Providing sufficient accuracy at low-computational cost, not surprisingly, density-functional theory ( DFT ) is a standard tool in theoretical material science. Similarly, quantum-electrodynamical density-functional theory ( QEDFT ) provides a promising route to describe experiments at the interface of quantum optics and quantum chemistry. Formally, in the Kohn-Sham ( KS ) formulation of DFT and QEDFT the exact many-particle Schrödinger equation is replaced by a set of coupled non-linear one-particle differential equations. The electron-electron, the electron-ion and the electron-photon interactions of the interacting system are mimicked by their corresponding effective KS potential with an, in principle unknown exchange-correlation ( xc ) term. In practice, DFT and QEDFT require approximations for this xc potential. The quality of which fully determines how much the DFT and QEDFT observables deviate from the exact ones. Understanding the shortcomings and improving the quality of approximate functionals makes the study of the exact functional of model systems indispensable. Here, we employ exact diagonalization for the Hamiltonian of different types of real-space lattice models in one spatial dimension to access the ground-state wave function, and thus to explicitly construct the exact functionals of DFT and QEDFT . To study the functional behavior of purely electronic systems, we consider two singlet-state electrons moving in an external potential with two connected atomic wells of variable depth and distance. To study coupled electron-photon systems, we consider the Rabi Hamiltonian with variable electron-photon coupling strength. By varying the model parameters, we systematically construct the functionals for systems from weak to strong coupling between the building blocks of the global system. Using the concept of the effective density-to- potential map, we discuss the intrasystem steepening which is an exact condition on the xc functional that captures the static correlation present in the system and carve out how it relates to the well-known intersystem derivative discontinuity. We further construct arbitrary observables as functionals of the internal variables which are directly affected by such features of the effective density-to-potential map. By constructing the reduced dipole map in the four-particle case of coupled electron-photon system, we show that the density-to-potential map helps to identify and understand recently discovered physical effects in the field of strongly coupled electron-photon systems. This work provides a contribution to the development of approximate function- als required in practical density-functional calculations of arbitrary experimentally accessible observables. Both the intrasystem steepening and the intersystem derivative discontinuity play a crucial role for systems with highly-localized electrons. Including both features in approximate functionals will improve the accu- racy of DFT evaluation of observables for such systems. Those results provide novel insight into the properties of the exact xc potentials in QEDFT . This work opens a new route to construct new functionals using coarse-grained or embed- ding approaches within the general and exact frameworks of DFT and QEDFT.
Providing sufficient accuracy at low-computational cost, not surprisingly, density-functional theory ( DFT ) is a standard tool in theoretical material science. Similarly, quantum-electrodynamical density-functional theory ( QEDFT ) provides a promising route to describe experiments at the interface of quantum optics and quantum chemistry. Formally, in the Kohn-Sham ( KS ) formulation of DFT and QEDFT the exact many-particle Schrödinger equation is replaced by a set of coupled non-linear one-particle differential equations. The electron-electron, the electron-ion and the electron-photon interactions of the interacting system are mimicked by their corresponding effective KS potential with an, in principle unknown exchange-correlation ( xc ) term. In practice, DFT and QEDFT require approximations for this xc potential. The quality of which fully determines how much the DFT and QEDFT observables deviate from the exact ones. Understanding the shortcomings and improving the quality of approximate functionals makes the study of the exact functional of model systems indispensable. Here, we employ exact diagonalization for the Hamiltonian of different types of real-space lattice models in one spatial dimension to access the ground-state wave function, and thus to explicitly construct the exact functionals of DFT and QEDFT . To study the functional behavior of purely electronic systems, we consider two singlet-state electrons moving in an external potential with two connected atomic wells of variable depth and distance. To study coupled electron-photon systems, we consider the Rabi Hamiltonian with variable electron-photon coupling strength. By varying the model parameters, we systematically construct the functionals for systems from weak to strong coupling between the building blocks of the global system. Using the concept of the effective density-to- potential map, we discuss the intrasystem steepening which is an exact condition on the xc functional that captures the static correlation present in the system and carve out how it relates to the well-known intersystem derivative discontinuity. We further construct arbitrary observables as functionals of the internal variables which are directly affected by such features of the effective density-to-potential map. By constructing the reduced dipole map in the four-particle case of coupled electron-photon system, we show that the density-to-potential map helps to identify and understand recently discovered physical effects in the field of strongly coupled electron-photon systems. This work provides a contribution to the development of approximate function- als required in practical density-functional calculations of arbitrary experimentally accessible observables. Both the intrasystem steepening and the intersystem derivative discontinuity play a crucial role for systems with highly-localized electrons. Including both features in approximate functionals will improve the accu- racy of DFT evaluation of observables for such systems. Those results provide novel insight into the properties of the exact xc potentials in QEDFT . This work opens a new route to construct new functionals using coarse-grained or embed- ding approaches within the general and exact frameworks of DFT and QEDFT.
The last two decades have witnessed increasing experimental interest in the study and control of many-electron systems strongly interacting with quantum electromagnetic fields. This includes notable experiments in the areas of cavity and circuit quantum electrodynamics, quantum computing via photon mediated atom entanglement, electromagnetically induced transparency, quantum plasmonics, quantum simulators and chemistry. The description of realistic coupled matter-photon systems requires combining electronic structure methods from material science with quantum-optical methods. In this work, we propose a formally exact and numerically feasible approach, termed quantum electrodynamical time-dependent density functional theory (QED-TDDFT), that generalizes successful TDDFT for electronic structure calculations, to the electron-photon coupling. We develop a general framework for the description of coupled matter-photon systems in all possible realizations of interest, ranging from the fully relativistic case, to well-known quantum-optical model (e.g., the Rabi model). Of particular interest for condensed-matter applications is the density functionalisation of non-relativistic QED. The underlying theory in fact corresponds to standard non-relativistic quantum mechanics for the electrons, but with additional terms, which correct for relativity and electron-photon interactions. However, any application of density-functional theory requires approximations to the exchange-correlation (xc) functional. In this work we construct such an approximation for the description of electron-photon interaction effects in an optical cavity (the theory of reference is QED-TDDFT for many-electron systems coupled to cavity photons). The derived electron-photon optimized effective potential (OEP) reduces in the static limit to the Lamb shift of the ground state. The OEP is then tested on the Rabi model from weak to strong coupling regimes. The results are promising and open a path for describing complex strongly coupled matter-photon systems. Further, we consider the inclusion of relativistic spin-spin effects into the xc functional of standard spin density functional theory. In this way we provide quantum corrections that are essential for the description of magnetic inhomogeneities at the nanoscale. Such an approach improves over the current (classical) micromagnetic treatment of domain walls and skyrmions for spintronic applications.
We present results of a photon-free exchange-correlation functional within the local density approximation (pxcLDA) for quantum electrodynamics density functional theory (QEDFT) that efficiently describes the electron density of many-electron systems across weak to strong light-matter coupling. Building on previous work [I-Te. Lu et al., Phys. Rev. A 109, 052823 (2024)] that captured electron-photon correlations via an exchange-correlation functional derived from the nonrelativistic Pauli-Fierz Hamiltonian and tested on one-electron systems, we use a simple procedure to compute a renormalization factor describing electron-photon correlations and inhomogeneity in the weak-coupling regime by comparing it with quantum electrodynamics coupled-cluster, and previous QEDFT optimized effective potential methods. Across various atoms and molecules, pxcLDA reproduces cavity-modified densities in close agreement with these references. The renormalization factor approaches unity as the system size or collective coupling increases, reflecting an electron-photon exchange-dominated behavior and improved accuracy for larger systems. This approach now offers a practical route to applying QEDFT functionals based on electron density to realistic electron systems.
In the present work, we employ exact diagonalization for model systems on a real-space lattice to explicitly construct the exact density-to-potential and graphically illustrate the complete exact density-to-wavefunction map that underly the Hohenberg–Kohn theorem in density functional theory. Having the explicit wavefunction-to-density map at hand, we are able to construct arbitrary observables as functionals of the ground-state density. We analyze the density-to-potential map as the distance between the fragments of a system increases and the correlation in the system grows. We observe a feature that gradually develops in the density-to-potential map as well as in the density-to-wavefunction map. This feature is inherited by arbitrary expectation values as functional of the ground-state density. We explicitly show the excited-state energies, the excited-state densities, and the correlation entropy as functionals of the ground-state density. All of them show this exact feature that sharpens as the coupling of the fragments decreases and the correlation grows. We denominate this feature as intra-system steepening and discuss how it relates to the well-known inter-system derivative discontinuity. The inter-system derivative discontinuity is an exact concept for coupled subsystems with degenerate ground state. However, the coupling between subsystems as in charge transfer processes can lift the degeneracy. An important conclusion is that for such systems with a near-degenerate ground state, the corresponding cut along the particle number N of the exact density functionals is differentiable with a well-defined gradient near integer particle number.
Accurate descriptions of the electronic properties of molecules, solids and other materials are crucial in the development of many modern technologies, and are therefore the subject of much scientific research. In tandem with experimental research, computational simulations play a huge role. Density functional theory (DFT) is the most popular method for performing ground-state electronic structure calculations due to its accuracy in many situations along with its computational efficiency. Its time-dependent extension (TDDFT) allows the modelling of the dynamics of systems of interacting electrons. (TD)DFT is an exact theory in principle, but requires approximations in practice, specifically to the many-body effects of electron exchange and correlation (xc), which are described by some unknown universal functional. Despite much success, it often proves inadequate in many scenarios of increasing interest. Therefore, we aim to contribute to the development of improved approximate density functionals. \n \nIn this thesis, we take the approach of studying small model systems for which we can compute the exact functionals. These systems are designed to exhibit the key features present in realistic quantum systems, such as atoms and molecules, which the exact universal (TD)DFT functional is capable of describing, but are often ill-described by the commonly used approximations. By analysing the nature of these exact functionals, we gain valuable insight into their fundamental properties, together with identifying where the current approximations are lacking, thereby allowing us to suggest how these approximations may be improved.
Aiming to combine density functional theory (DFT) and wave-function theory, we study a mapping from the many-body interacting system to an effectively interacting Kohn-Sham system instead of a noninteracting Kohn-Sham system. Because a ground state of effectively interacting systems requires having a solution for the correlated many-body wave functions, this provides a natural framework to many-body wave-function theories such as the configuration interaction and the coupled-cluster method in the formal theoretical framework of DFT. Employing simple one-dimensional two-electron systems---namely, the one-dimensional helium atom, the hydrogen molecule, and the heteronuclear diatomic molecule---we investigate properties of many-body wave functions and exact exchange-correlation potentials of effectively interacting Kohn-Sham systems. As a result, we find that the asymptotic behavior of the exact exchange-correlation potential can be controlled by optimizing that of the effective interaction. Furthermore, the typical features of the exact noninteracting Kohn-Sham system, namely, a spiky feature and a step feature in the exchange-correlation potential for the molecular dissociation limit, can be suppressed by a proper choice of the effective interaction. These findings open a possibility to construct numerically robust and efficient exchange-correlation potentials and functionals based on the effectively interacting Kohn-Sham scheme.
Most realistic calculations of moderately correlated materials begin with a ground-state density functional theory (DFT) calculation. While Kohn-Sham DFT is used in about 40,000 scientific papers each year, the fundamental underpinnings are not widely appreciated. In this chapter, we analyze the inherent characteristics of DFT in their simplest form, using the asymmetric Hubbard dimer as an illustrative model. We begin by working through the core tenets of DFT, explaining what the exact ground-state density functional yields and does not yield. Given the relative simplicity of the system, almost all properties of the exact exchange-correlation functional are readily visualized and plotted. Key concepts include the Kohn-Sham scheme, the behavior of the XC potential as correlations become very strong, the derivative discontinuity and the difference between KS gaps and true charge gaps, and how to extract optical excitations using time-dependent DFT. By the end of this text and accompanying exercises, the reader will improve their ability to both explain and visualize the concepts of DFT, as well as better understand where others may go wrong.