In most of published articles, we may find three different kinds of formulae for determining anisotropic polarizability of molecule. So I need to find the formula which is a correct one from the rest of the other. Can anybody help me to find this? See the attachment for showing formulae.
Mr. Karthick,
The anisotropic invariant (beta0 in my attachment below or delta(alpha) in your one) is given by the "Formula 2" (your attachment) obtained for a system of (alpha)ij, where i and j are the orientations with coordinates (x,y,z) (Eq. 1 in my atatchment). Matrix form is also given.
This system is described towards another principle one with principle axis k (X, Y, Z).
It is the respective axis (k) coinciding with the direction of the induced dipole moment.
Eq. 1 give the relationship between the ith or jth orientations towards the k using the cosines of the angles between these axes,
The anisotropic invariant is given by Eq. 2, i.e. Eq. 2 (my attachments) is = Formula 2 in yours.
When the relative orinetation of both these coordinate systems vanishes that Formula 2 is expressed by Eq. 3 in my attachment. Meaning that Formula 2 at (alpha)xx = (alpha)X using the coordinates of the mentioned principle axis resulted to Formula 1 in your attachment.
Formula 3 most probably is obtained by solving the Eq. 1 for optically inactive compound at (alpha)yx=(alpha)yz. See the matrix representation of Eq. 1 (attachment 1), accounting for that (alpha)ij = (alpha)ji
Dear Bojidarka B. Ivanova
Thank you for your kind help.
by
T.karthick
The polarizability tensor is a second-rank symmetric tensor whose components transform in the same way as the quadratic forms x2, y2, z2, xy, yz and zx (see Group Theory in Quantum Mechanics, by Volker Heine, p. 240). The polarizability tensors corresponding to the 32 crystallographic point-groups are listed in Table 3.4 (p. 66) of the book by Richard C. Powell (Symmetry, Group Theory, and the Physical Properties of Crystals, Springer, 2010). Table 7.8 (pp. 189 - 192) of the same book provides the list of the irreducible representations of the Raman active modes and their corresponding Raman tensors associated with the above-mentioned 32 point groups.
The polarizability tensor is said to be isotropic if its usual matrix representation is diagonal, with the three diagonal elements equal. Formula 1 properly measures anisotropy in the polarizability tensor only if this tensor is diagonal with unequal diagonal elements; its fails to account for non-vanishing off-diagonal elements of the actual polarizability tensor. Formula 2 overcomes this shortcoming. Formula 3 is limited by the same reasoning as above. It is however not as limited as Formula 1. A cursory review of the above-mentioned Tables 3.4 and 7.8 clarifies why. Note that if one changes the pre-factors 6 in Formula 2 into arbitrary finite positive numbers (say all into 7, or into 7, 8 and 9), the resulting formula would remain a valid measure for the anisotropy of the polarizability tensor.
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