Set, of identical nuclei

Figure 6.1 8 Illustration of the four types of sets of identical nuclei which are possible in the C2 point group...

The physical content of the requirement (38) is that any electronic variable in the problem should have exactly the same relationship to the nuclear variables as does any other electronic variable. The physical content of the requirement (39) is that every member of a set of identical nuclei must enter into the definition of te in the same way. Thus (38) is satisfied by requiring all the columns of Vne to be identical and from now on a typical column will be denoted v. If the entries in v are identical for identical nuclei then (39) is also satisfied. 

If there are n electrons in a molecule there are n ways of permuting them and we can form the permutation group (or symmetric group) 6 of degree n and order n that contains all the electron permutations. The molecular Hamiltonian is invariant to the elements of this group. Similarly, there can be sets of identical nuclei in a molecule and the Hamiltonian is invariant to the relevant identical-nucleus permutation groups. For example, the ethanol molecule... 

Simultaneously the eigenfunctions will provide irreps for the permutation group S of the system. This group comprises the direct product of the permutation group permutation groups S. for each set of identical nuclei i comprising Ai members. The physically realisable irreps of this group are restricted by the requirement that, when spin is properly incorporated into... 

For a set of equivalent nuclei in general site the matrices 11(G) are identical with the right regular representation matrices2 lf the nuclear position vectors of all K nuclei of a SRM are included in the basis Xkd), 11(G) denotes a K by K permutation matrix. In addition to the matrix groups (2.49) and (2.49 ) the set... 

The problem is conventionally sidestepped by assuming that nuclear and electronic motions are decoupled, but despite many efforts this condition has never been shown to yield a rigid molecular shape either. The insurmountable problem is permutational invariance. In molecular-orbital calculations that decouple electronic from nuclear motion the nuclei are identified in order to support the definition of molecular structure, but then permutation of identical nuclei implies rearrangement of bonds and a new set of calculated electronic energies. There is little hope of ever overcoming these problems ... 

Let P be any permutation of positions and spins of identical nuclei, or any product of sudi permutations. Let E be the identity, E the inversion of all particle positions with respect to the center of mass, and P" the product PE = E P. Then the Molecular Symmetry Group is the set of ... 

Fig. 4 Schematic diagram of fullerene-70 (based on the diagram in ref. 10). The five sets of identical carbon nuclei a-e lie in the vertical planes as indicated...

If a solution of the electronic problem is to be used in a solution of the full problem, then the solution must be one invariant under the permutation of identical nuclei. The direct integral form is indifferent to whether individual nuclei are identified, and it is thus perfectly possible to regard formally identical nuclei as distinguishable particles simply by a suitable labeling of the points specified by b. However, it is equally possible that the direct integral properly reflects the permu-tational symmetry by requiring that if h results from a permutation of identical nuclei specified by b both sets be included in the same way in the direct integral. 

The number of lines in the ESR spectrum of a radical is a clue to its identity. Of further assistance is the intensity pattern in the spectrum. For n equivalent nuclei with 7 = 1/2, the n + 1 lines have intensities proportional to the binomial expansion of order n. Table 13.2 lists the relative intensities up to n = 5. The intensity patterns for sets of equivalent nuclei with 7 > 1/2 are handled in a similar fashion. Figure 13.5 shows the ESR spectrum of diphenylpicrylhydrazyl (DPPH) free radical in benzene. Under low-resolution conditions, the proton hyperfine coupling is not observed and the five-line spectrum with the intensity distribution 1 2 3 2 1 results from the interaction of the unpaired electron with two (effectively equivalent) (7=1) nuclei for nuclei with 7 > 1/2, the intensity distribution is more complex than a simple binomial expansion, and is beyond the scope of this text. 

The GED approach uses a mixed QC ansatz to construct a separable model for the electronuclear problem. As it corresponds to the case of a set of identical fermions, the electronic part is fully quantized in the GED scheme. The PCB is endowed with a mass distribution allowing for a correspondence with the external potential created by nuclei. The harmonic oscillator model of nuclear dynamics then follows as a quantum extension of the latter model, one where mass fluctuations are described by normal modes related... 

Furthermore, the chemical identity of the nuclei plays no direct part in determining the nodal patterns of the orbitals that they generate. The important feature is the number and kind of atomic orbitals that they contribute. Changing nuclei distorts and displaces the nodes, but they will still be between certain atoms in each orbital. Thus the MO s formed by two sets of different nuclei will be topologically equivalent if the same atomic orbitals are used. 

---