Many-electron molecular wave functions

Many-electron molecular wave functions (i) Introduction 

The other approach to the electronic wave function and its links with the molecular parameters is the ab initio calculation we often use this term or mode of description elsewhere in this book. An ab initio calculation is one which uses only the fundamental laws of physics and the values of the fundamental constants. It is easy to define, but efforts to achieve success for molecules with many electrons, even diatomic molecules, have occupied the attention of many for the past seventy years. The electronic wave function produced by an ab initio calculation is often too complicated algebraically to provide something which can reasonably be described as a physical picture. In this situation the value of a molecular parameter, determined by experiment, is best regarded as a benchmark which can be used to test the accuracy of an ah initio calculation. 

The development of ab initio methods, which has come to be known as quantum chemistry, is one of the outstanding cumulative intellectual and technical achievements of the past fifty years. Fifty years ago, at the time of the publication of Herzberg s classic book [14], quantum mechanics had been applied to the calculation of the wave functions of the very simplest molecules, but for most systems the problems were considered intractable. At the turn of the millenium we have come a long way, but many difficult problems remain. Progress has been closely related to the development of the digital computer, and the technical achievements in the use of computers to solve problems in quantum mechanics have been impressive. There has always, however, been an accompanying and continual need for intellectual advances to make use of the technology. This section describes the nature of the problems to be solved, and some of the methods which have been developed to tackle them. 

Within the Bom-Oppenheimer approximation [28], the electron distribution is supposed to depend only on the instantaneous positions of the nuclei and not on their velocities. The Schrodinger equation for the electrons in the field of the fixed nuclei, which we have met before, and was derived in chapter 2, is 

The electronic wave function Pelec(r, R) depends on both the electronic coordinates, 

The Coulomb potential energy, V, is the sum of three different types of term, as follows  

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The most suitable many-electron molecular wave functions are the antisymmetrized products (Slater s determinants) of mono-electronic wave functions, and it may be shown that any antisymmetric wave function T = T (a i, x, , x ) may always be expanded in a series of such determinants ... 

DFT has led to a substantial simplification of quantum-chemical computations. Like the Hellmann-Feynman theorem it expresses the reasonable assumption of a reciprocal relationship between potential energy and electron density in a molecule. In principle this relationship means that all ground-state molecular properties may be calculated from the ground-state electron density p(x, y, z), which is a function of only three coordinates, instead of a many-parameter molecular wave function in configuration space. The formal theorem behind DFT which defines the electronic energy as a functional of the density function provides no guidance on how to establish the density function p r) without resort to wave mechanics. 

Figure 4.1 Progression from atomic orbitals (AO) (basis functions), to molecular orbitals (MO), to Slater determinants (SD) and to a many-electron (ME) wave function...

Point (3) above requires some amplification. At the quantitative level, the ultimate aim of either a VB or an MO calculation is to obtain the total molecular wave function. Such a function will lead to an electron density map for the molecule which should yield information about its bonding and insights into its reactivity. The function may also be manipulated in order to calculate various molecular constants whose theoretical values can be compared with experimental ones, if available. The kind of function we are talking about is a many-electron function it contains the coordinates of all the electrons in the molecule, and is usually expressed as a product of one-electron functions (i.e. orbitals). In MO theory, these are the MOs. The constraints of symmetry and orthogonality ensure that these MOs are amenable in themselves to quantum-mechanical manipulations. In VB theory, however, the one-electron functions are localised bond orbitals which are not quite respectable and are not immediately amenable to manipulation. The total molecular wave function obtained from a VB calculation is not necessarily inferior to its MO counterpart however, its factorisation into one-electron functions is designed to preserve the useful and successful notion of the localised electron-pair bond. This has the disadvantage that the one-electron functions are less useful for quantum-mechanical purposes. 

For larger molecules it is assumed that a molecular wave function, , is an anti-symmetric product of atomic wave functions, made up by linear combination of single-electron functions, called orbitals. The Hamiltonian operator, H which depends on the known molecular geometry, is readily derived and although eqn. (3.37) is too complicated, even for numerical solution, it is in principle possible to simulate the operation of H on d>. After variational minimization the calculated eigenvalues should correspond to one-electron orbital energies. However, in practice there are simply too many electrons, even in moderately-sized molecules, for this to be a viable procedure. 

Many of the principles and techniques for calculations on atoms, described in section 6.2 of this chapter, can be applied to molecules. In atoms the electronic wave function was written as a determinant of one-electron atomic orbitals which contain the electrons these atomic orbitals could be represented by a range of different analytical expressions. We showed how the Hartree-Fock self-consistent-field methods could be applied to calculate the single determinantal best energy, and how configuration interaction calculations of the mixing of different determinantal wave functions could be performed to calculate the correlation energy. We will now see that these technques can be applied to the calculation of molecular wave functions, the atomic orbitals of section 6.2 being replaced by one-electron molecular orbitals, constructed as linear combinations of atomic orbitals (l.c.a.o. method). 

Spin-Orbit Matrix and Electric Field Gradient Tensor Derived from Many-Electron Molecular Orbital Wave Functions... 

In the following three sections we shall discuss four applications of quantum mechanics to miscellaneous problems, selected from the very large number of applications which have been made. These are the van der Waals attraction between molecules (Sec. 47), the symmetry properties of molecular wave functions (Sec. 48), statistical quantum mechanics, including the theory of the dielectric constant of a diatomic dipole gas (Sec. 49), and the energy of activation of chemical reactions (Sec. 50). With reluctance we omit mention of many other important applications, such as to the theories of the radioactive decomposition of nuclei, the structure of metals, the diffraction of electrons by gas molecules and crystals, electrode reactions in electrolysis, and heterogeneous catalysis. 

It can now be predicted with confidence that machine calculations will lead gradually toward a really fundamental quantitative understanding of the rules of valence and the exceptions to these toward a real understanding of the dimensions and detailed structures, force constants, dipole moments, ionization potentitils, and other properties of stable molecules and equally unstable radicals, anions, and cations, and chemical reaction intermediates toward a basic understanding of activated states in chemical reactions, and of triplet and other excited states which are important in combustion and explosion processes and in photochemistry and in radiation chemistry and also of intermolecular forces further, of the structure and stability of metals and other solids of those parts of molecular wave functions which are important in nuclear magnetic resonance, nuclear quadrupole coupling, and other interaction involving electrons and nuclei and of very many other aspects of the structure of matter which are now understood only qualitatively or semi-empirically. 

Note that while in Fig. 1.1 the so called many-electronic molecular orbitals are about, in Eq. 1.50 the uni- (or i-) electronic molecular orbitals are assumed, each of them moving with its Hamiltonian that is effective since it contains also the influence of the all other electrons in the system upon it. Even more, these molecular mono-orbitals are further decomposed in mathematical atomic orbitals, see 4> s in Eq. 1.53 that are essentially not so different from the individual (p s wave-functions of the N-electrons considered in Eig. 1.1. The actual atomic orbitals are mathematical rather physical objects (for which reason they are often called as basis set) and reflect the atomic participation in the molecular or bonding system rather that the total or valence number of electrons in the system. As such, viewed as the linear combination over the atomic orbitals, the resulted MO-LCAO wave-function... 

Application of valence bond theory to more complex molecules usually proceeds by writing as many plausible Lewis structures as possible which correspond to the correct molecular connectivity. Valence bond theory assumes that the actual molecule is a hybrid of these canonical forms. A mathematical description of the molecule, the molecular wave function, is given by the sum of the products of the individual wave functions and weighting factors proportional to the contribution of the canonical forms to the overall structure. As a simple example, the hydrogen chloride molecule would be considered to be a hybrid of the limiting canonical forms H—Cl, H Cr, and H C1. The mathematical treatment of molecular structure in terms of valence bond theory can be expanded to encompass more complex molecules. However, as the number of atoms and electrons increases, the mathematical expression of the structure, the wave function, rapidly becomes complex. For this reason, qualitative concepts which arise from the valence bond treatment of simple molecules have been applied to larger molecules. The key ideas that are used to adapt the concepts of valence bond theory to complex molecules are hybridization and resonance. In this qualitative form, valence bond theory describes molecules in terms of orbitals which are mainly localized between two atoms. The shapes of these orbitals are assumed to be similar to those of orbitals described by more quantitative treatment of simpler molecules. 

The notation just introduced is rather more than a convenient shorthand for specifying which orbitals are occupied and by how many electrons. It expresses the fact that the MO approximation to the molecular wave function is a product of one-electron wave functions, i.e. orbitals, each taken to a power equal to the number of electrons occupying it. We recall that the irreducible representation of a product of coordinates is the direct product of their irreps extending the same idea to the product of orbitals, we see that the irrep of an electron configuration is simply the direct product of the irreps of its occupied... 

By using as a basis a set of many-electron functions of this type, it is possible to find a sufficiently approximate molecular wave function by the method of the configurations interaction, which consists in properly selecting a certain number of states and then writing down the total wave function as a linear combination of the functions of all the states ... 

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