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arxiv: 2606.05148 · v1 · pith:LEBSZK2Nnew · submitted 2026-06-03 · ⚛️ physics.chem-ph · cond-mat.str-el· quant-ph

Variational low-energy subspaces for chemically accurate excited states

Clemens Giuliani , Rocco Martinazzo , Giuseppe Carleo , Riccardo Rossi This is my paper

Pith reviewed 2026-06-28 03:22 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.str-elquant-ph
keywords excited statesvariational methodsquantum chemistrySlater determinantschemical accuracynon-orthogonal determinantspotential energy surfaces
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The pith

Reformulating excited-state optimization as iterated ground-state problems on low-energy subspaces enables automatic, constraint-free calculation of multiple chemically accurate states.

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

The paper shows that variational optimization of excited states can be recast as repeated ground-state-like problems acting on a low-energy subspace of the electronic Hamiltonian. When this principle is applied to non-orthogonal Slater determinants, the resulting EXIDOS method simultaneously optimizes several excited states while taking only the number of states and determinants per state as input. No explicit orthogonality constraints or imposed spin or point-group symmetries are required. Benchmarks on N2, CO, HCl, NH3, C2 and ethylene recover chemical accuracy for charge-transfer, Rydberg, double-excitation, avoided-crossing and conical-intersection states when compared with full configuration interaction and established quantum-chemistry methods.

Core claim

Variational excited-state optimization reduces to an iterated ground-state-like problem for a low-energy subspace of the electronic Hamiltonian; applying the principle to non-orthogonal Slater determinants produces the automatic EXIDOS method that delivers chemically accurate excited states for a range of molecular systems without symmetry or orthogonality constraints.

What carries the argument

The low-energy subspace of the electronic Hamiltonian, which converts excited-state variational search into repeated ground-state optimizations.

If this is right

  • Multiple excited states of different characters are optimized in a single run without prior knowledge of their symmetry or orbital character.
  • Chemical accuracy is obtained for charge-transfer states, Rydberg states, double excitations, extended potential-energy curves, avoided crossings and conical intersections.
  • The approach remains fully variational and requires only the number of target states and determinants per state as external parameters.
  • The method scales more favorably than traditional high-level excited-state techniques while avoiding manual state selection.

Where Pith is reading between the lines

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

  • The automatic character of the procedure may allow routine mapping of entire potential-energy surfaces in photochemical applications where state character changes with geometry.
  • Because no orthogonality is enforced by hand, the same framework could be combined with other variational ansatzes that naturally produce non-orthogonal states.
  • The subspace construction may generalize to larger active spaces or embedding schemes once the determinant optimization step is replaced by a more scalable solver.

Load-bearing premise

The reformulation of excited-state optimization as iterated ground-state problems on low-energy subspaces remains valid when the trial functions are non-orthogonal Slater determinants that carry no explicit orthogonality or symmetry constraints.

What would settle it

A full-configuration-interaction benchmark on any of the tested molecules in which EXIDOS energies for one or more states deviate by more than chemical accuracy (approximately 1 kcal/mol) when run with the stated number of states and determinants and without manual symmetry input.

Figures

Figures reproduced from arXiv: 2606.05148 by Clemens Giuliani, Giuseppe Carleo, Riccardo Rossi, Rocco Martinazzo.

Figure 1
Figure 1. Figure 1: Automatic optimization of the low-energy subspace. (a-b) EXIDOS(256,32) calculation of LiH spectrum, oscillator strength f and dipole moment µ, compared with FCI. (c-d) All-electron EXIDOS(24, 256) and EXIDOS(33, 256) results for the N2 and CO molecules, respectively, are compared with the benchmark results from Ref. [23, 24]. We note that panel (c-d) includes additional low-lying triplet states absent fro… view at source ↗
Figure 2
Figure 2. Figure 2: (a) EXIDOS potential energy curve of the lowest states of C2. We show an all-electron EXIDOS (24,256) calculation in the aug-cc-pVDZ basis. (b) Symmetry retrieval from operator measurements. We show the expectation value of the projector PˆΓ onto each irreducible representation Γ of a term symbol as function of the state index at equilibrium geometry for C2 (R = 1.248), computed using the techniques descri… view at source ↗
Figure 3
Figure 3. Figure 3: Excited states with different electronic character. (a) Charge-transfer excitation in HCl. A frozen-core EXIDOS(5,256) calculation reproduces the excitation energy, dipole moments µ and oscillator strength f of the 1Σ + → 1Π transition, compared with CCSDTQ [27]. Our calculation also includes the lowest 3Π state absent from the benchmark. The density difference plot illustrates the charge redistribution fo… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Bond torsion of the ethylene molecule starting from the planar geometry. (b) Bond pyramidization of the ethylene molecule starting from the twisted-orthogonal geometry. Results are obtained from an all-electron EXIDOS simulation of K = 10 states (16 for a few geometries) with ND = 256 determinants in the aug-cc-pVDZ basis, and the geometries are from Ref. [29]. We compare with MR-CISD in the same basis… view at source ↗
read the original abstract

Accurate electronic excited states are essential for photochemistry, spectroscopy and non-adiabatic molecular dynamics, but high-level calculations often scale steeply and require prior knowledge of the target state's character or symmetry. Here we show that variational excited-state optimization can be reformulated as an iterated ground-state-like problem for a low-energy subspace of the electronic Hamiltonian. Applying this variational principle to non-orthogonal Slater determinants leads to EXIDOS, an automatic method for excited state calculations controlled only by the number of states and determinants per state. EXIDOS optimizes multiple excited states simultaneously, without explicit orthogonality constraints or imposed spin and point-group symmetries. Benchmarks against FCI and state-of-the-art quantum chemistry methods show chemical accuracy for a multitude of states in N$_2$ and CO, charge-transfer states in HCl, Rydberg states in NH$_3$, double excitations and extended potential-energy curves in C$_2$, and avoided crossings and conical intersections in ethylene. These results establish EXIDOS as a low-scaling, fully variational route to chemically accurate 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.

Referee Report

2 major / 2 minor

Summary. The manuscript presents EXIDOS, a method reformulating variational excited-state optimization as an iterated ground-state-like problem over a low-energy subspace of the electronic Hamiltonian. Applied to non-orthogonal Slater determinants without explicit orthogonality, spin, or point-group symmetry constraints, it claims to achieve chemical accuracy for multiple states in N2 and CO, charge-transfer states in HCl, Rydberg states in NH3, double excitations and potential-energy curves in C2, and avoided crossings/conical intersections in ethylene, as shown in benchmarks against FCI and other quantum chemistry methods.

Significance. If the reformulation is variationally sound, this constitutes a notable advance as a fully variational, low-scaling route to chemically accurate excited states that requires only the number of states and determinants per state as input, eliminating the need for prior state character or symmetry knowledge. The breadth of benchmark systems (including challenging cases like double excitations and conical intersections) would support broad applicability in photochemistry and non-adiabatic dynamics if the underlying principle holds without artifacts from the optimization procedure.

major comments (2)
  1. [Abstract and variational reformulation section] The central reformulation (abstract and the section deriving the subspace variational principle): no explicit derivation or proof is supplied showing that the low-energy subspace energy functional remains bounded, unique, and automatically enforces state separation when trial functions are non-orthogonal Slater determinants without imposed orthogonality constraints. This is load-bearing for the claim that chemical accuracy on ethylene conical intersections and C2 double excitations follows from the variational principle rather than procedure-specific details.
  2. [Benchmark results section] Benchmark section (results on N2, C2, ethylene): the reported chemical accuracy is asserted without accompanying discussion of convergence criteria, implementation of the iterated subspace optimization, or analysis of possible failure modes (e.g., mixing or collapse) under the relaxed symmetry conditions. This undermines assessment of whether the results generalize beyond the tested cases.
minor comments (2)
  1. [Abstract] The acronym EXIDOS is used in the abstract without expansion or definition; introduce it with its full meaning on first use.
  2. [References] Ensure all compared state-of-the-art methods (e.g., those benchmarked against FCI) have complete and up-to-date citations in the reference list.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its potential significance. We respond to the two major comments below and will incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [Abstract and variational reformulation section] The central reformulation (abstract and the section deriving the subspace variational principle): no explicit derivation or proof is supplied showing that the low-energy subspace energy functional remains bounded, unique, and automatically enforces state separation when trial functions are non-orthogonal Slater determinants without imposed orthogonality constraints. This is load-bearing for the claim that chemical accuracy on ethylene conical intersections and C2 double excitations follows from the variational principle rather than procedure-specific details.

    Authors: We agree that the manuscript presents the subspace variational principle at a conceptual level without a self-contained mathematical derivation or proof of boundedness, uniqueness, and automatic state separation for non-orthogonal Slater determinants. This omission weakens the link between the variational principle and the reported accuracies. In the revised manuscript we will add an explicit derivation (as a new subsection or appendix) that proves the low-energy subspace functional is bounded from below by the sum of the lowest eigenvalues, is unique under the stated conditions, and enforces separation without explicit orthogonality constraints. This addition will demonstrate that the chemical accuracy on the cited systems follows directly from the variational principle. revision: yes

  2. Referee: [Benchmark results section] Benchmark section (results on N2, C2, ethylene): the reported chemical accuracy is asserted without accompanying discussion of convergence criteria, implementation of the iterated subspace optimization, or analysis of possible failure modes (e.g., mixing or collapse) under the relaxed symmetry conditions. This undermines assessment of whether the results generalize beyond the tested cases.

    Authors: We concur that the benchmark section lacks sufficient detail on convergence criteria, the precise implementation of the iterated subspace optimization, and analysis of failure modes such as mixing or collapse when symmetry constraints are relaxed. These omissions limit evaluation of robustness and generalizability. In the revised manuscript we will expand the benchmark section to include: explicit convergence thresholds and stopping criteria; a step-by-step description of the iterated optimization procedure; and a dedicated analysis of potential failure modes, supported by additional numerical diagnostics on the N2, C2, and ethylene systems showing that mixing or collapse did not occur. revision: yes

Circularity Check

0 steps flagged

No significant circularity; reformulation is presented as independent variational principle

full rationale

The paper introduces a reformulation of excited-state optimization as an iterated ground-state problem on a low-energy subspace, then applies it to non-orthogonal Slater determinants to define EXIDOS. The abstract and description frame this as a new variational principle controlled only by number of states and determinants, with external benchmarks against FCI and other methods. No equations or steps are shown that reduce the claimed accuracy or the subspace minimum to fitted inputs, self-definitions, or self-citation chains by construction. The central claim rests on the validity of the reformulation itself rather than on any renaming or imported uniqueness theorem from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the variational principle extends directly to low-energy subspaces for excited states when using non-orthogonal determinants. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Variational excited-state optimization can be reformulated as an iterated ground-state-like problem for a low-energy subspace of the electronic Hamiltonian.
    This is the key reformulation stated in the abstract that enables the EXIDOS method.

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Reference graph

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    Numerical stability To ensure numerical stability we normalize all states|Ψa⟩, and for the calculation of the matrix elements we also normalize the orbitals of the individual determinants|Φik⟩and use a sufficiently large Krylov space to render the Lanczos solver numerically stable. Appendix D: Dipole moment and oscillator strentgh We compute the dipole mo...