Polarizabilities Cartesian form

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). 

Laplace s equation, V V = 0, means that the number of unique elements needed to evaluate an interaction energy can be reduced. For the second moment this amounts to a transformation into a traceless tensor form, a form usually referred to as the quadrupole moment [5]. Transformations for higher moments can be accomplished with the conditions that develop from further differentiation of Laplace s equation. With modern computation machinery, such reduction tends to be of less benefit, and on vector machines, it may be less efficient in certain steps. We shall not make that transformation and instead will use traced Cartesian moments. It is still appropriate, however, to refer to quadrupoles or octupoles rather than to second or third moments since for interaction energies there is no difference. Logan has pointed out the convenience and utility of a Cartesian form of the multipole polarizabilities [6], and in most cases, that is how the properties are expressed here. 

The ordinary polarizability describes the dipole induced in a molecule by a uniform external field E = —S/V. In conventional cartesian form ... 

Equation (18) is valid when the polarizability of the dielectric is proportional to the electrostatic field strength [4]. The operator V in the Cartesian coordinate system has the form V = dldx,dldy,dldz). 

The response equations are usually solved in some iterative manner, in which the explicit construction of Q is avoided, being replaced by the repeated construction of matrix-vector products of the form Q where v is some trial vector . In general, the solution of one set of response equations is considerably cheaper than the optimization of the wave function itself. Moreover, since the properties considered in this chapter involves at most three independent perturbations (corresponding to the three Cartesian components of the external field), the solution of the full set of equations needed for the evaluation of the molecular dipole-polarizability and magnetizability tensors is about as expensive as the calculation of the wave function in the first place. 

Let us use these selection rules for investigating the main features of the Rayleigh and pure rotational Raman scattering by spherical-top molecules in the lowest vibronic states (Ogurtsov et al., 1978). The polarizability tensors dif2i and can be expanded into components of irreducible tensor operators that in cubic groups transform as E, T2, and T, respectively. Here the behavior of the operators dir 71 with respect to time reversal 0 has to be taken into consideration. To do this, we use the explicit form of the operator djj(a>) in Cartesian coordinates ... 

Using the explicit form of the relations between Cartesian dy(co) and spherical components of polarizability and Eq. (133), we obtain... 

Here is the dipole-quadrupole polarizability, describing the dipole induced by the field gradient (and also the quadrupole induced by a uniform field) and is the dipole-octopole polarizability, describing the dipole induced by the second derivative of the field (the third derivative of the potential), and again we note in passing the cumbersome nature of the cartesian notation, which requires a new symbol for each of these effects. If we are interested in higher moments induced by external fields, further polarizabilities arise for instance the quadrupole induced by an external field takes the form... 

Casimir-Polder formulation of the dispersion energy [31], which for dipole polarizabilities in the cartesian notation takes the form ... 

Equation [18] is valid when the polarizability of the dielectric is proportional to the electrostatic field strength. " The operator V in the Cartesian coordinate system has the form (didx, didy, d/dz). When one deals with a system composed of a macromolecule immersed in an aqueous medium containing a dissolved electrolyte, the partial charges of each atom of the macromolecule can be described as fixed charges charges of the dissolved electrolyte can be described as mobile charges with density determined by a Boltzmann s distribution, and Eq. [18] can be written in the following form, known as the Poisson-Boltzmann equation ... 

Since is a vector described by three Cartesian components Ej, Ej, Ei ), a, p, and Y are tensors of second, third, and fourth ranks, respectively a = a,y, p = Py, , and y= jijkh are, respectively, linear, second-order, and third-order molecular polarizabilities. The induced electric polarization P, defined as the dipole moment per unit volume, is therefore of the form... 

Methyl chloride has Raman-active vibrations belonging to A and the doubly degenerate E, syimnetry species. The E-vibrations are non-totally synunetric and contain contributions from compensatory molecular rotation. Geometry parameters, definition of synunetry coordinates and the orientation of the molecule in Cartesian space are given in Table 3.8 and Fig. 3.7. The equilibrium molecular polarizability tensor of CH3CI employed in the calculations has the following form [289] ... 

The heavy-isotope approach to evaluate rotational contributions to polarizability derivatives [288] will be illustrated with calculations on a series of molecules consisting of acetonitrile (C3V synunetiy), dichloromethane (C2v symmetry) and acetone (C2v symmetry). Structural parameters and polarizability tensors employed in die calculations are surtunarized in Table 9.1. Since the axes of the Cartesian reference systems (Fig. 9.1) are chosen to coincide with the respective inertial axes, the static polarizability tensors acquire sirtqile diagonal form. The symmetry coordinates corresponding to vibrations which may crmtain contributions from compensatory molecular rotation for the three molecules are given in Tables 9.2, 9.3 and 9.4, respectively. The following heavy isotopes are employed ... 

Each bond polarizability tensor is a 3 x3 square matrix which, in die case of an ordinaiy Raman experiment, has six independent elements. If, however, a special set of local bond Cartesian axes is chosen, the bond polarizability tensor can be presented in a simple diagonal form. Thus, the number of parameters characterizing electro-optica] properties of each molecular bond can be reduced to three. A local bond Cartesian system can be defined as follows (1) One of the coordinate axes is directed along the bond its unit vector is denoted by e lk) (2) The other two axes are chosen in such a w that they are perpendicular to the longitudinal vector and, at die same time, are at right angle one to another. Their particular orientation in space depends on the bond site symmetry. The respective unit vectors can be designated as e2W and e3(k). 

In this section the general equations (9.33) and (9.34) of VOTR are applied in interpreting Raman intensities of SO2. As was pointed out in Section 8.11, the Raman intensity experiment for bent XY2 molecules is not favorable in deteimining a complete set of molecular polarizability derivatives with respect to normal coordinates. That is why, in order to overcome the indeterminacy problem, a set of polarizability derivatives, with respect to symmetry coordinates for SO2 evaluated by means of ab initio MO calculations, is used [301]. The do/dSj derivatives forming the ag matrix are computed by applying the numerical differentiation approach. Other entries needed in solving the problem are taken from the same source [301]. Structural parameters for the sulfur dioxide molecule are given in Table 9.5. The Cartesian reference system and definition of internal coordinates and unit bond vectors are shown in Fig. 9.2. The a tensor employed in the calculations is as follows (in units of or rad" ) [301],... 

To each bond k of a molecule a local Cartesian coordinate system with z-axis directed along the bond direction is assigned. The other two bond coordinate axes are perpendicular to the z-axis and their particular orientation depends on the local symmetry of the bond. A bond polarizability tensor is presented in the form... 

The symmetry coordinates employed have their usual form. Cartesian reference system and numbering of atoms and bonds are shown in Fig. 3.6. The atomic polarizability tensor ox obtained as a standard output of the ab initio calculations is as follows (in units of 10-30 C.mA0 ... 

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