Schrodinger equation, theory

We have already seen examples of semiempirical methods, in Chapter 4 the simple Hiickel method (SHM, Erich Hiickel, ca. 1931) and the extended Hiickel method (EHM, Roald Hoffmann, 1963). These are semiempirical ( semi-experimental ) because they combine physical theory with experiment. Both methods start with the Schrodinger equation (theory) and derive from this a set of secular equations which may be solved for energy levels and molecular orbital coefficients (most efficiently... 

Such a fundamental theory does exist for chemistry quantum mechanics. The dependence of the property of a compound on its three-dimensional structure is given by the Schrodinger equation. Great progress has been made both in the de-... 

Statistical mechanics is the mathematical means to calculate the thermodynamic properties of bulk materials from a molecular description of the materials. Much of statistical mechanics is still at the paper-and-pencil stage of theory. Since quantum mechanicians cannot exactly solve the Schrodinger equation yet, statistical mechanicians do not really have even a starting point for a truly rigorous treatment. In spite of this limitation, some very useful results for bulk materials can be obtained. 

The Schrodinger equation contains the essence of all chemistry. To quote Dirac The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known. [P.A.M. Dirac, Proc. Roy. Soc. (London) 123, 714 (1929)]. The Schrodinger equation is... 

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. 

Reproducing the exact solution for the relevant n-electron problem a method ought to yield the same results as the exact solution to the Schrodinger equation to the greatest extent possible. What this means specifically depends on the theory underlying the method. Thus, Hartree-Fock theory should be (and is) able to reproduce the exact solution to the one electron problem, meaning it should be able to treat cases like HeH ... 

There is an equivalent way of generating solutions to the electronic Schrodinger equation which conceptually is much closer to the experimentalists language, known as Valence Bond (VB) theory. We will start by illustrating the concepts for the H2 molecule, and note how it differ from MO methods. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

The Rayleigh-Schrodinger Perturbation Theory (see [2]) leads then to the following system of linear equations for the determination of cj (j=l,. ..M) ... 

The perturbation theory is the convenient starting point for the determination of the polarizability from the Schrodinger equation, restricted to its electronic part and the electric dipole interaction regime. The Stark Hamiltonian —p. describes the dipolar interaction between the electric field and the molecule represented by its... 

In order to apply quantum-mechanical theory to the hydrogen atom, we first need to find the appropriate Hamiltonian operator and Schrodinger equation. As preparation for establishing the Hamiltonian operator, we consider a classical system of two interacting point particles with masses mi and m2 and instantaneous positions ri and V2 as shown in Figure 6.1. In terms of their cartesian components, these position vectors are... 

Perturbation theory provides a procedure for finding approximate solutions to the Schrodinger equation for a system which differs only slightly from a system for which the solutions are known. The Hamiltonian operator H for the system of interest is given by... 

The mathematical procedure that we present here for solving equation (9.15) is known as Rayleigh-Schrodinger perturbation theory. There are other procedures, but they are seldom used. In the Rayleigh-Schrodinger method, the eigenfunctions tpn and the eigenvalues E are expanded as power series in A... 

The first two chapters serve as an introduction to quantum theory. It is assumed that the student has already been exposed to elementary quantum mechanics and to the historical events that led to its development in an undergraduate physical chemistry course or in a course on atomic physics. Accordingly, the historical development of quantum theory is not covered. To serve as a rationale for the postulates of quantum theory, Chapter 1 discusses wave motion and wave packets and then relates particle motion to wave motion. In Chapter 2 the time-dependent and time-independent Schrodinger equations are introduced along with a discussion of wave functions for particles in a potential field. Some instructors may wish to omit the first or both of these chapters or to present abbreviated versions. 

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