Properties in the Matrix Approximations

The calculation of first-order properties in the atom-centered matrix approximations is almost absurdly simple, particularly for contracted basis sets. What we have done in the approximations is to expand the molecular wave function in a set of atomic positive-energy one-particle functions. So all we need do for the first-order properties is to evaluate the primitive property integrals and transform them. In an uncontracted basis, the electric and magnetic property matrices are given by 

The problem arises because the magnetic property operator is an odd (off-diagonal) operator, that is, it connects large and small components. One way to solve this problem is to convert it to an even operator using the approach outlined in section 13.7. The magnetic operator correct to 0(c°) is 

This is essentially the nonrelativistic operator multiplied by fi. We must transform this operator to the modified Dirac representation, in which we have 

The magnetic perturbation Hamiltonian Y now behaves like an electric perturbation, which is what we want for the approximations of this chapter. We can evaluate the matrix elements of this new operator in the same way as we did for the electric perturbations. There is a relativistic correction to the magnetic operator M, but it requires a second transformation of the perturbed Hamiltonian, and is discussed in section 13.7. As stated for the approximations to the energy, these approximations are useful where high accuracy is not required. 

---