Conditions for Optimal Choice of Approximate Wavefunctions and the Hypervirial Theorem

The hypervirial theorem for the operator L does not guarantee the independence of the expectation value L from all essential errors of the approximate wavefunction. The more complete system of conditions includes the matrix elements of higher commutators (and anticommutators) L with the Hamiltonian. These conditions prevent the effect of approximate wavefunction errors and may be useful for evaluations of expectation values and transition probabilities. The problem is considered in terms of dynamics or statics. These treatments are shown to be equivalent only for the exact solution.