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Excited State Densities from Time-Dependent Density Functional Response Theory

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Pith's one-line read This paper derives a real-space formula for excited-state densities from TDDFT response, including, for the first time, densities of states with double-excitation character.

desk verdict A sound and useful derivation of real-space TDDFT excited-state densities for non-adiabatic kernels, with clean 1D demonstrations; the main soft spots are presentational and scope, not load-bearing. read the letter →

arxiv 2506.05082 v2 pith:R7ZCFRC4 submitted 2025-06-05 physics.chem-ph

Anna Baranova , Neepa T. Maitra This is my paper
classification physics.chem-ph
keywords excited-statedensitiesTDDFTlinearresponsedoubleexcitationsnon-adiabatickernelssmall-matrixapproximationdressedcharge-transferexchange-correlation
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

The paper derives a real-space formula for excited-state densities from time-dependent density functional theory (TDDFT) response: the density difference of an excited state is the functional derivative of the TDDFT excitation energy with respect to the external potential, evaluated at the response eigenvalue. The derivation goes beyond the adiabatic approximation, so it covers frequency-dependent exchange-correlation kernels and, for the first time, yields densities of states with double-excitation character. The exact expressions reduce to practical approximations, and numerical tests on one-dimensional two-electron models show that the dressed TDDFT approximations give good double-excitation densities while adiabatic approximations fail. The paper also studies local and charge-transfer excitations, finding that a good Kohn-Sham ground state is essential and that exact-exchange (EXX) performs better than LDA in the tested cases.

What carries the argument

The central object is the TDDFT response matrix $\Omega_{qq'}(\omega)$ in the single-excitation basis, whose eigenvalues are squared excitation frequencies. The excited-state density is obtained from the functional derivative $\delta\Omega/\delta v_{\rm ext}$, which contains orbital-relaxation terms, a static inverse $(1 - f_{HXC}\chi_S)^{-1}$ that resums the density response, and the functional derivative $\tilde g_{XC}$ of the frequency-dependent kernel. For double excitations, the machinery switches to the dressed-kernel expressions DSMA/DSPA, where the excitation energy of a three-state subspace (ground, single, double) is differentiated with respect to the external potential.

What would settle it

Compute the double-excitation density for a model in which a single double state couples significantly to two different single excitations, and compare DSPA/DSMA densities with the exact density: if the density error grows while the energy stays accurate, the three-state truncation premise fails.

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Extended reading notes

Core claim

The central claim is that the density of any excited state obtainable from TDDFT linear response is fixed by the response matrix through $\Delta n_I(\mathbf r) = \delta \omega_I / \delta v_{\rm ext}(\mathbf r)$, and that this object can be evaluated directly in real space once the exchange-correlation kernel and its functional derivatives are known. Equations (15)-(16) give this density difference for general, possibly frequency-dependent kernels, and therefore include states of double-excitation character that are inaccessible to the adiabatic approximation. In the dressed small-matrix (DSMA) and dressed single-pole (DSPA) approximations, the resulting densities of double-excitation states agree closely with exact densities in the model systems, while the adiabatic SMA/SPA densities fail when the state is not single-excitation dominated.

Load-bearing premise

The exact claims assume the exchange-correlation kernel and its functional derivatives are known; the practical double-excitation densities assume the dressed three-state model, with one single excitation coupled to one double excitation, is a faithful description of the state.

Editorial extensions

If this is right

  • The exact real-space formula provides a direct route to excited-state densities without solving Z-vector equations, and it remains valid when the kernel depends on frequency.
  • For single-excitation-dominated states, the small-matrix approximation gives densities that improve on the bare Kohn-Sham density, with corrections that sum over all orbitals.
  • For double-excitation states, adiabatic TDDFT cannot produce the density at all; DSPA/DSMA supply it and reproduce exact densities in the tested one-dimensional models.
  • Accurate Kohn-Sham orbitals and gaps are a prerequisite: LDA's underestimated gap produces divergent small-matrix densities for charge transfer in the soft double well, while EXX orbitals do not.
  • The same energy-derivative idea can be applied to other response methods such as pp-RPA and BSE, and to non-equilibrium dynamics through response-reformulated TDDFT.

Reading between the lines

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

  • If the exact kernel were available, the real-space formulas would give exact excited-state densities for any state, including states beyond double excitations; the practical bottleneck is the missing general frequency-dependent kernel and its density derivatives.
  • The DSMA/DSPA density inherits the three-state truncation, so systems with strong coupling of one double state to several single excitations are the natural stress test; the density may be less robust than the energy even when the energy is accurate.
  • Because the density difference is expressed as a sum over all Kohn-Sham orbitals, the method suggests a diagnostic for when a single-transition picture is sufficient, and could be combined with orbital downfolding or embedding.
  • The model results point to hybrid functionals with a large fraction of exact exchange as the practical target for molecules, with excited-state dipole and quadrupole moments the next quantities to benchmark.
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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

0 major / 7 minor

Summary. The manuscript derives a real-space expression for the excited-state density difference, Δn_I(r) = δω_I/δv_ext(r), starting from the TDDFT linear-response eigenvalue equation. Equations (11)–(16) give the functional derivative of the response matrix including frequency-dependent (non-adiabatic) kernels, and the authors then specialize to the small-matrix approximation (SMA) and its single-transition limit (STL). They implement the expressions for one-dimensional two-electron models (a helium-like system, two double-well charge-transfer models, and a harmonic-plus-barrier model with double excitations), using LDA and EXX kernels and, for double-excitation states, the dressed DSMA and DSPA approximations. The central formal claim is that Eq. (15) is exact in principle once the exact kernel and its functional derivatives are known, and that the dressed TDDFT approximations provide accurate densities for states of double-excitation character, while adiabatic approximations fail. The numerical results also show the importance of an accurate ground-state Kohn-Sham potential, with EXX outperforming LDA, and identify a divergence in the first-order Taylor expansion of (1 − fHXCχS)^{-1} for the LDA charge-transfer case.

Significance. If correct, the formal result extends the Furche-Ahlrichs density-matrix route to non-adiabatic kernels and provides a TDDFT-based real-space density for double-excitation states, which was previously unavailable. The derivation is internally consistent: the SI proves that the STL limit reproduces the Furche-Ahlrichs result within the adiabatic approximation, and the DSMA/DSPA densities are parameter-free predictions in the sense that all Hamiltonian matrix elements are computed from Kohn-Sham orbitals rather than fitted. The numerical demonstrations are honest about their scope: they are restricted to one-dimensional two-electron models, and the conclusions explicitly call for real-molecule benchmarking. The main caveat is that the practical double-excitation route relies on a truncated three-state dressed kernel, so the resulting densities inherit the truncation error of that model; this is a scope limitation rather than an inconsistency in the formal derivation.

minor comments (7)
  1. [Section 2.2, after Eq. (33)] The sentence 'In the limit cos θ = 1, the state indexed by + reduces to a pure double excitation while that indexed by − reduces to a pure single excitation' is inconsistent with Eq. (33), which gives G²₊ = 1/2(1+cosθ) and G²₋ = 1/2(1−cosθ). For cosθ = 1, Eq. (33) implies G²₊ = 1 (pure single) and G²₋ = 0 (pure double). Please correct the assignment of the + and − labels in this limit.
  2. [Eq. (25)] The denominator ν_q + 4fHXC,qq in the second term should be explicitly identified as the static kernel fHXC,qq(ω=0), since frequency-dependent fHXC,qq(ω_I) appears elsewhere in the same equation; as written, the notation is ambiguous.
  3. [SI, Eq. (43)] The factor multiplying the orbital-relaxation terms is written as (1 − fHXCχS)(x, r), but it should be (1 − fHXCχS)^{-1}(x, r), matching Eq. (35) of the main text.
  4. [Section 2, text following Eq. (15)] The sentence 'with the normalization Eq. 12 the derivative of Eq. 11 because the derivative on the right hand side of the resulting equation Eq. 15 no longer includes the variation of the solution point ωI under vext' is grammatically incomplete and should be rewritten. A one-line algebraic demonstration of how the prefactor in Eq. (13) cancels when using Eq. (14) would improve clarity.
  5. [Figs. 3(c) and 4(c)] The label 'STL a ro x' in the figure captions appears to be a typo for 'STL approx'; please correct it.
  6. [Fig. 6 caption] The word 'exciatiotion' should be 'excitation'.
  7. [Eq. (31)] Please ensure the prefactors in Eq. (31) are typeset unambiguously as (1/2)(1±cosθ) and (1/2)(1∓cosθ), rather than 1/[2(1±cosθ)] and 1/[2(1∓cosθ)], to avoid confusion with the coefficients appearing in Eq. (38).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the excited-state densities are functional derivatives of the TDDFT energy tested against exact wavefunction benchmarks; the dressed-kernel self-citations are externally supported tools, not load-bearing reductions.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. The central real-space formula (Eqs. 15-16) follows by differentiating the Casida eigenvalue equation: after imposing the normalization condition of Eq. 12, the ∂Ω/∂ω² contribution cancels identically, leaving Δn_I(r) = (1/2ω_I) G_I† δΩ(ω)/δv|_ω G_I, and Eq. 16 is obtained from the static chain-rule identity χ_S^{-1}χ = (1 − f_HXC χ_S)^{-1}; this is internal algebra, not an input recycled as an output. The excited-state densities are genuine parameter-free predictions: the KS orbitals (exact-inverted, EXX, or LDA) and the DSMA/DSPA Hamiltonian matrix elements (H_qD, H_DD, H_00 from Slater-Condon rules) are computed from the model systems, while the target densities in Figs. 1-7 are exact wavefunction densities from Octopus, so every comparison is against an external benchmark. For the double-excitation states, the DSMA/DSPA densities are functional derivatives of the dressed energies of Refs. 31 and 33, with no parameter fitted to the exact densities compared in Figs. 5-7; the paper's own observation that an accurate dressed frequency does not guarantee an accurate density (Fig. 5(e), γ = 1) shows the density is not forced by construction. The dressed-kernel references overlap with the authors but carry independent weight: they were assessed externally on 28 organic chromophores (Ref. 42) and against high-level data for trans-butadiene (Ref. 34), and no uniqueness theorem is invoked to exclude alternatives. The honest limitations — the three-state truncation of the dressed model, the one-dimensional test systems, and the divergence of the first-order Taylor expansion of (1 − f_HXC χ_S)^{-1} for the LDA charge-transfer case (Sec. 3.2) — are scope conditions about the approximations, not circular steps.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The ledger is light: the paper introduces no free parameters fitted to the exact densities or energies; every input is fixed by the model Hamiltonians, standard functionals (LDA, EXX), or Hamiltonian integrals computed from KS orbitals. The load-bearing assumptions are the standard response-theory identities (Dyson equation, perturbation theory) and two approximations specific to this work: the 3-state truncation of the dressed kernel for double-excitation densities, and the first-order Taylor expansion of the static inverse operator in the SMA numerics, which the paper itself shows diverges in one of its test cases. Domain assumptions about exact KS inversion for the 1D models are stated in Sec. 3.

assumptions (6)
  • standard math The TDDFT linear-response eigenvalue problem (Eq. 3) and the Casida normalization condition (Eq. 12) are valid for the systems studied.
    The whole derivation of Eq. 13 uses these as starting points. The systems are spin-saturated, two-electron 1D models where the Casida matrix form applies. Invoked in Sec. 2, Eqs. 3-13.
  • standard math The static response functions satisfy the Dyson equation chi = chi_S + chi_S f_HXC chi, so chi^{-1}_S chi = (1 - f_HXC chi_S)^{-1}.
    Used throughout the derivative chain-rule and in the final real-space formulas. See SI Sec. 1, Eqs. (5)-(8).
  • standard math First-order non-degenerate perturbation theory describes the response of KS orbitals and orbital energies to the KS potential (Eq. 13 of SI).
    Used to evaluate delta nu_q / delta v_S and delta phi / delta v_S, the key ingredients of Eq. 16. Assumes a non-degenerate KS spectrum; the model systems are chosen accordingly.
  • ad hoc to paper The truncated 3-state Hamiltonian subspace (ground, single q, double D) used in DSMA/DSPA contains all configurations that materially contribute to the energy and density of the states studied.
    Eqs. 28-31 and the model construction in Sec. 3.3 rely on the single excitation q being well separated and only one double D coupling. This truncation is asserted, not derived, and limits the double-excitation density claim to such cases.
  • ad hoc to paper The first-order Taylor expansion of (1 - f_HXC chi_S)^{-1} and of chi (Eqs. 26-27) is accurate enough for the SMA densities.
    The paper itself shows this expansion diverges for the LDA charge-transfer case (Sec. 3.2, Fig. 3c), so the SMA numerical results rest on this approximation, which is uncontrolled in general.
  • domain assumption Exact ground-state KS potentials and orbitals are obtainable by inverting the exact ground-state density of the 1D models.
    Sec. 3 states the exact KS potential and orbitals are obtained from inverting the ground-state KS equation; this presumes the 1D density-to-potential map is computationally tractable and the inversion is unique.

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Pith. "Pith review of Excited State Densities from Time-Dependent Density Functional Response Theory." pith.science (2026). https://pith.science/paper/R7ZCFRC4

@misc{pith2026250605082,
  author       = {Pith},
  title        = {Pith review of: Excited State Densities from Time-Dependent Density Functional Response Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R7ZCFRC4}},
  note         = {Machine review of arXiv:2506.05082}
}
read the original abstract

While the variational principle for excited-state energies leads to a route to obtaining excited-state densities from time-dependent density functional theory, relatively little attention has been paid to the quality of the resulting densities in real space obtained with different exchange-correlation functional approximations, nor how non-adiabatic approximations developed for energies of states of double excitation character perform for their densities. Here we derive an expression directly in real space for the excited-state density, that includes the case of non-adiabatic kernels, and consequently is able, for the first time, to yield densities of states of double-excitation character. Under some well-defined simplifications, we compare the performance of the local-density approximation and exact-exchange approximation, which are in a sense at opposite extremes of the fundamental functional approximations, on local and charge-transfer excitations in one-dimensional model systems, and show that the dressed TDDFT approach gives good densities of double-excitations.

Figures

Figures reproduced from arXiv: 2506.05082 by the authors.

Figure 1
Figure 1. Excited-state density differences for 1D Helium: TDD [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Excited-state density differences for 1D Helium: com [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Excited-state density differences for the CT excita [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Excited-state density differences for the CT excita [PITH_FULL_IMAGE:figures/full_fig_p023_4.png]
Figure 5
Figure 5. Figure 5: Excited state densities of the Harmγ model with the set of exact orbitals and EXX kernel: Panel (a) First excitation with γ = 0, showing close agreement of exact, adiabatic SMA and SPA densities; Panels (b) and (c), Second and third excitations showing close agreement …
Figure 6
Figure 6. Figure 6: Excited state densities of the Harmγ model with the set of exact orbitals and LDA kernel: Panel (a) First excitation with γ = 0, where ASPA outperforms ASMA but still exhibiting error at the origin; Panels (b) and (c), Second and third excitations showing closer agreem…
Figure 7
Figure 7. Figure 7: Excited-state density differences for the second an [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

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