Atomic polar tensors dipole moment

The derivative of the dipole moment with respect to the coordinates determines the intensity of IR absorptions (Section 10.1.5). A central quantity in this respect is the Atomic Polar Tensor (APT), which for a given atom is defined as... 

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. 

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... 

APT Atomic Polar Tensor. An analytic means of considering the effect of atomic motion upon the dipole moment of a given system. 

Derivatives of the dipole moment with respect to Qj can be expressed within a Cartesian reference frame via a similarity transformation, introducing atomic polar tensors (APTs) [13, 14], The connection between the latter and the electric shielding is obtained by means of the Hellmann-Feynman theorem. Within the Born-Oppenheimer approximation and allowing for the dipole length formalism, the perturbed Hamiltonian in the presence of a static external electric field E is given by Eqs. (6) and (35). 

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. 

In an entirely different approach, Cioslowski has defined atomic charge based on the atomic polar tensor.This method makes no recourse to the nature of the atomic orbitals or the volume of an atom in a molecule. The method makes use of the dipole moment and the derivatives of the dipole moment, which are obtained from the wavefunction. 

Given the perturbed density matrices, the complete expression for the dipole moment derivative may be assembled. The derivatives will generally be with respect to all possible Cartesian displacements of the nuclei, so the result of such a calculation is that all the atomic polar tensors for the molecule will be obtained simultaneously. 

In order to provide some insights into the intensity enhancement undergone by the H—Y stretch of the proton donor molecule in a H-bond [162, 163], Zilles and Person made use of atomic polar tensors in their study of the water dimer [164]. These quantities are defined in terms of the effect on the dipole moment of a small displacement of each atomic center of the system under examination. Further information arises from a partitioning into charge, charge flux, and overlap contributions (CCFO). Even though their work was limited to a rather small basis set (4-3IG), they were able to extract some very useful information. Comparison of certain elements of their calculated spectra with experimental data led them to-believe that the 4-3IG basis set correctly reproduces the essential ingredients. 

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]. 

In practice, the electric and magnetic dipole transition moments are usually expressed as summations of atomic properties, namely the atomic polar tensor (APT), Pgp, and atomic axial tensor (AAT), respectively. In the... 

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. 

At variance with the preceding methods, Cioslowski defines the atomic charge from the atomic polar tensor. This method requires the knowledge of the molecular dipole moment, and, more exactly, its derivatives, which can be evaluated readily from the wavefunction. The charge borne by atom I is defined as the trace of the first-rank atomic polar tensor ... 

Analysis of the derivatives of dipole moments with respect to nuclear displacements (the atomic polar tensors) gives rise to yet another definition of atomic charges. The GAPT (generalized atomic polar tensor) atomic charges ate... 

C. Foroutan-Nejad, to be published. The structures of all molecules were optimized under and C2v symmetry at the B3LYP/def2-TZVPPD level using the Gaussian 09 rev D suite of programs [33]. Atomic polar tensor (APT) atomic charges were computed as one third of the sum of the first derivatives of the molecular electric dipole moment with respect to the three coordinates of each nucleus [34]. 

Px is 3x(3N) array of atomic polar tensors as originally developed 1 Biarge, Herranz and Morcillo [108], It contains derivatives of die Cartesian cmnponents of die molecular dipole moment with respect to atonic Cartesian displacement coordinates. The derivation of Px from experimental dipole moment derivatives will be discussed in die following part. 

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... 

A quantity called effective atomic charge can be derived from tire atomic polar tensors. Originally it has been defined by King, et al. [105] from considerations of intensity sum rules using Cartesian coordinate representation of the dipole moment derivatives. 

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