Polarizability tensor, definition

Measurements of dipole moments, Kerr constants, and dielectric absorption have been employed (81RCR336) widely to obtain information on the conformational equilibrium in acyl heterocycles. Details on conformer structures and populations depend on the choice of additive scheme, group moments, or polarizability tensor in the case of Kerr constants. Several early conclusions, especially for furan- and thiophene-2-carboxaldehyde, appeared contradictory, owing to the choice of these quantities. A more precise definition of polarizability tensors for several heterocycles and a choice of group moments and additive schemes tested on a large amount of available experimental results and supported by accurate theoretical calculations have led to more confidence in the use of experimental dipole moments and Kerr constants in conformational analysis. A limitation of the method is that the... 

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). 

Let us denote by at, the polarizability tensor associated with the element j, and by P the polarizability tensor associated with the element /. By definition [see (6.3.74)]... 

When a is independent of E, then m((,) is obviously linearly dependent on E. Generally, however, the polarizability tensor reflecting charge dis-placeability may depend on E. Thus a more general definition of the polarizability is given by... 

Equation (IIIB-83) is identical to (MFK-Equations 20, 21). It is also identical to Kirkwood s (1937) equation. Using the dipole approximation to ViQaiiot nd the definition of a polarizability tensor (Equation IIIB-68), we see that (IIIB-83c) can be written formally as a function of at (v) and a/ (v). These are tensors for groups t and j whose principal values and principal directions are frequency dependent. If we know the a s at each frequency we know the ji s and we can use (III B-83c) as it stands. 

The first calculations of the r-electron polarizabilities of long-chain molecules employed the free-electron model and simple Huckel theoryIn both cases the computed polarizability tensor elements were proportional to the cube of the length of the chain, a / P. In these calculations (and in the calculation of p by Zyss, who showed also a definite dependence of the elements of P on chain length), no electron-electron interaction was taken into account. In order to take Coulomb interaction at least partially into account one must perform Hartree-Fock calculations on the ab initio level. Such calculations would be expected to affect (most probably decrease somewhat) the dependence on /. 

In the previous section we have defined the tensor components aap, - a,p-y and Cap -ys of the electric dipole, dipole uadrupole and quadrupole-quadrupole polarizability tensors as derivatives of the energy E , ) in the presence of a field and field gradient, Eqs. (4.65) to (4.67), or alternatively as derivatives of the perturbation dependent electric dipole p , ) and quadrupole moment 0(5,f), Eqs. (4.46) to (4.48), see also Table B.l. Furthermore, we have seen in Sections 3.3 and 4.3 that the electronic contributions to the electric dipole and quadrupole moments can be expressed as expectation values of the electric dipole and quadrupole moment operators, j2 Ro) and Ro) for the electrons, respectively. Both definitions can be used to derive quantum mechanical expressions for the polarizabilities. 

Here, the inverse of the positive-definite tensor tff is given simply by the distributed polarizabilities tensor of/, and we have included explicit sums over... 

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] ... 

In this section an example of calculations employing the APZT formalism is presented. Atomic polarizability tensors for formaldehyde-do and foimaldehyde-d2 are evaluated. The initial data are taken from RHF/6-31G(d,p) db initio MO calculations [332]. Molecular geometry, static polarizability tensor and definition of symmetry coordinates are given in Table 9.11. The orientation of the molecule in the Cartesian space, definition of internal coordinates and numbering of atoms are shown in Fig. 3.1. 

The discussion presented above outlines some definite trends of changes in the effective induced bond chaiges evaluated in analyzing oh in///o calculated atonuc polarizability tensors. are closely related with the polarizability properties of the respective bonds and depend strongly on bond lengths, atomic polarizabilities, bond multiplicity and conjugation with other bonds. 

In the OKE response only the off-diagonal elements, i j, of the dielectric tensor are involved, so the isotropic polarizability, a, does not contribute. Furthermore, the experimental q wave-vector value is zero, hence the translational part of (2.27) is not relevant. Concerning the OKE response the susceptibility tensor definition can be simplified as... 

The fundamental scattering mechanism responsible for ROA was discovered by Atkins and Barron (1969), who showed that interference between the waves scattered via the polarizability and optical activity tensors of the molecule yields a dependence of the scattered intensity on the degree of circular polarization of the incident light and to a circular component in the scattered light. Barron and Buckingham (1971) subsequently developed a more definitive version of the theory and introduced a definition of the dimensionless circular intensity difference (CID),... 

Order parameters are usually derived from the measured spectral splittings through relationships such as Equation (2.269-b) thus the availability of good estimates of the magnetic tensors is an essential requirement to obtain accurate order parameters. The theoretical results suggest that the magnetic tensors obtained from calculations in a solvent, introduced by the polarizable continuum model, should definitely be a better choice than the tensors derived, as customary, from solid-state data or from calculations for molecules in a vacuum. 

The polarizability a being a symmetric tensor, six components can be separated by using a suitable experimental set-up. By definition. 

Electric and magnetic properties of microsystems. Definition of multipoles electrostatics of permanent multipoles interaction energies for two multipoles induced molecular multipoles interaction energies of induced multipoles. Tables of point groups tensor elements of multipoles vector elements of multipoles tensor elements of polarizabilities. 

In general, the polarizability is a tensor whose invariants, trace and anisotropy, give rise to polarized and fully depolarized light scattering, respectively. Collision-induced light scattering is caused by the excess polarizability of a collisional pair (or a larger complex of atoms or molecules) that arises from the intermolecular interactions. In Section I.l, we are concerned with the definition, measurement, and computation of interaction-induced polarizabilities and their invariants. 

General Theory. In addition to the work already quoted, a few papers considering various aspects of the general theory of polarizabilities are also included. We mention the efforts to consider the effects of the polarizability of relatively crude models (such as metallic spheres) [101, 103]. The tensor representations of molecular polarizabilities have been carefully discussed [84, 141, 362]. More general procedures applicable to a variety of systems (including big molecules and molecular complexes) that are often based on classical electrostatics have been proposed [4, 21, 118, 119, 127, 135, 136]. Various books and articles are concerned with the definition of the terms in common use [12, 13, 20, 91, 109]. Other articles deal with relevant general properties of matter [22, 113] and molecules in electric and magnetic fields [31,239], triplet polarizabilities [10,61, 102,126,208], and the calculations of polarizabilities of molecules in the liquid. 

Some molecular tensors (electric dipole polarizability, electric and magnetoelectric shielding) are origin independent, as can be immediately found by inspection of definitions (87)-(112). Other tensors depend on the origin assumed for the multipole expansion. For instance, in a change of origin... 

The definition of corresponding molecular tensors arrived at by continuous transformation of the origin of the current density, which formally sets its diamagnetic term to zero (abbreviated as CTOCD-DZ), is discussed in Section 3. Section 4 deals with the problem of origin dependence of magnetizability polarizabilities of a molecule in the presence of uniform EFG. 

This general complex transition polarizability has no definite symmetry with respect to permutation of the tensor subscripts, unlike the ordinary polarizability which has a pure symmetric real part (2.13a) and a pure antisymmetric imaginary part (2.13b). 

Notation. The symbols a, f, y are used throughout to denote the electric field polarizability, first and second hyperpolarizabilities respectively, suitably qualified by frequency factors where necessary. The magnetizability is denoted by y and the nuclear screening tensor by a. The numerous but well-known acronyms specifying the computational procedures are used without definition. The possibly rather less well-known acronyms for the principal gauge invariant procedures are given in Table 1. 

In practice, the calculation involves three steps the calculation of the property for three systems A- B, A and B. If the desired goal is the calculation of the interaction-induced mean dipole polarizability a) or second dipole hyperpolarizability (y), one must first obtain the Cartesian components of both tensors for all three systems. The general definition of the mean is... 

These expansions serve mainly as definitions of the polarizabilities and hyperpolarizabilities as proportionality constants in the correction terms to the permanent moments. The dipole polarizability a is a second-rank tensor with nine cartesian components the dipole-quadrupole polarizability and first dipole... 

Table B.l Definitions of tensor components of the electric polarizabilities and hyperpolarizabilities as derivatives of components of the field-dependent electric dipole Ha S,S) and quadrupole Qji(S, ) moments or of the field-dependent energy E , ). All derivatives have to be evaluated at zero field and field gradient.

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