Pseudoatom model

Alternative and much more elegant methods are those using aspherical pseudoatoms least squares refinements. These refinements permit access to the positional and thermal variables of the atoms as well as to the electron density parameters. Several pseudoatoms models of similar quality exist (9-12) and are compared in reference [7J],... 

A. Deconvolution between Thermal Motion Parameters and Deformation Density Parametrization of the Pseudoatom Model... 

Having secured a set of n values for phosphorus, the pseudoatom model was fitted to the four simulated data sets to test the effectiveness of the pseudoatoms model s formal deconvolution of multipolar valence density features from thermal vibrations smearing. Results are illustrated in Figure 4 as maps of the model static deformation densities ... 

Maps from the simulated dynamic data and from the static simulated data are hardly distinguishable from one another or from the theoretical density mapped directly from the extended basis wave function (see Figure 3). Furthermore, the statistical agreements factors are excellent [/ (F)] = 3.1, 3.0, 3.2, and 3.1%o, respectively, for the static and 75 K, 150 K, and room temperature simulated data. This shows that the pseudoatom model effectively recovers the theoretical electron density that comes from the dynamic structure factors. A full report, including a refinement against real room temperature structure factors of H3P04, is given in references 19 and 20. 

Fig. 5.36. Electron-density distributions in borates — deformation density maps calculated for BlOHlj-. (a) static model map calculated from pseudoatom model (b) theoretical deformation density map from ab initio Hartree-Fock configuration-interaction calculation (after Gajhede et al., 1986 reproduced with the publisher s permission).

A temperature dependent X—X+N study (100, 135, 170, and 205 K) on naphthalene [66] addresses the problem of thermal de-convolution, that is, the efficiency of the pseudoatom model to decouple density deformations due to chemical bonding from those due to nuclear motion. The authors analyze the self-consistency of multipole populations, extracted from different temperature XRD data, in terms of statistical distances d ) in the parameter space of the same refinement model ... 

Such observations immediately raise the question how reliable are projections of crystal-field effects onto multipoles The analyses of wavefunction-simulated X-ray data of small model compounds have revealed that the interaction density (8p = p(crystal) — p(isolated molecule)) manifests itself in low-order structure factors, and only to an extent that is comparable with the experimental noise [80]. Nevertheless, the multipole refinement was shown to retrieve this low signal (about 1% in F) successfully. A related study on urea, however, demonstrated that this is not the case if random errors of magnitude comparable with the effect of interaction density are added to the theoretical data [81]. The result also implies that indeterminacies associated with the interpretation of non-centrosymmetric structures can severely limit the pseudoatom model in distinguishing between noise and physical effects [82, 83]. 

Of particular relevance to the measurability of crystal-field effects is a recent study that analyzed the experimental BCP properties of 9-ethynyl-9-fluorenol (crystallizes in the centrosymmetric space group C2/c with two molecules in the asymmetric unit) in relation to the extent to which constraints (local pseudo-symmetry and chemical similarity) were imposed on the pseudoatom model... 

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]. 

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... 

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. 

In general, these models describe the continuous electron density of the unit cell as a sum over pseudoatom densities centered at the nuclear sites ... 

To take in account the nonspherical shape of the valence electron distribution, the K model has been improved by the addition of multipole parameters [lib]. Then, the pseudoatomic density is written (Molly program),... 

How extensively parametrized a pseudoatom multipole model is necessary for reproducing crystallographic informations of the electron density in molecules containing first, second-row elements, and first-row transition metals ... 

Pseudoatom multipole modeling reproduces accurately the deformation density within less than 0.05 e A-3 in the bonding and intermolecular regions calculations on theoretical thermally smeared structure factors show that deconvolution between density and thermal parameters is effective. The accuracy obtained from a high-resolution accurate diffraction experiment compares with extended triple- + polarization ab initio SCF calculations. These experiments are tractable for mole-... 

As an example of a pseudopotential simulation of boundary conditions for clusters representing some ionic compounds, let us consider the MgO case [79], which has been considered previously [80], [81] within a bare cluster model. The MgO crystal (the rock-salt lattice) was simulated by a cluster MggOg (see Fig. 3) placed in several embeddings comprised of a different number of pseudoatoms possessing the effective potentials and placed at the lattice sites. 

Three clusters were chosen (I) MggOg embedded in the field of the first cationic coordination sphere (each anion of the cluster has coordination number 6) (II) MggOg embedded in the whole first coordination sphere of the cluster plus the second cationic sphere (III) is obtained from (II) by deleting all the pseudoatoms attached above the upper plane of the MggOg cluster. Embedded clusters (I) and (II) could be considered as models of the MgO bulk, whereas (Ill)as modeling the MgO surface. The number of embedding centers is 21, 58, and 41 in (I), (II), and (III), respectively. 

Pool et al studied the adsorption of alkyl thiols on gold surfaces, both without and with a solvent. As solvent they considered w-hexane. For the simulations they used a force field with a so called united-atom model. This means that the thiol heads, the CH2 and CH3 groups, and the gold atoms are treated as individual united atoms . The interactions between these united pseudoatoms are described in terms of the force fields. Subsequently, molecular dynamics simulations were carried through for the different systems. 

In the most widely applied model [52], the atomic decomposition of the ED is retained, but each nucleus-centered spherical density is supplemented by non-spherical functions. The pseudoatom formalism is a finite nucleus-centered multipole expansion of the molecular (crystalline) ED about each nucleus. A pseudo atom is defined as... 

In view of the above comments, error estimates are usually made on the basis of overall reproducibility of, and matching between independent experimental or theoretical results, rather than on the basis of the precision reachable with a particular measurement and refinement model. There are several approaches that allow us to gain quantitative information on experimental reproducibility and uncertainties. These include the pseudoatom interpretation of error free, theoretical data [56, 57, 70-72], comparative analysis of experimental data sets in terms of different constrained models [73], theory versus experimental comparison of results obtained for the same system [74-76], systematic studies on a series of related compounds [77], and the simultaneous analysis of data collected at different temperatures [66]. 

A thorough investigation of simulated structure factor data of ammonia provides us with quantitative figures for the accuracy of BCP properties achievable with the pseudoatom projection [72]. The Pbcp> Pbcp> tBCp(H) indices of the static model density obtained by the multipole refinement of HF/6-31IG static structure factors deviate from the correct values (derived... 

In molecular mechanics a chemical bond is considered to be composed of two spheres attached by a spring. Modeling of M-olefin systems presents a simple problem Where do we anchor the metal (Strictly speaking, the metal should be anchored to the center of the olefin C=C bond, but there is no atom at the C=C centroid to anchor the metal.) One approach is to bond the metal to both carbon atoms in the olefin. This creates a metallocycle, which is not a realistic model for olefin binding. An alternate approach is to define a pseudoatom (an atom with... 

Figure 5 Bonding models for an Ti -olefin interaction, (a) Shows the actual bonding in the complex, (b) a molecular mechanics model of the metallo-cycle, (c) shows how the two halves of the olefin can rotate relative to each other if a pseudoatom, D, interrupts the bonding, and (d) shows a molecular mechanics model that is used in the literature. (From Ref. 23.)...

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