Pith. sign in

REVIEW 2 major objections 6 minor 89 references

Linear-Response Quantum-Electrodynamical Density Functional Theory Based on Two-Component X2C Hamiltonians

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A single X2C decoupling matrix, fixed at the ground state, suffices for relativistic linear-response QEDFT spectra and cuts cost more than fivefold.

desk verdict A solid extension of X2C to linear-response QEDFT: the fixed-U argument is standard and safe, and the two applications are useful, but the evidence would be stronger with quantitative benchmarks, timings, and code/data. read the letter →

arxiv 2507.07198 v1 pith:27ZQ4LRF submitted 2025-07-09 physics.chem-ph

Lukas Konecny , Valeriia P. Kosheleva , Michael Ruggenthaler , Michal Repisky , Angel Rubio This is my paper
classification physics.chem-ph
keywords exacttwo-component(X2C)transformationquantum-electrodynamicaldensityfunctionaltheorylinearresponseQEDFTrelativisticmethodspolaritonicchemistrycavityQEDatomicmean-fieldX2C(amfX2C)collectivestrongcoupling
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

Linear-response QEDFT describes molecular spectra under strong coupling to cavity photons, and its relativistic four-component form is accurate but expensive for heavy-element systems. This paper argues that the exact two-component (X2C) transformation collapses that cost: under the dipole and weak-field approximations, the X2C decoupling matrix only has to be computed once at the ground-state self-consistent-field stage, after which the entire linear-response calculation runs in a two-component regime with the same matrix. The argument rests on an order-of-magnitude estimate that the neglected off-diagonal electron-photon couplings scale as $\mathcal{O}(g_\alpha \lambda l c^{-2})$ or $\mathcal{O}(\lambda l c^{-2})$, whereas the diagonal blocks scale as $c^2$. In benchmarks, two-component spectra match the four-component references while being more than fivefold faster, and the method is applied to two-dimensional spectra of mercury porphyrin and collective coupling in a chain of AuH molecules.

What carries the argument

The carrying object is the X2C decoupling matrix $\mathbf{U}$, a unitary matrix parameterized by a single matrix $\mathbf{R}$ as a product of a decoupling and a renormalization block, that block-diagonalizes the four-component Dirac-Coulomb Fock matrix. In an orthonormal kinetically balanced basis $\mathbf{R}$ scales as $c^{-1}$, and this scaling, together with dipole-matrix scaling $\tilde{\mathbf{P}}^{2c}_{\alpha,\mu\nu} \sim l$ and photon displacement $q_\alpha(t) \sim \lambda$, feeds the estimates in Eqs. (33)-(37) that make the time- and field-dependence of $\mathbf{U}$ negligible. The machinery then produces the two-component linear-response eigenvalue equation (62), whose structure mirrors four-component QEDFT and can be solved with the same iterative Davidson-type solver.

What would settle it

Compute the same cavity spectrum with the full four-component Hamiltonian and with the two-component method for a molecule whose excited states are near-degenerate and strongly coupled to the mode (for example the mercury porphyrin B/N/L region at $g_\alpha=0.02$ au), and check whether the differences in the polaritonic splittings are larger than the target accuracy; if the neglected couplings change an avoided crossing by a noticeable amount, the core claim is false.

Watch

Extended reading notes

Core claim

The paper's central claim is that a single X2C decoupling matrix $\mathbf{U}$, fixed by the ground-state SCF solution, can be applied throughout a relativistic linear-response QEDFT calculation without re-deriving the transformation at each frequency or field strength. The transformed electron-photon Fock matrix and current are then built entirely from picture-changed two-component quantities, and the coupled electron-photon system is solved as the generalized eigenvalue equation (62) with electronic response blocks $\mathbf{A}^{2c}$, $\mathbf{B}^{2c}$, self-energy blocks $\boldsymbol{\Delta}^{2c}$, $\boldsymbol{\Delta}'^{2c}$, and photon coupling blocks $\boldsymbol{\Gamma}^{2c}$. The neglected off-diagonal blocks of the untransformed Fock matrix are estimated to be of order $\mathcal{O}(g_\alpha \lambda l c^{-2})$ for the electron-photon coupling and $\mathcal{O}(\lambda l c^{-2})$ for the external-field coupling, against diagonal blocks of order $c^2$. Using the atomic mean-field X2C Hamiltonian, the calculated spectra for mercury porphyrin reproduce reference four-component QEDFT spectra in the free molecule and in cavities, and the AuH chain calculation shows collective strong coupling locally modifying the impurity molecule's properties.

Load-bearing premise

The load-bearing premise is that the neglected off-diagonal couplings are tiny compared with the large diagonal terms of the Dirac Hamiltonian because they scale as the perturbation strength times inverse powers of the speed of light; this is an order-of-magnitude estimate rather than a proven bound, and it could fail for nearly degenerate levels where small couplings still cause avoided crossings.

Editorial extensions

If this is right

  • Relativistic polaritonic spectra can be computed in a two-component basis of half the dimension, more than fivefold faster than four-component QEDFT, making parameter scans such as two-dimensional cavity-frequency spectra practical.
  • A single X2C decoupling matrix from the ground-state SCF is reused for all photon frequencies, external-field perturbations, and coupling strengths in the linear-response regime, so no per-mode block diagonalization is needed.
  • The amfX2C-based QEDFT spectra match the reference four-component spectra for mercury porphyrin, including off-resonant coupling signatures and avoided-crossing polaritonic branches.
  • Explicit multi-molecule ensembles of heavy-element systems become feasible, as shown by the AuH chain, where collective strong coupling locally alters the chemical properties of the perturbed molecule.
  • The two-component eigenvalue equation keeps the same block structure as the four-component one, so existing linear-response solvers and self-energy treatment transfer directly to the relativistic two-component setting.

Reading between the lines

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

  • Editorial inference: the scaling argument justifies dropping off-diagonal couplings against the $c^2$ diagonal, but not against spectral gaps; for near-degenerate polaritonic levels the neglected terms could still shift avoided crossings, so a systematic 2c-versus-4c scan across dense spectral regions would be a sharper test.
  • Editorial inference: the same once-only decoupling logic should extend to real-time QEDFT and to higher-order response under weak driving, but the perturbation estimate would need to be re-derived order by order rather than assumed.
  • Editorial inference: using mmfX2C instead of amfX2C in the same framework would separate errors from the atomic picture-change approximation from errors of the decoupling itself, quantifying how much accuracy the cheaper atomic mean-field models cost.
  • Editorial inference: with explicit heavy-element ensembles now affordable, the framework could be pointed at spin-orbit-driven phenomena such as singlet-triplet transitions modified by collective coupling, which nonrelativistic treatments cannot capture.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. The manuscript extends linear-response quantum-electrodynamical density functional theory (QEDFT) to two-component exact two-component (X2C) Hamiltonians. It derives how the four-component electron-photon Hamiltonian is transformed by the X2C decoupling matrix and argues that under the weak-field and dipole approximations a single decoupling matrix obtained from the ground-state SCF procedure can be used throughout the subsequent linear-response steps. The resulting two-component linear-response eigenvalue equation has the same structure as the four-component QEDFT equation, with electron-photon coupling blocks expressed in terms of picture-changed dipole integrals. The implementation covers amfX2C, eamfX2C, and mmfX2C Hamiltonian models, although the presented benchmarks use amfX2C. Applications include 2D spectra of a mercury porphyrin complex in a Fabry-Pérot cavity and collective coupling effects in a chain of AuH molecules with a bond-length-distorted impurity.

Significance. If the fixed-U approximation is accurate, this is a valuable methods contribution: it brings relativistic polaritonic spectra of large, heavy-element systems into reach at substantially reduced cost compared with four-component QEDFT. The derivation is algebraically explicit, the working equation is presented in a form ready for implementation, and the benchmark against independent four-component QEDFT reference results for mercury porphyrin is a genuine external check. The explicit treatment of collective coupling in an AuH chain is a relevant demonstration of the method's intended use. The main limitation is that the central approximation is justified by an order-of-magnitude scaling argument and the benchmark evidence is largely visual rather than quantitative.

major comments (2)
  1. [II B 2, Eqs. (33)-(37)] The central approximation of the paper is the statement that a single ground-state decoupling matrix U can be used throughout the linear-response QEDFT calculation. The justification in Eqs. (33)-(37) is an order-of-magnitude estimate in which the off-diagonal blocks are compared with the c^2 diagonal blocks of the Dirac operator. For linear-response spectra, however, the relevant energy scale is the excitation-energy differences, and near-degenerate polaritonic branches can be sensitive to small couplings. I therefore recommend adding a quantitative test of the fixed-U approximation, for example a small system in which the calculation is repeated with the off-diagonal blocks retained or with a time-dependent U, and reporting the resulting peak shifts. This would elevate the central approximation from a plausible estimate to a demonstrated one.
  2. [IV B] The AuH chain is used to make a physical claim about collective strong coupling locally modifying chemical properties, but no reference calculation is provided for this system or for any collective-coupling setup. Given that this is the first demonstration of the method in the collective regime, I ask for at least one quantitative validation, such as a smaller chain where four-component QEDFT is feasible or a comparison with the nonrelativistic limit, so that the accuracy in this regime is not simply asserted.
minor comments (6)
  1. [IV A, Fig. 1(c)] The statement that amfX2C 'precisely reproduce[s]' the four-component spectra is based on visual overlap; please provide a numerical comparison of peak positions and intensities.
  2. [IV A] The claimed 'more than fivefold speed-up' is not supported by timing data; please specify hardware, wall-clock times, and the basis and SCF settings used for the comparison.
  3. [III and IV] The implementation is described as including amfX2C, eamfX2C, and mmfX2C, but all presented calculations use amfX2C; please clarify whether the other two models were validated and, if so, where.
  4. [Data Availability] The data availability statement says data are available 'upon reasonable request'; for a methods paper, depositing input files and a reproducible workflow would substantially strengthen the presentation.
  5. [Throughout] There are several typos and grammatical slips, for example 'T wo' in the title, 'accessibke' in Section IV A, and a duplicated 'the' in the Fig. 2 caption.
  6. [Eqs. (56) and (57)] In Eq. (56) the response kernel is written as A^{2c}_{ai,bj}(ω), but the definition in Eq. (57) has no frequency dependence; please make the notation consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fixed-X2C-matrix linear-response derivation is self-contained and benchmarked against independent four-component calculations.

full rationale

The paper's central claim is that a single ground-state X2C decoupling matrix U suffices for the subsequent linear-response QEDFT calculation. This is supported by an in-paper scaling analysis in Eqs. (30)-(37), where the relevant order estimates (R ~ c^{-1}, ~C_S^- ~ c^{-1}, ~P ~ l, q ~ lambda) are stated and applied directly rather than imported solely from a self-citation. The derivation of the two-component eigenvalue equation (62) proceeds algebraically from the four-component equations of motion by transforming every term with the same U; no parameter is fitted to the benchmark quantities, and no output quantity is defined in terms of an input in a way that would make the comparison circular. The claim that amfX2C spectra 'precisely reproduce' the reference results is validated by comparison with four-component QEDFT calculations from Ref. 56, which are external reference calculations rather than fitted inputs. Self-citations to Refs. 46, 54, 55, and 68 identify the amfX2C model and prior X2C response implementations, but these are model choices and prior framework context, not load-bearing replacements for a derivation step: the fixed-U reasoning is reproduced in the present text, and the amfX2C picture-change corrections are not tuned to the molecules studied here. The order-of-magnitude estimates underlying the neglect of off-diagonal blocks are heuristic and may be a correctness risk for near-degenerate polaritonic levels, but that is an evidence-quality concern, not circularity. No circular step meeting the required standard of a quoted equation-level reduction was found.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central claim relies on standard X2C domain assumptions, the weak-field and dipole regime, and the amfX2C atomic picture-change corrections borrowed from prior work; no genuinely new entities are introduced. The phenomenological coupling strength and broadening are honest free parameters of the applications. The order-of-magnitude estimates in Eq. (37) are the weakest axiomatic link because they substitute for a rigorous error bound.

free parameters (4)
  • coupling strength gα = 0.01, 0.02 (porphyrin); 0.0033, 0.005, 0.01 (AuH)
    Phenomenological cavity coupling parameters chosen by hand for the applications, not derived from first principles. They control the Rabi splittings and are part of the central results but are standard inputs in cavity QED calculations.
  • broadening parameter γ = 0.027 eV
    Empirical broadening used to convert line spectra into band spectra in the porphyrin 2D scans, stated explicitly as a convenience.
  • cavity mode frequency scan range = 3 to 4.35 eV
    The frequency range for the 2D spectra is chosen by hand to cover the B, N, and L bands; this is a presentation choice, not a fitted constant, but it does shape the 2D figures.
  • bond length distortion of impurity = 1.56057 Å for AuH vs equilibrium 1.52385 Å
    Taken from the same percentage distortion as in Ref. 76, so an existing model choice rather than fitted here.
assumptions (7)
  • domain assumption X2C decoupling is unitary and the R matrix scales as c^{-1} in an orthonormal RKB basis
    Used in the central scaling argument of Sec. II B 2 (Eqs. 26-27, 33). This is standard X2C background from Refs. 35-37, not proved in this paper.
  • domain assumption Dipole and long-wavelength approximations for the light-matter interaction
    Invoked at the start of Sec. II A through the Hamiltonian in Eq. (1) and used in the estimate ω l c^{-1} << 1 in Sec. II B 2.
  • domain assumption Weak-field approximation λ < 1 with the external field E(t) = λ E0 e cos(ωt)
    Used to justify the first-order Taylor expansion of U and the O(λ) estimates in Eqs. (30)-(37).
  • domain assumption Adiabatic approximation for the xc kernel and neglect of the photon-electron xc kernel (pRPA)
    Stated in Sec. II B 3 after Eq. (57): the kernel is made ω-independent and the photon random phase approximation is assumed. These are standard TDDFT/QEDFT approximations.
  • domain assumption Non-collinear parametrization of xc potentials and kernels according to Scalmani-Frisch and Cherry et al.
    Used in the construction of the xc contributions in Eqs. (19) and (41); stated in Sec. II A.
  • domain assumption The 4c QEDFT formulation and the Hamiltonian of Eq. (1) are correct
    The derivation starts from the 4c QEDFT of Ref. 56 and assumes its validity, with a step-by-step derivation of the Hamiltonian deferred to that reference.
  • domain assumption amfX2C and eamfX2C picture-change corrections can be constructed from atomic contributions
    The central approximation of Eq. (47), taken from Ref. 46; its accuracy for molecules in cavities is assumed rather than re-derived here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Linear-Response Quantum-Electrodynamical Density Functional Theory Based on Two-Component X2C Hamiltonians." pith.science (2026). https://pith.science/paper/27ZQ4LRF

@misc{pith2026250707198,
  author       = {Pith},
  title        = {Pith review of: Linear-Response Quantum-Electrodynamical Density Functional Theory Based on Two-Component X2C Hamiltonians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27ZQ4LRF}},
  note         = {Machine review of arXiv:2507.07198}
}
read the original abstract

Linear-response quantum electrodynamical density functional theory (QEDFT) enables the description of molecular spectra under strong coupling to quantized photonic modes, such as those in optical cavities. Recently, this approach was extended to the relativistic domain using the four-component Dirac-Coulomb Hamiltonian. To provide a computationally efficient yet accurate alternative-particularly for modeling 2D spectra or collective coupling for large, heavy-element systems-this article introduces a two-component linear-response QEDFT method based on exact two-component (X2C) Hamiltonian models. We derive how the parent four-component Hamiltonian for coupled electron-photon systems undergoes the X2C transformation. Moreover, we show that, under common weak-field and dipole approximations, it suffices to apply the X2C transformation only during the ground-state self-consistent field procedure, with the subsequent calculations performed fully in the two-component regime using the same X2C decoupling matrix. The current implementation includes the atomic mean-field (amfX2C), extended atomic mean-field (eamfX2C), and molecular mean-field (mmfX2C) Hamiltonian models. Benchmark calculations demonstrate that the X2C approach closely reproduces reference four-component results, enabling us to efficiently tackle systems that would be otherwise computationally too expensive. As applications, we compute 2D spectra of a mercury porphyrin complex in a Fabry-Perot cavity, demonstrating off-resonant coupling and the appearance of multiple polaritonic branches. We also study a chain of AuH molecules, showing that collective coupling can locally modify chemical properties of a molecule with a perturbed bond length.

Figures

Figures reproduced from arXiv: 2507.07198 by the authors.

Figure 3
Figure 3. FIG. 3: Calculated (amfX2C) transition densities for different states in a chain of AuH molecules in a cavity with coupling [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4: Calculated (amfX2C) transition density on the [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

89 extracted references · 68 canonical work pages

  1. [1]

    author author F. J. \ Garcia-Vidal , author C. Ciuti ,\ and\ author T. W. \ Ebbesen ,\ title title Manipulating matter by strong coupling to vacuum fields , \ @noop journal journal Science \ volume 373 ,\ pages eabd0336 ( year 2021 ) NoStop

    2021

  2. [2]

    author author J. A. \ Hutchison , author T. Schwartz , author C. Genet , author E. Devaux ,\ and\ author T. W. \ Ebbesen ,\ title title Modifying chemical landscapes by coupling to vacuum fields , \ @noop journal journal Angew. Chem. Int. Ed. \ volume 51 ,\ pages 1592--1596 ( year 2012 ) NoStop

    2012

  3. [3]

    Sandik , author J

    author author G. Sandik , author J. Feist , author F. J. \ Garc \' a-Vidal ,\ and\ author T. Schwartz ,\ title title Cavity-enhanced energy transport in molecular systems , \ @noop journal journal Nat. Mater. \ ,\ pages 1--12 ( year 2024 ) NoStop

    2024

  4. [4]

    Ruggenthaler , author D

    author author M. Ruggenthaler , author D. Sidler ,\ and\ author A. Rubio ,\ title title Understanding polaritonic chemistry from ab initio quantum electrodynamics , \ @noop journal journal Chem. Rev. \ volume 123 ,\ pages 11191--11229 ( year 2023 ) NoStop

    2023

  5. [5]

    Foley , Jonathan J

    author author I. Foley , Jonathan J. , author J. F. \ McTague ,\ and\ author I. DePrince , A. Eugene ,\ title title Ab initio methods for polariton chemistry , \ https://doi.org/10.1063/5.0167243 journal journal Chem. Phys. Rev. \ volume 4 ,\ pages 041301 ( year 2023 ) NoStop

    doi:10.1063/5.0167243

  6. [6]

    Ruggenthaler , author F

    author author M. Ruggenthaler , author F. Mackenroth ,\ and\ author D. Bauer ,\ title title Time-dependent kohn-sham approach to quantum electrodynamics , \ @noop journal journal Phys. Rev. A \ volume 84 ,\ pages 042107 ( year 2011 ) NoStop

    2011

  7. [7]

    Ruggenthaler , author J

    author author M. Ruggenthaler , author J. Flick , author C. Pellegrini , author H. Appel , author I. V. \ Tokatly ,\ and\ author A. Rubio ,\ title title Quantum-electrodynamical density-functional theory: Bridging quantum optics and electronic-structure theory , \ @noop journal journal Phys. Rev. A \ volume 90 ,\ pages 012508 ( year 2014 ) NoStop

    2014

  8. [8]

    author author I. V. \ Tokatly ,\ title title Time-dependent density functional theory for many-electron systems interacting with cavity photons , \ @noop journal journal Phys. Rev. Lett. \ volume 110 ,\ pages 233001 ( year 2013 ) NoStop

    2013

Show all 89 references
  1. [9]

    Ruggenthaler ,\ title title Ground-state quantum-electrodynamical density-functional theory , \ @noop journal journal arXiv:1509.01417 \ ( year 2015 ) NoStop

    author author M. Ruggenthaler ,\ title title Ground-state quantum-electrodynamical density-functional theory , \ @noop journal journal arXiv:1509.01417 \ ( year 2015 ) NoStop

    2015 arXiv

  2. [10]

    Penz , author E

    author author M. Penz , author E. I. \ Tellgren , author M. A. \ Csirik , author M. Ruggenthaler ,\ and\ author A. Laestadius ,\ title title The Structure of the Density-Potential Mapping. Part II: Including Magnetic Fields , \ https://doi.org/10.1021/acsphyschemau.3c00006 jou...

    doi

  3. [11]

    Flick , author D

    author author J. Flick , author D. M. \ Welakuh , author M. Ruggenthaler , author H. Appel ,\ and\ author A. Rubio ,\ title title Light--matter response in nonrelativistic quantum electrodynamics , \ @noop journal journal ACS Photonics \ volume 6 ,\ pages 2757--2778 ( year 201...

    2019

  4. [12]

    Yang , author Q

    author author J. Yang , author Q. Ou , author Z. Pei , author H. Wang , author B. Weng , author Z. Shuai , author K. Mullen ,\ and\ author Y. Shao ,\ title title Quantum-electrodynamical time-dependent density functional theory within Gaussian atomic basis , \ @noop journal jo...

    2021

  5. [13]

    Buchholz , author I

    author author F. Buchholz , author I. Theophilou , author S. E. \ Nielsen , author M. Ruggenthaler ,\ and\ author A. Rubio ,\ title title Reduced density-matrix approach to strong matter-photon interaction , \ @noop journal journal ACS Photonics \ volume 6 ,\ pages 2694--2711 ...

    2019

  6. [14]

    author author T. S. \ Haugland , author E. Ronca , author E. F. \ Kj nstad , author A. Rubio ,\ and\ author H. Koch ,\ title title Coupled cluster theory for molecular polaritons: Changing ground and excited states , \ @noop journal journal Phys. Rev. X \ volume 10 ,\ pages 04...

    2020

  7. [15]

    Buchholz , author I

    author author F. Buchholz , author I. Theophilou , author K. J. H. \ Giesbertz , author M. Ruggenthaler ,\ and\ author A. Rubio ,\ title title Light–Matter Hybrid-Orbital-Based First-Principles Methods: The Influence of Polariton Statistics , \ https://doi.org/10.1021/acs.jctc...

    doi

  8. [16]

    McTague \ and\ author J

    author author J. McTague \ and\ author J. J. \ Foley ,\ title title Non-Hermitian cavity quantum electrodynamics--configuration interaction singles approach for polaritonic structure with ab initio molecular Hamiltonians , \ @noop journal journal J. Chem. Phys. \ volume 156 ( ...

    2022

  9. [17]

    Mordovina , author C

    author author U. Mordovina , author C. Bungey , author H. Appel , author P. J. \ Knowles , author A. Rubio ,\ and\ author F. R. \ Manby ,\ title title Polaritonic coupled-cluster theory , \ https://doi.org/10.1103/PhysRevResearch.2.023262 journal journal Phys. Rev. Res. \ volu...

    doi

  10. [18]

    author author A. E. \ DePrince ,\ title title Cavity-modulated ionization potentials and electron affinities from quantum electrodynamics coupled-cluster theory , \ @noop journal journal J. Chem. Phys. \ volume 154 ( year 2021 ) NoStop

    2021

  11. [19]

    Monzel \ and\ author S

    author author L. Monzel \ and\ author S. Stopkowicz ,\ title title Diagrams in Polaritonic Coupled Cluster Theory , \ @noop journal journal J. Phys. Chem. A \ volume 128 ,\ pages 9572--9586 ( year 2024 ) NoStop

    2024

  12. [20]

    K \'e na-Cohen \ and\ author S

    author author S. K \'e na-Cohen \ and\ author S. Forrest ,\ title title Green polariton photoluminescence using the red-emitting phosphor PtOEP , \ @noop journal journal Phys. Rev. B \ volume 76 ,\ pages 075202 ( year 2007 ) NoStop

    2007

  13. [21]

    Stranius , author M

    author author K. Stranius , author M. Hertzog ,\ and\ author K. B \"o rjesson ,\ title title Selective manipulation of electronically excited states through strong light--matter interactions , \ @noop journal journal Nat. Commun. \ volume 9 ,\ pages 2273 ( year 2018 ) NoStop

    2018

  14. [22]

    Eizner , author L

    author author E. Eizner , author L. A. \ Mart \' nez-Mart \' nez , author J. Yuen-Zhou ,\ and\ author S. K \'e na-Cohen ,\ title title Inverting singlet and triplet excited states using strong light-matter coupling , \ @noop journal journal Sci. Adv. \ volume 5 ,\ pages eaax44...

    2019

  15. [23]

    Hertzog , author M

    author author M. Hertzog , author M. Wang , author J. Mony ,\ and\ author K. B \"o rjesson ,\ title title Strong light--matter interactions: a new direction within chemistry , \ @noop journal journal Chem. Soc. Rev. \ volume 48 ,\ pages 937--961 ( year 2019 ) NoStop

    2019

  16. [24]

    Xu , author W

    author author X. Xu , author W. Yao , author D. Xiao ,\ and\ author T. F. \ Heinz ,\ title title Spin and pseudospins in layered transition metal dichalcogenides , \ @noop journal journal Nat. Phys. \ volume 10 ,\ pages 343--350 ( year 2014 ) NoStop

    2014

  17. [25]

    \ Yang , author R

    author author S.-H. \ Yang , author R. Naaman , author Y. Paltiel ,\ and\ author S. S. \ Parkin ,\ title title Chiral spintronics , \ @noop journal journal Nat. Rev. Phys. \ volume 3 ,\ pages 328--343 ( year 2021 b ) NoStop

    2021

  18. [26]

    Spohn ,\ @noop title Dynamics of charged particles and their radiation field \ ( publisher Cambridge university press ,\ year 2004 ) NoStop

    author author H. Spohn ,\ @noop title Dynamics of charged particles and their radiation field \ ( publisher Cambridge university press ,\ year 2004 ) NoStop

    2004

  19. [27]

    Hiroshima ,\ @noop title Ground states of quantum field models: perturbation of embedded eigenvalues ,\ Vol

    author author F. Hiroshima ,\ @noop title Ground states of quantum field models: perturbation of embedded eigenvalues ,\ Vol. volume 35 \ ( publisher Springer ,\ year 2019 ) NoStop

    2019

  20. [28]

    author author L. H. \ Ryder ,\ @noop title Quantum field theory \ ( publisher Cambridge university press ,\ year 1996 ) NoStop

    1996

  21. [29]

    Mandl \ and\ author G

    author author F. Mandl \ and\ author G. Shaw ,\ @noop title Quantum field theory \ ( publisher John Wiley & Sons ,\ year 2010 ) NoStop

    2010

  22. [30]

    Takaesu ,\ title title On the spectral analysis of quantum electrodynamics with spatial cutoffs

    author author T. Takaesu ,\ title title On the spectral analysis of quantum electrodynamics with spatial cutoffs. I. \ @noop journal journal J. Mat. Phys. \ volume 50 ( year 2009 ) NoStop

    2009

  23. [31]

    author author J. C. \ Baez , author I. E. \ Segal ,\ and\ author Z. Zhou ,\ @noop title Introduction to algebraic and constructive quantum field theory ,\ Vol. volume 47 \ ( publisher Princeton University Press ,\ year 2014 ) NoStop

    2014

  24. [32]

    Saue ,\ title title Does chemistry need more physics? \ @noop journal journal arXiv preprint arXiv:2504.19003 \ ( year 2025 ) NoStop

    author author T. Saue ,\ title title Does chemistry need more physics? \ @noop journal journal arXiv preprint arXiv:2504.19003 \ ( year 2025 ) NoStop

    2025 arXiv

  25. [33]

    author author J. L. \ Heully , author I. Lindgren , author E. Lindroth , author S. Lundqvist ,\ and\ author A. M. \ Martensson-Pendrill ,\ title title Diagonalisation of the Dirac Hamiltonian as a basis for a relativistic many-body procedure , \ @noop journal journal J. Phys. ...

    1986

  26. [34]

    Douglas--Kroll the Easy Way

    author author H. J. A. \ Jensen ,\ @noop note "Douglas--Kroll the Easy Way", The Conference Talk, REHE 2005, M\"ulheim (Germany) NoStop

    2005

  27. [35]

    Kutzelnigg \ and\ author W

    author author W. Kutzelnigg \ and\ author W. Liu ,\ title title Quasirelativistic theory equivalent to fully relativistic theory , \ https://doi.org/10.1063/1.2137315 journal journal J. Chem. Phys. \ volume 123 ,\ pages 241102 ( year 2005 ) NoStop

    doi

  28. [36]

    Ilias \ and\ author T

    author author M. Ilias \ and\ author T. Saue ,\ title title An infinite-order two-component relativistic hamiltonian by a simple one-step transformation , \ https://doi.org/10.1063/1.2436882 journal journal J. Chem. Phys. \ volume 126 ,\ pages 064102 ( year 2007 ) NoStop

    doi

  29. [37]

    Liu \ and\ author W

    author author W. Liu \ and\ author W. Kutzelnigg ,\ title title Quasirelativistic theory. II . Theory at matrix level , \ https://doi.org/10.1063/1.2710258 journal journal J. Chem. Phys. \ volume 126 ,\ pages 114107 ( year 2007 ) NoStop

    doi

  30. [38]

    Sikkema , author L

    author author J. Sikkema , author L. Visscher , author T. Saue ,\ and\ author M. Ilia s ,\ title title The molecular mean-field approach for correlated relativistic calculations , \ https://doi.org/10.1063/1.3239505 journal journal J. Chem. Phys. \ volume 131 ,\ pages 124116 (...

    doi

  31. [39]

    Peng , author W

    author author D. Peng , author W. Liu , author Y. Xiao ,\ and\ author L. Cheng ,\ title title Making Four- and Two-Component Relativistic Density Functional Methods Fully Equivalent Based on the Idea of from Atoms to Molecule , \ @noop journal journal J. Chem. Phys. \ volume 1...

    2007

  32. [40]

    Liu \ and\ author D

    author author W. Liu \ and\ author D. Peng ,\ title title Exact two-component Hamiltonians revisited , \ https://doi.org/10.1063/1.3159445 journal journal J. Chem. Phys. \ volume 131 ,\ pages 031104 ( year 2009 ) NoStop

    2009 doi

  33. [41]

    Peng , author N

    author author D. Peng , author N. Middendorf , author F. Weigend ,\ and\ author M. Reiher ,\ title title An efficient implementation of two-component relativistic exact-decoupling methods for large molecules , \ @noop journal journal J. Chem. Phys. \ volume 138 ,\ pages 184105...

    2013

  34. [42]

    Filatov , author W

    author author M. Filatov , author W. Zou ,\ and\ author D. Cremer ,\ title title Spin-orbit coupling calculations with the two-component normalized elimination of the small component method , \ @noop journal journal J. Chem. Phys. \ volume 139 ,\ pages 014106 ( year 2013 ) NoStop

    2013

  35. [43]

    Konecny , author M

    author author L. Konecny , author M. Kadek , author S. Komorovsky , author O. L. \ Malkina , author K. Ruud ,\ and\ author M. Repisky ,\ title title Acceleration of Relativistic Electron Dynamics by Means of X2C Transformation: Application to the Calculation of Nonlinear Optic...

    2016

  36. [44]

    author author J. J. \ Goings , author J. M. \ Kasper , author F. Egidi , author S. Sun ,\ and\ author X. Li ,\ title title Real time propagation of the exact two component time-dependent density functional theory , \ @noop journal journal J. Chem. Phys. \ volume 145 ,\ pages 1...

    2016

  37. [45]

    Liu \ and\ author L

    author author J. Liu \ and\ author L. Cheng ,\ title title An atomic mean-field spin-orbit approach within exact two-component theory for a non-perturbative treatment of spin-orbit coupling , \ https://doi.org/10.1063/1.5023750 journal journal J. Chem. Phys. \ volume 148 ,\ pa...

    doi

  38. [46]

    Knecht , author M

    author author S. Knecht , author M. Repisky , author H. J. A. \ Jensen ,\ and\ author T. Saue ,\ title title Exact two-component hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple , \ @noop journal journal J. Chem. Phys. \ volu...

    2022

  39. [47]

    Zhang \ and\ author L

    author author C. Zhang \ and\ author L. Cheng ,\ title title Atomic mean-field approach within exact two-component theory based on the dirac--coulomb--breit hamiltonian , \ @noop journal journal J. Phys. Chem. A \ volume 126 ,\ pages 4537--4553 ( year 2022 ) NoStop

    2022

  40. [48]

    Ehrman , author E

    author author J. Ehrman , author E. Martinez-Baez , author A. J. \ Jenkins ,\ and\ author X. Li ,\ title title Improving one-electron exact-two-component relativistic methods with the Dirac--Coulomb--Breit-parameterized effective spin--orbit coupling , \ @noop journal journal ...

    2023

  41. [49]

    Blume \ and\ author R

    author author M. Blume \ and\ author R. E. \ Watson ,\ title title Theory of spin-orbit coupling in atoms i. derivation of the spin-orbit coupling constant , \ https://doi.org/10.1098/rspa.1962.0207 journal journal Proc. R. Soc. Lond. A \ volume 270 ,\ pages 127--143 ( year 19...

    1962

  42. [50]

    Blume \ and\ author R

    author author M. Blume \ and\ author R. E. \ Watson ,\ title title Theory of spin-orbit coupling in atoms, II . comparison of theory with experiment , \ https://doi.org/10.1098/rspa.1963.0036 journal journal Proc. R. Soc. Lond. A \ volume 271 ,\ pages 565--578 ( year 1963 ) NoStop

    1963

  43. [51]

    author author J. C. \ Boettger ,\ title title Approximate Two-Electron Spin-Orbit Coupling Term for Density-Functional-Theory DFT Calculations Using the Douglas-Kroll-Hess transformation , \ https://doi.org/10.1103/PhysRevB.62.7809 journal journal Phys. Rev. B \ volume 62 ,\ p...

    doi

  44. [52]

    author author B. A. \ He , author C. M. \ Marian , author U. Wahlgren ,\ and\ author O. Gropen ,\ title title A Mean-Field Spin-Orbit Method Applicable to Correlated Wavefunctions , \ https://doi.org/10.1016/0009-2614(96)00119-4 journal journal Chem. Phys. Lett. \ volume 251 ,...

    doi

  45. [53]

    van Wüllen \ and\ author C

    author author C. van Wüllen \ and\ author C. Michauk ,\ title title Accurate and Efficient Treatment of Two-Electron Contributions in Quasirelativistic High-Order Douglas-Kroll Density-Functional Theory , \ https://doi.org/10.1063/1.2133731 journal journal J. Chem. Phys. \ vol...

    doi

  46. [54]

    Konecny , author S

    author author L. Konecny , author S. Komorovsky , author J. Vicha , author K. Ruud ,\ and\ author M. Repisky ,\ title title Exact two-component TDDFT with simple two-electron picture-change corrections: X-ray absorption spectra near L-and M-edges of four-component quality at t...

    2023

  47. [55]

    Moitra , author L

    author author T. Moitra , author L. Konecny , author M. Kadek , author A. Rubio ,\ and\ author M. Repisky ,\ title title Accurate Relativistic Real-Time Time-Dependent Density Functional Theory for Valence and Core Attosecond Transient Absorption Spectroscopy , \ @noop journal...

    2023

  48. [56]

    Konecny , author V

    author author L. Konecny , author V. P. \ Kosheleva , author H. Appel , author M. Ruggenthaler ,\ and\ author A. Rubio ,\ title title Relativistic linear response in quantum-electrodynamical density functional theory , \ https://doi.org/10.1103/ttc3-867m journal journal Phys. ...

    doi

  49. [57]

    u bener , author C. Sch \

    author author M. K. \ Svendsen , author M. Ruggenthaler , author H. H \"u bener , author C. Sch \"a fer , author M. Eckstein , author A. Rubio ,\ and\ author S. Latini ,\ title title Theory of quantum light-matter interaction in cavities: Extended systems and the long waveleng...

    2023 arXiv

  50. [58]

    Horak , author D

    author author J. Horak , author D. Sidler , author T. Schnappinger , author W.-M. \ Huang , author M. Ruggenthaler ,\ and\ author A. Rubio ,\ title title Analytic model reveals local molecular polarizability changes induced by collective strong coupling in optical cavities , \...

    2025

  51. [59]

    Sch\"afer , author M

    author author C. Sch\"afer , author M. Ruggenthaler , author V. Rokaj ,\ and\ author A. Rubio ,\ title title Relevance of the quadratic diamagnetic and self-polarization terms in cavity quantum electrodynamics , \ @noop journal journal ACS Photonics \ volume 7 ,\ pages 975–990...

    2020

  52. [60]

    Visscher \ and\ author K

    author author L. Visscher \ and\ author K. G. \ Dyall ,\ title title Dirac--Fock Atomic Electronic Structure Calculations Using Different Nuclear Charge Distributions , \ @noop journal journal At. Data Nucl. Data Tables \ volume 67 ,\ pages 207--224 ( year 1997 ) NoStop

    1997

  53. [61]

    Malkin , author M

    author author E. Malkin , author M. Repisky , author S. Komorovsky , author P. Mach , author O. L. \ Malkina ,\ and\ author V. G. \ Malkin ,\ title title Effects of finite size nuclei in relativistic four-component calculations of hyperfine structure , \ @noop journal journal ...

    2011

  54. [62]

    author author R. E. \ Stanton \ and\ author S. Havriliak ,\ title title Kinetic balance: A partial solution to the problem of variational safety in dirac calculations , \ @noop journal journal J. Chem. Phys. \ volume 81 ,\ pages 1910--1918 ( year 1984 ) NoStop

    1910

  55. [63]

    Scalmani \ and\ author M

    author author G. Scalmani \ and\ author M. J. \ Frisch ,\ title title A new approach to noncollinear spin density functional theory beyond the local density approximation , \ https://doi.org/ https://doi.org/10.1021/ct300441z journal journal J. Chem. Theory Comput. \ volume 8 ...

    doi

  56. [64]

    Komorovsky , author P

    author author S. Komorovsky , author P. J. \ Cherry ,\ and\ author M. Repisky ,\ title title Four-component relativistic time-dependent density-functional theory using a stable noncollinear DFT ansatz applicable to both closed-and open-shell systems , \ @noop journal journal J...

    2019

  57. [65]

    Rokaj , author D

    author author V. Rokaj , author D. M. \ Welakuh , author M. Ruggenthaler ,\ and\ author A. Rubio ,\ title title Light--matter interaction in the long-wavelength limit: no ground-state without dipole self-energy , \ @noop journal journal Journal of Physics B: Atomic, Molecular ...

    2018

  58. [66]

    \ Lu , author M

    author author I.-T. \ Lu , author M. Ruggenthaler , author N. Tancogne-Dejean , author S. Latini , author M. Penz ,\ and\ author A. Rubio ,\ title title Electron-photon exchange-correlation approximation for quantum-electrodynamical density-functional theory , \ @noop journal ...

    2024

  59. [67]

    Repisky , author S

    author author M. Repisky , author S. Komorovsky , author M. Kadek , author L. Konecny , author U. Ekstr\" o m , author E. Malkin , author M. Kaupp , author K. Ruud , author O. L. \ Malkina ,\ and\ author V. G. \ Malkin ,\ title title ReSpect : Relativistic spectroscopy DFT pro...

    doi

  60. [68]

    Repisky , author S

    author author M. Repisky , author S. Komorovsky , author L. Konecny , author M. Kadek , author T. Moitra , author M. Joosten , author D. Misenkova , author R. Vikhamar-Sandberg , author M. Kaupp , author K. Ruud , et al. ,\ title title X2C Hamiltonian Models in ReSpect: Bridgi...

    2025 arXiv

  61. [69]

    Konecny , author M

    author author L. Konecny , author M. Repisky , author K. Ruud ,\ and\ author S. Komorovsky ,\ title title Relativistic four-component linear damped response TDDFT for electronic absorption and circular dichroism calculations , \ @noop journal journal J. Chem. Phys. \ volume 15...

    2019

  62. [70]

    author author M. E. \ Casida ,\ title title Time-dependent density functional response theory for molecules , \ in\ @noop booktitle Recent Advances In Density Functional Methods: (Part I) \ ( publisher World Scientific ,\ year 1995 )\ pp.\ pages 155--192 NoStop

    1995

  63. [71]

    author author M. E. \ Casida ,\ title title Time-dependent density-functional theory for molecules and molecular solids , \ @noop journal journal J. Mol. Struct.: THEOCHEM \ volume 914 ,\ pages 3--18 ( year 2009 ) NoStop

    2009

  64. [72]

    author author C. A. \ Ullrich ,\ @noop title Time-dependent density-functional theory: concepts and applications \ ( publisher OUP Oxford, R ,\ year 2011 ) NoStop

    2011

  65. [73]

    author author E. R. \ Davidson ,\ title title The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices , \ https://doi.org/https://doi.org/10.1016/0021-9991(75)90065-0 journal journal J. Comput. Phys. \ volume...

    doi

  66. [74]

    Olsen , author P

    author author J. Olsen , author P. J rgensen ,\ and\ author J. Simons ,\ title title Passing the one-billion limit in full configuration-interaction (FCI) calculations , \ @noop journal journal Chem. Phys. Lett. \ volume 169 ,\ pages 463--472 ( year 1990 ) NoStop

    1990

  67. [75]

    Norman , author K

    author author P. Norman , author K. Ruud ,\ and\ author T. Saue ,\ @noop title P rinciples and P ractices of M olecular P roperties \ ( publisher Wiley-VCH ,\ address Chichester ,\ year 2018 ) NoStop

    2018

  68. [76]

    Sidler , author C

    author author D. Sidler , author C. Sch\"afer , author M. Ruggenthaler ,\ and\ author A. Rubio ,\ title title Polaritonic chemistry: Collective strong coupling implies strong local modification of chemical properties , \ @noop journal journal J. Phys. Chem. Lett. \ volume 12 ,...

    2020

  69. [77]

    Schnappinger , author D

    author author T. Schnappinger , author D. Sidler , author M. Ruggenthaler , author A. Rubio ,\ and\ author M. Kowalewski ,\ title title Cavity Born--Oppenheimer Hartree--Fock Ansatz: Light--Matter Properties of Strongly Coupled Molecular Ensembles , \ @noop journal journal J. ...

    2023

  70. [78]

    author author K. G. \ Dyall \ and\ author A. S. P. \ Gomes ,\ title title Revised relativistic basis sets for the 5d elements hf--hg , \ @noop journal journal Theor. Chem. Acc. \ volume 125 ,\ pages 97--100 ( year 2010 ) NoStop

    2010

  71. [79]

    author author T. H. \ Dunning Jr ,\ title title Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen , \ @noop journal journal J. Chem. Phys. \ volume 90 ,\ pages 1007--1023 ( year 1989 ) NoStop

    1989

  72. [80]

    author author K. G. \ Dyall ,\ https://doi.org/10.5281/zenodo.7574629 title Dyall dz, tz, and qz basis sets for relativistic electronic structure calculations , \ ( year 2023 a ),\ note zenodo archive doi:10.5281/zenodo.7574629 https://zenodo.org/records/7574629 NoStop

    2023

  73. [81]

    author author K. G. \ Dyall ,\ https://doi.org/10.5281/zenodo.7606547 title Dyall double-zeta, triple-zeta, and quadruple-zeta basis set archive files , \ ( year 2023 b ),\ note zenodo archive doi:10.5281/zenodo.7606547 https://zenodo.org/records/7606547 NoStop

    2023

  74. [82]

    author author J. C. \ Slater ,\ title title A simplification of the hartree-fock method , \ @noop journal journal Phys. Rev. \ volume 81 ,\ pages 385 ( year 1951 ) NoStop

    1951

  75. [83]

    author author S. H. \ Vosko , author L. Wilk ,\ and\ author M. Nusair ,\ title title Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis , \ @noop journal journal Can. J. Phys. \ volume 58 ,\ pages 1200--1211 ( ...

    1980

  76. [84]

    author author A. D. \ Becke ,\ title title Density-functional exchange-energy approximation with correct asymptotic behavior , \ @noop journal journal Phys. Rev. A \ volume 38 ,\ pages 3098--3100 ( year 1988 ) NoStop

    1988

  77. [85]

    Lee , author W

    author author C. Lee , author W. Yang ,\ and\ author R. G. \ Parr ,\ title title Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density , \ @noop journal journal Phys. Rev. B \ volume 37 ,\ pages 785--789 ( year 1988 ) NoStop

    1988

  78. [86]

    author author P. J. \ Stephens , author F. J. \ Devlin , author C. F. \ Chabalowski ,\ and\ author M. J. \ Frisch ,\ title title Ab Initio Calculation of Vibrational Absorption and Circular Dichroism Spectra Using Density Functional Force Fields , \ @noop journal journal J. Ph...

    1994

  79. [87]

    author author A. G. \ Avramenko \ and\ author A. S. \ Rury ,\ title title Cavity polaritons formed from spatially separated quasi-degenerate porphyrin excitons: Structural modulations of bright and dark state energies and compositions , \ @noop journal journal J. Phys. Chem. C...

    2022

  80. [88]

    Sun , author B

    author author S. Sun , author B. Gu ,\ and\ author S. Mukamel ,\ title title Polariton ring currents and circular dichroism of Mg-porphyrin in a chiral cavity , \ @noop journal journal Chem. Sci. \ volume 13 ,\ pages 1037--1048 ( year 2022 ) NoStop

    2022

  81. [89]

    \ Huber \ and\ author G

    author author K.-P. \ Huber \ and\ author G. Herzberg ,\ @noop title Constants of diatomic molecules \ ( publisher Springer US ,\ year 1979 )\ pp.\ pages 8--689 NoStop

    1979

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.