Molecular symmetry point groups

Molecular point-group symmetry can often be used to determine whether a particular transition s dipole matrix element will vanish and, as a result, the electronic transition will be "forbidden" and thus predicted to have zero intensity. If the direct product of the symmetries of the initial and final electronic states /ei and /ef do not match the symmetry of the electric dipole operator (which has the symmetry of its x, y, and z components these symmetries can be read off the right most column of the character tables given in Appendix E), the matrix element will vanish. 

How do CMOs and LMOs differ The CMOs are symmetry-adapted eigenfunctions of the Fock (or Kohn-Sham) operator F, necessarily reflecting all the molecular point-group symmetries of F itself,26 whereas the LMOs often lack... 

In addition, group theory can be used to assess when transition dipole moments must be zero. The product of the irreducible representations of the two wave functions and the dipole moment operator within the molecular point group symmetry must contain the totally symmetric representation for the matrix element to be non-zero (note that, if the molecule does not contain an inversion center, the operator r does not belong to any single irrep, except for the trivial case of Ci symmetry see Appendix B for more details). A consequence of this consideration is that, for instance, electronic transitions between states of the same symmetry are forbidden in molecules possessing inversion centers. 

The excitation and fluorescence spectra of some crown ether complexes have been interpreted in terms of the molecular point group symmetry and vibronic coupling.452,453 This approach tends to be limited to europium(III) complexes. 

Thus, of the three normal modes of water, two will have A and one B2 symmetry. Let us stress again this information could be derived purely from the molecular point-group symmetry. 

M. Haser, J. Chem. Phys., 95, 8259 (1991). Molecular Point-Group Symmetry in Electronic Structure Calculations. 

Typical infrared or Raman assignments of partial ionicity also depend on preserving the molecular point group symmetry The number and types of normal frequencies are then fixed and their q-dependence can be monitored, again assuming crystal perturbations to be small. However, the point group symmetry of phenazines changes with ionicity. While no systematic spectroscopic data is available, we would not expect studies to be nearly as successful as for TCNQ, TTF and other D21, donors or acceptors. 

The least restricted model [Ref. (5) p. 199] is a first-order perturbation model upon a basis which is not an I basis, but which transforms as an I basis under the molecular point-group symmetry. This model is satisfactory when only one I electron is present in the partially filled shell, but gives rise to a tremendous number of interelectronic repulsion parameters for configurations with > / [Ref. (5) p. 229]. In this case it is therefore of little practical use. 

A second-rank tensor quantity such as the polarizability can be written in the form of Eq. (5.2.7) for tedrahedral or higher molecular point group symmetries. 

As we shall see in a later chapter, if we make the further constraint on the SCF method that the molecular orbitals are symmetry-adapted, then the above technique can be extended in such a way that only the symmetry-distinct integrals need be computed and stored What is more, the restriction to permutational symmetries can be removed so that the full molecular point group symmetry can be used, which may involve operations which induce linear combinations among the basis functions. 

The described algorithm is clearly not dependent on the number of spin-orbital groups. This allows one further efficiency in that Abelian molecular point group symmetry... 

The 3x3 transposed Jacobian matrix = Va/ , evaluated at the stagnation point To, has real elements. It represents a nonsymmetric tensor in the absence of molecular point group symmetry. Within the linear approximation [91], only the first term in the expansion is considered and the description of the field about a stagnation point amounts to solving a system of three coupled linear differential equations whose corresponding matrix is given by the transposed Jacobian matrix. 

In all the above treatment, no explicit use has been made of either molecular point-group symmetry or of the equivalent positions within the unit cell. These will have to take care of themselves in the calculation. The Bloch function approach accounts for translational symmetry only, and therefore the independent part of the whole calculation is the unit cell contents. A combination like 6.13 must be done for each of the Nbs atomic orbitals in the unit cell, and the resulting total energies are referred to the molecular contents of the whole unit cell. Nevertheless, molecular symmetry and equivalent positions within the cell will determine the asymmetric unit of the Brillouin zone, thus reducing the amount of independent k-space to be sampled. 

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