Continuum orbital expansion

For simplicity, the constant of proportionality which takes care of different normalizations of the radial functions involved has been omitted, the symmetry of the wavefunctions with respect to an interchange of r2 and r2 is not incorporated explicitly, and the presence of continuum orbitals is indicated only through the integral symbol.) The dntrmtr 2 coefficients then describe the quality of this expansion if only certain truncated sets of basis functions are selected, because for a complete basis set... 

On the right hand side, the ion state is still antisymmetric in the first N — 1 electrons, but the continuum orbital is occupied exclusively by the iVth. That it is possible to write this may be seen by expanding the left hand side as a sum of N terms, in which the continuum orbital is occupied by a different electron in turn. Each term must have the same value, since all electrons are equivalent in the state t>. Thus we have N equal terms, with a normalization factor of N K Replacing the ground state i in Eq. (10) by its expansion according to Eqs. (8) and (9), and using the fact that all the ionised states are orthogonal, we find ... 

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 a calculation of this matrix element one first changes the order of orbitals in such a way that the two different orbitals in the determinantal wavefunctions are at the same positions. Since in the expansion of the continuum function into partial waves, equ. (3.5a), only S is allowed, one gets... 

Expansion in a finite set of discrete pseudostates evidently converges much faster than expansion in true eigenstates of Hq (which would involve also continuum states). The same behaviour is observed for the expansion in eigenstates of Fq (the one-particle Fock Hamiltonian), so that convergence of conventional TDHF methods may prove rather slow, especially with large bases of atomic orbitals... 

The description of the photoionization process by means of a method based on the Density Functional Theory (DFT) is reviewed. The present approach is based on a basis set expansion in B-spline functions, which are particularly suited to deal with the boundary conditions of the continuum states. Both Kohn-Sham (KS) and its extension to the Time Dependent (TD-DFT) formalism are considered. The computational aspects of the method are described the implementations for atoms, for molecules in One Centre Expansion (OCE) and for molecules with the Linear Combination of Atomic Orbital (LCAO) scheme. The applications of the method are discussed, from atoms to large fullerenes, with comparison with available experimental data. 

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