Electronic wave functions Electron-repulsion potentials

From electronic structure theory it is known that the repulsion is due to overlap of the electronic wave functions, and furthermore that the electron density falls off approximately exponentially with the distance from the nucleus (the exact wave function for the hydrogen atom is an exponential function). There is therefore some justification for choosing the repulsive part as an exponential function. The general form of the Exponential - R Ey w function, also known as a ""Buckingham " or ""Hill" type potential is... 

The correlation of electron motion in molecular systems is responsible for many important effects, but its theoretical treatment has proved to be very difficult. Thus many quantum valence calculations use wave functions which are adjusted to optimize kinetic energy effects and the potential energy of interaction of nuclei and electrons but which do not adequately allow for electron correlation and hence yield excessive electron repulsion energy. This problem may be subdivided into cases of overlapping and nonoverlapping electron distributions. Both are very important but we shall concern ourselves here with only the nonoverlapping case. 

The energy due to the external potential is determined simply by the density and is therefore independent of the wave function generating that density. Hence, it is the same for all wave functions integrating to a particular density and we can separate it from the kinetic and electron-electron repulsion contributions... 

To understand how Kohn and Sham tackled this problem, we go back to the discussion of the Hartree-Fock scheme in Chapter 1. There, our wave function was a single Slater determinant SD constructed from N spin orbitals. While the Slater determinant enters the HF method as the approximation to the true N-electron wave function, we showed in Section 1.3 that 4>sd can also be looked upon as the exact wave function of a fictitious system of N non-interacting electrons (that is electrons which behave as uncharged fermions and therefore do not interact with each other via Coulomb repulsion), moving in the elfective potential VHF. For this type of wave function the kinetic energy can be exactly expressed as... 

Qualitatively, internal orbitals are contracted towards the nucleus, and the Radon core shrinks and displays a higher electron density. External electrons are much more efficiently screened from the nucleus, and the repulsion described by the (Hartree-Fock) central potential U(rj) in Eq. (7) of the preceding subsection is greatly enhanced. In fact, wave functions and eigenstates of external electrons calculated with the relativistic and the non-relativistic wave equations differ greatly. 

The Hamiltonian operator in Eq. 1 contains sums of different types of quantum mechanical operators. One type of operator in Ti gives the kinetic energy of each electron in by computing the second derivative of the electron s wave function with respect to all three Cartesian coordinates axes. There are also terms in H that use Coulomb s law to compute the potential energy due to (a) the attraction between each nucleus and each electron, (b) the repulsion between each parr of electrons, and (c) the repulsion between each pair of nuclei. 

Having solved for the electronic wave functions and energies, we use the electronic energy including nuclear repulsion U as the potential energy in the Schrodinger equation for nuclear motion ... 

At sufficiently close approach, overlap of electronic wave functions leads to a repulsive potential with a sharp spatial dependence, usually modeled as e or r" with n about 12. Together the repulsive and attractive forces create a potential well with a minimum at interatomic spacing of a few angstroms. 

It will now be shown that the existence of quantum potential energy eliminates the need to allow for repulsion between sub-electronic charge elements in an extended electron fluid. An electron, whatever its size or shape is described by a single wave function that fixes the electron density at any point... 

A common alternative is to synthesize approximate state functions by linear combination of algebraic forms that resemble hydrogenic wave functions. Another strategy is to solve one-particle problems on assuming model potentials parametrically related to molecular size. This approach, known as free-electron simulation, is widely used in solid-state and semiconductor physics. It is the quantum-mechanical extension of the classic (1900) Drude model that pictures a metal as a regular array of cations, immersed in a sea of electrons. Another way to deal with problems of chemical interaction is to describe them as quantum effects, presumably too subtle for the ininitiated to ponder. Two prime examples are, the so-called dispersion interaction that explains van der Waals attraction, and Born repulsion, assumed to occur in ionic crystals. Most chemists are in fact sufficiently intimidated by such claims to consider the problem solved, although not understood. 

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