Spherical multipole expansion

A further advantage of this approach to distributed multipoles is that, unlike some conventional multipole expansions (Stone, 2005 Popelier et al., 2001b), the (spherical) multipole expansion obtained from Hermite Gaussians in this way is intrinsically finite of order t+ + V (i.e., the highest angular momentum in the ABS) as shown in (Cisneros et al., 2006a), similar to the multipoles obtained by Volkov and Coppens (Volkov and Coppens, 2004). 

The expansion can be truncated after a finite number of, say, L terms. For the spherical multipole expansion, L = 15-21 is known to provide accuracies on the order of 10 a.u. and better in the total energy.Instead of taking into account all M field terms of the individual charges <7, the total field is then given by the L terms of the multipole expansion of the total charge distribution Q, as... 

The spherical multipole expansion as derived can be cast into a different form that achieves higher efficiency for computer implementations and finally decouples the angular parts of the electron coordinates. 

The spherical cavity, dipole only, SCRF model is known as the OnMger model.The Kirkwood model s refers to a general multipole expansion, if the cavity is ellipsoidal the Kirkwood—Westheimer model arise." A fixed dipole moment of yr in the Onsager model gives rise to an energy stabilization. 

In connection with electronic strucmre metlrods (i.e. a quantal description of M), the term SCRF is quite generic, and it does not by itself indicate a specific model. Typically, however, the term is used for models where the cavity is either spherical or ellipsoidal, the charge distribution is represented as a multipole expansion, often terminated at quite low orders (for example only including the charge and dipole terms), and the cavity/ dispersion contributions are neglected. Such a treatment can only be used for a qualitative estimate of the solvent effect, although relative values may be reasonably accurate if the molecules are fairly polar (dominance of the dipole electrostatic term) and sufficiently similar in size and shape (cancellation of the cavity/dispersion terms). 

Multipole Expansion for a Spherical Cavity170. For a molecule embedded in a polarizable medium this method requires little more computational effort than the calculations for an isolated molecule. For a cavity with a radius a, the electrostatic potential is164 ... 

Pole Figure and Multipole Expansion. Pole figures, g(other functions that are suitably expressed in spherical polar coordinates are favorably expanded... 

One point of particular interest is that it is not clear from the electrostatics-only models whether non-electrostatic phenomena affect the aqueous tautomeric equilibria. For instance, the DO results of Wong et al. [297] would suggest there are differentiating non-electrostatic phenomena, while the results of Young et al. [195] for a multipole expansion in a spherical cavity suggest that there are not. Since the SMI, SM2, and SM3 GB/ST models use Mulliken charges rather than... 

For a given value of n, the functions httk are identical to a sum of spherical harmonics with l = n, n — 2, n — 4,..., (0,1) for n > 1. The relationships are summarized in Table 3.8. For n = 0,1, the Hirshfeld functions are identical to the spherical harmonics with / = 0, 1, but, starting with the n = 2 functions, lower-order spherical harmonics are included for each n value. Unlike the spherical harmonics, the hnl functions are therefore not mutually orthogonal. As the radial functions in Eq. (3.48) contain the factor r", quite diffuse s, p, and d functions are included in the n = 2, 3, and 4 sets. For n <4 there are 35 deformation functions on each atom, compared with 25 valence-shell density functions with / < 4 in the multipole expansion of Eq. (3.35). 

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. 

Since the spherical core- and valence-scattering factors in the multipole expansion are based on theoretical wave functions, expressions for the corresponding density functions are needed in the analytical evaluation of the integrals in Eqs. (8.35)... 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

One of the simplest orientational-dependent potentials that has been used for polar molecules is the Stockmayer potential.48 It consists of a spherically symmetric Lennard-Jones potential plus a term representing the interaction between two point dipoles. This latter term contains the orientational dependence. Carbon monoxide and nitrogen both have permanent quadrupole moments. Therefore, an obvious generalization of Stockmayer potential is a Lennard-Jones potential plus terms involving quadrupole-quadrupole, dipole-dipole interactions. That is, the orientational part of the potential is derived from a multipole expansion of the electrostatic interaction between the charge distributions on two different molecules and only permanent (not induced) multipoles are considered. Further, the expansion is truncated at the quadrupole-quadrupole term. In all of the simulations discussed here, we have used potentials of this type. The components of the intermolecular potentials we considered are given by ... 

On the U(l) level, the plane wave is subjected to a multipole expansion in terms of the vector spherical harmonics, in which only two physically significant values of M in Eq. (761) are assumed to exist, corresponding to M = +1 and — 1, which translates into our notation as follows ... 

Here, r denotes the position vector of the charges qt with respect to the center of the sphere, and r, the distance from the center. By applying the dielectric scaling function for dipoles (Eq. (2.3)), which—as we have seen in Fig. 2.1—is also a good approximation for most other multipole orders, it was immediately clear that the idea of using a scaled conductor instead of the EDBC leads to a considerable simplification of the mathematics of dielectric continuum solvation models, with very small loss of accuracy. It may also help the finding of closed analytic solutions where at present only multipole expansions are available, as in the case of the spherical cavity. Thus the Conductor-like Screening Model (COSMO) was bom. 

We will use the basis vectors (1) where > i2 and apply equation (4) when needed. For two tlu electrons, our basis (1) consists of 15 different state vectors I/) (for two holes, the fivefold hu degeneracy leads to 45 states). In the following we will study the intramolecular correlations of electrons (holes) within a multipole expansion of the two-body Coulomb potential V(r, f) = 1/lr —1 (charge e = unity). In terms of real spherical harmonics YJ, where r stands for m = 0,... 

Although it is possible to determine the complete electron density distribution using the Fourier transform of the observed structure factors, Eq. (1), the errors inherent in the structure factor amplitudes and, in the case of non-centrosymmetric structures, the errors in their phases introduce significant noise and bias into the result. Because of this, it has become normal practice to model the electron density by a series of pseudo-atoms consisting of a frozen, spherical core and an atom centered multipole expansion to represent the valence electron density [2,17]. 

To negate this observation, the conviction that covalent interaction mandates an excess bonding density in all cases, prompted the formulation of aspherical atomic densities to reflect the requirements of bonding theory. By multipole expansion of atomic densities, based on real spherical harmonics, in line with traditional models of orbital hybridization, the mandated deformation densities are retrieved. Increased flexibility of the model by the introduction of scaling parameters further ensures the elimination of any discrepancies with the theory. However, it is debatable whether this exercise proofs anything other than the power of well-chosen parameters to improve the fit between incompatible data sets. 

The spherical form of the multipole expansion is very useful if we are looking for the explicit orientational dependence of the interaction energy. However, in some applications the use the conceptually simpler Cartesian form of the operators V1a 1b may be more convenient. Moreover, unlike the spherical derivation, the Cartesian derivation is very simple, and can be followed by everybody who knows how to differentiate a function of x, y and z 149. To express the operator V,, in terms of Cartesian tensors we have to define the reducible, with respect to SO(3), tensorial components of multipole moments,... 

By using the multipole expansion, we in fact replace the exact radial expansion coefficients A (7 ) in Eq. (1-124) by the approximate coefficients A poi(-K), which are power series in R 1. Closed expressions for the latter have been given149 161 in terms of the irreducible spherical tensors of multipole moments and polarizabilities. 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

Fig. 3 A comparison of different coarse grain lipid models. The Shelley model " of DMPC, and Marrink and Essex models of DPPC are compared to their atomistic equivalents (for ease of comparison, hydrogen atoms of the atomistic models are not shown). Solid lines represent harmonic bonds connecting CG particles, and the CG particle types for the Shelley and Marrink models are labelled (the labels are the same as those used in the main text). The point charges (represented by + and —) and point dipoles (represented by arrows) are shown for the Essex model (the charges and dipoles are located at the centre of their associated CG particle). The Shelley and Marrink models use LJ particles (represented by spheres), while the Essex model uses a combination of LJ particles (spheres) and Gay-Berne particles (ellipsoids). Finally, the blob model proposed by Chao et al is also shown for comparison. This model represents groups of atoms as rigid non-spherical blobs that use interaction potentials based on multipole expansions.

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