Coulomb operator 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 Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

Calculation of the nonlogarithmic polarization operator contribution is quite straightforward. One simply has to calculate two terms given by ordinary perturbation theory, one is the matrix element of the radiatively corrected external magnetic field, and another is the matrix element of the radiatively corrected external Coulomb field between wave functions corrected by the external magnetic field (see Fig. 9.13). The first calculation of the respective matrix elements was performed in [34]. Later a number of inaccuracies in [34] were uncovered [22, 23, 40, 43, 44, 45] and the correct result for the nonlogarithmic contribution of order a Za) EF to HFS is given by... 

Classical anharmonic spring models with or without damping [9], and the corresponding quantum oscillator models seem well removed from the molecular problems of interest here. The quantum systems are frequently described in terms of coulombic or muffin tin potentials that are intrinsically anharmonic. We will demonstrate their correspondence after first discussing the quantum approach to the nonlinear polarizability problem. Since we are calculating the polarization of electrons in molecules in the presence of an external electric field, we will determine the polarized molecular wave functions expanded in the basis set of unperturbed molecular orbitals and, from them, the nonlinear polarizability. At the heart of this strategy is the assumption that perturbation theory is appropriate for treating these small effects (see below). This is appropriate if the polarized states differ in minor ways from the unpolarized states. The electric dipole operator defines the interaction between the electric field and the molecule. Because the polarization operator (eq lc) is proportional to the dipole operator, there is a direct link between perturbation theory corrections (stark effects) and electronic polarizability [6,11,12]. 

For d = 3, the solution of Eq. (7) is the familiar Coulomb wave function in 3 dimensions. For d 3 this is not true. This represents an additional problem since the non-relativistic calculations have to be done without an explicit knowledge of the wave function. Fortunately, cancelation of all divergences can be ensured on the operator level using the Schrodinger equation in d-dimensions. Once divergences are canceled, the limit d —> 3 can be taken and a non-trivial (but now finite) matrix elements can be easily computed. 

One guesses at an initial set of wave functions, , and constructs the Hartree-Fock Hamilton S which depends on the through the definitions of the Coulomb and exchange operators, (/ and One then calculates the new set of , and compares it (or the energy or the density matrix) to the input set (or to the energy or density matrix computed from the input set). This procedure is continued until the appropriate self-consistency is obtained. 

The method works as follows. The mass velocity, Darwin and spin-orbit coupling operators are applied as a perturbation on the non-relativistic molecular wave-functions. The redistribution of charge is then used to compute revised Coulomb and exchange potentials. The corrections to the non-relativistic potentials are then included as part of the relativistic perturbation. This correction is split into a core correction, and a valence electron correction. The former is taken from atomic calculations, and a frozen core approximation is applied, while the latter is determined self-consistently. In this way the valence electrons are subject to the direct influence of the relativistic Hamiltonian and the indirect effects arising from the potential correction terms, which of course mainly arise from the core contraction. 

So far we have made no approximations to the Coulomb and exchange terms in the potential, which are still expressed in terms of the large and small components. In an actual calculation we would not have the large and small components, only the approximate transformed wave function. The potentials must therefore be expressed in terms of the transformed wave functions and the transformed operators, taking into account that these are not exact. 

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