Multipole moments, Cartesian

Cartesian multipole moments, 729 catalyzed muon fusion, 327, 328... 

Hellmann-Feynman theorem (p. 618) Cartesian multipole moments (p. 624) dipole, quadrupole, octupole moments (p. 624)... 

The parentheses [] mean the corresponding Cartesian multipole moment. When computing the Fock matrix correction, the first multipole moment (] stands for the multipole moment of the charge distribution XpXq Ihe second, of the unit cell. For example, for the correction C Jq(N) is equal to... 

Let us consider again the x component of the Cartesian multipole-moment integrals (9.3.11) ... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

In Cartesian coordinates, the expectation values of multipole moment operators are computed as... 

With this notation, the electric charge qo of a monopole equals Qoo-Cartesian dipole components px, py, pz, are related to the spherical tensor components as Ql0 = pz, Qi i = +(px ipy)/y/2, with i designating the imaginary unit. Similar relationships between Cartesian and spherical tensor components can be specified for the higher multipole moments (Gray and Gubbins 1984). 

The connection between the covariant cyclic and cartesian coordinates of the vector J yields Eq. (A.6), whilst (A.5) makes it possible to form the vector itself out of the components (J)q. As follows from (2.18), the components of the multipole moment pq characterize the preferred orientation of the angular momentum J in the molecular ensemble. Fig. 2.3(a, b) shows the probability density p(0, [Pg.30]

The operator V,, b is physically interpreted as representing the interaction of the instantaneous moment with respect to center A with the instantaneous 2>B moment with respect to center B and can be expressed in terms of irreducible spherical or reducible Cartesian tensor operators of multipole moments. The operator V,, can be written as... 

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). 

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... 

In Appendix X available at booksite.elsevier.com/978-0-444-59436-5 on p. el69, the definition of the polar coordinate-based multipole moments is reported. The number of independent components of such moments is equal to the number of independent Cartesian compments and equals (2/ -I- 1) for I = 0,1,2,... with the consecutive I pertaining, respectively, to the monopole (or charge) (2/ - - 1 = 1), dipole (3), quadmpole (5), octupole (7), etc. (in agreement with what we have found a while before for the particular mrxnmtsl. 

The spherical harmonics are quite appropriate to express the explicit orientational dependence of the interaction, but in the chemical practice it is customary to introduce a linear transformation of the complex spherical functions Y into real functions expressed over Cartesian coordinates, which are easier to visualize. In Table 8.3 we report the expressions of the multipole moments. 

Table 8.3. Multipole moments expressed with the aid of real Cartesian harmonics...

In spite of the disadvantages of the Cartesian formulation, it is preferred by many workers because the alternative, the spherical tensor formulation, is perceived as mathematically difficult. There is undoubtedly some truth in this view. Moreover the spherical-tensor formulation deals in complex quantities which are more difficult to comprehend than the cartesian-tensor components. However the power and versatility of the spherical tensor approach should not be abandoned lightly, and the main purpose of the present paper is to show that it is possible to combine the best features of the cartesian and spherical-tensor methods. We will show that this hybrid approach leads to very compact expressions for the electrostatic energy and related quantities such as the induction and dispersion energies, and that these can be expressed entirely in terms of real multipole moments referred to molecule-fixed coordinate systems. The transformation between molecule-fixed and space-fixed coordinates can be carried out once and for all, and the analogues in this method of the interaction tensors contain the necessary orientational information. 

Table 1. Definitions of the multipole moments to rank 4, and equivalent expressions in Cartesian tensor notation...

It is apparent that the 7 take the place in this formulation of the interaction tensors T of the conventional Cartesian formulation, but it should be emphasized once again that all the formulae given here refer to multipole moment components in the local, molecule-fixed frame of each molecule, whereas the corresponding Cartesian formulae deal in space-fixed components throughout and require a separate transformation between molecule-fixed and space-fixed frames. ( Space-fixed is perhaps a misleading term here, since the calculation is commonly carried out in a coordinate system with one of its axes along the intermolecular vector. However, the point is that in the Cartesian tensor notation there has to be a common set of axes for the system as a whole, and this can be the molecule-fixed frame for at most one of the molecules involved.)... 

---