Effective utilization of off-diagonal hypervirial relations considering diagonal hypervirial relation: Harmonic oscillator case

The effective utilization of hypervirial relations is scrutinized to improve the approximate excited-state functions in the harmonic oscillator system. A new method is presented which simultaneously employs the off-diagonal hypervirial relations with the diagonal hypervirial relation. In order to use these relations effectively, the following points are pointed out: (i) the presented method is useful to get better reasonable results for the excitation energies and the state functions; (ii) the ground state given must satisfy the virial theorem; (iii) in the hypervirial operator used here as x^m, the smaller integers of m's present better results; and (iv) the employment of the comparatively small number of trial basis functions of the type exp (?γ|x|) is sufficient for reproducing the exact excited state. Especially among them, condition (ii) plays an important role. Applying all the proposals to the first and the second excited states, one gets a highly improved excitation energy, state function, and other physical quantities (e.g., transition moment and oscillator strength). The presented method is also found to be more effective than the employment of only the off-diagonal hypervirial relations or the method of the scaling operation.