Reflection Through An Inversion Center: U and G Symmetry
Taking the molecule center of mass as origin of coordinates, consider the change of all electrons' position from (xi, yi, zi) to (−xi, −yi, −zi). If the resulting wave function is unchanged, it is said to be gerade (German for even); if the wave function changes sign then it is said to be ungerade (odd). For a molecule with a center of inversion, all orbitals will be symmetric or antisymmetric. The resulting wavefunction for the whole multielectron system will be gerade if an even number of electrons is in ungerade orbitals, and ungerade if there is an odd number of electrons in ungerade orbitals, independently of the number of electrons in gerade orbitals.
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