Approximate methods of quantum mechanics

For many applications the full molecular Hamiltonian is not necessary it is sufficient to include only relevant energy terms into the model Hamiltonian. One of the model Hamiltonians is the spin Hamiltonian which includes only the angular momentum operators in their mutual interaction (orbit-orbit, spin-orbit, spin-spin interactions) as well as their interaction with a magnetic field (the Zeeman terms orbit-magnetic field and spin-magnetic field). 

---

Some investigations have been inspired by another special circumstance concerning the structure of the fundamental heteroaromatic rings like the parent aromatic homocyclic hydrocarbons, these structures are readily amenable to theoretical treatment by the approximation methods of quantum mechanics. Quantitative studies are clearly desirable in this connection for a reliable test of the theory and, indeed, they have been utilized to this end. ... 

APPROXIMATE METHODS OF QUANTUM MECHANICS 1.3.1 Linear variation method... 

The linear variation method and the perturbation method for stationary states belong to the most important approximate methods of quantum mechanics. The latter exists in several variants. 

We shall examine the simplest possible molecular orbital problem, calculation of the bond energy and bond length of the hydrogen molecule ion Hj. Although of no practical significance, is of theoretical importance because the complete quantum mechanical calculation of its bond energy can be canied out by both exact and approximate methods. This pemiits comparison of the exact quantum mechanical solution with the solution obtained by various approximate techniques so that a judgment can be made as to the efficacy of the approximate methods. Exact quantum mechanical calculations cannot be carried out on more complicated molecular systems, hence the importance of the one exact molecular solution we do have. We wish to have a three-way comparison i) exact theoretical, ii) experimental, and iii) approximate theoretical. 

APPROXIMATION METHODS IN QUANTUM MECHANICS The square of Eq. (43) is given by... 

In principal one can calculate the electronic energy as a function of the Cartesian coordinates of the three atomic nuclei of the ground state of this system using the methods of quantum mechanics (see Chapter 2). (In subsequent discussion, the terms coordinates of nuclei and coordinates of atoms will be used interchangeably.) By analogy with the discussion in Chapter 2, this function, within the Born-Oppenheimer approximation, is not only the potential energy surface on which the reactant and product molecules rotate and vibrate, but is also the potential... 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

The elements of o(co) are determined from studies of molecular refraction, anisotropic li t scattering, the Kerr effect, and optical birefiringence. Frequency-dependent polarizabilities can also be calculated directly to various orders of approximation by methods of quantum mechanics. Research has of late been developing on the spectra of depolarized li t in simple gases and liquids, " leading to the detemunation of pair polarizabOity of atoms or molecules - as wdl as of the polarizability anisotropies induced by multiple correlations. ... 

This approach is named after its founders—Wentzel (1926), Kramers (1926), Brillouin (1926a,b), and Jeffreys (1925). The WKBJ method is one of the powerful approximate approaches of quantum mechanics. Although in the present discussion we are concerned only with its application to obtaining approximate eigenvalues for bound states of the onedimensional Schrodinger equation, such application does not cover all of its range and its force. 

In contrast to force-field calculations in which electrons are not explicitly addressed, molecular orbital calculations, use the methods of quantum mechanics to generate the electronic structure of molecules. Fundamental to the quantum mechanical calculations that are to be performed is the solution of the Schrodinger equation to provide energetic and electronic information on the molecular system. The Schrodinger equation cannot, however, be exactly solved for systems with more than two particles. Since any molecule of interest will have more than one electron, approximations must be used for the solution of the Schrodinger equation. The level of approximation is of critical importance in the quality and time required for the completion of the calculations. Among the most commonly invoked simplifications in molecular orbital theory is the Bom-Oppenheimer [13] approximation, by which the motions of atomic nuclei and electrons can be considered separately, since the former are so much heavier and therefore slower moving. Another of the fundamental assumptions made in the performance of electronic structure calculations is that molecular orbitals are composed of a linear combination of atomic orbitals (LCAO). 

The power of quantum mechanics is revealed by experimental confirmation of the predicted spectroscopic properties of atomic hydrogen. The reasonable expectation of successfully extending the method to many-electron atoms and molecules has been thwarted by mathematical complexity. It has never been possible to solve the wave equation for the motion of more than one particle. The most complex chemical system that has been solved (numerically) is for the single electron in the field of two protons, clamped in place, to define the molecular ion hJ. In order to apply the methods of quantum mechanics to any atom or molecule, apart from H and H, it is necessary to apply approximation methods or introduce additional assumptions based on chemical intuition. 

A is a complex function of the energies of bonds broken and formed in the reaction, so that, for most reactions, A is an experimental quantity only. However, in reactions in which one of the reactants is a free atom or a free radical, A can be estimated approximately from bond energies. Furthermore, A can be calculated for some more complex reactions by the methods of quantum mechanics, but the methpd is very laborious, and the results are subject to considerable error because of necessarily rough approximations. 

Approximation Methods in Quantum Mechanics, 1969 Nonlinear Plasma Theory, 1969 Quantum Kinematics and Dynamics, 1970 Statistical Mechanics A Set of Lectures, 1972 Photon-Hadron Interactions, 1972 Combinatorics and Renormalization in Quantum Field Theory, 1973... 

The VSEPR model is usually a satisfactory method for predicting molecular geometries. To understand bonding and electronic structure, however, you must look to quantum mechanics. We will consider two theories stemming from quantum mechanics valence bond theory and molecular orbital theory. Both use the methods of quantum mechanics but make different simplifying assumptions. In this section, we will look in a qualitative way at the basic ideas involved in valence bond theory, an approximate theory to explain the electron pair or covalent bond by quantum mechanics. 

The purpose of this chapter is to provide an introduction to tlie basic framework of quantum mechanics, with an emphasis on aspects that are most relevant for the study of atoms and molecules. After siumnarizing the basic principles of the subject that represent required knowledge for all students of physical chemistry, the independent-particle approximation so important in molecular quantum mechanics is introduced. A significant effort is made to describe this approach in detail and to coimnunicate how it is used as a foundation for qualitative understanding and as a basis for more accurate treatments. Following this, the basic teclmiques used in accurate calculations that go beyond the independent-particle picture (variational method and perturbation theory) are described, with some attention given to how they are actually used in practical calculations. 

The preferable theoretical tools for the description of dynamical processes in systems of a few atoms are certainly quantum mechanical calculations. There is a large arsenal of powerful, well established methods for quantum mechanical computations of processes such as photoexcitation, photodissociation, inelastic scattering and reactive collisions for systems having, in the present state-of-the-art, up to three or four atoms, typically. " Both time-dependent and time-independent numerically exact algorithms are available for many of the processes, so in cases where potential surfaces of good accuracy are available, excellent quantitative agreement with experiment is generally obtained. In addition to the full quantum-mechanical methods, sophisticated semiclassical approximations have been developed that for many cases are essentially of near-quantitative accuracy and certainly at a level sufficient for the interpretation of most experiments.These methods also are com-... 

Presents the basic theory of quantum mechanics, particularly, semi-empirical molecular orbital theory. The authors detail and justify the approximations inherent in the semi-empirical Hamiltonians. Includes useful discussions of the applications of these methods to specific research problems. 

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. 

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. 

Several examples of the application of quantum mechanics to relatively simple problems have been presented in earlier chapters. In these cases it was possible to find solutions to the Schrtidinger wave equation. Unfortunately, there are few others. In virtually all problems of interest in physics and chemistry, there is no hope of finding analytical solutions, so it is essential to develop approximate methods. The two most important of them are certainly perturbation theory and the variation method. The basic mathematics of these two approaches will be presented here, along with some simple applications. 

The previous subsections defined the AIMS method, the various approximations that one could employ, and the resulting different limits classical mechanics, Heller s frozen Gaussian approximation, and exact quantum mechanics. As emphasized throughout the derivation, the method can be computationally costly, and this is one of the reasons for developing and investigating the accuracy of various approximations. Alternatively, and often in addition, one could try to develop algorithms that reduce the computational cost of the method without compromising its accuracy. In this subsection we discuss two such extensions. Each of these developments has been extensively discussed in a publication, and interested readers should additionally consult the relevant papers (Refs. 125 and 41, respectively). We conclude this subsection with a discussion of the first steps... 

To make QM studies of chemical reactions in the condensed phase computationally more feasible combined quantum me-chanical/molecular mechanical (QM/MM) methods have been developed. The idea of combined QM/MM methods, introduced first by Levitt and Warshell [17] in 1976, is to divide the system into a part which is treated accurately by means of quantum mechanics and a part whose properties are approximated by use of QM methods (Fig. 5.1). Typically, QM methods are used to describe chemical processes in which bonds are broken and formed, or electron-transfer and excitation processes, which cannot be treated with MM methods. Combined QM and MM methods have been extensively used to study chemical reactions in solution and the mechanisms of enzyme-catalyzed reactions. When the system is partitioned into the QM and MM parts it is assumed that the process requiring QM treatment is localized in that region. The MM methods are then used to approximate the effects of the environment on the QM part of the system, which, via steric and electrostatic interactions, can be substantial. The... 

---