Lower Bounds for Nonrelativistic Atomic Energies

Abstract

A recently developed lower bound theory for Coulombic problems (E. Pollak, R. Martinazzo, J. Chem. Theory Comput. 2021Comput. , 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.

On the inclusion of cusp effects in expectation values with explicitly correlated Gaussians

“…Finally, we would like to mention that the p4 -and δ(r)-type singular operators appear not only in the perturbative relativistic theory, but also in lower-bound theory due to the appearance of the Ĥ2 operator [5,[29][30][31]. This fact contributes to the observation that the energy lower bounds typically converge slower to the exact energy [5,31], than the energy upper bound.…”

“…Finally, we would like to mention that the p4 -and δ(r)-type singular operators appear not only in the perturbative relativistic theory, but also in lower-bound theory due to the appearance of the Ĥ2 operator [5,[29][30][31]. This fact contributes to the observation that the energy lower bounds typically converge slower to the exact energy [5,31], than the energy upper bound. It would be interesting to use (generalize) the IT technique to the Ĥ2 expectation value and variance computations, which may speed up the convergence of the best energy lower bounds [32] and that would open the route to computation of rigorous theoretical error bars for numerically computed non-relativistic energies.…”

Vibronic mass computation for the $EF$-$GK$-$H\bar{H}$ $^1Σ_\text{g}^+$ manifold of molecular hydrogen

“…Minimization of W (k) with respect to the variation of φ and the λ i s assumes the fulfillment of the auxiliary orthogonality conditions, Eqs. (24), and…”

“…III has been implemented in the in-house developed computer program named QUANTEN (QUANTum mechanical description of Electrons and atomic Nuclei). QUANTEN has recent applications including non-relativistic energy upper and lower bounds, non-adiabatic, pre-Born-Oppenheimer, perturbative and variational relativistic computations [21][22][23][24][25][26][27][28][29][30][31][32]. The program contains a (stochastic and deterministic) non-linear variational engine and an integral library for variants of explicitly correlated Gaussian (ECG) functions.…”

Variational Dirac-Coulomb explicitly correlated computations for atoms and molecules

“…( 59), is implemented in the QUANTEN computer program [57] (for recent applications of the program see Refs. [7,10,19,[58][59][60][61][62]). QUANTEN was written using the Fortan90 programming language and contains several analytic ECG integrals.…”