Multipole series

The key element in London s approach is the expansion of the electrical potential energy in multipole series. Since neutral molecules or portions of molecules are involved, the leading term is that for dipole-dipole interaction. While attention has been given to higher-order terms, these are usually small, and the greater need seems to be for improved treatment of the dipole-dipole terms. London used second order perturbation theory in his treatment, but Slater and Kirkwood38,21 soon followed with a variation method treatment which yielded similar results. Other individual papers will be mentioned later, but the excellent review of Mar-genau26 should not be overlooked. 

Ferenczy GG (1991) Charges derived from distributed multipole series. J Comput Chem 12(8) 913-917... 

Note that failure to satisfy (5.20) has far more serious theoretical consequences than mere failure of convergence of the multipole series (5.19). This can be seen by examining excited-state solutions fi, and f " of (5.16a) and (5.16b), the higher unperturbed eigenfunctions of 7f(0) ... 

Young et al. [195] have provided a calculation in which they compared expanding the multipole series up to /= 6 in a spherical cavity of 3.8 A. These results may be compared directly to those of Wong et al. [297] at the identical level of theon asis set in order to assess the effect of including higher moments. In each case, the differential solvation free energy increases by about 40%. This illustrates nicely the relationship between cavity radius and model... 

The Coulombic term can be expanded into a sum of terms (multipole-multipole series), but it is generally approximated by the first predominant term representing the dipole-dipole interaction between the transition dipole moments Md and MA of the transitions D —- D and A A (the squares of the transition dipole moments are proportional to the oscillator strengths of these transitions) ... 

The integration is over the whole charge cloud. It is common practice to express this in terms of a multipole series (Jackson 1984),... 

Let r stand for the radius vector of the ith electron with respect to the center of mass of the molecule. Assuming that all r, are much less than b, we can expand the Coulomb interaction operator into a multipole series. Keeping only the dipole terms, we get the following expression for P0n(b)... 

Examination of the first two columns of Table 4 indicates that for H-bonds that are stretched somewhat beyond their equilibrium geometry, the electrostatic term furnishes an excellent estimate of the full interaction energy, and the former is in turn nicely reproduced by a truncated multipole series. The other contributions, EX, POL, CT, and MIX all make smaller contributions which cancel to a large extent. The same is not true when the H-bond is compressed. Exchange repulsion grows rapidly and cannot be ignored. Moreover, the multipole series deviates significantly from the full ES component. 

Examination of the various contributors to ES reveals that the dipole-dipole R"3 term is particularly sensitive to angular distortions. Whereas the dipole-quadrupole interactions contained in the R 4 term are also sizable, it is important to note that they behave differently depending upon which molecule is rotated. That is, the R 4 term produces a net stabilization if the donor is turned but adds to the destabilization of R-3 if the rotation occurs in the acceptor. Overall, the multipole series, truncated at R 5, provides a reasonable approximation of the full ES distortion energy, particularly at the longer distance. 

Besides the penetration effect, this also may be assigned to an inevitable truncation of higher terms in the multipole expansion. It should be noted at this point that the convergence of the multipole series may be improved. Various approaches based on the decomposition of the molecular charge density into smaller distributions and procedure to generate high-order moments were suggested by Mezei and Camp-beir ". ... 

Figure i. 4 Distance dependence of the multipole series of the electrostatic interaction energy, truncated at various orders, for the water dimer. Data from. The values for the series truncated at differ only very slightly from the series and so are not shown explicitly. 

Because of its importance in the water dimer, as well as in a number of other H-bonded systems, the electrostatic interaction was partitioned into a multipole series in powers of 1/R, consisting of terms corresponding to interactions between dipole, quadrupole, and so... 

Reaction Fields from Higher Order Multipolar Expansions Generalizations of the Born—Kirkwood—Onsager model have appeared which extend the multipole series to arbitrarily high order.20,62,144,234-236 ybis approach yields... 

We now show some representative results which illustrate the applicability of different methods for computing the intermolecular potential. We start with the long range part by looking at the first and second order multipole series (16), (20) and (21). T e lowest term(s) in the first order series can be easily checked by... 

A multipole series should strictly be truncated at a given power, w = -(/j + /z + 1), of the intersite separation R, rather than by only using the multipoles Qik up to a given value of / on each molecule. Hence, to include the important quadrupole-quadrupole interactions (the terms in Q k Q2k )i the series should be taken up to R and thus also should include the hexadecapole charge Q k QIq and Qlo Qik) and octopole-dipole (Ql Qik and Q k Qlk ) terms. 

A resonance interaction can occur between two oscillators when one of them is in an excited state. The energy of this interaction is determined by that part of the total Hamiltonian that represents all pairwise coulom-bic interactions between electrons and nuclei in the two groups. At sufficiently large distances (probably over 3 A) these interactions can be expanded in a multipole series, of which the first important term for a neutral system is that due to transition dipole coupling (TDC). Higher transition multipoles may be important in some cases (Cheam and Krimm, 1985), but we treat here only the TDC case (Krimm and Abe, 1972 Moore and Krimm, 1975 Cheam and Krimm, 1984c). 

In the study of weakly bound systems the range of distances around the van der Waals minimum is usually the one of primary interest. Here the performance of the long-range terms in their multipole-expanded forms is somewhat problematic. First, the multipole series may not converge sufficiently fast or not at all and, secondly, penetration effects become non-negligible. 

In summary, the multipole series may provide an estimate of the full ES interaction with a certain measure of accuracy, particularly if the two subunits are well separated. This concept offers the opportunity to gain insight into the nature of the ES force, based on the direction and magnitudes of some of the lower moments of the individual molecules. It also has a predictive capability with regard to the anisotropy of the full ES interaction, i.e., its sensitivity to angular aspects of the intermolecular geometry. 

In practice, site multipole series for the electric potential are truncated, usually at quadrupoles or less. If higher terms in the series are not actually calculated, the degree of convergence is uncertain. The PD method, discussed in the next section, avoids this problem by optimizing for the best possible model of the electric potential with a given number of multipole terms. Thus in the PD method convergence is always the best possible for a given set of multipole parameters and potential points. 

---