The Hellmann-Feynman theorem is extended to account for degenerate states. Given a point $\lambda ={\lambda }_0$ in parameter space where the energy $E\left({\lambda }_0\right)$ is n-fold degenerate, it is shown that the corresponding n forces (slopes) are obtained by diagonalizing the derivative of the Hamiltonian, $-\partial H\left(\lambda \right)/\partial \lambda {|}_{\lambda ={\lambda }_0},$ in the subspace of degenerate eigenstates. Such a rotation within the subspace of degenerate eigenfunctions is easy and simple to apply in practical calculations and should be performed separately for each independent direction in parameter space for which the forces are to be calculated.

• Received 2 August 2002

DOI:https://doi.org/10.1103/PhysRevB.68.033105