Coulomb functions numerical calculation

The Darwin potential generates the logarithmic correction to the nonrela-tivistic Schrodinger-Coulomb wave function in (3.65), and the result in (3.97) could be obtained by taking into account this correction to the wave function in calculation of the contribution to the Lamb shift of order a Za.ym. This logarithmic correction is numerically equal 14.43 kHz for the IS -level in hydrogen, and 1.80 kHz for the 2S level. 

For certain mathematical functions and operations it is necessary for the physicist to know their context, definition and mathematical properties, which we treat in the book. He does not need to know how to calculate them or to control their calculation. Numerical values of functions such as sinx have traditionally been taken from table books or slide rules. Modern computational facilities have enabled us to extend this concept, for example, to Coulomb functions, associated Legendre polynomials, Clebsch—Gordan and related coefficients, matrix inversion and diagonali-sation and Gaussian quadratures. The subroutine library has replaced the table book. We give references to suitable library subroutines. 

Here, emphasis is given to the application of few-state models in the description of the near-resonant vacancy exchange between inner shells. It is well known that the quantities relevant for inner-shell electrons may readily be scaled. Therefore, the attempt is made to apply as much as possible analytic functional forms to describe the characteristic quantities of the collision system. In particular, analytic model matrix elements derived from calculations with screened hydrogenic wave functions are applied. Hydrogenic wave functions are suitable for inner shells, since the electrons feel primarily the nuclear Coulomb field of the collision particles. Input for the analytic expressions is the standard information about atomic ionization potentials available in tabulated form. This procedure avoids a fresh numerical calculation for each new collision system. 

Numerical calculations of the Coulomb interaction O Eq. 23.68 are not trivial. They were simplified in Allan and Delerue (2008) by using a screened Coulomb potential involving one-electron wave functions which were derived for bulk semiconductor materials (Landsberg 1991). To calculate the impact ionization interaction from an initial state of Ret ri)Ye mi 0u(l>i)Re r2)Ye mi(d2,2) to a final state Re, ri)Ye mA i>h)Re,(r2)Ye, (02. 2) (the first term one in O Eq. 23.69), we notice... 

In this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

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