Approximations to the Many-Electron Wave Function

The use oE self-consistent field molecular orbital theory (SCF-MO) implies a number of approximations which are necessary, since the Schrodinger equation is generally unsolvable for three or more bodies. Thus SCF MO s represent an optimum approximation to the many-electron wave function. 

In the case of the simplest approximation of the many-electron wave function, which is a single Slater determinant that manifests the basis of the Hartree-Fock model, the electronic energy simplifies to... 

Problem 11-1. Consider three levels of approximation (a) Exact many-electron wave function, (b) Hartree-Fock wave function, (including all electrons), (c) Simple LCAO-MO valence electron wave function. For each of the following molecular properties, would you expect the Hartree-Fock approximation to give a correct prediction (to within 1% in the cases of quantitative predictions) Would you expect the LCAO-MO approximation to give a correct prediction ... 

Since in Eq. 1 is the wave function for all of the electrons in the molecule, it is simplest to begin trying to find 4/ by assuming that it can be approximated as the product of one-electron wave functions, one wave function for each of the electrons in a molecule. These one-electron wave functions are called orbitals, and they are distinguished from the many-electron wave function by using a lower case psi (v[/) for the former and an upper case psi (th) for the latter. 

Unfortunately, if a single configuration is used to approximate the many-electron wave function, electrons of opposite spin remain uncorrelated. The tacit assumption that electrons of opposite spin move independently of each other is, of course, physically incorrect, because, in order to minimize their mutual Coulombic repulsion energy, electrons of opposite spin do certainly tend to avoid each other. Therefore, a wave function, T, that consists of only one configuration will overestimate the Coulombic repulsion energy between electrons of opposite spin. 

We turn now to the interaction energy e2/r12 between electrons and consider first its effect on the Fermi surface. The theory outlined until this point has been based on the Hartree-Fock approximation in which each electron moves in the average field of all the other electrons. A striking feature of this theory is that all states are full up to a limiting value of the energy denoted by F and called the Fermi energy. This is true for non-crystalline as well as for crystalline solids for the latter, in addition, occupied states in fc-space are separated from unoccupied states by the "Fermi surface . Both of these features of the simple model, in which the interaction between electrons is neglected, are exact properties of the many-electron wave function the Fermi surface is a real physical quantity, which can be determined experimentally in several ways. 

Answer. Orbitals are one-electron wave functions, ). The fact that electrons are fermions requires that each electron be described by a different orbital. The simplest form of a many-electron wave function, T(l, 2,..., Ne), is a simple product of orbitals (a Hartree product), 1(1) 2(2) 3(3) NfNe). However, the fact that electrons are fermions also imposes the requirement that the many-electron wave function be antisymmetric toward the exchange of any two electrons. All of the physical requirements, including the indistinguishability of electrons, are met by a determinantal wave function, that is, an antisymmetrized sum of Hartree products, ( 1,2,3,..., Ne) = 1(1) 2(2) 3(3) ( ). If (1,2,3,...,Ne) is taken as an approximation of (1,2,..., Ne), i.e., the Hartree-Fock approximation, and the orbitals varied so as to minimize the energy expectation value,... 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

In the MO approach appropriate to outer s and p electrons, the simple formalism does not distinguish between a covalent-ionic band and a metallic band. The use of determinantal (antisymmetrized) wave functions automatically introduces correlations between electrons of parallel spin. Traditionally the many-electron wave function has, at best, been represented by a single Slater determinant of one-electron wave functions (Hartree-Fock approximation), whereas the true wave function would be given by a series of such determi-... 

In the most commonly utilized approximation, the many-electron wave functions are written in terms of products of one-electron wave functions similar to the solutions obtained for the hydrogen atom. These one-electron functions used to construct the many-electron wave function are called atomic orbitals. They are also called hydrogen-like orbitals since they are one-electron orbitals and also because their shape is similar to that of the hydrogen atom orbitals. 

In the DFT, as in the Hartree-Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistency. The many-electron wave function is still written as a Slater determinant. However, the wave functions used to construct the Slater determinant are not the one-electron wave functions of the Hartree-Fock approximation. In the DFT, these wave functions have no individual meaning. They are merely used to construct the total electron-charge density. The difference between the Hartree-Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange correlation potential, Vxc[ ](t), a functional derivative of the exchange correlation energy, Exc, that, in turn, is a functional (a function of a function) of the electron density. In DFT, the Schrodinger equation is expressed as ... 

A necessary and reasonable approximation in the generation of wave functions for many-electron systems is to represent the many-electron wave function using products of one-electron wave functions or orbitals, c >,. To ensure compliance with the Pauli exclusion principle, the wave... 

When approximate solutions are extended to heavier atoms and to molecules, it is necessary to be content with a rather lower accuracy (e.g, 1 to 2%). The most fruitful approximation method is that of Hartree (1928), later justified and refined by Slater (1930) and Fock (1930). This is suggested by lirst neglecting the electron repulsion and observing that the probability function P (1, 2,. .. N) must then be approximated by the product (1) Pg(2). .. Pj (N), for then the probability of any configuration of the electrons is a product of the probabilities of N independent events. Since P = Pf, this means the many-electron wave function must also be a product... 

The distribution of electric charge in a molecule is intimately related to its structure and reactivity. Knowledge of the distribution gives us a feeling for the physical and chemical properties of the molecule and provides a valuable assessment of the accuracy of approximate molecular wavefunctions. The charge distribution in the nth stationary state is determined by the many-electron wave function 0 of the free molecule. If the molecule interacts with an external electric perturbation E, the wave function determines the distortion and,... 

As a first step in obtaining an approximate solution to the molecular Schroedinger equation, we agree to regard the many electron wave function P(r) as having been broken up into molecular orbitals y/i... 

In the 1930s Douglas Hartree and his father William Hartree used a mechanical analog computer to explore the idea that an electron in an atom moves partly in the attractive potential of the nucleus and partly in an averaged repulsive potential due to all the other electrons. Later, V. Fock added to the Hartrees model an exchange term due to the effects of the antisymmetry of the many-electron wave function. The Hartree-Fock approximation is still a basic tool of quantum chemistry. 

As an approximation to the exact many-electron wave function of the real system, we will use the Slater determinant built from the occupied KS MO s [64]. Although it is only an approximation to the exact many-electron wave function of the real system, it seems to be a reasonable one especially if one is interested in calculations of one-electron matrix elements only (in the case of a local multiplicative operator such an approximation yields the exact values of the matrix elements). To describe an excited state corresponding to the transition of an electron from the occupied MO k into the virtual MO a , we will use the many-electron wave function of the excited state in the form of a Slater determinant that differs from the ground state determinant by replacing the occupied MO k by the virtual MO a . 

Through adiabatic approximation, motions of the electrons and the nuclei are handled separately. The crucial problem is that the many-electron wave functions are difficult to express and calculate. When a solid system s electron numbers exceed 1000 (the solid with volume 1 cm contains 10 electrons), the multi-electron wave function will be an unreasonable scientific concept [2, 3]. For many-electron system, Eq. (6.2) still cannot be solved strictly due to the electron-electron repulsion operator -3-y in terms of potential energy. To solve the... 

The two-electron interaction operators g i,j) in the many-electron Hamiltonian are the reason why a product ansatz for the electronic wave function Yg/( r, ) that separates the coordinates of the electrons is not the proper choice if the exact function is to be obtained in a single product of one-electron functions. Instead, the electronic coordinates are coupled and the motion of electrons is correlated. It is therefore natural to assume that a suitable ansatz for the many-electron wave function requires functions that depend on two coordinates. Work along these lines within the clamped-nuclei approximation has a long history [320-328]. The problem, however, is then what functional form to choose for these functions. First attempts in molecular quantum mechanics used simply terms that are linear in the interelectronic distance Tij = ki — Tjl [329-333], while it could be shown that an exponential ansatz is more efficient [334-337]. 

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