Central multipole expansion

A central multipole expansion therefore provides a way to calculate the electrostatic interaction between two molecules. The multipole moments can be obtained from the wave-function and can therefore be calculated using quantum mechanics (see Section 2.7.3) or can be determined from experiment. One example of the use of a multipole expansion is... 

The dispersion forces arise from a purely quantum mechanical effect, and thus are difficult to envisage. The sum over all the excited states of both A and B shows that the dispersion arises from correlated distortions in the two molecular charge densities. Application of the central multipole expansion produces the usual series. 

The electrostatic term (Eei) describes the electrostatic energy between molecules A and B with nondeformed electronic structure. Using classical electrostatics, the electron density of a molecule can be expanded in a series of multipoles centered on one point, usually the center of mass of the molecule [1], For quantitative studies, where more accuracy is required, the multipole expansion is done on all atoms of the interacting molecules (the so-called distributed multipole expansion [29]). However, for qualitative analysis, the central multipole expansion provides sufficient accuracy. One then uses the multipole values found in the literature for isolated molecules [1], The electrostatic energy in the central multipole expansion can be written as a series, whose leading terms up to the dipole level are ... 

The polarization term (Ep) takes into account the electrostatic effect of the mutual polarization of the electronic density of the interacting molecules. Notice that Eei + Ep is the true electrostatic energy between two molecules, and that Ep is usually smaller than Eei. Within the central multipole expansion Eq. (1.2.6) provides an analytical expression (up to the dipole moment) [1] ... 

Spherical harmonic functions are important in many problems in Chemistry and Physics. Spherical harmonic functions are central in discussions of rotation, motion in a central potential, multipole expansion, cluster bonding, spherical wave expansions and many more topics. The calculation of symmetrized powers of representations give a way of obtaining the... 

Figure 2 Examples of global and local axis systems, (a) Molecular axis system for a homonuclear diatomic. Wth this system, all central multipoles with k 0 otl odd will be zero, and no S functions with k 0 oxl odd can appear in a molecule-molecule expansion of U(R, Q). The atomic multipoles Q q (all / 0 allowed) on the two atoms will be related by Qio = (-l) Qio- ( ) Local atomic axis system for a homonuclear diatomic molecule. With this definition QJq = Qio- (c) Molecular axis system for water. The nonzero atomic multipole moments for the O atom would be QoO> QlO) QzOj Qz2c = (Q22 + Qz-2)1 > QsOJ Q32c = (Qs2 + on the hydrogen atoms Qjo = Qoo. Qio = Qio> Qiic = (-Q11 + Qi-i) = -Qiic...

Thus, in the limiting case, where the expansion of the electrostatic interaction operator in terms of the multipoles (see (19.6)) includes only the central-symmetric part (i.e. only the terms with k = 0), dependent on the term in (18.52) is only the summand with the operator T2.1116 eigenvalues of the operator T2, according to (18.28), are equal to T(T + 1), i.e. in this approximation we obtain the spectrum of energy levels rotational with respect to isospin. 

An alternative approach was to include explicit, higher order electrostatic moments in the pairwise interactions. This approach has not been extensively developed for use in molecular simulations because of the complex set of moments needed to obtain sensible results, particularly to mimic hydrogen bonding. A notable exception is the polarizable electropole model, which relies on a central polarizability as well as higher order moments to capture the electrostatic part of the interactions." The computational effort required for a multipole-based representation of the electrostatics is much greater than is involved in the use of distributed charges to represent the electrostatic interactions. If, on the other hand, the number of partial charge sites is substantially increased, a local expansion of multipole moments can become computationally economical. ... 

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