Quantum Mechanical Expression for the Magnetizability

In the previous section we have defined the cartesian components of the magnetizability tensor as second derivatives of the energy E B) in the presence of a magnetic induction B, Eq. (5.39), or alternatively as first derivatives of the magnetic-field-dependent electronic magnetic dipole moment ma B), Eq. (5.32). Both definitions can be used to derive quantum mechanical expressions for the magnetizability. 

Let us start with the first definition as derivative of the energy. 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 magnetic induction. This leads us immediately to the second-order correction to the energy, because the first-order correction depends only hnearly on the fields. We can therefore express the magnetizability as 

One should recall that the magnetic perturbations enter the molecular Hamiltonian in Eq. (2.101) in the form of the vector potential A and that A contributes both to and to. Therefore, we have to use the full expression for the second-order energy correction including the ( q° term for the magnetizability and in 

In Section 5.3 we have obtained already the appropriate expression for in Eq. (5.20). In the same way we can now derive the second-order perturbation Hamiltonian for the case of an external magnetic induction as [see Exercise 5.5] 

These definitions are collected in the last column of Table B.2 of Appendix B. 

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