Tensors axial

Lazzeretti and coworkers175 calculated nuclear electric and electromagnetic shielding tensors for 1 and oxirane. These properties are related to atomic polar tensors and atomic axial tensors used by infrared and VCD spectroscopists. The authors demonstrated that they could obtain fairly accurate sum rules for atomic polar tensors and atomic axial tensors with relatively little computational effort. 

Pseudoscalars with the property T T (where the positive sign applies to proper rotations and the negative sign applies to improper rotations) are also called axial tensors of rank 0, 7 (0)ax. A quantity T with three components 7 7 2 7) that transform like the coordinates x x2 x3 of a point P, that is like the components of the position vector r, so that... 

D. Yang and A. Rauk, Chem. Phys., 178,147 (1993). Sum Rules for Atomic Polar and Axial Tensors from Vibronic Coupling Theory. 

The tensors are termed atomic axial tensors (AATs) 7 and are the electronic and nuclear components. Here, (dpG/dXXa)0 is the same derivative which occurred already in Equation (2.86). The electronic AAT, 7 is the overlap integral with the derivative (dif/G/dHp)0. The latter is defined via... 

To visualize an object with Dm symmetry, imagine a cylinder whose outside is covered with n slanted striations, as illustrated at the top of Figure 8. The two constructions shown (D symmetry) are enantiomorphs whose sense of chirality is related to the way in which the striations are slanted. As n approaches infinity, the symmetry of the constructions approaches Z) in the limit, infinitely many C2 axes are embedded in a plane perpendicular to the C axis. This is the symmetry of a stationary cylinder undergoing a twisting motion, as indicated by the arrows on the cylinders at the bottom of Figure 8, and of an axial tensor of the second rank.41 It is also the helical symmetry of a nonpolar object undergoing a screw displacement, that is, of an object whose enantiomorphism and sense of chirality are T-invariant. 

One question to be settled is how the formal definition of tensor involves the quality of the vectors projecting onto Y, that is how are products of type ET Y F affected by whether vectors E and F are axial or polar [see Ref. 152, p. 98]. The latter authors, Pake and Estle, make the statement that for example g is not a true tensor due to this characteristic. Our thought is that both axial and polar tensors are tensors as defined by mathematicians. Then too, the question arises if Y is a tensor, how does one decide in practice whether it is an axial tensor or a pseudotensor ... 

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... 

Cheeseman JR, Frisch MJ, Devlin FJ etal (1996) Ab initio calculation of atomic axial tensors and vibrational rotational strengths using density functional theory. Chem Phys Lett 252 211-220... 

Tensors may be isotropic (no orientation dependence, as in a sphere), axial (two principal values equal, as in an oblate or prolate spheroid), or asymmetric (no special symmetry). Axial tensors vary as j (3 cos2 0 - 1), with 0 being the angle between the tensor axis and B0. 

This theory was first formulated by Stephens (1985). In his approach to go beyond the Bom-Oppenheimer approximation, Stephens mixed excited electronic states with the ground state. Though that approach seemed to call for the difficult calculation of excited electronic states, he finally arrived at expressions, that only involve ground state properties. As an example to the application of the theory Jalkanen et al. (1989b) report nuclear shielding tensors, atomic polar and axial tensors, as well as IR and VCD intensities of the ammonia isotopomer N H (NHDT) using different basis sets. 

The pnZ effect acts to convert the isotropic giv into a (essentially axial) tensor with the following components ... 

Some of those have long been known a is the electric dipole polarizability (a symmetric polar tensor of dimension P) in length formalism, k (an asymmetric axial tensor of dimension P t) is related to the optical activity, A is the mixed dipole-quadrupole polarizability, x is the magnetic susceptibility (or magnetizability), written as a sum of diamagnetic and paramagnetic components. 

As written, the CIDs (2.3) and (2.5) apply to Rayleigh scattering. The same expression can be used for Raman optical activity if the property tensors are replaced by corresponding vibrational Raman transition tensors. This enables us to deduce the basic symmetry requirements for natural vibrational ROA 15,5) the same components of aap and G p must span the irreducible representation of the particular normal coordinate of vibration. This can only happen in the chiral point groups C , Dn, O, T, I (which lack improper rotation elements) in which polar and axial tensors of the same rank, such as aaP and G (or e, /SAv6, ) have identical transformation properties. Thus, all the Raman-active vibrations in a chiral molecule should show Raman optical activity. 

Direct calculations of vibrational optical activity are younger, but once theoretical obstacles to them were solved, they turned out to be easier despite the fact that such a calculation is a many step procedure and its protocol is rather complicated. The use of this procedure for absolute configuration determination is described in Section 8.5.1. A fundamental advantage of VCD calculations consists in the fact that we are dealing with molecules usually in their well-defined electronic state -the ground state. For the calculation of vibrational optical activity, we must calculate atomic polar and axial tensors [16, 92]. 

The calculation involves optimization of the molecular structure, computation of vibrational modes (by far the most demanding part computationally), computation of atomic polar and axial tensors and of all the sums leading to dipole and rotational strengths. As the last step, the theoretical VCD curve is simulated by using the empirical values for bandwidths. The quantum chemical part of the calculation... 

---