Local multipole expansions

We have reported this discussion in some detail because it combines several points treated separately by many authors. These remarks shed some light on the difficulty of getting local multipole expansions which are reliable, easy to handle, and at the same time transferable from molecule to molecule. We have in fact considered, until now, the problem of getting multipole expansions from an already known (r) function, without touching the more important problem of formulating local expansions transferable from molecule to molecule. We shall see later how these problems may be partially solved. 

A method concerned with MEP calculations on large molecules, and based on a LO description of the charge distribution, was proposed by Pullman s group [177-178]. This method, called LMTP (Localized MulTiPole expansion) uses the centroids of each LO as an expansion centre. The number of expansion centres is lower than the OMPT method elaborated by the same group (see Sect. 6.1). Expansions limited to the quadrupole, or also including the octupole term are... 

Etchebest, C., R. Lavery, and A. Pullman. 1982. The Calculations of Molecular Electrostatic Potential from a Multipole Expansion Based on Localized Orbitals and Developed at Their Centroids Accuracy and Applicability for Macromolecular Computations. Theor. Chim. Acta 62, 17. 

The static monopole relaxation diagram clearly describes the response to a delocalized core hole while the dynamic relaxation (fluctuation) process describes the response to the dipole moment of the hole. Together, the two diagrams (Figs. 38c,e) describe the response of the system to a hole localized to an atomic core orbital on either nucleus41, and individually they represent a multipole expansion of the hole. 

Since the single-center multipole expansion of the interaction energy is divergent, one could use a kind of multicenter expansion. One can hope that the multipole expansion will provide better results if multipole moments and polarizabilities localized at various points of a molecule are used instead of global multipole moments and polarizabilities. This idea forms the basis of the so-called distributed multipole analysis of the electrostatic, induction, and dispersion interactions between molecules187 195. 

The electrostatic, induction, and dispersion terms can be expanded in a convergent series closely related to the multipole expansion, but fully accounting for the charge-overlap effects, the so-called bipolar expansion introduced by Buehler and Hirschfelder199,200. In the local coordinate systems with the origins located at the centers of masses of the monomers A and B, separated by the distance R, and with their x and y axes parallel and aligned along the z axes, the distance between two particles in space can be expressed as follows,... 

Multipoles. One can express [15] the multipole expansion of the energy of a charge distribution in an external field by defining the free energy G of a localized charge distribution p(r), which is placed in an external potential (which has no charge distribution associated with it) ... 

The best strategy to be followed in order to get accurate sets of A values has not been defined, so at present more or less complex statistical elaborations of some data are used. Among the numerical data that have been used we mention solvation and solvent transfer energies, intrinsic solute properties (electron isodensity surfaces, isopotential electronic surfaces, multipole expansions of local charge distribution), isoenergy surfaces for the interaction with selected probes (water, helium atoms), Monte Carlo simulations with solutes of various nature. All these sets of data deserve comments, that are here severely limited not to unduly extend this Section. 

Due to the non-local character of the Coulomb operator, the decomposition for the electrostatic energy is more complex. In order to distinguish between local and global terms, we need to introduce atom-dependent screening densities, (hard) and (soft), that generate the same multipole expansion as the local density ha — nA A n, where is the nuclear charge of atom A. 

The theory of van der Waals (vdW) surface interactions is presented here in terms of correlation-self energies of the constituent parts involved in the interaction due to their mutual polarization in the electrostatic limit. In this description the van der Waals interactions are exhibited using the dynamic, nonlocal and inhomogeneous screening functions of the constituent parts. In regard to the van der Waals interaction of a single molecule and a substrate, this problem is substantially the same as that of the van der Waals interaction of an atom and a substrate, in which the atomic aspects of the problem are subsumed in a multipole expansion based on spatial localization of the atom/molecule. As we (and others) have treated this in detail in the past we will not discuss it further in this paper. Here, our attention will be focussed on the van der Waals interaction of an adsorbate layer with a substrate, with the dielectric properties of the adsorbate layer modeled as a two-dimensional plasma sheet, and those of the substrate modeled by a semi-infinite bulk plasma. This formulation can be easily adapted to an... 

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