Multipolar expansion

Bell R J 1970 Multipolar expansion for the non-additive third-order interaction energy of three atoms J. 

When atoms occupy highly symmetrical sites, a further limitation of the current multipolar expansions is the limited order of the spherical harmonics employed, that do not usually extend past the hexadecapolar level (/ = 4). Only two multipolar studies published to date used spherical harmonics to orders higher than 1 = 4 graphite [15] and crystalline beryllium [16]. In the latter work, the most significant contribution to the valence density was indeed shown to be given by a pole of order 1 = 6. 

To check this prediction, a number of MaxEnt charge density calculations have been performed with the computer program BUSTER [42] on a set of synthetic structure factors, obtained from a reference model density for a crystal of L-alanine at 23 K. The set of 1500 synthetic structure factors, complete up to a resolution of 0.555 A [45], was calculated from a multipolar expansion of the density, with the computer program VALRAY[ 46],... 

We have described in this paper the first implementation of this Bayesian approach to charge density studies, making joint use of structural models for the atomic cores substructure, and MaxEnt distributions of scatterers for the valence part. Used in this way, the MaxEnt method is safe and can usefully complement the traditional modelling based on finite multipolar expansions. This supports our initial proposal that accurate charge density studies should be viewed as the late stages of the structure determination process. 

Qa and Qc are the charges obtained from the multipolar expansion of the interacting A and C molecular charge distributions, NyAL and NyAL being their respective number of valence electrons. Wa and Wc are the A and C atoms effective van der Waals radii. Kac is a proportionality factor tabulated upon the atomic numbers of the A and C atoms, a is a constant fixed to 12.35. The same treatment is applied to the others terms of the repulsion energy. 

Piquemal J-P, Gresh N, Giessner-Prettre C (2003) Improved formulas for the calculation of the electrostatic contribution to intermolecular interaction energy from multipolar expansion of the electronic distribution. J Phys Chem A 107 10353... 

The electrostatic energy is calculated using the distributed multipolar expansion introduced by Stone [39,40], with the expansion carried out through octopoles. The expansion centers are taken to be the atom centers and the bond midpoints. So, for water, there are five expansion points (three at the atom centers and two at the O-H bond midpoints), while in benzene there are 24 expansion points. The induction or polarization term is represented by the interaction of the induced dipole on one fragment with the static multipolar field on another fragment, expressed in terms of the distributed localized molecular orbital (LMO) dipole polarizabilities. That is, the number of polarizability points is equal to the number of bonds and lone pairs in the molecule. One can opt to include inner shells as well, but this is usually not useful. The induced dipoles are iterated to self-consistency, so some many body effects are included. 

The probability of finding the system in the state vRjn> has an oscillatory time dependence. For off-resonance conditions, the system presents a line width at half maximum equal to 4 ll/fi. This matrix element can be expanded in a multipolar expansion, the first term being the electric dipole approximation [45, 152, 154],... 

We recall that in the multipolar expansion, the 3d density is expressed in terms of the density-normalized spherical harmonic functions dlmp as... 

In the 1970s, among many other approaches, the method of multipolar expansion of atomic electron density was recognized as the most applicable and accurate. 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

Most of the quantum chemical calculations of the nuclear shielding constants have involved two classes of solvation models, which belong to the second group of models (n), namely, the continuum group (i) the apparent surface charge technique (ASC) in formulation C-PCM and IEF-PCM, and (ii) models based on a multipolar expansion of the reaction filed (MPE). The PCM formalism with its representation of the solvent field through an ASC approach is more flexible as far as the cavity shape is concerned, which permits solvent effects to be taken into account in a more accurate manner. 

The inclusion of the environment effects for non-linear optical (NLO) properties is presented within the PCM (Cammi Mennucci) and the multipolar expansion (Agren Mikkelsen) solvation models. In the first contribution the attention is focused on the connection between microscopic effective properties and macroscopic NLO susceptibilities, whereas in the latter contribution the analysis is extended to treat heterogeneous dielectric media. 

We have thus provided an accurate determination of the spherically averaged interaction, the leading term in the multipolar expansion. For the water-He and water-Ar systems there is full agreement with previous differential scattering [76]... 

As emphasized, the Born—Kirkwood—Onsager (BKO) approach includes only the solute s monopole and dipole interaction with the continuum. That is, the full classical multipolar expansion of the total solute charge distribution is truncated at the dipole term. This simplification of the electronic distribution fails most visibly for neutral molecules whose dipole moments vanish as a result of symmetry. A distributed monopole or distributed dipole model is more... 

Reaction Fields from Higher Order Multipolar Expansions Generalizations of the Born—Kirkwood—Onsager model have appeared which extend the multipole series to arbitrarily high order.20,62,144,234-236 ybis approach yields... 

The aqueous solvation free energies of the four tautomers available to the 5-(2H)-isoxazolone system have also been studied using a variety of continuum models (Table 7). Hillier and co-workers - " have provided data at the ab initio level using the Born-Kirkwood-Onsager model, the classical multipolar expansion model (up to I = 7), and an ab initio polarized continuum model. We examined the same BKO model with a different cavity radius and the AMl-SMl and AMl-SMla o- models, and Wang and Ford have performed calculations with the AMl-PCM model. 

The multipolar expansion of the electrostatic potential due to a charge distribution p(r) provides a convenient guide to the dependence of the interaction energy on the separation between A and B. This dependence is summarized in Table 2 for electrostatic, polarization and dispersion forces between multipoles. 

Extracting the dependence on R g the multipolar expansion can be written in a more compact form ... 

For further reading on multipolar expansion of interaction energy, we refer to [1,3,4,14]. ... 

Several methods can be distinguished within the framework of the perturbative approach. Some [29-37] are based on a multipolar expansion of the operator i.e. the interaction potential of the two species, others rely on the linear response theory [38,39]. 

On the other hand, the perturbative approach for long-range interactions, that decompose the energy into terms of clear physical meaning is quite helpful in the development of a model. Hence, from the multipolar expansion of each term, one is able to know the form of the potential dependence on the distance. As to the short range part, the perturbative approach also indicates that the above long-range part must be supplemented by repulsive terms, that are well described by exponential functions [45,75]. 

An alternative to the approach outlined above, based on perturbation theory and multipolar expansions, is the inversion of experimental data [7,82]. In practice, this technique is restricted to monatomic systems, where interaction energies depend on a single geometric variable. Moreover, it is often impossible to gain a complete picture of the potential function by inversion of data corresponding to just one quantity, e.g. second virial coefficient [7]. 

With large molecules, the number of sites is often smaller than that of atoms and the multipoles are located at the center of groups of atoms. For instance, the site-site description of the interactions in hydrocarbons with, say, a LJ potential, can either place an interaction center on each nucleus or just on the carbon of CH3 or CH2 groups (united atom approach). From this point of view, different treatment may be applied to the interaction terms in multipolar expansions. 

From a practical point of view, the number of sites and of functions per site should be kept as small as possible to reduce computational times. This is the main reason behind the success of empirical and semiempirical models based on LJ functions plus electrostatic terms corresponding to multipolar expansions with very few sites. Conversely, potentials derived from accurate ab initio calculations have been less widely used in view of their complexity, not compensated by real advantages, unless they include many-body terms. 

Electrostatic and induction energies are evaluated by multicenter multipolar expansions truncated at quadrupolar level, with an accurate description of the charge distribution and polarizability calculated ab initio at HF level on the monomer. The electrostatic interaction is then fitted by a Coulomb potential among two positive point charges on the hydrogens and two negative close to... 

The more recent ASP-W2 and ASP-W4 improve the accuracy of the description of multipoles, which is at multireference Cl level instead of MP2. All three potentials share the same form of repulsion and polarization term, the former using three sites with exponential functions while the latter is described by anisotropic dipolar and quadrupolar polarizabilities centered on the oxygen. Electrostatic interactions are modeled by a single-center (APS-W) or three-center (ASP-W2 and ASP-W4) multipolar expansion up to quadrupolar term for ASP-W and APS-W2 and up to hexadecapolar term for APS-W4. 

One may use multipolar expansions for an approximate calculation of MEP, and it is often convenient to use fractional charges on atoms obtained from ab initio or semiempirical quantum chemical MO population analyses [288,300]. 

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