Vibrational atomic polar tensors

This model has the advantage that the atomic polar tensor elements can be determined at the equilibrium geometry from a single molecular orbital calculation. Coupled with a set of trajectories (3R /3G)o obtained from a normal coordinate analysis, the IR and VCD intensities of all the normal modes of a molecule can be obtained in one calculation. In contrast, the other MO models require a separate MO calculation for each normal mode, since the (3p,/3G)o contributions for each unit are determined by finite displacement of the molecule along each normal coordinate. Both the APT and FPC models are useful in readily assessing how changes in geometry or refinements in the vibrational force field affect the frequencies and intensities of all the vibrational modes of a molecule. 

Rotational and Dipole Strength Calculations for the CH-Stretching Vibrations of L-alanine Using the Localized Molecular Orbital, Nonlocalized Molecular Orbital, Atomic Polar Tensor, and Fixed Partial Charge Models ... 

Pq, is expressed in Cartesian coordinates. These polar tensors T), can be derived from experimental intensities by elementary coordinate transformation. If the axes x, y, and z are chosen such that the bonds are oriented along one of the axes, then the derivatives can be used to interpret the changes of the electron clouds during a vibration. Besides, considering the definitions of the axes, it is possible to transfer atomic polar tensors between similar molecules and to estimate their intensities (Person and Newton, 1974 Person and Overend, 1977). 

Because the intensity of a given vibrational mode is connected with the changing molecular dipole moment associated with that particular motion of the atoms, analysis of these intensities offers valuable insights into charge redistributions within the system. One can partition the dipole changes into contributions from various atoms using an atomic polar tensor (APT) formalism - " which is defined for an atom a as... 

From infrared and Raman spectra of bicyclobutane and appropriate deuterated isomers a new vibrational assignment was made with the help of the spectrum calculated using the 6-3IG basis set. A normal coordinate analysis furnished atomic polar tensors and related properties. The results were compared with similar data for cyclopropane and [l.l.l]propellane. 

T. B. Freedman and L. A. Nafie, / Chem. Phys, 78, 27 (1983). Vibrational Optical Activity Calculations Using Infrared and Raman Atomic Polar Tensors. Ibid., 79, 1104 (1983). 

Erratum Vibrational Optical Activity Calculations Using Infrared and Raman Atomic Polar Tensors. 

Whereas the dipole moment gradient (the atomic polar tensor) is well defined and it is the same quantity that determines conventional infrared absorption spectra (see the chapter on vibrational spectroscopy), the gradient of the magnetic dipole moment is zero within the Born-Oppenheimer approximation. This is due to the fact that the magnetic dipole moment for a closed-shell molecule is quenched (since it corresponds to an expectation value of an imaginary operator), making the rotational strength in Eq. 2.150 zero. 

In our presentation of the atomic polar tensor formulation we shall follow the notation introduced by Person and Newton [33] since it is now generally accepted. The dipole moment changes induced by vibrational distortions are represented as functions of individual atom displacements... 

Equation (4.16) is a basis for applications of the APT formulation in predicting vibrational intensities by transferring atomic polar tensors between molecules having the same atoms in a similar environment. 

The total number of elements 1 of the atomic polar tensor of a molecule is, therefore, equal to the number of Cartesian symmetry coordinates in the infrared active species. The set of Cartesian symmetry coordinates describes, in the general case, vibrational distortions as well as translations and rotations belonging to the same symmetry species as the infrared active modes. The translational and rotational conditions can be explicitly written as shown in Table 4.3. The important conclusion is that the net number of independent atomic polar tensor elements is exactly equal to the number of infrared active modes. In the case of AB2 (C2v) molecule 1 = 3+5 = S. For such molecules, however, there are three translational and two rotational ctmditions relating the APT elements as shown in Table 4.3. Subtracting these from 1 yields exactly the number of infrared active vibrations of the molecule. 

Coordinate definitions, Lg and P matrices for H2O used in evaluating the atomic polar tensor elements are as given in section 3.3. With the aid of relation (4.14) die Pg matrix is transformed into vibrational polar tensor, while the rotational polar tensor is calculated using a permanent dipole moment value of-1.85 D [34]. The two submatrices obtained are as follows (in units of D A l) ... 

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]. 

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. 

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 structure of the C" matrix will be shown later in this section. D(v) matrices are numerically equivalent with the respective Px(v) arrays less the atomic polar tensor for the first atom. The elements of D(v) have dimension of electric charge. The charge fluctuations reflected in D(v) are determined, as already emphasized, fi om vibrational distortions of the respective bonds. The elements of D(v) are expected to be connected with the electronic structure of the bonds undergoing vibrational distortions. More polar bonds should give rise to higher values for the elements of D(v) and, also, to higher integrated intensities of the respective infi ared absorption bands. 

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. 

Hansen AE, Stephens PJ, Bouman TD (1991) Theory of vibrational circular-dichroism -formalisms for atomic polar and axial tensors using noncanonical orbitals. J Phys Chem... 

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