Pseudoatoms

A modification of the united-atom approach, called the anisotropic united-atom (AUA) model was the focus of extensive work by Karabomi et al. [362-365]. As in the other models of hydrocarbon chains described so far, the AUA approach to monolayers was preceded by work on alkanes [367]. hi the AUA model the interaction site is located at the geometrical mean of the valence electrons of the atoms it represents, while the pseudoatom itself is located at the carbon atom position. The movement of each interaction center depends on the conformation of the molecule as a whole. 

According to the aspherical-atom formalism proposed by Stewart [12], the one-electron density function is represented by an expansion in terms of rigid pseudoatoms, each formed by a core-invariant part and a deformable valence part. Spherical surface harmonics (multipoles) are employed to describe the directional properties of the deformable part. Our model consisted of two monopole (three for the sulfur atom), three dipole, five quadrupole, and seven octopole functions for each non-H atom. The generalised scattering factors (GSF) for the monopoles of these species were computed from the Hartree-Fockatomic functions tabulated by Clementi [14]. 

In the final stages of the refinement the positional parameters of the H atoms were kept fixed, and these atoms too were described with multipoles, up to the dipole level. For both poles of the H pseudoatoms the radial functions were again single exponentials, with n = 0, 1 for monopole and dipole respectively, and the a value was 2.48 bohr1. 

Stewart, R.F. (1976) Electron population analysis with rigid pseudoatoms, Acta Crystallogr., A, 32, 565-574. 

The least-squares Molly program based on the Hansen-Coppens model [10] was used to determine atomic coordinates, thermal parameters and multipolar density coefficients in scolecite. In the Hansen-Coppens model, the electron density of unit cell is considered as the superposition of the pseudo-atomic densities. The pseudoatom electron density is given by... 

NOEs that involve groups of protons with degenerate chemical shifts, in particular methyl groups, may be referred to pseudoatoms located in the center of the protons that they represent, and the upper bound is increased by a pseudoatom correction equal to the proton-pseudoatom distance [27, 28]. Another method that usually incurs a smaller loss of information [29] is to treat NOEs for groups of protons with degenerate chemical shifts as ambiguous distance constraints (see Eq. (13) below). 

Figure 1,8 Electron density plots along (110) plane of BeO (A) effective total electron density (pseudoatom approximation) (B) total electron density of lAM (C) deformation density (pseudoatom-IAM). From Downs (1992). Reprinted with permission of Springer-Verlag, New York.

CCP4 MTZ format for maps. A key advance in map representation beyond the conventional isosurface contours was extraction of ridge lines or a skeleton (Greer, 1974, 1985). This provides a pseudoatom structure with atoms at local density peaks, connected through maximum density. Each program has its own format for this data structure. 

Some minor discrepancies between theory and experiment on tetrafluoroterephthalonitrile remain to be resolved. The peak densities in the bonds are slightly but systematically lower in the theoretical than in the experimental maps. Analysis of the second moments of the pseudoatoms from the Hirshfeld space partitioning (chapter 6) indicate a greater contraction into the molecular plane in the theoretical than in the experimental study. Whether such discrepancies are artifacts of the refinement model, the result of inter-molecular interactions, or have another origin, is a question of considerable interest. 

The atomic electrostatic moments of an atom are obtained by integration over its charge distribution. As the multipole formalism separates the charge distribution into pseudoatoms, the atomic moments are well defined. 

The first term in Eq. (7.39) represents the sum over the dipole moments of the individual subunits, each referred to its own origin the second term represents the effect of the origin shift. As expected, Eq. (7.39) for the sum over subunit dipole moments is identical in form to Eq. (7.23) for the sum over pseudoatom dipole moments. 

A related theoretical approach to charge density transferability has been developed by Mezey and collaborators (Walker and Mezey 1993,1994). But rather than composing a molecule of standard pseudoatoms, the density of large molecules, including proteins, is constructed from the density of a number of standard theoretical fragments. The fragment densities are defined by the distribution... 

For a general formulation of the Zintl-Klemm concept, consider an intermetallic AmX phase, where A is the more electropositive element, t3 pically an alkali or an alkaline earth metal. Both A and X, viewed as individual atoms, are assumed to follow the octet rule leading to transfer of electrons from A to X, i.e., A AF, X —> X , so that mp = nq. The anionic unit X arising from this electron transfer is considered to be a pseudoatom, which exhibits a structural chemistry closely related to that of the isoelectronic elements [11]. Since bonding also is possible in the cationic units, the numbers of electrons involved in A-A and X-X bonds of various types (caa and exx> respectively) as well as the number of electrons e not involved in localized bonds can be generated from the numbers of valence electrons on A and X, namely and ex, respectively, by the following equations of balance ... 

Several formulations were proposed [65, 66], but the intuitive notation introduced by Hansen and Coppens [67] afterwards became the most popular. Within this method, the electron density of a crystal is expanded in atomic contributions. The expansion is better understood in terms of rigid pseudoatoms, i.e., atoms that behave stmcturally according to their electron charge distribution and rigidly follow the nuclear motion. A pseudoatom density is expanded according to its electronic stiucture, for simplicity reduced to the core and the valence electron densities (but in principle each atomic shell could be independently refined). Thus,... 

However, second-order perturbation theory can compare one bulk situation with another, since errors in the absolute prediction of the binding energy (usually residing in the energy of the individual pseudoatoms, = 0))... 

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