Rotational polar tensor

The relation between rototranslational and Cartesian coordinates is given by the matrix P [Eq. (2.12)]. The product PpP is termed a rotational polar tensor... 

The entire representation of (dp/dQt)o in Eq. (4.5) in terms of Cartesian displacement coordinates is simply the sum of vibrational and rotational polar tensors... 

Px is a sum of two terms — vibrational polar tensor (PgBg) and rotational polar tensor (PpP). The rotational polar tensor is evaluated using an equilibrium dipole moment value of -1.87 D [34]. The PsBs and PpP matrices are given below. 

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) ... 

It is easily seen that some elements of Px. i e. all second row elements (pyy), come directly from the rotational polar tensor. Since the non-zero elements of Pp 3 are derived from the Cartesian components of the equilibrium dipole moment, it is evident that the final Px matrix also contains such non-vibrational contributions. Therefore, the term Pp3 does not simply correct for rotational contributions to dipole moment derivatives in the sense discussed by Crawford [35,36]. It directly introduces terms arising from the permanent dipole moment into the Px matrix. 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

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 ... 

For less symmetric molecules one has to resort to computer programs [164] to solve the Woessner equations. The orientation of the rotational diffusion tensor is usually defined by assuming that its principal axes coincide with those of the moment of inertia tensor. This assumption is probably a good approximation for molecules of low polarity containing no heavy atoms, since under these conditions the moment of inertia tensor roughly represents the shape of the molecule. 

This study is the first where semiquantitative use of relaxation data was made for conformational questions. A similar computer program was written and applied to the Tl data of several small peptides and cyclic amino acids (Somorjai and Deslauriers, 1976). The results, however, are questionable since in all these calculations it is generally assumed that the principal axis of the rotation diffusion tensor coincides with the principal axis of the moment of inertia tensor. Only very restricted types of molecules can be expected to obey this assumption. There should be no large dipole moments nor large or polar substituents present. Furthermore, the molecule should have a rather rigid backbone, and only relaxation times of backbone carbon atoms can be used in this type of calculation. 

The calculated Euler angles (a = 50°, /3 = 60°, and y = 40°), which determine the relative orientation between the principal-axis system of the rotational diffusion tensor and that of the moment of inertia tensor, indicate a significant shift between the two tensors. This result is expected because of the fact that molecule 31 contains a number of polar groups and hydrogen-bonding centers, leading to strong intermolecular interactions. 

The electric field polarization is conveniently described in the LF. The MF spherical tensor components of the electric field polarization tensor are related to the components in the LF through a rotation... 

In Eq.(23), J and K are the rotational quantum numbers for the total angular momentum and the component projected to the molecular principal axis, p is the anisotropy of the Raman polarization tensor, pj is the thermal rotational distribution function in the initial state, and N specifies the selection rule of the rotational Raman transitions. (J)=0 if J<0 and 4(J)=1 or all other values of J,... 

Dipole moment derivatives. Rotational contributions to the derivatives with respect to the symmetry coordinates (see [12]) were calculated for HOF and DOF [3, 13]. The derivatives of the components of fx with respect to atomic Cartesian coordinates ( atomic polar tensor [27]) were calculated for both the F and the H atom from different contributions (i.e., atomic charge, charge flux [28], and overlap) [29]. 

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. 

The derivation of the induced contribution, on the other hand, is very similar to the derivation for the magnetizability. We could start from the definition of the rotational g tensor as first derivative of the rotational magnetic moment, Eq. (6.8), which would then be the induced contribution to it, and use the response theory formalism of Section 3.11. Using Eq. (3.116) we could express the derivatives of the induced rotational magnetic moment in terms of a polarization propagator and ground-state expectation value. Here we will, however, make use of the definition as second... 

Corresponding relations for the other diagonal components can be obtained by cyclic permutations of the coordinate triple xyz to yzx or zxy. Since the quadrupole moment of a polar molecule depends on the origin of the coordinate system it is important to remember that the centre of nuclear masses is automatically chosen as the origin in Eq. (6.41). Similar to the previously derived relations between the rotational g tensor and other molecular properties, Eqs. (6.30) and (6.38), one has to take care of vibrational corrections to the properties in Eq. (6.41), when it is applied to measured rotational g tensors and magnetizabilities. 

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

With the addition of the Ppp term the atomic polar tensor matrix becomes mass independent since it refers to a space-fixed coordinate system. Thus, the transformation of measured infrared intensities of different isotopic species of a molecule widi identical symmetry will result in proximately the same Px mahix, within the experimental uncertainties. This is an important feature of the atomic polar tensor representation of infrared band intensities. The troublesome problem of rotational correction terms is treated in a straightforward and general way. The treatment, however, introduces some difficulties in the physical interpretation of the elements of atomic polar tensors. These will be discussed later in conjunction with some examples of calculations. 

Following such an approach Decius and Mast [117] have tabulated the structure of atomic polar tensors for a number of simple molecules representative of several point groups. Translational and rotational dependencies arising from Eqs. (4.18) and (4.20) are also explicitly defined. The structures obtained are summarized in Table 4.3. 

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. 

Person and Kubulat [86,106] have enqdiasized the inqiortance of analyzing die vibradonal part of atomic polar tensors. Both experimental and theoretical calculations have shown that rotational contributions to atomic polar tmisors can be quite significant and, sometimes, overwhelming. For the water molecule, for instance, it accounts for two diirds of the total intensity [86]. The rotational terms, therefore, dominate the... 

Some of the difficulties encountered may be avoided by transferring vibrationa] polar tensors only, while calculating the rotational part for the particular molecule considered if the dipole moment is known. These options appear not to have been pursued. 

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