Operator antisymmetric, electric dipole

A) with respect to the operation of inversion about the origin of the system. The electric dipole operator is antisymmetric (A) with respect to inversion at a point of symmetry. The electric quadrupole operator is inversion symmetric (S). A transition is allowed if the product function in the expression for transition moment is symmetric for electric dipole radiation and antisymmetric for electric quadrupole radiation. 

The i f f) contribution gives rise to the familiar electric dipole absorption responsible for a large part of electronic spectroscopy. The second term in the matrix element may be divided into a symmetric part, identifiable with the electric quadrupole interaction, and an antisymmetric part which is the magnetic dipole operator ... 

The electric dipole operator is antisymmetric with respect to the inversion operator, i, and thus it connects states of opposite inversion symmetry. To see this note that. 

The electric dipole moment operator A is a hermitian and real operator, whereas is hermitian and purely imaginary. The linear response function of such operators is thus purely imaginary and according to Eq. (3.113) antisymmetric with respect to a change in the sign of the frequency uti. We can therefore rewrite the expansion as... 

Mof may vanish for reasons of symmetry then the transition between 0) and f) is said to be symmetry forbidden. For example, if the molecule possesses a plane of symmetry, the product of (0, M and f) will be either symmetric or antisymmetric with respect to the plane if it is antisymmetric, the integral vanishes. The electric dipole operator M corresponds to the sum of charges on particles in the molecule times their coordinates and is therefore antisymmetric with respect to any symmetry plane, thus (0 and I f) must have different symmetry properties with respect to the plane in order for the integral to be nonzero. 

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