Linear species point groups

In Section 4.3.f it was shown that there are 3N — 5 normal vibrations in a linear molecule and 3N — 6 in a non-linear molecule, where N is the number of atoms in the molecule. There is a set of fairly simple rules for determining the number of vibrations belonging to each of the symmetry species of the point group to which the molecule belongs. These rules involve the concept of sets of equivalent nuclei. Nuclei form a set if they can be transformed into one another by any of the symmetry operations of the point group. For example, in the C2 point group there can be, as illustrated in Figure 6.18, four kinds of set ... 

The main difference between hydrogen bond and the halogen bond lies in the propensity of the hydrogen bond to be non-linear [28,29], when symmetry of the complex is appropriate (molecular point group Cs or Ci). In so far as complexes B- ClF are concerned, the nuclei Z Cl - F, where Z is the acceptor atom/centre in B, appear to be nearly collinear in all cases, while the nuclei Z- H - Cl in complexes B- HC1 of appropriate symmetry often show significant deviations from collinearity. This propensity for the hydrogen-bonded species B- HC1 to exhibit non-linear hydrogen bonds can be understood as follows. 

Examining the six quadratic functions displayed in Fig. 6.3.3, it is clear that x2 + y2, x2 - y2, and z2 are symmetric with respect to all four operations of the C2v point group. Hence they belong to the totally symmetric species Ai, as do x2 and y2 since they are merely linear combinations of x2 + y2 and x2 -y2. Similarly, the xy function is symmetric with respect to E and C2 and antisymmetric with respect to the reflection operations, and hence it has A2 symmetry. The symmetries of the xz and yz functions can be determined easily in an analogous manner. 

The chalcogenocyanate ions are linear triatomic species belonging to the point group The three normal modes of vibration shown in Fig. 1 are both infrared and Raman active. The vibrations are commonly described as though group frequencies existed unmixed in these ions,... 

Coo signifies the presence of an 00-fold axis of rotation, i.e. that possessed by a linear molecule (Figure 3.7) for the molecular species to belong to the Coov point group, it must also possess an infinite number of planes but no o-ji plane or inversion centre. These criteria are met by asymmetrical diatomics such as HF, CO and [CN] (Figure 3.7a), and linear polyatomics (throughout this book, polyatomic is used to mean a species containing three or more atoms) that do not possess a centre of symmetry, e.g. OCS and HCN. 

Fig. 3.7 Linear molecular species can be classified according to whether they possess a centre of symmetry (inversion centre) or not. All linear species possess a Coo axis of rotation and an infinite number of a., planes in (a), two such planes are shown and these planes are omitted from (b) for clarity. Diagram (a) shows an as5mmetrical diatomic belonging to the point group Coov, and (b) shows a symmetrical diatomic belonging to the point group Dooh-...

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. 

The two bonds must be considered together. Their sum and difference are S5nnmetry adapted linear combinations. These are now shown in Fig. 2 as localized MO s corresponding to bonds between O and H. In the Czv point group, the symmetries are clearly a and i>i. The sum and difference of lone pair orbitals concentrated on O would correspond to a and h% symmetries. For convenience F. 3 shows the S3mametry species of the Cse point group. Only the behavior with respect to two mirror planes is necessary to classify all four species. There is also a two-fold axis at the intersection of the two planes. 

Symmetrical diatomics (e.g. H2, [02] ) and linear polyatomics that contain a centre of symmetry (e.g. [N3], CO2, HC=CH) possess a plane in addition to a Cqo axis and an infinite number of a-y planes (Figure 4.7). These species belong to the point group. 

When a molecule has doubly degenerate symmetry species, three adiabatic electronic states can intersect conically. Three-state CIs can occur e.g. for E and II states of linear molecules or for non-degenerate and degenerate states in other point groups, as in the PIT effect. These multiple CIs cause a huge increase of NA effects. 

In other words, in order to factor the secular equation, the coordinates are formed into linear combinations such that each combination (or new coordinate) belongs to one of the symmetry species of the molecular point group. In the water illustration, Si and S2 are of species A, while S3 is Bi. When only real one-dimensional species occur, the proof is immediate that no cross terms will occur in cither the kinetic or potential energies between two coordinates, S<" and S<" say, of different symmetry species r( i ) and FThere will always be some operation B of the group for which... 

In order to illustrate the vibrational motions of a molecule belonging to a non-commutative symmetry point group, we return to the considerations of Section 2.3.2 and once more use as our example the square-planar complex, NiFj. A non-linear penta-atomic molecule has nine independent vibrational coordinates, distributed among the symmetry species of T>4h. These can be fully specified by standard methods [7], but the following simple qualitative considerations allow us to conclude that there are seven in-plane and two out-of-plane vibrations. Fig. 4.10 depicts several of the in-plane modes the motion of the nickel atom to conserve the center of mass is implied. 

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