Nuclear-electronic calculations need uncontracted basis sets on the quantum protons (2503.03966v2)

Abstract: An attractive way to calculate nuclear quantum effects is to describe select nuclei quantum mechanically at the same level as the electrons, requiring the solution of coupled Schr\"odinger equations for the electrons and the quantum nuclei. This is commonly known as the nuclear-electronic orbital (NEO) method, but it also has many other names. Two types of basis sets are required: a nuclear basis set is required in addition to the usual electronic basis set. In this work, we demonstrate that while existing nuclear basis sets are sufficient for NEO density-functional calculations, many sets producing proton affinities converged within 0.1 kcal/mol of the complete basis set limit, NEO calculations should always use uncontracted electronic basis sets on the quantum protons, since the contraction coefficients in typical electronic basis sets have been derived for point nuclear charge distributions. Uncontracting the basis sets on the quantized protons leads to significantly faster convergence to the basis set limit, leading to improvements of 18 kcal/mol and 10 kcal/mol in proton affinities employing double-$\zeta$ aug-pc-1 and triple-$\zeta$ aug-pc-2 electronic basis sets, respectively, with little effect on the computational effort. The partially uncontracted aug-pc-3 electronic basis set already affords proton affinities converged beyond 0.1 kcal/mol from the complete basis set limit. Similar results are also obtained with Dunning's correlation-consistent cc-pVXZ basis sets, as well as the Karlsruhe def2-XZP basis sets, albeit at a somewhat slower rate of convergence. As the protonic basis sets yield fully converged values, we find the protonic basis sets to be unnecessarily large for ground state density functional calculations, as the error in the protonic basis set is not balanced with that for typical electronic basis sets.

Enhancements in NEO Calculations through Uncontracted Basis Sets on Quantum Protons

The paper under discussion explores advancements in the computational approach called the nuclear-electronic orbital (NEO) method, primarily focusing on the multicomponent quantum mechanical treatment of specific nuclei, such as protons, within molecular systems. This method departs from the traditional Born-Oppenheimer approximation by treating selected nuclei, typically protons, with the same quantum mechanical rigor as electrons, thereby incorporating nuclear quantum effects directly into electronic structure calculations.

A salient technical innovation presented in the research emphasizes the need for employing uncontracted electronic basis sets specifically on quantum-proton treatments within NEO calculations. Traditionally, contracted basis sets are utilized for point nuclear charge distributions, which implicitly assumes classical, static nuclei. However, this assumption does not hold in NEO methodologies, as quantum nuclei are distributed over a finite region rather than concentrated points, influencing electron localization and energy convergence. The paper demonstrates that uncontracting said basis sets leads to substantially more rapid convergence towards the complete basis set limit with minimal computational overhead.

The authors evaluated 13 molecular systems—ranging from simple diatomics like \ce{N2} to more complex molecules such as \ce{NH3}—to calculate proton affinities using density functional theory within the NEO framework. Remarkably, results yielded an 18 kcal/mol and 10 kcal/mol improvement in proton affinity calculations with uncontracted basis sets when employing double-$\zeta$ and triple-$\zeta$ configurations, respectively. Overall, precision enhancements reached 0.1 kcal/mol convergence within the complete basis set limit for some combinations, showcasing the efficacy of the uncontracted basis approach.

From a theoretical standpoint, these findings illustrate an important paradigm shift in quantum chemistry regarding the treatment of nuclear degrees of freedom within electronic structure calculations, particularly relevant for systems where nuclear quantum effects like tunneling and delocalization are prominent. The methodological advances discussed could spur further theoretical developments in fields beyond traditional chemistry, including condensed-matter physics and materials science, where intricate nuclear-electronic interactions play critical roles.

As we peer into future developments in computational quantum chemistry, this work implicitly raises several questions and potential research pathways. One pertinent area involves the identification and optimization of new basis set families that strike an optimal balance between computational efficiency and accuracy for both electronic and nuclear wave functions. Another potential avenue includes refining NEO approaches to accommodate a broader spectrum of quantum particles, possibly extending the methodology's applicability to exotic systems involving muons or positrons.

In conclusion, by addressing hitherto overlooked nuances in the simultaneous quantum mechanical treatment of electrons and nuclei, this paper offers tangible enhancements to the precision and efficiency of NEO methods. Future explorations could integrate these basis set strategies into broader multiscale simulations, potentially bridging the gap between molecular and macroscopic descriptions of quantum systems and leading to more comprehensive models of complex phenomena.