Optimization of the temple lower bound

Abstract

The Temple formula is perhaps the most common method used in the uncommon endeavor of calculating a lower bound to the ground-state energy of an atomic or molecular system. We generalize the Temple formula by introducing a parameter that can be varied to optimize the lower bound. This generalization does not require any information that is not already used for the traditional Temple lower bound. Examples with the helium cation and neutral atom show that improvement is greatest when the approximate wave function poorly approximates the true groundstate wave function. The examples also show that in some cases the traditional Temple lower bound may already be optimal so that our generalization gives no improvement.

Lower Bounds for Coulombic Systems

“…Lower bound methods abound, starting with Temple’s seminal expression derived in 1928 . Landmarks in the derivation of lower bounds are Weinstein’s lower bound of 1934 , and Lehmann’s optimization of Temple’s lower bound presented in 1949–50. , Especially Lehmann’s expression has turned out to be quite accurate in different settings, however not so for Coulombic systems, as exemplified by computations on the He − and Li atoms.…”

Upper and lower bounds for tunneling splittings in a symmetric double-well potential

“…For many years, lower bounds were not sufficiently accurate. [48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63] Especially when considering tunneling doublets, their accuracy was too poor, the gap between the lower bounds and the true energies was typically much larger than the actual energy differences between the levels. Upper and lower bounds for level differences are only meaningful if the accuracy of the upper and lower bounds is comparable and tighter than the energy splitting between the doublet states.…”

An Improved Lower Bound to the Ground-State Energy

“…The variational procedure optimizes this wave function such that, for a given basis set, it leads to the "best" Temple lower bound. 11 Marmorino and co-workers 6,11,13,14 have discussed various scenarios, perhaps the most relevant one in the context of this letter, 6 shows that an improved bound may be obtained by including the third moment of the Hamiltonian. Yet this lower bound is implicit, not explicit.…”