Atomic Polar Tensors APT

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... 

With quantum-mechanical methods, the second derivatives of the energy could be used directly for the FF and atomic polar tensors (APT) for the dipole derivatives. Both are standardly computed in most quantum-chemistry programs but for accurate results, moderately large basis sets and/or some accommodation for correlation interaction is needed. Until recently, this has restricted most ab initio studies to modest-sized molecules. 

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. 

The second-rank molecular tensors, Pajg, are termed atomic polar tensors (APTs). Separating electronic and nuclear parts ... 

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... 

Atomic Polar Tensor (APT), which for a given atom is defined as... 

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). 

Measurement of infrared intensities yields the dipole derivatives with respect to the normal coordinates. However, it is more useful if these can be expressed in some other coordinate system, such as the symmetry or internal coordinates of the molecule, or Cartesian coordinates. One form of analysis that is particularly suitable for comparison with ab initio results uses the atomic polar tensors (APT). - This particular quantity is just the set of derivatives of the dipole, usually in Cartesian coordinates (despite its name). 

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... 

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... 

Antarafacial, Woodward-Hoffmann rules, 357 Antisyimnetrizing operator, 59 Atom types, in force field, 7, 8 Atomic basin, 223 Atomic units, 54, 415 Atomic Natural Orbitals (ANO), 161 Atomic orbital (AO), 65, 150 Atomic Polar Tensor (APT), use for atomic charges, 226... 

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]. 

The Raman (Eq. 2) and ROA (Eq. 3) invariants require the evaluation of the first-order derivatives of three polarizability tensors (Bo/O/f, ), (6A/6R ), and (0G /07 ia)- A review by Buckingham [34] defines all these polarizabilities. The dipole strength and rotational strength entering into the IR (Eq. 4) and VCD (Eq. 5) intensities require the calculation of the atomic polar tensors (APTs, Pm = and of the atomic axial tensors (AATs,... 

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