Quantum Electrodynamic Coupled Cluster (QED-CC)

• Quantum electrodynamic coupled cluster (QED-CC) is a unified many-body framework that nonperturbatively treats both electron correlation and electron–photon interactions.
• It retains the exponential parametrization, systematic improvability, and size extensivity of traditional CC methods while expanding the excitation space to include mixed matter–photon states.
• QED-CC has been implemented in various computational schemes, demonstrating accurate predictions of cavity effects on ground and excited molecular properties.

Quantum electrodynamic coupled cluster (QED-CC) is a many-body wave-function-based framework that extends coupled-cluster theory to coupled matter–photon systems, most commonly molecules strongly coupled to a quantized cavity mode, by treating electron correlation and electron–photon correlation on the same ab initio footing (Liebenthal et al., 2021). In contemporary cavity-QED quantum chemistry, it is typically formulated in a mixed electronic–photonic Hilbert space with the Pauli–Fierz Hamiltonian in the dipole approximation and length gauge (Liebenthal et al., 2023, Pathak et al., 2024). The same label has also been used for electrodynamical coupled-cluster formalisms that start directly from quantum electrodynamics in Coulomb gauge and derive Lamb, Breit, hyperfine, and vacuum-polarization effects within relativistic many-electron theory (Datta, 2019, Datta, 2024).

1. Conceptual scope and defining features

In the cavity-QED setting, QED-CC generalizes ordinary coupled cluster by making photons part of the cluster expansion rather than treating the cavity as an external perturbation. The framework is intended to describe ordinary electronic correlation, photon-induced renormalization of the electronic ground state, mixed electron–photon excitations, and the dependence of these effects on cavity frequency and polarization (Pathak et al., 2024). A recurring point in the literature is that this is not restricted to explicitly driven transitions or to the preparation of polaritonic excited states: the cavity vacuum itself can modify the ground state, and QED-CC is designed to capture that modification nonperturbatively (Pathak et al., 2024).

The formal attraction of QED-CC is that it preserves the standard coupled-cluster structure—exponential parametrization, systematic improvability, and size extensivity—while enlarging the excitation manifold to include bosonic and mixed fermion–boson sectors (Liebenthal et al., 2023). One early application emphasized that this differs from polaritonic coupled-cluster variants that resemble configuration interaction in the photon sector and are not size-extensive with respect to photons; by contrast, QED-CC was presented there as fully size-extensive (III, 2020).

A broader usage of the term persists outside cavity quantum chemistry. In electrodynamical coupled-cluster theory based on standard QED Hamiltonians in Coulomb gauge, the objective is not cavity-modified molecular structure but a unified treatment of relativistic correlation and radiative effects. In that line of work, a radiative cluster is used to generate Lamb, Breit, and hyperfine interactions, while an extended matter cluster with pair modifications produces vacuum-polarization contributions (Datta, 2024). The coexistence of these usages means that “QED-CC” denotes a family of coupled-cluster constructions rooted in quantized electromagnetic degrees of freedom rather than a single universally fixed formalism.

2. Hamiltonians, gauges, and reference states

The dominant cavity-QED formulation employs the single-mode Pauli–Fierz Hamiltonian in the length gauge, within the dipole and Born–Oppenheimer approximations. A common form is

$$\hat{H}_{\rm PF} = \hat{H}_{\rm e} + \omega_{\rm cav}\hat{b}^\dagger \hat{b} -\sqrt{\frac{\omega_{\rm cav}}{2}}\,(\bm{\lambda}\cdot \hat{\bm{\mu}})(\hat{b}^\dagger+\hat{b}) +\frac{1}{2}(\bm{\lambda}\cdot \hat{\bm{\mu}})^2,$$

where $\hat{H}_{\rm e}$ is the electronic Hamiltonian, $\omega_{\rm cav}$ the cavity frequency, $\hat{b}^\dagger,\hat{b}$ photon creation and annihilation operators, and $\bm{\lambda}$ the light–matter coupling vector (Liebenthal et al., 2023). In coherent-state formulations, the dipole operator is shifted by its mean-field expectation value, giving

$$\hat{H}_{\rm CS} = \hat H_{\rm e} + \omega_{\rm cav}\hat b^\dagger \hat b -\sqrt{\frac{\omega_{\rm cav}}{2}}\,\big(\bm\lambda\cdot[\bm\mu-\langle\bm\mu\rangle]\big)(\hat b^\dagger+\hat b) +\frac12\big(\bm\lambda\cdot[\bm\mu-\langle\bm\mu\rangle]\big)^2.$$

This shift is central in practical QED-HF and post-HF implementations (Liebenthal et al., 2023).

The dipole self-energy term is repeatedly identified as essential. In one implementation study it was described as necessary for boundedness and origin invariance for neutral systems, while the bilinear term proportional to $(b^\dagger+b)$ was emphasized as the part that directly mediates electron–photon coupling (Pathak et al., 2024). The coherent-state transformation also changes the mean-field reference. A standard reference is a direct product of an electronic Hartree–Fock determinant and a photon vacuum,

$$|R\rangle = |0^{\rm e}\rangle \otimes |0^{\rm p}\rangle,$$

or, equivalently, a coherent-state-transformed zero-photon state (Liebenthal et al., 2021, Liebenthal et al., 2023).

A practical distinction developed in later work is between “relaxed” and “unrelaxed” formulations. In the relaxed version, the mean-field reference is obtained self-consistently with $\hat H_{\rm CS}$, so cavity-induced orbital relaxation is included at the SCF level and post-SCF energies are origin invariant. In the unrelaxed version, the orbitals come from the bare electronic Hamiltonian and the coherent-state transformation is omitted; unrelaxed QED-CC energies for neutral species were found to be only modestly origin dependent, whereas unrelaxed mean-field energies showed severe origin dependence (Liebenthal et al., 2023). This distinction became important both numerically and conceptually.

3. Cluster parametrization and truncation hierarchies

The defining ansatz is the usual exponential coupled-cluster wavefunction on the combined electron–photon space,

$$|\Psi_{\rm CC}\rangle = e^{\hat T}|0^{\rm e}0^{\rm p}\rangle,$$

with a cluster operator that contains purely electronic, purely photonic, and mixed electron–photon excitations (Liebenthal et al., 2021). A compact general form used in several papers is

$\hat{H}_{\rm e}$0

where $\hat{H}_{\rm e}$1 labels electronic excitation operators and $\hat{H}_{\rm e}$2 the photon number (Pavošević et al., 2021).

At the one-photon level, the widely used QED-CCSD-1 truncation augments electronic singles and doubles by a pure one-photon operator and mixed electron–photon excitations up to one photon: $\hat{H}_{\rm e}$3 This structure appears, with minor notational variation, across ground-state, excited-state, and implementation papers (Liebenthal et al., 2021, Liebenthal et al., 2023).

A later ExaChem/TAMM implementation introduced a systematic $\hat{H}_{\rm e}$4 hierarchy in which $\hat{H}_{\rm e}$5 is the highest electronic excitation rank and $\hat{H}_{\rm e}$6 the highest photon number. In that notation,

$\hat{H}_{\rm e}$7

and QED-CCSD(2,2) includes explicit pure-photon and mixed two-photon sectors in addition to ordinary electronic CCSD (Pathak et al., 2024).

The literature therefore uses several closely related truncation labels:

Model	Included sectors	Note
QED-CCSD-1	electronic singles/doubles, one-photon, mixed up to one photon	Standard cavity-QED truncation (Liebenthal et al., 2023)
QED-CCSD(2,0)	electronic CCSD with QED-modified integrals	No explicit photon amplitudes (Pathak et al., 2024)
QED-CCSD(2,1)	$\hat{H}_{\rm e}$8	One-photon explicit treatment (Pathak et al., 2024)
QED-CCSD(2,2)	QED-CCSD(2,1) plus $\hat{H}_{\rm e}$9	Adds two-photon sectors (Pathak et al., 2024)
CCSD-12-SD	$\omega_{\rm cav}$0	Adds pure two-photon excitations (Monzel et al., 2024)

For cavity models with two perpendicular polarizations, the ansatz must be extended further. In the unpolarized Fabry–Pérot formulation, the photon operators come in pairs for the two transverse polarizations, and the cluster operator is truncated at the CCSD-12-SD level while preserving the symmetry of the unpolarized cavity (Monzel et al., 25 Jul 2025). The same work stresses that the two-polarization extension is not just a refinement of a single-polarization model: it restores the correct symmetry of the cavity.

In several implementations the formal asymptotic scaling remains that of ordinary CCSD. QED-CCSD(2,1) and QED-CCSD(2,2) were reported to retain formal $\omega_{\rm cav}$1 scaling, although with larger prefactors because of the additional tensor blocks (Pathak et al., 2024). Diagrammatic work on CCSD-1-SD and CCSD-12-SD reached the same conclusion for single-mode cases, with the two-photon $\omega_{\rm cav}$2 sector not changing the overall scaling (Monzel et al., 2024).

4. Ground-state equations, EOM formalisms, and particle-number sectors

The projected similarity-transformed Schrödinger equation retains its standard non-Hermitian structure. For ground-state QED-CC, amplitudes are obtained from equations of the form

$\omega_{\rm cav}$3

or equivalently

$\omega_{\rm cav}$4

for all included electron–photon excitation manifolds (Liebenthal et al., 2023, Pathak et al., 2024). The resulting residual equations couple electronic, bosonic, and mixed amplitudes nonlinearly through the similarity-transformed Hamiltonian.

Excited states are commonly treated by equation-of-motion QED-CC. In the particle-conserving sector, the right and left EOM states are

$\omega_{\rm cav}$5

with $\omega_{\rm cav}$6 and $\omega_{\rm cav}$7 containing both electronic and photonic components (Liebenthal et al., 2023). Diagrammatic derivations have emphasized that the EOM operator spans pure electronic, pure photonic, and mixed electron–photon excitations, enabling the direct description of polaritonic excited states (Monzel et al., 2024).

A major generalization is EOM-EA-QED-CC, which extends equation-of-motion coupled cluster to the particle-nonconserving electron-attachment sector. In EOM-EA-QED-CCSD-1, the right operator contains both electron-attached configurations and electron-attached configurations with one photon: $\omega_{\rm cav}$8 This construction explicitly spans the $\omega_{\rm cav}$9-electron polaritonic manifold rather than restricting the theory to a fixed electron number (Liebenthal et al., 2021). The distinction from standard EOM-EA-CC is threefold: the Hamiltonian is the Pauli–Fierz cavity-QED Hamiltonian, the state space includes photon-containing configurations, and the method can describe cavity-modified electron affinities, photon admixture, and Rabi splittings in electron-attached states (Liebenthal et al., 2021).

An implementation detail became particularly important for electron attachment. The coherent-state basis should be defined using the QED-HF dipole of the target $\hat{b}^\dagger,\hat{b}$0-electron manifold rather than that of the $\hat{b}^\dagger,\hat{b}$1-electron reference; otherwise electron affinities can be substantially in error (Liebenthal et al., 2021). This finding made explicit that coherent-state choices are not merely notational conveniences in particle-nonconserving sectors.

5. Implementations and representative numerical results

QED-CC has moved from formal derivation to multiple software implementations. A 2024 implementation of QED-CCSD for electronic and bosonic amplitudes, including individual and mixed excitation processes, was reported in ExaChem on top of the Tensor Algebra for Many-body Methods infrastructure. TAMM was described there as providing distributed tensor storage, tensor operation scheduling, support for real and complex algebra, CPU and GPU execution, and MPI plus Global Arrays for scalable distributed-memory execution; the reported setup used integrals from libint2.9.0, an integral threshold of $\hat{b}^\dagger,\hat{b}$2, a linear dependence threshold of $\hat{b}^\dagger,\hat{b}$3, an SCF density threshold of $\hat{b}^\dagger,\hat{b}$4, an SCF residual norm threshold of $\hat{b}^\dagger,\hat{b}$5, DIIS with subspace size 5, and no frozen electrons (Pathak et al., 2024).

Other implementations targeted complementary parts of the theory. EOM-EA-QED-CC was implemented as a Psi4 plugin with working equations generated by a modified p^\dagger q operator-manipulation library (Liebenthal et al., 2021). Earlier QED-CCSD-1 applications to sodium halides were implemented in the open-source hilbert plugin for Psi4 using density fitting and a $\hat{b}^\dagger,\hat{b}$6-dressed Hamiltonian (III, 2020). Analytical ground-state gradients at the QED-CCSD-1 level were later implemented in a development branch of eT 2.0 together with the geomeTRIC optimizer and a Cholesky-based treatment of two-electron integrals (Lexander et al., 2024).

Validation studies have been correspondingly varied. For water, QED-CCSD(2,2) in ExaChem/TAMM agreed with a developmental code from Flick et al. in ground-state and correlation energies across multiple basis sets, matching to many digits; for H$\hat{b}^\dagger,\hat{b}$7, QED-CCSD(2,2) matched QED-FCI exactly wherever comparison was possible, and origin invariance was confirmed by shifting the molecular coordinate origin without changing QED-HF or QED-CCSD(2,2) energies (Pathak et al., 2024).

Applications to ground-state molecular properties highlighted selective cavity sensitivity. In NaF, NaCl, NaBr, and NaI, ionization potentials were found to be only weakly affected by cavity coupling, whereas electron affinities decreased systematically with coupling; under experimentally accessible parameters, the electron affinities could be reduced by as much 0.22 eV, or about 50%, for NaF (III, 2020). Orientation scans showed that the largest cavity-induced electron-affinity shifts occurred at perpendicular rather than parallel alignment in that series (III, 2020).

Excited-state applications emphasized polaritonic structure rather than only energetic shifts. For MgF, EOM-EA-QED-CCSD-1 predicted a Rabi splitting of 1.20 eV near equilibrium for polarization along the molecular axis and 1.17 eV for a perpendicular polarization case; photon weights around 0.60 and 0.35 were reported for the two strongly mixed states at $\hat{b}^\dagger,\hat{b}$8 Å, and near stretched geometries the similarity-transformed Hamiltonian developed an accidental same-symmetry conical intersection with complex-valued eigenvalues (Liebenthal et al., 2021). In comparisons of relaxed and unrelaxed formulations, QED-EOM-CC excitation energies and Rabi splittings were usually similar at experimentally relevant couplings, with differences in Rabi splittings below about 9.3 meV in most cases, but growing to about 72.2 meV at strong coupling near an avoided crossing in LiF (Liebenthal et al., 2023).

6. Symmetry, analytical gradients, and the coherent-state problem

As the formalism expanded, derivation and implementation issues became topics in their own right. A diagrammatic notation generalizing Kucharski–Bartlett coupled-cluster diagrams to photons introduced wavy photon lines alongside electronic particle and hole lines, and showed how QED-CC energy and amplitude equations can be obtained from the Baker–Campbell–Hausdorff expansion in direct analogy with ordinary CC (Monzel et al., 2024). In coherent-state HF references, bubble diagrams generated by Wick reordering of the bilinear light–matter term vanish because the dipole expectation value is subtracted (Monzel et al., 2024).

Symmetry plays a central computational and interpretive role. In cavity calculations, the relevant symmetry is that of the combined molecule–cavity system, not the isolated molecule. Point-group implementations use direct-product decomposition to block amplitudes and Hamiltonian matrices, reduce computational cost, and target specific polaritonic states (Monzel et al., 2024). In the two-polarization treatment of unpolarized Fabry–Pérot cavities, this symmetry analysis was extended to two degenerate transverse modes; avoided crossings were shown to occur only when states of the same irreducible representation are coupled, and the additional polarization introduced extra photonic branches relative to linearly polarized models (Monzel et al., 25 Jul 2025).

Analytical gradients made QED-CC usable for cavity-modified structures. A Lagrangian/Z-vector derivation of ground-state QED-CC gradients at the QED-CCSD-1 level led to a Cholesky-based implementation in which gradient evaluation remained below 10% of total calculation time in the reported benchmarks. Geometry optimizations then showed cavity-induced orientation effects: cyclooctatetraene rotated so that the plane of the boat became perpendicular to the cavity polarization, cis-azobenzene adopted a finite angle to the field while changing internal torsion, and porphine rotated so that its molecular plane became perpendicular to the polarization vector (Lexander et al., 2024).

The coherent-state transformation has become the principal conceptual controversy. Practical studies had already shown that the relaxed coherent-state-transformed Hamiltonian guarantees origin invariance for post-SCF calculations and usually gives results close to unrelaxed QED-CC for neutral molecules (Liebenthal et al., 2023). A later analysis argued, however, that because the coherent-state displacement does not commute with the polaritonic cluster operator, a fully consistent QED-CC theory should transform not only the Hamiltonian but also the cluster operator, the deexcitation operator, and the Lagrangian (Fischer, 20 Feb 2026). In that formulation, both the correlation energy and the ground state are renormalized by dipole-dependent terms; the correction is small at large cavity frequencies, but the fully transformed theory exhibits a divergent zero-frequency limit for molecules with a non-vanishing molecular dipole moment, and origin invariance is broken for charged systems (Fischer, 20 Feb 2026). The literature therefore distinguishes a practically successful Hamiltonian-level coherent-state treatment from a stricter operator-level consistency analysis.

7. Related formulations and broader methodological landscape

Within the cavity-QED hierarchy itself, simplified correlation models have been connected back to QED-CC. A 2026 analysis generalized the standard electronic CCD/RPA equivalence to cavity QED and showed that QED-dRPA is exactly equivalent, at the level of ground-state correlation energy, to a QED direct-ring CCD model containing double electron excitations, coupled single electron excitation/single photon creation, and double photon creation (DePrince et al., 10 Feb 2026). This result places QED-RPA inside the QED-CC hierarchy and shows that the two-photon sector $\hat{b}^\dagger,\hat{b}$9, though often numerically small at realistic couplings, is part of the correct ring-limit structure (DePrince et al., 10 Feb 2026).

Quantum-computing variants have also been built directly on the cavity-QED coupled-cluster framework. QED-UCC replaces the nonvariational exponential by a unitary ansatz,

$\bm{\lambda}$0

and is optimized variationally within VQE; the same work introduced QED-EOM in the qubit basis and classical reference models QED-CCSD-1 and QED-CCSD-2, the latter explicitly including two-photon terms (Pavošević et al., 2021). That paper explicitly states that it is not a conventional purely classical QED-CC development, but it shows how the cavity-QED coupled-cluster ansatz can be transplanted into quantum-algorithmic settings (Pavošević et al., 2021).

Outside cavity quantum chemistry, electrodynamical coupled-cluster theory based on the standard QED Hamiltonian in Coulomb gauge remains a separate but relevant branch. There the formalism is built from a radiative cluster, pure matter clusters, and pair modifications; averaging over photon states generates Lamb, Breit, and hyperfine interactions, while de-excitations into negative-energy states generate vacuum-polarization or pair energy (Datta, 2024). Closed-shell variants use a Dirac–Fock picture, while open-shell variants employ MCDF or MCSCF-type references so that QED and electron correlation are treated on the same footing in multireference situations (Datta, 2024). Earlier work in the same line had already formulated a two-step QED-based CC framework that begins from a covariant formalism and equal-time approximation rather than from a preassembled relativistic electronic Hamiltonian (Datta, 2019).

A more distant relative is the light-front coupled-cluster method applied to QED. That approach uses the exponential coupled-cluster philosophy in a light-front Hamiltonian field-theory setting, truncating the operator $\bm{\lambda}$1 rather than the Fock basis, but it is explicitly not the usual equal-time quantum-chemical QED-CC framework (Chabysheva et al., 2012). Its inclusion underscores a broader methodological point: the coupled-cluster exponential remains a flexible organizing principle across cavity QED, relativistic electrodynamics, and nonperturbative quantum field theory, but the physical content of “QED-CC” depends strongly on the Hamiltonian, gauge, and Hilbert space to which it is applied.