Self-repulsion energy, wave function calculations

Cl calculations can be used to improve the quality of the wave-function and state energies. Self-consistent field (SCF) level calculations are based on the one-electron model, wherein each electron moves in the average field created by the other n-1 electrons in the molecule. Actually, electrons interact instantaneously and therefore have a natural tendency to avoid each other beyond the requirements of the Exclusion Principle. This correlation results in a lower average interelectronic repulsion and thus a lower state energy. The difference between electronic energies calculated at the SCF level versus the exact nonrelativistic energies is the correlation energy. 

In molecular orbital (MO) theory, which is the most common implementation of QM used by chemists, electrons are distributed around the atomic nuclei until they reach a so-called self-consistent field (SCF), that is, until the attractive and repulsive forces between all the particles (electrons and nuclei) are in a steady state, and the energy is at a minimum. An SCF calculation yields the electronic wave function 4C (the electronic motion being separable from nuclear motion thanks to the Born-Oppenheimer approximation). This is the type of wave function usually referred to in the literature and in the rest of this chapter. 

Quantum mechanics (QM) can be further divided into ab initio and semiempiri-cal methods. The ab initio approach uses the Schrodinger equation as the starting point with post-perturbation calculation to solve electron correlation. Various approximations are made that the wave function can be described by some functional form. The functions used most often are a linear combination of Slater-type orbitals (STO), exp (-ax), or Gaussian-type orbitals (GTO), exp (-ax2). In general, ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. 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, and these in turn are used to calculate a new repulsion energy. The process is continued until convergence occurs and self-consistency is achieved. 

From the above discussion it follows that the calculation of the Coulomb repulsion energy represents the most demanding computational task in Eq. 16.18. The introduction of the variational approximation of the Coulomb potential (Dunlap et al. 1979 Mintmire and Dunlap 1982 Mintmire et al. 1982) reduces the formal scaling of this term to x M, where M is the number of auxiliary functions which is usually three to five times N. This technique is nowadays used in most LCGTO-DFT programs. It is identical to the so-called resolution of the identity (RI) (Flores-Moreno and Ortiz 2009 Hamel et al. 2001 Vahtras et al. 1993) that cropped up in wave function methods, too. The variational approximation of the Coulomb potential, as implemented in deMon2k, is based on the minimization of the following self-interaction term ... 

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