Symmetry-adapted tensors

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

In this section, we follow the symmetry-adapted approach put forward by Acevedo et al. [10], and introduce the vibronic crystal coupling constants Av y(i, t), the tensor operators 0 (Txr i, t) and the general symmetry-adapted coefficients to give a master formula to evaluate the relevant reduced matrix elements as given below ... 

Although the spherical form of the multipole expansion is definitely superior if the orientational dependence of the electrostatic, induction, or dispersion energies is of interest, the Cartesian form171-174 may be useful. Mutual transformations between the spherical and Cartesian forms of the multipole moment and (hyper)polarizability tensors have been derived by Gray and Lo175. The symmetry-adaptation of the Cartesian tensors of quadrupole, octupole, and hexadecapole moments to all 51 point groups can be found in Ref. (176) while the symmetry-adaptation of the Cartesian tensors of multipole (hyper)polarizabilities to simple point groups has been considered in Refs. (172-175). 

SYMMETRY-ADAPTED STRAIN, SYMMETRY-BREAKING STRAIN, NON-SYMMETRY-BREAKING STRAIN AND SOME TENSOR FORMALITIES... 

In these formulae the symmetry adaptation coefficients, 3/- and 6y-symbols occur along with the set of crystal field parameters Dq, Ds, Dt, Da and Dt. In addition, there are the reduced matrix elements of the n-electron unit tensor operators, evaluable with the help of the coefficients of fractional parentage (dn lvSL dnvSL) as follows... 

The molecular point group G char acterizes the symmetry exhibited by the location of nuclei. TURBOMOLE has been designed from the very beginning to take symmetry properly into account. This has two advantages it guarantees an accurate representation of the symmetry behavior of computed molecular properties (electronic states, vibrational modes, polarizability tensors, etc.), and it reduces computational work which is typically proportional to G , the inverse of the order of the point group G. The proper symmetry behavior is achieved in TURBOMOLE by a transformation of the original GTO basis into a symmetry adapted basis which is then used to represent all MOs. 

Prove that when the operators of the unitary group U(n) and the permutation group S v act on the symmetry-adapted hmctions (10.3.1) the two types of operation commute and hence that the CFs displayed schematically in (10.3.2) behave as indicated in the text. [Hint The basic tensor products are Qk, with K = kyk2 - ks, and the unitary operator U induces a transformation of the IV-index tensor components in which... 

Fig. 6.4 Schematic drawing of the bent stnicture of the NO-Na adsorption complex in Na-LTA zeolite. The z principal axes of the g, A( Na), and g( Na) tensors lie within the Na+-N-0 complex plane and form an angle with the Na -N(0) bond direction of 38, 3, and 8°, respectively. The unpaired electron is localized mainly in the anti-bonding 2pii molecular orbital of the NO molecule (refer to Fig. 6.2), which is also within the plane of the complex. The cation at site S2 is coordinated to the framework oxygens. Of, in the stx-membered rings in a trigoneil symmetry, refer to Fig. 6.1. Only the three oxygens in the first coordination sphere are shown. The figure is adapted from [26] with permission from the American Chemical Society...

Fig. 6.7 Orientation of principal axes frames of the g, A(N), and A(Cu) tensors of the Cu(I)-NO moiety. The A(Cu) frame is rotated by the angle, p, about the common x principal axis, but the AjjCCu) principal axis is not necessarily parallel to the Cu—N bond. The y-z plane of the g tensor is spanned by the N—O bond and the symmetry axis of the 2ptiy orbital of the NO molecule. See Table 6.3 for ESR parameters and structural data. The figure is adapted from [31] with permission from the American Chemical Society...

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