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. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT 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 SCLBT 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 SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.

Lower Bounds for Nonrelativistic Atomic Energies

“…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 that would obviate the need for an explicit calculation of the H 2 matrix was recently explored, 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.…”

“…These difficulties have been addressed in a recent series of papers. ,− A central aspect which has led to significant improvement is combining lower bound theory with basis sets created by the Lanczos method . Due to the resulting tridiagonal representation of the Hamiltonian the computation of the variance needs as input only matrix elements of the Hamiltonian itself.…”

“…The advantage of this manipulation is that from its definition (eq ) one may rewrite the residual energy in terms of the overlaps and the exact eigenenergies as where δ jk is the Kronecker delta. Equation may be used to obtain lower bounds to the residual energy. , For example, for the ground state we can write so that the Temple lower bound for the ground state energy may be expressed as where ε 2,– denotes a lower bound to the first excited state energy which can be obtained by other means, such as by using the Weinstein- or the Bazley-type lower bounds. Indeed, for the He atom, Bazley already found a lower bound energy for the first excited state of −2.165 E h to be compared with the numerically exact energy of −2.146 E h .…”

“…Obtaining lower bounds to the residual energy of excited states is more involved, though straightforward when using a Lanczos basis set. A detailed derivation and discussion may be found in refs − .…”

“…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 , leads to gap ratios which are of the order of unity, and sometimes even less.…”