Single-center expansion

Stener and co-workers [59] used an alternative B-spline LCAO density functional theory (DFT) method in their PECD investigations [53, 57, 60-63]. In this approach a normal LCAO basis set is adapted for the continuum by the addition of B-spline radial functions. A large single center expansion of such... 

One way around the slow convergence of single-center expansions is a multi-center multipole expansion [225-229], Several workers have explored the utility ofDME within the SCRF framework [112, 164, 171], Of course, when the multipoles do not reside at atomic positions, it is clear that calculation of such quantities as analytic energy derivatives will become more difficult. 

Since the hydrogenlike Sturmian basis functions form a complete set, the term %i,o,o(xy-R) can be represented as a single-center expansion in terms of functions localized at the origin ... 

Gianturco, F.A., Uiompson, D.G. and Jain, A. (1995). Electron scattering from polyatomic molecules using a single-center expansion formulation, in Computational Methods for Electron-Molecule Collisions, eds. W.M. Huo and F.A. Gianturco (Plenum, New York), pp. 75-118. 

Since dispersion has a weaker anisotropy than other terms, it is also quite sufficient to use a single-centered expansion... 

The key is that a single-center expansion of the transition density, implicit in a multipolar expansion of the Coulombic interaction potential, cannot capture the complicated spatial patterns of phased electron density that arise because molecules have shape. The reason is obvious if one considers that, according to the LCAO method, the basis set for calculating molecular wavefunctions is the set of atomic orbital basis functions localized at atomic centers a set of basis functions localized at one point in a molecule is unsatisfactory. 

In the next section we will discuss the approach we have developed for obtaining the molecular Hartree-Fock continuum orbitals. We will discuss how our approach is based on the Schwinger variational method and how in its present form it can be viewed as a hybrid method that uses both the basis-set expansion techniques of quantum chemistry and the numerical single-center expansion techniques of atomic collision physics. We will then discuss the results of applications of this approach to study shape resonances in the photolonlzatlon of several molecules, e.g., N2, CO, CO2, C2H2, and C2N2. These results will also be compared with available experimental data and with the results of studies of these same systems by different methods and models. 

For the expansion of the Hartree-Fock molecular orbitals we have used either Slater or Cartesian Gaussian functions. In addition to these basis functions we can also include spherical Gaussian functions in the initial scattering basis. A detailed discussion of the single-center expansion of Slater and Cartesian Gaussian functions has been given by Harris and Michels (19) and by Fliflet and McKoy (20), respectively. Spherical Gaussian functions, i.e.. 

These single-center expansions must be truncated at some suitable maximum value. These maximum values, the step sizes on the grid. 

This is a multi-center expansion of the wave function in terms of the incoming waves of the system. It can also be expressed as a single center expansion... 

Abstract, In this paper I give an overview of the current status of knowledge of the four-body potential energy function and dynamics of the HF dimer. The discussion of potential energy functions includes both single-center expansions and multi-site functions. The discussion of d)mamics includes both intramolecular processes of the van der Waals dimer and diatom-diatom energy transfer collisions. 

The photoelectron orbital is expanded in a single-center expansion using the spherical harmonics, Yim 0,4>), as... 

Electrostatic energy The generalization of the single-center expansion is straightforward and we get... 

The factors fU i" (J, 1), called the reaction-field factors, appear to be dependent on the dielectric constant of the solvent, and on the shape of the cavity only. They can be determined fully analytically in the cases of a single center expansion and of a cavity with a regular shape, such as a sphere, a spheroid or an ellipsoid. The case of the sphere is particularly simple and corresponds to the well-known Kirkwood model of solvation the reaction-field factors are scalar quantities which depend on I and on the radius a of the sphere ... 

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