The Hellmann-Feynman theorem that governs the linear changes in the energy with respect to a parameter of the Hamiltonian is extended to functions of the Hamiltonian. The extension, applicable to a Hamiltonian with both discrete and continuous spectrum, is of particular importance for semi-empirical theories of molecular rate processes. Nonlinear dependence on a parameter is also considered and a power series expansion is provided. 'Integral' forms of these results are also considered. The basic tool employed is the spectral representation of functions of operators. The methods are illustrated by various aspects of excitation propagation in pure and impure linear chains, calculated by means of a variational derivation of time-dependent Huckel theory.