A Tight Lower Bound to the Ground-State Energy.

Ninety years ago Temple ( Proc. R. Soc. (London) 1928 , A119 , 276 ) derived a lower bound for the ground-state energy. The bound was tested and invariably found to be poor as compared to the upper bound obtained through the Rayleigh Ritz procedure due to the fact that it is based also on the second moment of the Hamiltonian. In this paper we (a) improve upon Temple's lower bound estimate for the overlap squared of the true ground-state wave function with the approximate one and (b) describe in detail and generalize our recent improvement on the Temple lower bound based on utilization of higher-order basis functions derived by the Arnoldi algorithm. Both improvements combined lead to a lower bound on the ground-state energy whose accuracy is better than that of the Temple lower bound. This is exemplified by considering the ground-state energy of a quartic potential where one finds that the improvements lead to a lower bound whose quality is comparable to that of the upper bound. The applicability of the method to atoms and molecules is discussed.