Effective molecular symmetry group

When correlating the number of infrared-active CO-stretching modes with molecular structure, the phase in which measurements are made has some bearing on the effective molecular symmetry and must therefore be considered. Measurement of the infrared spectra of compounds in the gas phase is ideal because mtermolecular interactions can be neglected the selection rules which determine the number and activity of the CO-stretching modes are those associated with the point group of the isolated molecule. Because of the limited volatility of many carbonyl complexes and their tendency to decompose at higher temperatures, gas-phase measurements have been limited chiefly to the binary carbonyls (44). It... 

According to Equation 1.44, the normal coordinates of an excited electronic state q relative to those of the ground electronic state q are rotated (rotation matrix W) and displaced by the vector k. This rotation is called the Duschinsky rotation or Duschinsky mixing effect [41—44] (of the vibrational modes among each other). This mixing effect is subject to symmetry rules of the molecular symmetry group. Since in the most common instances vibrational modes of the same symmetry are mixed with each other (Equations 1.29-1.31 and 1.37), the matrix W assumes the... 

The irreducible representations of a symmetry group of a molecule are used to label its energy levels. The way we label the energy levels follows from an examination of the effect of a synnnetry operation on the molecular Sclnodinger equation. 

It should be noted that the trace of a matrix that represents a given geo] operation is equal to 2 cos y 1, the choice of signs is appropriate to or improper operations. Furthermore, it should be noted that the aim direction of rotation has no effect on the value of the trace, as a inverse sense corresponds only to a change in sign of the element sin y. TE se operations and their matrix representations will be employed in the following chapter, where the theory of groups is applied to the analysis of molecular symmetry. 

In effect, the division by two is the result of the molecular symmetry, as specified by the character table for the group 0. In general it is useful to define a symmetry number a (= 2 in this case), as shown below. The well-known example of the importance of nuclear spin is that of ortho- arid para-hydrogen (see Section 10.9.5). 

Key words Point groups, space groups, molecular symmetry, buckminsterfullerene, Jahn-Teller effect, chirality, dissymmetry, molecular packing, quasicrystals... 

Problem 10-5. In a homonuclear diatomic molecule, taking the molecular axis as z, the pair of LCAO-MO s tpi = 2p A + PxB and tp2 = 2 PyA + 2 PyB forms a basis for a degenerate irreducible representation of D h, as does the pair 3 = 2pxA PxB and 4 = PxA — PxB Identify the symmetry species of these wave functions. Write down the four-by-four matrices for the direct product representation by examining the effect of the group elements on the products 0i 03, 0i 04, V 2 03) and 02 04- Verify that the characters of the direct product representation are the products of the characters of the individual representations. 

We start with the 3N unit displacement vectors, where there are three mutually perpendicular vectors on each nucleus. Consider the effect of a symmetry operation on one of these 3N vectors. Each vector either will remain on the same nucleus, with or without its direction changed, or it will be moved to an identical nucleus, with or without its direction changed. (We consider the nuclei as fixed while examining the effect of a symmetry operation on a displacement vector see Section 6.3.) Since any vector on a nucleus can be expressed as some linear combination of the three mutually perpendicular unit displacement vectors on that nucleus, a symmetry operation will send each of the 3N displacement vectors into some linear combination of these 3 N vectors. Therefore, the unit displacement vectors form a basis for some SW-dimensional representation of the molecular point group we shall call this representation r3Jv. 

The theoretical basis for a conserved quantity is the commutation of an effective Hamiltonian with the elements of some symmetry group. If this condition exists, then the irreducible representations of the group are good quantum numbers, i.e., are conserved. Conversely, the existence of good quantum numbers implies a Hamiltonian which commutes with an appropriate group. The most general molecular A-electron Hamiltonians... 

Given the illustrations of the normal modes of a molecule, it is possible to identify their symmetry species from the character table. Each non-degenerate normal mode can be regarded as a basis, and the effects of all symmetry operations of the molecular point group on it are to be considered. For instance, the v mode of XeF4 is invariant to all symmetry operations, i.e. R(v 1) = (l)vj for all values of R. Since the characters are all equal to 1, v belongs to symmetry species Aig. For V2, the symmetry operation C4 leads to a character of -1, as shown below ... 

Symmetry considerations are instrumental in a qualitative discussion of spin-orbjt effects. Qualitatively, a phenomenological Hamiltonian of the form Aso Z matrix representation of the usual molecular point group. The same is true for the spatial and spin wave... 

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