Integrals multipole expansion

Consequently, we introduce the second approximation which is to use an approximate electrostatic potential in Eq.(4-21) to determine inter-fragment electronic interaction energies. Thus, the electronic integrals in Eq. (4-21) are expressed as a multipole expansion on molecule J, whose formalisms are not detailed here. If we only use the monopole term, i.e., partial atomic charges, the interaction Hamiltonian is simply given as follows ... 

Each coefficient of the multipole expansion is computed by a numerical integration - after aligning and normalizing the found orientation distribution. 

Since the spherical core- and valence-scattering factors in the multipole expansion are based on theoretical wave functions, expressions for the corresponding density functions are needed in the analytical evaluation of the integrals in Eqs. (8.35)... 

One of the nagging features of these expressions is that the radial integral from the multipole expansion introduces a factor of r21, and thus the dimensions of B(E, l) and Bsp(E, l) depend on l. 

The expressions are particularly useftd in the evaluation of integrals over products of rotational matrices, as we shall see. They are widely used in many branches of physics and chemistry from multipole expansions through to statistical mechanical averaging. 

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 multipole expansion has already been used in certain quantum chemical calculations [59-65]. As localized orbitals are concentrated in certain spatial region, they can also be represented by their multipole moments. In the following we investigate whether the Coulomb integrals in terms of localized orbitals can be substituted by the multipole expansion of electric moments. 

As it can be seen Coulomb integrals cannot always be regularly approximated by using multipole expansion. The above values resulted in our calculations are aimed to demonstrate their transferable properties. 

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