Quantum Mechanical Expressions for Polarizabilities

In the previous section we have defined the tensor components aap, - a,p-y and Cap -ys of the electric dipole, dipole uadrupole and quadrupole-quadrupole polarizability tensors as derivatives of the energy E , ) in the presence of a field and field gradient, Eqs. (4.65) to (4.67), or alternatively as derivatives of the perturbation dependent electric dipole p , ) and quadrupole moment 0(5,f), Eqs. (4.46) to (4.48), see also Table B.l. Furthermore, we have seen in Sections 3.3 and 4.3 that the electronic contributions to the electric dipole and quadrupole moments can be expressed as expectation values of the electric dipole and quadrupole moment operators, j2 Ro) and Ro) for the electrons, respectively. Both definitions can be used to derive quantum mechanical expressions for the polarizabilities. 

Let us start with the first definition as derivatives of the energy, Eqs. (4.65) to (4.67). Again we will use the perturbation theory expression for the perturbed energy, Eq. (3.15), but differentiate it now twice with respect to the appropriate components of the field or field gradient. This leads us immediately to the second-order correction to the energy, because the first-order correction depends only linearly on the fields. We can therefore express the polarizabilities as 

These definitions are also collected in the last column of Table B.l of Appendix B. 

However, in Section 4.3 we have seen that the derivatives of the first-order perturbation Hamiltonian, with respect to a component of the electric field a and 

One should note that there is no contribution from the nuclear operators Q Ro) and Q Ko) and that the polarizabilities are independent of the origin Rq- The explanation for that is that neither Cl (Ro) nor Rq act on the electronic wavefunctions 

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