Self-repulsion energy

This corresponds to determining a set of LMOs which maximize the self-repulsion energy. 

In addition, the functional must somehow cancel the fictitious repulsion energy between an electron and itself, which arises if the electron density, due to all the electrons, is used to compute the Coulombic energy of a single electron. As discussed in Section 3.2.1, in HF theory cancellation of the self-repulsion energy results from the presence of the exchange operator in T. If this effect of Kj, in the Fock operator is not mirrored exactly by the functional chosen, the cancellation of the self-repulsion energy will not occur. 

Owing to their transferability and high self-repulsion energies, localized molecular orbitals should be useful in discussions of the correlation problem 27> in quantum mechanics 28>. Indeed, it is found that the correlation energy (the best SCF energy less the observed energy) is remarkably constant for electronic configurations that maintain well-... 

Edminton-Ruedenberg Maximization of the Coulomb self-repulsion energy [236]... 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

It should be noted that several of the proposed functionals violate fundamental restrictions, such as predicting correlation energies for one-electron systems (for example P86 and PW91) or failing to have the exchange energy cancel the Coulomb self-repulsion (Section 3.3, eq. (3.32)). One of the more recent functionals which does not have these problems is due to Becke (B95), which has the form... 

There are several things known about the exact behavior of Vxc(r) and it should be noted that the presently used functionals violate many, if not most, of these conditions. Two of the most dramatic failures are (a) in HF theory, the exchange terms exactly cancel the self-interaction of electrons contained in the Coulomb term. In exact DFT, this must also be so, but in approximate DFT, there is a sizeable self-repulsion error (b) the correct KS potential must decay as 1/r for long distances but in approximate DFT it does not, and it decays much too quickly. As a consequence, weak interactions are not well described by DFT and orbital energies are much too high (5-6 eV) compared to the exact values. 

On length scales larger than the thermal blob size in athermal and good solvents, the excluded volume repulsion energy is larger than the thermal energy kT and the polymer is a swollen chain of Njgj- thermal blobs (Fig. 3.14). The end-to-end distance of this chain is determined as a self-... 

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