Multicenter multipole expansion

One of the advantages of the molecular (one center) and multicenter multipole expansions of the MEP is that, through truncating the series after some terms, one can get an analytical expression. The molecular multipole expansion, in contrast with the multicenter multipole expansion, diverges at distances of chemical interest. The various multicenter multipole expansion [87, 89, 93-96] practically equivalent to each other [97]. 

However, there are several problems with the multicenter multipole expansions as well, such as (i) how many centers are to be taken, (ii) where to place... 

It should stressed that unlike in the case of the electrostatic energy, the expressions for the long-range coefficients and the angular function Aj defining the multicenter multipole expansions of the induction and dispersion energies are different. This difference is due to the fact that in the multicenter expansions the products of the D functions, (wj) ) and ( )... 

G. Jansen, C. Hattig, B. A. Hess, and J. G. Angyan, Mol. Phys., 88,69 (1996). Intermolecular Interaction Energies by Topologically Partitioned Electric Properties. 1. Electrostatic and Induction Energies in One-Center and Multicenter Multipole Expansions. 

Whitehead C, Breneman C, Sukumar N, Ryan M. Transferable atom equivalent multicentered multipole expansion method. J Comput Chem 2003 24 512-29. 

The electric potential resulting from any charge distribution can be represented by a convergent multicenter multipole expansion.Molecular moments of isolated molecules can be obtained experimentally, such as from the Stark effect on the microwave spectra of gas molecules. Studies of van der Waals clusters and molecular association in general require a multicenter model for short-range electrostatic interaction. Mulder and Huiszoon, for instance, found that a molecular multipole expansion was not satisfactory for the electrostatic interaction of... 

Fig. 1 illustrates torsional potentials in HS-SH and HSe-SeH molecules obtained from moment truncated multicenter multipole expansion combined with Frozen Fragment (FF) approximation [13] and compared with corresponding SCF results. Apparently the torsional potentials are controlled in this case solely by quadrupole moments (M—k -l- -m- -2). Dominant contribution of electrostatic effects in torsional potentials seems to be limited to systems with unusually long bonds [13,26], where also delocalization is excluded. The leading role of quadrupole term could explain why d polarization functions are sometimes required to obtain proper results in conformational analysis in torsional angle space. 

Figure 1. Torsional potentials calculated in SCF approach with 6-31G basis set and within moment truncated frozen fragment multicenter multipole expansion for HS-SH (a) and HSe-SeH (b) molecules (angle 0 corresponds to cis conformation).

The powerful computational techniques, developed in the 1980s to obtain accurate polarization propagators (see Dynamic Properties), can be utilized via equation (21) in calculations of dispersion energies at finite distances. Equation (21) forms also a basis for the rigorous multicenter multipole expansion of the dispersion energy (see Section 4.4). 

In many applications, especially involving distributed or multicenter multipole expansions, see Section 4.4, it is convenient to express the functions Qj (r) in local, body-fixed coordinate systems associated with monomers A and B. The corresponding form of the multipole expansion is ... 

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. 

Inserting for each pair (a, b) the multipole expansion of r 1 with respect to centers located at sites a and b, cf. Eq. (1-141), and using Eqs. (1-142) and (1-145) one gets the following expression for the multicenter distributed multipole expansion of the electrostatic energy ... 

The power of multicenter multipolc expansions could be demonstrated in the studies of mframolecular interactions, whereas the conventional multipole expansions based on molecular moments may even fail to converge in complexes with intermolecular distances shorter than 10 A [25]. 

The multicenter expansion of the induction energy in terms of the distributed multipole moments and polarizabilities can be obtained is a similar way starting from Eq. (1-87) rewritten as follows,... 

Very simple models are often used for water and other small polar molecules, with a single LJ site and a few charge sites whose size and location is optimized to reproduce the first multipoles, usually dipole and quadrupole. Accurate multicenter expansion would require higher order multipoles, in addition to point charges [9] (note that the distribution of multipoles is not unique [1,29,30,33-37]). 

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