Multi-center integrals

For both types of orbitals, the coordinates r, 0, and (j) refer to the position of the electron relative to a set of axes attached to the center on which the basis orbital is located. Although Slater-type orbitals (STOs) are preferred on fundamental grounds (e.g., as demonstrated in Appendices A and B, the hydrogen atom orbitals are of this form and the exact solution of the many-electron Schrodinger equation can be shown to be of this form (in each of its coordinates) near the nuclear centers), STOs are used primarily for atomic and linear-molecule calculations because the multi-center integrals < XaXbl g I XcXd > (each... 

Because the calculation of multi-center integrals that are inevitable for ab initio method is very difficult and time-consuming, Hyper-Chem uses Gaussian Type Orbital (GTO) for ab initio methods. In truly reflecting a atomic orbital, STO may be better than GTO, so HyperChem uses several GTOs to construct a STO. The number of GTOs depends on the basis sets. For example, in the minimum STO-3G basis set HyperChem uses three GTOs to construct a STO. 

With development of solid-state detectors, relative x-ray intensities, such as, KP/Ka ratios, have been measured and compiled in the tabulated or graphical forms [2-4]. However, these values are still considered as an atomic property and compared with the theoretical calculations for free atoms [5]. This is because for calculation of x-ray emission rates in molecules it is necessary to perform multi-center integration for molecular wave function. Such calculations are tedious and require a lot of computation time and large memory capacity. 

The main difficulty in MO calculations consists in evaluation of the multi-center integrals. The DV-integration method is one of the powerful methods to avoid such difficulty and has been successfully used in the MO method. When this technique is applied to the calculations of the transition matrix elements, it makes the evaluations of x-ray emission rates in complex molecules easier. 

The first-principles calculation of the multi-center integral for the dipole-matrix element above is feasible by the use of DV numerical integration procedure( ... 

The calculation of the matrix elements (38) and (39) is for small elementary cells the most time-consuming part of the (R)FPLO approach. For the overlap matrix S, one- and two-center integrals have to be provided while the Hamiltonian matrix requires the calculation of one-, two- and three-center integrals. As both the orbital and potential functions involved are well localized, only a limited number of multi-center integrals have to be calculated. The one- and two-center-integrals are further simplified by the application of angular momentum rules to one- and two-dimensional integrations, respectively. There are however two points which make the calculation of these matrix elements (in principle) much more involved for the relativistic approach. At first, the... 

So far, approximations in the calculation of the matrix elements (66) were avoided. The computation of the one-center integrals is fast and there is neither a need nor an obvious scheme for simplifications. For small elementary cells and moderate numbers of fc-points, the numerical effort of the (R)FPLO approach is primarily determined by the multi-center integrals, especially by the three-center integrals. Given a certain geometry, the effort mainly depends on the number of different radial functions that have to be considered. 

We take advantage of the first point and neglect all multi-center integrals between small components. With this approximation, roughly a factor of two... 

Fig. 4. Total energy of fee Au vs. lattiee eonstant, using the Perdew-Wang 92 version of LDA [25] and a valence basis consisting of 5p, 6s, 6p, 5d and 5f states. Results with and without contributions of the small components to the multi-center integrals are compared. The related energy difference is shown in the upper panel at enlarged scale. The experimental value for the lattice constant is indicated by the vertical line.

Fig. 5. Computational effort (lower panel) and accuracy (upper panel) vs. maximal allowed error for the fit of radial wave functions in multi-center-integrals for CoPt with magnetic moments along the (OOl)-direction.

Yet, these formulas are not congenial with our atom-by-atom descriptions. This is primarily due to a host of two-electron Coulomb and exchange terms - such as valence-other-valence and core-other-valence multi-center integrals -which, besides being intrinsically complex, require beforehand specification of the boundaries delimiting the individual atoms in a molecule. Hence the idea of bypassing this sort of problem in favor of a considerably simplified approach, one that highlights the role of the electrostatic potentials. 

Note that the one-electron Hamiltonian effective matrix components differ from those of Eq. (43) in what they truly represent. In this form, it represents the kinetic energy plus the interaction of a single electron with the core electrons around all nuclei present. The other integrals appearing in Eq. (52) are generally called two-electron-multi-centers integrals and are written as ... 

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