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REVIEW 3 major objections 5 minor 61 references

Redundant parameter dependencies in truncated classic and quantum Linear Response and Equation of Motion theory

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Truncated linear-response and equation-of-motion excitation energies depend on wave-function parameters that are redundant for the ground-state energy, and the paper proposes a constrained trace-optimization of the response Hessian to…

desk verdict A clean, credible demonstration that truncated LR/EOM excitation energies depend on redundant orbital and cluster parameters, with a remedy that is plausible but only lightly validated. read the letter →

arxiv 2506.06063 v1 pith:VGW7SI44 submitted 2025-06-06 physics.chem-ph quant-ph

Erik Rosendahl Kjellgren , Peter Reinholdt , Karl Michael Ziems , Stephan P. A. Sauer , Sonia Coriani , Jacob Kongsted This is my paper
classification physics.chem-phquant-ph
keywords linearresponsetheoryequationofmotionredundantparametersexcitationenergiesunitarycoupledclusterorbitalrotationtraceoptimizationquantum
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes that truncated linear response (LR) and equation-of-motion (EOM) excitation energies are not uniquely defined: they change when the wave function is varied along directions that leave the ground-state energy unchanged. The dependence is shown for a helium atom in a 6-31G basis with four different parametrizations of the excitation operators, and persists for lithium hydride in STO-3G when the response is truncated to singles or singles and doubles, even when the ground state is the full configuration interaction solution. The authors trace the mechanism to the fact that the sum of truncated excitation energies equals the trace of the response matrix on a subspace, which is not invariant under the unitary transformations generated by the redundant parameters, unlike the full-space trace. As a remedy, they minimize this subspace trace subject to keeping the ground-state energy fixed, implemented as a penalty-function optimization, and show it improves the low-lying spectra. They further show that restricting the trace optimization to a few target states (constrained state-averaged UCC) can improve small LR expansions.

What carries the argument

The load-bearing object is the subspace trace of the response Hessian, $\mathrm{tr}\{[A]_{SS}\} = \sum_{I\in SS} \langle I| U^\dagger H U |I\rangle - E_0$, which for state-transfer and self-consistent (eigenfunction) parametrizations equals the sum of the truncated excitation energies. Under redundant orbital rotations and cluster amplitudes, the full trace is conserved by unitarity but the subspace trace is not, and this is what makes truncated spectra ambiguous. The proposed remedy is a constrained optimization (Eqs. 48-49) that minimizes this subspace trace subject to the ground-state energy staying fixed, implemented with a large penalty parameter $K$; restricting the subspace to a few target states yields the constrained state-averaged UCC (CSA-UCC) variant used to improve small LR expansions.

What would settle it

A computation showing that a truncated LR excitation energy of an FCI ground state is invariant under a redundant orbital rotation that preserves the ground-state energy to machine precision would falsify the central claim of parameter dependence.

Watch

Extended reading notes

Core claim

The central claim is that for a unitary-parametrized wave function (UCC or a quantum circuit), any truncation of the LR/EOM excitation space makes the excitation energies functions of parameters that are redundant for the ground-state energy. For the state-transfer parametrization, and for the self-consistent parametrization when the wave function is an eigenfunction of the Hamiltonian, the response matrix reduces to a unitary transformation of the Hamiltonian shifted by the ground-state energy. Because a unitary transformation conserves the total trace but not the trace over a truncated subspace, the sum of the truncated excitation energies, $\sum_i \varepsilon_i = \mathrm{tr}\{[H(U_\kappa,U_\theta)]_{SS}\}$, varies with redundant orbital rotations $\kappa$ and cluster amplitudes $\theta$. The authors prove that individual excitation energies must therefore depend on these redundant parameters, and they demonstrate the effect numerically: for He/6-31G the single-excitation energy sweeps over a wide range as one redundant orbital rotation is varied, and for LiH/STO-3G the difference between trace-maximized and trace-minimized parameters grows with wave-function flexibility, reaching tens of Hartrees in the subspace trace for oo-UCCSDTQ.

Load-bearing premise

The remedy rests on the hope that minimizing the subspace trace of the response matrix, subject only to keeping the ground-state energy fixed, actually improves the truncated excitation spectrum; this proxy leaves the ground-state wave function itself unconstrained and is only tested on two small molecules.

Editorial extensions

If this is right

  • Truncated LR/EOM spectra are not well-defined quantities until the redundant parameters are fixed or optimized; different but equally valid ground-state parameter sets yield different excitation energies.
  • Because quantum circuit ansätze are unitary, the same redundant-parameter dependence applies to qLR/qEOM implementations on quantum computers.
  • The constrained trace optimization of the Hessian, with the ground-state energy held fixed, provides a practical way to remove the ambiguity.
  • Targeting only a few states in the trace optimization (constrained state-averaged UCC) can improve the accuracy of small LR expansions, reducing the number of excitation operators needed.
  • Orbital localization or other uses of redundant rotations should be combined with truncated LR only with caution, since the resulting spectra depend on the chosen orbitals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The subspace-trace proxy assumes that lowering the sum of truncated excitation energies (or a selected-state sum) improves the individual states that matter; for general systems the correlation between trace minimization and per-state accuracy is not guaranteed and would need testing on larger molecules.
  • Because the constraint fixes only the ground-state energy, the optimization can move along near-null directions that leave the energy essentially unchanged but distort the response; the paper's numerical evidence is limited to two small systems, so the robustness of the remedy for realistic molecules remains open.
  • The subspace-trace viewpoint suggests a diagnostic: one could estimate the sensitivity of any truncated LR spectrum by computing the variance of the subspace trace over redundant parameter directions, without running full optimizations.
  • The same redundant-parameter ambiguity should affect other response properties (transition moments, polarizabilities) computed from truncated LR, not just excitation energies, though the paper does not demonstrate this.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper shows that truncated linear response (LR) / equation-of-motion (EOM) excitation energies depend on orbital rotation and UCC cluster parameters that are redundant with respect to the ground-state energy. For the state-transfer (st) and self-consistent (sc*) parametrizations, the authors derive a trace identity: the sum of truncated excitation energies equals the subspace trace of a unitarily transformed Hamiltonian, which is not conserved under redundant unitary rotations. They demonstrate this dependence numerically for He/6-31G (with naive, projected, self-consistent, and state-transfer parametrizations) and for LiH/STO-3G with st-LRSD and st-LRS. As a remedy, they propose a ground-state-constrained trace optimization of the Hessian (Eq. 49), including a state-averaged variant (CSA-UCC), and report improved spectra for LiH/STO-3G.

Significance. If the central claim holds, it is an important and non-obvious result: truncated LR/EOM spectra are not well-defined until redundant parameters are fixed, with direct consequences for orbital-optimized and quantum-computing formulations (oo-qLR, qEOM, sc-qEOM). The trace identity for the st/sc* parametrizations is derived cleanly and is a useful diagnostic. The paper also ships no code or data, but the analytical argument is self-contained. The proposed trace-optimization remedy is, however, only a heuristic proxy, validated on two small systems, and the numerical evidence contains some ambiguities; this weakens the otherwise credible contribution.

major comments (3)
  1. [Section III and Section IV.B] The description of the post-optimization energy minimization is methodologically unclear and potentially load-bearing. After the penalty-function optimization with K=10^12, the text states that 'an energy minimization was performed with respect to the ground state, to ensure that the ground-state wave function was correct.' If this energy minimization moves along flat directions of the ground-state energy surface (which are precisely the redundant directions identified in Section II.C), the final parameters may no longer be the trace-optimized ones, and the traces in Table I and the spectra in Figs. 3-5 might not correspond to the reported θ_opt/κ_opt. The paper does not report whether the trace is preserved after this step, nor does it justify why this additional minimization is needed given that the penalty already enforces the energy to 1e-8 Hartree. The authors should either remove this step, demonstrate that it does not change the trace, or explain how the final wave function is unambiguously defined.
  2. [Section II.D, Eq. (49), and Section IV.B] The claim that trace optimization 'alleviates' the redundant-parameter problem rests on a heuristic objective that is not theoretically justified. Minimizing the subspace trace of the transformed Hamiltonian is equivalent to minimizing the sum of the truncated excitation energies for the st/sc* parametrizations, but there is no argument that this moves individual excitation energies toward the FCI values. The numerical support is limited to LiH/STO-3G, and for oo-UCCSD the reported difference between bad and opt traces is only 2.27e-3 Hartree (Table I), so the improvement is negligible in that case. The abstract's statement that the problem 'can be alleviated' is therefore only empirically demonstrated for two small molecules and a specific proxy; the paper should be more cautious in its wording or provide additional evidence that the trace objective correlates with spectral accuracy beyond these examples.
  3. [Table I] Table I reports the subspace Hamiltonian and Hessian traces to only two decimal places, which makes it impossible for the reader to verify the quantitative differences quoted in the text. For oo-UCCSD, the text states that the difference between the bad-parameter and opt-parameter traces is 2.27e-3 Hartree, but the table lists both values as -274.60 and 113.93. For oo-UCCSDT and oo-UCCSDTQ the differences (0.71 and 28.25 Hartree) are visible, but the oo-UCCSD result is not. The table should include enough significant digits to substantiate all quoted differences.
minor comments (5)
  1. [Section II.C, Eq. (41)] The notation (θ_red, κ_red) is used in Eq. (41) but the subscript 'red' is not explicitly defined; it should be stated that these denote redundant parameters as defined in Eqs. (14)-(15).
  2. [Section II.C, Eqs. (14)-(15)] The notation min_{θ,κ\κ_pq} is somewhat awkward; consider using a clearer formulation such as minimization over all parameters except κ_pq, with κ_pq free.
  3. [Figures 1 and 2] The y-axis labels '1 [Hartree]' and 'LRS_1 + LRD_1 [Hartree]' are informal; using ε_1 and ε_1^S + ε_1^D (or a textual description) would improve readability.
  4. [Section II.B, Eq. (35)] The symbol 'sc*' is introduced in Eq. (35) without an explicit definition of the asterisk; the text should state that sc* refers to the self-consistent parametrization evaluated for a wave function that is an eigenfunction of the Hamiltonian.
  5. [References] Reference [61] (Grimsley and Evangelista) is a duplicate of Reference [28]; this should be merged or cross-referenced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the redundant-parameter dependence is derived from the paper's own trace identities, and the trace-optimization remedy is a variational proposal rather than a fitted prediction.

full rationale

The central derivation is self-contained. For the state-transfer (and sc*) parametrization, the paper obtains A^st_IJ = <HF|G_I† U† H U G_J|HF> - delta_IJ E0 (Eqs. 29-31 and 42), so the response matrix is literally a submatrix of a unitarily transformed Hamiltonian. Eq. 44 uses the trace invariance of the full unitary transform; Eq. 45 shows that the subspace trace is not invariant; Eq. 46 identifies the subspace trace of the Hessian with the sum of the truncated excitation energies. The conclusion that the excitation energies depend on redundant parameters therefore follows mathematically from the paper's own working equations, not from any fitted input. The numerical He/6-31G and LiH/STO-3G results are external demonstrations that could in principle conflict with the analytic argument, and they do. The proposed remedy in Eqs. 48-49 is a variational optimization of a subspace trace subject to a ground-state energy constraint; it does not fit any excitation-energy datum, and although the objective is the sum of certain excitation energies, the reported spectral improvements are individual peak positions and intensities that are not equal to the minimized objective by construction. Self-citations, notably Refs. 31 and 36, provide parametrization definitions, context, and an explanation of metric singularities, but the equations needed for the main claim are stated in the paper itself and do not reduce to a same-author citation chain. The main caveats identified by the skeptic -- limited validation of the trace-optimization remedy to He and LiH, and the energy-only constraint not uniquely selecting a truncated ground-state branch -- are correctness and generalizability concerns, not circularity. No qualifying circular step is present.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation is almost self-contained: it uses standard trace identities and the explicit forms of the state-transfer and self-consistent response matrices. No physical constants are fitted, and no new particles or forces are introduced. The only hand-chosen numerical parameter is the penalty factor K; the CSA-UCC target-state selection is a modeling choice rather than a fitted parameter.

free parameters (1)
  • Penalty factor K = 1e12
    Chosen by hand in Eq. 49 to enforce the ground-state energy constraint in the constrained trace optimization; not fitted to data but a method parameter. The reported results could depend on this choice.
assumptions (4)
  • domain assumption For a unitary coupled cluster wavefunction, linear response and equation of motion formalisms are identical (Ref. 41).
    Used to transfer all conclusions from LR to EOM in Section II B.
  • domain assumption The state-transfer and self-consistent (in the eigenfunction limit) parametrizations reduce the response matrix to A_IJ = <HF| G_I^dagger U^dagger H U G_J |HF> - delta_IJ E0 with B=0 and Sigma=identity (Eqs. 29-37 and 42).
    Load-bearing for the trace argument and for the trace optimization objective.
  • standard math A unitary transformation conserves the trace of the full Hamiltonian but not necessarily the trace of a sub-block (Eqs. 44-45).
    Basis for the conclusion that subspace traces, and therefore truncated sums of excitation energies, depend on redundant parameters.
  • ad hoc to paper The constrained optimization in Eq. 49 finds global or near-global minima (supported by 500 random restarts) and the penalty factor K=10^12 enforces the ground-state energy to within 1e-8 Hartree.
    Numerical reliability assumption; if the optimization misses the global solution, the reported traces and spectra in Tables I-II and Figures 3-5 could differ.

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Cite this review

Pith. "Pith review of Redundant parameter dependencies in truncated classic and quantum Linear Response and Equation of Motion theory." pith.science (2026). https://pith.science/paper/VGW7SI44

@misc{pith2026250606063,
  author       = {Pith},
  title        = {Pith review of: Redundant parameter dependencies in truncated classic and quantum Linear Response and Equation of Motion theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VGW7SI44}},
  note         = {Machine review of arXiv:2506.06063}
}
read the original abstract

Extracting molecular properties from a wave function can be done through the linear response (LR) formalism or, equivalently, the equation of motion (EOM) formalism. For a simple model system, He in a 6-31G basis, it is here shown that calculated excitation energies depend on the specifically chosen orbitals, even when the ground-state is the FCI solution, if the LR is truncated to a singles expansion. This holds for naive, projected, self-consistent, and state-transfer parametrizations of the LR operators. With a focus on the state-transfer parameterization, this problem is shown to also hold for more complicated systems, and is also present when the LR is truncated to singles and doubles. This problem can be alleviated by performing a ground-state constrained trace optimization of the Hessian matrix before performing the LR calculation. It is finally shown that spectra can be further improved for small LR expansions by targeting only a few states in the constrained trace optimization using constrained state-averaged UCC.

Figures

Figures reproduced from arXiv: 2506.06063 by the authors.

Figure 1
Figure 1. FIG. 1. Helium. Lowest excitation energy ( [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The singlet single excitation energy and singlet double excitation energy of the He atom [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Calculated electronic spectra of LiH in the STO-3G basis using different levels of theory. [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. LiH/STO-3G. Spectra calculated with state-transfer LR singles (st-LRS) using various [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. LiH/STO-3G. Spectra calculated with state-transfer LR singles (st-LRS) using various [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

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