Dipole operator directions, definition

Figure 1. Definition of dipole operator directions and axis system (a) long-axis chelate transitions (b) short-axis chelate transitions.

An element of an electrostatic moment tensor can only be nonzero if the distribution has a component of the same symmetry as the corresponding operator. In other words, the integrand in Eq. (7.1) must have a component that is invariant under the symmetry operations of the distribution, namely, it is totally symmetric with respect to the operations of the point group of the distribution. As an example, for the x component of the dipole moment to be nonzero, p(r)x must have a totally symmetric component, which will be the case if p(r) has a component with the symmetry of x. The symmetry restrictions of the spherical electrostatic moments are those of the spherical harmonics given in appendix section D.4. Restrictions for the other definitions follow directly from those listed in this appendix. 

Now consider the quantum-mechanical definition of the electric dipole moment. Suppose we apply a uniform external electric field E to an atom or molecule and ask for the effect on the energy of the system. To form the Hamiltonian operator, we first need the classical expression for the energy. The electric field strength E is defined as E = F/Q, where F is the force the field exerts on a charge Q. We take the z direction as the direction of the applied field E = The potential energy V is [Eq. (4.24)]... 

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