Molecular modelling Schrodinger equation

It is straightforward to write down and solve the many-electron Schrodinger equation if it is assumed that the electrons do not interact, or interact only to a very small extent. Indeed, it is on this premise that the fabric of modem qualitative molecular orbital theory is based. For the two electrons in a helium atom [Z = 2] for example, this independent particle model Schrodinger equation is simply... 

Various kinds of mixed quantum-classical models have been introduced in the literature. We will concentrate on the so-called quantum-classical molecular dynamics (QCMD) model, which consists of a Schrodinger equation coupled to classical Newtonian equations (cf. Sec. 2). 

A theoretical model should be uniquely defined for any given configuration of nuclei and electrons. This means that specifying a molecular structure is all that is required to produce an approximate solution to the Schrodinger equation no other parameters are needed to specify the problem or its solution. 

The orbital model would be exact were the electron repulsion terms negligible or equal to a constant. Even if they were negligible, we would have to solve an electronic Schrodinger equation appropriate to CioHs " " in order to make progress with the solution of the electronic Schrodinger equation for naphthalene. Every molecular problem would be different. 

Exact solutions to the electronic Schrodinger equation are not possible for many-electron atoms, but atomic HF calculations have been done both numerically and within the LCAO model. In approximate work, and for molecular applications, it is desirable to use basis functions that are simple in form. A polyelectron atom is quite different from a one-electron atom because of the phenomenon of shielding", for a particular electron, the other electrons partially screen the effect of the positively charged nucleus. Both Zener (1930) and Slater (1930) used very simple hydrogen-like orbitals of the form... 

Most semi-empirical models are based on the fundamental equations of Hartree-Fock theory. In the following section, we develop these equations for a molecular system composed of A nuclei and N electrons in the stationary state. Assuming that the atomic nuclei are fixed in space (the Born-Oppenheimer approximation), the electronic wavefunction obeys the time-independent Schrodinger equation ... 

Quantum mechanical methods follow a similar path, except that the starting point is the solution of the Schrodinger equation for the system under investigation. The most successful and widely used method is that of Density Functional Theory. Once again, a key point is the development of a realistic model that can serve as the input to the computer investigation. Energy minimization, molecular dynamics, and Monte Carlo methods can all be employed in this process. 

Molecular Orbital Models. Methods based on writing the many-electron solution of the Electronic Schrodinger Equation in terms of a product of one-electron solutions (Molecular Orbitals). 

As a result of these assumptions, qualitative molecular orbital models can be developed in which one assumes that each mo (f>i obeys a one-electron Schrodinger equation... 

We know that not all solids conduct electricity, and the simple free electron model discussed previously does not explain this. To understand semiconductors and insulators, we turn to another description of solids, molecular orbital theory. In the molecular orbital approach to bonding in solids, we regard solids as a very large collection of atoms bonded together and try to solve the Schrodinger equation for a periodically repeating system. For chemists, this has the advantage that solids are not treated as very different species from small molecules. 

An overview of the approaches that have been taken to linking different theoretical and computational modeling descriptions is also provided in Fig. 2 The first principles (QC) descriptions are based on the Schrodinger equation and the Bom Oppenheimer approximation as realized in most chemical applications by density functional [14] or Hartree-Fock [15] methods. Molecular dynamics (MD) methods [16], based on classical Newtonian me-... 

The basis of ab initio modeling of materials is the time-independent Schrodinger equation in which the state of a molecular system is described with a wavefunction ... 

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