Point group symmetry correction

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

Since Eqs. (B.7) and (B.8) are equal, this implies that a rotational symmetry element of direct space is also a rotational symmetry element of reciprocal space. This result must be correct since X-ray scattering is a physical property of the crystal, it must at least have the point-group symmetry of the crystal. 

R. B. Shirts, Correcting two long-standing errors in point group symmetry character tables. J. Chem. Educ. 84, 1882-4 (2007). 

This treatment of symmetry demonstrates that the threshold above which significant autocorrelation is estimated to lie must be revised upward in line with the point group symmetry of the structure (van Heel et al, 2000 Orlova et al, 1997). Interestingly this results in the correct significance level for an icosahedral object being similar to the 0.5 correlation coefficient FSC criterion. 

So far spin has not been considered explicitly. It is usual to ignore it in discussions of point group symmetry because the total spin operator is invariant under aU point group transformations and the axis orientation is defined by the fixed frame choice so there is a fixed internal choice for the z-axis. Thus it is regarded as sufficient to require that the spatial part of any trial function has the correct point group symmetry and then to form properly antisymmetric functions from the spatial parts and the spin eigenfunctions in the manner outlined earlier. If orbitals are used to construct the spatial part, then it is usual simply to extend the symmetry orbitals to become symmetry spin-orbitals. Returning to the functional form used in O Eq. 2.116 this would become... 

If the symmetries of the two adiabatic functions are different at Rq, then only a nuclear coordinate of appropriate symmeti can couple the PES, according to the point group of the nuclear configuration. Thus if Q are, for example, normal coordinates, xt will only span the space of the totally symmetric nuclear coordinates, while X2 will have nonzero elements only for modes of the correct symmetry. 

---