Rotation-free atomic polar tensor

EFFECTIVE BOND CHARGES FROM ROTATION-FREE ATOMIC POLAR TENSORS 

Among the different models for interpretation of vibrational absorption intensities the atomic polar tensor formulation is by far the simplest to ply in transforming die experimental dp/dQi dipole derivatives into quantities associated with molecular subunits, atoms in molecules in the particular case. Besides, the transformation does not involve urmecessary approximations and assumptions. The APT formulation provides also the possibility to directly compare experimental data and theoretical ab initio results. The physical interpretation of atomic polar tensors is, however, hampered by die redundancies between the elements of atomic polar tensors as expressed by Eqs. (4.18) and (4.19). Rotational atomic polar tensors associated with the permanent dipole moment value can make, in the general case, substantial contributions to APT elements. 

In the present section a theoretical framework for analysis of vibrational intensities recendy developed by Galabov et al. [146] is presented. Fully corrected for rotational contributions atomic polar tensors are transformed into quantities termed effective bond charges. The effective bond charges are expected to reflect in a generalized manner, polar properties of the valence bonds in molecules. Aside from die usual harmonic approximation no other constraints are imposed on the dipole moment functirm. 

The starting molecular quantities in evaluating effective bond charges are atrnnic polar tensors as erqiressed in the Px matrix. From Px one easily obtains the vilnational polar tensor matrix Vx from the relation 

If experimental data are treated, Vx is calculated from die relation (4.93). hi Eq. (4.93) Ps is the array of dipole nuMiient derivatives with respect to symmetry coordinates [Eq. (3.4)]. As underlined earlier, Vx refers to a molecule-fixed reference cocHrdinate system as also do the experimental dp /dQj derivatives ( = x, y, z). The array Vx may contain implicidy contributions originating from die compensatory molecular rotation in the case of polar molecules. These contributions are also present in the P matrix. RotatirmaUy corrected P may, however, be used to derive a fully rotation-free atomic polar tensor matrix. This is achieved through the equation 

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IV. Effective Bond Charges from Rotation-Free Atomic Polar Tensors.131... 

S is the matrix of rotational correction expressed in terms of symmetry coordinates [Eqs. (3.5) and (3.11)]. The elements of Px(v) are determined by purely vibrational distortions. From the rotation-free atomic polar tensor an invariant with respect to Cartesian axes reorientation can be deduced from the trace of the product Px( )(v). P x( >(v)... 

The expression Pg Bg - Rg Bg appearing in the right-hand side of Eq. (5.8) represents a vibrational polar tensor corrected for contributions arising from the compensatoiy molecular rotation accompanying some vibrational modes. This relation was used in section 4.4 to obtain rotation-free atomic polar tensor Px(v) [Eq. (4.143)]. As already mentioned, in contrast to the usual atomic polar tensors Px, the rotation-free tensor Px(v) refers to a molecule-fixed Cartesian system. Because of the presence of the term Rg Bg, the elements of Px(v) will be the same for all isotopes of the molecule with identical symmetry. 

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