Quantum Electrodynamical DFT

• Quantum Electrodynamical Density Functional Theory is a framework that extends traditional DFT by explicitly including quantized photon fields to model light–matter interactions.
• It uses coupled, non-linear equations linking electron densities and photon coordinates, enabling ab initio predictions in regimes like cavity QED and strong coupling.
• The method applies Kohn–Sham constructions and variational principles to capture non-classical effects, guiding advances in polaritonic chemistry and quantum optics.

Quantum Electrodynamical Density Functional Theory (QEDFT) generalizes the concepts of Kohn–Sham density functional theory to systems where quantum matter and quantized electromagnetic fields (photons) are coupled. Unlike conventional DFT, which neglects the quantum nature of light, QEDFT incorporates photon degrees of freedom explicitly into the ab initio description of electronic and optical structure. This allows for predictive calculations of chemical, material, and photonic properties in regimes where quantum light–matter interactions are significant, such as in optical or infrared cavities, under strong or ultrastrong coupling, or when quantum field fluctuations affect observable molecular/solid-state phenomena. QEDFT unifies and extends many methods from quantum optics, electronic structure theory, and many-body QED, providing a flexible framework for modeling polaritonic chemistry, collective excitations, and cavity-induced modifications of fundamental properties.

1. Theoretical Foundations and Formalism

QEDFT extends the foundational principles of DFT to systems interacting with quantized radiation fields. The starting point is a fully relativistic QED Hamiltonian, typically in Coulomb gauge, to describe electrons and positrons interacting with the photon field (Ruggenthaler et al., 2014). The canonical variables are the polarization of the electronic Dirac field and the spatially varying vector potential of the electromagnetic field. By taking suitable non-relativistic limits, the formalism reduces to the Pauli–Fierz (PF) Hamiltonian, which is the standard model for electrons interacting with quantized light in the long-wavelength regime relevant for molecular and condensed phase cavity QED:

$$H = \sum_{i} \left[-\frac{\hbar^2}{2m}\nabla_i^2 + v_\mathrm{ext}(\mathbf{r}_i, t)\right] + \sum_{i<j} \frac{e^2}{4\pi\varepsilon_0 |\mathbf{r}_i-\mathbf{r}_j|} + \sum_\alpha \frac{1}{2}\left[\hat{p}_\alpha^2 + \omega_\alpha^2\left(\hat{q}_\alpha - \frac{\lambda_\alpha \cdot \mathbf{R}}{\omega_\alpha}\right)^2\right] + \frac{j_\mathrm{ext}^\alpha(t)}{\omega_\alpha}\hat{q}_\alpha$$

where $\mathbf{R} = \sum_i \mathbf{r}_i$ and $\hat{q}_\alpha$ are photon-mode displacement operators.

In the density-functional reformulation, the many-body state is uniquely determined (for suitable initial conditions) by a pair of reduced variables: the electronic density (or its relativistic generalization, polarization) $n(\mathbf{r}, t)$ and the expectation value of the photon coordinate (or equivalently, the vector potential) $q_\alpha(t)$, together with the initial state (Ruggenthaler et al., 2014, Ruggenthaler, 2015). The evolution of the system is governed by two coupled, non-linear equations for these quantities, eliminating the need to propagate the full electron-photon wavefunction.

A crucial result is the generalized Hohenberg–Kohn theorem: for well-chosen variables (charge density and vector potential in non-relativistic limit; polarization and photon vector potential in the relativistic case), there is a one-to-one mapping between sets of external fields (scalar potential and transverse current, or their relativistic analogs) and the internal observables (Ruggenthaler, 2015). This mapping underpins the universal QED density functional $F[n, A]$ and provides a variational principle for the ground-state energy: $$E[v, j] = \inf_{n, A} \left\{ F[n, A] + \int d^3r\, n(\mathbf{r}) v(\mathbf{r}) - \int d^3r\, A(\mathbf{r})\cdot j(\mathbf{r})/(\mu_0 c^2) \right\}$$

2. Kohn–Sham Constructions and Hierarchy of Approximations

QEDFT introduces a hierarchy of Kohn–Sham (KS) like constructions for light–matter coupled systems. In the KS framework, the interacting electron-photon system is mapped onto a noninteracting reference system consisting of noninteracting electrons and photons, driven by effective potentials constructed such that the chosen densities and photon expectations are reproduced (Ruggenthaler et al., 2014). The KS potentials are, in general, nonlocal and time-dependent, incorporating the effects of both electron–electron and electron–photon interactions.

In the most general case, the KS construction involves solving:

• A time-dependent single-electron KS equation with an effective potential $v_s(\mathbf{r}, t)$, which contains external, Hartree, and exchange–correlation (xc) terms that include photon-induced contributions.
• A photon mode equation where the displacement coordinate $q_\alpha(t)$ evolves under an effective current.

For practical implementations, the reduction to the nonrelativistic limit (Pauli–Fierz Hamiltonian) is key. Further model simplifications, such as restricting to a limited number of photon modes (cavity QED), lead to a spectrum of density-functional-type theories aligned with the complexity of the physical situation. Standard electronic DFT emerges as a limiting case when photon quantization is neglected.

Explicit Kohn–Sham schemes at the model Hamiltonian level have been worked out for realistic settings—including lattice Hubbard models coupled to single or few photonic modes—and for few-site matter-photon systems, all highlighting the structure of the exact QED Kohn–Sham potentials (Ruggenthaler et al., 2014).

3. Mathematical Structure, Observables, and Analytical Insights

The QEDFT framework is constructed to guarantee v-representability and the differentiability of the universal functional under regularity conditions, leveraging constrained-search (Levy–Lieb) functionals in the "density-pair" $(\sigma, \xi)$, where $\sigma$ is the matter polarization and $\xi$ is the photonic displacement (Bakkestuen et al., 18 Sep 2024, Bakkestuen et al., 22 Nov 2024). For model systems such as the quantum Rabi or Dicke models, it was shown that pure-state v-representability and functional differentiability can be established, providing explicit analytical expressions for the functionals and their derivatives (Kohn–Sham potentials). This is in contrast to standard DFT, where non-differentiability of the universal functional is generic.

Tables of Internal vs External Variables (Model Case)

External	Internal (Density)
Scalar Potential $v$	Magnetization $\sigma$ (Pauli Z exp.)
External Current $j$	Photon Displacement $\xi$

The adiabatic connection formalism allows for a near-explicit expression of the universal functional as a sum of noninteracting (decoupled) contributions plus an integral over the electronic-photonic coupling (Bakkestuen et al., 22 Nov 2024): $$F_{\mathrm{LL}}^\lambda(\sigma, \xi) = \text{noninteracting functional} + \lambda g \sigma\xi + g \int_0^\lambda \langle \hat{\sigma}_z \hat{x} \rangle_{\psi^\nu} d\nu$$ where the last term encodes photon-induced correlation.

4. Applications and Implications

QEDFT provides an ab initio route to paper chemical and material properties under strong light-matter coupling, including the ground-state and excited-state structure of molecules and extended systems inside optical cavities (Ruggenthaler et al., 2014, Flick et al., 2017). Applications include:

• Calculation of cavity-modified chemical reactivity and reaction rates (polaritonic chemistry)
• Modeling of photon-induced changes in collective electronic states and phase behavior in solids
• Prediction of spectroscopic signatures arising from electron-photon entanglement and nonclassical light features
• Systematic derivation of model Hamiltonians and clarification of the classical-to-quantum crossover in light–matter interactions

The formalism captures both mean-field (i.e., classical dipole interaction) and genuine quantum effects, such as photon-induced steps and peaks in the KS potentials (Flick et al., 2015), and can accommodate both static and time-dependent scenarios. It is naturally suited to provide benchmark data for less rigorous models (e.g., few-level Jaynes–Cummings or Dicke Hamiltonians) and enables the systematic testing of v-representability, universality, and the accuracy of approximate functionals in the QED setting (Bakkestuen et al., 18 Sep 2024, Bakkestuen et al., 22 Nov 2024).

5. Limiting Cases, Model Reductions, and Computational Strategies

By hierarchical reduction—from the fully relativistic QED Hamiltonian to nonrelativistic, minimal-coupling, and finally cavity-constrained limits—QEDFT retains full generality while enabling tractable ab initio calculations in regimes of physical interest (Ruggenthaler et al., 2014):

Level	Description	Basic Variables
Relativistic QED	Full quantum field theory	Polarization, Vector Potential
Pauli–Fierz limit	Non-relativistic, quantized light	Current, Vector Potential
Restricted cavity QED	Few modes, e.g., optical microcavities	Electron density, $q_\alpha$
Atomic/molecular DFT	Classical, no photon quantization	Electronic density

Numerical implementation relies on reformulating the coupled equations for the internal variables as nonlinear equations for the KS system, often solved via fixed-point iteration (e.g., in time-dependent settings, as in (Flick et al., 2015)). Rigorous existence and uniqueness proofs guide functional construction and benchmarking.

6. Challenges and Future Directions

Open problems include the development of reliable approximations for the exchange–correlation functionals involving electron-photon interactions—especially for regimes beyond second-order perturbation theory, strong coupling, or including dissipative and open-system effects. The inclusion of dissipative photon baths, extension to fully periodic and extended systems, and development of new observables sensitive to genuine quantum field effects (e.g., photon number statistics, joint density-photon correlators) are areas of active research (Bakkestuen et al., 22 Nov 2024, Ruggenthaler et al., 2014).

The QEDFT framework outlined in (Ruggenthaler et al., 2014) and its descendants sets the stage for systematic inclusion of quantum light–matter correlation and effects that go beyond what can be described by classical field theories or standard electronic-structure methods alone. The mathematically rigorous model studies (Bakkestuen et al., 18 Sep 2024, Bakkestuen et al., 22 Nov 2024) not only provide benchmarks but also illuminate the formal underpinnings of functional construction, v-representability, and density–potential mappings in the presence of quantized electromagnetic fields.