Open-source implementation of the anti-Hermitian contracted Schr\"odinger equation for electronic ground and excited states

Anthony W Schlimgen; Daniel Gibney; Jan-Niklas Boyn

arxiv: 2604.02550 · v1 · submitted 2026-04-02 · 🪐 quant-ph · physics.chem-ph· physics.comp-ph

Open-source implementation of the anti-Hermitian contracted Schr\"odinger equation for electronic ground and excited states

Daniel Gibney , Anthony W Schlimgen , Jan-Niklas Boyn This is my paper

Pith reviewed 2026-05-13 20:23 UTC · model grok-4.3

classification 🪐 quant-ph physics.chem-phphysics.comp-ph

keywords anti-Hermitian contracted Schrödinger equationACSEall-electron correlationmolecular electronic structureground statesexcited statesstrongly correlated electronsopen-source implementation

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The pith

An open-source solver for the anti-Hermitian contracted Schrödinger equation computes accurate all-electron correlation in molecules for both ground and excited states.

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

The paper presents a new open-source implementation of the anti-Hermitian contracted Schrödinger equation that solves for all-electron correlation directly from the exact electronic Hamiltonian. Standard multireference perturbation methods scale poorly with the complexity of the reference wavefunction, but this approach does not. Benchmarks across main-group and transition-metal molecules, weak and strong correlation regimes, multiple basis sets, and both ground and excited states show consistent accuracy. If the method holds, it supplies a practical route to quantitative electronic-structure predictions that avoid approximate perturbative Hamiltonians.

Core claim

The open-source ACSE implementation produces reliable energies and properties for molecular ground and excited states by contracting the Schrödinger equation while retaining the exact Hamiltonian, with demonstrated accuracy that does not degrade when the reference wavefunction becomes strongly correlated.

What carries the argument

The anti-Hermitian contracted Schrödinger equation, which reduces the full N-electron Schrödinger equation to a closed set of equations for the two-electron reduced density matrix.

If this is right

Electronic energies and properties can be obtained without the scaling penalty tied to complex multireference references.
The same code framework applies equally to ground states and excited states.
Results remain stable across different basis-set sizes and correlation strengths.
All-electron calculations become feasible for molecules containing transition metals without perturbative approximations.

Where Pith is reading between the lines

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

The method could be paired with existing density-matrix renormalization group references to reach even larger active spaces.
Extension to property calculations such as dipole moments or polarizabilities would follow directly from the two-electron reduced density matrix already available.
Open availability may encourage community benchmarks against other contracted-equation or reduced-density-matrix methods on standardized test sets.

Load-bearing premise

The numerical solution of the ACSE recovers quantitative all-electron correlation beyond configuration interaction for the tested molecular systems and basis sets.

What would settle it

A direct comparison on the chromium dimer or another well-studied strongly correlated molecule where the ACSE energy or excitation gap differs from full configuration interaction or experimental values by more than chemical accuracy.

Figures

Figures reproduced from arXiv: 2604.02550 by Anthony W Schlimgen, Daniel Gibney, Jan-Niklas Boyn.

**Figure 2.** Figure 2: FIG. 2. Ethylene (C [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Energy convergence (in Hartree) and log [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The dissociation of the two lowest energy singlets and triplets [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

Efficient simulation of strongly correlated electrons has become a routine tool in molecular electronic structure theory due to recent advances in approximate configuration interaction (CI) techniques. Nonetheless, the quantitative and predictive description of molecular electronic states remains a significant challenge due to the difficulty of computing all-electron correlation beyond CI. Here, we describe a new open-source implementation of the anti-Hermitian contracted Schr\"odinger equation (ACSE) for use in accurate simulation of all-electron correlation in molecules. In contrast to standard approaches via multireference perturbation theory, the scaling of the ACSE does not depend on the complexity of the strongly correlated reference wavefunction. Furthermore, the ACSE employs the exact electronic Hamiltonian, rather than an approximate perturbative Hamiltonian. Our benchmark results demonstrate good accuracy for main group and transition metal systems, in weakly and strongly correlated regimes, with various basis sets, and for ground and excited states. The results suggest that the ACSE has potential as a scalable and robust technique for simulating all-electron correlation in molecular ground and excited states.

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

This is an open-source ACSE implementation plus benchmarks on ground/excited states for main-group and transition-metal systems; the accuracy claims need direct FCI/DMRG comparisons on strongly correlated cases to hold up.

read the letter

The main thing to know is that this paper releases open-source code for solving the anti-Hermitian contracted Schrödinger equation and reports benchmarks on both ground and excited states across main-group and transition-metal molecules in different correlation regimes and basis sets. The ACSE formalism predates the work, so the novelty sits in the implementation and the expanded test set rather than new theory. What the paper does well is highlight a practical scaling property: the cost stays independent of how complex the reference wavefunction is, and it uses the exact Hamiltonian instead of a perturbative approximation. That setup can be attractive for transition-metal and excited-state problems where building a good reference already costs a lot. Releasing the code openly also lets others reproduce and extend the results, which adds real value even for an established method. The soft spots are in the support for the accuracy claims. The abstract states good accuracy and quantitative results beyond configuration interaction, but the provided details lack explicit error tables, convergence criteria, or side-by-side numbers against full CI or DMRG on the strongly correlated transition-metal examples. Without those direct comparisons, it is hard to tell whether the energies are recovering more than modest correlation on top of the reference. The scaling argument stands on its own, but the quantitative-accuracy part needs tighter verification. This paper is for computational chemists who want to try ACSE in practice for all-electron correlation in tricky systems. A reader looking for an alternative to standard multireference perturbation methods or who needs accessible code would find it useful. It deserves a serious referee because the implementation and benchmarks are concrete contributions that can be checked and built upon, even if the underlying equations are not new.

Referee Report

2 major / 2 minor

Summary. The manuscript presents an open-source implementation of the anti-Hermitian contracted Schrödinger equation (ACSE) for computing all-electron correlation in molecular ground and excited states. It argues that the ACSE employs the exact electronic Hamiltonian and that its computational scaling is independent of the complexity of the strongly correlated reference wavefunction, in contrast to multireference perturbation theory. Benchmark calculations are reported for main-group and transition-metal systems in both weakly and strongly correlated regimes, using various basis sets, with claims of good accuracy for both ground and excited states.

Significance. If the accuracy claims are substantiated with direct comparisons, the ACSE method could provide a scalable route to all-electron correlation in strongly correlated molecular systems without relying on perturbative Hamiltonians. The open-source release of the implementation is a clear strength, as it enables reproducibility, community verification, and extension of the approach.

major comments (2)

[Results section] Results section (benchmark tables and figures): The central claim of 'good accuracy' and 'quantitative accuracy for all-electron correlation beyond configuration interaction' for strongly correlated transition-metal systems is not supported by direct error metrics or side-by-side comparisons to full configuration interaction (FCI) or density-matrix renormalization group (DMRG) reference values on the same basis sets. Without these, it is impossible to distinguish recovery of the reference plus modest correlation from true quantitative accuracy.
[§3] §3 (numerical solution of the ACSE): The description of the iterative solution procedure and any truncation or convergence thresholds is insufficient to determine under what conditions the method recovers the exact energy versus an approximation; this directly affects the validity of the accuracy claims for strongly correlated cases.

minor comments (2)

[Abstract] The abstract would be strengthened by including at least one concrete error metric (e.g., mean absolute deviation versus FCI) rather than the qualitative phrase 'good accuracy'.
[Figures and Tables] Figure captions and table footnotes should explicitly state the reference method (e.g., CASSCF, CASCI) used for each benchmark entry to allow immediate assessment of the correlation recovered.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments, which have helped us improve the clarity and rigor of the manuscript. We have revised the results section to include direct error metrics and comparisons where feasible, and we have substantially expanded the description of the numerical procedure in §3. Our point-by-point responses follow.

read point-by-point responses

Referee: [Results section] Results section (benchmark tables and figures): The central claim of 'good accuracy' and 'quantitative accuracy for all-electron correlation beyond configuration interaction' for strongly correlated transition-metal systems is not supported by direct error metrics or side-by-side comparisons to full configuration interaction (FCI) or density-matrix renormalization group (DMRG) reference values on the same basis sets. Without these, it is impossible to distinguish recovery of the reference plus modest correlation from true quantitative accuracy.

Authors: We agree that explicit error metrics and side-by-side comparisons strengthen the claims. For the smaller main-group systems, we have added FCI reference values (computed in the same basis sets) together with mean absolute errors and maximum deviations for the ACSE energies. For the transition-metal systems, where FCI is intractable, we now report comparisons to DMRG results from the literature performed on identical basis sets and active spaces, including quantitative error statistics. These additions demonstrate that the ACSE recovers correlation energy beyond the reference wavefunction in both regimes. The revised tables and figures include these metrics explicitly. revision: yes
Referee: [§3] §3 (numerical solution of the ACSE): The description of the iterative solution procedure and any truncation or convergence thresholds is insufficient to determine under what conditions the method recovers the exact energy versus an approximation; this directly affects the validity of the accuracy claims for strongly correlated cases.

Authors: We acknowledge that the original description was too brief. We have rewritten §3 to provide a complete account of the iterative solver: the reconstruction of the 3-RDM from the 2-RDM, the specific truncation applied to the 3-RDM (retaining only connected diagrams up to a user-specified rank), the convergence threshold (energy change < 10^{-8} E_h and 2-RDM change < 10^{-6}), and the exact conditions under which the ACSE solution becomes exact (when the 3-RDM reconstruction is exact). All benchmark calculations now state the precise thresholds employed, allowing readers to assess when the reported energies are exact versus approximate. revision: yes

Circularity Check

0 steps flagged

No circularity: ACSE implementation uses exact Hamiltonian with independent benchmarks

full rationale

The paper presents an open-source implementation of the anti-Hermitian contracted Schrödinger equation (ACSE) for ground and excited states. The core method employs the exact electronic Hamiltonian rather than an approximate perturbative one, and the scaling is stated to be independent of reference wavefunction complexity. Benchmark results are numerical comparisons to external data across systems and basis sets; no equations, parameters, or accuracy claims reduce by construction to fitted inputs from the same dataset or to self-citations that bear the load of the central result. The derivation chain is self-contained against external benchmarks, with no self-definitional loops, renamed known results, or ansatz smuggling identified.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the prior derivation of the ACSE from the electronic Schrödinger equation and on the assumption that the numerical solver converges to the exact correlation energy for the tested systems.

axioms (1)

domain assumption The anti-Hermitian contracted Schrödinger equation, when solved numerically, recovers all-electron correlation beyond configuration interaction for molecular systems.
This is the foundational premise of the ACSE approach invoked throughout the abstract.

pith-pipeline@v0.9.0 · 5494 in / 1142 out tokens · 51082 ms · 2026-05-13T20:23:11.855053+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Valdemoro Here we show the explicit expansion of the first 3-RDM term in Eq. 5 using the Valdemoro reconstruction, ∑ pqr Kkp rq 3Di j p rql = 1 9 ∑ pqr Kkp rq (Mi j rq 1D p l −M i j lq 1D p r −M i j rl 1D p q −M ip rq 1D j l +M ip lq 1D j r +M ip rl 1D j q +M j p rq 1D i l −M j p lq 1D i r −M j p rl 1D i q). (D1) This is simplified by recognizing that som...

work page
[2]

ˆAis the an- tisymmetry operator which here permutes all indices excepta

Nakatsuji-Yasuda The NY approximation to the 3-cumulant is, 3∆i j p rql = 1 6 ∑ a σa ˆA(2∆ia rq 2∆ j p al ) (D5) whereσ a is 1 ifais an occupied orbital and -1 if it is an unoccupied orbital in the Hartree-Fock reference. ˆAis the an- tisymmetry operator which here permutes all indices excepta. 13 Explicitly, the resulting equation is, 3∆i j p rql = 1 6 ∑...

work page

[1] [1]

Valdemoro Here we show the explicit expansion of the first 3-RDM term in Eq. 5 using the Valdemoro reconstruction, ∑ pqr Kkp rq 3Di j p rql = 1 9 ∑ pqr Kkp rq (Mi j rq 1D p l −M i j lq 1D p r −M i j rl 1D p q −M ip rq 1D j l +M ip lq 1D j r +M ip rl 1D j q +M j p rq 1D i l −M j p lq 1D i r −M j p rl 1D i q). (D1) This is simplified by recognizing that som...

work page

[2] [2]

ˆAis the an- tisymmetry operator which here permutes all indices excepta

Nakatsuji-Yasuda The NY approximation to the 3-cumulant is, 3∆i j p rql = 1 6 ∑ a σa ˆA(2∆ia rq 2∆ j p al ) (D5) whereσ a is 1 ifais an occupied orbital and -1 if it is an unoccupied orbital in the Hartree-Fock reference. ˆAis the an- tisymmetry operator which here permutes all indices excepta. 13 Explicitly, the resulting equation is, 3∆i j p rql = 1 6 ∑...

work page