Bond charge tensors

An alternative roach for analysis of vibrational intensities has been put forward by Migrants and Averbukh [116,129,130]. An extensive review on die mediod has been published by Ruppredit [37], Hereafter, we shall follow with few exceptions the notation used by Riqiprecht which is closer to die notation used so far. Instead, on die basis of atomic Cartesian displacement coordinates, the dp/dQi quantities are transformed into the coordinate space of bond displacement vectors. The change of dipole moment is defined as 

The summation is over the number of bonds N-1, with k a bond index. XgO ) may be expressed as 

The elements aie in units of electric charge. l (k) are termed bond chaise tensms. All DO ) matrices fimn the molecular bond charge tensor matrix D 

The dipole momoit derivatives widi respect to normal coordinates are related to the elemmits of the D matrix by die general equation 

In the Mayants-Averbukh formulation the rotational terms are not treated separately as in the APT method. An augmented Pq matrix is used that comprises both Pq and the three dipole moment derivatives with respect to the rotational coordinates. The resulting matrix is similar to the (Pq Pp) matrix [Eqs. (3.1), (4.8), (4.37)]. The three zero elements in Pp are deleted since translational motion is eliminated through the definition of X 0c). For consistency of notation this matrix is denoted as Pg. It can be expressed as 

---

In general terms, there is a considerable similarity between the APT and BCT (bond charge tensor) formulations of irffiared intensities. Formulas cormecting the elements of Px and D matrices have been derived [131]. The audiors have shown that if the coordinate system and numbering of atoms are conveniently chosen the following relation holds [131]... 

Translational motion does not affect the elements of bond charge tensors. The values are, however, dependent on rotational contributions. The change of dipole moment with infinitesimal rotation of the molecule has been shown to produce six relations among the elements of the D matrix as shown by Averbukh [118]. The general expression is as follows... 

The problem with rotational contributions to intensities is dealt with by eliminating the rotational terms from both sides of the resulting linear equations. As a consequence, the parameters obtained are determined from purely vibrational distortions in the molecules. As noted in an early review Overend [16], subtraction of contributions to dipole moment derivatives arising from the compensatory molecular rotation present in particular modes of polar molecules is required to consider tire quantities obtained as purely intramolecular parameters that depend solely on the electronic structure of molecules. A satisfactory treatment of rotational contributions is implicit in the valence optical scheme. In contrast, in atomic polar tensors and bond charge tensors, due to the requirement that intensities are expressed on the basis of parameters referring to space-fixed Cartesian systems, a considerable amount of rotational intensity is introduced into the respective tensor elements, as shown by Person and Kubulat [86]. 

Bond displacement coordinates are defined by relations (4.96) ai (4.97). By e q)ressing the btMid coordinates as differences between the respective Cartesian displacements of the two atoms forming a txnid die translational motion is eliminated. Thus, there are no redundancies associated widi translational modem between die elements of bond charge tensors [129]. The elements of D matrix [Eq. (4.98)] are expressed in terms of a space-fixed Cartesian reference system. The elements of D matrix may contain considerable contribudons associated with the equilibrium dipole moment value. The resulting implicit redundancies are expressed by six reladtms [Eq. (4.118)]. 

One result of studying nonlinear optical phenomena is, for instance, the determination of this susceptibility tensor, which supplies information about the anharmonicity of the potential between atoms in a crystal lattice. A simple electrodynamic model which relates the anharmonic motion of the bond charge to the higher-order nonlinear susceptibilities has been proposed by Levine The application of his theory to calculations of the nonlinearities in a-quarz yields excellent agreement with experimental data. 

Next we will introduce the optical dielectric impermeability tensor of a crystal. The coefficients (17, ) of this tensor depend on the distribution of bond charges in the material [15,71]. The 17, are found by taking the reciprocal of the relative permittivity or dielectric constant [71]. The 17, have been defined in terms of the refractive index of the crystal as [71]... 

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

IV. EFFECTIVE BOND CHARGES FROM ROTATION-FREE ATOMIC POLAR TENSORS... 

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... 

Of particular interest are die invariants with respect to reorientation of the Cartesian reference system of the tensors D ((v). An effective bond charge has been defined [146]... 

VI. Effective Induced Bond Charges From Atomic Polarizability Tensors.261... 

VI. EFFECTIVE INDUCED BOND CHARGES FROM ATOMIC POLARIZABILITY TENSORS... 

In diis section a method for interpretation of Raman intensities based on further transformations of atomic polarizability tensors is presented. The formulation was recently proposed by Ehidev and Galabov [333], A new molecular quantity - effective induced bond charge, Ok introduced. The effective induced bond charges are obtained from rotation-free atomic polarizability tensors following the strate as outlined by Galabov, Dudev and nieva [146] in the infrared case (Section 4.IV). The Ok parameters are expected to be associated with polarizability properties of valence bonds. 

A representative series of molecules is selected to determine the trends of changes of the effective induced bond charges as defined by Eq. (9.102). The formulation developed has been applied in interpreting atomic polarizability tensors evaluated by HF/6-311+G(d,p) ab initio MO calculations [333]. A series of 17 molecules containing various bonds in different environment have been studied. The molecules are grouped as follows ... 

Its magnitude is governed by the amount of electronic and nuclear nuclear charge that lies along the z axis along the C—bond. The shape of the interaction (electric field gradient tensor) is described by the asymmetry parameter (ti), which... 

---