Permutational symmetry levels

Besides the elementary properties of index permutational symmetry considered in eq. (7), and intrinsic point group symmetry of a given tensor accounted for in eqs. (8)-(14), much more powerful group-theoretical tools [6] can be developed to speed up coupled Hartree-Fock (CHF) calculations [7-11] of hyperpolarizabilities, which are nowadays almost routinely periformed in a number of studies dealing with non linear response of molecular systems [12-35], in particular at the self-consistent-field (SCF) level of accuracy. 

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

Longuet-Higgins phase-based treatment, two-dimensional two-surface system, scattering calculation, 154-155 three-state molecular system, 134-137 two-state molecular system, single conical intersection solution, 98-101 permutational symmetry, degenerate/near-degenerate vibrational levels, 730-733 Polyene molecules ... 

Such localized states as under discussion here may arise in a system with local permutational symmetries [Aa] and [AB], If [Aa] + [S] and [Ab] = [5], the outer direct product [Aa] 0 [AB] gives rise to a number of different Pauli-allowed [A], If the A and B subsystems interact only weakly, these different spin-free [A] levels will be closely spaced in energy. The extent of mixing of these closely spaced spin-free states under the full Hamiltonian, H = HSF + f2, may then be large. Thus, systems which admit a description in terms of local permutational symmetries may in some cases readily undergo spin-forbidden processes, such as intersystem crossing. 

With the permutation-symmetry-corrected results(17) it appears that we cannot successfully fit our QEO data to a two-level model, at least for centrosymmetric structures. We have only one data point (9) to make this judgement, however, and... 

Symmetry considerations play a role on several levels in the analysis of Hartmann-Hahn experiments. In the presence of rotational symmetry and permutation symmetry, the effective Hamiltonian often can be simplified by using symmetry-adapted basis functions (Banwell and Primas, 1963 Corio, 1966). For example, any zero-quantum mixing Hamiltonian can be block-diagonalized in a set of basis functions that have well defined magnetic quantum numbers. Block-diagonalization of the effective Hamiltonian simplifies the analysis of Hartmann-Hahn experiments (Muller and... 

It would appear that identical particle permutation groups are not of help in providing distinguishing symmetry labels on molecular energy levels as are the other groups we have considered. However, they do provide very useful restrictions on the way we can build up the complete molecular wavefimction from basis functions. Molecular wavefunctions are usually built up from basis functions that are products of electronic and nuclear parts. Each of these parts is further built up from products of separate uncoupled coordinate (or orbital) and spin basis functions. When we combine these separate functions, the final overall product states must conform to the permutation symmetry mles that we stated above. This leads to restrictions in the way that we can combine the uncoupled basis functions. 

The character tables of these groups are given in table A14.6 and table A1.4.7I. If there were no restriction on permutation symmetry we might think that the energy levels of the H2 molecule could be of any one of the following four symmetry... 

However, drastic consequences may arise if the nuclear spin is 0 or In these cases, some rovibronic states cannot be observed since they are symmetry forbidden. For example, in the case of C 02, the nuclei are spinless and the nuclear spin function is symmetric under permutation of the oxygen nuclei. Since the ground electronic state is only even values of J exist for the ground vibrational level (vj, V3) = (OO O), where (vi,V2,V3) are the... 

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