Central multipoles

Standard theorem of electrostatics that states that the potential due to a charge distribution, at any point at a distance R from the center of charges, can be expanded in a series of terms in the inverse powers of R, called multipoles [7]  

Equation 4.2 implies that if R is large, the potential may be taken in a first approximation as just the first non-zero term in the expansion, because further terms depend on increasing inverse powers of distance. So for a non-ionic molecule with non-zero dipole the monopole term is zero, while quadrupole and higher multipole terms are much smaller than the dipole term. At large intermolecular distance, therefore, the electric potential of a molecule is to a good approximation described by just the dipole term. For condensed phases, where molecules may come very close to one another, the dipolar approximation is unsatisfactory. 

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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... 

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...

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. 

For k 0. Eq. (5) reduces to the Laplace equation, which is used to calculate the potential inside the particle that is modeled as dielectric sphere with central multipoles. For this case the solution in spherical coordinates is given by a series of Legendre polynomials Pi (x) and related powers of r,... 

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] ... 

In the above formulation Ry is the distance between centroids, while the overall distance dependence is still of the 12-6 type. Both well depth and equilibrium separation depend on the distance and relative orientation of the two ellipsoids, and two extra parameters, p. and v, add flexibility to the model by exponential scaling of the implied dot products between orientation vectors. All in all, a GB potential depends on four parameters, the length/breadth ratios for and a, plus p, and v. Figure 13.8 shows the nice result an orientation dependence of the potential. In addition, the ellipsoids or the discoids can be made electrically active by imposing some central multipoles along the molecular envelope the calculation of the multipole energy then is easily accomplished by standard electrostatic formulas (see Section 4.2 and ref. [30]). For two dipoles di and dj at a distance R, for example, the total simulation energy of the liquid crystalline sample then becomes ... 

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