1) are j 1/ 7. which counts each interaction twice 

(5-3) we have merely grouped H into two one-electron operators and one two-electron operator. There is no way to separate this hamiltonian completely into a sum of one-electron operators without loss of rigor. However, if we wish to approximate the hamiltonian for helium in such a way that it becomes separable, we might try simply ignoring the interelectronic repulsion term  

If we do this, our approximate hamiltonian /4pprox treats the kinetic and potential energies of each electron completely independently of the motion or position of the other. For this reason, such a treatment falls within the category of independent electron approximations.  

Within the Born-Oppenheimer approximation discussed earlier, you can solve an electronic Schrodinger equation 

Electrons are identical, and each term in this sum is essentially the same operator. You can then solve an independent-electron Schrodinger equation for a wave function /, describing an individual electron  

Multiple solutions /j and 8j are possible for this last equation. The wave functions for individual electrons, /j, are called molecular orbitals, and the energy, 8j, of an electron in orbital /j is called the orbital energy. 

The molecular orbitals describe an electron such that the values of /j (r) dr at a point r describe the probability of the electron being in a small volume dr around that point. The total probability of finding the electron somewhere is 

One of the reasons why so much progress has been possible in the analysis of band structures is the fact that many-body effects are, for many purposes, negligible. Most of the effects which demand the consideration of many-body effects are confined to very low temperatures (superconductivity, Kondo effect, etc.) or very near thresholds (soft X-ray edge singularities, etc.). A band structure calculated in the Hartree approximation, with corrections for exchange and correlation within this self-consistent independent-electron formalism, suffices for the interpretation of many properties. 

Reversing the roles of the primed and unprimed systems and adding the two resultant inequalities leads to 

This result can be used to develop a set of self-consistent one-electron equations which are exactly equivalent to the full many-body problem. There are two snags in this procedure. First, the result is purely formal—it does not tell us how to actually construct the 

The idea of a local density correction predates the Sham-Kohn work. In particular, Slater proposed a local exchange potential of the form  

A different approach, usually associated with the work of Hubbard, and used especially in the discussion of interatomic forces, results in a formalism which may most easily be conceived as 

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Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

This is the one-electron approximation, also called the independent electron approximation and hence the le superscript, where a Hamiltonian Hq of an A e-electron system can be expressed as the sum of Ne one-electron Hamiltonians and the Schrodinger equation to be solved becomes ... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

The independent-electron approximation was discussed in the previous chapter. The molecular wave functions, ifi, are solutions of the Hartree-Fock equation, where the Fock operator operates on tfi, but the exact form of the operator is determined by the wave-function itself. This kind of problem is solved by an iterative procedure, where convergence is taken to occur at the step in which the wave function and energy do not differ appreciably from the prior step. The effective independent-electron Hamiltonian (the Fock operator) is denoted here simply as H. The wave functions are expressed as linear combinations of atomic functions, x-... 

CO. For that matter, in regards to predicting the type of electrical behavior, one has to be careful not to place excessive credence on actual electronic structure calculations that invoke the independent electron approximation. One-electron band theory predicts metallic behavior in all of the transition metal monoxides, although it is only observed in the case of TiO The other oxides, NiO, CoO, MnO, FeO, and VO, are aU insulating, despite the fact that the Fermi level falls in a partially hUed band. In the insulating phases, the Coulomb interaction energy is over 4 eV whereas the bandwidths have been found to be approximately 3 eV, that is, U > W. 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

The independent-electron approximation allows for a distinction of target electrons and projectile-centered electrons which screen the projectile... 

These theorems are further discussed in [21]. They show that, in the independent electron approximation, we can think, not only of independent electrons, but also of independent vacancies, which behave like electrons and even give rise to very nearly the same multiplet structure. This simplifies X-ray spectroscopy enormously, and further emphasises the significance of closed shells. 

Centrifugal barriers have a profound effect on the physics of many-electron atoms, especially as regards subvalence and inner shell spectra. One aspect not discussed above is how energy degeneracies arising from orbital collapse can lead to breakdown of the independent electron approximation and the appearance of multiply excited states. Similarly, we have not discussed multiple ionisation (the ejection of several electrons by a single photon) enhanced by a giant resonance. Both issues will be considered in chapter 7. 

Instead, what is usually done is to patch up the independent electron approximation by applying corrections to it using perturbation theory. This purely mechanical approach is referred to as configuration mixing, and leads to the following picture or classification of double excitations. 

As long as n and remain good quantum numbers, the independent particle model and the central field approximation both apply, and quantum chaos does not arise. We can thus identify two situations where chaos could emerge the first is a complete breakdown in the independent electron approximation (due, for example, to strong correlations) and the second is a distortion of the central field approximation (due, for example, to a strong external field). 

The independent-electron approximation, implicit in Hartree-Fock (HF) theory and the Koopmans approximation which derives from it, is shown to be an acceptable model for the description of outer-valence electron ionization in molecules, as illustrated in Figure 3 for CH3SH, but is inadequate for ionization of the inner-valence electrons. 

The theoretical description of atoms and molecules has to rely on approximate solutions to Schrodinger s equation. For the standard methods in current use, the starting approximation treats the electrons as if they were independent particles. The advantage of this approach is the ease with which it can be formulated even for very large systems [1]. However, the correlation of electronic motion often has a major role, particularly in chemical bonding and reactivity. The independent-electron approximation does not provide a qualitative model for correlation effects, nor an efficient basis for evaluating numerical contributions from correlation. 

The Hartree-Fock approximation. The independent electron approximation, which includes the electron-electron Coulomb interaction and the exchange interaction between electrons with parallel spins. 

Let us consider the pressure of electron gas within the framework of the independent electron approximation. 

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