Explicit Expressions for Electric and Magnetic Properties

Having considered the general expressions for first- and second-order molecular properties, we now restrict ourselves to properties associated with the application of static uniform external electric and magnetic fields. For such perturbations, the Hamiltonian operator may be written in the manner (in atomic units) 

the first summation is over all electrons and the second summation over all nuclei. The positions of the electrons are given by r,- and the charges and positions of the nuclei by Zi and R/, respectively. In Eq. 54, the p,-and st- are the conjugate momenta and spins of the electrons. We have also introduced the operators 

Having set up the Hamiltonian, we may calculate the first- and second-order properties in the eigenvector representation. For the permanent electric and magnetic dipole moments, we obtain 

Whereas the permanent electric dipole moment vanishes for molecules belonging to certain points groups (e.g., for all molecules that possess a center of inversion), the permanent magnetic dipole moment vanishes for all closed-shell systems. To see how the vanishing of the magnetic dipole moment comes about, we first note that 

Let us now consider the second-order molecular properties. The static electric dipole-polarizability tensor is given by the expression 

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