Schrodinger equation computational chemistry

T. E. Simos and P. S. Williams, On finite difference methods for the solution of the Schrodinger equation. Computers Chemistry, 1999, 23, 513-554. 

The quantum mechanical methods described in this book are all molecular orbital (MO) methods, or oriented toward the molecular orbital approach ab initio and semiempirical methods use the MO method, and density functional methods are oriented toward the MO approach. There is another approach to applying the Schrodinger equation to chemistry, namely the valence bond method. Basically the MO method allows atomic orbitals to interact to create the molecular orbitals of a molecule, and does not focus on individual bonds as shown in conventional structural formulas. The VB method, on the other hand, takes the molecule, mathematically, as a sum (linear combination) of structures each of which corresponds to a structural formula with a certain pairing of electrons [16]. The MO method explains in a relatively simple way phenomena that can be understood only with difficulty using the VB method, like the triplet nature of dioxygen or the fact that benzene is aromatic but cyclobutadiene is not [17]. With the application of computers to quantum chemistry the MO method almost eclipsed the VB approach, but the latter has in recent years made a limited comeback [18],... 

Two equivalent formulations of QM were devised by Schrodinger and Heisenberg. Here, we will present only the Schrodinger form since it is the basis for nearly all computational chemistry methods. The Schrodinger equation is... 

The description of electronic distribution and molecular structure requires quantum mechanics, for which there is no substitute. Solution of the time-independent Schrodinger equation, Hip = Eip, is a prerequisite for the description of the electronic distribution within a molecule or ion. In modern computational chemistry, there are numerous approaches that lend themselves to a reasonable description of ionic liquids. An outline of these approaches is given in Scheme 4.2-1 [1] ... 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

A second fundamental classification of quantum chemistry calculations can be made according to the quantity that is being calculated. Our introduction to DFT in the previous sections has emphasized that in DFT the aim is to compute the electron density, not the electron wave function. There are many methods, however, where the object of the calculation is to compute the full electron wave function. These wave-function-based methods hold a crucial advantage over DFT calculations in that there is a well-defined hierarchy of methods that, given infinite computer time, can converge to the exact solution of the Schrodinger equation. We cannot do justice to the breadth of this field in just a few paragraphs, but several excellent introductory texts are available... 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

Computational methods typically employ the Born-Oppenheimer approximation in most electronic structure programs to separate the nuclear and electronic parts of the Schrodinger equation that is still hard enough to solve approximately. There would be no potential energy (hyper)surface (PES) without the Born-Oppenheimer approximation -how difficult mechanistic organic chemistry would be without it ... 

The first problem is the subject of modern quantum chemistry. Many efficient methods and computer programs are nowadays available to solve the electronic Schrodinger equation from first principles without using phenomenological or experimental input data (so-called ab initio methods Lowe 1978 Carsky and Urban 1980 Szabo and Ostlund 1982 Schmidtke 1987 Lawley 1987 Bruna and Peyerimhoff 1987 Werner 1987 Shepard 1987 see also Section 1.5). The potentials Vfe(Q) obtained in this way are the input for the solution of the nuclear Schrodinger equation. Solving (2.32) is the subject of spectroscopy (if the motion is bound) and... 

Abstract A historical view demystifies the subject. The focus is strongly on chemistry. The application of quantum mechanics (QM) to computational chemistry is shown by explaining the Schrodinger equation and showing how this equation led to the simple Hiickel method, from which the extended Hiickel method followed. This sets the stage well for ab initio theory, in Chapter 5. 

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