Quantum Mechanical Expressions for Electric Moments

Let us look first at the transition of the original definitions as integrals over the charge density, Eqs. (4.5), (4.6) and (4.8), to quantum mechanics that we will illustrate for the example of the electric dipole moment. In the Born-Oppenheimer approximation, Section 2.2, the electrons in a molecule form a continuous charge distribution whereas the discrete nuclear charges are located at fixed points Rk- The expression, Eq. (4.5) for the a-component of the electric dipole moment can therefore be rewritten as 

The transition to quantum mechanics can now be made by inserting the quantum mechanical expression Eq. (2.23) 

Recalling the properties of the Dirac S function in three dimensions, Eq. (2.18), and the fact that the Dirac S function is a symmetrical function of its argument 

In complete analogy we can derive quantum mechanical expressions for the components of the second electric moment tensor Qap and the quadrupole moment tensor Qati 

In the second approach we will use the fact that the moments are defined as derivatives of the energy of a molecule in the presence of an inhomogeneous electric field, Eqs. (4.19), (4.20) and (4.21). In order to apply these definitions we need to find an expression for the energy of a molecule in the presence of an inhomogeneous electric field. Here, we are using perturbation theory as developed in Section 3.2. The first step is thus to define the perturbation Hamiltonian operators and to derive explicit expressions for them in terms of components of the electric field a(Ro) and field gradient tensor a/3(Ro)- The electric field and field gradient enter the molecular Hamiltonian in the form of the scalar potential From Eq. (4.15) we can see 

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