Schrodinger equation multi-electron approximation

This equation as well as multi-partiele Schrodinger equation and all approximate equations which follow are given in so-called atontic units. This allows fundamental constants as well as the mass of the electron to be folded in. 

Since the exact solution of Schrodinger s equation for multi-electron, multi-nucleus systems turned out to be impossible, efforts have been directed towards the determination of approximate solutions. Most modern approaches rely on the implementation of the Born-Oppenheimer (BO) approximation, which is based on the large difference in the masses of the electrons and the nuclei. Under the BO approximation, the total wave-function can be expressed as the product of the electronic il/) and nuclear (tj) wavefunctions, leading to the following electronic and nuclear Schrodinger s equations ... 

Thus, the method described above allows us to obtain a number of new physical results partially presented in this communication. These calculations are carried out in the Hartree-Fock approximation for multi-electron systems and are exact solutions of the Schrodinger equation for the single-electron case. As the following development of the method we plan to implement the configuration interaction approach in order to study correlation effects in multi-electron systems both in electric and magnetic fields. 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

Under adiabatic approximation, the Schrodinger equation solution of multi-particle system can be written as the product of a nuclear wave function R) and an electronic wave function The electron wave function > i(f,R is... 

Schrodinger s equation cannot be solved exactly, especially for multi-electron atoms, mainly because of the effects determined by the attraction and rejection of electrons phenomena. However, by using different methods of approximation, one can obtain satisfactory results. 

From a computational perspective, the ideal would be to solve Schrodinger s equation one electron at a time, to give N one-electron functions. These equations would then be summed to generate the complete multi-electron solution. But the electron-electron repulsion (term (iv)) presents us with a serious problem it states that the behavior of each electron in the system influences that of all the others. It is this correlated behavior that means that we cannot describe each electron individually without making some more approximations, and it is for this reason that we cannot obtain exact solutions to the Schrodinger equation for multi-electron systems. 

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