Multipole moments derivatives

Thanks to efficient recurrence formulae, multipole moments and multipole moment derivatives can be calculated at very high order with a low computational cost. The calculation of reaction field factors, however, may become computationally expensive at high order due to the increasing number of linear equations to be solved. Thus, in practice, the multipole moment expansion is cut off at a maximum value of f (/max), usually taken around 6. In order to get an order of magnitude of the error introduced by the truncation, let us consider Kirkwood s equations [5] for the free energy of a charge distribution of charges q, and r, in a spherical cavity of radius a ... 

Gaussian also predicts dipole moments and higher multipole moments (through hexadecapole). The dipole moment is the first derivative of the energy with respect to an applied electric field. It is a measure of the asymmetry in the molecular charge distribution, and is given as a vector in three dimensions. For Hartree-Fock calculations, this is equivalent to the expectation value of X, Y, and Z, which are the quantities reported in the output. 

Expression (4.31) is widely applied to calculate the error in properties derived from the least-squares variables. We will use it in chapter 7 for the calculation of the standard deviations of the electrostatic moments derived from the parameters of the multipole formalism. 

As noted above, in the traceless definition the /th-order multipoles are the sole contributors to the /th electrostatic moments. This implies that the traceless moments derived from the total density p(r) and from the deformation density Ap(r) are identical, that is, ,mp(p) = 0,mp(Ap) for / > 2. 

To evaluate this expression for distributions expressed in terms of their multipolar density functions, the potential <1> and its derivatives must be expressed in terms of the multipole moments. The expression for charge distribution has been given in chapter 8 [Eq. (8.54)]. Since the potential and its derivatives are additive, a sum over the contributions of the atom-centered multipoles is again used. The resulting equation contains all pairwise interactions between the moments of the distributions A and B, and is listed in appendix J. 

Born s idea was taken up by Kirkwood and Onsager [24,25], who extended the dielectric continuum solvation approach by taking into account electrostatic multipole moments, Mf, i.e., dipole, quadrupole, octupole, and higher moments. Kirkwood derived the general formula ... 

Recently, Sokalski et al. presented distributed point charge models (PCM) for some small molecules, which were derived from cumulative atomic multipole moments (CAM Ms) or from cumulative multicenter multipole moments (CMMMs) [89,90] (see Sect. 3.2). For this method the starting point can be any atomic charge system. In their procedure only analytical formulas are used,... 

A factor -2 included in the last term here compensates for the use of Rydberg units and for the omission of the negative electronic charge in potential functions derived from Eq. (7.14). Hence the electrostatic multipole moments of atomic cell r/( are... 

A = J —J f. Here the Dyakonov tensor <1>q characterizes the polarization of the registered light. In deriving Eq. (2.24) the properties of the 17-functions were used, as previously in the case of Eq. (2.21) and according to (A.13) and (B.4). It may thus be seen from (2.24) that only the multipole moments bpQ of rank K < 2 have any direct effect on the intensity and polarization of molecular fluorescence. This latter assertion also holds in the case where multipole moments of rank higher than K = 2 are created in the excited state (6) in the case of absorption of sufficiently intensive light. 

The spherical form of the multipole expansion is very useful if we are looking for the explicit orientational dependence of the interaction energy. However, in some applications the use the conceptually simpler Cartesian form of the operators V1a 1b may be more convenient. Moreover, unlike the spherical derivation, the Cartesian derivation is very simple, and can be followed by everybody who knows how to differentiate a function of x, y and z 149. To express the operator V,, in terms of Cartesian tensors we have to define the reducible, with respect to SO(3), tensorial components of multipole moments,... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

In Eq. (18), we recognize the first quantization fcth-order electronic multipole moment operator (r - Ro). An analogous expansion of r - Rm 1 would yield the nuclear multipole moment operator (Rm — Rq). The higher-order multipole moments generally depend on the choice of origin however, to simplify the notation, we omit any explicit reference to this dependence. The partial derivatives in Eq. (18) are elements of the so-called interaction tensors defined as... 

Knowing the molecular permanent multipole moments and transition moments (or closure moments derived from sum rules, such as (36)), the computation of the fust and second order interaction energies in the multipole expansion becomes very easy. One just substitutes all these multipole properties into the expressions (16), (20), (21) and (22), together with the algebraic coefficients (24) (tabulated up to terms inclusive in ref. in a somewhat different form ), and one calculates the angular functions (lb) for given orientations of the molecules. 

The power series in Eqn. (2) has as many partial derivative values as there are multipole moments. They can also be arranged in a first-degree Cartesian... 

The electrostatic interaction, which is defined as the classical Coulombic interaction between the undistorted charge distributions of the isolated molecules, is the easiest to derive from wavefunctions. When there is no overlap of the charge distributions of the molecules, all that is required is a representation of the molecular charge density. The traditional, and simplest, representation of the molecular charge distribution is in terms of the total multipole moments. The first nonvanishing multipole moment could often be derived from experi-... 

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