Quantum Field Approaches to Chemical Systems

Open figure viewer Quantum-matter theory (QMT), based on the Schrödinger or Dirac equations, is firmly established for both intra- and intermolecular interactions. However, there are two key issues with QMT. First, its applicability to large molecular complexes is hindered by the relatively high computational cost of the calculations required to achieve high accuracy. Second, fields are also quantum objects that produce many intriguing effects beyond standard QMT approaches to molecular systems. This review focuses on recent developments in quantum-field theory (QFT) approaches to both covalent and noncovalent interactions for molecules in vacuum and subject to environments such as cavities and solvents. QFT provides a rich playground for novel chemical theories and insights. For example, chemical reactions and van der Waals interactions can be manipulated by cavities, boundaries, and optical excitations; novel interactions emerge when molecules interact with quantized fields; systems with millions of atoms could soon be treated with coarse-grained QFT formalisms; and unexpected scaling laws for atomic and molecular properties can emerge when QFT is applied to sets of chemical systems. This review sets the stage for an exciting QFT-driven path for further development of chemical theory. In the following, we first summarize the prominent experimental evidence demonstrating the impact of the quantum vacuum on chemical systems. We then present a perspective on QED and QFT methodologies in quantum chemistry. We begin by discussing the conventional quantum-chemistry approach to describing the electronic structure of molecular systems. Subsequently, we explore the advantages of employing field-theoretical methods, such as second quantization and the introduction of particle density (as a scalar field) and polarization tensor fields, in quantum chemistry. We then delve into the description of atoms and molecules in molecular QED, highlighting how QED effects influence noncovalent and covalent interactions (e.g., rotational and vibrational properties of bonds). Finally, we emphasize the importance of QED effects in precision experiments and the key considerations for developing predictive theories for both strong and weak QED effects. Before proceeding, we define our terminology to distinguish general theoretical frameworks from specific physical interactions. We use the term 'quantum field theory (QFT)' to denote the overarching conceptual framework, as well as the mathematical techniques inspired by field theory, such as second quantization, the representation of matter as continuous fields (e.g., scalar density fields, or vector fields describing matter polarization), and the handling of variable particle numbers in matter and fields. We also use QFT to describe phenomena inherently driven by the quantization of Fermionic matter fields, such as vacuum polarization (fluctuations of electron–positron or other particle–antiparticle pairs), which contributes to the Lamb shift. Conversely, we strictly use the words 'quantum electrodynamics (QED)' when referring to the specific subset of QFT that couples matter to the quantized electromagnetic field. Therefore, phenomena driven directly by photon exchange or electromagnetic vacuum fluctuations, such as finite speed-of-light retardation, polaritonic chemistry in cavities, and spontaneous emission, are explicitly termed QED effects. To provide a practical roadmap for these physical boundaries, ( Figure 2 ) presents a decision-tree guide that illustrates when standard QMT is sufficient and when specific QFT or QED frameworks become essential. In quantum chemistry, approaches that combine density-functional theory with many-body dispersion (MBD) methods ( 39, 42, 85, 86 ) successfully capture these cooperative effects for large molecular assemblies. However, when the electromagnetic field itself is quantized, such techniques no longer apply directly. From a QFT standpoint, matter is naturally described as a field whose excitations constitute the molecular structure, so many-body correlations arise intrinsically within the formalism. A comprehensive description of molecules interacting with quantized electromagnetic fields–whether in cavity vacuum conditions or under external driving–thus becomes a problem of interacting quantum fields (see Figure 1 d for an illustration of such a chemical QFT end goal). The advantage of this viewpoint is that the theoretical framework remains unified and scalable, regardless of whether one studies a small organic molecule or a complex macromolecular environment. As experimental control of large molecular systems with confined or structured light continues to advance, the need for a QFT-grounded, many-body–consistent description of molecular systems becomes increasingly apparent. In addition to external electromagnetic fields, collective interactions among neighboring atoms and molecules strongly influence the molecular response properties. As systems grow in size, many-body interactions–particularly van der Waals dispersion forces–modify the local electronic environment and can significantly enhance molecular polarizabilities. This enhancement is a genuine many-body effect: in one-dimensional carbyne chains, for example, the polarizability of each carbon atom increases by a factor of 50 along the extended structure due to cooperative dispersion interactions. ( 83, 84 ) Similar trends are observed in other extended materials, such as graphene, where atoms near the interior of a sheet exhibit larger in-plane polarizabilities than those at the edges. ( 83 ) Consequently, atoms or molecules embedded in large, correlated matter systems interact more strongly with the fluctuating electromagnetic field than their isolated counterparts, thereby amplifying QED effects and radiative corrections. Any realistic treatment of light–matter coupling in extended systems must therefore incorporate these many-body enhancements in the matter subsystem. An unambiguous signature of strong coupling is Rabi splitting in the absorption spectrum of materials inside cavities as illustrated in ( Figure 1 c), where the material's excitation and the cavity photons are tuned into resonance, forming upper and lower polariton states. ( 57−61 ) The formation of polaritons has profound implications, offering novel pathways to influence chemical reactivity ( 62−67 ) and conductivity, ( 68−71 ) shifting conical intersections, ( 72−74 ) or modifying molecular bond lengths. ( 55, 75, 76 ) Beyond coupling photonic states to molecular electronic or vibrational states, the electromagnetic nature of the fluctuating field also enables matter-field interactions via the magnetic degrees of freedom of these systems. Coupling the quantized field to molecular spins, which are the engines of their magnetic properties, alters spin interactions (e.g., via field-mediated spin–orbit coupling) and effectively retunes the molecules' magnetic responses, thereby directly influencing their aromaticity. ( 53, 54, 77 ) A natural framework accounting for such interactions is relativistic quantum electrodynamics, in which spin is inherently incorporated into the quantum-mechanical description of matter through the Dirac equation or its multiple-Fermion extensions. ( 78 ) Another advantage of employing relativistic QED is its ability to achieve precision in describing atomic and molecular systems at the level of fundamental constants, where relativistic and radiative effects become inseparable and measurable in the domain of precision spectroscopy. ( 79−82 ) The inclusion of quantized electromagnetic fields and their interactions with matter fundamentally modify the electronic structure and response properties of molecules. ( 45−56 ) The significance of such alterations depends on the strength of the photon-matter coupling, which can range from weak to strong and ultrastrong. In the weak-coupling regime, the bare states of atoms and molecules provide a sufficiently accurate description of matter, and the quantized field acts as a perturbation that yields radiative corrections. Well-known examples of phenomena arising from weak atom-field couplings include spontaneous emission, the Lamb shift, Casimir-Polder interactions, and the electron's anomalous magnetic moment. ( 1−5 ) In contrast, strong and ultrastrong atom-field couplings lead to the formation of photon-matter hybrid states, known as polaritons, ( 51−53, 56 ) which give rise to modified potential-energy surfaces and reshaped chemical landscapes. The ability to engineer these surfaces─for example, in optical cavities by tuning properties such as cavity frequency and coupling strength─is the central concept underpinning the emerging field of polariton chemistry. ( 51, 53, 56 ) Importantly, QED has proven extraordinarily successful in practice. It not only offers conceptual completeness, but also delivers numerical predictions that agree with experiments across a vast range of scales, establishing it as the most accurate physical theory currently available. The QED methodology provides conceptual clarity, enabling a deeper understanding and predictive modeling of quantum correlations beyond traditional particle-like treatments. Recent advances, exemplified by methods such as quantum-electrodynamical density functional theory (QEDFT) ( 30, 34−38 ) and coupled quantum Drude oscillator models (cQDO), ( 39−44 ) bridge the gap between highly accurate but computationally demanding techniques (e.g., CC theory and QMC methods) and more efficient but approximate DFT approaches. Quantum electrodynamics (QED) overcomes these deficiencies by quantizing both matter and the radiation field within a unified dynamical framework. This methodology provides a rigorous foundation for electron–photon interactions, fully accounting for processes mediated by virtual and real photons. For instance, incorporating QED effects modifies fundamental interatomic dimer potentials, such as altering the vdW dispersion interaction scaling from R –6 to R –7 due to retardation ( 33 ) at large distances R, or to R –1 and R –2 due to externally induced excitations of the fluctuating electromagnetic field. ( 26 ) Molecular Interactions from a Quantum Field Theoretical Perspective. (a) Molecular interactions between two polarizable entities and their corresponding QED Feynman diagrams (up to the fourth-order perturbation theory) in the minimal-coupling formalism. Here, H ^ f is the Hamiltonian of the fluctuating electromagnetic field (EMF), H ^ mol denotes the atomic/molecular Hamiltonians, V dd is the dipolar Coulomb coupling, and the remaining sum describes atom–field interactions (see Section S3 of the Supporting Information for more details about the Hamiltonian of a system of interacting matter and a quantized electromagnetic field). The diagrams represent processes up to fourth order in perturbation theory. (b) The well-known effects of the vacuum EMF on atoms/molecules, dispersive macroscopic bodies, or the combination of both. The presence of macroscopic bodies and boundaries alters EMF fluctuations, thereby influencing molecular interactions. In macroscopic QED, electric, magnetic, and geometric properties of the boundaries and macroscopic bodies or the bulk materials are encoded into classical Green functions, and the EMF is expressed in terms of that. (c) Inside optical cavities and resonators, strong matter-field couplings are reachable. In such situations, strong coupling between matter and the cavity EMF gives rise to hybrid states that combine properties of molecular and field states, known as polaritons. (d) A quantum-field-theoretical description of matter as a field interacting with other fields (e.g., EMF) allows for the full account of the many-body nature of molecular interactions under the influence of external fields and boundaries and enables the development of computational methods scalable from simple atomic dimers to complex biological macromolecules. Molecular Interactions from a Quantum Field Theoretical Perspective. (a) Molecular interactions between two polarizable entities and their corresponding QED Feynman diagrams (up to the fourth-order perturbation theory) in the minimal-coupling formalism. Here, H ^ f is the Hamiltonian of the fluctuating electromagnetic field (EMF), H ^ mol denotes the atomic/molecular Hamiltonians, V dd is the dipolar Coulomb coupling, and the remaining sum describes atom–field interactions (see Section S3 of the Supporting Information for more details about the Hamiltonian of a system of interacting matter and a quantized electromagnetic field). The diagrams represent processes up to fourth order in perturbation theory. (b) The well-known effects of the vacuum EMF on atoms/molecules, dispersive macroscopic bodies, or the combination of both. The presence of macroscopic bodies and boundaries alters EMF fluctuations, thereby influencing molecular interactions. In macroscopic QED, electric, magnetic, and geometric properties of the boundaries and macroscopic bodies or the bulk materials are encoded into classical Green functions, and the EMF is expressed in terms of that. (c) Inside optical cavities and resonators, strong matter-field couplings are reachable. In such situations, strong coupling between matter and the cavity EMF gives rise to hybrid states that combine properties of molecular and field states, known as polaritons. (d) A quantum-field-theoretical description of matter as a field interacting with other fields (e.g., EMF) allows for the full account of the many-body nature of molecular interactions under the influence of external fields and boundaries and enables the development of computational methods scalable from simple atomic dimers to complex biological macromolecules. On a more fundamental level, interactions between particles are consequences of their coupling to the quantum fluctuations of vacuum fields. Such fluctuations occur at various scales, but the most relevant to atomic and molecular forces are vacuum electromagnetic field fluctuations. ( 1−6 ) However, QMT treats atoms and molecules as closed quantum-mechanical systems and assumes that electromagnetic fields are fixed classical perturbations. ( 5 ) Although this simplification makes the theory convenient, it also means that the field is not part of the quantum system. Therefore, this semiclassical theory cannot consistently describe essential features, such as vacuum fluctuations and emergent phenomena, including finite speed-of-light effects in dispersion interactions (see Figure 1 a), the Lamb shift ( Figure 1 b), spontaneous photon emission, the well-known Casimir effect between macroscopic ensembles of atoms and molecules ( Figure 1 b), and the intricate interplay between electronic and photonic degrees of freedom. These phenomena become increasingly pronounced in large systems, strongly coupled molecular ensembles, and in external electromagnetic fields or optical cavity environments. ( 5, 26−32 ) The limitations of semiclassical theory become particularly severe when dealing with modern coherent light sources and precision spectroscopy, where the quantized nature of the vacuum field assumes a central role. Despite substantial progress in QMT method development, many fundamental questions remain about the completeness of QMT based on electronic Hamiltonians for large molecular systems, ( 22−24 ) and several puzzles persist even when comparing apparently well-established benchmark QM calculations. ( 25 ) For example, reference CC and QMC calculations are in mutual agreement on the binding energies of molecular complexes containing up to approximately 100 atoms; however, these 'benchmark' methods begin to disagree when molecules form supramolecular complexes with a significant contribution from van der Waals (vdW) dispersion interactions to their stability. ( 25 ) For a cycloparaphenylene-ring/buckyball complex containing 132 atoms, the minimal disagreement between the state-of-the-art CC and QMC calculations reaches 31 kJ/mol. ( 25 ) Identifying the reasons behind this unsettling discrepancy requires an improved understanding of electron correlations in many-particle systems over a wide range of spatial scales. Within QMT, interactions in matter are typically separated into intra- and intermolecular components and are widely studied under different categories, defined by bond types and energy scales. ( 14−16 ) However, the applicability of QMT to large molecular systems is limited by the immense computational cost of the calculations required to achieve the desired accuracy. Furthermore, molecules in cavities or subject to excitations (electric and magnetic fields as well as light) often require ad hoc approaches beyond the static Schrödinger equation. Painstaking quantum mechanical (QM) calculations based on coupled-cluster (CC) theory up to triple excitations and/or the quantum Monte Carlo (QMC) method can now reach an unprecedented accuracy of 1 kJ/mol (0.25 kcal/mol or 10 meV) per molecule for systems with a few dozen atoms or molecular crystals. ( 17−19 ) For several important classes of molecular systems, much more efficient semilocal density-functional theory (DFT), including nonlocal many-body dispersion interactions, can achieve predictive accuracy comparable to that of CC or QMC methods when compared with experimental reference data. ( 20, 21 ) However, DFT methods are routinely applicable to systems with only 100–1000 atoms. Our ability to model and understand chemistry is intimately tied to the development of quantum mechanics for interacting matter. Quantum-matter theory (QMT), based on the Schrödinger or Dirac equations, is the foundation of current computational methods and conceptual tools for modeling and understanding molecules and materials. The basis for QMT is the Hamiltonian of coupled nuclear and electronic degrees of freedom. Nuclear and electronic charges interact via the instantaneous Coulomb potential, giving rise to many-particle states, including molecular orbitals, molecular vibrations, and molecular plasmons (collective oscillations of the electron density). As many physicists and chemists know, this is an approximation. ( 1−6 ) The quantum vacuum, as the ground state of interacting fluctuating fields of various types, is the fundamental object endowed with its own dynamical degrees of freedom. Quantum field theory (QFT) is the fundamental approach that provides tools for quantizing both matter and fields. ( 1−3, 7 ) QFT approaches are not widely used in chemistry because treating the bound states in atoms and molecules is nontrivial in this theory. However, this is slowly changing in the 21st century, with many groups contributing to the brewing of a 'chemical QFT revolution'. The importance of QFT techniques is also highlighted by significant advances in quantum information and quantum computing applied to molecular systems. ( 8−13 ) It is important to stress that while the aforementioned experimental evidence focuses on specific atomic or molecular benchmarks, QED effects are fundamental and universal; they apply to all atoms, molecules, and materials characterized by many interacting electrons and nuclei. However, the precise scaling of QED phenomena for systems of increasing size remains an open question. In many-body systems, simple dipole-coupling models often fail to capture the nonadditive nature of the interaction, much as the vdW dispersion energy deviates from the standard pairwise interatomic R –6 power law in extended nanostructures. ( 83 ) From a QED perspective, rough scaling arguments based on the p ^ · A ^ interaction term (the coupling between momenta in matter and the vector potential of the EMF) suggest that the coupling strength is mediated by the square of the transition dipole moment (| μ | 2 ). Hence, in large polarizable systems, the delocalization of electrons and the emergence of collective modes can lead to a significant enhancement of this effective coupling, potentially making QED effects far more dominant than in isolated atoms. Identifying and extracting these contributions from experimental measurements remains a formidable challenge, as we currently lack a practical many-body theory capable of disentangling pure vacuum effects from intricate interparticle correlations that are omnipresent in chemical systems. In addition to electronic transitions, strong coupling of molecules to cavity vacuum fields can also occur via their vibrational degrees of freedom, a phenomenon termed Vibrational Strong Coupling (VSC). ( 51, 52 ) By tuning a cavity to resonate with specific bond vibrations, VSC hybridizes molecular vibrational modes with the vacuum field, fundamentally restructuring the ground-state potential energy surface and thereby modifying chemical kinetics and product selectivity without external optical pumping. A fundamental example of this 'chemistry in the dark' is the deprotection of 1-phenyl-2-trimethylsilylacetylene (PTA), where resonant coupling to its Si–C stretching mode at 860 cm –1 produced a Rabi splitting of 98 cm –1 and reduced the reaction rate by a factor of 5.5. ( 51, 52 ) Such modifications, which may also include rate enhancement through solvent coupling or the realization of mode-selective chemistry, ( 52 ) are typically enabled by collective coupling that scales with √N for N molecules. ( 52, 95 ) These effects are frequently attributed to dynamical caging by the cavity field and have recently been extended to the relativistic regime to control formally forbidden singlet–triplet transitions in heavy atoms such as mercury, providing a noninvasive approach to manipulate intersystem crossing and phosphorescence. ( 38, 78 ) In complex chemical systems (more intricate than small molecules or single atoms) confined within cavities, as shown in ( Figure 1 c), strong coupling between molecules and quantized fields has provided compelling experimental evidence that QED effects alter macroscopic chemical properties. The most direct indicator of strong coupling is the Rabi splitting of the molecular absorption spectrum into upper and lower polariton branches (P + and P – ), separated by the Rabi frequency ℏΩ R . A prominent example of Rabi splitting was observed with cyanine dye molecules and J-aggregates (TDBC) in Fabry–Pérot cavities. In these systems, substantial Rabi splittings of up to approximately 1 eV, or about 0.25 of the resonance frequency, have been measured at room temperature ( 51, 56, 93 ) (see Figure 5 b). These significant Rabi splittings, resulting from strong coupling of molecular electronic states to a cavity's vacuum field, have been shown to influence photochemical reactivity. For example, the photoisomerization reaction between spiropyran and mecrocyanine was modified by resonant coupling to a Fabry-Pérot cavity, resulting in a Rabi splitting of ℏΩ R = 700 meV, which could create a polaritonic barrier that effectively inhibits the reaction rate. ( 52, 62 ) Other studies have used plasmonic nanoantennas to generate a strong cavity field that couples to dye molecules, inducing large Rabi splitting that allows excited-state populations to decay faster than they would undergo triplet-state reactions, thereby suppressing the photobleaching of organic chromophores. ( 52, 94 ) Beyond isolated atoms, precision spectroscopy of small molecules provides a rigorous testbed for QED. Recent measurements of the hydrogen molecule and molecular helium ion ( 4 He + 2 ) have reached unprecedented accuracy, challenging theoretical models. For example, the lowest-energy rotational interval of 4 He + 2 has been measured as 70.937589 cm –1 , establishing a benchmark for three-body QED calculations. ( 79, 91 ) Discrepancies between theoretical predictions and experimental results in these systems often indicate the need for higher-order QED corrections. In the HD molecule, precision measurements have revealed deviations of 1.4σ to 1.9σ relative to current theoretical values, underscoring the complexity of nonadiabatic and relativistic QED effects in chemical bonds. ( 92 ) Furthermore, the proton radius puzzle, a 4% discrepancy in the proton charge radius measured via electronic versus muonic hydrogen spectroscopy, highlights the sensitivity of chemical systems to vacuum field interactions. ( 82 ) These findings confirm that even seemingly simple molecular properties are fundamentally connected to fluctuations in the underlying quantum fields. ( 79 ) Another notable early achievement of QED was its explanation of spontaneous emission. Semiclassical theory suggests that an atom not subjected to an external electromagnetic field does not radiate. Nevertheless, it is well established that a free atom in an excited state will eventually decay to its ground state by emitting a photon. In QED, where the vacuum is characterized by a fluctuating electromagnetic field, this emission is not entirely spontaneous, but is instead stimulated by the vacuum field. ( 2, 4 ) The recognition that spontaneous emission can be significantly enhanced or suppressed by the surrounding environment, as proposed by Purcell and termed the Purcell effect, inspired the concept of engineering the vacuum to control matter's behavior. This insight ultimately led to the development of Cavity-QED, ( 90 ) a field that has become prominent among physicists and chemists for investigating both weak and strong matter-field interactions. Beyond EMF, the uncertainty principle in quantum mechanics permits fluctuations in all other quantum fields, including both bosonic and Fermionic ones. Within quantum field theory, massive Fermionic fields are a natural generalization of the massless electromagnetic field. While photons can be created and annihilated in QED, general field theories also allow for the creation and annihilation of other massive or massless particles, such as Dirac Fermions. ( 1−3, 7 ) Early evidence for Fermionic vacuum effects was provided by Heisenberg and Euler, who examined the low-energy effective action of QED in a constant electromagnetic background and demonstrated that virtual electron-positron pairs polarize the vacuum. ( 2 ) Subsequent calculations and measurements of the Lamb shift ( 46, 50 ) confirmed that vacuum polarization induces spectral shifts, such as the 2S 1/2 –2P 1/2 splitting of approximately 27 MHz in hydrogen. Although this represents only a fraction of the total Lamb shift in an ordinary hydrogen atom, it is crucial to ensure that theory and experiment match. These effects are particularly significant in muonic atoms, where vacuum polarization dominates, leading to a substantial increase in the vacuum-polarization contribution to the Lamb shift. For example, in muonic helium, vacuum polarization accounts for approximately 90% of the Lamb shift. ( 89 ) From a historical perspective, the most remarkable evidence that a comprehensive theory was required beyond QMT (Schrödinger or Dirac equation for matter) emerged with the observation of splitting in the 2S 1/2 –2P 1/2 energy levels of the hydrogen atom. ( 2, 4, 45−50 ) The QMT, even when employing the Dirac equation, predicts degenerate 2S 1/2 and 2P 1/2 levels for hydrogen. ( 87 ) Lamb and Retherford, using microwave spectroscopy, ( 45 ) measured a small energy difference between these states, with the 2S 1/2 state having a slightly higher energy than the 2P 1/2 state. ( 46−48 ) The inability of the QMT to account for the Lamb shift arises from its neglect of the interaction between the bound electron and the fluctuating electromagnetic field (EMF). One interpretation of the Lamb shift attributes it to the atom's coupling to the quantized EMF fluctuations, as illustrated in ( Figure 1 b), which induce variations in the electron's position and thereby modify the potential it experiences. ( 2 ) Experimental measurements demonstrated that these fluctuations produce energy shifts that are most pronounced in S states, splitting degenerate levels, with the largest observed shift of 1057.8298 MHz for the 2S 1/2 –2P 1/2 transition (roughly 10 –4 kcal/mol). Although this shift appears insignificant in a hydrogen atom, it is worth mentioning that a muonic hydrogen exhibits a Lamb shift of approximately 4.8 kcal/mol due to the significantly higher mass of the muon. ( 88 ) The scaling of the Lamb shift for many-body systems beyond atoms remains unknown at present. The standard formulation of quantum chemistry models electrons as point-like quantum particles and nuclei as classical charges, interacting via the electrostatic Coulomb potential. While wave function-based methods built on this picture have been highly successful, the particle-oriented perspective exhibits intrinsic limitations, particularly in its ability to provide a unified and scalable framework spanning systems from small molecules to large biomolecular complexes in realistic environments. Subsection 3.1 reviews the strengths and limitations of particle-based descriptions of electronic degrees of freedom. Subsection 3.2 introduces a field-theoretical formulation in which both matter and interactions are described by the Schrödinger field for many-body Fermionic systems and the quantized electromagnetic field. Finally, subsection 3.3 presents the conceptual foundations of a more fundamental and unifying framework for nonrelativistic quantum electrodynamics (meaning that matter degrees of freedom are treated non relativistically) in which electronic matter is described by fields and interactions are mediated by fully quantum dynamical degrees of freedom of the electromagnetic field. At the same time, the Hamiltonian in eq 1 is structurally limited by its exclusive reliance on electrostatic Coulomb interactions. Here, the EMF does not appear as an independent dynamical quantum object, and energy/momentum exchange via its quantized excitations is excluded at the Hamiltonian level. These limitations are evident when matter degrees of freedom interact with real EMF excitations such as photons and/or electric and magnetic fields. This occurs, for example, with energy transfer in photosynthetic molecular complexes. ( 4, 113−117 ) However, EMF excitations are also relevant even without external photons. Intrinsic quantum EMF excitations are carried by 'virtual' photons, which can affect both intramolecular and intermolecular interactions. ( 4, 5 ) Examples include large-scale charge displacements over long distances ( 84, 118 ) in supramolecules or biomolecular complexes, or when EMF–matter coupling is enhanced by specific setups (e.g., molecules in optical cavities), ( 31, 52, 119 ) or when high computational accuracy is required to predict electronic, optical, or vibrational molecular spectra. ( 79 ) Moreover, leading-order quantum electrodynamic effects for relativistic bound electrons─most notably the Lamb shift─have been conjectured, on the basis of rough estimates, to contribute on the order of ∼1 kcal/mol to the energetics of small molecules containing heavy elements. ( 120 ) The same effects have been shown to produce measurable corrections to predicted single and double ionization energies of 1s and 2s orbitals for elements from the third row of the periodic table. ( 121 ) Taken together, these considerations motivate a reformulation in which both matter and electromagnetic degrees of freedom are described within a QFT framework. In the next section, we introduce the formalism of quantum fields, with particular emphasis on the Hamiltonian formulation of the electromagnetic field; the physical consequences and applications of this extended description are discussed in Sections 4 and 5 . In first quantization, spin statistics are not encoded in the observable algebra and must instead be imposed by antisymmetrizing the electronic wave function. As a result, the many-electron wave function is defined on a 4N e -dimensional configuration space, severely constraining the construction of systematically improvable and scalable ansätze. ( 103, 104 ) This leads to exponential scaling in wave function-based methods like Full Configuration Interaction, as well as expressivity limitations in Variational Quantum Monte Carlo. ( 105 ) These challenges are partially alleviated by the growing functional expressivity offered by the neural-network wave function ansätze. ( 106−108 ) Further complications arise when the electron number is not fixed, for instance, when exploring molecular chemical space. In this case, the wave function must then be defined over configuration spaces of varying dimensionality. Therefore, projection procedures are required to evaluate real-space observables, including the electron density, whose topological properties encode essential information about chemical bonding (see ref ( 109 ) and references therein). In addition, particle-index-based partitions implicit in first-quantized formulations can lead to ambiguities when applying quantum-information measures to systems of identical Fermions ( 110 ) (see Section 4.1 for further details). Moreover, first-quantization picture is not suited for describing relativistic effects for electrons in atoms and molecules. Such relativistic effects have been shown to be particularly relevant for core properties of heavy atoms. For instance, relativistic effects on the ionization energy and electron affinity of the gold atom are comparable in magnitude to electron correlation effects, as estimated at the coupled- cluster CCSD(T) level. ( 111 ) A consistent description of these effects, and the development of effective relativistic quantum Hamiltonians, require field-based approaches for matter. ( 112 ) These limitations motivate alternative representations of many-body quantum systems of matter particles in terms of fields defined in real or momentum space. In the first-quantized representation, the positions and momenta {} of each particle are promoted to Hermitian operators {} acting on the Hilbert space of square-integrable functions over real space. These operators satisfy the canonical commutation relations [] = iℏδand [] = [] = 0 where k, klabel particles and x, xlabel Cartesian coordinates. Neglecting the trivial Coulomb repulsion between nuclei, the molecular Hamiltonian takes the formHere,denotes the single-particle kinetic energy operator,the electron–electron Coulomb interaction, andthe electron–nuclear interaction potential, whereand Zdenotes the position and atomic number of the A-th nucleus, respectively. Equivalent expressions for the nucleus–electron and electron–electron interactions arewhere V) = (4πε∥)is the kernel modeling electrostatic Coulomb interactions,) = −e∑) is the electronic density operator, and ρ) = e∑) is the nuclear density. Note that self-interaction terms are unphysical and must be removed via regularization of the electronic density product, i.e.,:) ≔) – (−e))δ). It should be noted that all Coulomb interactions in the system can be rewritten in terms of the interaction between the electronic charge density field) = −e∑) and the electrostatic potential, t)where) ]is a functional of the total charge density operator) =) + ρ) that satisfies the Poisson equations, i.e.Such a formulation of the Coulomb Hamiltonian suggests that at a more fundamental level the interactions between matter degrees of freedom should be described as interactions between fields, e.g., the charge density field and the EMF. A key structural distinction between QFT and first-quantized quantum mechanics is particularly relevant for quantum-chemical applications across multiple length scales. In first quantization, spatial coordinates are promoted to operators, and no c-number variables directly represent real-space configurations. In QFT, by contrast, field amplitudes are quantized while spatial coordinates remain c-numbers, enabling a direct real-space description of structure formation and collective behavior, even for systems with arbitrary and fluctuating particle numbers. ( 126 ) Correspondingly, field operators are labeled by spatial coordinates or mode indices rather than by particle indices. As a result, observables such as densities and currents admit a direct real-space representation, without the need for projection procedures. Within this formalism, single-particle orbitals emerge as a choice of basis for expanding the field operators, rather than as explicit building blocks of the many-body wave function. The specific field used to represent the matter degrees of freedom depends on the physical regime: relativistic electrons are described by the Dirac field, whereas nonrelativistic electrons are described by the Schrödinger field (a concise summary of the principal conceptual aspects of quantum field theory is provided in Section S1 of the Supporting Information). As this review focuses primarily on a nonrelativistic treatment of matter degrees of freedom, selected applications of nonrelativistic field-based approaches to quantum chemistry are reviewed in Section 4 . In a field-theoretic formulation, quantum degrees of freedom are described by fields defined over space. Depending on their nature, such fields may transform as scalars, vectors, or spinors, and their dynamics is governed by partial differential equations in real space. ( 125−128 ) For practical purposes, a basis set of square-integrable field modes { ϕ k ( r )} k=1, ... +∞ is introduced, typically chosen as eigenfunctions of a linear operator describing the corresponding free field, allowing a generic field expansion, i.e., Φ ( r ,t) = ∑ k c k (t) ϕ k ( r ). Quantization is achieved by promoting the classical field amplitudes c k to operators acting on a Fock space, introducing annihilation and creation operators { c ^ k , c ^ † k } k and a vacuum state |0⟩ defined by c ^ k |0⟩ = 0. Acting on the vacuum, c ^ † k creates a single quantum of excitation in mode k. The states of the field are then represented in the occupation-number basis |n 1 , n 2 , ..., n k , ...⟩, defined as eigenstates of the number operators n ^ k = c ^ † k c ^ k . In QFT, the Heisenberg picture is often preferred: fields are treated as dynamical variables, while asymptotic quantum states (in Fock space) are time-independent. Although the interaction picture is commonly used for practical computations in chemical applications (e.g., perturbative response and Fermi's golden rule), care is needed. Unlike quantum mechanics, the vacuum states of free and interacting Hamiltonians are generally unitarily inequivalent in the continuum limit─a consequence of Haag's theorem. ( 128, 129 ) This formal subtlety underlines why QFT requires careful handling of interactions, even in nonrelativistic chemical settings. Spin–statistics are encoded directly in the operator algebra: bosonic fields satisfy canonical commutation relations, while Fermionic fields obey canonical anticommutation relations. This construction is referred to as second quantization ( 130 ) and forms the basis of field-theoretic approaches to quantum many-body systems. In the following sections, we illustrate the impact of field-inspired methods in quantum chemistry. We show how second quantization provides a natural framework for quantum-information analyses of electronic structure, yielding direct insight into covalent bonding through field-theoretical observables such as the energy–stress tensor. We further review how effective field-based Hamiltonians enable coarse-grained descriptions of electronic density and correlations ( Section 4.2 ), supporting multiscale modeling and the derivation of scaling laws linking microscopic electronic correlations to macroscopic physicochemical properties. It is worth emphasizing that, in both the minimal-coupling and multipolar formulations, the matter-EMF coupling can be rewritten just in terms of the matter density fields rather than in terms of individual particles. In a first-quantized description, these fields reduce to singular distributions supported at the particle coordinates, i.e., Dirac delta functions. By contrast, within a second-quantized formulation of matter, the electric charge and current densities become operator-valued fields and admit a mode expansion in terms of the underlying matter field operators. The particle-based description is recovered as a particular limiting case of this field-theoretic formulation, corresponding to states with fixed and pointwise localized particle content. It is worth noting that the constitutive relations defining the polarization and magnetization fields do not uniquely fix these fields themselves, but only the associated charge and current densities. As a consequence, the polarization and magnetization fields are defined only up to transformations that leave the physical sources invariant. ( 133 ) Such a gauge freedom reflects the fact that different choices of polarization and magnetization fields can represent the same underlying charge distribution. Formally, it constitutes the mathematical counterpart of the arbitrariness inherent in the choice of reference point (or reduction pole) used in the multipole expansion of an extended electronic charge distribution. For systems composed of spatially localized and distinguishable assemblies of bound charges─such as atoms, molecules, or molecular aggregates─it is often advantageous to adopt an alternative description of the EMF–matter interaction, inspired by the multipolar framework. In this approach, the matter degrees of freedom are described not by individual particle coordinates, but by macroscopic polarization and magnetization fields. This provides a mathematically rigorous formulation of the intuitive idea: that confined charge distributions can be effectively represented through their multipole moments (dipoles, quadrupoles, etc.). This motivates the use of the multipolar coupling Hamiltonian, obtained by applying the Power–Zienau–Woolley (PZW) unitary transformation to the combined matter–EMF system. ( 4, 5, 133 ) The PZW transformation recasts the EMF–matter interaction in terms of collective polarization and magnetization fields associated with spatially localized subsystems. For each localized charge set ξ, the interaction is described by the polarization field operator P ^ ξ ( r ,t), defined such that ρ ρ ^ ξ ( r ,t) = −∇· P ^ ξ ( r ,t), together with the magnetization field operator M ^ ξ ( r ,t) satisfying ∇ × M ^ ξ ( r ,t) = ∂ t P ^ ξ ( r ,t) – J ^ ξ ( r ,t) and the diamagnetic susceptibility tensor O ^ ξ ( r , r ′) (see Section S3 of the Supporting Information). The multipolar gauge, introducing a description of the quantum electronic degrees of freedom in terms of molecular polarization field, provides a rigorous and conceptually consistent framework for embedding and developing approaches to intermolecular noncovalent dispersion interactions. ( 134, 135 ) This section reviews representative applications of field-oriented descriptions of electronic quantum matter in chemistry. Subsection 4.1 revisits the Schrödinger field formalism─namely, the occupation-number representation of spin orbitals─in quantum chemistry, with applications to the analysis of chemical bonding, including approaches based on quantum information theory. Subsection 4.2 examines effective field-theoretical models of electronic degrees of freedom, with particular emphasis on density functional theory and the second-quantized formulation of many-body dispersion interactions. Finally, Subsection 4.3 discusses how field- oriented formulations of the electronic structure problem enable the derivation of scaling laws for key physicochemical observables, such as molecular polarizability, across a broad range of molecular sizes. The field-theoretical approaches for describing electronic quantum states allow one to adapt methods derived from continuous mechanics and quantum fluid mechanics to gain a deeper understanding of chemical bonds. A remarkable example in this sense is the 'rigged' QED formalism for describing covalent and hydrogen bonding (see refs ( 161, 162 ) and references therein for a comprehensive perspective). In this approach─particularly within the primary rigged QED framework tailored for quantum chemistry─both electrons and nuclei are treated as quantum Schrödinger fields coupled to the quantized electromagnetic field. Evaluating these fields in the ground state yields classical field representations, from which the real-space energy projection─the system's rank-2 stress–energy tensor─is derived. Analyzing the flow lines of its principal eigenvector reveals a characteristic spindle-shaped pattern signature of covalent bonding (see panel (b) of Figure 3 ). Recent extensions have captured subtle features of hydrogen bonding, ( 163 ) underscoring the power of quantum field-based descriptions of matter to uncover the field-theoretic roots of molecular interactions. Field-inspired approaches are effective in describing the real-space embedding of electronic structure properties beyond the local scale of covalent bonds. Applications of QFT-based methods to Quantum-information and electronic-stress analyses of molecular bonding. (a) Single-orbital entanglement (quantum relative entropy) in the Complete Active Space calculations correlating 10 electrons in 10 π orbitals for C 10 H 12 . The orbital numbering follows the upper panel (canonical from self-consistent Hartree–Fock/from Pipek-Mezey (PM) localization ( 136 )/atomic by Jacobi-rotation of PM π-orbitals ( 137 )). Colors indicate no Superselection Rules (SSR) (all colors), Parity-SSR (black/dark gray), and Number-SSR (black). Reproduced from ref ( 138 ). Available under a CC-BY 4.0 license. Copyright 2022 L. Ding, S. Knecht, Z. Zimboras, and C. Schilling. Published by IOP Publishing Ltd. (b) Bonding/antibonding characterization in H 2 -like dimers via eigenvalues and flow lines of the electronic stress tensor τ ↔ e ( r ) ⁠: the 1sσ state exhibits tensile stress (>0) at the bond midpoint and a spindle structure in the stress lines, whereas 1sσ*shows compressive stress (<0) and a disrupted stress topology consistent with antibonding character. Reproduced with permission from ref ( 139 ). Copyright 2018 John Wiley and Sons. Applications of QFT-based methods to Quantum-information and electronic-stress analyses of molecular bonding. (a) Single-orbital entanglement (quantum relative entropy) in the Complete Active Space calculations correlating 10 electrons in 10 π orbitals for C 10 H 12 . The orbital numbering follows the upper panel (canonical from self-consistent Hartree–Fock/from Pipek-Mezey (PM) localization ( 136 )/atomic by Jacobi-rotation of PM π-orbitals ( 137 )). Colors indicate no Superselection Rules (SSR) (all colors), Parity-SSR (black/dark gray), and Number-SSR (black). Reproduced from ref ( 138 ). Available under a CC-BY 4.0 license. Copyright 2022 L. Ding, S. Knecht, Z. Zimboras, and C. Schilling. Published by IOP Publishing Ltd. (b) Bonding/antibonding characterization in H 2 -like dimers via eigenvalues and flow lines of the electronic stress tensor τ ↔ e ( r ) ⁠: the 1sσ state exhibits tensile stress (>0) at the bond midpoint and a spindle structure in the stress lines, whereas 1sσ*shows compressive stress (<0) and a disrupted stress topology consistent with antibonding character. Reproduced with permission from ref ( 139 ). Copyright 2018 John Wiley and Sons. Recent research highlights the second-quantized framework as a natural, powerful setting for applying quantum information tools to quantum chemistry. Central to this is the concept of orbital entanglement: ( 110, 138, 153−155 ) in identical-particle systems, subsystems are defined not by partitioning particles, but by partitioning field modes─e.g., electronic spin–orbitals. Operators for a single Fermion (e.g., an electron) do not form a valid subsystem in many-body Fermionic systems ( 110 ) from which it follows that first-quantized reduced density matrices (RDMs)─obtained by tracing out electron positions─are conceptually problematic. In contrast, the algebra of creation/annihilation operators for a single orbital defines a proper subsystem, enabling consistent 1- and 2-orbital RDMs. Some care is required in dealing with Fermionic RDMs, as creation and annihilation operators for different orbitals do not commute in general─a feature consistent with the microcausality principle in nonrelativistic settings. ( 156 ) This implies that superselection rules must be applied to the RDM, forbidding superpositions of Fermionic states with different particle-number parity─the so-called parity superselection rule (P-SSR)─or preserving the total number of Fermions─the number superselection rule (N-SSR). ( 154, 155 ) Within this framework, quantum information methods yield new insights─e.g., via orbital entanglement estimators applied across spin–orbital bases to identify those best describing covalent bonds, bridging molecular orbital and valence bond theories. Atomic orbital bases maximizing orbital entanglement estimators are most suited (see panel (a) of Figure 3 ) for characterizing covalent bonds. Based on this, a new method assesses active orbital space quality in Complete Active Space Configuration Interaction, ( 157, 158 ) yielding results analogous to first-quantized CC via Domain-Based Local Pair Natural Orbital. ( 159, 160 ) Such a second-quantized description of the nonrelativistic electronic wave function forms the foundation of modern electronic structure methods, including Hartree–Fock theory and the CC approach, ( 144, 145 ) for which second quantization constitutes a fundamental basis. Recently, renewed interest in field-based formulations of CC theory has highlighted the connections to algebraic geometry, ( 146, 147 ) offering new perspectives on the structure and solvability of the coupled-cluster equations. Moreover, the second-quantization formulation of the electronic Hamiltonian in quantum chemistry enables the determination of ground-state properties using quantum computational methods. ( 148−152 ) In particular, the Jordan–Wigner transformation provides a mapping from Fermionic modes to qubit operators, thereby allowing the quantum chemistry Hamiltonian to be expressed as a sum of qubit operations. These examples indicate that a field-based formulation offers a natural and systematic framework for the construction of effective Hamiltonians describing matter degrees of freedom across different length and energy scales. In this context, a compelling extension of the MBD framework is to introduce an effective Hamiltonian for quantum electronic charge-density fluctuations defined with respect to a reference semiclassical electronic density obtained from DFT. Conceptually, this approach parallels semiclassical approximations in the path-integral formulation of quantum field theory, where the dynamics of fluctuations are treated perturbatively around a saddle-point configuration. This formulation would transcend the QDO paradigm by enabling an optimized and self-consistent determination of the spatial distribution, parametrization, and coupling constants of the effective oscillators, rather than prescribing them through semiempirical models. Applications of QFT-based methods to Quantum-information and electronic-stress analyses of molecular bonding. (a) Second Quantized Many-Body (SQ-MBD) decomposition of the Many-Body dispersion energy into intrafragment (U MBD ), interfragment pair (V MBD ), and per-fragment (E Frag MBD ) contributions, shown per residue in crambin (meV). ( 42 ) Reproduced from ref ( 42 ). Available under a CC-BY 4.0 license. Copyright 2023 M. Gori, P. Kurian, and A. Tkatchenko. Published by Springer Nature. (b) Reference van der Waals radii R ref vdW ( 140 ) versus electromagnetic field (EMF)-dressed radii for 72 elements, and corresponding polarizabilities (in atomic units); noble gases (red), transition metals (open symbols), and other elements (green). The plot illustrates the scaling relation between static polarizability A and the EMF-dressed van der Waals radius R f , with a prefactor involving the fine-structure constant α. Reproduced from ref ( 141 ). Available under a CC-BY 4.0 license. Copyright 2021 A. Tkatchenko, D. Fedorov, and M. Gori. Published by American Chemical Society. Applications of QFT-based methods to Quantum-information and electronic-stress analyses of molecular bonding. (a) Second Quantized Many-Body (SQ-MBD) decomposition of the Many-Body dispersion energy into intrafragment (U MBD ), interfragment pair (V MBD ), and per-fragment (E Frag MBD ) contributions, shown per residue in crambin (meV). ( 42 ) Reproduced from ref ( 42 ). Available under a CC-BY 4.0 license. Copyright 2023 M. Gori, P. Kurian, and A. Tkatchenko. Published by Springer Nature. (b) Reference van der Waals radii R ref vdW ( 140 ) versus electromagnetic field (EMF)-dressed radii for 72 elements, and corresponding polarizabilities (in atomic units); noble gases (red), transition metals (open symbols), and other elements (green). The plot illustrates the scaling relation between static polarizability A and the EMF-dressed van der Waals radius R f , with a prefactor involving the fine-structure constant α. Reproduced from ref ( 141 ). Available under a CC-BY 4.0 license. Copyright 2021 A. Tkatchenko, D. Fedorov, and M. Gori. Published by American Chemical Society. A field-theoretic formulation of electronic degrees of freedom grants direct access to real-space densities of physical observables, enabling spatially resolved characterization of intra- and intermolecular interactions. Central among these is the electronic charge density which underpins most molecular properties. The electron density is both computationally tractable and experimentally measurable─notably via X-ray diffraction and quantum tomography─and its geometric and topological features have been systematically analyzed to decode chemical bonding mechanisms. ( 109, 164 ) The centrality of the expectation value of the charge density field ρ el ( r ) = ⟨ ρ ρ ^ el ( r )⟩ GS in the ground state is formalized by the Hohenberg–Kohn theorem: a one-to-one mapping exists between the nuclear potential U ^ n–el ( r ) and ρ el ( r ), derived from the Hamiltonian in eq 1 . This implies an energy functional E[ρ el ] = F[ρ el ] + U el–n [ρ el ], where F[ρ el ] = T s [ρ el ] + E H [ρ el ] + E xc [ρ el ] is universal─encoding kinetic, Hartree, and exchange–correlation contributions─yet remains unknown in exact form. In Kohn–Sham DFT, the exchange-correlation functional E xc [ρ el ], in principle, accounts for all beyond-Hartree correlations. Its form, especially for what concerns long-range component─critical for nonlocal effects─remains the principal modeling challenge, motivating advanced approximations. QFT methods provide a first-principles route to F[ρ el ]: DFT arises naturally as an effective low-energy field theory for the density operator ( 165, 166 )─in the sense of an effective action for the density–enabling perturbative constructions of the exchange-correlation functional E xc . Nonperturbative approaches, inspired by functional renormalization, have recently yielded 3D universal functionals. ( 167 ) Despite decades of progress in constructing increasingly accurate energy density functionals, no universally applicable functional yet exists that reliably captures the contribution of electronic density correlations to interaction energies─particularly in noncovalent, many-body regimes. A promising strategy consists in defining model integrable Hamiltonians encoding electronic density correlations via physically motivated representations. A prominent example is the Many Body Dispersion (MBD) model. ( 39, 168−170 ) In this framework, the electronic response─derived from a chosen density functional approximation (DFA)─is mapped onto a system of quantum Drude oscillators (QDOs), which interact via a dipole–dipole electrostatic potential. The MBD Hamiltonian thus provides a low-energy effective description of long-range electronic correlations, typically manifesting at length scales ≫5 Å, through the collective normal modes of the coupled QDO system. This formalism yields computationally efficient, physically transparent expressions for the MBD energy─a contribution systematically absent in semilocal DFAs─enabling systematic studies of dispersion effects in large molecular complexes and extended materials. Moreover, MBD has been validated as a quantitative proxy for charge density distortion relative to high-accuracy references (e.g., CCSD(T)), when initialized with DFA-derived polarizabilities. ( 118 ) A second-quantized formulation of MBD, inspired by quantum field theory, was recently introduced. ( 42 ) It maps collective MBD modes─and the ground state of the interacting QDO system─onto atomic QDO displacements and noninteracting eigenstates via a Bogoliubov transformation. This allows one to derive a fragment-resolved decomposition of the MBD energy (see panel (a) of Figure 4 ), and provides a rigorous basis for applying quantum information tools to quantify correlation pathways in the MBD-induced charge density. ( 42 ) The results presented in this section demonstrate the power of the QFT formalism and techniques in providing a unified understanding of quantum chemistry, as described by eq 1 , across multiple length scales. However, as systems are scaled to larger and more complex structures, coupling to the quantized radiative electromagnetic field can no longer be neglected. In the next section, we discuss quantum-chemical applications in which treating the electromagnetic field as a dynamical quantum degree of freedom is essential for reproducing experimental observations. For this purpose it is necessary to consider the framework of nonrelativistic QED where matter and radiative dynamical EMF fields are mutually coupled. These examples show that scaling laws for both covalent and noncovalent interactions arise naturally when the electronic structure is formulated within a field-theoretic framework. A central challenge moving forward is the development of fully first-principles, QFT-based approaches capable of deriving scaling relations for a broad range of observables across molecular systems spanning vastly different length scales. A particularly important direction is the extension of scaling laws linking the real-space geometry of the electronic charge density to extensive response properties, such as the static polarizability. These properties govern long-range interactions and collective phenomena in large molecular assemblies, including biomolecular complexes comprising millions of atoms in realistic aqueous environments. Understanding how electronic and nuclear charge distributions in real space determine electronic-structure properties is therefore essential to develop accurate coarse-grained interaction models and to build chemical fragment databases to train machine-learning force fields with quantum-chemical accuracy. ( 175, 176 ) Field-theoretic methods provide a natural route to derive scaling laws for extensive response properties, due to flexibility of these methods in describing systems with different number of particles. In particular, recent work has shown that the static polarizability A of systems governed by central potentials scales as A ∝ L 4 / a 0 ⁠, where L is the characteristic length scale of the ground-state charge density and a 0 the Bohr radius, rather than following a naive volume scaling. ( 173, 174 ) This result underpins the theoretical derivation of the empirical relation R vdW ∼ A 1 / 7 for vdW radii ( 141 ) (see panel (b) of Figure 4 ). Importantly, the proportionality constant in this scaling law encodes the fine-structure constant α, revealing that vdW radii originate from a quantum electrodynamical dressing of electronic degrees of freedom by low-energy virtual photons. This perspective underlies field-theoretic and field-inspired approaches to chemical bonding. A notable example is Alchemical Perturbation Density Functional Theory (APDFT), which exploits functional derivatives of the ground-state DFT energy and electron density with respect to nuclear density to define an alchemical potential. ( 171, 172 ) This observable quantifies the response of electronic structure to changes in nuclear charge and provides chemically meaningful insight into bonding trends, such as variations in acidity or reactivity. By enabling the exploration of entire families of isoelectronic compounds without explicit enumeration, APDFT illustrates how field-theoretic concepts allow systematic navigation of chemical space. A central limitation of first-quantized quantum mechanics in the study of scaling laws is its reliance on a fixed-particle-number description. When the number of electrons changes across systems─an unavoidable situation when exploring chemical space─scaling relations for quantum-mechanical observables such as energies, multipole moments, or polarizabilities cannot be formulated in a unified manner. By contrast, QFT-inspired approaches operate within an occupation-number representation, naturally embedding Hilbert spaces corresponding to different particle numbers and thereby providing a consistent framework for analyzing trends across families of molecular systems. The realization that fluctuating electromagnetic fields mediate molecular interactions has revealed new opportunities to tailor material behavior by controlling those fluctuations through spatial confinement or external fields. The electronic structure and properties of the molecules 'dressed' by the fluctuating electromagnetic field in general deviate from those of the isolated molecules. To understand the physical concept of a dressed state, one can consider the paradigmatic shift in the energy levels of a hydrogen atom known as the Lamb shift. Although conceptually distinct, both the Lamb shift and polaritonic level restructuring originate from the coupling of matter to the quantized electromagnetic field and can be viewed as manifestations of electromagnetic dressing in different coupling regimes. Understanding such states requires a theoretical framework in which both the field and matter degrees of freedom are treated quantum-mechanically and evolve under the same fundamental principles. In this section, we discuss noncovalent and covalent interactions in QED and explain why a quantum electrodynamical extension of conventional quantum chemistry is essential for studying the mediation and modification of these molecular interactions via vacuum fluctuations. Atoms and molecules are constantly subject to quantum vacuum fluctuations arising from all underlying fields, mainly the electromagnetic field, but also the Dirac electron-positron field. The fluctuating EMF interacts with matter and can lead to feedback mechanisms, whereby the field alters the properties of matter and matter responds by exciting the field. The strength of these interactions is determined by how the field–matter coupling compares with the relevant loss and decoherence rates in both subsystems, i.e., relaxation and dephasing of the material excitations and photon loss of electromagnetic modes. When the rate of coherent field–matter energy exchange is slower than these decay processes, the system is in the weak-coupling regime. Such interactions can be studied perturbatively by treating them as perturbations to the isolated field and matter systems. In contrast, when the energy exchange between field and matter becomes faster than the losses of the combined system, the matter-field system must be described by hybridized states, ( 90 ) which can decouple after some time (e.g., in a cavity). Hence, the dynamics of each entity in the combined system become strongly interdependent, leading to hybrid states known as polaritons. The rapid, repetitive exchange of energy that occurs before energy leaks out of the system is called Rabi oscillation. Instead of the molecule's original energy levels, the hybrid state splits into two distinct levels (Upper and Lower Polaritons), separated by a gap called the Rabi frequency. ( 51, 52 ) By combining quantum mechanical descriptions of the electromagnetic field, for example, as found in cavity quantum electrodynamics, with the complete molecular characterization based on potential energy surfaces used in chemistry, these hybrid states are studied. In such studies, the goal is to generalize well-developed concepts from quantum chemistry to field-matter hybrids. Weak matter-field coupling is the fundamental mechanism behind numerous physical phenomena, including the Lamb shift and spontaneous emission. In molecular QED, intermolecular interactions are mediated through the vacuum electromagnetic field. Resonant energy transfer, vdW dispersion interactions, and scattering phenomena are examples of field-mediated interactions between atomic and molecular entities. In these cases, matter excitations are transferred to the electromagnetic field with probabilities that depend on the field's density of states. Consequently, they are influenced by the environment through its impact on the density of states of the field. Therefore, an advantage of considering these interactions from the QED perspective is that any change in the environment in which the systems interact, or the application of external fields to atoms and molecules, can be accounted for by examining how these changes affect fluctuations of the vacuum electromagnetic field. For example, imposing macroscopic boundaries on a system induces reflections of the vacuum field from these boundaries, thus altering the distribution of the field's modes and, consequently, modifying the molecular interactions. Such scenarios are the focus of macroscopic QED. (6) On the other hand, variations in matter systems, particularly changes in the electric and magnetic properties of atoms or molecules, can also alter how these entities interact with the vacuum field, thereby influencing molecular interactions. For instance, since chiral molecules exhibit distinct interactions with electromagnetic fields compared to nonchiral ones, molecular interactions within chiral systems also differ. Another example is when interacting atoms are initially excited. In such cases, in addition to the vdW dispersion interaction, resonance contribution is also present due to the exchange of real photons between the molecules. (177, 178) H ^ 0 = H ^ atom + H ^ field (see H ^ atom and cv field in the minimal-coupling formalism of QED) are expressed as the product states of the atoms and the field, i.e. | Ψ ( 0 ) ⟩ = | ψ ( 0 ) ( r 1 ) ⟩ | ψ ( 0 ) ( r 2 ) ⟩ | n k 1 λ 1 , n k 2 λ 2 , ... ⟩ (15) k i λ i denote the occupation numbers of the field's modes with frequency ω i = ck i and polarization λ i , and |ψ(0)(r i )⟩ are the eigenkets of isolated atoms. Using these states, perturbation calculations can be performed to determine the energy of the total system under the perturbation Hamiltonian ( E (0) f = ∑ ∞ i=1 ℏck i . Nevertheless, it is evident that the interaction energy between the two atoms, which is dependent on the interatomic distance R, is finite and can be obtained by renormalizing the energy of the interacting matter-field system with respect to the energy of the noninteracting system (R → ∞). To evaluate the interaction energy between atoms, one must solve the equation of motion for the entire matter-field system, which is challenging even in the simplest scenario of two interacting atoms. Consequently, approximate methods and simplified physical models are necessary to address this problem. Among the approximate approaches in the weak coupling regime, perturbation theory stands out as one of the most widely employed, within which atom-field and atom–atom couplings are considered perturbatively. However, this approach is only practical for a small number of atoms, thereby avoiding higher-order perturbations. In the case of two atoms, the states of the unperturbed system with(see Section 3.3 of the main text and Section S3 of the Supporting Information for the definition ofand cvin the minimal-coupling formalism of QED) are expressed as the product states of the atoms and the field, i.e.where ndenote the occupation numbers of the field's modes with frequency ω= ckand polarization λ, and |ψ)⟩ are the eigenkets of isolated atoms. Using these states, perturbation calculations can be performed to determine the energy of the total system under the perturbation Hamiltonian ( eq 8 ). As is well-known in quantum field theory and QED, the energy of the matter-field system is infinite due to the fact that the energy of the fluctuating field in its vacuum state is the sum of zero-point energy of an infinite number of modes,= ∑ℏck. Nevertheless, it is evident that the interaction energy between the two atoms, which is dependent on the interatomic distance R, is finite and can be obtained by renormalizing the energy of the interacting matter-field system with respect to the energy of the noninteracting system (R → ∞). An elegant approach to perturbative evaluation of interaction energies in QED and quantum field theory is the use of Feynman diagrams. In the minimal-coupling formalism of molecular QED, typical Feynman diagrams representing the interaction of two atoms resemble those depicted in (Figure 1a). (179) Vertices in these diagrams denote atom–atom or atom-field interactions accompanied by virtual transitions of the subsystems, thereby indicating the perturbation order for each scenario depicted in the diagram. Consequently, diagrams (i) and (ii) represent second-order perturbation, (iii) and (iv) represent third-order perturbation, and diagram (v) represents fourth-order perturbation. Diagram (i) illustrates an interaction between the atoms from second-order perturbation with the instantaneous dipolar Coulomb coupling V dd . This diagram corresponds to the same physical process responsible for London dispersion interaction in semiclassical theory. In contrast, in diagram (ii), the interaction is solely attributed to the term A ^ 2 and the exchange of two virtual photons that are emitted or absorbed simultaneously at the atomic centers. The third-order diagrams (iii) and (iv) depict interactions arising from mixed processes. In diagram (iii), an instantaneous dipolar Coulomb coupling (V dd ) is followed by the exchange of a virtual photon between the atoms due to two consecutive atom-field couplings p ^ · A ^ at the two atomic centers. The physical process underlying diagram (iv) involves the exchange of two virtual photons between the two atoms. In the first atom-field coupling ( p ^ · A ^ ) at the atomic center B, a virtual photon is emitted. Subsequently, atom A interacts with the field via A ^ 2, which involves simultaneous emission and absorption of two virtual photons [Note that A ^ 2 includes combinations of creation and annihilation operators of photons ( a ^ † n and a ^ n , respectively) such as a ^ † n a ^ n and a ^ n a ^ † n , enabling two-photon processes through A ^ 2 atom-field couplings. For more details about these bosonic operators and their commutation relations, refer to Section S1 of the Supporting Information]. Finally, the remaining photon is absorbed in another p ^ · A ^ interaction at atom B. The fourth-order diagram (v) is the result of four consecutive p ^ · A ^ interactions (two at each atom), which lead to the exchange of two virtual photons between the atoms. Since the dipole polarizability of an atom is proportional to e2, or equivalently to the dimensionless fine-structure constant α, the power of e2 or α can be regarded as a measure of the magnitude of atom–atom and atom-field interactions. Considering the order of perturbations and the physical processes associated with each diagram in (Figure 1a), it is evident that all five diagrams exhibit the same power of e4 (or α2), indicating that they possess comparable physical significance, despite originating from different perturbation orders. Naturally, there are additional diagrams, but all of them can be classified into one of these five categories. By summing over all possible diagrams, the total interaction energy between the two atoms can be determined. Including all possible diagrams is essential to ensure that the interaction energy is computed from a physical picture consistent with QED principles. In particular, in diagrams (i) and (iii), the two atoms interact via an instantaneous dipolar Coulomb coupling, which, at first, might seem unphysical within the QED framework. However, considering every relevant contribution of the same order with respect to the fine structure constant (α), all instantaneous contributions to the interaction energy cancel out, (33) and the remaining expression is fully retarded, thereby complying with the relativistic nature of the fluctuating electromagnetic field. This cancellation of instantaneous interactions indicates that, in molecular QED, all interactions are effectively mediated through the field and the exchange of transverse virtual photons. The mediation of intermolecular interactions by the fluctuating electromagnetic field explains the retardation effects on these interactions. For two atoms to interact, they must interact with the field and borrow polarization and energy from it in the form of virtual transitions. Then, the borrowed polarization and energies enable the atoms to interact via the exchange of virtual photons. However, the borrowing process is subject to the uncertainty principle. The borrowed energy must be returned within a time frame that does not violate the energy-time uncertainty. This means that, at large interatomic distances, where virtual photons have to travel for a long time, those virtual photons belonging to the electromagnetic field's modes with higher frequencies contribute less compared to those with lower frequencies. Conversely, high-frequency modes contribute more significantly to the interaction when the interatomic distance is small. (4) In the nonretarded limit, where the interatomic distance R is significantly smaller than the characteristic atomic wavelengths, i.e., R ≪ λ, the general expression for the interaction energy can be approximated to reproduce the London dispersion interaction (R–6). In the opposite limit, where R ≫ λ, the interaction energy is given by the Casimir-Polder expression (R–7). (33) An alternative formalism of molecular QED can be obtained by applying a quantum canonical transformation (the PZW transformation), to the minimal-coupling Hamiltonian (eq 8). As shown in Section S3 of the Supporting Information, in the resulting Hamiltonian, matter-field interactions arise from the coupling of matter's electric and magnetic multipole moments to the electric and magnetic fields; hence, the formalism is referred to as multipolar coupling. In dipole approximation, the interaction Hamiltonian is given by H ^ mult int = −∑ ξ x3b5–1 0 μ μ ^ ξ · D ^ ⊥(R ξ ), where μ μ ^ ξ is the electric dipole moment of atom ξ and D ^ ⊥ is the transverse component of the microscopic displacement field (there is also another term in the multipolar Hamiltonian that must be taken into account when computing self-energies, e.g., Lamb shift, but does not play a role in the interaction between atoms or molecules (4, 5)). In this formalism, the interaction Hamiltonian consists solely of couplings between matter and the field, with no direct couplings among the matter components; therefore, the retarded nature of the interactions is evident from the Hamiltonian. Additionally, the multipolar form of matter-field interactions facilitates the interpretation of the physical behavior of the resulting interaction energies, as they have counterparts in classical electromagnetism. It is seen that in multipolar formalism, the dispersion interaction between two neutral nonpolar atoms is obtained from fourth-order perturbation theory with the interaction Hamiltonian H ^ mult int . (4, 5) As noted earlier, one advantage of the QED perspective on intermolecular interactions is that it enables us to understand how physical conditions influence these interactions by considering their effects on the quantum-mechanical fluctuations of the electromagnetic field. Introducing media or boundaries, such as macroscopic bodies, disrupts and alters these fluctuations. Scattering of the electromagnetic field at these boundaries alters the distribution of modes and their fluctuations, leading to new interactions between atoms. In macroscopic QED, these medium-induced effects are analyzed using classical Green's functions that incorporate all the geometric and electromagnetic properties of the environment in which the atoms interact. (6) An alternative approach to modifying the fluctuating electromagnetic field is to apply an external dynamic field, such as a monochromatic laser beam. This external field can be regarded as an excitation of the vacuum field, thereby producing real photons in addition to the virtual photons that atoms use for interaction. In an intense field, dispersion interactions undergo significant changes and scale with the interatomic distance R, unlike the well-known Casimir-Polder and London dispersion energies. These changes depend on the polarization of the applied field and the orientation of the interacting pair relative to the direction of the field's propagation. When averaged over all directions of R, an attractive dispersion interaction ∝R–1 is observed in the near zone. Conversely, in the far zone, the dispersion interaction is proportional to R–2k f sin(2k f R), where ck f represents the frequency of the applied field. Consequently, the interaction in the far zone can be either attractive or repulsive depending on the ratio k f /R. (26) Instead, if the external field is static, the dynamics of the electromagnetic field remain unchanged, but the atoms acquire additional static dipole moments that allow new channels of interaction with the vacuum field and with one another. In the multipolar-coupling scheme, this corresponds to extending the interaction Hamiltonian to H ^ mult int = −∑ ξ x3b5–1 0 [ μ μ ^ (f) ξ + μ μ ^ ξ (s)]· D ^ ⊥(R ξ ), where μ μ ^ (s) ξ is the external-field–induced dipole and μ μ ^ (f) ξ the intrinsic fluctuating dipole. Perturbation theory then yields interaction-energy contributions ΔE = ΔE(2) ss + ΔE(4) fs + ΔE(4) ff + ..., where each term corresponds to a distinct atom–field coupling channel. The ss channel describes the interaction of two induced dipoles mediated by the vacuum field, giving a field-induced electrostatic term ∝R–3 that may be attractive or repulsive. The mixed fs channel corresponds to the coupling of a fluctuating dipole on one atom with an induced dipole on the other, producing an always-attractive polarization interaction ∝R–6. Finally, the ff channel represents the coupling of two fluctuating dipoles and produces the familiar dispersion interaction ∝R–6 (nonretarded) or ∝R–7 (retarded). The many-body character of these forces and the interplay between them imply that external static fields can be used to tailor noncovalent interactions (for example, in a benzene dimer as shown in (Figure 5a)), (180−184) enabling applications such as tunning materials' properties using external fields (185−187) and field-assisted exfoliation of layered nanostructures. (188, 189) Figure 5 View LargeDownload to Slide Figure 5. Atoms and Molecules Coupled with Quantum Electromagnetic Field (EMF): (a) interplay between molecular forces induced by an external static field and dispersion force due to the vacuum EMF in two configurations of a benzene dimer demonstrates the possibility of tailoring molecular forces through external fields. Many-body characteristics of dispersion and field-induced polarization forces (taken into account using the MBD method (85)) result in an intricate interplay between these forces and the field-induced electrostatic force. Reproduced from ref (181). Available under a CC-BY 4.0 license. Copyright 2022 M. R. Karimpour, D. V. Fedorov, and A. Tkatchenko. Published by American Chemical Society. (b) Electronic strong-coupling of molecules with the cavity EMF yields the matter-field hybridization and formation of upper and lower polaritons that are spectroscopically observable. Reproduced from ref (51). Copyright 2016 American Chemical Society. (c) Vibrational strong-coupling of molecules with the cavity EMF: (1) formation of vibrational polariton states | ± ⟩from hybridization of molecular and cavity excitations. (2) Suppression of reaction rates as a function of cavity photon frequency inside and outside the cavity, with the IR spectrum of the uncoupled molecule. (3) Enhancement of reaction rates under vibrational strong coupling. (4) Cavity-induced mode selectivity in a reaction with two competing products. Reproduced from ref (52). Available under a CC-BY 4.0 license. Copyright 2023 A. Mandal, M. A.D. Taylor, B. M. Weight, E. R. Koessler, X. Li, and P. Huo. Published by American Chemical Society. [Panel (c.2) is reproduced with permission from ref (63). Copyright 2016 John Wiley and Sons. Panel (c.3) is reproduced with permission from ref (65). Copyright 2019 John Wiley and Sons. Panel (c.4) is reproduced with permission from ref (64). Copyright 2019 The American Association for the Advancement of Science.]. Figure 5 View LargeDownload to Slide Figure 5. Atoms and Molecules Coupled with Quantum Electromagnetic Field (EMF): (a) interplay between molecular forces induced by an external static field and dispersion force due to the vacuum EMF in two configurations of a benzene dimer demonstrates the possibility of tailoring molecular forces through external fields. Many-body characteristics of dispersion and field-induced polarization forces (taken into account using the MBD method (85)) result in an intricate interplay between these forces and the field-induced electrostatic force. Reproduced from ref (181). Available under a CC-BY 4.0 license. Copyright 2022 M. R. Karimpour, D. V. Fedorov, and A. Tkatchenko. Published by American Chemical Society. (b) Electronic strong-coupling of molecules with the cavity EMF yields the matter-field hybridization and formation of upper and lower polaritons that are spectroscopically observable. Reproduced from ref (51). Copyright 2016 American Chemical Society. (c) Vibrational strong-coupling of molecules with the cavity EMF: (1) formation of vibrational polariton states | ± ⟩from hybridization of molecular and cavity excitations. (2) Suppression of reaction rates as a function of cavity photon frequency inside and outside the cavity, with the IR spectrum of the uncoupled molecule. (3) Enhancement of reaction rates under vibrational strong coupling. (4) Cavity-induced mode selectivity in a reaction with two competing products. Reproduced from ref (52). Available under a CC-BY 4.0 license. Copyright 2023 A. Mandal, M. A.D. Taylor, B. M. Weight, E. R. Koessler, X. Li, and P. Huo. Published by American Chemical Society. [Panel (c.2) is reproduced with permission from ref (63). Copyright 2016 John Wiley and Sons. Panel (c.3) is reproduced with permission from ref (65). Copyright 2019 John Wiley and Sons. Panel (c.4) is reproduced with permission from ref (64). Copyright 2019 The American Association for the Advancement of Science.]. Close Figure 5. In addition to these external influences, the structure of the matter system itself plays a crucial role in the resulting QED effects, since molecular vdW interactions inherently reflect the collective response of many atoms to vacuum fluctuations. These interactions cannot be obtained by summing pairwise contributions (85) because each atom couples not only to the vacuum field but also to the fluctuating dipoles of its neighbors. As more atoms or molecules are added, the pattern of electromagnetic fluctuations changes, which alters both the interactions among particles and the overall coupling of the system to the quantum vacuum field. Many-body effects are known to delocalize the charge density along symmetry axes in extended systems such as macromolecules and nanostructures, (84) leading to substantial enhancements in atomic polarizabilities and, consequently, atom-field couplings. The resulting increase in matter-field coupling amplifies QED effects and makes explicit many-body dispersion essential in any realistic QED description of large molecular assemblies. Since solving the fully coupled atom–field dynamics is generally implausible, practical treatments rely on controlled approximations, numerical many-body methods, and simplified models, such as a QED extension of MBD. (43, 190)