Group Symmetry Perturbation

Any perturbation from ideal space-group symmetry in a crystal will give rise to diffuse scattering. The X-ray diffuse scattering intensity at some point (hkl) in reciprocal space can be written as... 

Selection rules also arise on considering the point-group symmetry of tfo(Qeq). In the case of electric dipole radiation the perturbation Y, which describes the interaction with the radiation field, may be expressed in terms of the x, y, z components of the dipole moment operator r. The operators (t) transform as the x, y, or z components of r. 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

Since our i-basis has even inversion symmetry, the matrix elements connected with the perturbation from a given water molecule are independent of whether this molecule is situated on one or on the other side of the central ion. This means that if we want to discuss the perturbation from our six water molecules with octahedrally positioned ligators (point group symmetry Tn), we can as well take into account only three of them, nos. 1, 2, and 3, say, and eventually multiply all perturbation matrix elements by two. One may say that the holohedrized symmetry (9, 21, 22, 23) of the three water molecules around the central ion is T1. ... 

M. Kollwitz, M. Haser, and J. Gauss,/. Chem. Phys., 108, 8295 (1998). Non-Abelian Point Group Symmetry in Direct Second-Order Many-Body Perturbation Theory Calculations of NMR Chemical Shifts. 

Typical infrared or Raman assignments of partial ionicity also depend on preserving the molecular point group symmetry The number and types of normal frequencies are then fixed and their q-dependence can be monitored, again assuming crystal perturbations to be small. However, the point group symmetry of phenazines changes with ionicity. While no systematic spectroscopic data is available, we would not expect studies to be nearly as successful as for TCNQ, TTF and other D21, donors or acceptors. 

The least restricted model [Ref. (5) p. 199] is a first-order perturbation model upon a basis which is not an I basis, but which transforms as an I basis under the molecular point-group symmetry. This model is satisfactory when only one I electron is present in the partially filled shell, but gives rise to a tremendous number of interelectronic repulsion parameters for configurations with > / [Ref. (5) p. 229]. In this case it is therefore of little practical use. 

Miller-Bravais indices Pearson classification point groups reducing cell method Schoenflies notation space groups stereographic projection symmetry perturbations Weiss zone law Wulff map... 

The perturbation of the PMD symmetry is accompanied by a decrease in the charge alternation and by the appearance of bond alternation from one end group to another. The bond alternation ampHtude has been revealed to be proportional to the asymmetry degree, which can be calculated as the difference of topological indexes = 4>gj — 4>gg. The effect is maximum if Tgj > 45° and Tgg < 45°. If A4>j2 = 90°, the ideal polyene state is... 

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