Double group symmetry

H is even with respect to the interchange of any particle index (double symmetry group for each pair). As a result of group theory we immediately derive that the corresponding eigenstates are either even or odd ... 

Figure 16, Relation between symmetry in (a) the single space and (b) the double space of a system whose molecular symmetry group has a a plane in the single space, and is isomorphic with C2v in the double space.

To see why this is so, let us attempt to apply the procedure of Section II.B to a bound-state wave function. This is illustrated schematically in Fig. 19. It is clear immediately that we cannot construct an unsymmetric in the double space, because each bound-state eigenfunction must be an irreducible representation of the double-space symmetry group. Thus a bound-state function in the double space is necessarily symmetric or antisymmetric under R2k, and is thus either a Fq or a Fn function. For a Fq function, we have Fn = 0 (since and Fn cannot form a degenerate pair), which implies [from Eq. (6)] that... 

This definition is because there are cases in which the Hamiltonian symmetry group has more elements (double) than the point symmetry group of the active center. Those cases deal with rare earth ions with half-integer J values for instance, the Nd + ion. They will be treated in Section 7.7. 

VIII. Double Point-Group Symmetry and Selection Rules.26... 

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. 

Vm. DOUBLE POINT-GROUP SYMMETRY AND SELECTION RULES... 

However, if Q(Qea) does not involve nuclear spin interactions, another symmetry group for tf(Qea) may be found. We let be the point group which acts on electronic spin coordinates and which is isomorphic to sp. The double group of is denoted and is obtained from double group contains twice as many elements as The two groups sp and may be combined to give an inner subdirect product which is denoted sp < a-i and is defined to contain elements of the form... 

In the derivation of these spin-interaction selection rules the harmonic approximation was made. In taking nuclear vibration into account2,77 these selection rules are often broken. In addition to coupling with the internal vibrational modes of a molecule, coupling with the phonon modes in the solid state may be important.124 Some use of double point group symmetry will be found in Sections IX, XI, and XII. 

The 3B1 state of methylene gives rise to three double-point group symmetries... 

Let be the vibronic ground state localized in the left well of the double-well potential, and IP2 be the right-side ground state. The reference symmetry group of the undistorted molecule AB2 is D. An off-center displacement of the central atom A along the axis B - B reduces this symmetry to This means that the local symmetry... 

It is noticeable that the NRG theory developed here furnishes two different groups for the double rotation in planar pyrocatechin (or acetone), and the double rotation and wagging mode in non-planar pyrocatechin (or pyramidal acetone). The group structures, however, are seen to be the same. The Longuet-Higgins theory yields indeed the same Molecular Symmetry Groups for both pyramidal uid planar systems. As a result, the NRG theory is seen to furnish a more det ulled information about the dynamics of the non-rigid systems. 

It may be concluded that, when the external and internal motions may be considered separely, the local full NRG is isomorphic to the direct product of the restricted NRG by the symmetry point group of the molecule in its most symmetric configuration (20). This conclusion holds in the case of synunetric linear molecules, in which the reference frame is an axis, except that double valued groups have to be used, but it does not hold in the case of centro-symmetric molecules in whidi the reference frame is a single atom. 

Furthermore, a rather bold but chemically more feasible mechanism can account for the proposed migration of the acetoxy group. This is double, symmetry allowed, suprafacial-suprafacial, Claisen-type, 3,3-sigmatropic shift. 

The remaining groups preserve double electron degeneracy and are subjects to further JT symmetry descent (moreover, the symmetry group is not feasible for coronene). There are two possible ways to JT stable C2h group in Bg electronic state ... 

Planar C2v structure with Ai electronic state (C model) cannot be explained by JT symmetry descent from parent group and must be explained by JT symmetry descent of parent D h symmetry group (see Fig. 4) with double electron degeneracy by the symmetry descent path (Scheme 6b)... 

As is outlined in Appendix A, the U can straightforwardly be determined, making use of the determinantal form (10) of Hso- In particular, an associated unitary 2x2 matrix can be found for each of the 12 elements of D3. The resulting group of order 24, the so-called spin double group is the symmetry group of the SO operator (10). 

Using the methods discussed in the tutorial (Sect. 2), the group of symmetry operations of Hso can be constructed explicitly. As is shown in Appendix B, the symmetry group of Hso of (53) is T, the spin double group of the tetrahedral rotation group, is of order 48. T also is the symmetry group of the total Hamiltonian H = Hes + Hso-... 

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