Comparison of an improved self-consistent lower bound theory with Lehmann’s method for low-lying eigenvalues

Abstract

Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds

Lower Bounds for Nonrelativistic Atomic Energies

“…The diagonalization of the Lehmann equation becomes equivalent to the solution for the zeros of a sum of rational terms. Third, our recent self-consistent lower bound theory 40,42 leads to gap ratios which are of the order of unity, and sometimes even less.…”

“…Equation 2.18 may be used to obtain lower bounds to the residual energy. 40,42 For example, for the ground state we can write…”

“…This increases the computational expense involved in the practical application of the Pollak− Martinazzo theory, for example, in quantum chemistry. An extrapolation procedure based on eq 2.9 that would obviate the need for an explicit calculation of the H 2 matrix was recently explored, 42 and initial results for the hydrogen atom using an odd Gauss−Hermite basis are promising. Further work is necessary to test this strategy for Gaussian-type basis functions, the mathematical properties of which differ from those of orthogonal polynomials.…”

Lower bounds on par with upper bounds for few-electron atomic energies

“…In the past few years, a novel class of lower bound methods [35,36] based on the Lánczos construction of basis sets has been proposed. A self-consistent lower bound theory (SCLBT) [37,38] was developed and successfully applied to quartic [38] and double-well [39] potentials as well as lattice models [37,40]; however, these methods are typically not applicable for Coulomb-interacting systems due to the divergence of matrix elements of cubic and higher powers of the Coulombic Hamiltonian, unless one can devise basis sets, which like the true eigenfunctions, prevent such divergence and still the basis is in principle complete.…”