Coulombic repulsion wave function calculations

Functionals. The difference between the Fock operator, in wave function based calculations and the analogous operator in DFT calculations is that the Coulomb and exchange operators in T are replaced in DFT by a functional of the electron density. In principle, this functional should provide an exact formula for computing the Coulombic interactions between an electron in a KS orbital and all the other electrons in a molecule. To be exact, this functional must include corrections to the Coulombic repulsion energy, computed directly from the electron density, for exchange between electrons of the same spin and correlation between electrons of opposite spin. 

The energies of the d-d-excitations in this model are obtained by diagonalizing the matrix of the Hamiltonian constructed in the basis of rid-electronic wave functions (nd is the number of d-electrons). Matrix elements of the Hamiltonian are expressed through the parameters describing the crystal field and those of the Coulomb repulsion of d-electrons, which are Slater-Condon parameters Fk, k = 0,2,4, or the Racah parameters A, B, and C. In the simplest version of the CFT these quantities are considered empirical parameters and determined by fitting the calculated excitation energies to the experimental ones. 

Non dynamical electron correlation is the part of the total correlation that is taken into account in a CASSCF calculation that correlates the valence electrons in valence orbitals. Physically, the non dynamical electron correlation is a Coulomb correlation that permits the electrons to avoid one another and reduce their mutual repulsion as much as possible with respect to a given zero order electronic structure defined by the Hartree-Fock wave function. In VB terms, the non dynamical correlation ensures a correct balance between the ionic and covalent components of the wave function for a given electronic system. The dynamical correlation is just what is still missing to get the exact nonrelativistic wave function. 

The theorem just proved shines in its simplicity. People thought that the wave function, usually a very complicated mathematical object (that depends on 3N space and N spin coordinates) is indispensable for computing the properties of the system. Moreover, the larger the system, the worse the difficulties in calculating it (recall Chapter 10 with billions of excitations, nonlinear parameters, etc.). Besides, how can we interpret such a complex object This is a horribly complex problem. And it turns out that everything about the system just sits in p(r), a function of position in our well-known 3-D space. It turns out that information about nuclei is hidden in such a simple object. This seems trivial (cusps), but it also includes much more subtle information about how electrons avoid each other due to Coulombic repulsion and the Pauli exclusion principle. 

The Hartree-Fock (HF) method is widely used in electronic structure calculations, which is based on the following assumptions (1) Bom-Oppenheimer approximation, (2) the many-electron Hamiltonian is replaced with an effective one-electron Hamiltonian which acts on orbitals (one-electron wave functions), (3) the Coulomb repulsion between electrons is represented in an averaged way. 

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