Distributed multipolar expansion

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. 

Freitag, M. A., Gordon, M. S., Jensen, J. H., Stevens, W. J. (2000). Evaluation of Charge Penetration between Distributed Multipolar Expansions,/ Chem. Phys., 112, 7300-7306. 

The first approach is based on distributed multipolar expansions in general cavities [152] and has been recently reviewed [153]. 

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... 

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... 

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. 

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... 

To illustrate how the multipolar expansion is related to a distribution of charges in a system, let us consider the simple case of a molecule with two charges and electrostatic potential at point P (a distance r from the origin, Vi from charge qi and V2 from charge q2) is then given by ... 

The eleefrostatie eonfribution arises from the interaetion of the unpolarized eharge distribution of the moleeules. This interaction can be analyzed using a multipolar expansion of the eharge distribution of die interacting subsystems whieh usually is eut off in the first term... 

In order to avoid the laborious evaluation of the above integrals we propose a rational approximation which consists of a multipolar expansion the elementary charge distributions around atomic centres. This goal can be attained in two steps. First a Ruedenberg-type approximation [169] is introduced to reduce all multicentre integrals to at most two-centre ones and then the G(r, r ) functions are developed in bipolar series with respect to atomic centres. 

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