Quantum Mechanical Expression for the Magnetic Moment

Using the definition of the magnetic dipole moment given in Eq. (5.10) and the definition of the vector potential given in Eq. (5.12) the expansion of the energy can be written as 

From this equation it can be seen that as an alternative to Eq. (5.10) the magnetic dipole moment can also be defined as the derivative of the potential energy with respect to the field induction Ba 

Quantum mechanical expressions for the permanent magnetic dipole moment can be derived in exactly the same way as the corresponding formulas for the electric dipole moment, in Section 4.3. We will therefore skip most of the derivations here and only discuss the final equations. There is, however, one interesting difference. 

But first, we need to derive explicit expressions for the first-order perturbation Hamiltonian operator for the case of a static and homogeneous magnetic induction B. The corresponding vector potential at the position of electron i can be obtained from the general expression in Eq. (2.121) 

Inserting this vector potential in the general expression for the molecular Hamiltonian, Eq. (2.101), we can write the first-order perturbation Hamiltonian as [see Exercise 5.4] 

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