Abstract
A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput.2021, 17, 1535) is further developed and applied to the highly accurate calculation of the ground-state energy of two- (He, Li+, and H–) and three- (Li) electron atoms. The method has been implemented with explicitly correlated many-particle basis sets of Gaussian type, on the basis of the highly accurate (Ritz) upper bounds they can provide with relatively small numbers of functions. The use of explicitly correlated Gaussians is developed further for computing the variances, and the necessary modifications are here discussed. The computed lower bounds are of submilli-Hartree (parts per million relative) precision and for Li represent the best lower bounds ever obtained. Although not yet as accurate as the corresponding (Ritz) upper bounds, the computed bounds are orders of magnitude tighter than those obtained with other lower bound methods, thereby demonstrating that the proposed method is viable for lower bound calculations in quantum chemistry applications. Among several aspects, the optimization of the wave function is shown to play a key role for both the optimal solution of the lower bound problem and the internal check of the theory.
Highlights
The challenge of obtaining lower bounds for atomic and molecular energies has a long history
The Temple and Weinstein−Stevenson expressions result in “poor” numerical lower bounds in the sense that the gap ratio, the ratio of the deviation of the upper and the lower bound from the exact value,/(εj,+ − εj), is all too often orders of magnitude larger than unity. (Lower and upper bounds are denoted with − and + subscripts respectively throughout.)
In the second section of this paper we briefly review the Temple, Lehmann, and Pollak− Martinazzo (PM) lower bound theories and extend the latter so that it may be used in conjunction with eigenfunctions of Lehmann’s equation and the associated diagonal matrix elements and variances
Summary
The challenge of obtaining lower bounds for atomic and molecular energies has a long history. A meaningful choice within the context of lower bound theory is provided by the solutions of the Lehmann equation, eq 2.21.18,19,69 To achieve this goal, we constructed a not necessarily orthonormal basis, using the (normalized) Lehmann eigenfunctions |Ωj(L)⟩, j = 1, ···, L With these functions we computed the diagonal matrix elements (expectation values) of the Hamiltonian and their associated variances, eqs 2.31 and 2.32. We have experimented with the Lehmann pole ρ, eq 2.21, and the energy parameter ε, which is the input to the PM equation, eq 2.30, and set them equal to the (estimated) relevant lower bounds for higher excited states (see Table 2) This approach did not lead to any significant improvement for the Temple bound. A better understanding of the interplay of the ground- and excited-state properties affecting the PM lower bound will help to develop an efficient and systematic basis generation and selection procedure
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