Atomic polar tensor model

An approximate form of the NMO model, the atomic polar tensor (APT) model, has also proved effective (42). Rather than considering local contributions, this model considers contributions from the derivative of the total molecular dipole moment with respect to Cartesian displacement of a given nucleus, (3p/9R )o. The latter ate the elements of the atomic polar tensor for atom n. 

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 ... 

It is clear that the above results are contrary to the simple and common model of fixed atomic point charges that predicts an isotropic response to an external field. The source of the problem is that the atomic polar tensor is not isotropic. This tensor has nine elements (for each nucleus), corresponding to the derivatives of each component of the dipole moment with respect to the three Cartesian coordinates of the nucleus. The properties of these tensors and their interpretation as atomic charges have been extensively discussed in the literature in connection with infrared intensities.It is well known that the anisotropy of these tensors can be explained as follows. If we write the total molecular dipole moment as formed from point charges, then a typical element of the atomic polar tensor is, for example. 

Approximate models include the coupled oscillator (CO), the fixed partial charge (FPC), the chaige flow (CF), the ring current, the dynamic polarizability (DP), the bond dipole (BD), the localized molecular orbital (LMO) the nonlocalized molecular orbital (NMO), the atomic polar tensor (APT), and the locally distributed origins gauge... 

A modification of the CCFO model was proposed by Gussoni et al. [126,127]. Basically, it resets to expressing atomic polar tensors in terms of two sub-matrices rather than three as in the original formulation. A polar tensor of an atom a is represented as... 

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. 

The development of the methods described in Section 9.2 was an important step in modeling polarization because it led to accurate calculations of molecular polarizability tensors. The most serious issue with those methods is known as the polarization catastrophe since they are unable to reproduce the substantial decrease of the total dipole moment at distances close to contact as obtained from ab initio calculations. As noted by Applequist et al. [49], and Thole [50], a property of the unmodified point dipole is that it may originate infinite polarization by the cooperative interaction of the two induced dipoles in the direction of the line connecting the two. The mathematical origins of such singularities are made more evident by considering a simple system consisting of two atoms (A and B) with isotropic polarizabilities, aA and c b. The molecular polarizability, has two components, one parallel and one perpendicular to the bond axis between A and B,... 

Most of the four above-mentioned properties for Raman spectra can be explained by using a simple classical model. When the crystal is subjected to the oscillating electric field = fioc " of the incident electromagnetic radiation, it becomes polarized. In the linear approximation, the induced electric polarization in any specific direction is given by Pj = XjkEk, where Xjk is the susceptibility tensor. As for other physical properties of the crystal, the susceptibility becomes altered because the atoms in the solid are vibrating periodically around equilibrium positions. Thus, for a particular... 

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