Computational quantum mechanics approximate orbital theories

A key question about the use of any molecular theory or computer simulation is whether the intermolecular potential model is sufficiently accurate for the particular application of interest. For such simple fluids as argon or methane, we have accurate pair potentials with which we can calculate a wide variety of physical properties with good accuracy. For more complex polyatomic molecules, two approaches exist. The first is a full ab initio molecular orbital calculation based on a solution to the Schrddinger equation, and the second is the semiempirical method, in which a combination of approximate quantum mechanical results and experimental data (second virial coefficients, scattering, transport coefficients, solid properties, etc.) is used to arrive at an approximate and simple expression. 

The theory of the chemical bond is one of the clearest and most informative examples of an explanatory phenomenon that probably occurs in some form or other in many sciences (psychology comes to mind) the semiautonomous, nonfundamental, fundamentally based, approximate theory (S ANFFBAT for short). Chemical bonding is fundamentally a quantum mechanical phenomenon, yet for all but the simplest chemical systems, a purely quantum mechanical treatment of the molecule is infeasible especially prior to recent computational developments, one could not write down the correct Hamiltonian and solve the Schrodinger equation, even with numerical methods. Immediately after the introduction of the quantum theory, systems of approximation began to appear. The Born Oppenheimer approximation assumed that nuclei are fixed in position the LCAO method assumed that the position wave functions for electrons in molecules are linear combinations of electronic wave functions for the component atoms in isolation. Molecular orbital theory assumed a characteristic set of position wave functions for the several electrons in a molecule, systematically related to corresponding atomic wave functions. 

Frequently the work involved conjugated molecules to which Electronic population analysis was usually added to the energy calculations and a theoretical dipole moment was obtained that could be compared with the experimental data. With the advent of NMR. and ESR. spectroscopy other observables became available, and theory was successfully applied to the interpretation of these spectra. However, very little was done in the field of real chemistry, that is, in the study of reaction mechanisms and reaction rates. Over the last decade the availability of large electronic computers, the introduction of approximate but reliable quantum mechanical methods which include all the electrons, or at least all valence electrons in a molecular system and the discovery of the rules of orbital symmetry have led to a significant change of the situation. 

We then present ab initio molecular orbital theory. This is a well-defined approximation to the full quantum mechanical analysis of a molecular system, and also the basis of an array of powerful and popular computational approaches. Molecular orbital theory relies upon the linear combination of atomic orbitals, and we introduce the mathematics and results of such an approach. Then we discuss the implementation of ab initio molecular orbital theory in modern computational chemistry. We also describe a number of more approximate approaches, which derive from ab initio theory, but make numerous simplifications that allow larger systems to be addressed. Next, we provide an overview of the theory of organic TT systems, primarily at the level of Hiickel theory. Despite its dramatic approximations, Hiickel theory provides many useful insights. It lies at the core of our intuition about the electronic structure of organic ir systems, and it will be key to the analysis of pericyclic reactions given in Chapter 15. 

The term "ab initio means "from first principles" it does not mean "exact" or "true". In ab initio molecular orbital theory, we develop a series of well-defined approximations that allow an approximate solution to the Schrodinger equation. We calculate a total wavefunc-tion and individual molecular orbitals and their respective energies, without any empirical parameters. Below, we outline the necessary approximations and some of the elements and principles of quantum mechanics that we must use in our calculations, and then provide a summary of the entire process. Along with defining an important computational protocol, this approach will allow us to develop certain concepts that will be useful in later chapters, such as spin and the Born-Oppenheimer approximation. 

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