Multipole formalism

Atomic density functions are expressed in terms of the three polar coordinates r, 6, and density functions are products of r-dependent radial functions and 8- and -dependent angular functions. The angular functions are the real spherical harmonic functions ytm (8, ), but with a normalization suitable for density functions, further discussed below. The functions are well known as they describe the angular dependence of the hydrogenic s, p, d,f... orbitals. 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

To obtain the atomic form factor according to the multipole formalism, we apply the Fourier transform... 

Expression (4.31) is widely applied to calculate the error in properties derived from the least-squares variables. We will use it in chapter 7 for the calculation of the standard deviations of the electrostatic moments derived from the parameters of the multipole formalism. 

The nature of the charge density parameters to be added to those of the structure refinement follows from the charge density formalisms discussed in chapter 3. For the atom-centered multipole formalism as defined in Eq. (3.35), they are the valence shell populations, PLval, and the populations PUmp of the multipolar density functions on each of the atoms, and the k expansion-contraction parameters for... 

Atomic Electrostatic Moments in Terms of the Parameters of the Multipole Formalism... 

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. 

If the moments are referred to the nuclear position, only the electronic part of the charge distribution contributes to the integral. According to the multipole formalism of Eq. (3.32),... 

This chapter deals with the evaluation of the electrostatic potential and its derivatives by X-ray diffraction. This may be achieved either directly from the structure factors, or indirectly from the experimental electron density as described by the multipole formalism. The former method evaluates the properties in the crystal as a whole, while the latter gives the values for a molecule or fragment lifted out of the crystal. 

The expressions given here are valid for the multipole formalism of Hansen and Coppens, as described by Eq. (3.35). With other formalisms, similar expressions are used. Experimental molecular potentials reported in the literature include those of imidazole (Spackman and Stewart 1984, pp. 302-320), phosphorylethanolamine (Swaminathan and Craven 1984), alloxan (Swaminathan and Craven 1985), and parabanic acid (He et al. 1988). 

In the following section we discuss application of the multipole formalism to a series of Fe(II) porphyrins. A next step towards derivation of the electronic wavefunction of a transition metal complex may be based on the LCAO formalism for the molecular orbitals. Test calculations with such a formalism, using a theoretical data set, are described in the final part of this article. 

The requirement I > 2 can be understood from the symmetry considerations. The case of no restoring force, 1=1, corresponds to a domain translation. Within our picture, this mode corresponds to the tunneling transition itself. The translation of the defects center of mass violates momentum conservation and thus must be accompanied by absorbing a phonon. Such resonant processes couple linearly to the lattice strain and contribute the most to the phonon absorption at the low temperatures, dominated by one-phonon processes. On the other hand, I = 0 corresponds to a uniform dilation of the shell. This mode is formally related to the domain growth at T>Tg and is described by the theory in Xia and Wolynes [ 1 ]. It is thus possible, in principle, to interpret our formalism as a multipole expansion of the interaction of the domain with the rest of the sample. Harmonics with I > 2 correspond to pure shape modulations of the membrane. 

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]. 

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

The scaling factor Sj can take any value between 0 and 1 and is applied to site j. The superscripts p and m indicate permanent and mutual induction, respectively. Equation (9-19) can be solved iteratively using similar procedures to those used to solve Eq. (9-3). The formal permanent moments can be calculated by subtracting induced moments from moments from ab initio calculations. For any conformation of a given compound the atomic multipoles can be determined from Distributed Multipole Analysis (DMA) [51]. 

The interaction in the coordination sphere is described by a 13-term function considering multipole-, polarization-, dispersion-, and repulsion interactions. They succeeded in reproducing free energies of solvation of a large variety of cations to within standard deviations of a few percent. This formalism has been applied by... 

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. 

The coefficients n, have to obey the condition n, f, imposed by Poisson s electrostatic equation, as pointed out by Stewart (1977). The radial dependence of the multipole density deformation functions may be related to the products of atomic orbitals in the quantum-mechanical electron density formalism of Eq. (3.7). The ss, sp, and pp type orbital products lead, according to the rules of multiplication of spherical harmonic functions (appendix E), to monopolar, dipolar, and quadrupolar functions, as illustrated in Fig. 3.6. The 2s and 2p hydrogenic orbitals contain, as highest power of r, an exponential multiplied by the first power of r, as in Eq. (3.33). This suggests n, = 2 for all three types of product functions of first-row atoms (Hansen and Coppens 1978). 

The Hirshfeld functions give an excellent fit to the density, as illustrated for tetrafluoroterephthalonitrile in chapter 5 (see Fig. 5.12). But, because they are less localized than the spherical harmonic functions, net atomic charges are less well defined. A comparison of the two formalisms has been made in the refinement of pyridinium dicyanomethylide (Baert et al. 1982). While both models fit the data equally well, the Hirshfeld model leads to a much larger value of the molecular dipole moment obtained by summation over the atomic functions using the equations described in chapter 7. The multipole results appear in better agreement with other experimental and theoretical values, which suggests that the latter are preferable when electrostatic properties are to be evaluated directly from the least-squares results. When the evaluation is based on the density predicted by the model, both formalisms should perform well. 

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. 

The atom-centered multipole expansion used in the density formalisms described in chapter 3 implicitly assigns each density fragment to the nucleus at which it is centered. Since the shape of the density functions is fitted to the observed density in the least-squares minimalization, the partitioning is more flexible than that based on preconceived spherical atoms. 

For the static density, the zero term in the potential can be expressed in terms of the multipole coefficients of the aspherical-atom formalism. Substituting for atom j at... 

A (truncated) multipole expansion is a computationally convenient single-center formalism that allows one to quantitatively compute die degree to which a positive or negative test charge is attracted to or repelled by die molecule that is being represented by the multipole expansion. This quantity, die molecular electrostatic potential (MEP), can be computed exactly for any position r as... 

A third possibility that has received extensive study in the SCRF regime is one that has seen less use at the classical level, at least within the context of general cavities, and that is representation of the reaction field by a multipole expansion. Rinaldi and Rivail (1973) presented this methodology in what is arguably the first paper to have clearly defined the SCRF procedure. While the original work focused on ideal cavities, this group later extended the method to cavities of arbitrary shape. In formalism, Eq. (11.17) is used for any choice of cavity shape, but the reaction field factors f must be evaluated numerically when the cavity is not a sphere or ellipsoid (Dillet et al. 1993). Analytic derivatives for this approach have been derived and implemented (Rinaldi et al. 2004). 

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