Molecule centered expansions

For the purposes of this review it is convenient to focus attention on that class of molecules in which the valence electrons are easily distinguished from the core electrons (e.g., -n electron systems) and which have a large number of vibrational degrees of freedom. There have been several studies of the photoionization of aromatic molecules.206-209 In the earliest calculations either a free electron model, or a molecule-centered expansion in plane waves, or coulomb functions, has been used. Only the recent calculation by Johnson and Rice210 explicitly considered the interference effects which must accompany any process in a system with interatomic spacings and electron wavelength of comparable magnitude. The importance of atomic interference effects in the representation of molecular continuum states has been emphasized by Cohen and Fano,211 but, as far as we know, only the Johnson-Rice calculation incorporates this phenomenon in a detailed analysis. 

Molecule-Centered Expansions Versus Atom-Atom Models... 

At the restoration stage, a one-center expansion in the spherical harmonics with numerical radial parts is most appropriate both for orbitals (spinors) and for the description of external interactions with respect to the core regions of a considered molecule. In the scope of the discussed two-step methods for the electronic structure calculation of a molecule, finite nucleus models and quantum electrodynamic terms including, in particular, two-electron Breit interaction may be taken into account without problems [67]. 

One-center expansion was first applied to whole molecules by Desclaux Pyykko in relativistic and nonrelativistic Hartree-Fock calculations for the series CH4 to PbH4 [81] and then in the Dirac-Fock calculations of CuH, AgH and AuH [82] and other molecules [83]. A large bond length contraction due to the relativistic effects was estimated. However, the accuracy of such calculations is limited in practice because the orbitals of the hydrogen atom are reexpanded on a heavy nucleus in the entire coordinate space. It is notable that the RFCP and one-center expansion approaches were considered earlier as alternatives to each other [84, 85]. 

The scientific interests of Huzinaga are numerous. He initially worked in the area of solid-state theory. Soon, however, he became interested in the electronic structure of molecules. He studied the one-center expansion of the molecular wavefunction, developed a formalism for the evaluation of atomic and molecular electron repulsion integrals, expanded Roothaan s self-consistent field theory for open-shell systems, and, building on his own work on the separability of many-electron systems, designed a valence electron method for computational studies on large molecules. 

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. 

By weighting atoms by atomic charges, the first-order moment of charges is the —> dipole moment in neutral molecules. Moreover, for molecules with zero net charge and nonvanishing dipole moment, the center-of-dipole was defined as the appropriate molecule center for multipolar expansions to obtain rotational invariance [Silverman and Platt, 1996]. 

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. 

In a larger molecule convergence will be slower. On the right side of Fig. 47 similar results are reported for the aziridine molecule. The expansion containing terms up to octopole is sufficiently accurate for distances more than 3 A from the center of the molecule. Another test is to analyze the errors introduced by using the multipolar expansion. 

The curves refer to H2O (a), NHs(b), aziridine (c) (solid curves). For ease of visualization, rj is reported on a logarithmic scale, while distances are measured in fractions of the medium van der Waals molecular radius of the molecule considered. The rj values rapidly decrease with increasing distance and, in the case of the two small molecules, they are less than 10 % at a distance equal to 2 VdW. With aziridine, it is necessary to go to distances larger than 1.4 i VdW in order to have comparable rj values. Fig. 48 gives analogous results obtained from a many-center expansion of the b) type (dashed curves). As was to be expected, at... 

Molecular calculations. Molecular relativistic ab initio DF codes with electron correlation are still in development (see, for example. Refs. 86,87 and the corresponding chapters in this issue). Correlation effects are included there at the Cl [88], MBPT (the seccmd order Moller-Plesset, MP2 [89,90]), or the CCSD levels [91,92]. They are too computer time intensive and still not sufficiently economic to be applied to the heaviest element systems in a routine manner, especially to those studied experimentally. DF molecular codes, some without correlation, were recently used for small molecules of the heaviest elements. The main aim of those calculations was to study relativistic and correlation effects on some model systems like lllH, 117H, 113H, (113)2 or II4H4 [93-99]. Some pioneer calculations by PyykkO for Rfitt and SgHe using the one-center expansion DF method should also be mentioned here [100-102]. 

The number of sites reflects the possibility we have examined, and advocated, of using many-center expansions to improve the representation. Each expansion center will be a site. There are models with one, two, and more sites. This sequence of increasing complexity reaches the number of heavy atoms of the molecule and then the whole number of atoms, including hydrogens. It is not limited to the nuclei as expansion sites. There are potentials introducing other locations of sites, in substitution or in addition to the nuclei. For example potentials widely used in simulations adopt for water a four-site model other potentials (rarely used in simulations) prefer to use the middle of the bonds instead of (or in addition to) nuclei. Each site of the molecule must be combined with the sites of the second (and other) molecule to give the potential. 

In the case of atoms and molecules with central symmetry, in which case a one-center expansion is feasible, these equations are solved numerically. In the case of molecules with no central symmetry and in the case of crystals, this procedure is not possible and one must expand the molecular orbitals (MOs) or crystal orbitals (COs) as a linear combination of some basis functions (LCAO expansion). This was first conducted for molecules by Malli and Oreg< and applications can be found for diatomic or linear molecules, the only exception being the HjCO molecule, which was treated by Aoyama et... 

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