Multipole model

Distributed multipole models for Nj and HF. (Figure adapted from Stone A j and M Alderton 19S5. ibuted Multipole Analysis Methods and Applications. Molecular Physics 56 3 047-1064.)... 

Ljii.iiitiJtTi mechanical calculation on this molecule used 1000 basis functions). However, dis-nilnited multipole models have not yet been widely incorporated into force fields, not least because of the additional computational effort required. It can be complicated to calculate llic atomic forces with the distributed multipole model in particular, multipoles that are lull located on atoms generate torques, which must be analysed further to determine the roi es on the nuclei. 

Williams D E 1991. Net Atomic Charge and Multipole Models for the Ah Initio Molecula Electric Potential. In Lipkowitz K B and D B Boyd (Editors). Reviews in Computational Chemistr Volume 2. New York, VCH Publishers, pp. 219-271. 

Many of the mesogenic molecules are stericaUy asymmetric, which is determined by the fractures and bending of the molecular core as well as by the presence of the tail chains of different nature, including the branched, biforked or polyphilic moieties (Fig. 2c-f). In terms of the multipole model of molecular asymmetry introduced by Petrov and Derzhanski [34], we can speak about longitudinal or transverse steric dipoles or multipoles (Fig. 3). 

Most of the relevant features of the charge density distribution can be elegantly elucidated by means of the topological analysis of the total electron density [43] nevertheless, electron density deformation maps are still a very effective tool in charge density studies. This is especially true for all densities that are not specified via a multipole model and whose topological analysis has to be performed from numerical values on a grid. 

A preliminary least-squares refinement with the conventional, spherical-atom model indicated no disorder in the low-temperature structure, unlike what had been observed in a previous room-temperature study [4], which showed disorder in the butylic chain at Cl. The intensities were then analysed with various multipole models [12], using the VALRAY [13] set of programs, modified to allow the treatment of a structure as large as LR-B/081 the original maximum number of atoms and variables have been increased from 50 to 70 and from 349 to 1200, respectively. The final multipole model adopted to analyse the X-ray diffraction data is described here. 

Table 2. Experimental (final multipole model) and theoretical (6-31G ) values of mk(r) and of r(Omin) - rttast om in the most meaningful planes of LR-B/081. 

Table 3. Spherical valence populations of the nitromalonamide multipole model.

The combination of the neutron structural model with a multipole model of the X-ray data measured at a matching temperature, has enabled us to obtain detailed information about the electron density distribution. This has revealed new information about the bonding in m-enol systems. 

Williams, D. E. 1991. Net Atomic Charge and Multipole Models for the Ab Initio Molecular Electrostatic Potential. In Reviews in Computational Chemistry. K. B. Lipkowitz and D. B. Boyd, eds. VCH Publisher, New York. 

Sharma, R.R., Das, T.P., and Orbach, R. 1966. Zero-field splitting of S-state ions. I. Point-multipole model. Physical Review 149 257-269. 

Donald E. Williams, Net Atomic Charge and Multipole Models for the Ab... 

Amongst known structural models which could be used for the reconstruction of ESP and electron density the multipole model proposed by Hansen and Coppens is the most suitable for EDSA [5] ... 

Table 4. Results of the multipole model refinement of germanium (single crystal) with electron diffraction data...

The multipole model reduces the crystal electron density to a number of parameters, which can be fitted to experimental structure factors. For CU2O, structure factors for the (531) and higher-order reflections out to (14,4,2) were taken from X-ray measurements. Weak (ooe) (with o for odd and e for even) and very weak (eeo) reflections were also taken from X-ray work. Fig. 6 shows a three-dimensional plot of the difference between the static crystal charge density obtained from the multipole fitting to... 

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

When observed structure factors are used, the thermally averaged deformation density, often labeled the dynamic deformation density, is obtained. An attractive alternative is to replace the observed structure factors in Eq. (5.8) by those calculated with the multipole model. The resulting dynamic model deformation map is model dependent, but any noise not fitted by the muitipole functions will be eliminated. It is also possible to plot the model density directly using the model functions and the experimental charge density parameters. In that case, thermal motion can be eliminated (subject to the approximations of the thermal motion formalism ), and an image of the static model deformation density is obtained, as discussed further in section 5.2.4. 

In the multipole-model description, the charge density is a sum of atom-centered density functions. The moments of the entire distribution are obtained as the sum over the individual atomic moments plus contributions due to the shift to a common origin. 

Stewart s conclusion underscores the need for short-wavelength, low-temperature studies, if very high accuracy electrostatic properties are to be evaluated by Fourier summation. But, as pointed out by Hansen (1993), the convergence can be improved if the spherical atoms subtracted out are modified by the k values obtained with the multipole model. Failure to do this causes pronounced oscillations in the deformation density near the nuclei. For the binuclear manganese complex ( -dioxo)Mn(III)Mn(IV)(2,2 -bipyridyl)4, convergence of the electrostatic potential at the Mn nucleus is reached at 0.7 A" as checked by the inclusion of higher-order data (Frost-Jensen et al. 1995). 

The dipolar terms contribute to the electric field. With the density deformation functions of the multipole model (chapter 3) and Eq. (8.36), one obtains... 

Many of the currently available studies of metal-metal bonding were completed before the multipole model and the topological analysis of the total density were fully developed. For this reason, the discussions reported below focus on the deformation density distributions, and their comparison with theoretical results, though a more quantitative analysis is now possible and would be of considerable interest. 

Table 3.7 also lists ternary spectral moments for a few systems other than H2-H2-H2. For the H2-He-He system, the pairwise-additive dipole moments are also known from first principles. The measured spectral moments are substantially greater than the ones calculated with the assumption of pairwise additivity - just as this was seen in pure hydrogen. For the other systems listed in the Table, no ab initio dipole surfaces are known and a comparison with theory must therefore be based on the approximate, classical multipole model. 

K. L. C. Hunt. Classical multipole models Comparison with ab initio and experimental results. In G. Birnbaum, ed., Phenomena Induced by Intermolec. Interactions, p. 1, Plenum Press, New York, 1985. 

Williams DE. Net atomic charge and multipole models for the ab initio molecular electric potential. In Lipkowitz KB, Boyd DB, eds. Reviews in Computational Chemistry. Vol. 2. New York VCH, 1991 219-271. 

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