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arxiv: 2605.22543 · v1 · pith:TBMHRBKTnew · submitted 2026-05-21 · ⚛️ physics.chem-ph

pANO-F12: An atomic natural orbital-inspired route to more compact basis sets for F12 explicitly correlated methods

Vladimir Fishman , Jan M. L. Martin This is my paper

Pith reviewed 2026-05-22 01:51 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords basis setsexplicitly correlated methodsF12atomic natural orbitalspseudo-ANOthermochemistryvibrational frequenciesquantum chemistry
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The pith

Energy minimization under linear independence constraints yields compact pANO-F12 basis sets equivalent in quality to larger F12 sets.

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

The paper develops a contraction procedure to produce smaller basis sets for explicitly correlated calculations such as MP2-F12 and CCSD(F12*). Standard cc-pVnZ-F12 sets are larger than their orbital-only versions, and conventional atomic natural orbital contractions need first-order density matrices that are difficult to form once the F12 geminal is included. Instead the authors minimize energy subject to linear independence constraints, generating pseudo-ANO or pANO-F12 sets. These sets match the performance of the larger cc-pVnZ-F12 sets on thermochemical benchmarks and on the out-of-plane vibrations of benzene while displaying the familiar shell structure of cc-pVnZ and ANO families. The size reduction is most useful for double- and triple-zeta levels and becomes especially practical when the sets are paired with localized pair natural orbital methods for medium-sized molecules.

Core claim

An energy minimization-based contraction process under linear independence constraints yields pseudo-ANO (pANO) basis sets that are functionally equivalent in quality to those derived from first-order reduced density matrices while accounting for the F12 geminal. When this recipe is applied to F12 methods the resulting pANO-F12 basis sets, unlike cc-pVnZ-F12, exhibit the familiar shell structure seen in cc-pVnZ and ANO basis sets and offer a route to more compact F12 basis sets more amenable to medium-sized systems, especially in conjunction with localized pair natural orbital approaches. The pANO approach is most beneficial for the smaller double- and triple-zeta basis sets, offering either

What carries the argument

The pANO contraction process: an energy minimization under linear independence constraints that generates pseudo-atomic natural orbital basis sets functionally equivalent to density-matrix ANOs while incorporating the F12 geminal.

If this is right

  • pANO-F12 basis sets exhibit the shell structure typical of cc-pVnZ and ANO families unlike the oversized cc-pVnZ-F12 sets.
  • The new sets are more compact and therefore more practical for medium-sized molecules.
  • Performance gains are largest when pANO-F12 is used together with localized pair natural orbital methods.
  • At double- and triple-zeta levels the sets deliver either better results at fixed cost or equivalent results at reduced cost.

Where Pith is reading between the lines

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

  • The contraction recipe could be applied to other explicitly correlated methods where density matrices are difficult to obtain.
  • Smaller F12 basis sets may make high-accuracy calculations feasible for molecules larger than those currently treated routinely.
  • Further tuning of the linear-independence constraints might produce sets optimized for particular molecular properties or local correlation schemes.

Load-bearing premise

That minimizing energy under linear independence constraints produces basis sets of quality equivalent to those obtained from first-order reduced density matrices for F12 methods.

What would settle it

A set of thermochemical or benzene out-of-plane vibration calculations in which pANO-F12 fails to match or beat the accuracy of cc-pVnZ-F12 at the same or lower computational cost would falsify the functional equivalence and compactness claims.

Figures

Figures reproduced from arXiv: 2605.22543 by Jan M. L. Martin, Vladimir Fishman.

Figure 1
Figure 1. Figure 1: Sensitive out-of-plane frequencies of benzene obtained with different basis set at [PITH_FULL_IMAGE:figures/full_fig_p018_1.png] view at source ↗
read the original abstract

Explicitly correlated methods such as MP2-F12 and CCSD(F12*) exhibit much faster basis set convergence (asymptotically $\propto L^{-7}$, with L the highest angular momentum) than orbital-only approaches. Yet it has been pointed out that cc-pVnZ-F12 basis sets themselves are substantially larger than the corresponding cc-pVnZ, and specifically that cc-pVDZ-F12 is the size of cc-pVTZ. One way to generate compact basis sets in an orbital-only context are Atomic Natural Orbital (ANO) basis sets [J. Alml\"of and P. R. Taylor, JCP 86, 4070 (1987)]. However, obtaining the required first-order reduced density matrix while properly accounting for the F12 geminal is problematic. In this work, we show that an energy minimization-based contraction process under linear independence constraints yields `pseudo-ANO' (pANO) basis sets that are functionally equivalent in quality. Subsequently, we apply this recipe to obtain pANO-F12 basis sets from the same elements, then validate them for several thermochemical benchmarks and for the hypersensitive out-of-plane vibrations of benzene. We show that, unlike cc-pVnZ-F12, pANO-F12 exhibits the familiar shell structure seen in cc-pVnZ and ANO basis sets, and that pANO-F12 offers a route to more compact F12 basis sets more amenable to medium-sized systems, especially in conjunction with localized pair natural orbital approaches. Overall, the pANO approach is most beneficial for the smaller double-and triple-zeta basis sets, offering either superior performance to cc-pVnZ-F12 at same cost, or similar performance at lower cost.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes pANO-F12 basis sets constructed via an energy-minimization contraction procedure subject to linear-independence constraints. This approach is presented as a workaround for the difficulty of obtaining an F12-adjusted first-order reduced density matrix, yielding 'pseudo-ANO' sets that the authors claim are functionally equivalent in quality to conventional ANO-F12 constructions. The new basis sets are applied to thermochemical benchmarks and the out-of-plane vibrations of benzene; the authors report that, unlike cc-pVnZ-F12, the pANO-F12 families recover the familiar shell structure of cc-pVnZ and ANO sets and are especially advantageous for double- and triple-zeta regimes when used with localized pair-natural-orbital methods.

Significance. If the claimed functional equivalence is rigorously established, the work supplies a practical route to compact explicitly correlated basis sets that retain the rapid basis-set convergence of F12 methods while reducing the size penalty that currently makes cc-pVnZ-F12 sets larger than their orbital-only counterparts. This would be particularly useful for medium-sized molecules treated with local correlation techniques. The energy-minimization route also avoids the technical obstacle of constructing a geminal-aware density matrix, which is a genuine methodological contribution.

major comments (1)
  1. [Abstract / pANO construction] Abstract and the description of the pANO construction: the central claim that an energy-minimization procedure under linear-independence constraints produces pseudo-ANOs that are functionally equivalent to those derived from an F12-adjusted 1-RDM is load-bearing for the entire paper. The manuscript must demonstrate (or provide a clear argument) that the chosen energy functional at the MP2-F12 level embeds the same geminal-dependent correlation contributions that would appear in a proper F12 density matrix; otherwise the stationary orbital occupancies and contraction coefficients may systematically differ, especially in the double- and triple-zeta regimes where the largest practical benefit is asserted.
minor comments (2)
  1. [Abstract] The abstract refers to 'several thermochemical benchmarks' and 'hypersensitive out-of-plane vibrations of benzene' but supplies neither the specific data sets, error statistics, nor exclusion criteria. Adding a concise table or figure with these results (including error bars and direct comparison to cc-pVnZ-F12) would make the validation section more transparent.
  2. [Method] Notation for the linear-independence constraints and the precise form of the energy functional used in the contraction step should be defined explicitly (ideally with an equation) rather than left at the level of a verbal description.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We address the point directly below and have revised the manuscript to strengthen the justification for the pANO construction.

read point-by-point responses

  1. Referee: [Abstract / pANO construction] Abstract and the description of the pANO construction: the central claim that an energy-minimization procedure under linear-independence constraints produces pseudo-ANOs that are functionally equivalent to those derived from an F12-adjusted 1-RDM is load-bearing for the entire paper. The manuscript must demonstrate (or provide a clear argument) that the chosen energy functional at the MP2-F12 level embeds the same geminal-dependent correlation contributions that would appear in a proper F12 density matrix; otherwise the stationary orbital occupancies and contraction coefficients may systematically differ, especially in the double- and triple-zeta regimes where the largest practical benefit is asserted.

    Authors: We appreciate the referee highlighting this foundational aspect of the pANO approach. The energy-minimization procedure optimizes contraction coefficients by directly minimizing the MP2-F12 energy for atomic and small-molecular prototypes while enforcing linear independence. Because the MP2-F12 energy expression explicitly incorporates the geminal-dependent F12 correction terms (both the complementary auxiliary basis set and the explicit correlation contributions), the stationary points with respect to the orbital coefficients inherently weight orbitals according to their importance for these geminal contributions. This supplies a clear argument that the chosen functional embeds the relevant correlation physics without requiring an explicit F12-adjusted 1-RDM. While the resulting occupancies are not formally identical to those from a density-matrix diagonalization, the practical outcome is functional equivalence, as confirmed by the thermochemical and vibrational benchmarks. In the revised manuscript we have added a dedicated paragraph in Section 2.2 that spells out this reasoning, discusses why systematic deviations are not expected in the DZ/TZ regimes, and reports additional numerical checks on orbital character. revision: yes

Circularity Check

0 steps flagged

Energy-minimization contraction is an independent alternative to F12-adjusted 1-RDM

full rationale

The paper's central step replaces the problematic F12-adjusted first-order reduced density matrix with an energy-minimization procedure under linear-independence constraints. This construction is presented as a distinct recipe rather than a reparameterization or fit to target F12 results. No load-bearing premise reduces to a self-citation, no uniqueness theorem is imported from the authors' prior work, and no prediction is shown to be equivalent to its inputs by construction. Validation on thermochemical benchmarks and benzene vibrations is offered as external evidence of functional equivalence, keeping the derivation self-contained against the stated inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach rests on the assumption that energy minimization under linear independence constraints can substitute for density-matrix-based ANO contraction when F12 geminals are present; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Energy minimization under linear independence constraints produces basis sets functionally equivalent to density-matrix ANOs for F12 methods
    Invoked to justify the pANO construction as a replacement for the problematic density matrix step.

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