Reaching precise proton affinities in non-Born–Oppenheimer calculations

An attractive way to model nuclear quantum effects is to describe select nuclei quantum mechanically at the same level as the electrons. This non-Born-Oppenheimer (non-BO) method is known by many names, including the nuclear-electronic orbital and the multicomponent method. Two basis sets are typically used for such calculations: a nuclear basis set and an electronic basis set. In this work, we investigate the convergence of non-BO proton affinities (PAs) with respect to the protonic and electronic basis sets. PAs are a sensitive measure of the proton and electron densities. We demonstrate that most protonic basis sets are sufficient for non-BO density-functional calculations of PAs, resulting in convergence to within 0.1 kcal/mol of the complete protonic basis set limit. This indicates that the truncation error is dominated by the electronic basis and that smaller protonic basis sets could be developed. We show that non-BO calculations should use uncontracted electronic basis sets on the quantum protons. The contraction coefficients in typical electronic basis sets have been derived for point nuclear charge distributions, and uncontracting the electronic basis set on the quantized proton leads to significantly faster convergence to the electronic basis set limit. Uncontraction leads to results of at least one ζ-level higher quality with negligible additional computational cost in multiple diffuse basis set families: Jensen's polarization consistent aug-pc-n basis sets, Dunning's correlation-consistent aug-cc-pVXZ basis sets, as well as the Karlsruhe def2-XZPD basis sets. Specifically, the aug-pc-3 electronic basis set already affords PAs converged beyond 0.1 kcal/mol when uncontracted on the quantum proton.