Induced dipole Cartesian components

Table 4.5. Cartesian dipole components of the H2-He system, for three inter-nuclear spacings r after [279]. The center-of-mass of the H2 molecule is at the origin and the He atom is at z = R. A positive induced dipole moment p2 corresponds to the polarity H He+ (py — 0).

Fig. 12.4. The direction of the induced dipole moment may differ from the direction of the electric field applied (due to the tensor character of the polarizability and hyperpolarizabiUties). Exan )le the vinyl molecule in a planar conformation. Assume the following Cartesian coordinate system x (horizontal in the Figure plane), y (vertical in the Figure plane) and z (perpendicular to the Figure plane), and the external electric field = (O, y, O). The component x of the induced dipole moment is equal to [within the accuracy of linear terms, Eq. (12.19)] fJ-ind x = l x — I Ox ° xy y< J-ind,y ( yy y> Mind.r diiySy. Due to the symmetry plane z = 0 of the molecule (cf. p. 704) oizy = ce x = 0, and similarly for the hyperpolarizabilities, we have As we can see, despite the field having its x component equal to zero, the induced dipole moment x component does not equal to zero x 7 0)-...

In a molecule which has tetrahedral symmetry (CCI4) or octahedral symmetry (SFg) an external electric field will generate an induced dipole moment whose direction is the same as that of the field regardless of the orientation of the molecule (see Fig. 1.33). Such a molecule is said to be isotropic. In such a case when the electric field vector is resolved into cartesian coordinate components, Eq. (1.110) becomes... 

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... 

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... 

Electric dipole polarizability a is a second-order tensorial (3x3 matrix) quantity whose element describes the change of dipole moment component p,- induced by a change of electric field in direction j, where ij = x,y,z denote arbitrarily chosen Cartesian directions. Formally, if p, is the field-free dipole moment, and Ap,y is the change in dipole moment induced by a static electric field AFj in the j direction, then OLy can be evaluated as the limiting ratio... 

Polarizability of an Isolated Molecule.—In Cartesian tensor notation, the components of the molecular dipole moment p, induced by an electric field E, can be written as ... 

The simplest approach to the theory of light scattering in general and Raman scattering in particular is to introduce an oscillating electric dipole moment induced in the molecule by the electric field vector JE of the light wave. In cartesian tensor notation, the -component is written... 

For example the dipole polarizability given in (2.33) has spherical tensor components OQQdl) and a2K(11) The dipole-quadrupole polarizability (A .gY in Cartesian notation), which describes the quadrupole moment induced by an electric field or the dipole moment induced by an electric field gradient, has components 0 (12), 021(02) and 02k(12). The polarizabilities are even (g) or odd (u) under inversion according as 1+1 is even or odd. This information is then sufficient, with the help of Table 3, to determine the transformation properties in the molecular symmetry group. Any component which transforms according to the totally symmetric representation may have a non-zero value. 

Thus light of a particular frequency can simultaneously induce a dipole moment in a molecule and then couple with the dipole components to result in light absorption Raman spectra are observed within the spectram of light scattered from an intense source. Induced vibrational transitions are observed with a dispersive device (monochrometer) and some sort of electronic detection (in the visible range) at 9(f from the light source (laser) beam. Remarkably, C. V. Raman first observed this effect with a handheld spectroscope in 1928 for which he received the Nobel Prize in 1930. Thus we can examine the symmetry properties of second-order combinations of the Cartesian coordinates (in column 2 ) and use them to indicate a yes/no answer as to whether a given molecular vibration will occur in Raman spectroscopy. 

---