Schrodinger equation independent particle model

In order to define orbitals in a many-electron system, two approaches are possible, which we may refer to as constructive and analytic . The first approach is more common one makes the ad hoc postulate that every electron can be associated with one orbital and the total wave function can be constructed from these orbitals. Then, one is led to an effective one-electron Schrodinger equation from one electron in the field of the other electrons. The underlying model is the independent particle model (IPM). When following the constructive way, one does not know a priori whether the model is a good approximation to the actual physical situation one only knows that it cannot be rigorously correct. The merit of this approach is its relative simplicity from both the mathematical and physical points of view. 

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... 

Unfortunately, the determination of exact solutions of the SchrOdinger equation is intractable for almost all systems of practical interest. On the other hand, independent particle models are not sufficiently accurate for most studies of molecular structure. In particular, the Hartree-Fock model, which is the best independent particle model in the variational sense, does not support sufficient accuracy for many applications. Some account of electron correlation effects has to be included in the theoretical apparatus which underpins practical computational methods. Although the energy associated with electron correlation is a small fraction of the total energy of an atom or molecule, it is of the same order as most energies of chemical interest. However, such theories may not be true many-body theories. They may contain terms which scale non-linearly with electron number and are therefore unphysical and should be discarded. Any theory which contains such unphysical terms is not acceptable as a true many-body method. Either the theory is abandoned or corrections, such as that of Davidson [7] which is used in limited configuration interaction studies, are made in an attempt to restore linear scaling. 

The central field approximation and the simplifications which result from it allow one to construct a highly successful quantum-mechanical model for the AT-electron atom, by using Hartree s principle of the self-consistent field (SCF). In this method, one equation is obtained for each radial function, and the system is solved iteratively until convergence is obtained, which leaves the total energy stationary with respect to variations of all the functions (the variational principle ). The Hartree-Fock equations for an AT-electron system are equivalent to several one electron radial Schrodinger equations (see equation (2.2)), with terms which make the solution for one orbital dependent on all the others. In essence, the full AT-electron problem is approximated by a smaller number of coupled one-electron problems. This scheme is sometimes (somewhat inappropriately) referred to as a one-electron model in fact, the Hartree-Fock equations are a genuine AT-electron theory, but describe an independent particle system. 

Although a separation of electronic and nuclear motion provides an important simplification and appealing qualitative model for chemistry, the electronic Schrodinger equation is still formidable. Efforts to solve it approximately and apply these solutions to the study of spectroscopy, stmcture and chemical reactions form the subject of what is usually called electronic structure theory or quantum chemistry. The starting point for most calculations and the foundation of molecular orbital theory is the independent-particle approximation. 

The theoretical description of atoms and molecules has to rely on approximate solutions to Schrodinger s equation. For the standard methods in current use, the starting approximation treats the electrons as if they were independent particles. The advantage of this approach is the ease with which it can be formulated even for very large systems [1]. However, the correlation of electronic motion often has a major role, particularly in chemical bonding and reactivity. The independent-electron approximation does not provide a qualitative model for correlation effects, nor an efficient basis for evaluating numerical contributions from correlation. 

Nilsson (1955) extended the single-particle shell model to deformed potentials. The solutions of the Schrodinger equation then depend on deformation also. In the independent-partide model (Wagemans 1991) the sum of the single-particle energies of an even-even nucleus is given by... 

To obtain further information on the nature of the Dirac wave function, we can solve the equation for a simple model system. The simplest case is the time-independent equation for a free particle. In the nonrelativistic case the Schrodinger equation for a free particle moving along the x axis is... 

In this section we solve the time-independent Schrodinger equation for the two simplest model systems the particle in a box and the free particle. This analysis will show how the wave function and the values of the energy are determined by the Schrodinger equation and the three conditions obeyed by the wave function. 

In 1926 the physicist Llewellyn Thomas proposed treating the electrons in an atom by analogy to a statistical gas of particles. No electron-shells are envisaged in this model which was independently rediscovered by Italian physicist Enrico Fermi two years later, and is now called the Thomas-Fermi method. For many years it was regarded as a mathematical curiosity without much hope of application since the results it yielded were inferior to those obtained by the method based on electron orbitals. The Thomas-Fermi method treats the electrons around the nucleus as a perfectly homogeneous electron gas. The mathematical solution for the Thomas-Fermi model is universal , which means that it can be solved once and for all. This should represent an improvement over the method that seeks to solve Schrodinger equation for every atom separately. Gradually the Thomas-Fermi method, or density functional theories, as its modem descendants are known, have become as powerful as methods based on orbitals and wavefunctions and in many cases can outstrip the wavefunction approaches in terms of computational accuracy. 

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