Breakdown of the Hellmann-Feynman theorem: Degeneracy is the key - art. no. 033110

摘要

The Hellmann-Feynman theorem is a powerful and popular method to efficiently calculate forces in a variety of dynamical processes, but its validity has rarely been addressed. Here a surprising failure of this theorem is reported. The forces calculated by the theorem can be more than fifty times smaller than the forces calculated by the finite differential method. Numerical evidence shows that the energy-level degeneracy is the main reason. An analytical proof reveals that although eigenvalues do not depend on a linear combination of degenerate wave functions, forces do sensitively depend on it, which leads to ill-defined forces. A scheme is proposed to overcome this difficulty. [References: 20]