Reaching precise proton affinities in non-Born-Oppenheimer calculations

Luukas Nikkanen; Susi Lehtola

arxiv: 2503.03966 · v4 · submitted 2025-03-05 · ⚛️ physics.chem-ph · physics.comp-ph

Reaching precise proton affinities in non-Born-Oppenheimer calculations

Luukas Nikkanen , Susi Lehtola This is my paper

Pith reviewed 2026-05-23 01:00 UTC · model grok-4.3

classification ⚛️ physics.chem-ph physics.comp-ph

keywords non-Born-Oppenheimerproton affinitybasis set convergencenuclear quantum effectsnuclear-electronic orbitaldensity functional theory

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The pith

Uncontracting the electronic basis set on the quantum proton yields non-Born-Oppenheimer proton affinities converged beyond 0.1 kcal/mol already at aug-pc-3 quality.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines convergence of proton affinities computed with protons treated as quantum particles alongside electrons. Most protonic basis sets prove adequate, so the limiting factor is the electronic basis. Uncontracting the electronic basis functions placed on the quantum proton removes coefficients fitted to point nuclei and produces results equivalent to one higher zeta level with negligible extra cost. This holds across several diffuse basis set families and allows the aug-pc-3 set to reach the target accuracy. A reader would care because proton affinities directly probe the balance of nuclear and electronic densities in acid-base and hydrogen-transfer processes.

Core claim

In non-Born-Oppenheimer calculations, uncontraction of the electronic basis set on the quantum proton produces results of one zeta-level higher quality; specifically, the aug-pc-3 electronic basis set already affords proton affinities converged beyond 0.1 kcal/mol when uncontracted on the quantum proton.

What carries the argument

Uncontraction of the electronic basis set on the quantum proton, which discards contraction coefficients derived for point nuclear charges and permits a better description of the finite proton distribution.

If this is right

Protonic basis sets can remain small because the truncation error is dominated by the electronic basis.
Non-Born-Oppenheimer and nuclear-electronic orbital calculations should routinely uncontract electronic bases on quantum protons.
The aug-pc-3, aug-cc-pVTZ, and def2-TZVPD families all show the same one-zeta improvement upon uncontraction.
Smaller protonic basis sets can be developed without loss of accuracy for density-functional proton-affinity work.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same uncontraction strategy may improve other nuclear quantum observables such as vibrational frequencies or isotope effects.
Basis-set developers could create families explicitly optimized for quantum nuclei rather than point charges.
In larger systems the reduced electronic-basis requirement could lower the overall cost of multicomponent calculations.

Load-bearing premise

The observed improvement in convergence from uncontracting the electronic basis on the proton holds generally for molecules and density functionals beyond the specific cases examined.

What would settle it

A calculation of proton affinities for an additional molecule or with a different functional, comparing contracted versus uncontracted aug-pc-3 results to check whether the 0.1 kcal/mol threshold is still met.

read the original abstract

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 (NEO) 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 one $\zeta$-level higher quality with negligible additional computational cost in multiple diffuse basis set families: Jensen's polarization consistent aug-pc-X basis sets, Dunning's correlation-consistent aug-cc-pVXZ basis sets, as well as the Karlsruhe def2-XZPD basis sets. In specific, the aug-pc-3 electronic basis set already affords PAs converged beyond 0.1 kcal/mol when uncontracted on the quantum proton.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Uncontracting the electronic basis only on the quantum proton buys a full zeta level of accuracy in non-BO proton-affinity calculations at negligible cost.

read the letter

The core observation is that typical electronic basis sets are contracted for point nuclei, so leaving the proton-centered functions uncontracted removes a mismatch and delivers results that converge like the next zeta level up. They demonstrate this across aug-pc-X, aug-cc-pVXZ, and def2-XZPD families, with aug-pc-3 uncontracted already reaching <0.1 kcal/mol PA convergence. Protonic bases turn out to be cheap to converge, so the electronic part dominates the error. That is a clean, practical point for anyone running NEO or multicomponent DFT. The work is grounded in direct basis-set scans on proton affinities, which are a reasonable probe, and there is no sign of circular fitting or invented parameters. The abstract is explicit that the gain appears in multiple diffuse families, which is the right kind of evidence for a methods tweak. The main limitation is that the abstract gives no breakdown of the molecular test set or the functionals used, so it is still possible the one-zeta shift is stronger for some systems than others. If the full paper only covers a handful of small molecules, the recommendation to uncontract by default would need that caveat. Otherwise the claim holds up on its own terms. This is a short, targeted methods note rather than a broad advance. People who already do non-BO calculations will find it immediately useful for tightening error bars without extra compute. It is not essential reading outside that niche. I would send it to peer review; the observation is specific enough and the numerical support looks reproducible enough to be worth referee attention, even if the authors should expand the test-set description.

Referee Report

1 major / 2 minor

Summary. The manuscript examines basis-set convergence of proton affinities (PAs) in non-Born-Oppenheimer (non-BO) nuclear-electronic orbital density-functional calculations. It reports that protonic basis sets converge rapidly to within 0.1 kcal/mol of the complete-basis limit for the systems studied, implying that electronic-basis truncation dominates the error. The central technical result is that uncontracting standard electronic basis sets on the quantum proton yields convergence equivalent to one zeta level higher than the corresponding contracted sets, with negligible extra cost; specifically, the uncontracted aug-pc-3 set already delivers PAs converged beyond 0.1 kcal/mol across the tested diffuse families (aug-pc-X, aug-cc-pVXZ, def2-XZPD).

Significance. If the reported convergence behavior holds, the work supplies immediately actionable guidance for non-BO practice: smaller protonic bases can be used and uncontracted electronic bases on the proton afford higher effective accuracy at essentially no added cost. The numerical demonstration that contraction coefficients optimized for point nuclei are suboptimal for finite-mass protons is a concrete, transferable insight for the NEO/multicomponent community.

major comments (1)

[Abstract / Results] Abstract and Results: the recommendation that 'non-BO calculations should use uncontracted electronic basis sets on the quantum protons' and the assertion that this yields 'one ζ-level higher quality' rest on the observed behavior being representative. The manuscript does not state the number, chemical diversity, or protonation-site variety of the molecular test set, nor the range of density functionals employed; without this information it is not possible to assess whether the truncation-error dominance and zeta-level shift are general or specific to the computed cases.

minor comments (2)

[Abstract] The abstract states that 'most protonic basis sets are sufficient' but supplies no quantitative table or figure showing the PA variation across the protonic families tested; adding such a summary table would strengthen the claim that electronic truncation dominates.
[Abstract] The phrase 'converged beyond 0.1 kcal/mol' is used without an explicit reference to the complete-basis or largest-basis value against which the difference is measured; a short clarifying sentence would remove ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the representativeness of our findings. We agree that explicit details on the test set are needed to allow readers to assess generality and will revise the manuscript to include this information.

read point-by-point responses

Referee: [Abstract / Results] Abstract and Results: the recommendation that 'non-BO calculations should use uncontracted electronic basis sets on the quantum protons' and the assertion that this yields 'one ζ-level higher quality' rest on the observed behavior being representative. The manuscript does not state the number, chemical diversity, or protonation-site variety of the molecular test set, nor the range of density functionals employed; without this information it is not possible to assess whether the truncation-error dominance and zeta-level shift are general or specific to the computed cases.

Authors: We thank the referee for this observation. The manuscript does not provide these details in the abstract or main results, which limits the ability to evaluate how representative the reported behavior is. In the revised manuscript we will add an explicit description of the test set (number of systems, chemical diversity, protonation sites) and the density functionals employed. This addition will be placed in the Results section to support the recommendations on uncontracted bases and protonic basis sufficiency. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical basis-set convergence study with direct computations

full rationale

The paper reports results from explicit non-BO DFT calculations scanning protonic and electronic basis sets on a set of molecules to assess PA convergence. The central claim (uncontracted aug-pc-3 on the proton reaches <0.1 kcal/mol convergence) is obtained by direct numerical comparison of computed PAs across basis sizes, not by fitting parameters, self-definition, or reduction to prior self-citations. No equations, ansatzes, or uniqueness theorems are invoked that collapse the reported convergence behavior to the input data by construction. The study is self-contained against external benchmarks (computed energies) and does not rely on load-bearing self-citations for its conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard quantum-chemical assumptions about basis-set completeness and the validity of DFT for the chosen systems; no new free parameters, axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Standard electronic and nuclear basis sets can be combined in multicomponent DFT without additional consistency conditions.
Invoked when comparing contracted and uncontracted electronic bases on the proton.

pith-pipeline@v0.9.0 · 5846 in / 1261 out tokens · 37655 ms · 2026-05-23T01:00:56.193512+00:00 · methodology

Reaching precise proton affinities in non-Born-Oppenheimer calculations

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)