Encoding electronic ground-state information with variational even-tempered basis sets

TL;DR

The paper addresses the challenge of designing system-oriented basis sets that encode electronic ground-state information without empirical contractions. It introduces variational even-tempered basis sets, first in a reduced formalism with $\zeta_m = αβ^m$ and two-level optimization, and then extends to molecules via correlated centers and two-level optimization (including an $\alpha$ bootstrap). Key contributions include demonstrating stable, rapid HF energy convergence for hydrogen, competitive dissociation curves for H$_2$ against high-level basis sets, and improved RHF energies for H$_4$ using nested, augmented centers. The work suggests that data-free, system-aware discretizations can approach CBS accuracy with fewer functions, while outlining practical directions for extending angular momentum, center correlations, and nesting strategies to broaden applicability.

Abstract

We propose a system-oriented basis-set design based on even-tempered basis functions to variationally encode electronic ground-state information into molecular orbitals. First, we introduce a reduced formalism of concentric even-tempered orbitals that achieves hydrogen energy accuracy on par with the conventional formalism, with lower optimization cost and improved scalability. Second, we propose a symmetry-adapted, even-tempered formalism specifically designed for molecular systems. It requires only primitive S-subshell Gaussian-type orbitals and uses two parameters to characterize all exponent coefficients. In the case of the diatomic hydrogen molecule, the basis set generated by this formalism produces a dissociation curve more consistent with cc-pV5Z than cc-pVTZ at the size of aug-cc-pVDZ. Finally, we test our even-tempered formalism against several types of tetra-atomic hydrogen molecules for ground-state computation and point out its current limitations and potential improvements.

Figures (16)

• Figure 1: The atomic hydrogen ground-state energy error of the reduced variational even-tempered basis set $\tilde{\mathcal{G}}^{\rm r}_M$ (the solid colored diamonds) with respect to $M$ (equivalent to basis set size in this case) and $\alpha$. (a) shows the logarithmic errors, where the open diamonds with black strokes represent the conventionally optimized even-tempered basis set $\tilde{\mathcal{G}}^{\rm o}_M$ at different $M$ (incremented by one). (b) shows the relation between the error and optimized parameter $\tilde{\beta}$ for each configuration of $\tilde{\mathcal{G}}^{\rm r}_M$. For each color group, the data points in (b) follow the same correspondence to $M$ as the ones in (a), respectively.
• Figure 2: The errors of the exponents $\left\{\zeta_m\vert m\right\}$ from $\tilde{\mathcal{G}}^{\rm r}_M$ with respect to the optimal exponents $\left\{\alpha_{m'}\vert m'\right\}$ from $\tilde{\mathcal{G}}^{\rm o}_M$. In (a), each colored diamond represents the mean absolute error of $\left\{\zeta_m\vert m\right\}$ (rearranged in ascending order to match $\left\{\alpha_{m'}\vert m'\right\}$) from $\tilde{\mathcal{G}}^{\rm r}_M(\alpha)$. (b) shows that for $M\!=\!6$, the rearranged exponents of $\tilde{\mathcal{G}}^{\rm r}_M(\alpha)$ converge to their respective optimal values (the open diamonds chained by a dotted line) as $\alpha$ increases exponentially from $1$ to $128$.
• Figure 3: The scaling of the overlap-matrix ($\bm{S}$) condition number for various even-tempered basis set configurations tested against the hydrogen atom. In (a), the colored diamonds represent the results for $\tilde{\mathcal{G}}^{\rm r}_M(\alpha)$, and the open diamonds (marked as "Optimal") represent the results for $\tilde{\mathcal{G}}^{\rm o}_M$. The light green stroke passes through the data points of $\tilde{\mathcal{G}}^{\rm r}_M(\{\alpha \!=\! 2^{M+1}\})$. (b) shows how the $\bm{S}$ condition number of $\tilde{\mathcal{G}}^{\rm r}_6(\alpha)$ is suppressed as $\alpha$ increases, where the dashed horizontal line represents the condition number for $\tilde{\mathcal{G}}^{\rm o}_{6}$.
• Figure 4: The numerically-stable values of $(\alpha,\,\tilde{\beta})$ with respect to basis degree $M$ of the even-tempered basis sets for: (a) the atomic hydrogen; (b) the diatomic hydrogen (H$_{2}$) molecule at the bond length of $1.4$ a.u.
• Figure 5: The convergence characteristics of a variationally optimized (through Algorithm \ref{['alg:mop2']}) reduced even-tempered basis set $\tilde{\mathcal{G}}^{\rm r}_M$ with respect to the basis degree $M$ in the case of H$_2$ at a bond length of $1.4$ a.u. (a) shows the restricted closed-shell Hartree--Fock energy (blue dashed line) for $\tilde{\mathcal{G}}^{\rm r}_M$ and the condition number of the corresponding overlap matrix (orange line). (b) shows the optimized values of the smallest basis-function exponent $\alpha\tilde{\beta}$ (blue dashed line) and the growth rate $\tilde{\beta}$ (orange dashed line). For (a) and (b), the vertical dashed lines cross the data points corresponding to the steps when $\alpha$ is updated.