Two-Component Linear-Response QEDFT

• Two-component LR-QEDFT is a framework that treats electronic density and photon displacements as coupled variables to capture matter–photon interactions.
• It utilizes a mixed susceptibility matrix combined with Kohn–Sham mappings and Dyson-like equations to derive polaritonic excitation spectra.
• Recent extensions incorporate relativistic and exact two-component approaches, enabling scalable and precise simulations for heavy-element cavity systems.

Searching arXiv for papers directly relevant to two-component linear-response QEDFT and its foundations. Two-component linear-response quantum-electrodynamical density-functional theory (LR-QEDFT) is the linear-response formulation of QEDFT in which matter and photonic degrees of freedom are treated on equal footing through coupled basic variables and coupled susceptibilities. In the nonrelativistic Pauli–Fierz setting these variables are typically the electronic density $n(\mathbf r,t)$ and mode-resolved photon displacements $q_\alpha(t)$, while minimal models replace them by a two-level polarization $\sigma$ and a photon displacement $\xi$. The formalism yields mixed matter–photon response functions, generalized Kohn–Sham mappings, and polaritonic excitation spectra; recent work extends the same structure to relativistic four-component and exact two-component (X2C) Hamiltonians for heavy-element systems in cavities (Welakuh et al., 2022, Konecny et al., 2024, Konecny et al., 9 Jul 2025).

1. Basic variables, Hamiltonians, and the meaning of “two-component”

In nonrelativistic cavity QED, the standard starting point is the Pauli–Fierz Hamiltonian in the long-wavelength dipole approximation. A representative length-gauge form for $N$ electrons and $M$ photon modes is

$$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$

with electronic density $n(\mathbf r,t)$ and photon displacements $q_\alpha(t)$ as the basic internal variables, conjugate to an external scalar potential $v_{\mathrm{ext}}(\mathbf r,t)$ and external currents $q_\alpha(t)$0 (Welakuh et al., 2022). In this sense, “two-component” refers to the coupled matter–photon variable set.

Minimal QEDFT models make this structure fully explicit. For the quantum Rabi model, the internal Hamiltonian is

$q_\alpha(t)$1

and the external fields enter through

$q_\alpha(t)$2

The basic densities are then

$q_\alpha(t)$3

namely a matter polarization and a photon displacement. For the Dicke extension with $q_\alpha(t)$4 two-level systems, the matter variable becomes the vector polarization $q_\alpha(t)$5, coupled to the same $q_\alpha(t)$6 (Bakkestuen et al., 2024).

Recent relativistic work introduces a second use of “two-component.” In relativistic cavity QED, four-component Dirac–Kohn–Sham QEDFT has been reduced to exact two-component Hamiltonians via X2C decoupling, so that LR-QEDFT can be two-component both in the matter–photon sense and in the relativistic electronic sense (Konecny et al., 9 Jul 2025).

2. Kohn–Sham mapping and the coupled response structure

The central LR-QEDFT object is a matrix of coupled susceptibilities. In the length gauge, the first-order responses of density and photon displacements are

$q_\alpha(t)$7

$q_\alpha(t)$8

These four response functions are the matter–matter, matter–photon, photon–matter, and photon–photon blocks of the two-component response theory (Welakuh et al., 2022).

The corresponding Kohn–Sham system consists of noninteracting electrons in an effective scalar potential and harmonic-oscillator photon modes driven by effective currents. In the ground state, representative equations are

$q_\alpha(t)$9

$\sigma$0

so the photonic coordinate is fixed self-consistently by the matter dipole (Welakuh et al., 2022).

In linear response, the interacting and Kohn–Sham susceptibilities are connected by a Dyson-like matrix equation,

$\sigma$1

where $\sigma$2 contains the blocks $\sigma$3, $\sigma$4, $\sigma$5, and $\sigma$6. The adiabatic-connection fluctuation-dissipation formulation of QEDFT expresses the correlation energy in terms of the same coupled response functions, making these blocks the central objects of both ground-state functionals and response theory (Flick, 2021).

3. Exact constraints from minimal models

The quantum Rabi model supplies an unusually explicit exact reference for two-component LR-QEDFT. For regular ground-state densities $\sigma$7, the mapping

$\sigma$8

is injective on ground states and, through constrained search, bijective between $\sigma$9 and the regular density domain. The Levy–Lieb functional is

$\xi$0

and the conjugate potentials follow from

$\xi$1

At zero coupling, the constrained search is analytic: $\xi$2 with Kohn–Sham potentials

$\xi$3

This provides an exactly solvable Kohn–Sham reference for the coupled matter–photon problem (Bakkestuen et al., 2024).

A decisive structural result is the exact displacement rule

$\xi$4

so the entire $\xi$5-dependence is explicit, linear plus quadratic, and all nontrivial correlation resides in $\xi$6. The interaction contribution can be written as

$\xi$7

which depends only on $\xi$8. In this model the exchange energy vanishes, and the many-body correction is purely correlation (Bakkestuen et al., 2024).

The adiabatic connection is “almost explicit”: $\xi$9 with the only non-explicit term $N$0 bounded by

$N$1

For LR-QEDFT, this implies a highly constrained static kernel structure: cross matter–photon terms come from the explicit $N$2 contribution, the photon–photon correlation kernel vanishes, and the nontrivial density dependence is confined to the matter sector. In the minimal Rabi setting,

$N$3

while the matter–matter kernel is determined by the second derivative of the correlation functional (Bakkestuen et al., 2024).

4. Response formalisms: Casida-like matrices, Sternheimer equations, and real-time propagation

Two-component LR-QEDFT has been formulated both as a generalized eigenvalue problem and as a frequency-dependent Sternheimer scheme. In the Sternheimer formulation, the density response is written solely in terms of occupied Kohn–Sham orbitals,

$N$4

with first-order orbital corrections satisfying

$N$5

$N$6

The photonic response is obtained algebraically from the density fluctuation. This avoids explicit sums over unoccupied states and, as stated explicitly, “the arbitrarily many but finite photon modes that can be included does not add to this scaling” (Welakuh et al., 2022).

The same theory can be written in Casida-like form. In relativistic QEDFT, the coupled first-order equations for electronic amplitudes and photon coordinates lead to a generalized eigenvalue problem with electronic blocks $N$7, dipole-self-energy blocks $N$8, and electron–photon coupling blocks $N$9. The eigenvalues are polaritonic excitation energies, and the eigenvectors contain both electronic excitation/de-excitation amplitudes and photon creation/annihilation amplitudes (Konecny et al., 2024).

A complementary route is real-space, real-time propagation on a tensor product of a Fock-space and real-space grid. In that approach the Pauli–Fierz Hamiltonian is propagated directly after a weak delta-kick, and the absorption cross section is extracted from the Fourier transform of the time-dependent dipole moment. This yields cavity-modified absorption spectra, Rabi splittings, and, beyond linear response, high-harmonic generation in cavities (Malave et al., 2022).

Related cavity-QED Hartree–Fock response theory uses the same electronic–photonic block structure for frequency-dependent linear and quadratic response functions. A plausible implication is that this mean-field structure provides a direct template for QEDFT implementations in which Hartree–Fock kernels are replaced by QEDFT kernels (Yuwono et al., 23 Jun 2026).

5. Exchange–correlation structure and practical approximations

The ACFD formulation of QEDFT decomposes the correlation energy into

$M$0

where $M$1 is associated with the explicit electron–photon interaction and is expressed through mixed density–photon response functions, while $M$2 contains Coulomb and dipole-self-energy contributions through the density–density response. At lowest order in the light–matter coupling, the leading electron–photon contribution is

$M$3

and the associated exchange term is

$M$4

From this, a gradient-based density functional $M$5 was constructed; within that approximation the functional has no explicit $M$6-dependence, so $M$7 and the xc content resides in the electronic block $M$8 (Flick, 2021).

A complementary route starts from the local-force equation for the Pauli–Fierz Hamiltonian and defines an explicit electron–photon exchange potential $M$9. For one electron and one dressed mode,

$$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$0

while the homogeneous-electron-gas limit yields a pxLDA functional. In weak coupling, a renormalization factor $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$1 is introduced so that

$$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$2

thereby incorporating part of the electron–photon correlation contribution (Lu et al., 2024).

For LR-QEDFT, these constructions imply two common approximation strategies. One is to let xc corrections modify mainly the electronic kernel $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$3, while keeping the mixed matter–photon blocks at mean-field level. The other, suggested by the exact Rabi-model functional, is that in minimal coupled models the photon dependence can be explicit and simple, with the complicated part concentrated in matter correlation (Bakkestuen et al., 2024).

6. Relativistic four-component and exact two-component LR-QEDFT

Relativistic LR-QEDFT starts from the QED Hamiltonian in Coulomb gauge with minimally coupled Dirac fields and quantized photons, followed by a long-wavelength approximation and a length-gauge transformation. The basic variables are the electronic density $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$4 and photon displacement coordinates $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$5, and the Kohn–Sham system consists of four-component Dirac electrons in an effective scalar potential plus independent photon modes obeying driven harmonic-oscillator equations (Konecny et al., 2024).

The relativistic linear-response equations lead to a generalized Casida-like eigenvalue problem in which the electronic blocks are built from four-component spinors and the photon blocks encode cavity frequencies. A central conceptual result is the appearance of a cavity-mediated spin–orbit-like interaction,

$$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$6

with the cavity electric field operator expressed through photon coordinates and collective polarization. Benchmark calculations for Zn, Cd, Hg, and mercury porphyrin show that a relativistic treatment enables polaritonic mixing of formally forbidden singlet–triplet transitions with cavity modes (Konecny et al., 2024).

The exact two-component extension applies X2C decoupling to the parent four-component electron–photon Hamiltonian. Under common weak-field and dipole approximations, it is sufficient to perform the X2C transformation only during the ground-state self-consistent field procedure, after which all linear-response calculations can be carried out fully in the two-component regime using the same decoupling matrix. The current implementation includes the atomic mean-field, extended atomic mean-field, and molecular mean-field X2C Hamiltonian models, and benchmark calculations show that the X2C approach closely reproduces reference four-component results (Konecny et al., 9 Jul 2025).

At the response level, the resulting two-component polaritonic eigenvalue problem preserves the same block structure as the four-component theory, with all quantities replaced by their picture-changed two-component counterparts. Applications include 2D spectra of a mercury porphyrin complex in a Fabry–Perot cavity, showing off-resonant coupling and multiple polaritonic branches, and a chain of AuH molecules, showing that collective coupling can locally modify chemical properties of a molecule with a perturbed bond length (Konecny et al., 9 Jul 2025).

7. Applications, limitations, and open directions

Two-component LR-QEDFT is used to compute cavity-modified absorption spectra, polaritonic splittings, dispersive response, and response to structured photonic environments. In molecular cavities it reproduces lower and upper polariton branches and their oscillator-strength redistribution; in coupling to continua it captures the crossover from Lorentzian line shapes to Fano resonances when the molecule interacts strongly with a continuum of modes (Welakuh et al., 2022). In relativistic settings it describes exciton polaritons involving singlet–triplet transitions and spin–orbit effects, and in heavy-element systems it makes 2D cavity spectra and collective-coupling calculations computationally feasible (Konecny et al., 2024, Konecny et al., 9 Jul 2025).

The formalism also has sharply defined limitations. Exact minimal-model results are available only for highly truncated systems such as one two-level system plus one photon mode or $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$7 two-level systems plus one mode; those studies treat ground-state DFT and static adiabatic connections explicitly, but do not derive frequency-dependent xc kernels (Bakkestuen et al., 2024). Practical ab initio implementations usually employ adiabatic electronic xc kernels and photon RPA, so explicit photon xc effects and nonadiabatic memory are neglected (Flick, 2021, Konecny et al., 2024). The real-space tensor-product approach scales poorly with the number of photon modes because the Fock-space dimension grows rapidly, while the Sternheimer route was developed precisely to avoid that bottleneck (Malave et al., 2022, Welakuh et al., 2022).

Open problems identified in the literature include the construction of frequency-dependent kernels $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$8, $$\begin{aligned} \hat{H}(t)&=\sum_{i=1}^{N}\left(\frac{\hat{p}_{i}^{2}}{2m} + v_{\textrm{ext}}(\hat{r}_i,t)\right) + \sum_{i>j}^{N}w(|\hat{r}_{i}-\hat{r}_{j}|) \ &\quad +\sum_{\alpha=1}^{M}\frac{1}{2}\left[\hat{p}^2_{\alpha} +\omega^2_{\alpha}\left(\hat{q}_{\alpha} - \frac{\boldsymbol{\lambda}_{\alpha}}{\omega_{\alpha}}\cdot \hat{R}\right)^2\right] + \sum_{\alpha=1}^{M}\frac{j_{\textrm{ext}}^{(\alpha)}(t)}{\omega_\alpha}\hat{q}_\alpha , \end{aligned}$$9 consistent with exact static limits, the extension to multimode and beyond-dipole settings, the treatment of non-regular densities and degeneracies in Dicke-like models, and the systematic benchmarking of photon-free, adiabatic, OEP-based, or ACFD-inspired approximations against exact or four-component references (Bakkestuen et al., 2024, Konecny et al., 9 Jul 2025). A plausible implication of the present body of work is that progress will continue along two complementary lines: exact constraint building from minimal two-component models, and scalable relativistic implementations for chemically realistic cavity systems.