Coulomb approximation method

Table 12.5 The oscillator strengths of the El transitions in Eul (Table 12.4) theoretical data - the Coulomb approximation method (columns A, B and C are corresponding to the gauges of the photon propagator Coulomb, Babushkin, Landau), multiconfiguration DF method (column D), experimental data (columns El, E2)...

In Table 12.5, we present our results of calculating (colimm F) the oscillator sttengths of the electric dipole transitions (listed in Table 12.4). Here, the optimized DF scheme within the REA has been used. For comparison, there are also listed the results of calculations within the Coulomb approximation method (columns... 

Model potential methods and their utilization in atomic structure calculations are reviewed in [139], main attention being paid to analytic effective model potentials in the Coulomb and non-Coulomb approximations, to effective model potentials based on the Thomas-Fermi statistical model of the atom, as well as employing a self-consistent field core potential. Relativistic effects in model potential calculations are discussed there, too. Paper [140] has examples of numerous model potential calculations of various atomic spectroscopic properties. 

The problem of finding the ground-state properties of a system consisting of more than one electron is very important in the study of atoms, molecules and solids. In order to obtain the ground-state properties, one has to solve the Schrodinger equation for the system under investigation. Since no exact solution exists to this problem for Coulomb systems, many different approximate methods have been developed for approaching this subject. 

An alternative to the perturbation theory approach is the approximate method of Gordon and Kim.60 In this method the electron density is first calculated as the sum of the densities of the separate atoms and the energy is then obtained as the sum of a Coulomb term calculated exactly, and kinetic energy, exchange, and correlation terms calculated from the free electron gas model. Though it worked well for larger... 

Lindhard J, Nielsen V, Scharff M, Thomsen PV (1963) Integral equations governing radiation effects (Notes on atomic collisions (III). KDan Vidensk Matematisk-fysiske Meddelelser 33 1-42 Lindhard J, Nielsen V, Scharff M (1968) Approximation method in classical scattering by screened coulomb fields (Notes on atomic colhsiorts (I). K Dan Vidensk Selsk Matematisk-fysiske Meddelelser... 

Lindhard, I, Nielsen, V., Scharff, M. Approximation method in classical scattering by screened coulomb fields (notes on atomic collisions I). Mat. Fys. Medd. Dan. Vidensk. Selsk. 36(10) (1968)... 

One approximate method for obtaining Eq. 7.182 in a central field form wa.s introduced by Hartree and named the self-consistent field approach. This method regards each electron in a many-electron atom as moving in the temporarily fixed field of the remaining electrons. The system can now be described in terms of one-electron wavefunctions (or orbitals) j(rj). The non-Coulomb potential energy for the jth electron is then V y(ry) atid this contains the other electronic coordinates only as parameters. Vjirj) can be chosen to be spherically symmetric. The computational procedure is to solve the Schrodinger equation for every electron in its own central field and then to make the wavefunctions, so found, self-consistent with their potential fields. The complete wavefunction for the system is a product of the one-electron functions. 

Picklesimer et al. (PTTW) [Pi 84] suggested another approximation method aimed at avoiding the sharp cut-off problem. They separated the full optical potential into three parts corresponding to Coulomb due to a point charge, remaining Coulomb due to the finite charge distribution, and nuclear, as follows ... 

For very small molecular systems both the solute and the solvent molecules can be treated using accurate levels of theory. However, when the solvation effects are studied, the number of molecules is usually large (refer to the explicit solvent molecules in Fig. 17.2), and the employment of approximate methods becomes mandatory. A very cost-effective and reliable approach is to treat the solute at a high level of theory, usually based on QM methods, and the solvent molecules at a lower level of theory, usually based on MM methods. Since only non-covalent bonds couple the different layers, the interaction energy between the solute (QM subsystem) and the solvent (MM subsystem) contains only Coulomb (Cou) and van der Waals (vdW) contributions ... 

The solutions to this approximation are obtained numerically. Fast Fourier transfonn methods and a refomuilation of the FINC (and other integral equation approximations) in tenns of the screened Coulomb potential by Allnatt [M are especially useful in the numerical solution. Figure A2.3.12 compares the osmotic coefficient of a 1-1 RPM electrolyte at 25°C with each of the available Monte Carlo calculations of Card and Valleau [ ]. 

In the connnonly used atomic sphere approximation (ASA) [79], the density and the potential of the crystal are approximated as spherically synnnetric within overlapping imifiBn-tin spheres. Additionally, all integrals, such as for the Coulomb potential, are perfonned only over the spheres. The limits on the accuracy of the method imposed by the ASA can be overcome with the fiill-potential version of the LMTO (FP-LMTO)... 

Nevertheless, the examination of the applicability of the crude BO approximation can start now because we have worked out basic methods to compute the matrix elements. With the advances in the capacity of computers, the test of these methods can be done in lower and lower cost. In this work, we have obtained the formulas and shown their applications for the simple cases, but workers interested in using these matrix elements in their work would find that it is not difficult to extend our results to higher order derivatives of Coulomb interaction, or the cases of more-than-two-atom molecules. 

In Chapter IX, Liang et al. present an approach, termed as the crude Bom-Oppenheimer approximation, which is based on the Born-Oppen-heimer approximation but employs the straightforward perturbation method. Within their chapter they develop this approximation to become a practical method for computing potential energy surfaces. They show that to carry out different orders of perturbation, the ability to calculate the matrix elements of the derivatives of the Coulomb interaction with respect to nuclear coordinates is essential. For this purpose, they study a diatomic molecule, and by doing that demonstrate the basic skill to compute the relevant matrix elements for the Gaussian basis sets. Finally, they apply this approach to the H2 molecule and show that the calculated equilibrium position and foree constant fit reasonable well those obtained by other approaches. 

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