Irreps

As a first application, consider the case of a single particle with spin quantum number S. The spin functions will then transform according to the IRREPs of the 3D rotational group SO(3), where a is the rotational vector, written in the operator form as [36]... 

Using D2h symmetry and labeling the orbitals (fi-fi2) as shown above proeeed by using the orbitals to define a redueible representation.whieh may be subsequently redueed to its irredueible eomponents. Use projeetors to find the SALC-AOs for these irreps. 

The number of irredueible representations may be found by using the following formula nirrep = g Xred )5(irrep ) ... 

Other decarboxylations reported by Takahashi have proved irrep-roducible. The reaction between 2,2 -biphenyldicarboxylic acid anhydride in alkali and mercuric oxide in acetic acid was claimed to give 2 -mercurio-2-biphenylcarboxylate by hemidecarboxylation [Eq. (84)] (97) but yielded instead mercuric 2,2 -biphenyldicarboxylate and negligible decarboxylation [Eq. (85)] (98). Similarly, the reaction between sodium 2,3-pyridinedicarboxylate and mercuric oxide in acetic acid, reported to give... 

Irreducible representations (IRREPs), permutational symmetry degenerate/near-degenerate vibrational levels, 728-733... 

The eigenvalues of this matrix have the form of Eq. (68), but this time the matrix elements are given by Eqs. (84) and (85). The symmetry arguments used to determine which nuclear modes couple the states, Eq. (81), now play a crucial role in the model. Thus the linear expansion coefficients are only nonzero if the products of symmetries of the electronic states at Q0 and the relevant nuclear mode contain the totally symmetric irrep. As a result, on-diagonal matrix elements are only nonzero for totally symmetric nuclear coordinates and, if the electronic states have different symmetry, the off-diagonal elements will only... 

The spectra appear to be uncorrelated otherwise note however, that the spectrum for each sub-block alone are correlated following CUE statistics, which gives rise to the deviations from purely Poisson behaviour in P(s) (cf. dashed curve) as well as in the behaviour of the form factor for r < 3/24 which is dominated by the sub-spectra of the three dimensional irreps. 

For spin-free fe-particle excitation operators and density matrices, linear combinations that transform as irrep of the symmetric group Sk can be defined in an analogous way [15]. 

The presence of symmetry in a molecule can be used to great advantage in electronic structure calculations, although some care is required to avoid possible pitfalls that are simultaneously introduced (Appendix B provides a brief overview of nomenclature (e.g., the term irrep . 

The simplest approach, of course, is to maintain the minimum-determinantal description and reoptimize all of the orbitals. In practice, however, such an approach is practical only in instances where die ground-state and the excited-state wave functions belong to different incduciblc representations of die molecular point group (cf. Section 6.3.3). Otherwise, the variational soludon for die excited-state wave function is simply to collapse back to the ground-state wave function And, even if the two states do differ in symmetry, the desired excited state may not be the lowest energy such state widiiii its irrep, to which variational optimization will nearly always lead. 

Figure 14.4 Valence MO occupations for three different electronic states of fluorovinylidene. The triplet states belong to different ineducible representations of the molecular point group because the singly occupied orbitals in which they differ belong alternatively to either the a or a" irreps...

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. 

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