Atom-centered distributed multipole

In the multipole-model description, the charge density is a sum of atom-centered density functions. The moments of the entire distribution are obtained as the sum over the individual atomic moments plus contributions due to the shift to a common origin. 

To evaluate this expression for distributions expressed in terms of their multipolar density functions, the potential <1> and its derivatives must be expressed in terms of the multipole moments. The expression for charge distribution has been given in chapter 8 [Eq. (8.54)]. Since the potential and its derivatives are additive, a sum over the contributions of the atom-centered multipoles is again used. The resulting equation contains all pairwise interactions between the moments of the distributions A and B, and is listed in appendix J. 

The electrostatic interaction between two nonoverlapping charge distributions A and B, consisting of NA and NB atoms, respectively, and each represented by their atom-centered multipole moments, is given by (using the Einstein summation convention for the indices a, / , y) (Buckingham 1978)... 

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]. 

As mentioned after Equation [24], atom-centered monopoles in principle generate the higher multipoles required to describe the electronic distribution (although, of course, a finite number n of charges can give at most n nonvanishing multipole moments), and as noted by Dillet et al., the distributed monopole term provides the vast majority of the polarization effect (albeit not all). We note this only for comparative purposes, though, since calculation of the ENP terms does not actually involve the multipole moments explicitly. 

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 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 ... 

Indeed, this idea lies at the heart of the venerable notion of assigning partial charges to the atoms in a molecule [17]. One may think of this practice as a representation of the true charge distribution of the molecule by a series of distributed multipoles (in this case, limited to monopoles) at various sites, namely, atomic centers. Even if limited to monopoles, the act of spreading them out over the entire molecule is equivalent in some sense to simulation of high orders of molecular center-based multipoles. 

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... 

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. 

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... 

The two-center two-electron repulsion integrals ( AV Arr) represents the energy of interaction between the charge distributions at atom Aand at atom B. Classically, they are equal to the sum over all interactions between the multipole moments of the two charge contributions, where the subscripts I and m specify the order and orientation of the multipole. MNDO uses the classical model in calculating these two-center two-electron interactions. 

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