Symmetry Conditions for Resonance Hybrids

It remains to state the symmetry conditions to be obeyed by resonance hybrids. For the total wave function to be able to split into different adiabatic states, it should be decomposable into independent parts. If the nuclear framework has some sort of spatial symmetry, it is easy to know the possible structures of the resonance hybrids. However, we must distinguish between two different situations point group and accidental degeneracies [25,26]. 

The real Hilbert space is always partitioned into a direct sum of subspaces, each representing a different energy eigenvalue of the spectrum of the hamiltonian operator  

Point groups decomposable in direct product forms Point groups not decomposable in direct product forms 

Possible point group decompositions in direct products  

Good examples are the core hole excited states of homonuclear molecules. When one electron is removed from a core orbital, the original Dooh symmetry is lowered to C v The D h group can be decomposed into two CooV components related by a C, or Cs operation, so it is fair to consider that the core-hole excited states are described by resonance between the two structures. The adiabatic subsystems have, by definition, zero overlap in the real space. Their interaction is defined only in complex space through the explicit overlap between the many-electron states. 

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