Potential functions quantum-mechanical problem

The main handicap of MD is the knowledge of the function [/( ). There are some systems where reliable approximations to the true (7( r, ) are available. This is, for example, the case of ionic oxides. (7( rJ) is in such a case made of coulombic (pairwise) interactions and short-range terms. A second example is a closed-shell molecular system. In this case the interaction potentials are separated into intraatomic and interatomic parts. A third type of physical system for which suitable approaches to [/( r, ) exist are the transition metals and their alloys. To this class of models belong the glue model and the embedded atom method. Systems where chemical bonds of molecules are broken or created are much more difficult to describe, since the only way to get a proper description of a reaction all the way between reactant and products would be to solve the quantum-mechanical problem at each step of the reaction. 

Kby the potential function V = in one dimension. The corse-g quantum-mechanical problem, which leads to the wavefunctions panted in Table 5-1, yielded the expression, for the energy s hv° (v + )... 

The basic quantum-mechanical problem is to formulate the wave-like description of an electron moving in the electrostatic field of the nuclei and other electrons. If the potential energy of an electron at a point x, y, z) is V %, y, z), this is accomplished by solving the well-known Schrodinger equation for a wave function w(%, y. z)... 

The potential energy of the system U(q) can be determined from the solution of the corresponding quantum-mechanical problem for electrons (in this case, it is called first-principles MD), or by using empirical energy functionals (they will be discussed in detail below). 

The quantum mechanical problem involves the solution of the Schrddinger equation in its appropriate form, given a potential energy function V x) which descnbes the system concerned. In its simplest (one-dimensional) form the Schrddinger equation is... 

To some extent the pair function methods seem to fall between two stools. They attempt to utilize chemical intuition in solving the complex quantum mechanical problem, and they do indeed possess an appealing simplicity of interpretation in terms of electron-pair bonds, especially at the GVB or separated pair level. The price that must be paid for a wavefunction of such an easily interpretable form is that such methods are unlikely, in general, to yield chemically accurate potential energy surfaces. 

For application in the quantum mechanical problem of determining the atomic or molecular electronic structure we need the potential energy function V r) = often simply called the potential for brevity. 

The standard quantum mechanical problem of the harmonic oscillator may be used to demonstrate the diffusion quantum Monte Carlo method. The system is illustrated in Figure 2. The potential energy is given by the function V = Vikx. The potential energy may be shifted by an arbitrary constant energy to make V negative in the central region near x = 0 and positive away from the center. 

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