Atom-centered distributed multipole expansion

The electrostatic term (Eei) describes the electrostatic energy between molecules A and B with nondeformed electronic structure. Using classical electrostatics, the electron density of a molecule can be expanded in a series of multipoles centered on one point, usually the center of mass of the molecule [1], For quantitative studies, where more accuracy is required, the multipole expansion is done on all atoms of the interacting molecules (the so-called distributed multipole expansion [29]). However, for qualitative analysis, the central multipole expansion provides sufficient accuracy. One then uses the multipole values found in the literature for isolated molecules [1], The electrostatic energy in the central multipole expansion can be written as a series, whose leading terms up to the dipole level are ... 

Note that a distinction is made between electrostatic and polarization energies. Thus the electrostatic term, Ue e, here refers to an interaction between monomer charge distributions as if they were infinitely separated (i.e., t/°le). A perturbative method is used to obtain polarization as a separate entity. The electrostatic and polarization contributions are expressed in terms of multipole expansions of the classical coulomb and induction energies. Electrostatic interactions are computed using a distributed multipole expansion up to and including octupoles at atom centers and bond midpoints. The polarization term is calculated from analytic dipole polarizability tensors for each localized molecular orbital (LMO) in the valence shell centered at the LMO charge centroid. These terms are derived from quantum calculations on the... 

A solution to this problem is to use a multi-centered multipole expansion, more commonly called a distributed multipole expansion. In principle the centers could be arbitrarily chosen, but it is convenient to use the atomic nuclei as centers. The distributed multipole expansion of the intermolecular interaction operator V is... 

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]. 

By summing over the AOs localized at each atomic center and integrating these over r, we can reduce Eq. (12) to a distribution of TD monopoles located at each atomic center, and a distribution of overlap-densities from the second term on the right-hand side of Eq. (12). When the overlap densities arise from overlap of AOs on different atomic centers, the resulting TD monopole can arbitrarily be placed halfway between the two atomic centers. More sophisticated reductions of Eq. (12) represent each TD monopole as a multipole expansion about the atomic center. 

The solute charge distribution can be represented by atom centered point charges or as multipole expansions. Of course, if the solute is treated quantum mechanically the charge distribution can be obtained directly from its wave function. Depending on the solvation model, the electrostatic potential derived from the wave function is fitted to atomic charges or multipoles that are then used to construct the solvent reaction field. 

The types of expression in Eqs. (2a) and (2b) are generally valid when the optical fields are weak compared to the electric field that binds the electrons in the material and when the coefficients of the various terms in Eqs. (2a) and (2b) are constant over the range of frequencies contained in the individual incident and generated fields. In addition, the wavelength of the radiation must be long compared to the dimension of the scattering centers (the atoms and molecules of the nonlinear medium), so that the charge distributions can be accounted for with a multipole expansion. 

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