Vibrational atomic polarizability tensor

In these equations oq and as are the matrices containing molecular polarizability derivatives with respect to normal and symmetry vibrational coordinates, respectively [Eqs. (8.41) and (9.3)], and Op is an array comprising polarizability derivatives with respect to molecular translations and rotations. The matrix product ag Bg represents die so-called vibrational atomic polarizability tensor Vq accounting for the changes in molecular polarizability with molecular vibrations. The Vq tensor for the entire molecule can be expressed as a juxtaposhion of individual atomic tensors ... 

In vibrational analysis it is of particular importance to operate widi indepoident quantities that are associated with vibrational motions of die molecule only. Atomic polarizability tensors [Eq. (9.76)] can be corrected for non-vibradonal contributions following the procedure outlined below. The first step is to eliminate the contribution from the rotational atomic polarizability tensor [Eq. (9.78)]. After subtractmg Rq from both sides of Eq. (9.76) the vibrational atomic polarizability tensor is obtained... 

The atomic polarizability tensor ax for SO2 evaluated by employing Eq. (9.76) is shown as follows together with the vibrational and rotational polarizability tensors (in units A ) ... 

The atomic polarizability tensor of an atom a is defined by expression (9.73). Arranged in a row all ax tensors form the polarizability tensor of the molecule, ax [E<1-(9.74)]. Its elements can be obtained via relations (9.75) and (9.76). As was discussed in Section 9.IV, atomic polarizability tensors are sum of two arrays (i) vibrational and (ii) rotational polarizability tensors [Eq. (9.79)]. The elements of atomic polarizability tensors are interconnected by the dependency conditions (9.84) and (9.85). The presence of such relations hampers die physical interpretation of these quantities. 

Fig. 5. HF dependence of the dipole and quadrupole polarizability tensor (in atomic units) on the vibrational quantum number v. Calculated from the ioo > 9552 polarizability radial function and the potential energy curve.

X HE VIBRATIONAL SPECTRUM of any material consists of two parts the infrared (IR) and Raman spectra. IR spectroscopy is sensitive to the changes in dipole moment that occur during the vibrations of atoms that are forming chemical bonds. Raman spectroscopy detects the polarizability tensor changes of the electron clouds that surround these atoms. These apparent differences in the physical principles of both effects have led to the development of these two distinctly different techniques. IR and Raman spectra complement each other. Because they are sensitive to the vibrations of atoms, they are called vibrational spectra. 

Table 16-1. Energies, electric dipolar moments, net atomic populations, vibrational polarizabilities and mean vibrational molecular polarization, magnetizability and contributions thereto, isotropic g tensor and nuclear and electronic paramagnetic and diamagnetic contributions thereto, principal moments of inertia and rotational parameters calculated for H2 C N2 in seven structural isomers... 

As the molecule vibrates (undergoes atom displacements)) the electronic charge distribution and, hence, the polarizability (a) varies in time. The polarizability is related to the electron density of the molecule and is often visualized in three dimensions as an ellipsoid and represented mathematically as a symmetric second-rank tensor. The time-dependent amplitude (Q ) of a normal vibrational mode executing simple harmonic motion is written in terms of the equilibrium amplitude Q , the normal mode frequency o), and time t). 

Other possible basis vectors include true spatial vectors such as a dipole-moment component (e.g. or a displacement of an atom along a direction defined by a bond (i.e. a bond stretching motion). More elaborate basis vectors are also useful one very important one consists of the set of three Cartesian-axis displacements of each of the N atoms in a molecule, and has a dimensionality of 3N. We will see this basis vector in action in Section 8.5.3, where we use symmetry to characterize a set of molecular fundamental vibrational modes. We can also use the wavefunction of a molecular orbital as a basis vector, and so classify that orbital according to the effects of the various symmetry operations acting on that wave-function (Section 9.3). Even a tensor such as the molecular polarizability can be used as a basis vector. Throughout this book we use a number of subscript labels for the different basis vectors we discuss. The convention we use is as follows. 

The a tensor is the polarizability of the material. Considering its modulation by the normal atomic vibrations of the material, for small amplitude oscillations near the equilibrium, the polarizability dependence on the normal coordinate Q associated with a normal mode of vibration can be written as ... 

---