Independent electron theory

Conjugated molecules have attracted the attention of chemists and physicists since the early days of quantum mechanics. A remarkable example is given by Hiickel who developed in the thirties the first independent-electron theory applicable to polyenes [1], In the sixties, J.A. Pople and S.H. Walm-... 

This theoretical approach is essentially an independent-electron theory. The importance and role of electron correlations in explaining the properties of CPs have been hotly debated. We return to this below. A balanced review focused on PA can be found in Ref. 37. 

Fig. 13. The difference between the experimental ionisation potentials and the behaviour expected for the two classical scaling laws [see eqn (12)] is plotted as a function of the mean coordination number z, which increases to the left. The zero of the two curves has been shifted by 1 eV. The open circles joined by the straight, dashed line are from the tight-binding calculation of rdf. (8). The experimental results show a much more rapid decrease than calculate in the independent electron theory. This suggests that the transition at z 10.6 is indeed driven by electronic correlation, as in a Mott transition in bulk material. The insert shows the bandgap e expected for an independent electron theory (a) and one with enough correlation (b) to induce the discontinuous Mott transition as discussed in the text. The similarity to the theoretical and experimental data near z = 10.6 is obvious.

In summary, ionisation potentials, dissociation and cohesive energies for mercury clusters have been determined. The mass spectrum of negatively charged Hg clusters is reported. The influence of the transition from van der Waals (n < 13), to covalent (30 < n < 70) to metallic bonding (n > 100) is discussed. A cluster is defined to be metallic , if the ionisation potential behaves like that calculated for a metal sphere. The difference between the measured ionisation potential and that expected for a metallic cluster vanishes rather suddenly around n 100 Hg atoms per cluster. Two possible interpretations are discussed, a rapid decrease of the nearest-neighbour distance and/or the analogue of a Mott transition in a finite system. Electronic correlation effects are strong they make the experimentally observed transitions van der Waals/covalent and covalent/metallic more pronounced than calculated in an independent electron theory. 

Since both k and all the y/ s can be related to probabilities (refer back to Section 14.1.1), the orbital approximation defines a total probability as a product of individual probabilities. This is only true in probability theory if the individual events (y/ s) are independent, such as each flipping of a coin to get "heads" or "tails" is an independent event. With respect to electrons, this means that the probability that electron 1 will exist at a certain position in space is completely independent of the positions of electrons 2, 3,4, etc. The electrons movements are therefore not correlated, a severe approximation (see the discussion of electron correlation given later). This is therefore called an independent electron theory. 

When we first introduced the "orbital approximation" and noted that, at least for electrons of opposite spin, it is an independent electron theory, we mentioned that the electrons are therefore not correlated. Hence, in the HF-SCF method each electron moves in a general field created by all the other electrons. In reality, however, the motions of electrons are pairwise correlated to keep the electrons apart when one electron "moves left", a nearby electron will move out of its way, because like charges repel. This correlation of electron motions is a stabilizing effect, and thus Escf is always too high, even at the so-called Hartree-Fock limit (the result you would get with a basis set of infinite size). The error associated with the Hartree-Fock limit is called the correlation energy (Eq. 14.36). 

The simplest approximation to the Schrodinger equation is an independent-electron approximation, such as the Hiickel method for Jt-electron systems, developed by E. Hiickel. Later, others, principally Roald Hoffmann of Cornell University, extended the Hiickel approximations to arbitrary systems having both n and a electrons—the Extended Hiickel Theory (EHT) approximation. This chapter describes some of the basics of molecular orbital theory with a view to later explaining the specifics of HyperChem EHT calculations. 

HyperChem currently supports one first-principle method ab initio theory), one independent-electron method (extended Hiickel theory), and eight semi-empirical SCFmethods (CNDO, INDO, MINDO/3, MNDO, AMI, PM3, ZINDO/1, and ZINDO/S). This section gives sufficient details on each method to serve as an introduction to approximate molecular orbital calculations. For further details, the original papers on each method should be consulted, as well as other research literature. References appear in the following sections. 

It is traditional to divide quantum-mechanical molecular models into three broad bands depending on their degree of sophistication. There are sublevels within each band, and a great deal of jargon accompanied by acronyms. Many authors speak of the level of theory . The Hiickel independent electron model of Chapter 7 typifies the lowest level of theory, and authors sometimes refer to these models as empirical . The Hamiltonian is not rigorously defined, and neither are the basis functions. Nevertheless, these models have been able to produce impressive predictions and rationalizations. 

An alternative approach to conventional methods is the density functional theory (DFT). This theory is based on the fact that the ground state energy of a system can be expressed as a functional of the electron density of that system. This theory can be applied to chemical systems through the Kohn-Sham approximation, which is based, as the Hartree-Fock approximation, on an independent electron model. However, the electron correlation is included as a functional of the density. The exact form of this functional is not known, so that several functionals have been developed. 

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,... 

In most cases, the orbital relaxation contribution is negligible and the Fukui function and the FMO reactivity indicators give the same results. For example, the Fukui functions and the FMO densities both predict that electrophilic attack on propylene occurs on the double bond (Figure 18.1) and that nucleophilic attack on BF3 occurs at the Boron center (Figure 18.2). The rare cases where orbital relaxation effects are nonnegligible are precisely the cases where the Fukui functions should be preferred over the FMO reactivity indicators [19-22], In short, while FMO theory is based on orbitals from an independent electron approximation like Hartree-Fock or Kohn-Sham, the Fukui function is based on the true many-electron density. 

We have not mentioned open shells of electrons in our general considerations but then we have not specifically mentioned closed shells either. Certainly our examples are all closed shell but this choice simply reflects our main area of interest valence theory. The derivations and considerations of constraints in the opening sections are independent of the numbers of electrons involved in the system and, in particular, are independent of the magnetic properties of the molecules concerned simply because the spin variable does not occur in our approximate Hamiltonian. Nevertheless, it is traditional to treat open and closed shells of electrons by separate techniques and it is of some interest to investigate the consequences of this dichotomy. The independent-electron model (UHF - no symmetry constraints) is the simplest one to investigate we give below an abbreviated discussion. 

The 1947 Nobel Prize in chemistry was to be awarded to Robinson to honor his work in the synthesis of natural products, investigations that he pursued all through his career, and a field in which his wife both collaborated and worked independently.87 Yet, in his memoirs, Robert Robinson wrote that he considered the development of an electronic theory of reaction mechanisms "my most important contribution to knowledge."88 This suggests the seriousness with which he viewed scientific theories and his belief that scientific glory and reputation rest on theories, not discoveries. Let us turn now to these theories. 

Ingold, "Principles of an Electronic Theory," 227, 265270 see also Ingold, Structure and Mechanism, esp. 200203. "The order in which groups arrange themselves with respect to their polarization will bear no simple relation to the order of their polarizability, and the relative importance of the two contributory effects, dependent respectively on their two independent electrical characteristics, must necessarily vary with the nature of the reaction" (Ingold, "Principles of an Electronic Theory," 228). 

However, by the 1920s, as we have seen, a self-conscious use of electron theory in a dynamical interpretation of the old static chemical molecule recovered dynamical theoretical foundations for organic chemists in what became the disciplinary specialization of physical organic chemistry. The theory of electron valency and organic reaction mechanisms, in particular, the theory of mesomerism, developed as a new theoretical chemistry, a little prior to wave mechanics, along a largely independent track. 

Part A. Quantum-Mechanical Theory of Diffusion Independent Electron Transfer in Biological Systems by Ephraim Buhks (University of Delaware)... 

Beyond these well established characteristic features however the total width of the electron distributions is much narrower for the argon target than for neon or helium. These patterns are due to the signatures of the initial state wave function. In figure 3, a systematic discrepancy between experiment and theory appears at higher electron energies. This discrepancy occurs due to one of the basic postulates of the CDW-EIS model, namely that it is based on the independent electron model which considers there... 

Band theory is a one-electron, independent particle theory, which assumes that the electrons are distributed amongst a set of available stationary states following the Fermi-Dirac statistics. The states are given by solutions of the Schrodinger equation... 

Now we have written down a wave function appropriate for use in the case where H = h(i). In HF theory, we make some simplifications so many-electron atoms and molecules can be treated this way. By tacitly assuming that each electron moves in a percieved electric field generated by the stationary nuclei and the average spatial distribution of all the other electrons, it essentially becomes an independant-electron problem. The HF Self Consistent Field procedure (SCF) will be bent on constructing each x(x) to give the lowest energy. 

Orbital interaction theory forms a comprehensive model for examining the structures and kinetic and thermodynamic stabilities of molecules. It is not intended to be, nor can it be, a quantitative model. However, it can function effectively in aiding understanding of the fundamental processes in chemistry, and it can be applied in most instances without the use of a computer. The variation known as perturbative molecular orbital (PMO) theory was originally developed from the point of view of weak interactions [4, 5]. However, the interaction of orbitals is more transparently developed, and the relationship to quantitative MO theories is more easily seen by straightforward solution of the Hiickel (independent electron) equations. From this point of view, the theoretical foundations lie in Hartree-Fock theory, described verbally and pictorially in Chapter 2 [57] and more rigorously in Appendix A. 

One of the most interesting of these properties is the small temperature-independent paramagnetism shown by many metals, including the alkali metals. It was the discussion of this phenomenon by Pauli1 in 1987 that initiated the development of the modern electronic theory of metals. The fundamental concept is that there exists in a metal a continuous or partially continuous set of energy levels for the free electrons. At the absolute zero the electrons (N in number) would... 

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