Point-group operations

Nuclear pemuitations in the N-convention (which convention we always use for nuclear pemuitations) and rotation operations relative to a nuclear-fixed or molecule-fixed reference frame, are defined to transfomi wavefunctions according to (equation Al.4.56). These synnnetry operations involve a moving reference frame. Nuclear pemuitations in the S-convention, point group operations in the space-fixed axis convention (which is the convention that is always used for point group operations see section Al.4.2,2 and rotation operations relative to a space-fixed frame are defined to transfomi wavefiinctions according to (equation Al.4.57). These operations involve a fixed reference frame. 

A detailed discussion of the relation between MS group operations and point group operations is given in section... 

The equivalence of the pairs of Cartesian coordinate displacements is a result of the fact that the displacement vectors are connected by the point group operations of the C2v group. In particular, reflection of Axr through the yz plane produces - Axr, and reflection of AyL through this same plane yields AyR. 

Just as the individual orbitals formed a basis for aetion of the point-group operators, the eonfigurations (N-orbital produets) form a basis for the aetion of these same point-group operators. Henee, the various eleetronie eonfigurations ean be treated as funetions on whieh S operates, and the maehinery illustrated earlier for deeomposing orbital symmetry ean then be used to earry out a symmetry analysis of eonfigurations. 

Since the point group operations of SF commute with the permutations of 5 F,... 

Whatever method is used in practice to generate spin eigenfunctions, the construction of symmetry-adapted linear combinations, configuration state functions, or CSFs, is relatively straightforward. First, we note that all the methods we have considered involve 7V-particle functions that are products of one-particle functions, or, more strictly, linear combinations of such products. The application of a point-group operator G to such a product is... 

If a tensor T represents a physical property of a crystal, it must be invariant under the operations of the point group of the crystal. But if T is invariant under the generators of the point group, it is certainly invariant under any of the point group operators and so it will be sufficient to examine the effect of the group generators on T. 

MR T k(K), where the set of matrices r k(R) form a PR of the point-group operators R that form the point group P(k) of the little group G(k). 

Benzene is another molecule where molecular symmetry can be useful for construction of the ground and covalent excited states (5,6,16,17,24). The molecule possesses five covalent VB Rumer structures, two are the Kekule structures, and the three Dewar types, shown in Fig. 7.4a. It is seen that the Kekule structures are mutually transformable by the D6h point group operations i, C2, and av. Consequently, we can make two linear combinations the positive one is the totally symmetric Aig state, which gives rise to the ground state, while the negative one is the B2U state, which corresponds to the excited state. This is shown in the VB mixing diagram in Fig. 7.4b. 

We exemplify the procedure of determining the spinor transformation properties under molecular point group operations for the Czv double group. Other double groups can be treated analogously. The character tables of the 32 molecular double groups may be found, e.g., in Ref. 68. 

The advantage of the symmetrized SF tensor is that, since the two sides of equation (51) must transform in the same way under all point group operations, the element will be non-zero only if the product of co-ordinates QrQtQt... transforms in the same way as the co-ordinate Si. 

A typical characteristic of hypersymmetry operations is that they exercise their influence in well-defined discrete domains. These domains do not overlap—they do not even touch each other. The usual hypersymmetry elements lead to point-group properties. This means that no infinite molecular chains could be selected, for example, to which these hypersymmetry operations would apply. They affect, instead, pairs of molecules or very small groups of molecules. Thus, they can really be considered as local point-group operations. These hypersymmetry elements, accordingly, divide the whole crystalline system into numerous small groups of molecules, or transform the crystal space into a layered structure. 

A crystal is similar to three-dimensional wallpaper, in that it is an endless repetition of some motif (a group of atoms or molecules). The motif is created by point group operations, while the wallpaper, which we caU the space lattice, is generated by translation of the motif, either with or without rotation or reflection. The symmetry of the motif is the true point group symmetry of the crystal and the S5mmetry of the crystal s external (morphological) form can be no higher than the point symmetry of the lattice. The... 

A set of a-type atomic orbitals transform amongst themselves under point group operations like scalar quantities. Hence, to consttuct descent-in-symmetry-type LCAO cluster orbitals, we simply follow the same reasoning as for the linear and cychc polyenes, above, and define (unnormalized) cr-type cluster orbitals as... 

Figure 2.33. Illustration of crystallographic point group operations. Shown are (a) rotation axis, (b) rotation-inversion axis, and (c) mirror plane.

Show the relationship between dj and 4 crystallographic point group operations. 

Because the individuals building the twin or allotwin are related by point group operations, the A-B-C sequence has no influence on the composite lattice and the two or three portions can be described as juxtaposed and non-mixed for example, Zj... 

As stated earlier, we can derive this same result by considering linear combinations of local bond quantities transforming as a vector, under point group operations [i.e., in the same way as the pair of coordinates (jc, y) associated with the degenerate species (for in-plane vibrations)]. Thus we are left to determine two arbitrary parameters y, fB. The first one is typically fixed by a normalization procedure, while /3 is obtained by explicitly accounting for the observed slope of the infrared intensities. 

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