Multipole approximation

Since empirical force fields do not accurately estimate the true interatomic forces, it is difficult a priori to say how accurate the fast multipole approximation to the exact Coulomb potential and forces (exact in terms of the sum over partial charges) should be. Probably a good rule is to make sure that at each atom the approximate electrostatic force is within a few percent relative error of the true electrostatic force, obtained by explicitly summing over all atom pairs, i.e., IF — FJ < 0.05 F , for all atoms i, where F is the... 

However, unlike point charges, the continuous charge distributions that occur in quantum chemistry have varying extents and the applicability of the multipole approximation is not only limited by the distance but also by the extent or diffuseness of the charge distribution. This additional complexity makes a transfer of the concepts of the fast multipole method to applications in quantum chemistry less straightforward. Therefore it should come as no surprise that several adaptations to extend the applicability of the FMM to the Coulomb problem with continuous charge distributions have been suggested. These lead to... 

Like (5.15b), the multipole approximation (5.19) is dependent on the long-range assumption (5.14) both approximations fail (for different reasons) if the molecular wavefunctions f and fi" overlap appreciably. 

Specifically, the eleven profiles include the obvious single transitions, i.e., the rotovibrational transitions in just one of the two colliding H2 molecules these are the Si(0), Si(l), and Qi (1) transitions in one of the two interacting molecules. Double transitions in both collisional partners are also taking place, such as the simultaneous transitions gi(l)+So(0) (which occur near the Si(0) transition frequency) and Q (1) + So(l) (near Si(l)), Fig. 3. 32. Intensities of all these lines are known from theory (classical multipole approximation, Chapter 6) their superposition reproduces the measurement closely, Fig. 3.33. 

Fig. 4.2. The principal induced dipole components of H2-H2 ground state vibrational averages are shown in the left-hand plot. Overlap (dotted), multipole-induced terms (solid lines) classical multipole approximation (dashed). Right-hand plot ditto, but vibrational transition elements, v = 0 - v = 1. Molecule 2 undergoes the vibrational transition [281].

The radial part of the electrostatic energy in the multipole approximation is given... 

The radial component of the second-order induction energy in the multipole approximation can be written as 14946i,i64,i65,i66... 

We wish to end this section by saying that the variation-perturbation approach as discussed above, introduces a natural hierarchy of gradually more and more sophisticated models starting from the crude evaluation of the electrostatic energy in the distributed multipole approximation, and ending with the inclusion of the intramolecular and dispersion contributions at the MP2 or even more correlated level. 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic serie Still, for the long range the multipole approximation to can be quite... 

The other two components of the octahedral Ai Ti transition, due to the single-orbital promotions, dyz - dy2 j,2 andd x - d2i xJ, give rise analogously to the rotational strength components, Rg and Rg, respectively. The forms of the latter two components are obtained in the point-multipole approximation by cyclic permutation of the coordinates in Eqs. (21) and (22). 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic series . Still, for the long range the multipole approximation to AE can be quite accurate, if properly truncated (for instance, after the smallest term). For shorter distances, the penetration between the molecular charge clouds becomes significant, the screening of the nuclei by the electrons becomes incomplete even for neutral molecules, and the power law for AE 5 is modified by contributions which increase exponentially with decreasing R. These penetration contributions we define as ... 

Armed with a multipole approximation of the full charge distribution of the two molecules in question, one can evaluate the correct electrostatic interaction as a series of terms, in inverse powers of R, the intermolecular separation [9,11]. We will use the notation that i represents the order of any nonzero 2i moments of the first molecule (i = 0 for monopole, 1 for dipole, 2 for quadrupole, etc.). Terms expressing the interaction of the ith moment of one with the /th moment of the other are collected in the inverse nth power of R where n — i + j + 1 [13], For example, the... 

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