Quantum mechanics A mathematical method

Quantum mechanics A mathematical method of treating particles on the basis of quantum theory, which assumes that energy (of small particles) is not infinitely divisible. 

In applying quantum mechanics to real chemical problems, one is usually faced with a Schrodinger differential equation for which, to date, no one has found an analytical solution. This is equally true for electronic and nuclear-motion problems. It has therefore proven essential to develop and efficiently implement mathematical methods which can provide approximate solutions to such eigenvalue equations. Two methods are widely used in this context- the variational method and perturbation theory. These tools, whose use permeates virtually all areas of theoretical chemistry, are briefly outlined here, and the details of perturbation theory are amplified in Appendix D. 

Quantum mechanics (QM) is the correct mathematical description of the behavior of electrons and thus of chemistry. In theory, QM can predict any property of an individual atom or molecule exactly. In practice, the QM equations have only been solved exactly for one electron systems. A myriad collection of methods has been developed for approximating the solution for multiple electron systems. These approximations can be very useful, but this requires an amount of sophistication on the part of the researcher to know when each approximation is valid and how accurate the results are likely to be. A significant portion of this book addresses these questions. 

A fully theoretical calculation of a potential energy surface must be a quantum mechanical calculation, and the mathematical difflculties associated with the method require that approximations be made. The first of these is the Bom-Oppenheimer approximation, which states that it is acceptable to uncouple the electronic and nuclear motions. This is a consequence of the great disparity in the masses of the electron and nuclei. Therefore, the calculation can proceed by fixing the location... 

Ab initio molecular orbital theory is concerned with predicting the properties of atomic and molecular systems. It is based upon the fundamental laws of quantum mechanics and uses a variety of mathematical transformation and approximation techniques to solve the fundamental equations. This appendix provides an introductory overview of the theory underlying ab initio electronic structure methods. The final section provides a similar overview of the theory underlying Density Functional Theory methods. 

The idea in perturbation methods is that the problem at hand only differs slightly from a problem which has already been solved (exactly or approximately). The solution to the given problem should therefore in some sense be close to the solution of the already known system. This is described mathematically by defining a Hamilton operator which consists of two part, a reference (Hq) and a perturbation (H )- The premise of perturbation methods is that the H operator in some sense is small compared to Hq. In quantum mechanics, perturbational methods can be used for adding corrections to solutions which employ an independent particle approximation, and the theoretical framework is then called Many-Body Perturbation Theory (MBPT). 

The mathematical relationship (4) is the one Bohr was able to deduce. Current quantum mechanical methods also deduce this relationship, of course, but with a model that is in fundamental discord with the one used by Bohr. 

It is to be especially emphasized that problems relating to choice among various alternative structures are usually not solved directly by the application of the rules resulting from the quantum mechanics nevertheless, the interpretation of valence in terms of quantities derived from the consideration of simpler phenomena and susceptible to accurate mathematical investigation by known methods now makes it possible to attack them with a fair assurance of success in many cases. 

This paper is dedicated to Gaston Berthier, from whom I have learned a lot. Although Berthier s publications have mostly dealt with applications of quantum mechanical methods to chemical problems, he never liked black boxes or unjustified approximations even if they appeared to work. The question why the quantum chemical machinery does so well although it often lies on rather weak grounds has concerned him very much. I am therefore convinced that he will appreciate this excursion to applied mathematics. 

We begin with the assumption that you have a background in some part of the life sciences or related fields, and that your familiarity with quantum mechanics and the related mathematics (together abbreviated as QM) may be limited or even nonexistent. It is possible to apply biomolecular EPR spectroscopy in your field of research ignoring the QM part, however, for a full appreciation of the method and to develop skills for its all-round applicability, the QM has to be mastered too. 

In order to describe the hydrogen molecule by quantum mechanical methods, it is necessary to make use of the principles given in Chapter 2. It was shown that a wave function provided the starting point for application of the methods that permitted the calculation of values for the dynamical variables. It is with a wave function that we must again begin our treatment of the H2 molecule by the molecular orbital method. But what wave function do we need The answer is that we need a wave function for the H2 molecule, and that wave function is constructed from the atomic wave functions. The technique used to construct molecular wave functions is known as the linear combination of atomic orbitals (abbreviated as LCAO-MO). The linear combination of atomic orbitals can be written mathematically as... 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

Many other approaches for finding a correct structural model are possible. A short description of ab-initio, density functional, and semiempirical methods are included here. This information has been summarized from the paperback book Chemistry with Computation An Introduction to Spartan. The Spartan program is described in the Computer Software section below.65 Another description of computational chemistry including more mathematical treatments of quantum mechanical, molecular mechanical, and statistical mechanical methods is found in the Oxford Chemistry Primers volume Computational Chemistry,52... 

The purpose of the following sections is not to describe the calculations but to give an idea of the basics of the methods, particularly that of atomistic simulation, because quantum mechanical calculations remain largely mathematical in nature and often cannot be described in visual terms. The starting point is a simple estimation of defect interaction energy. 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

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