Many-electron problem

At this point, it is appropriate to make some conmrents on the construction of approximate wavefiinctions for the many-electron problems associated with atoms and molecules. The Hamiltonian operator for a molecule is given by the general fonn... 

Another fundamental deficiency in the 1-electron theory is its failure to account for the significant role played by electron-electron interaction in the charge-transfer process. An approximate solution to this difficult many-electron problem appeared towards the end of the sixties (Edwards and Newns 1967, Grimley 1967, Newns 1969), which tackled it by adopting a modified version of the work of Anderson (1961) on dilute magnetic impurities... 

Brout, R., Phys. Rev. 115, 824 (1959) Brout, R., and Carruthers, P., Lectures on the Many-Electron Problem, Interscience Publishers, New York, 1963. 

If many atoms are bound together, for example in a crystal, their atomic orbitals overlap and form energy bands with a high density of states. Different bands may be separated by gaps of forbidden energy for electrons. The calculation of electron levels in the periodic potential of a crystal is a many-electron problem and requires several approximations for a successful solution. 

It was fairly straightforward to modify Bohr s model to include the idea of energy sublevels for the hydrogen spectrum and for atoms or ions with only one electron. There was a more fundamental problem, however. The model still could not explain the spectra produced by many-electron atoms. Therefore, a simple modification of Bohr s atomic model was not enough. The many-electron problem called for a new model to explain spectra of all types of atoms. However, this was not possible until another important property of matter was discovered. 

A solution to the many-electron problem, including correlation, can be obtained directly, without the use of the SCF approximation. 

J. E. Harriman, Limitation on the density-equation approach to many-electron problems. Phys. Rev. A 19, 1893 (1979). 

Let us start by considering the general many-electron problem of Ng valence electrons, which contribute to chemical bonding, and A ion ions, which contain the nuclei and the tightly bound core electrons. The positions of the electrons and ions are given by r, and Rj, respectively, referred to the same arbitrary origin. This problem can be described quantum-mechanically, in the absence of external fields, by the Hamilton operator Ho ... 

So far we have studied the general many-electron problem of electrons and Nion fixed or very slow ions. Let us now apply the above-developed formalism to molecules and solids. 

Methods are introduced for generating many-electron Sturmian basis sets using the actual external potential experienced by an N-electron system, i.e. the attractive potential of the nuclei. When such basis sets are employed, very few basis functions are needed for an accurate representation of the system the kinetic energy term disappears from the secular equation solution of the secular equation provides automatically an optimal basis set and a solution to the many-electron problem is found directly, including electron correlation, and without the self-consistent field approximation. In the case of molecules, the momentum-space hyperspherical harmonic methods of Fock, Shibuya and Wulfman are shown to be very well suited to the construction of many-electron Sturmian basis functions. 

Exchange/Correlation Functional. A function of the Electron Density and perhaps as well the gradient of the Electron Density. The functional form derives from exact solution of the Schrodinger Equation for an idealized many-electron problem. Used in Density Functional Models. 

For this many-electron problem, the contact term for calculating A will be... 

This book explores the connections between the theory of hyperspherical harmonics, momentum-space quantum theory, and generalized Sturmian basis functions and introduces methods which may be used to solve many-electron problems directly, without the use of the self-consistent-field approximation. ... 

In the Time Dependent Density Functional Theory (TDDFT) [16], the correlated many-electron problem is mapped into a set of coupled Schrodinger equations for each single electronic wavefunctions (o7 (r, t),j= 1, ), which yields the so-called Kohn-Sham equations (in atomic units)... 

Ehrenreich, H. and Cohen, M.H. (1959). Self-consistent field approach to the many-electron problem, Phys. Rev. 115, 786-790. 

In the previous section we briefly described the ab initio QC methods and the problems arising when they are applied to the modeling of complex systems. These problems cannot be considered as merely technical ones even if the computer power is sufficient and the required solution of the many electron problem can be obtained by brute force, the problem of the status of the result produced by the uncertainty introduced by poorly defined junction between the quantum and classical regions may still be important. Pragmatically, however, the resource requirements may have already... 

The orbital approach to the problem of electronic structure reduces a many-electron problem to a corresponding number of one-electron problems. The Hartree-Fock solution represents the best attainable description of the electronic structure of a many-electron system in terms of the one-electron orbital approach. 

An efficient way to solve a many-electron problem is to apply relativistic effective core potentials (RECP). According to this approximation, frozen inner shells are omitted and replaced in the Hamiltonian hnt by an additional term, a pseudopotential (UREP)... 

Sharp RT, Horton GG (1953) A variational approach to the unipotential many-electron problem, Phys Rev, 30 317-317... 

Goedecker S, Umrigar CJ (1998) Natural orbital functional for the many-electron problem, Phys Rev Lett, 81 866-869... 

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