# Reduced Density-Matrix Approach to Strong Matter-Photon Interaction

- Florian Buchholz
*****Florian BuchholzTheory Department, Max Planck Institute for the Structure and Dynamics of Matter - Luruper Chaussee 149, 22761 Hamburg, Germany*****E-mail: [email protected]More by Florian Buchholz, - Iris Theophilou
*****Iris TheophilouTheory Department, Max Planck Institute for the Structure and Dynamics of Matter - Luruper Chaussee 149, 22761 Hamburg, Germany*****E-mail: [email protected]More by Iris Theophilou, - Soeren E. B. NielsenSoeren E. B. NielsenTheory Department, Max Planck Institute for the Structure and Dynamics of Matter - Luruper Chaussee 149, 22761 Hamburg, GermanyMore by Soeren E. B. Nielsen,
- Michael Ruggenthaler
*****Michael Ruggenthaler*****E-mail: [email protected]More by Michael Ruggenthaler, and - Angel Rubio
*****Angel RubioCenter for Computational Quantum Physics (CCQ), Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, United States*****E-mail: [email protected]More by Angel Rubio

## Abstract

We present a first-principles approach to electronic many-body systems strongly coupled to cavity modes in terms of matter–photon one-body reduced density matrices. The theory is fundamentally nonperturbative and thus captures not only the effects of correlated electronic systems but accounts also for strong interactions between matter and photon degrees of freedom. We do so by introducing a higher-dimensional auxiliary system that maps the coupled fermion-boson system to a dressed fermionic problem. This reformulation allows us to overcome many fundamental challenges of density-matrix theory in the context of coupled fermion-boson systems and we can employ conventional reduced density-matrix functional theory developed for purely fermionic systems. We provide results for one-dimensional model systems in real space and show that simple density-matrix approximations are accurate from the weak to the deep-strong coupling regime. This justifies the application of our method to systems that are too complex for exact calculations and we present first results, which show that the influence of the photon field depends sensitively on the details of the electronic structure.

^{5}× 8 Byte ≈ 75 Gigabyte of working memory, which is close to the edge of current high performance technology). Instead, reformulations of the many-body problem in terms of reduced quantities, like the electron density (44,45) or the Green’s function, (46−48) have been shown to provide accurate results for relatively low computational costs. Thus, working with reduced quantities seems to be a natural choice also for coupled light–matter systems. (49) Recently, quantum-electrodynamical density-functional theory (QEDFT) was introduced as an extension to pure electronic density-functional theory (DFT). (50−53) First calculations showed the feasibility of QEDFT, (54,55) leading to the possibility to perform full first-principles calculations of real molecules coupled to cavity modes. (56,57) However, standard approximations in DFT (usually based on a noninteracting auxiliary Kohn–Sham system) become inaccurate if applied to strongly correlated systems. This is very well studied in terms of the strong-correlation regime in electronic systems (58) and also observed for the existing QEDFT functionals that approximate the electron–photon interaction. (56) Consequently, to study novel effects arising in the ultrastrong (59) or deep-strong coupling regime (60) of light and matter from first-principles, that is, without resorting to simplified few-level systems, one needs to develop new functionals for the combined matter–photon systems or explore alternative many-body methods.

*N*-representability conditions) are not trivial. (65−67) For RDMs in matter–photon systems, these conditions are entirely unexplored. Nevertheless, we manage to overcome this difficulty by mapping the original system to a higher-dimensional auxiliary system that allows for an effective description of the problem by fermions. Hence, the corresponding RDMs connect to the auxiliary wave functions under

*N*-representability conditions of fermionic systems. Despite being fermions, the newly introduced quasi-particles will depend on electronic and bosonic degrees of freedom. This construction was recently introduced by some of the authors in ref (68) and used to construct a DFT scheme specifically for the strong coupling regime of light–matter systems. In the auxiliary system, the Hamiltonian consists of only 1- and 2-body terms for the new quasi-particles, thus, it has the same structure as the conventional electronic Hamiltonian for molecular systems, which allows us to apply electronic RDMFT without major modifications. We will present some results for model systems with the simplest known RDMFT functional, the Müller functional, (62) and show that this dressed RDMFT is accurate from the weak to the strong coupling regime. Then, we will present two examples highlighting that how matter reacts to the interaction with photons depends strongly on the system. For instance, for the same coupling strength, the repulsion between the particles can be locally suppressed in one system but enhanced in another. We finish with commenting on some open issues and challenges for future applications.

## Physical Setting

*d*of the electronic system (λ ≫

*d*) such that the dipole approximation (here in the Coulomb gauge) is valid. (69,70) In the case of the dipole approximation, where every photon mode couples to all Fourier components of the charge current of the electronic subsystem, (52,53) an effective description with only a few modes is usually sufficient. The continuum of modes is then effectively taken into account by using, instead of the bare mass, the physical mass of the electrons. (27,57) Since we focus on equilibrium situations, the openness of the cavity can be neglected.

*Ĥ*

_{e}=

*T̂*+

*V̂*+

*Ŵ*, used to describe the uncoupled matter system consisting of

*N*electrons in an external potential

*v*(

**r**) interacting via the Coulomb repulsion

*w*(

**r**,

**r**′). The third sum describes

*M*photon modes that are characterized by their elongation

*p*

_{α}, frequency ω

_{α}, and polarization vectors

**λ**

_{α}. The polarization vectors include already the effective coupling strength (35) and couple to the total dipole

**D̂**= ∑

_{k=1}

^{N}

**r**

_{k}of the electronic system. The sum can be decomposed in a purely photonic part , the dipole self-interaction (note that this term is necessary for the existence of a ground state (70))

*Ĥ*

_{d}= ∑

_{α=1}

^{M}1/2(

**λ**

_{α}·

**D̂**)

^{2}, which we split for later convenience in its one-body

*Ĥ*

_{d}

^{(1)}= ∑

_{α=1}

^{M}∑

_{k=1}

^{N}1/2(

**λ**

_{α}·

**r**

_{k})

^{2}and two-body part

*Ĥ*

_{d}

^{(2)}= ∑

_{α=1}

^{M}∑

_{k≠l}1/2(

**λ**

_{α}·

**r**

_{k})(

**λ**

_{α}·

**r**

_{l}), and the bilinear interaction

*Ĥ*

_{I}= −∑

_{α=1}

^{M}ω

_{α}

*p*

_{α}

**λ**

_{α}·

**D̂**. Some comments on this Hamiltonian are appropriate. Since we work in Coulomb gauge, the dipole-self-energy term arises only for the transversal but not for the longitudinal part of the field. (71) However, if we assume a cavity then the Coulomb interaction is modified. (72) We can easily incorporate this into our framework since

*w*(

**r**,

**r**′) is completely at our disposal and we do not rely on any kind of Coulomb approximation. Thus, we can also treat the influence of, for example, a plasmonic environment. (73)

*N*+

*M*coordinates(2)where σ

_{k}is the electronic spin degree of freedom. The wave function Ψ is, as usual, antisymmetric with respect to the exchange of any two electron coordinates

**r**

_{j}σ

_{j}↔

**r**

_{k}σ

_{k}, and also depends on

*M*photon-mode displacement coordinates

*p*

_{α}.

*p*

_{α}with

*p*

_{β}. The bosonic symmetry instead refers to the exchange of mode excitations, which are interpreted as photons in the number-state representation. To see this, we first use the ladder operators and to represent the sum of harmonic Hamiltonians, that is,(3)Eigenstates of the individual

*â*

_{α}

^{+}

*â*

_{α}in the above representation are given by multiple applications of creation operators to the vacuum state |0⟩, that is, |φ

_{α}

^{n}⟩ = (

*â*

_{α}

^{+})

^{n}|0⟩. (Note that, as in refs (47and74), we chose here explicitly a non-normalized basis {|φ

_{α}

^{n}⟩} of the

*n*-photon sector, with ⟨φ

_{α}

^{n}|φ

_{α}

^{n}⟩ =

*n*!, which will be convenient for the discussion of bosonic reduced density matrices in the next section. The missing normalization factor is shifted to the resolution of identity, that is, = 1/

*N*

_{b}!|

*α*

_{1},...,

*α*

_{Nb}⟩⟨

*α*

_{1},...,

*α*

_{Nb}|, where |α

_{1}, ..., α

_{Nb}⟩ =

*â*

_{α1}

^{+}···

*â*

_{αNb}

^{+}|0⟩, as defined later in the text.) These eigenstates are connected to the displacement representation by φ

_{α}

^{n}(

*p*

_{α}) = ⟨

*p*

_{α}|φ

_{α}

^{n}⟩ = ⟨

*p*

_{α}|(

*â*

_{α}

^{+})

^{n}|0⟩. We can then express an

*M*-mode eigenfunction as ϕ

_{n1, ..., nM}(

*p*

_{1}, ...,

*p*

_{M}) = ⟨

*p*

_{1}···

*p*

_{M}|ϕ

_{n1, ..., nM}⟩ = ⟨

*p*

_{1}···

*p*

_{M}| (

*â*

_{1}

^{+})

^{n1}···(

*â*

_{M}

^{+})

^{nM}|0⟩. In this form it becomes clear that a multimode eigenstate |ϕ

_{n1, ..., nM}⟩ can be considered to consist of

*N*

_{b}=

*n*

_{1}+ ... +

*n*

_{M}photons (mode excitations). We can associate every such multimode eigenstate with a specific photon-number sector, that is, the zero-photon sector is merely one-dimensional and corresponds to |ϕ

_{01, ..., 0M}⟩ ≡ |0⟩, the single-photon sector is

*M*-dimensional and corresponds to the span of

*â*

_{α}

^{+}|0⟩ ≡ |α⟩ for all α and so on. For the multiphoton sectors we see due to the commutation relations of the ladder operators the bosonic exchange symmetry appearing, for example,

*â*

_{α1}

^{+}

*â*

_{α2}

^{+}|0⟩ =

*â*

_{α2}

^{+}

*â*

_{α1}

^{+}|0⟩ ≡ |α

_{1}, α

_{2}⟩ for α

_{1}, α

_{2}∈ {1, ...,

*M*}. It is no accident that the bosonic symmetry becomes explicit in this representation since the different modes α determine how the photon wave functions look in real space (see also the discussion in Supporting Information, section 5). A general photon state can therefore be represented by a sum over all photon-number sectors as where Φ̃(

*α*

_{1}, ...,

*α*

_{n}) = ⟨

*α*

_{1}, ...,

*α*|Φ⟩.

_{n}## Reduced Density Matrices for Coupled Light–Matter Systems

_{e}(

**r**

_{1}σ

_{1}, ...,

**r**

_{N}σ

_{N}) of

*N*electrons (see Table 1), the expectation value of a general (nonlocal)

*q*-body operator

*Ô*(

**r**

_{1}, ...

**r**

_{q};

**r**

_{1}

^{′}, ...

**r**

_{q}

^{′}), which is given by

*O*= ⟨ψ

_{e}|

*Ô*ψ

_{e}⟩, can be determined via the electronic (spin-summed; we define here only the spin-summed version of the

*q*-body RDM (

*q*RDM), because in this work, we do not consider explicitly spin-dependent quantities; for instance, if we included magnetic fields in the Hamiltonian, the situation would change)

*q*RDM (note that there are different conventions in the literature for the normalization of the

*q*RDM; we followed ref (75))(5)For instance, the well-known electronic 1RDM, that we denote in the following by γ

_{e}(

**r**,

**r**′) = Γ

_{e}

^{(1)}(

**r**;

**r**′), is sufficient to calculate all electronic single-particle observables such as the kinetic energy. A prominent example of a higher-order operator in the electronic case is the two-body Coulomb interaction among the

*N*electrons. To calculate its expectation value, we need to consider the diagonal of the 2RDM Γ

^{(2)}(

**r**

_{1},

**r**

_{2};

**r**

_{1},

**r**

_{2}). In the chosen normalization, all RDMs satisfy the sum-rule . So for instance, the 1RDM integrates to the particle number ∫d

^{3}

*r*γ

_{e}(

**r**,

**r**) =

*N*, the 2RDM integrates to two times the number of pairs, and so on. Additionally, higher and lower order RDMs are connected via

Ψ(r_{1}σ_{1}, ..., r_{N}σ_{N};p_{1}, ..., p_{M}) | electron-photon many-body state |

ψ_{e}(r_{1}σ_{1}, ..., r_{N}σ_{N}) | purely electronic many-body state |

ψ_{b}(α_{1}, ..., α_{Nb}) | photonic many-body state in mode representation with fixed particle number |

Φ(α_{1},α_{2}, ...) | photonic many-body state in Fock space |

ϕ_{e}^{i} (r)/ϕ_{b}^{i}(α) | electronic/photonic natural orbital |

*N*

_{b}boson state in the (symmetrized) mode-representation ψ

_{b}(α

_{1}, ..., α

_{Nb}) (see Supporting Information, section 5, for more details) leads to the corresponding bosonic

*q*RDM,(6)According to the electronic case, we denote the 1RDM by γ

_{b}(α, β) = Γ

_{b}

^{(1)}(α; β). However, in the specific case of photons, where the number of particles is undetermined and we work with Fock-space wave functions |Φ⟩, we need to consider a Fock-space 1RDM of the formIn an according manner one can define a bosonic Fock-space

*q*RDM via Γ

_{b}

^{(q)}(α

_{1}, ..., α

_{q}; α

_{1}

^{′}, ..., α

_{q}

^{′}) = ⟨Φ|

*â*

_{α1′}

^{+}···

*â*

_{αq′}

^{+}

*â*

_{αq}···

*â*

_{α1}Φ⟩.

_{e}

^{(q)}≡ d

*, as well as Γ*

^{M}p_{b}

^{(q)}≡ ⟨Ψ|

*â*

_{α1′}

^{+}···

*â*

_{αq′}

^{+}

*â*

_{αq}···

*â*

_{α1}Ψ⟩.

*Ĥ*

_{e}and the single-particle part of the dipole self-energy

*Ĥ*

_{d}

^{(1)}are given in terms of the 1RDM byHere we have denoted on the left-hand side the explicit dependence of the expectation value on the 1RDM, and the subscript |

_{r′=r}indicates that

**r**′ is set to

**r**after the application of the semilocal single-particle operator (−1/2∇

_{r}

^{2}+

*v*(

**r**)). The expectation value of the electronic interaction energy

*Ŵ*and the two-body part of the dipole self-energy

*Ĥ*

_{d}

^{(2)}are given in terms of the (diagonal) of the 2RDM byHence, for the electronic operator expectation values little changes in comparison to a purely fermionic problem, except that we have a coupled electron-boson wave function and the extra contributions of the dipole self-energy. For the purely bosonic part of the coupled Hamiltonian, we findHowever, the bilinear coupling term is not given in a simple RDM form but becomesA new reduced quantity appears that mixes light and matter degrees of freedom and can be interpreted as a 3/2-body operator Γ

_{e,b}

^{(3/2)}(α;

**r**,

**r**′). (To see this in a simple manner, we also lift the continuous fermionic problem into its own Fock space and introduce genuine field operators ψ̂

_{e}

^{†}(

**r**σ) and ψ̂

_{e}(

**r**σ) with the usual anticommutation relations. Similar to the discussed bosonic case, the electronic RDMs can then be written in terms of strings of creation and annihilation field operators. (76) We re-express ⟨Ψ|[(

*â*

_{α}

^{+}+

*â*

_{α})

**λ**

_{α}·

**D̂**]Ψ⟩ = ∑

_{σ}∫d

^{3}

*r*⟨Ψ|[(

*â*

_{α}

^{+}+

*â*

_{α})ψ̂

_{e}

^{†}(

**r**σ)ψ̂

_{e}(

**r**σ)(

**λ**

_{α}·

**r**)]Ψ⟩. If we then define Γ

_{e,b}

^{(3/2)}(α;

**r**,

**r**′) = ∑

_{σ}⟨Ψ|[(

*â*

_{α}

^{+}+

*â*

_{α})ψ̂

_{e}

^{†}(

**r**σ)ψ̂

_{e}(

**r**′σ)]Ψ⟩, we can rewrite ⟨Ψ|[(

*â*

_{α}

^{+}+

*â*

_{α})

**λ**

_{α}·

**D̂**]Ψ⟩ = ∫d

^{3}

*r*(

**λ**

_{α}·

**r**)Γ

_{e,b}

^{(3/2)}(α;

**r**,

**r**).) The bilinear interaction term therefore creates/annihilates bosons by interacting with the electronic subsystem. The 3/2-body RDM has, in general, no simple connection to any

*q*RDM, even if we extend the definitions to include combined matter–boson

*q*RDMs. (Using the field-operator formulation, the usual

*q*RDMs consist of strings of particle-number-conserving combinations of electron and boson operators. Integrating/summing out a number-nonconserving part of it does not lead to a simple relation to half-body RDMs, in general.) One obvious reason is that

*q*RDMs conserve particle numbers, while half-body RDMs do not. Take, for instance, the Fock-space γ

_{b}(α, β) = ⟨Φ|

*â*

_{β}

^{+}

*â*

_{α}Φ⟩. In the special case that |Φ⟩ consists only of coherent states for each mode (which essentially means that we have treated the photons in mean field) and since the coherent states are eigenfunctions to the annihilation operators, we find γ

_{b}(α, β) =

*d*

_{β}

^{*}

*d*

_{α}, where

*d*

_{α}is the total displacement of the coherent state of mode α. In this case, we also know ⟨Φ|

*â*

_{β}

^{+}Φ⟩ =

*d*

_{β}

^{*}. If we now assume all but one mode, say mode 1, having zero displacement, then we only know γ

_{b}(1, 1) = |

*d*

_{1}|

^{2}from the bosonic 1RDM. We do, however, in general, not know what

*d*

_{1}

^{*}is. For other states, such a connection is even less explicit.

**r**

_{1}σ

_{1}, ...,

*p*

_{M}), the above reformulation seems to indicate that we can replace this by varying over (

**r**,

**r**′) for the diagonal of Γ

_{e}

^{(2)}and also for the 1RDM γ

_{e}, together with a variation over (α, β) for γ

_{b}and over (α,

**r**) for Γ

_{e,b}

^{(3/2)}. Such a reformulation is the basis of any RDMFT, and for electronic systems, the properties of RDMs have been studied for more than 50 years. (77) However, this seeming reduction of complexity is deceptive. In order to find physically sensible results, we cannot vary arbitrarily over the above RDMs, but need to ensure that they are consistent among each other and that they are all connected to a physical wave function. This is indicated in eq 7, where {γ

_{e}, Γ

_{e}

^{(2)}, γ

_{b}, Γ

_{e,b}

^{(3/2)}} → Ψ highlights that the RDMs are contractions of a common wave function. For systems with fixed particle numbers, it is, in principle, known how to restrict the set of trial RDMs to physical ones. The corresponding restrictions are called

*N*-representability conditions. (65−67) However, only for the 1RDM of ensembles (fermionic or bosonic) are the conditions simple. In this case, by diagonalizing the 1RDM in its eigenbasis γ

_{e/b}= ∑

*n*

_{i}

^{e/b}(ϕ

_{e/b}

^{i})*ϕ

_{e/b}

^{i}, where the ϕ

_{e/b}

^{i}are called the natural orbitals and the

*n*

_{e/b}

^{i}are the natural occupation numbers, the conditions are(8)for fermions and bosons, respectively. If the particle number

*N*

_{e/b}of one species of the system is conserved, the respective sum-rule(9)becomes a second part of the

*N*-representability conditions. Consequently, to define a proper RDM framework for coupled electron–boson problems, one would need to know the corresponding constraints that connect the wave function with all the necessary RDMs. A glance at the history of an important example, the search for the

*N*-representability conditions of the electronic 2RDM, suggests that finding similar conditions for the novel half-body RDMs together with connections between the fermionic and bosonic

*q*RDMs is a very challenging task. The electronic-2RDM problem was proposed in 1960 (78) and it took until 2012, to understand how to make the conditions explicit. (67)

## “Fermionization” of Matter–Photon Systems

*p*

_{α,2}, ...,

*p*

_{α,N}), where the number of these extra dimensions depends on the number of electrons

*N*. We therefore embed the physical configuration space in a higher-dimensional space, that is, we now consider wave functions depending on (

**r**

_{1}σ

_{1}, ...,

**r**

_{N}σ

_{N},

*p*

_{1}, ...,

*p*

_{M},

*p*

_{1,2}, ...,

*p*

_{1,N}, ...,

*p*

_{M,2}, ...,

*p*

_{M,N}). Second, we add for every photon mode α a linear operator(10)to the physical Hamiltonian of eq 1. This auxiliary Hamiltonian is a sum of quantum harmonic oscillators with respect to the auxiliary coordinates. The resulting Hamiltonian in the extended configuration space is

*Ĥ*′ =

*Ĥ*+ ∑

_{α=1}

^{M}Π̂

_{α}(we denote all quantities in the auxiliary space with a prime symbol). Here we see that the auxiliary degrees of freedom do not mix with the physical ones. This will allow in a very simple manner to embed but also to reconstruct the physical wave function. In the third step, we perform an orthogonal variable transformation of the photonic plus auxiliary coordinates to new coordinates (

*q*

_{α,1}, ...,

*q*

_{α,N}) such that(11)This whole procedure can be viewed as the inverse of a center-of-mass coordinate transformation. (80) In total, we find the auxiliary Hamiltonian in the higher-dimensional configuration space given as(12)where we inserted the definition of the dipole operator,

**D̂**= ∑

_{k=1}

^{N}

**r**

_{k}, and reordered the expressions, such that the terms with only one index and the terms with two different indices are grouped together. Introducing then a (3 +

*M*)-dimensional polaritonic vector of space and auxiliary photon coordinates

**z**= (

**r**,

*q*

_{1}, ...,

*q*

_{M}), we can rewrite the above Hamiltonian as(13)where we introduced the dressed Laplacian Δ′ = ∑

_{i=1}

^{3}+ ∑

_{α=1}

^{M}, the dressed local potential

*v*′(

**z**) =

*v*(

**r**) + and the dressed interaction kernel

*w*′(

**z**,

**z**′) =

*w*(

**r**,

**r**′) + ∑

_{α=1}

^{M}

*q*

_{α}

**λ**_{α}·

**r**′ −

*q*

_{α}′

**λ**_{α}·

**r**+

**λ**_{α}·

**r**λ

_{α}·

**r**′]. Note that the choice of the linear auxiliary operator (eq 10) and the coordinate transformation (eq 11) have a certain freedom. The operator of eq 10 must not contain physical coordinates, such that the physical system cannot be influenced by this auxiliary operator. Further, it must together with the transformation give rise to polaritonic 1- and 2-body terms, as shown in eq 12. These requirements are met using an orthonormal transformation together with the harmonic oscillators of eq 10. (Note that there are many different orthonormal transformations, but the exact choice is not important for the formalism. It only needs to include the first line of eq 11. One specific example of such a transformation is (68)

*p*

_{α,k}= (

*q*

_{1}+ ... +

*q*

_{k−1}− (

*k*− 1)

*q*

_{k}) for 2 ≤

*k*≤

*N*, alongside the first line of eq 11.) Since the operator of eq 10 only acts on the auxiliary coordinates, the normalized physical solution Ψ of the original (time-independent) Schrödinger equation

*E*

_{0}Ψ =

*ĤΨ*in the standard configuration space is connected to a new physical solution of the auxiliary Hamiltonian

*Ĥ*′ in a very simple manner, that is,Here the normalized solution Ψ′ of the auxiliary Schrödinger equation

*E*

_{0}

^{′}Ψ′ =

*Ĥ*′Ψ′ is found with χ being the ground state of ∑

_{α}Π̂

_{α}. The ground state χ is merely a tensor product of individual harmonic-oscillator ground states, and therefore, exchanging

*p*

_{α,i}with

*p*

_{α,j}does not change the total wave function Ψ′. If we rewrite this wave function in the new coordinatesand due to the fact that we constructed χ to be symmetric with respect to the exchange of

*q*

_{α,k}and

*q*

_{α,l}, we realize that Ψ′ is antisymmetric with respect to the exchange of (

**z**

_{k}σ

_{k}) and (

**z**

_{l}σ

_{l}). (Note that the auxiliary ground state of mode α is given as a Gaussian with respect to

*p*

_{α,i}

^{2}=

*q*

_{α,k}

^{2}– 1/

*N*(

*q*

_{α,k})

^{2}, where we used

*p*= 1/√

*N*

*q*

_{α,k}.) Thus Ψ′ is fermionic with respect to the polaritonic coordinates (

**z**σ) and can be represented by a sum of Slater determinants of (3 +

*M*)-dimensional polaritonic orbitals φ(

**z**σ). This makes the application of the usual fermionic many-body methods possible, and we can rely on fermionic

*N*-representability conditions. However, besides the extra dimensions and the new fermionic exchange symmetry, the physical wave functions of the dressed auxiliary space also have a further

*q*

_{α,k}↔

*q*

_{α,l}exchange symmetry. Simple approximations based on single polaritonic Slater determinants will violate this extra symmetry. We will remark on such violations when we introduce dressed RDMFT in the next section. To further see that indeed the constructed Ψ′ is a minimal-energy state in the extended space with the appropriate symmetries, we first point out that for any trial wave functionthe

*q*-exchange symmetry implies that the fermionic symmetry is in the (

**r**σ) coordinates. Then it holds since

*Ĥ*only acts on (

**r**

_{1}, ...,

*p*

_{M}), and ∑

_{α=1}

^{M}Π̂

_{α}only acts on

*p*

_{1,2}, ...,

*p*

_{M,N}so thatAlthough we constructed the auxiliary space explicitly in a way that the physical wave function Ψ can be reconstructed exactly from its dressed counterpart Ψ′, by integration of all auxiliary coordinates, this does not hold for all types of operators. (Note that

*Ĥ*′ also has many eigenstates that are not of the form Ψ′ = Ψχ, with Ψ antisymmetric under exchange of

**r**

_{k}σ

_{k}↔

**r**

_{l}σ

_{l}and χ symmetric under exchange of

*q*

_{α,k}↔

*q*

_{α,l}. In general, we thus have to enforce these properties to only retain the eigenstates of this form. However, for the ground state, it is sufficient to enforce the symmetry under exchange of

*q*

_{α,k}↔

*q*

_{α,l}, together with the antisymmetry under exchange of

**z**

_{k}σ

_{k}↔

**z**

_{l}σ

_{l}.) For operators that depend only on electronic coordinates, there is no difference, and we have ⟨Ψ|

*Ô*Ψ⟩ = ⟨Ψ′|

*Ô*Ψ′⟩. This is not surprising because the coordinate transformation (eq 11) acts only on the photonic part of the system. For photonic observables instead, the transformation changes the respective operators and, thus, the connection between physical and auxiliary space becomes nontrivial in general. However, at least for all observables that depend on photonic 1/2- or 1-body expressions, there is an analytical connection. For half-body operators, that is, any operator that depends only on the elongation of (

*q*

_{α,1}+ ... +

*q*

_{α,N}) and its conjugate , the coordinate transformation itself provides us with the connection. For 1-body operators, this becomes slightly more involved. For example, consider the mode energy operator , that we can straightforwardly generalize in the auxiliary space to . The connection between

*Ĥ*

_{ph}and

*Ĥ*

_{ph}

^{′}is given by the definition of the coordinate transformation (eq 11),Since the expectation value of Π̂

_{α}is known analytically, , we have . This can be generalized to any operator that contains terms of the form and , where α and β denote any two modes. The reason is that the transformation (eq 11) preserves the standard inner product of the Euclidean space of the mode plus extra coordinates. This transfers also to their conjugates and combinations of both. From the above, it is straightforward to derive also the expression for the occupation of mode α, . In the auxiliary system, we have(14)

## Dressed Reduced Density-Matrix Functional Theory

*Ĥ*′, given in eq 13. It consists of only polaritonic one-body terms

*ĥ*

^{(1)}(

**z**) = −1/2Δ′ +

*v*′(

**z**), and two-body terms

*ĥ*

^{(2)}(

**z**,

**z**′) =

*w*′(

**z**,

**z**′). It commutes with the polaritonic particle-number operator

*N̂*′ = ∫d

^{3+M}

*zn̂*(

**z**), where we used the definition of the polaritonic local density operator

*n̂*(

**z**) = ∑

_{i=1}

^{N}δ

^{3+M}(

**z**–

**z**

_{i}). This means that the auxiliary system has a constant polaritonic particle number

*N*. Additionally, the physical wave function of the dressed system Ψ′(

**z**

_{1}σ

_{1}, ...,

**z**

_{N}σ

_{N}) is per construction antisymmetric. This allows for the definition of a dressed (spin-summed) 1RDM(15)in accordance to eq 5. By construction, this auxiliary density matrix reduces to the physical electron density matrix via γ

_{e}(

**r**,

**r**′) = ∫d

^{M}

*q*γ(

**r**,

*q*

_{1}, ...,

*q*

_{M};

**r**′,

*q*

_{1}, ...,

*q*

_{M}). Furthermore, we introduce the (spin-summed) dressed 2RDM Γ

^{(2)}(

**z**

_{1},

**z**

_{2};

**z**

_{1}

^{′},

**z**

_{2}

^{′}) =

*N*(

*N*– 1)∑

_{σ1, ..., σN}∫d

^{(3+M)(N–2)}

*z*Ψ′*(

**z**

_{1}

^{′}σ

_{1},

**z**

_{2}

^{′}σ

_{2},

**z**

_{3}σ

_{3}, ...,

**z**

_{N}σ

_{N})Ψ′(

**z**

_{1}σ

_{1},

**z**

_{2}σ

_{2},

**z**

_{3}σ

_{3}, ...,

**z**

_{N}σ

_{N}). These dressed RDMs allow for expressing the energy expectation value of the dressed system byThus, we can define the variational principle for the ground state only with respect to well-defined reduced quantities,(16)To perform this minimization, we need to constrain the configuration space to the physical dressed RDMs that connect to an antisymmetric wave function with the extra

*q*-exchange symmetry by testing the appropriate

*N*-representability conditions of the dressed 2RDM and the dressed 1RDM. Besides the by now well-known conditions for the fermionic 2RDM (67) and the fermionic 1RDM, (65) we would in principle get further conditions to ensure the extra exchange symmetry. However, already for the usual electronic 2RDM the number of conditions grows exponentially with the number of particles, and it is out of the scope of this work to discuss possible approximations. The interested reader is referred to, for example, ref (82). Instead, we want to stick to the dressed 1RDM γ and approximate the 2-body part as a functional of γ. The mathematical justification of RDMFT is given by Gilbert’s theorem, (61) which is a generalization of the Hohenberg–Kohn theorem of DFT. (83) More specifically, Gilbert proves that the ground state energy of any Hamiltonian with only 1-body and 2-body terms is a unique functional of its 1RDM. Following this idea, we will express the ground-state energy of the dressed system as a partly unknown functional

*F*′ of only the system’s dressed 1RDM(17)For this minimization, we need a functional of the diagonal of the dressed 2RDM in terms of the dressed 1RDM as well as adhering to the corresponding

*N*-representability conditions when varying over γ. We also see now the advantage of the dressed RDMFT approach, which avoids the original variation over all wave functions as well as a variation over many different RDMs, as shown in eq 7. Instead, we only need the dressed 1RDM, which has a comparatively simple connection to fermionic wave functions (at least when we vary over ensembles, i.e., eqs 8 and 9). The price we pay for this is 2-fold: First, we have a new symmetry that will most likely lead to extra

*N*-representability conditions. Second, we need to increase the dimension of the natural orbitals by one for every photon mode. However, to capture the main physics of usual cavity experiments, often one effective mode is enough. Computations with four-dimensional dressed orbitals are numerically feasible. It is specifically such settings, where we envision a dressed RDMFT to be a reasonable alternative to other ab initio approaches to cavity QED. (30,40,41,52,55,68)

*N*-representability conditions (this approximation is similarly employed in electronic RDMFT; for details, we refer to, e.g., ref (84)) and ignore presently the extra exchange symmetry between the

*q*-coordinates (in all the numerical examples that we studied, this “fermion polariton approximation” was very accurate; one can find more details in the Supporting Information), and we employ simple approximations to the unknown part

*W*′[γ] that have been developed for the electronic case. To do so, we further, similarly to the electronic case, decompose

*W*′[γ] =

*E*

_{H}[γ] +

*E*

_{xc}[γ] into a classical Hartree part

*E*

_{H}[γ] = 1/2∫∫d

^{3+M}

*z*d

^{3+M}

*z*′γ(

**z**,

**z**)γ(

**z**′,

**z**′)

*w*′(

**z**,

**z**′) and an unknown exchange-correlation part

*E*

_{xc}[γ]. Almost all known functionals

*E*

_{xc}[γ] are expressed in terms of the eigenbasis and eigenvalues of the 1RDM. In our case, the dressed natural orbitals ϕ

_{i}(

**z**) and occupation numbers

*n*

_{i}are found by solving ∫d

^{3+M}

*z*′γ(

**z**,

**z**′)ϕ

_{i}(

**z**′) =

*n*

_{i}ϕ

_{i}(

**z**). The simplest approximation is to only retain the fermionic exchange symmetry and employ the Hartree–Fock (HF) functional(18)As the HF functional depends linearly on the natural occupation numbers, any kind of minimization will lead to the single-Slater-determinant HF ground state (which corresponds to occupations of 1 and 0). (85) We call this approximation dressed Hartree–Fock (dressed HF). We can go beyond the single Slater determinant in dressed RDMFT if we employ a nonlinear occupation-number dependence in the exchange-correlation functional. We here consider the Müller functional (62)(19)which has been rederived by Bjuise and Baerends. (86) The Müller functional has been studied for many physical systems (63,86) and gives a qualitatively reasonable description of electronic ground states. Additionally, it has many advantageous mathematical properties. (62,87) A thorough discussion of different functionals goes beyond the scope of this work, thus, we only want to remark that a variety of functionals were proposed after

*E*

_{M}[γ] and it is likely to have even better agreement with the exact solution by choosing more elaborate functionals.

## Numerical Implementation

*v*to its dressed counterpart

*v*′. This, together with a change of the electronic Coulomb interaction

*w*to its dressed counterpart

*w*′ already allows to perform dressed HF calculations, at least under the approximation of violating the additional symmetry constraint, discussed in the previous section. If the code one uses is furthermore able to perform RDMFT minimizations, it is straightforward to extend the implementation to also solve coupled electron-photon problems via dressed RDMFT from first principles. We have done so with the electronic-structure code Octopus, (88) and the implementation will be made available with the upcoming release.

*N*-representability, we first represent the occupation numbers as the squared sine of auxiliary angles, that is, 0 ≤

*n*

_{i}= 2 sin

^{2}(θ

_{i}) ≤ 2, to satisfy eq 8. (Note that the

*n*

_{i}is bounded by 2 because we employed a spin-restricted formulation. If we considered natural spin–orbitals instead, the upper bound would be 1). The second part of the conditions (eq 9), that is, ∑

_{i=1}

*n*

_{i}=

*N*, as well as the orthonormality of the dressed natural orbitals, that is, ∫d

^{3+M}

*zϕ*

_{i}

^{*}(

**z**)ϕ

_{j}(

**z**) = δ

_{ij}, are imposed via Lagrange multipliers as, for example, explained in ref (88). We have available two different orbital-optimization methods, a conjugate-gradient algorithm (we used this method only for some benchmark calculations, as it is not yet optimized for the needs of RDMFT) and an alternative method that was introduced by Piris et al. in ref (89). The latter expresses the ϕ

_{i}in a basis set and can use this representation to considerably speed up calculations in comparison to the conjugate-gradient algorithm. It was used for all results presented in this paper. However, it is not trivial to converge such calculations in practice and we developed a protocol to obtain properly converged results. The interested reader is referred to the Supporting Information, section 4.

## Numerical Results

*v*(

*x*) and coupled to one photon mode. We transfer the systems in the dressed basis, that leads to a dressed local potential . Specifically, we consider a one-dimensional model of a helium atom (He), that is, , a one-dimensional model of a hydrogen molecule (H

_{2}), that is,

*v*

_{H2}(

*x*) = − , first at its equilibrium position

*d*=

*d*

_{eq}= 1.628 au, later with varying

*d*, and a one-dimensional model of a Beryllium atom Be, that is, . For the latter, we consider a smaller softening parameter ϵ = 0.5 to make sure that all electrons are properly bound. We use the soft Coulomb interaction (90,91) for all test systems. For the two-electron examples, we set the photon frequency in resonance with the lowest excitations of the respective “bare” systems, so outside of the cavity. For that we calculate the ground and first excited state of each system with the exact solver and find the corresponding excitation frequencies ω

_{He}= 0.5535 au and ω

_{H2}= 0.4194 au. For Be instead, we choose ω

_{Be}= 3.0 au, which is not a resonance of the Be atom, for rather numerical than physical reasons. As resonance is not an important feature for ground state calculations, different choices of ω do not crucially change the physics of the investigated system. This is in contrast to the excited states and the ensuing Rabi splitting. (27) Instead, the chosen ω

_{Be}considerably enhances the numerical stability of the calculations, which has the following reason: At the current state of our implementation, we need to make use of a basis set that we generate by a preliminary calculation. To generate a basis that captures electronic and photonic parts of the system equally well, we need to make sure that the energy scales of both degrees of freedom are similar. This can be controlled easiest by varying ω. We want to stress that this basis-set issue is not a fundamental problem of the dressed orbital approach. On the one hand, we plan to control the photonic basis directly and on the other hand, we are working on optimizing an alternative conjugate-gradient routine that does not use a specific basis. Details can be found in the Supporting Information, section 2.3.1.

*N*= 2)-particles examples. In this setting, the dressed auxiliary system is four-dimensional (two particles with two coordinates each), which is still small enough to be solved exactly in a four-dimensional discretized simulation box, so that we can compare dressed RDMFT (with the Müller functional of eq 19), dressed HF (see eq 18) and the exact solutions. We used the box lengths of

*L*

_{x}=

*L*

_{q}= 16 au and spacings of d

*x*= d

*q*= 0.14 au to model the electronic

*x*and photonic coordinates

*q*of the two dressed particles in the exact routine. (We want to mention that the box length is not entirely converged with these parameters. In a (numerically very expensive) benchmark calculation, we observed a further decrease of energy with larger boxes (the calculations with respect to the spacing are converged), but the changes in energy and density are only of the order of 10

^{–5}or less. All the following results require a maximal precision of the order of 10

^{–2}in energy as well as in the density and thus we can safely use the given parameters. Details can be found in the Supporting Information, section 1). For dressed RDMFT and dressed HF instead, we considered two-dimensional simulation boxes for every dressed orbital and we set

*L*

_{x}=

*L*

_{q}= 20 au and d

*x*= d

*q*= 0.1 au. We obtained converged results for () natural orbitals for He (H

_{2}). Details about how we determined these parameters can be found in the Supporting Information.

*g*/ω (this quantity is typically used as a measure for the strength of the light–matter interaction, see, e.g., ref (28)) for He and H

_{2}, respectively. We thereby go from weak to deep-strong coupling with

*g*/ω = 1. (3) The deep-strong coupling regime has been reached in different systems like for instance for Landau polaritons. (60) For (organic) molecules the highest reported coupling strengths are in the ultrastrong regime of

*g*/ω ≈ 0.4. (3) We see that while dressed HF deviates strongly for large couplings, dressed RDMFT remains very accurate over the whole range of coupling strength. Still, a more severe test of the accuracy of our method is if instead of merely energies, we compare spatially resolved quantities like the ground-state density ρ(

*x*,

*q*) ≡ γ(

*x*,

*q*;

*x*,

*q*). To simplify this discussion, we separate the electronic and photonic parts of the two-dimensional density by integration, i.e., ρ(

*x*) = ∫d

*qρ*(

*x*,

*q*) and ρ(

*q*) = ∫d

*x*ρ(

*x*,

*q*). The exact reference solutions show that with increasing

*g*/ω, the electronic part of the density becomes more localized, while the photonic part becomes broadened (the reader is referred to the last two paragraphs of this section and Figures 10 and 11 for details about the effects of matter−photon coupling). This behavior is captured qualitatively with dressed HF as well as with dressed RDMFT. The latter performs for the electronic density considerably better over the whole range of coupling strength, whereas for the photonic densities, both levels of theory deviate in a similar way from the exact result. This is shown for

*g*/ω = 0.1 in Figure 3 for both test systems. Looking at the electronic densities, we can observe a feature that the ground state energy does not reveal. In some cases the effects of the two approximations are contrary to each other as we can see in the He case. Here, the dressed RDMFT electronic density is more localized around the center of charge than the exact reference and the electronic density of dressed HF less. In other cases instead, both theories overlocalize ρ(

*x*) (here visible for H

_{2}).

_{i}(

*x*,

*q*). Again, we separate electronic and photonic contributions and show their reduced electronic density ρ

_{i}(

*x*) = ∫d

*q*|ϕ

_{i}(

*x*,

*q*)|

^{2}. Figure 4 depicts the first three dressed natural orbital densities of dressed RDMFT in comparison with the exact ones for both test systems. While it holds that for both systems, the lowest natural orbital density of the dressed RDMFT approximation is almost the same as the exact one, and the second natural orbital density is only slightly different, the third natural orbital density of H

_{2}differs even qualitatively. For He, similar strong deviations are visible for the fourth natural orbital. However, as long as such strong deviations only occur for natural orbitals with small natural occupation numbers, like in these cases (H

_{2}:

*n*

_{1}= 1.878,

*n*

_{2}= 0.102,

*n*

_{3}= 0.015; He:

*n*

_{1}= 1.978,

*n*

_{2}= 0.020,

*n*

_{3}= 0.001), their (inaccurate) contribution to the density and total energy remains small.

_{i}(

*q*) = ∫d

*x*|ϕ

_{i}(

*x*,

*q*)|

^{2}, the first three of which are plotted in Figure 5, for He and H

_{2}. Here, the dressed RDMFT results even agree better with the exact solution than their electronic counterparts. Apparently, dressed RDMFT captures the photonic properties of the tested systems very accurately for the ultrastrong coupling regime. The accuracy drops with increasing

*g*/ω.

*N*

_{ph}as a function of the coupling strength

*g*/ω that we calculated by using eq 14, that is,

*N*

_{ph}= − , with the photon mode energy From weak to the beginning of the ultrastrong coupling regime (

*g*/ω ≈ 0.1), both dressed HF and dressed RDMFT capture

*N*

_{ph}well. For very large coupling strengths, the deviations to the exact mode occupation becomes sizable. This might sound counterintuitive, as the photonic density is described comparatively well. The reason is that the photon occupation, in contrast to the density, is mainly determined by the second and third natural orbital, because the first natural orbital resembles a photonic ground state with occupation number zero in the studied cases. Dressed HF does not consider a second orbital (the first instead is doubly occupied) and thus cannot capture the effect, and for dressed RDMFT, the error in the second and third natural orbital is much larger than in the first (see Figure 5). However, it is probable that this can be improved by better functionals.

*g*/ω for dressed HF and dressed RDMFT, respectively. Like in the two-electron systems, the deviation between both curves increases for larger

*g*/ω and, as expected, the dressed RDMFT energies are lower than the dressed HF results. Analyzing the ground-state densities, we see a similar trend as in the two-particle systems. With increasing

*g*/ω, the electronic (photonic) part of the density becomes more (less) localized, though the details differ as we show in the last part of this section (see Figure 11 and the corresponding part in the main text). Comparing dressed RDMFT with dressed HF, we observe that the variation of the electronic (photonic) density with increasing coupling strength is less (more) prounounced for dressed RDMFT, as Figure 8 shows. We conclude the survey of Be with the mode occupation under variation of the coupling strength (see Figure 9). We see that the value of

*g*/ω ≈ 0.5 separates two regions. For

*g*/ω < 0.5, dressed RDMFT finds a larger mode occupation than dressed HF, and for

*g*/ω > 0.5 instead, the dressed HF mode is more strongly occupied. We found similar behavior also for the two-particle systems, although the boundary between the two regions was considerably different there (He:

*g*/ω ≈ 0.8, H

_{2}:

*g*/ω ≈ 0.1, see Figure 6).

_{2}as an example for a chemical reaction, where we use

*v*

_{H2}(

*x*) with different

*d*. In Figure 10, we see the density of two H-atoms under variation of the distance

*d*with and without the (strong) coupling to the cavity. We see that the influence of the cavity mode strongly depends on the exact electronic structure. The interaction with the cavity mode can locally reduce or enhance the electronic repulsion due to the Coulomb interaction, where the exact interplay between both effects depends on the interatomic distance. Thus, we can observe a number of different effects like pure localization of the density toward the center of charge (

*d*= 1) or localization combined with a local enhancement of repulsion such that the density deviations exhibit a double peak structure (

*d*= 2). The local enhancement of electronic repulsion can grow so strong that the density at the center of charge is reduced but at the same time the density maxima shift closer to each other, which is an effective suppression of electronic repulsion (

*d*= 3). This interplay is reflected in the natural orbitals and occupation numbers. The coupling shifts a considerable amount of occupation from the first natural orbital to the second and third one. The contribution to the total density of the former (latter) has the character of enhanced (suppressed) electron repulsion. To show the potential of these effects, we present calculations in the deep-strong coupling regime with

*g*/ω = 1.0, where the effects reach the order of 10% of the unperturbed density, which is enormous. For smaller coupling strengths of the order of

*g*/ω = 0.1, these effects are as diverse, but naturally smaller with density deformations of the order of 10

^{–3}. However, as every observable depends on the density, such deviations are significant. Remarkably, dressed RDMFT reproduces the effects accurately.

*g*/ω = 0.8, we still observe

*n*

_{1}= 1.85. However, it should be noted that the good agreement of dressed RDMFT with the exact calculation in comparison to dressed HF is exactly because of the contribution of the second natural orbital, that is (still considerably) occupied with

*n*

_{2}= 0.14.) Things change for Be, where we have several dominant orbitals. With increasing coupling strength, we see like in the dissociation example a subtle interplay between suppression and local enhancement of the electronic repulsion, that depends on the coupling strength. Thus, for the same coupling strength, we can observe opposite (

*g*/ω = 0.1 and 0.4 in the plot) but also similar effects (

*g*/ω = 0.8 in the plot) in two systems that have almost the same “bare” density shape. Like in the dissociation example, this intricate behavior can be understood by the interplay of the different natural orbitals contributing to the electronic density. In this particular case, the main physics happens in the second and third natural orbital, where the former (with a double-peak structure) loses a considerable amount of occupation to the latter (with a triple peak structure) with increasing coupling strength.

## Conclusion

*N*electrons coupled to

*M*photon modes, (

*N*– 1)

*M*auxiliary coordinates, which allowed us to “fermionize” the coupled problem with respect to new polaritonic coordinates. The resulting dressed fermionic particles are governed by a Hamiltonian with only one-body and two-body terms and, thus, can be applied to any standard electronic-structure method. The extension is constructed in such a way that the auxiliary dimensions do not modify the original physical system, and the physical observables are easy to recover. Notably, besides the possibility to study modifications of electronic systems due to a cavity mode, dressed RDMFT offers also the possibility to calculate purely photonic observables like the mode occupation or fluctuations of the electric and magnetic field. We used this framework to develop and implement dressed RDMFT in the electronic-structure code Octopus (88) and tested it with the Hartree–Fock and Müller functional. For simple one-dimensional models of atoms and molecules, the obtained approximate results were in good agreement with the exact results from the weak to the deep-strong coupling regime. We then used our method to show that the modifications due to strong matter–photon coupling are far from trivial and depend on the detailed electronic structure. For a molecular as well as an atomic system we showed that strong coupling can locally enhance and suppress the Coulomb-induced repulsion between electrons. This behavior does not only depend on the strength of the matter–photon coupling but also on the details of the matter subsystem (e.g., the interatomic distance of the atoms of a molecule). We showed that our method allows to predict the structures accurately inside and outside of the cavity and furthermore extends the well-established tools of natural orbitals to analyze coupled light–matter systems.

*q*-exchange symmetry? First calculations for many particles indicate that it will become important to enforce this extra symmetry to stay accurate when going from the weak to the deep-strong coupling regime. Furthermore, it might become beneficial to avoid the “fermionization” that we employed, and then very interesting mathematical questions about

*N*-representability for coupled fermion-boson systems need to be addressed. Here, the understanding how to enforce the

*q*-exchange symmetries in the dressed formulation could be very useful.

## Supporting Information

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.9b00648.

Survey on the bosonic symmetry of the photon wave function. Details about the convergence study of the numerical examples shown in the paper. Protocol for the convergence of a dressed HF/RDMFT calculation (PDF)

## Terms & Conditions

Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

## Acknowledgments

F.B. would like to thank Nicole Helbig, Klaas Giesbertz, Micael Oliveira, and Christian Schäfer for stimulating and useful discussions. We acknowledge financial support from the European Research Council (ERC-2015-AdG-694097).

## References

This article references 91 other publications.

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Particular attention is given to so-called strong coupling limit, where the hybrid exciton-plasmon nature of the system response is strongly expressed. While traditional descriptions of such phenomena usually rely on anal. of the electromagnetic response of inhomogeneous dielec. environments that individually support plasmon and exciton excitations, here we explore also the consequences of a more detailed description of the mol. environment in terms of its quantum d. matrix (applied in a mean field approxn. level). Such a description makes it possible to account for characteristics that cannot be described by the dielec. response model: the effects of dephasing on the mol. response on one hand, and nonlinear response on the other. It also highlights the still missing important ingredients in the numerical approach, in particular its limitation to a classical description of the radiation field and its reliance on a mean field description of the many-body mol. system. We end our review with an outlook to the near future, where these limitations will be addressed and new novel applications of the numerical approach will be pursued.**3**Kockum, A. F.; Miranowicz, A.; De Liberato, S.; Savasta, S.; Nori, F. Ultrastrong coupling between light and matter.*Nat. Rev. Phys.*2019,*1*, 19– 40, DOI: 10.1038/s42254-018-0006-2**4**Hutchison, J. A.; Schwartz, T.; Genet, C.; Devaux, E.; Ebbesen, T. W. Modifying chemical landscapes by coupling to vacuum fields.*Angew. Chem., Int. Ed.*2012,*51*, 1592– 1596, DOI: 10.1002/anie.201107033[Crossref], [CAS], Google Scholar4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XkvVSjuw%253D%253D&md5=36ea0bf96a50529ee200f504802ab0caModifying Chemical Landscapes by Coupling to Vacuum FieldsHutchison, James A.; Schwartz, Tal; Genet, Cyriaque; Devaux, Eloise; Ebbesen, Thomas W.Angewandte Chemie, International Edition (2012), 51 (7), 1592-1596, S1592/1-S1592/3CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)Modifying chem. landscapes by coupling to vacuum fields is discussed.**5**Thomas, A.; George, J.; Shalabney, A.; Dryzhakov, M.; Varma, S. J.; Moran, J.; Chervy, T.; Zhong, X.; Devaux, E.; Genet, C.; Hutchison, J. A.; Ebbesen, T. W. Ground-State Chemical Reactivity under Vibrational Coupling to the Vacuum Electromagnetic Field.*Angew. Chem.*2016,*128*, 11634– 11638, DOI: 10.1002/ange.201605504**6**Firstenberg, O.; Peyronel, T.; Liang, Q. Y.; Gorshkov, A. V.; Lukin, M. D.; Vuletić, V. Attractive photons in a quantum nonlinear medium.*Nature*2013,*502*, 71– 75, DOI: 10.1038/nature12512[Crossref], [PubMed], [CAS], Google Scholar6https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsFaktrbO&md5=3df8aad7d0d12741f01e3e27e8f2bb48Attractive photons in a quantum nonlinear mediumFirstenberg, Ofer; Peyronel, Thibault; Liang, Qi-Yu; Gorshkov, Alexey V.; Lukin, Mikhail D.; Vuletic, VladanNature (London, United Kingdom) (2013), 502 (7469), 71-75CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)The fundamental properties of light derive from its constituent particles-massless quanta (photons) that do not interact with one another. However, it has long been known that the realization of coherent interactions between individual photons, akin to those assocd. with conventional massive particles, could enable a wide variety of novel scientific and engineering applications. Here we demonstrate a quantum nonlinear medium inside which individual photons travel as massive particles with strong mutual attraction, such that the propagation of photon pairs is dominated by a two-photon bound state. We achieve this through dispersive coupling of light to strongly interacting atoms in highly excited Rydberg states. We measure the dynamical evolution of the two-photon wavefunction using time-resolved quantum state tomog., and demonstrate a conditional phase shift exceeding one radian, resulting in polarization-entangled photon pairs. Particular applications of this technique include all-optical switching, deterministic photonic quantum logic and the generation of strongly correlated states of light.**7**Goban, A.; Hung, C. L.; Hood, J. D.; Yu, S. P.; Muniz, J. A.; Painter, O.; Kimble, H. J. Superradiance for Atoms Trapped along a Photonic Crystal Waveguide.*Phys. Rev. Lett.*2015,*115*, 063601, DOI: 10.1103/PhysRevLett.115.063601[Crossref], [PubMed], [CAS], Google Scholar7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsVaisrY%253D&md5=b20dba87cbcd98b45f1b94701a44581aSuperradiance for atoms trapped along a photonic crystal waveguideGoban, A.; Hung, C.-L.; Hood, J. D.; Yu, S.-P.; Muniz, J. A.; Painter, O.; Kimble, H. J.Physical Review Letters (2015), 115 (6), 063601/1-063601/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We report observations of superradiance for atoms trapped in the near field of a photonic crystal waveguide (PCW). By fabricating the PCW with a band edge near the D1 transition of at. cesium, strong interaction is achieved between trapped atoms and guided-mode photons. Following short-pulse excitation, we record the decay of guided-mode emission and find a superradiant emission rate scaling as ΓSR α NΓ1D for av. atom no. 0.19.ltorsim. N .ltorsim. 2.6 atoms, where Γ1D/Γ' =1.0±0.1 is the peak single-atom radiative decay rate into the PCW guided mode, and Γ" is the radiative decay rate into all the other channels. These advances provide new tools for investigations of photon-mediated atom-atom interactions in the many-body regime.**8**Coles, D. M.; Yang, Y.; Wang, Y.; Grant, R. T.; Taylor, R. A.; Saikin, S. K.; Aspuru-Guzik, A.; Lidzey, D. G.; Tang, J. K.-H.; Smith, J. M. Strong coupling between chlorosomes of photosynthetic bacteria and a confined optical cavity mode.*Nat. Commun.*2014,*5*, 5561, DOI: 10.1038/ncomms6561[Crossref], [PubMed], [CAS], Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjvFajsb0%253D&md5=0ada1595a0efdba9f15a6d20c4b8f243Strong coupling between chlorosomes of photosynthetic bacteria and a confined optical cavity modeColes, David M.; Yang, Yanshen; Wang, Yaya; Grant, Richard T.; Taylor, Robert A.; Saikin, Semion K.; Aspuru-Guzik, Alan; Lidzey, David G.; Tang, Joseph Kuo-Hsiang; Smith, Jason M.Nature Communications (2014), 5 (), 5561CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)Strong exciton-photon coupling is the result of a reversible exchange of energy between an excited state and a confined optical field. This results in the formation of polariton states that have energies different from the exciton and photon. We demonstrate strong exciton-photon coupling between light-harvesting complexes and a confined optical mode within a metallic optical microcavity. The energetic anti-crossing between the exciton and photon dispersions characteristic of strong coupling is obsd. in reflectivity and transmission with a Rabi splitting energy on the order of 150 meV, which corresponds to about 1000 chlorosomes coherently coupled to the cavity mode. We believe that the strong coupling regime presents an opportunity to modify the energy transfer pathways within photosynthetic organisms without modification of the mol. structure.**9**Orgiu, E.; George, J.; Hutchison, J. A.; Devaux, E.; Dayen, J. F.; Doudin, B.; Stellacci, F.; Genet, C.; Schachenmayer, J.; Genes, C. Conductivity in organic semiconductors hybridized with the vacuum field.*Nat. Mater.*2015,*14*, 1123– 1129, DOI: 10.1038/nmat4392[Crossref], [PubMed], [CAS], Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhsV2ltrvI&md5=ff034f20b52069c3e3d86e8352b3d786Conductivity in organic semiconductors hybridized with the vacuum fieldOrgiu, E.; George, J.; Hutchison, J. A.; Devaux, E.; Dayen, J. F.; Doudin, B.; Stellacci, F.; Genet, C.; Schachenmayer, J.; Genes, C.; Pupillo, G.; Samori, P.; Ebbesen, T. W.Nature Materials (2015), 14 (11), 1123-1129CODEN: NMAACR; ISSN:1476-1122. (Nature Publishing Group)Much effort over the past decades was focused on improving carrier mobility in org. thin-film transistors by optimizing the organization of the material or the device architecture. Here the authors take a different path to solving this problem, by injecting carriers into states that are hybridized to the vacuum electromagnetic field. To test this idea, org. semiconductors were strongly coupled to plasmonic modes to form coherent states that can extend over as many as 105 mols. and should thereby favor cond. Indeed the current does increase by an order of magnitude at resonance in the coupled state, reflecting mostly a change in field-effect mobility. A theor. quantum model confirms the delocalization of the wavefunctions of the hybridized states and its effect on the cond. The authors' findings illustrate the potential of engineering the vacuum electromagnetic environment to modify and to improve properties of materials.**10**Andrew, P.; Barnes, W. L. Energy transfer across a metal film mediated by surface plasmon polaritons.*Science*2004,*306*, 1002– 1005, DOI: 10.1126/science.1102992[Crossref], [PubMed], [CAS], Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXptFyjtbg%253D&md5=4c56f8c2952e56e2bd46d08b19e7cdd7Energy Transfer Across a Metal Film Mediated by Surface Plasmon PolaritonsAndrew, P.; Barnes, W. L.Science (Washington, DC, United States) (2004), 306 (5698), 1002-1005CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)Coupled surface plasmon polaritons (SPPs) are shown to provide effective transfer of excitation energy from donor mols. to acceptor mols. on opposite sides of metal films up to 120 nm thick. This variant of radiative transfer should allow directional control over the flow of excitation energy with the use of suitably designed metallic nanostructures, with SPPs mediating transfer over length scales of 10-7 to 10-4 meters. In the emerging field of nanophotonics, such a prospect could allow subwavelength-scale manipulation of light and provide an interface to the outside world.**11**Dicke, R. H. Coherence in spontaneous radiation processes.*Phys. Rev.*1954,*93*, 99– 110, DOI: 10.1103/PhysRev.93.99[Crossref], [CAS], Google Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaG2cXjsV2jtA%253D%253D&md5=45eea4ac630974b5996ac36a4951308dCoherence in spontaneous radiation processesDicke, R. H.Physical Review (1954), 93 (), 99-110CODEN: PHRVAO; ISSN:0031-899X.Energy levels corresponding to certain correlations between individual mols. were obtained from the consideration of a radiating gas as a single quantum-mech. system. Spontaneous emission of radiation in a transition between 2 such levels leads to the emission of coherent radiation. The discussion is limited first to a gas of dimension small compared with a wave length. Spontaneous radiation rates and natural line breadths are calcd. For a gas of large extent the effect of photon recoil momentum on coherence is calcd. The effect of a radiation pulse in exciting "super-radiant" states is discussed. The angular correlation between successive photons spontaneously emitted by a gas initially in thermal equil. is calcd.**12**Todorov, Y.; Andrews, A. M.; Colombelli, R.; De Liberato, S.; Ciuti, C.; Klang, P.; Strasser, G.; Sirtori, C. Ultrastrong light-matter coupling regime with polariton dots.*Phys. Rev. Lett.*2010,*105*, 1– 4, DOI: 10.1103/PhysRevLett.105.196402**13**Michetti, P.; Mazza, L.; La Rocca, G. C. In*Organic Nanophotonics: Fundamentals and Applications*; Zhao, Y. S., Ed.; Springer Berlin Heidelberg: Berlin, Heidelberg, 2015; pp 39– 68.Google ScholarThere is no corresponding record for this reference.**14**Cwik, J. A.; Kirton, P.; De Liberato, S.; Keeling, J. Excitonic spectral features in strongly coupled organic polaritons.*Phys. Rev. A: At., Mol., Opt. Phys.*2016,*93*, 1– 12, DOI: 10.1103/PhysRevA.93.033840**15**Galego, J.; Garcia-Vidal, F. J.; Feist, J. Suppressing photochemical reactions with quantized light fields.*Nat. Commun.*2016,*7*, 1– 6, DOI: 10.1038/ncomms13841**16**Feist, J.; Galego, J.; Garcia-Vidal, F. J. Polaritonic Chemistry with Organic Molecules.*ACS Photonics*2018,*5*, 205– 216, DOI: 10.1021/acsphotonics.7b00680[ACS Full Text ], [CAS], Google Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFGhsrnJ&md5=a208ab762f33f613e3144f69d88ad3f5Polaritonic Chemistry with Organic MoleculesFeist, Johannes; Galego, Javier; Garcia-Vidal, Francisco J.ACS Photonics (2018), 5 (1), 205-216CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)A review. The authors present an overview of the general concepts of polaritonic chem. with org. mols., i.e., the manipulation of chem. structure that can be achieved through strong coupling between confined light modes and org. mols. Strong coupling and the assocd. formation of polaritons, hybrid light-matter excitations, lead to energy shifts in such systems that can amt. to a large fraction of the uncoupled transition energy. This has recently been shown to significantly alter the chem. structure of the coupled mols., which opens the possibility to manipulate and control reactions. The authors discuss the current state of theory for describing these changes and present several applications, with a particular focus on the collective effects obsd. when many mols. are involved in strong coupling.**17**Feist, J.; Garcia-Vidal, F. J. Extraordinary exciton conductance induced by strong coupling.*Phys. Rev. Lett.*2015,*114*, 1– 5, DOI: 10.1103/PhysRevLett.114.196402**18**Herrera, F.; Spano, F. C. Cavity-Controlled Chemistry in Molecular Ensembles.*Phys. Rev. Lett.*2016,*116*, 1– 6, DOI: 10.1103/PhysRevLett.116.238301**19**Cirio, M.; De Liberato, S.; Lambert, N.; Nori, F. Ground State Electroluminescence.*Phys. Rev. Lett.*2016,*116*, 1– 7, DOI: 10.1103/PhysRevLett.116.113601**20**Kockum, A. F.; MacRì, V.; Garziano, L.; Savasta, S.; Nori, F. Frequency conversion in ultrastrong cavity QED.*Sci. Rep.*2017,*7*, 1– 13, DOI: 10.1038/s41598-017-04225-3[Crossref], [PubMed], [CAS], Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsFehtbjF&md5=446004959d31342ff1754300e64daea1Smoking induces DNA methylation changes in Multiple Sclerosis patients with exposure-response relationshipMarabita, Francesco; Almgren, Malin; Sjoeholm, Louise K.; Kular, Lara; Liu, Yun; James, Tojo; Kiss, Nimrod B.; Feinberg, Andrew P.; Olsson, Tomas; Kockum, Ingrid; Alfredsson, Lars; Ekstroem, Tomas J.; Jagodic, MajaScientific Reports (2017), 7 (1), 1-15CODEN: SRCEC3; ISSN:2045-2322. (Nature Research)Cigarette smoking is an established environmental risk factor for Multiple Sclerosis (MS), a chronic inflammatory and neurodegenerative disease, although a mechanistic basis remains largely unknown. We aimed at investigating how smoking affects blood DNA methylation in MS patients, by assaying genome-wide DNA methylation and comparing smokers, former smokers and never smokers in two Swedish cohorts, differing for known MS risk factors. Smoking affects DNA methylation genome-wide significantly, an exposure-response relationship exists and the time since smoking cessation affects methylation levels. The results also show that the changes were larger in the cohort bearing the major genetic risk factors for MS (female sex and HLA risk haplotypes). Furthermore, CpG sites mapping to genes with known genetic or functional role in the disease are differentially methylated by smoking. Modeling of the methylation levels for a CpG site in the AHRR gene indicates that MS modifies the effect of smoking on methylation changes, by significantly interacting with the effect of smoking load. Alongside, we report that the gene expression of AHRR increased in MS patients after smoking. Our results suggest that epigenetic modifications may reveal the link between a modifiable risk factor and the pathogenetic mechanisms.**21**De Liberato, S. Light-matter decoupling in the deep strong coupling regime: The breakdown of the purcell effect.*Phys. Rev. Lett.*2014,*112*, 1– 5, DOI: 10.1103/PhysRevLett.112.016401**22**De Liberato, S. Virtual photons in the ground state of a dissipative system.*Nat. Commun.*2017,*8*, 1– 6, DOI: 10.1038/s41467-017-01504-5[Crossref], [PubMed], [CAS], Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXos1KqtbY%253D&md5=289e8664a6fd1299b28e81a6c39f7e58Virtual photons in the ground state of a dissipative systemDe Liberato, SimoneNature Communications (2017), 8 (1), 1-6CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)Much of the novel physics predicted to be observable in the ultrastrong light-matter coupling regime rests on the hybridization between states with different nos. of excitations, leading to a population of virtual photons in the system's ground state. In this article, exploiting an exact diagonalisation approach, we derive both anal. and numerical results for the population of virtual photons in presence of arbitrary losses. Specialising our results to the case of Lorentzian resonances we then show that the virtual photon population is only quant. affected by losses, even when those become the dominant energy scale. Our results demonstrate most of the ultrastrong-coupling phenomenol. can be obsd. in loss-dominated systems which are not even in the std. strong coupling regime. We thus open the possibility to investigate ultrastrong-coupling physics to platforms that were previously considered unsuitable due to their large losses.**23**Gely, M. F.; Parra-Rodriguez, A.; Bothner, D.; Blanter, Y. M.; Bosman, S. J.; Solano, E.; Steele, G. A. Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics.*Phys. Rev. B: Condens. Matter Mater. Phys.*2017,*95*, 1– 5, DOI: 10.1103/PhysRevB.95.245115**24**De Bernardis, D.; Pilar, P.; Jaako, T.; De Liberato, S.; Rabl, P. Breakdown of gauge invariance in ultrastrong-coupling cavity QED.*Phys. Rev. A: At., Mol., Opt. Phys.*2018,*98*, 1– 16, DOI: 10.1103/PhysRevA.98.053819**25**Sánchez Muñoz, C.; Nori, F.; De Liberato, S. Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics.*Nat. Commun.*2018,*9*, 1924, DOI: 10.1038/s41467-018-04339-w[Crossref], [PubMed], [CAS], Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1Mfjs1aksw%253D%253D&md5=cbace44e5dcfb489cb04a9fd02c6d374Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamicsSanchez Munoz Carlos; Nori Franco; Nori Franco; De Liberato SimoneNature communications (2018), 9 (1), 1924 ISSN:.Recent technological developments have made it increasingly easy to access the non-perturbative regimes of cavity quantum electrodynamics known as ultrastrong or deep strong coupling, where the light-matter coupling becomes comparable to the bare modal frequencies. In this work, we address the adequacy of the broadly used single-mode cavity approximation to describe such regimes. We demonstrate that, in the non-perturbative light-matter coupling regimes, the single-mode models become unphysical, allowing for superluminal signalling. Moreover, considering the specific example of the quantum Rabi model, we show that the multi-mode description of the electromagnetic field, necessary to account for light propagation at finite speed, yields physical observables that differ radically from their single-mode counterparts already for moderate values of the coupling. Our multi-mode analysis also reveals phenomena of fundamental interest on the dynamics of the intracavity electric field, where a free photonic wavefront and a bound state of virtual photons are shown to coexist.**26**Jaako, T.; Xiang, Z. L.; Garcia-Ripoll, J. J.; Rabl, P. Ultrastrong-coupling phenomena beyond the Dicke model.*Phys. Rev. A: At., Mol., Opt. Phys.*2016,*94*, 1– 10, DOI: 10.1103/PhysRevA.94.033850**27**Schäfer, C.; Ruggenthaler, M.; Rubio, A. Ab initio nonrelativistic quantum electrodynamics: Bridging quantum chemistry and quantum optics from weak to strong coupling.*Phys. Rev. A: At., Mol., Opt. Phys.*2018,*98*, 043801, DOI: 10.1103/PhysRevA.98.043801**28**Schäfer, C.; Ruggenthaler, M.; Appel, H.; Rubio, A. Modification of excitation and charge transfer in cavity quantum-electrodynamical chemistry.*Proc. Natl. Acad. Sci. U. S. A.*2019,*116*, 4883– 4892, DOI: 10.1073/pnas.1814178116[Crossref], [PubMed], [CAS], Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXkvVCisb0%253D&md5=6e129c186de5d7ce049a3148ac79f815Modification of excitation and charge transfer in cavity quantum-electrodynamical chemistrySchAfer, Christian; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelProceedings of the National Academy of Sciences of the United States of America (2019), 116 (11), 4883-4892CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)Energy transfer in terms of excitation or charge is one of the most basic processes in nature, and understanding and controlling them is one of the major challenges of modern quantum chem. In this work, we highlight that these processes as well as other chem. properties can be drastically altered by modifying the vacuum fluctuations of the electromagnetic field in a cavity. By using a real-space formulation from first principles that keeps all of the electronic degrees of freedom in the model explicit and simulates changes in the environment by an effective photon mode, we can easily connect to well-known quantum-chem. results such as Dexter charge-transfer and FA~¶rster excitation-transfer reactions, taking into account the often-disregarded Coulomb and self-polarization interaction. We find that the photonic degrees of freedom introduce extra electron-electron correlations over large distances and that the coupling to the cavity can drastically alter the characteristic charge-transfer behavior and even selectively improve the efficiency. For excitation transfer, we find that the cavity renders the transfer more efficient, essentially distance-independent, and further different configurations of highest efficiency depending on the coherence times. For strong decoherence (short coherence times), the cavity frequency should be in between the isolated excitations of the donor and acceptor, while for weak decoherence (long coherence times), the cavity should enhance a mode that is close to resonance with either donor or acceptor. Our results highlight that changing the photonic environment can redefine chem. processes, rendering polaritonic chem. a promising approach toward the control of chem. reactions.**29**Galego, J.; Garcia-Vidal, F. J.; Feist, J. Cavity-induced modifications of molecular structure in the strong-coupling regime.*Phys. Rev. X*2015,*5*, 1– 14, DOI: 10.1103/PhysRevX.5.041022**30**Kowalewski, M.; Bennett, K.; Mukamel, S. Non-adiabatic dynamics of molecules in optical cavities.*J. Chem. Phys.*2016,*144*, 054309, DOI: 10.1063/1.4941053[Crossref], [PubMed], [CAS], Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xit1OitL4%253D&md5=18ee3f0139812477efa1846918c29a63Non-adiabatic dynamics of molecules in optical cavitiesKowalewski, Markus; Bennett, Kochise; Mukamel, ShaulJournal of Chemical Physics (2016), 144 (5), 054309/1-054309/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Strong coupling of mols. to the vacuum field of micro cavities can modify the potential energy surfaces thereby opening new photophys. and photochem. reaction pathways. While the influence of laser fields is usually described in terms of classical field, coupling to the vacuum state of a cavity has to be described in terms of dressed photon-matter states (polaritons) which require quantized fields. The authors present a derivation of the nonadiabatic couplings for single mols. in the strong coupling regime suitable for the calcn. of the dressed state dynamics. The formalism allows using quantities readily accessible from quantum chem. codes like the adiabatic potential energy surfaces and dipole moments to carry out wave packet simulations in the dressed basis. The implications for photochem. are demonstrated for a set of model systems representing typical situations found in mols. (c) 2016 American Institute of Physics.**31**Garcia-Vidal, F. J.; Feist, J. Long-distance operator for energy transfer.*Science*2017,*357*, 1357– 1358, DOI: 10.1126/science.aao4268[Crossref], [PubMed], [CAS], Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1SrtL3F&md5=c2605da7523151a23e42c70b52984a58Long-distance operator for energy transferGarcia-Vidal, Francisco J.; Feist, JohannesScience (Washington, DC, United States) (2017), 357 (6358), 1357-1358CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)There is no expanded citation for this reference.**32**Zeb, M. A.; Kirton, P. G.; Keeling, J. Exact states and spectra of vibrationally dressed polaritons.*ACS Photonics*2018,*5*, 249– 257, DOI: 10.1021/acsphotonics.7b00916[ACS Full Text ], [CAS], Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1OqsrjN&md5=6b9e2e10680c5a2cdd3f517ea7e1ce5fExact States and Spectra of Vibrationally Dressed PolaritonsZeb, M. Ahsan; Kirton, Peter G.; Keeling, JonathanACS Photonics (2018), 5 (1), 249-257CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)Strong coupling between light and matter is possible with a variety of org. materials. In contrast to the simpler inorg. case, org. materials often have a complicated spectrum, with vibrationally dressed electronic transitions. Strong coupling to light competes with this vibrational dressing and, if strong enough, can suppress the entanglement between electronic and vibrational degrees of freedom. By exploiting symmetries, the authors can perform exact numerical diagonalization to find the polaritonic states for intermediate nos. of mols. and use these to define and validate accurate expressions for the lower polariton states and strong-coupling spectrum in the thermodn. limit. Using this approach, vibrational decoupling occurs as a sharp transition above a crit. matter-light coupling strength. Also the polariton spectrum evolves with the no. of mols., recovering classical linear optics results only at large N.**33**Luk, H. L.; Feist, J.; Toppari, J. J.; Groenhof, G. Multiscale molecular dynamics simulations of polaritonic chemistry.*J. Chem. Theory Comput.*2017,*13*, 4324– 4335, DOI: 10.1021/acs.jctc.7b00388[ACS Full Text ], [CAS], Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1eks7nK&md5=9f428e53a85a5f9c1e7124ffb699a378Multiscale Molecular Dynamics Simulations of Polaritonic ChemistryLuk, Hoi Ling; Feist, Johannes; Toppari, J. Jussi; Groenhof, GerritJournal of Chemical Theory and Computation (2017), 13 (9), 4324-4335CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)When photoactive mols. interact strongly with confined light modes as found in plasmonic structures or optical cavities, new hybrid light-matter states can form, the so-called polaritons. These polaritons are coherent superpositions (in the quantum mech. sense) of excitations of the mols. and of the cavity photon or surface plasmon. Recent exptl. and theor. works suggest that access to these polaritons in cavities could provide a totally new and attractive paradigm for controlling chem. reactions that falls in between traditional chem. catalysis and coherent laser control. However, designing cavity parameters to control chem. requires a theor. model with which the effect of the light-matter coupling on the mol. dynamics can be predicted accurately. Here we present a multiscale quantum mechanics/mol. mechanics (QM/MM) mol. dynamics simulation model for photoactive mols. that are strongly coupled to confined light in optical cavities or surface plasmons. Using this model we have performed simulations with up to 1600 Rhodamine mols. in a cavity. The results of these simulations reveal that the contributions of the mols. to the polariton are time-dependent due to thermal fluctuations that break symmetry. Furthermore, the simulations suggest that in addn. to the cavity quality factor, also the Stokes shift and no. of mols. control the lifetime of the polariton. Because large nos. of mols. interacting with confined light can now be simulated in at. detail, we anticipate that our method will lead to a better understanding of the effects of strong coupling on chem. reactivity. Ultimately the method may even be used to systematically design cavities to control photochem.**34**del Pino, J.; Schröder, F. A.; Chin, A. W.; Feist, J.; Garcia-Vidal, F. J. Tensor network simulation of polaron-polaritons in organic microcavities.*Phys. Rev. B: Condens. Matter Mater. Phys.*2018,*98*, 165416, DOI: 10.1103/PhysRevB.98.165416**35**Ruggenthaler, M.; Tancogne-Dejean, N.; Flick, J.; Appel, H.; Rubio, A. From a quantum-electrodynamical light–matter description to novel spectroscopies.*Nat. Rev. Chem.*2018,*2*, 0118, DOI: 10.1038/s41570-018-0118[Crossref], [CAS], Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtF2lurjP&md5=e6cdcdad8e973727c57a498378a91eb1From a quantum-electrodynamical light-matter description to novel spectroscopiesRuggenthaler, Michael; Tancogne-Dejean, Nicolas; Flick, Johannes; Appel, Heiko; Rubio, AngelNature Reviews Chemistry (2018), 2 (3), 0118CODEN: NRCAF7; ISSN:2397-3358. (Nature Research)Insights from spectroscopic expts. led to the development of quantum mechanics as the common theor. framework for describing the phys. and chem. properties of atoms, mols. and materials. Later, a full quantum description of charged particles, electromagnetic radiation and special relativity was developed, leading to quantum electrodynamics (QED). This is, to our current understanding, the most complete theory describing photon-matter interactions in correlated many-body systems. In the low-energy regime, simplified models of QED have been developed to describe and analyze spectra over a wide spatiotemporal range as well as phys. systems. In this Review, we highlight the interrelations and limitations of such theor. models, thereby showing that they arise from low-energy simplifications of the full QED formalism, in which antiparticles and the internal structure of the nuclei are neglected. Taking mol. systems as an example, we discuss how the breakdown of some simplifications of low-energy QED challenges our conventional understanding of light-matter interactions. In addn. to high-precision at. measurements and simulations of particle physics problems in solid-state systems, new theor. features that account for collective QED effects in complex interacting many-particle systems could become a material-based route to further advance our current understanding of light-matter interactions.**36**Groenhof, G.; Toppari, J. J. Coherent light harvesting through strong coupling to confined light.*J. Phys. Chem. Lett.*2018,*9*, 4848– 4851, DOI: 10.1021/acs.jpclett.8b02032[ACS Full Text ], [CAS], Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsVKjtbfE&md5=f1f63c555637939a5fc7b8524a597ff8Coherent Light Harvesting through Strong Coupling to Confined LightGroenhof, Gerrit; Toppari, J. JussiJournal of Physical Chemistry Letters (2018), 9 (17), 4848-4851CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)When photoactive mols. interact strongly with confined light modes, new hybrid light-matter states may form: the polaritons. These polaritons are coherent superpositions of excitations of the mols. and of the cavity photon. Recently, polaritons were shown to mediate energy transfer between chromophores at distances beyond the Forster limit. Here we explore the potential of strong coupling for light-harvesting applications by means of atomistic mol. dynamics simulations of mixts. of photoreactive and non-photo-reactive mols. strongly coupled to a single confined light mode. These mols. are spatially sepd. and present at different concns. Our simulations suggest that while the excitation is initially fully delocalized over all mols. and the confined light mode, it very rapidly localizes onto one of the photoreactive mols., which then undergoes the reaction.**37**Vendrell, O. Collective Jahn-Teller interactions through light-matter coupling in a cavity.*Phys. Rev. Lett.*2018,*121*, 253001, DOI: 10.1103/PhysRevLett.121.253001[Crossref], [PubMed], [CAS], Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXltFWhu78%253D&md5=2588a03813dba311ef4680461d4eaa4fCollective Jahn-Teller Interactions through Light-Matter Coupling in a CavityVendrell, OriolPhysical Review Letters (2018), 121 (25), 253001CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)The ultrafast nonradiative relaxation of a mol. ensemble coupled to a cavity mode is considered theor. and by real-time quantum dynamics. For equal coupling strength of single mols. to the cavity mode, the nonradiative relaxation rate from the upper to the lower polariton states is found to strongly depend on the no. of coupled mols. The coupling of both bright and dark polaritonic states among each other constitutes a special case of (pseudo-)Jahn-Teller interactions involving collective displacements the internal coordinates of the mols. in the ensemble, and the strength of the first order vibronic coupling depends exclusively on the gradient of the energy gaps between mol. electronic states. For N>2 mols., the N-1 dark light-matter states between the two optically active polaritons feature true collective conical intersection crossings, whose location depends on the internal at. coordinates of each mol. in the ensemble, and which contribute to the ultrafast nonradiative decay from the upper polariton.**38**Reitz, M.; Sommer, C.; Genes, C. Langevin approach to quantum optics with molecules.*Phys. Rev. Lett.*2019,*122*, 203602, DOI: 10.1103/PhysRevLett.122.203602[Crossref], [PubMed], [CAS], Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtFKlsbfE&md5=dd16d30dffec617059d2846562fed4c4Langevin Approach to Quantum Optics with MoleculesReitz, Michael; Sommer, Christian; Genes, ClaudiuPhysical Review Letters (2019), 122 (20), 203602CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)We investigate the interaction between light and mol. systems modeled as quantum emitters coupled to a multitude of vibrational modes via a Holstein-type interaction. We follow a quantum Langevin equations approach that allows for anal. derivations of absorption and fluorescence profiles of mols. driven by classical fields or coupled to quantized optical modes. We retrieve anal. expressions for the modification of the radiative emission branching ratio in the Purcell regime and for the asym. cavity transmission assocd. with dissipative cross talk between upper and lower polaritons in the strong coupling regime. We also characterize the F.ovrddot.orster resonance energy transfer process between donor-acceptor mols. mediated by the vacuum or by a cavity mode.**39**Triana, J. F.; Sanz-Vicario, J. L. Revealing the Presence of Potential Crossings in Diatomics Induced by Quantum Cavity Radiation.*Phys. Rev. Lett.*2019,*122*, 063603, DOI: 10.1103/PhysRevLett.122.063603[Crossref], [PubMed], [CAS], Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXnvFSrsLs%253D&md5=53316bbfd4a8311f93286fe1ca9ce117Revealing the Presence of Potential Crossings in Diatomics Induced by Quantum Cavity RadiationTriana, Johan F.; Sanz-Vicario, Jose LuisPhysical Review Letters (2019), 122 (6), 063603CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)We propose an expt. to find evidence of the formation of light-induced crossings provoked by cavity quantum radiation on simple mols. by using state-of-the-art optical cavities, mol. beams, pump-probe laser schemes, and velocity mapping detectors for fragmentation. The procedure is based on prompt excitation and subsequent dissocn. in a three-state scheme of a polar diat. mol., with two Σ1 states (ground and first excited) coupled first by the UV pump laser and then by the cavity radiation, and a third fully dissociative state Π1 coupled through the delayed UV/V probe laser. The obsd. enhancement of photodissocn. yields in the Π1 channel at given time delays between the pump and probe lasers unambiguously indicates the formation of a light-induced crossing between the two Σ1 field-dressed potential energy curves of the mol. Also, the prodn. of cavity photons out of the vacuum field state via nonadiabatic effects represents a showcase of a mol. dynamical Casimir effect. To simulate the expt. outcome, we perform ab initio coherent quantum dynamics of the mol. LiF subject to external lasers and quantum cavity interactions in the strong coupling regime, using a product grid representation of the total polaritonic wave function for both vibrational and photon degrees of freedom.**40**Galego, J.; Climent, C.; Garcia-Vidal, F. J.; Feist, J. Cavity Casimir-Polder forces and their effects in ground state chemical reactivity.*Phys. Rev. X*2019,*9*, 1– 22, DOI: 10.1103/PhysRevX.9.021057**41**Csehi, A.; Kowalewski, M.; Halász, G. J.; Vibók, Á. Ultrafast dynamics in the vicinity of quantum light-induced conical intersections.*New J. Phys.*2019, na, DOI: 10.1088/1367-2630/ab3fcc**42**Martínez-Martínez, L. A.; Ribeiro, R. F.; Campos-González-Angulo, J.; Yuen-Zhou, J. Can Ultrastrong Coupling Change Ground-State Chemical Reactions?.*ACS Photonics*2018,*5*, 167– 176, DOI: 10.1021/acsphotonics.7b00610[ACS Full Text ], [CAS], Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1aksL%252FK&md5=3c3aa349f79c593e175d976bc588a60dCan Ultrastrong Coupling Change Ground-State Chemical Reactions?Martinez-Martinez, Luis A.; Ribeiro, Raphael F.; Campos-Gonzalez-Angulo, Jorge; Yuen-Zhou, JoelACS Photonics (2018), 5 (1), 167-176CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)Recent advancements on the fabrication of org. micro- and nanostructures have permitted the strong collective light-matter coupling regime to be reached with mol. materials. Pioneering works in this direction have shown the effects of this regime in the excited state reactivity of mol. systems and at the same time have opened up the question of whether it is possible to introduce any modifications in the electronic ground energy landscape which could affect chem. thermodn. and/or kinetics. In this work, we use a model system of many mols. coupled to a surface-plasmon field to gain insight on the key parameters which govern the modifications of the ground-state potential energy surface. Our findings confirm that the energetic changes per mol. are detd. by effects that are essentially on the order of single-mol. light-matter couplings, in contrast with those of the electronically excited states, for which energetic corrections are of a collective nature. Hence the prospects of ultrastrong coupling to change ground-state chem. reactions for the parameters studied in this model are limited. Still, we reveal some intriguing quantum-coherent effects assocd. with pathways of concerted reactions, where two or more mols. undergo reactions simultaneously and which can be of relevance in low-barrier reactions. Finally, we also explore modifications to nonadiabatic dynamics and conclude that, for our particular model, the presence of a large no. of dark states yields negligible effects. Our study reveals new possibilities as well as limitations for the emerging field of polariton chem.**43**Viehmann, O.; Von Delft, J.; Marquardt, F. Superradiant phase transitions and the standard description of circuit QED.*Phys. Rev. Lett.*2011,*107*, 1– 5, DOI: 10.1103/PhysRevLett.107.113602**44**Dreizler, E. G.*Density Functional Theory - An Approach to the Quantum Many-Body Problem*; Springer: Berlin Heidelberg, 1990.Google ScholarThere is no corresponding record for this reference.**45**Dreizler, R. M.; Engel, E.*Density Functional Theory: An Advanced Course*; Springer, 2011.Google ScholarThere is no corresponding record for this reference.**46**Fetter, A.; Walecka, J.*Quantum Theory of Many-Particle Systems*; Dover: Mineola, NY, 2003.Google ScholarThere is no corresponding record for this reference.**47**Stefanucci, G.; van Leeuwen, R.*Nonequilibrium many-body theory of quantum systems: a modern introduction*; Cambridge University Press, 2013.**48**Onida, G.; Reining, L.; Rubio, A. Electronic excitations: Density-functional versus many-body Green’s-function approaches.*Rev. Mod. Phys.*2002,*74*, 601– 659, DOI: 10.1103/RevModPhys.74.601[Crossref], [CAS], Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xlt1ymsL0%253D&md5=904c22dc306e014cab96b27d8d971951Electronic excitations: density-functional versus many-body Green's-function approachesOnida, Giovanni; Reining, Lucia; Rubio, AngelReviews of Modern Physics (2002), 74 (2), 601-659CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. Electronic excitations lie at the origin of most of the commonly measured spectra. However, the 1st-principles computation of excited states requires a larger effort than ground-state calcns., which can be very efficiently carried out within d.-functional theory. However, two theor. and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green's-function equations, starting with a 1-electron propagator and considering the electron-hole Green's function for the response. Key ingredients are the electron's self-energy Σ and the electron-hole interaction. A good approxn. for Σ was obtained with Hedin's GW approach, using d.-functional theory as a zero-order soln. First-principles GW calcns. for real systems were successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional deriv. of Σ. An alternative approach to calcg. electronic excitations is the time-dependent d.-functional theory (TDDFT), which offers the important practical advantage of a dependence on d. rather than on multivariable Green's functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-d. approxn. gave promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green's functions and the TDDFT approaches profit from mutual insight. This review compares the theor. and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.**49**Theophilou, I.; Buchholz, F.; Eich, F. G.; Ruggenthaler, M.; Rubio, A. Kinetic-Energy Density-Functional Theory on a Lattice.*J. Chem. Theory Comput.*2018,*14*, 4072– 4087, DOI: 10.1021/acs.jctc.8b00292[ACS Full Text ], [CAS], Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXht1Kgu73L&md5=cb188271a155f26e3b03de3813850e75Kinetic-Energy Density-Functional Theory on a LatticeTheophilou, Iris; Buchholz, Florian; Eich, F. G.; Ruggenthaler, Michael; Rubio, AngelJournal of Chemical Theory and Computation (2018), 14 (8), 4072-4087CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a kinetic-energy d.-functional theory and the corresponding kinetic-energy Kohn-Sham (keKS) scheme on a lattice and show that, by including more observables explicitly in a d.-functional approach, already simple approxn. strategies lead to very accurate results. Here, we promote the kinetic-energy d. to a fundamental variable alongside the d. and show for specific cases (anal. and numerically) that there is a one-to-one correspondence between the external pair of on-site potential and site-dependent hopping and the internal pair of d. and kinetic-energy d. On the basis of this mapping, we establish two unknown effective fields, the mean-field exchange-correlation potential and the mean-field exchange-correlation hopping, which force the keKS system to generate the same kinetic-energy d. and d. as the fully interacting one. We show, by a decompn. based on the equations of motions for the d. and the kinetic-energy d., that we can construct simple orbital-dependent functionals that outperform the corresponding exact-exchange Kohn-Sham (KS) approxn. of std. d.-functional theory. We do so by considering the exact KS and keKS systems and comparing the unknown correlation contributions as well as by comparing self-consistent calcns. based on the mean-field exchange (for the effective potential) and a uniform (for the effective hopping) approxn. for the keKS and the exact-exchange approxn. for the KS system, resp.**50**Ruggenthaler, M.; Mackenroth, F.; Bauer, D. Time-dependent Kohn-Sham approach to quantum electrodynamics.*Phys. Rev. A: At., Mol., Opt. Phys.*2011,*84*, 042107, DOI: 10.1103/PhysRevA.84.042107[Crossref], [CAS], Google Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsVWmt7fF&md5=6afdf8c78c3ecb60dfb2f6c712472dcaTime-dependent Kohn-Sham approach to quantum electrodynamicsRuggenthaler, M.; Mackenroth, F.; Bauer, D.Physical Review A: Atomic, Molecular, and Optical Physics (2011), 84 (4, Pt. A), 042107/1-042107/7CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We prove a generalization of the van Leeuwen theorem toward quantum electrodynamics, providing the formal foundations of a time-dependent Kohn-Sham construction for coupled quantized matter and electromagnetic fields. We circumvent the symmetry-causality problems assocd. with the action-functional approach to Kohn-Sham systems. We show that the effective external four-potential and four-current of the Kohn-Sham system are uniquely defined and that the effective four-current takes a very simple form. Further we rederive the Runge-Gross theorem for quantum electrodynamics.**51**Tokatly, I. V. Time-dependent density functional theory for many-electron systems interacting with cavity photons.*Phys. Rev. Lett.*2013,*110*, 1– 5, DOI: 10.1103/PhysRevLett.110.233001**52**Ruggenthaler, M.; Flick, J.; Pellegrini, C.; Appel, H.; Tokatly, I. V.; Rubio, A. Quantum-electrodynamical density-functional theory: Bridging quantum optics and electronic-structure theory.*Phys. Rev. A: At., Mol., Opt. Phys.*2014,*90*, 1– 26, DOI: 10.1103/PhysRevA.90.012508**53**Ruggenthaler, M. Ground-State Quantum-Electrodynamical Density-Functional Theory.*arXiv:1509.01417 [quant-ph]*2015, 1– 6Google ScholarThere is no corresponding record for this reference.**54**Flick, J.; Ruggenthaler, M.; Appel, H.; Rubio, A. Kohn–Sham approach to quantum electrodynamical density-functional theory: Exact time-dependent effective potentials in real space.*Proc. Natl. Acad. Sci. U. S. A.*2015,*112*, 15285– 15290, DOI: 10.1073/pnas.1518224112[Crossref], [PubMed], [CAS], Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvFalt7zL&md5=ab2f497f36a34a13df4f6731ea546c60Kohn-Sham approach to quantum electrodynamical density-functional theory: Exact time-dependent effective potentials in real spaceFlick, Johannes; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelProceedings of the National Academy of Sciences of the United States of America (2015), 112 (50), 15285-15290CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)The d.-functional approach to quantum electrodynamics extends traditional d.-functional theory and opens the possibility to describe electron-photon interactions in terms of effective Kohn-Sham potentials. In this work, we numerically construct the exact electron-photon Kohn-Sham potentials for a prototype system that consists of a trapped electron coupled to a quantized electromagnetic mode in an optical high-Q cavity. Although the effective current that acts on the photons is known explicitly, the exact effective potential that describes the forces exerted by the photons on the electrons is obtained from a fixed-point inversion scheme. This procedure allows us to uncover important beyond-mean-field features of the effective potential that mark the breakdown of classical light-matter interactions. We observe peak and step structures in the effective potentials, which can be attributed solely to the quantum nature of light; i.e., they are real-space signatures of the photons. Our findings show how the ubiquitous dipole interaction with a classical electromagnetic field has to be modified in real space to take the quantum nature of the electromagnetic field fully into account.**55**Flick, J.; Ruggenthaler, M.; Appel, H.; Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry.*Proc. Natl. Acad. Sci. U. S. A.*2017,*114*, 3026– 3034, DOI: 10.1073/pnas.1615509114[Crossref], [PubMed], [CAS], Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXjvF2htLs%253D&md5=c469fe9a59a87cc6422835a1b0824ea4Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistryFlick, Johannes; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelProceedings of the National Academy of Sciences of the United States of America (2017), 114 (12), 3026-3034CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)A review is provided of how well-established concepts in the fields of quantum chem. and material sciences have to be adapted when the quantum nature of light becomes important in correlated matter-photon problems. Model systems in optical cavities were analyzed, where the matter-photon interaction is considered from the weak- to the strong-coupling limit and for individual photon modes as well as for the multimode case. The authors identify fundamental changes in Born-Oppenheimer surfaces, spectroscopic quantities, conical intersections, and efficiency for quantum control. The authors conclude by applying the recently developed quantum-electrodynamical d.-functional theory to spontaneous emission and show how a straightforward approxn. accurately describes the correlated electron-photon dynamics. This work paves the way to describe matter-photon interactions from 1st principles and addresses the emergence of new states of matter in chem. and material science.**56**Flick, J.; Schäfer, C.; Ruggenthaler, M.; Appel, H.; Rubio, A. Ab Initio Optimized Effective Potentials for Real Molecules in Optical Cavities: Photon Contributions to the Molecular Ground State.*ACS Photonics*2018,*5*, 992– 1005, DOI: 10.1021/acsphotonics.7b01279[ACS Full Text ], [CAS], Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXkvFanug%253D%253D&md5=fc695d87a81e7fa21abf2b6797a5e419Ab Initio Optimized Effective Potentials for Real Molecules in Optical Cavities: Photon Contributions to the Molecular Ground StateFlick, Johannes; Schaefer, Christian; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelACS Photonics (2018), 5 (3), 992-1005CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)A simple scheme to efficiently compute photon exchange-correlation contributions due to the coupling to transversal photons as formulated in the newly developed quantum-electrodynamical d. functional theory (QEDFT) is introduced. The construction employs the optimized-effective potential (OEP) approach by the Sternheimer equation to avoid the explicit calcn. of unoccupied states. The efficiency of the scheme was demonstrated by applying it to an exactly solvable GaAs quantum ring model system, a single azulene mol., and chains of Na2, all located in optical cavities and described in full real space. While the 1st example is a 2-dimensional system and allows to benchmark the employed approxns., the latter 2 examples demonstrate that the correlated electron-photon interaction appreciably distorts the ground-state electronic structure of a real mol. By using this scheme, the authors not only construct typical electronic observables, such as the electronic ground state d., but also illustrate how photon observables, such as the photon no., and mixed electron-photon observables, e.g. electron-photon correlation functions, become accessible in a DFT framework. This work constitutes the 1st 3-dimensional ab initio calcn. within the new QEDFT formalism and thus opens up a new computational route for the ab initio study of correlated electron-photon systems in quantum cavities.**57**Flick, J.; Welakuh, D. M.; Ruggenthaler, M.; Appel, H.; Rubio, A. Light-Matter Response Functions in Quantum-Electrodynamical Density-Functional Theory: Modifications of Spectra and of the Maxwell Equations.*arXiv Preprint arXiv:1803.02519*2018, 1– 27Google ScholarThere is no corresponding record for this reference.**58**Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Challenges for Density Functional Theory.*Chem. Rev.*2012,*112*, 289– 320, DOI: 10.1021/cr200107z[ACS Full Text ], [CAS], Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs1GltrbE&md5=51a7564af74b194a423868c40e5bc3caChallenges for Density Functional TheoryCohen, Aron J.; Mori-Sanchez, Paula; Yang, WeitaoChemical Reviews (Washington, DC, United States) (2012), 112 (1), 289-320CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review includes the following topics: the entrance of DFT into chem., constructing approx. functionals and minimizing the total energy, insight into large systematic errors of functionals, and strong correlation.**59**Niemczyk, T.; Deppe, F.; Huebl, H.; Menzel, E. P.; Hocke, F.; Schwarz, M. J.; Zueco, D.; Hümmer, T.; Solano, E.; Marx, A.; Gross, R. Circuit quantum electrodynamics in the ultrastrong-coupling regime.*Nat. Phys.*2010,*6*, 772– 776, DOI: 10.1038/nphys1730[Crossref], [CAS], Google Scholar59https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXht1ehtbvM&md5=48ded8663f7ae7736c09d0937be3a636Circuit quantum electrodynamics in the ultrastrong-coupling regimeNiemczyk, T.; Deppe, F.; Huebl, H.; Menzel, E. P.; Hocke, F.; Schwarz, M. J.; Garcia-Ripoll, J. J.; Zueco, D.; Huemmer, T.; Solano, E.; Marx, A.; Gross, R.Nature Physics (2010), 6 (10), 772-776CODEN: NPAHAX; ISSN:1745-2473. (Nature Publishing Group)In circuit quantum electrodynamics (QED), where superconducting artificial atoms are coupled to on-chip cavities, the exploration of fundamental quantum physics in the strong-coupling regime has greatly evolved. In this regime, an atom and a cavity can exchange a photon frequently before coherence is lost. Nevertheless, all expts. so far are well described by the renowned Jaynes-Cummings model. Here, we report on the first exptl. realization of a circuit QED system operating in the ultrastrong-coupling limit, where the atom-cavity coupling rate reaches a considerable fraction of the cavity transition frequency . Furthermore, we present direct evidence for the breakdown of the Jaynes-Cummings model. We reach remarkable normalized coupling rates of up to 12% by enhancing the inductive coupling of a flux qubit to a transmission line resonator. Our circuit extends the toolbox of quantum optics on a chip towards exciting explorations of ultrastrong light-matter interaction.**60**Bayer, A.; Pozimski, M.; Schambeck, S.; Schuh, D.; Huber, R.; Bougeard, D.; Lange, C. Terahertz Light-Matter Interaction beyond Unity Coupling Strength.*Nano Lett.*2017,*17*, 6340– 6344, DOI: 10.1021/acs.nanolett.7b03103[ACS Full Text ], [CAS], Google Scholar60https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFelsbvK&md5=13d1e3958a3ae5177905d5ddce178da9Terahertz Light-Matter Interaction beyond Unity Coupling StrengthBayer, Andreas; Pozimski, Marcel; Schambeck, Simon; Schuh, Dieter; Huber, Rupert; Bougeard, Dominique; Lange, ChristophNano Letters (2017), 17 (10), 6340-6344CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)Achieving control over light-matter interaction in custom-tailored nanostructures is at the core of modern quantum electrodynamics. In strongly and ultrastrongly coupled systems, the excitation is repeatedly exchanged between a resonator and an electronic transition at a rate known as the vacuum Rabi frequency ΩR. For ΩR approaching the resonance frequency ωc, novel quantum phenomena including squeezed states, Dicke superradiant phase transitions, the collapse of the Purcell effect, and a population of the ground state with virtual photon pairs are predicted. Yet, the exptl. realization of optical systems with ΩR/ωc ≥ 1 has remained elusive. Here, we introduce a paradigm change in the design of light-matter coupling by treating the electronic and the photonic components of the system as an entity instead of optimizing them sep. Using the electronic excitation to not only boost the electronic polarization but furthermore tailor the shape of the vacuum mode, we push ΩR/ωc of cyclotron resonances ultrastrongly coupled to metamaterials far beyond unity. As one prominent illustration of the unfolding possibilities, we calc. a ground state population of 0.37 virtual photons for our best structure with ΩR/ωc = 1.43 and suggest a realistic exptl. scenario for measuring vacuum radiation by cutting-edge terahertz quantum detection.**61**Gilbert, T. L. Hohenberg-Kohn theorem for nonlocal external potentials.*Phys. Rev. B*1975,*12*, 2111– 2120, DOI: 10.1103/PhysRevB.12.2111**62**Müller, A. M. K. Explicit approximate relation between reduced two- and one-particel density matrices.*Phys. Lett. A*1984,*105*, 446, DOI: 10.1016/0375-9601(84)91034-X**63**Goedecker, S.; Umrigar, C. J. Natural orbital functional for the many-electron problem.*Phys. Rev. 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Our tests on atoms show that it yields on av. more accurate energies and charge densities than the Hartree-Fock method, the local d. approxn., and the generalized gradient approxns.**64**Sharma, S.; Dewhurst, J. K.; Shallcross, S.; Gross, E. K. U. Spectral density and metal-insulator phase transition in mott insulators within reduced density matrix functional theory.*Phys. Rev. Lett.*2013,*110*, 1– 5, DOI: 10.1103/PhysRevLett.110.116403**65**Coleman, A. J. Structure of Fermion Density Matrices.*Rev. Mod. Phys.*1963,*35*, 668– 686, DOI: 10.1103/RevModPhys.35.668**66**Klyachko, A. A. Quantum marginal problem and N-representability.*J. Phys.: Conf. Ser.*2006,*36*, 72, DOI: 10.1088/1742-6596/36/1/014**67**Mazziotti, D. A. Structure of Fermionic Density Matrices: Complete*N*-Representability Conditions.*Phys. Rev. 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Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.**68**Nielsen, S. E. B.; Schäfer, C.; Ruggenthaler, M.; Rubio, A. Dressed-Orbital Approach to Cavity Quantum Electrodynamics and Beyond.*arXiv preprint arXiv:1812.00388*2018, naGoogle ScholarThere is no corresponding record for this reference.**69**Grynberg, G.; Aspect, A.; Fabre, C.*Introduction to quantum optics: from the semi-classical approach to quantized light*; Cambridge University Press, 2010.**70**Rokaj, V.; Welakuh, D. M.; Ruggenthaler, M.; Rubio, A. Light–matter interaction in the long-wavelength limit: no ground-state without dipole self-energy.*J. Phys. B: At., Mol. Opt. Phys.*2018,*51*, 034005, DOI: 10.1088/1361-6455/aa9c99[Crossref], [CAS], Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsFynu7bL&md5=406a602d898e5ab2e1f6a85f8383ec7eLight-matter interaction in the long- wavelength limit: no ground-state without dipole self-energyRokaj, Vasil; Welakuh, Davis M.; Ruggenthaler, Michael; Rubio, AngelJournal of Physics B: Atomic, Molecular and Optical Physics (2018), 51 (3), 034005/1-034005/13CODEN: JPAPEH; ISSN:0953-4075. (IOP Publishing Ltd.)Most theor. studies for correlated light-matter systems are performed within the long- wavelength limit, i.e., the electromagnetic field is assumed to be spatially uniform. In this limit the so-called length-gauge transformation for a fully quantized light-matter system gives rise to a dipole self-energy term in the Hamiltonian, i.e., a harmonic potential of the total dipole matter moment. In practice this term is often discarded as it is assumed to be subsumed in the kinetic energy term. In this work we show the necessity of the dipole self-energy term. First and foremost, without it the light-matter system in the long-wavelength limit does not have a ground-state, i.e., the combined light-matter system is unstable. Further, the mixing of matter and photon degrees of freedom due to the length-gauge transformation, which also changes the representation of the translation operator for matter, gives rise to the Maxwell equations in matter and the omittance of the dipole self-energy leads to a violation of these equations. Specifically we show that without the dipole self-energy the so-called 'depolarization shift' is not properly described. Finally we show that this term also arises if we perform the semi-classical limit after the length-gauge transformation. In contrast to the std. approach where the semi-classical limit is performed before the length-gauge transformation, the resulting Hamiltonian is bounded from below and thus supports ground-states. This is very important for practical calcns. and for d.-functional variational implementations of the non-relativistic QED formalism. For example, the existence of a combined light-matter ground-state allows one to calc. the Stark shift non-perturbatively.**71**Spohn, H.*Dynamics of charged particles and their radiation field*; Cambridge university press, 2004.**72**Power, E. A.; Thirunamachandran, T. Quantum electrodynamics in a cavity.*Phys. Rev. A: At., Mol., Opt. Phys.*1982,*25*, 2473– 2484, DOI: 10.1103/PhysRevA.25.2473[Crossref], [CAS], Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xit1Gnu7c%253D&md5=d61053ce33b923c88c1a790685834d50Quantum electrodynamics in a cavityPower, E. A.; Thirunamachandran, T.Physical Review A: Atomic, Molecular, and Optical Physics (1982), 25 (5), 2473-84CODEN: PLRAAN; ISSN:0556-2791.The theory of interaction of atoms and mols. with radiation confined in a cavity is presented. The effect of the confinement on the field commutation relations is examd. and the modified commutation relations are derived for an arbitrary cavity. A canonical transformation on the minimal-coupling Hamiltonian for such systems leads to a new Hamiltonian contg. explicitly neither the interat. nor the atom image potentials. The new multipolar Hamiltonian is used to calc. energy shifts of an atom and a pair of atoms near a conducting wall. Changes in the rate of spontaneous emission from an excited atom are also evaluated.**73**Shahbazyan, T. V.; Stockman, M. I.*Plasmonics: theory and applications*; Springer, 2013.**74**Giesbertz, K. J. H.; Ruggenthaler, M. One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures.*Phys. Rep.*2019,*806*, 1– 47, DOI: 10.1016/j.physrep.2019.01.010**75**Bonitz, M.*Quantum Kinetic Theory*, 2nd ed.; Springer: Berlin, 2016.**76**van Leeuwen, R.; Stefanucci, G.*Nonequilibrium Many-Body Theory of Quantum Systems*; Cambridge University Press, 2013.Google ScholarThere is no corresponding record for this reference.**77**Coleman, A. J.; Yukalov, V. I.*Reduced density matrices: Coulson’s challenge*; Springer Science & Business Media, 2000; Vol. 72.**78**Coulson, C. Present State of Molecular Structure Calculations.*Rev. Mod. Phys.*1960,*32*, 170– 177, DOI: 10.1103/RevModPhys.32.170[Crossref], [CAS], Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF3cXovF2gtA%253D%253D&md5=5f84ea370cca7ff42889b1d5a03f3a39Present state of molecular structure calculationsCoulson, C. A.Reviews of Modern Physics (1960), 32 (), 170-7CODEN: RMPHAT; ISSN:0034-6861.There is no expanded citation for this reference.**79**Rokaj, V.; Penz, M.; Sentef, M. A.; Ruggenthaler, M.; Rubio, A. Quantum Electrodynamical Bloch Theory with Homogeneous Magnetic Fields.*Phys. Rev. Lett.*2019,*123*, 1– 6, DOI: 10.1103/PhysRevLett.123.047202**80**Watson, J. K. Simplification of the molecular vibration-rotation Hamiltonian.*Mol. Phys.*1968,*15*, 479– 490, DOI: 10.1080/00268976800101381[Crossref], [CAS], Google Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF1MXlt1Kjug%253D%253D&md5=220a477da0af4bbb6b19fbbf53e08e50Simplification of the molecular vibration-rotation hamiltonianWatson, James K. G.Molecular Physics (1968), 15 (5), 479-90CODEN: MOPHAM; ISSN:0026-8976.By use of the commutation relations and sum rules, the Darling-Dennison vibration-rotation Hamiltonian for a nonlinear mol. is rearranged. A simple expansion is given for the μαβ tensor in terms of the normal coordinates.**81**Piris, M. Global Method for Electron Correlation.*Phys. Rev. Lett.*2017,*119*, 1– 5, DOI: 10.1103/PhysRevLett.119.063002**82**Mazziotti, D. A. Two-electron reduced density matrix as the basic variable in many-electron quantum chemistry and physics.*Chem. Rev.*2012,*112*, 244– 262, DOI: 10.1021/cr2000493[ACS Full Text ], [CAS], Google Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtVOns7zN&md5=fb6dcf6b6470ff39479a446778190941Two-Electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and PhysicsMazziotti, David A.Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 244-262CODEN: CHREAY; ISSN:0009-2665. 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(American Institute of Physics)Functionals of the one-body reduced d. matrix (1-RDM) are routinely minimized under Coleman's ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any no. of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced d.-matrix functional theory calcns. In particular, we examine whether the std. minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addn. to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation nos. with those obtained by the enforcement of the ensemble conditions alone. (c) 2015 American Institute of Physics.**85**Lieb, E. H. Variational principle for many-fermion systems.*Phys. Rev. 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An approximate exchange-correlation hole density as a functional of the natural orbitals.*Mol. Phys.*2002,*100*, 401– 421, DOI: 10.1080/00268970110070243[Crossref], [CAS], Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XitFSitro%253D&md5=afe235742d04e5fe1951437f0faccde0An approximate exchange-correlation hole density as a functional of the natural orbitalsBuijse, M. A.; Baerends, E. J.Molecular Physics (2002), 100 (4), 401-421CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The Fermi and Coulomb holes that can be used to describe the physics of electron correlation are calcd. and analyzed for a no. of typical cases, ranging from prototype dynamical correlation to purely nondynamical correlation. Their behavior as a function of the position of the ref. electron and of the nuclear positions is exhibited. The notion that the hole can be written as the square of a hole amplitude, which is exactly true for the exchange hole, is generalized to the total holes, including the correlation part. An Ansatz is made for an approx. yet accurate expression for the hole amplitude in terms of the natural orbitals, employing the local (at the ref. position) values of the natural orbitals and the d. This expression for the hole amplitude leads to an approx. two-electron d. matrix that: (a) obeys correct permutation symmetry in the electron coordinates; (b) integrates to the exact one-matrix; and (c) yields exact correlation energies in the limiting cases of predominant dynamical correlation (high Z two-electron ions) and pure nondynamical correlation (dissocd. H2).**87**Frank, R. L.; Lieb, E. H.; Seiringer, R.; Siedentop, H. Müller’s exchange-correlation energy in density-matrix-functional theory.*Phys. Rev. A: At., Mol., Opt. Phys.*2007,*76*, 1– 16, DOI: 10.1103/PhysRevA.76.052517**88**Andrade, X. Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems.*Phys. Chem. Chem. Phys.*2015,*17*, 31371– 31396, DOI: 10.1039/C5CP00351B[Crossref], [PubMed], [CAS], Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjtVyktLw%253D&md5=2457aa1e92d75da4f45ac2d66297c5caReal-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systemsAndrade, Xavier; Strubbe, David; De Giovannini, Umberto; Larsen, Ask Hjorth; Oliveira, Micael J. T.; Alberdi-Rodriguez, Joseba; Varas, Alejandro; Theophilou, Iris; Helbig, Nicole; Verstraete, Matthieu J.; Stella, Lorenzo; Nogueira, Fernando; Aspuru-Guzik, Alan; Castro, Alberto; Marques, Miguel A. L.; Rubio, AngelPhysical Chemistry Chemical Physics (2015), 17 (47), 31371-31396CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new phys. models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calc. response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact soln. of the Schr.ovrddot.odinger equation for low-dimensionality systems.**89**Piris, M.; Ugalde, J. M. Iterative Diagonalization for Orbital Optimization in Natural Orbital Functional Theory.*J. Comput. Chem.*2009,*30*, 2078– 2086, DOI: 10.1002/jcc.21225[Crossref], [PubMed], [CAS], Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXptleqsbw%253D&md5=cd6c06c44b6d1b4861af90c14e21c39dIterative diagonalization for orbital optimization in natural orbital functional theoryPiris, M.; Ugalde, J. M.Journal of Computational Chemistry (2009), 30 (13), 2078-2086CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F. The off-diagonal elements of the latter are detd. explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be detd. from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calcd. with the Hartree-Fock orbitals after the occupation nos. have been optimized. The method has been tested on the G2/97 set of mols. for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off-diagonal elements of F. The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009.**90**Ruggenthaler, M.; Bauer, D. Rabi oscillations and few-level approximations in time-dependent density functional theory.*Phys. Rev. Lett.*2009,*102*, 2– 5, DOI: 10.1103/PhysRevLett.102.233001**91**Fuks, J. I.; Helbig, N.; Tokatly, I. V.; Rubio, A. Nonlinear phenomena in time-dependent density-functional theory: What Rabi oscillations can teach us.*Phys. Rev. B: Condens. Matter Mater. Phys.*2011,*84*, na, DOI: 10.1103/PhysRevB.84.075107

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**1**Ebbesen, T. W. Hybrid Light–Matter States in a Molecular and Material Science Perspective.*Acc. Chem. Res.*2016,*49*, 2403– 2412, DOI: 10.1021/acs.accounts.6b00295[ACS Full Text ], [CAS], Google Scholar1https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhslWhurfM&md5=112474dfc62bce8cf5ea2ba21fab9665Hybrid Light-Matter States in a Molecular and Material Science PerspectiveEbbesen, Thomas W.Accounts of Chemical Research (2016), 49 (11), 2403-2412CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)A review. The notion that light and matter states can be hybridized the way s and p orbitals are mixed is a concept that is not familiar to most chemists and material scientists. Yet it has much potential for mol. and material sciences that is just beginning to be explored. For instance, it has already been demonstrated that the rate and yield of chem. reactions can be modified and that the cond. of org. semiconductors and nonradiative energy transfer can be enhanced through the hybridization of electronic transitions. The hybridization is not limited to electronic transitions; it can be applied for instance to vibrational transitions to selectively perturb a given bond, opening new possibilities to change the chem. reactivity landscape and to use it as a tool in (bio)mol. science and spectroscopy. Such results are not only the consequence of the new eigenstates and energies generated by the hybridization. The hybrid light-matter states also have unusual properties: they can be delocalized over a very large no. of mols. (up to ca. 105), and they become dispersive or momentum-sensitive. Importantly, the hybridization occurs even in the absence of light because it is the zero-point energies of the mol. and optical transitions that generate the new light-matter states. The present work is not a review but rather an Account from the author's point of view that first introduces the reader to the underlying concepts and details of the features of hybrid light-matter states. It is shown that light-matter hybridization is quite easy to achieve: all that is needed is to place mols. or a material in a resonant optical cavity (e.g., between two parallel mirrors) under the right conditions. For vibrational strong coupling, microfluidic IR cells can be used to study the consequences for chem. in the liq. phase. Examples of modified properties are given to demonstrate the full potential for the mol. and material sciences. Finally an outlook of future directions for this emerging subject is given.**2**Sukharev, M.; Nitzan, A. Optics of exciton-plasmon nanomaterials.*J. Phys.: Condens. Matter*2017,*29*, 443003, DOI: 10.1088/1361-648X/aa85ef[Crossref], [PubMed], [CAS], Google Scholar2https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXnvVWktbo%253D&md5=43f317b326ec57cdc3f3372e22b1e6bcOptics of exciton-plasmon nanomaterialsSukharev, Maxim; Nitzan, AbrahamJournal of Physics: Condensed Matter (2017), 29 (44), 443003/1-443003/34CODEN: JCOMEL; ISSN:0953-8984. (IOP Publishing Ltd.)A review. This review provides a brief introduction to the physics of coupled exciton-plasmon systems, the theor. description and exptl. manifestation of such phenomena, followed by an account of the state-of-the-art methodol. for the numerical simulations of such phenomena and supplemented by a no. of FORTRAN codes, by which the interested reader can introduce himself/herself to the practice of such simulations. Applications to CW light scattering as well as transient response and relaxation are described. Particular attention is given to so-called strong coupling limit, where the hybrid exciton-plasmon nature of the system response is strongly expressed. While traditional descriptions of such phenomena usually rely on anal. of the electromagnetic response of inhomogeneous dielec. environments that individually support plasmon and exciton excitations, here we explore also the consequences of a more detailed description of the mol. environment in terms of its quantum d. matrix (applied in a mean field approxn. level). Such a description makes it possible to account for characteristics that cannot be described by the dielec. response model: the effects of dephasing on the mol. response on one hand, and nonlinear response on the other. It also highlights the still missing important ingredients in the numerical approach, in particular its limitation to a classical description of the radiation field and its reliance on a mean field description of the many-body mol. system. We end our review with an outlook to the near future, where these limitations will be addressed and new novel applications of the numerical approach will be pursued.**3**Kockum, A. F.; Miranowicz, A.; De Liberato, S.; Savasta, S.; Nori, F. Ultrastrong coupling between light and matter.*Nat. Rev. Phys.*2019,*1*, 19– 40, DOI: 10.1038/s42254-018-0006-2**4**Hutchison, J. A.; Schwartz, T.; Genet, C.; Devaux, E.; Ebbesen, T. W. Modifying chemical landscapes by coupling to vacuum fields.*Angew. Chem., Int. Ed.*2012,*51*, 1592– 1596, DOI: 10.1002/anie.201107033[Crossref], [CAS], Google Scholar4https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XkvVSjuw%253D%253D&md5=36ea0bf96a50529ee200f504802ab0caModifying Chemical Landscapes by Coupling to Vacuum FieldsHutchison, James A.; Schwartz, Tal; Genet, Cyriaque; Devaux, Eloise; Ebbesen, Thomas W.Angewandte Chemie, International Edition (2012), 51 (7), 1592-1596, S1592/1-S1592/3CODEN: ACIEF5; ISSN:1433-7851. (Wiley-VCH Verlag GmbH & Co. KGaA)Modifying chem. landscapes by coupling to vacuum fields is discussed.**5**Thomas, A.; George, J.; Shalabney, A.; Dryzhakov, M.; Varma, S. J.; Moran, J.; Chervy, T.; Zhong, X.; Devaux, E.; Genet, C.; Hutchison, J. A.; Ebbesen, T. W. Ground-State Chemical Reactivity under Vibrational Coupling to the Vacuum Electromagnetic Field.*Angew. Chem.*2016,*128*, 11634– 11638, DOI: 10.1002/ange.201605504**6**Firstenberg, O.; Peyronel, T.; Liang, Q. Y.; Gorshkov, A. V.; Lukin, M. D.; Vuletić, V. Attractive photons in a quantum nonlinear medium.*Nature*2013,*502*, 71– 75, DOI: 10.1038/nature12512[Crossref], [PubMed], [CAS], Google Scholar6https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXhsFaktrbO&md5=3df8aad7d0d12741f01e3e27e8f2bb48Attractive photons in a quantum nonlinear mediumFirstenberg, Ofer; Peyronel, Thibault; Liang, Qi-Yu; Gorshkov, Alexey V.; Lukin, Mikhail D.; Vuletic, VladanNature (London, United Kingdom) (2013), 502 (7469), 71-75CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)The fundamental properties of light derive from its constituent particles-massless quanta (photons) that do not interact with one another. However, it has long been known that the realization of coherent interactions between individual photons, akin to those assocd. with conventional massive particles, could enable a wide variety of novel scientific and engineering applications. Here we demonstrate a quantum nonlinear medium inside which individual photons travel as massive particles with strong mutual attraction, such that the propagation of photon pairs is dominated by a two-photon bound state. We achieve this through dispersive coupling of light to strongly interacting atoms in highly excited Rydberg states. We measure the dynamical evolution of the two-photon wavefunction using time-resolved quantum state tomog., and demonstrate a conditional phase shift exceeding one radian, resulting in polarization-entangled photon pairs. Particular applications of this technique include all-optical switching, deterministic photonic quantum logic and the generation of strongly correlated states of light.**7**Goban, A.; Hung, C. L.; Hood, J. D.; Yu, S. P.; Muniz, J. A.; Painter, O.; Kimble, H. J. Superradiance for Atoms Trapped along a Photonic Crystal Waveguide.*Phys. Rev. Lett.*2015,*115*, 063601, DOI: 10.1103/PhysRevLett.115.063601[Crossref], [PubMed], [CAS], Google Scholar7https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28XhsVaisrY%253D&md5=b20dba87cbcd98b45f1b94701a44581aSuperradiance for atoms trapped along a photonic crystal waveguideGoban, A.; Hung, C.-L.; Hood, J. D.; Yu, S.-P.; Muniz, J. A.; Painter, O.; Kimble, H. J.Physical Review Letters (2015), 115 (6), 063601/1-063601/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We report observations of superradiance for atoms trapped in the near field of a photonic crystal waveguide (PCW). By fabricating the PCW with a band edge near the D1 transition of at. cesium, strong interaction is achieved between trapped atoms and guided-mode photons. Following short-pulse excitation, we record the decay of guided-mode emission and find a superradiant emission rate scaling as ΓSR α NΓ1D for av. atom no. 0.19.ltorsim. N .ltorsim. 2.6 atoms, where Γ1D/Γ' =1.0±0.1 is the peak single-atom radiative decay rate into the PCW guided mode, and Γ" is the radiative decay rate into all the other channels. These advances provide new tools for investigations of photon-mediated atom-atom interactions in the many-body regime.**8**Coles, D. M.; Yang, Y.; Wang, Y.; Grant, R. T.; Taylor, R. A.; Saikin, S. K.; Aspuru-Guzik, A.; Lidzey, D. G.; Tang, J. K.-H.; Smith, J. M. Strong coupling between chlorosomes of photosynthetic bacteria and a confined optical cavity mode.*Nat. Commun.*2014,*5*, 5561, DOI: 10.1038/ncomms6561[Crossref], [PubMed], [CAS], Google Scholar8https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjvFajsb0%253D&md5=0ada1595a0efdba9f15a6d20c4b8f243Strong coupling between chlorosomes of photosynthetic bacteria and a confined optical cavity modeColes, David M.; Yang, Yanshen; Wang, Yaya; Grant, Richard T.; Taylor, Robert A.; Saikin, Semion K.; Aspuru-Guzik, Alan; Lidzey, David G.; Tang, Joseph Kuo-Hsiang; Smith, Jason M.Nature Communications (2014), 5 (), 5561CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)Strong exciton-photon coupling is the result of a reversible exchange of energy between an excited state and a confined optical field. This results in the formation of polariton states that have energies different from the exciton and photon. We demonstrate strong exciton-photon coupling between light-harvesting complexes and a confined optical mode within a metallic optical microcavity. The energetic anti-crossing between the exciton and photon dispersions characteristic of strong coupling is obsd. in reflectivity and transmission with a Rabi splitting energy on the order of 150 meV, which corresponds to about 1000 chlorosomes coherently coupled to the cavity mode. We believe that the strong coupling regime presents an opportunity to modify the energy transfer pathways within photosynthetic organisms without modification of the mol. structure.**9**Orgiu, E.; George, J.; Hutchison, J. A.; Devaux, E.; Dayen, J. F.; Doudin, B.; Stellacci, F.; Genet, C.; Schachenmayer, J.; Genes, C. Conductivity in organic semiconductors hybridized with the vacuum field.*Nat. Mater.*2015,*14*, 1123– 1129, DOI: 10.1038/nmat4392[Crossref], [PubMed], [CAS], Google Scholar9https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhsV2ltrvI&md5=ff034f20b52069c3e3d86e8352b3d786Conductivity in organic semiconductors hybridized with the vacuum fieldOrgiu, E.; George, J.; Hutchison, J. A.; Devaux, E.; Dayen, J. F.; Doudin, B.; Stellacci, F.; Genet, C.; Schachenmayer, J.; Genes, C.; Pupillo, G.; Samori, P.; Ebbesen, T. W.Nature Materials (2015), 14 (11), 1123-1129CODEN: NMAACR; ISSN:1476-1122. (Nature Publishing Group)Much effort over the past decades was focused on improving carrier mobility in org. thin-film transistors by optimizing the organization of the material or the device architecture. Here the authors take a different path to solving this problem, by injecting carriers into states that are hybridized to the vacuum electromagnetic field. To test this idea, org. semiconductors were strongly coupled to plasmonic modes to form coherent states that can extend over as many as 105 mols. and should thereby favor cond. Indeed the current does increase by an order of magnitude at resonance in the coupled state, reflecting mostly a change in field-effect mobility. A theor. quantum model confirms the delocalization of the wavefunctions of the hybridized states and its effect on the cond. The authors' findings illustrate the potential of engineering the vacuum electromagnetic environment to modify and to improve properties of materials.**10**Andrew, P.; Barnes, W. L. Energy transfer across a metal film mediated by surface plasmon polaritons.*Science*2004,*306*, 1002– 1005, DOI: 10.1126/science.1102992[Crossref], [PubMed], [CAS], Google Scholar10https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2cXptFyjtbg%253D&md5=4c56f8c2952e56e2bd46d08b19e7cdd7Energy Transfer Across a Metal Film Mediated by Surface Plasmon PolaritonsAndrew, P.; Barnes, W. L.Science (Washington, DC, United States) (2004), 306 (5698), 1002-1005CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)Coupled surface plasmon polaritons (SPPs) are shown to provide effective transfer of excitation energy from donor mols. to acceptor mols. on opposite sides of metal films up to 120 nm thick. This variant of radiative transfer should allow directional control over the flow of excitation energy with the use of suitably designed metallic nanostructures, with SPPs mediating transfer over length scales of 10-7 to 10-4 meters. In the emerging field of nanophotonics, such a prospect could allow subwavelength-scale manipulation of light and provide an interface to the outside world.**11**Dicke, R. H. Coherence in spontaneous radiation processes.*Phys. Rev.*1954,*93*, 99– 110, DOI: 10.1103/PhysRev.93.99[Crossref], [CAS], Google Scholar11https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaG2cXjsV2jtA%253D%253D&md5=45eea4ac630974b5996ac36a4951308dCoherence in spontaneous radiation processesDicke, R. H.Physical Review (1954), 93 (), 99-110CODEN: PHRVAO; ISSN:0031-899X.Energy levels corresponding to certain correlations between individual mols. were obtained from the consideration of a radiating gas as a single quantum-mech. system. Spontaneous emission of radiation in a transition between 2 such levels leads to the emission of coherent radiation. The discussion is limited first to a gas of dimension small compared with a wave length. Spontaneous radiation rates and natural line breadths are calcd. For a gas of large extent the effect of photon recoil momentum on coherence is calcd. The effect of a radiation pulse in exciting "super-radiant" states is discussed. The angular correlation between successive photons spontaneously emitted by a gas initially in thermal equil. is calcd.**12**Todorov, Y.; Andrews, A. M.; Colombelli, R.; De Liberato, S.; Ciuti, C.; Klang, P.; Strasser, G.; Sirtori, C. Ultrastrong light-matter coupling regime with polariton dots.*Phys. Rev. Lett.*2010,*105*, 1– 4, DOI: 10.1103/PhysRevLett.105.196402**13**Michetti, P.; Mazza, L.; La Rocca, G. C. In*Organic Nanophotonics: Fundamentals and Applications*; Zhao, Y. S., Ed.; Springer Berlin Heidelberg: Berlin, Heidelberg, 2015; pp 39– 68.Google ScholarThere is no corresponding record for this reference.**14**Cwik, J. A.; Kirton, P.; De Liberato, S.; Keeling, J. Excitonic spectral features in strongly coupled organic polaritons.*Phys. Rev. A: At., Mol., Opt. Phys.*2016,*93*, 1– 12, DOI: 10.1103/PhysRevA.93.033840**15**Galego, J.; Garcia-Vidal, F. J.; Feist, J. Suppressing photochemical reactions with quantized light fields.*Nat. Commun.*2016,*7*, 1– 6, DOI: 10.1038/ncomms13841**16**Feist, J.; Galego, J.; Garcia-Vidal, F. J. Polaritonic Chemistry with Organic Molecules.*ACS Photonics*2018,*5*, 205– 216, DOI: 10.1021/acsphotonics.7b00680[ACS Full Text ], [CAS], Google Scholar16https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFGhsrnJ&md5=a208ab762f33f613e3144f69d88ad3f5Polaritonic Chemistry with Organic MoleculesFeist, Johannes; Galego, Javier; Garcia-Vidal, Francisco J.ACS Photonics (2018), 5 (1), 205-216CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)A review. The authors present an overview of the general concepts of polaritonic chem. with org. mols., i.e., the manipulation of chem. structure that can be achieved through strong coupling between confined light modes and org. mols. Strong coupling and the assocd. formation of polaritons, hybrid light-matter excitations, lead to energy shifts in such systems that can amt. to a large fraction of the uncoupled transition energy. This has recently been shown to significantly alter the chem. structure of the coupled mols., which opens the possibility to manipulate and control reactions. The authors discuss the current state of theory for describing these changes and present several applications, with a particular focus on the collective effects obsd. when many mols. are involved in strong coupling.**17**Feist, J.; Garcia-Vidal, F. J. Extraordinary exciton conductance induced by strong coupling.*Phys. Rev. Lett.*2015,*114*, 1– 5, DOI: 10.1103/PhysRevLett.114.196402**18**Herrera, F.; Spano, F. C. Cavity-Controlled Chemistry in Molecular Ensembles.*Phys. Rev. Lett.*2016,*116*, 1– 6, DOI: 10.1103/PhysRevLett.116.238301**19**Cirio, M.; De Liberato, S.; Lambert, N.; Nori, F. Ground State Electroluminescence.*Phys. Rev. Lett.*2016,*116*, 1– 7, DOI: 10.1103/PhysRevLett.116.113601**20**Kockum, A. F.; MacRì, V.; Garziano, L.; Savasta, S.; Nori, F. Frequency conversion in ultrastrong cavity QED.*Sci. Rep.*2017,*7*, 1– 13, DOI: 10.1038/s41598-017-04225-3[Crossref], [PubMed], [CAS], Google Scholar20https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsFehtbjF&md5=446004959d31342ff1754300e64daea1Smoking induces DNA methylation changes in Multiple Sclerosis patients with exposure-response relationshipMarabita, Francesco; Almgren, Malin; Sjoeholm, Louise K.; Kular, Lara; Liu, Yun; James, Tojo; Kiss, Nimrod B.; Feinberg, Andrew P.; Olsson, Tomas; Kockum, Ingrid; Alfredsson, Lars; Ekstroem, Tomas J.; Jagodic, MajaScientific Reports (2017), 7 (1), 1-15CODEN: SRCEC3; ISSN:2045-2322. (Nature Research)Cigarette smoking is an established environmental risk factor for Multiple Sclerosis (MS), a chronic inflammatory and neurodegenerative disease, although a mechanistic basis remains largely unknown. We aimed at investigating how smoking affects blood DNA methylation in MS patients, by assaying genome-wide DNA methylation and comparing smokers, former smokers and never smokers in two Swedish cohorts, differing for known MS risk factors. Smoking affects DNA methylation genome-wide significantly, an exposure-response relationship exists and the time since smoking cessation affects methylation levels. The results also show that the changes were larger in the cohort bearing the major genetic risk factors for MS (female sex and HLA risk haplotypes). Furthermore, CpG sites mapping to genes with known genetic or functional role in the disease are differentially methylated by smoking. Modeling of the methylation levels for a CpG site in the AHRR gene indicates that MS modifies the effect of smoking on methylation changes, by significantly interacting with the effect of smoking load. Alongside, we report that the gene expression of AHRR increased in MS patients after smoking. Our results suggest that epigenetic modifications may reveal the link between a modifiable risk factor and the pathogenetic mechanisms.**21**De Liberato, S. Light-matter decoupling in the deep strong coupling regime: The breakdown of the purcell effect.*Phys. Rev. Lett.*2014,*112*, 1– 5, DOI: 10.1103/PhysRevLett.112.016401**22**De Liberato, S. Virtual photons in the ground state of a dissipative system.*Nat. Commun.*2017,*8*, 1– 6, DOI: 10.1038/s41467-017-01504-5[Crossref], [PubMed], [CAS], Google Scholar22https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXos1KqtbY%253D&md5=289e8664a6fd1299b28e81a6c39f7e58Virtual photons in the ground state of a dissipative systemDe Liberato, SimoneNature Communications (2017), 8 (1), 1-6CODEN: NCAOBW; ISSN:2041-1723. (Nature Research)Much of the novel physics predicted to be observable in the ultrastrong light-matter coupling regime rests on the hybridization between states with different nos. of excitations, leading to a population of virtual photons in the system's ground state. In this article, exploiting an exact diagonalisation approach, we derive both anal. and numerical results for the population of virtual photons in presence of arbitrary losses. Specialising our results to the case of Lorentzian resonances we then show that the virtual photon population is only quant. affected by losses, even when those become the dominant energy scale. Our results demonstrate most of the ultrastrong-coupling phenomenol. can be obsd. in loss-dominated systems which are not even in the std. strong coupling regime. We thus open the possibility to investigate ultrastrong-coupling physics to platforms that were previously considered unsuitable due to their large losses.**23**Gely, M. F.; Parra-Rodriguez, A.; Bothner, D.; Blanter, Y. M.; Bosman, S. J.; Solano, E.; Steele, G. A. Convergence of the multimode quantum Rabi model of circuit quantum electrodynamics.*Phys. Rev. B: Condens. Matter Mater. Phys.*2017,*95*, 1– 5, DOI: 10.1103/PhysRevB.95.245115**24**De Bernardis, D.; Pilar, P.; Jaako, T.; De Liberato, S.; Rabl, P. Breakdown of gauge invariance in ultrastrong-coupling cavity QED.*Phys. Rev. A: At., Mol., Opt. Phys.*2018,*98*, 1– 16, DOI: 10.1103/PhysRevA.98.053819**25**Sánchez Muñoz, C.; Nori, F.; De Liberato, S. Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamics.*Nat. Commun.*2018,*9*, 1924, DOI: 10.1038/s41467-018-04339-w[Crossref], [PubMed], [CAS], Google Scholar25https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A280%3ADC%252BC1Mfjs1aksw%253D%253D&md5=cbace44e5dcfb489cb04a9fd02c6d374Resolution of superluminal signalling in non-perturbative cavity quantum electrodynamicsSanchez Munoz Carlos; Nori Franco; Nori Franco; De Liberato SimoneNature communications (2018), 9 (1), 1924 ISSN:.Recent technological developments have made it increasingly easy to access the non-perturbative regimes of cavity quantum electrodynamics known as ultrastrong or deep strong coupling, where the light-matter coupling becomes comparable to the bare modal frequencies. In this work, we address the adequacy of the broadly used single-mode cavity approximation to describe such regimes. We demonstrate that, in the non-perturbative light-matter coupling regimes, the single-mode models become unphysical, allowing for superluminal signalling. Moreover, considering the specific example of the quantum Rabi model, we show that the multi-mode description of the electromagnetic field, necessary to account for light propagation at finite speed, yields physical observables that differ radically from their single-mode counterparts already for moderate values of the coupling. Our multi-mode analysis also reveals phenomena of fundamental interest on the dynamics of the intracavity electric field, where a free photonic wavefront and a bound state of virtual photons are shown to coexist.**26**Jaako, T.; Xiang, Z. L.; Garcia-Ripoll, J. J.; Rabl, P. Ultrastrong-coupling phenomena beyond the Dicke model.*Phys. Rev. A: At., Mol., Opt. Phys.*2016,*94*, 1– 10, DOI: 10.1103/PhysRevA.94.033850**27**Schäfer, C.; Ruggenthaler, M.; Rubio, A. Ab initio nonrelativistic quantum electrodynamics: Bridging quantum chemistry and quantum optics from weak to strong coupling.*Phys. Rev. A: At., Mol., Opt. Phys.*2018,*98*, 043801, DOI: 10.1103/PhysRevA.98.043801**28**Schäfer, C.; Ruggenthaler, M.; Appel, H.; Rubio, A. Modification of excitation and charge transfer in cavity quantum-electrodynamical chemistry.*Proc. Natl. Acad. Sci. U. S. A.*2019,*116*, 4883– 4892, DOI: 10.1073/pnas.1814178116[Crossref], [PubMed], [CAS], Google Scholar28https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXkvVCisb0%253D&md5=6e129c186de5d7ce049a3148ac79f815Modification of excitation and charge transfer in cavity quantum-electrodynamical chemistrySchAfer, Christian; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelProceedings of the National Academy of Sciences of the United States of America (2019), 116 (11), 4883-4892CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)Energy transfer in terms of excitation or charge is one of the most basic processes in nature, and understanding and controlling them is one of the major challenges of modern quantum chem. In this work, we highlight that these processes as well as other chem. properties can be drastically altered by modifying the vacuum fluctuations of the electromagnetic field in a cavity. By using a real-space formulation from first principles that keeps all of the electronic degrees of freedom in the model explicit and simulates changes in the environment by an effective photon mode, we can easily connect to well-known quantum-chem. results such as Dexter charge-transfer and FA~¶rster excitation-transfer reactions, taking into account the often-disregarded Coulomb and self-polarization interaction. We find that the photonic degrees of freedom introduce extra electron-electron correlations over large distances and that the coupling to the cavity can drastically alter the characteristic charge-transfer behavior and even selectively improve the efficiency. For excitation transfer, we find that the cavity renders the transfer more efficient, essentially distance-independent, and further different configurations of highest efficiency depending on the coherence times. For strong decoherence (short coherence times), the cavity frequency should be in between the isolated excitations of the donor and acceptor, while for weak decoherence (long coherence times), the cavity should enhance a mode that is close to resonance with either donor or acceptor. Our results highlight that changing the photonic environment can redefine chem. processes, rendering polaritonic chem. a promising approach toward the control of chem. reactions.**29**Galego, J.; Garcia-Vidal, F. J.; Feist, J. Cavity-induced modifications of molecular structure in the strong-coupling regime.*Phys. Rev. X*2015,*5*, 1– 14, DOI: 10.1103/PhysRevX.5.041022**30**Kowalewski, M.; Bennett, K.; Mukamel, S. Non-adiabatic dynamics of molecules in optical cavities.*J. Chem. Phys.*2016,*144*, 054309, DOI: 10.1063/1.4941053[Crossref], [PubMed], [CAS], Google Scholar30https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC28Xit1OitL4%253D&md5=18ee3f0139812477efa1846918c29a63Non-adiabatic dynamics of molecules in optical cavitiesKowalewski, Markus; Bennett, Kochise; Mukamel, ShaulJournal of Chemical Physics (2016), 144 (5), 054309/1-054309/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Strong coupling of mols. to the vacuum field of micro cavities can modify the potential energy surfaces thereby opening new photophys. and photochem. reaction pathways. While the influence of laser fields is usually described in terms of classical field, coupling to the vacuum state of a cavity has to be described in terms of dressed photon-matter states (polaritons) which require quantized fields. The authors present a derivation of the nonadiabatic couplings for single mols. in the strong coupling regime suitable for the calcn. of the dressed state dynamics. The formalism allows using quantities readily accessible from quantum chem. codes like the adiabatic potential energy surfaces and dipole moments to carry out wave packet simulations in the dressed basis. The implications for photochem. are demonstrated for a set of model systems representing typical situations found in mols. (c) 2016 American Institute of Physics.**31**Garcia-Vidal, F. J.; Feist, J. Long-distance operator for energy transfer.*Science*2017,*357*, 1357– 1358, DOI: 10.1126/science.aao4268[Crossref], [PubMed], [CAS], Google Scholar31https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1SrtL3F&md5=c2605da7523151a23e42c70b52984a58Long-distance operator for energy transferGarcia-Vidal, Francisco J.; Feist, JohannesScience (Washington, DC, United States) (2017), 357 (6358), 1357-1358CODEN: SCIEAS; ISSN:0036-8075. (American Association for the Advancement of Science)There is no expanded citation for this reference.**32**Zeb, M. A.; Kirton, P. G.; Keeling, J. Exact states and spectra of vibrationally dressed polaritons.*ACS Photonics*2018,*5*, 249– 257, DOI: 10.1021/acsphotonics.7b00916[ACS Full Text ], [CAS], Google Scholar32https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1OqsrjN&md5=6b9e2e10680c5a2cdd3f517ea7e1ce5fExact States and Spectra of Vibrationally Dressed PolaritonsZeb, M. Ahsan; Kirton, Peter G.; Keeling, JonathanACS Photonics (2018), 5 (1), 249-257CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)Strong coupling between light and matter is possible with a variety of org. materials. In contrast to the simpler inorg. case, org. materials often have a complicated spectrum, with vibrationally dressed electronic transitions. Strong coupling to light competes with this vibrational dressing and, if strong enough, can suppress the entanglement between electronic and vibrational degrees of freedom. By exploiting symmetries, the authors can perform exact numerical diagonalization to find the polaritonic states for intermediate nos. of mols. and use these to define and validate accurate expressions for the lower polariton states and strong-coupling spectrum in the thermodn. limit. Using this approach, vibrational decoupling occurs as a sharp transition above a crit. matter-light coupling strength. Also the polariton spectrum evolves with the no. of mols., recovering classical linear optics results only at large N.**33**Luk, H. L.; Feist, J.; Toppari, J. J.; Groenhof, G. Multiscale molecular dynamics simulations of polaritonic chemistry.*J. Chem. Theory Comput.*2017,*13*, 4324– 4335, DOI: 10.1021/acs.jctc.7b00388[ACS Full Text ], [CAS], Google Scholar33https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXht1eks7nK&md5=9f428e53a85a5f9c1e7124ffb699a378Multiscale Molecular Dynamics Simulations of Polaritonic ChemistryLuk, Hoi Ling; Feist, Johannes; Toppari, J. Jussi; Groenhof, GerritJournal of Chemical Theory and Computation (2017), 13 (9), 4324-4335CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)When photoactive mols. interact strongly with confined light modes as found in plasmonic structures or optical cavities, new hybrid light-matter states can form, the so-called polaritons. These polaritons are coherent superpositions (in the quantum mech. sense) of excitations of the mols. and of the cavity photon or surface plasmon. Recent exptl. and theor. works suggest that access to these polaritons in cavities could provide a totally new and attractive paradigm for controlling chem. reactions that falls in between traditional chem. catalysis and coherent laser control. However, designing cavity parameters to control chem. requires a theor. model with which the effect of the light-matter coupling on the mol. dynamics can be predicted accurately. Here we present a multiscale quantum mechanics/mol. mechanics (QM/MM) mol. dynamics simulation model for photoactive mols. that are strongly coupled to confined light in optical cavities or surface plasmons. Using this model we have performed simulations with up to 1600 Rhodamine mols. in a cavity. The results of these simulations reveal that the contributions of the mols. to the polariton are time-dependent due to thermal fluctuations that break symmetry. Furthermore, the simulations suggest that in addn. to the cavity quality factor, also the Stokes shift and no. of mols. control the lifetime of the polariton. Because large nos. of mols. interacting with confined light can now be simulated in at. detail, we anticipate that our method will lead to a better understanding of the effects of strong coupling on chem. reactivity. Ultimately the method may even be used to systematically design cavities to control photochem.**34**del Pino, J.; Schröder, F. A.; Chin, A. W.; Feist, J.; Garcia-Vidal, F. J. Tensor network simulation of polaron-polaritons in organic microcavities.*Phys. Rev. B: Condens. Matter Mater. Phys.*2018,*98*, 165416, DOI: 10.1103/PhysRevB.98.165416**35**Ruggenthaler, M.; Tancogne-Dejean, N.; Flick, J.; Appel, H.; Rubio, A. From a quantum-electrodynamical light–matter description to novel spectroscopies.*Nat. Rev. Chem.*2018,*2*, 0118, DOI: 10.1038/s41570-018-0118[Crossref], [CAS], Google Scholar35https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhtF2lurjP&md5=e6cdcdad8e973727c57a498378a91eb1From a quantum-electrodynamical light-matter description to novel spectroscopiesRuggenthaler, Michael; Tancogne-Dejean, Nicolas; Flick, Johannes; Appel, Heiko; Rubio, AngelNature Reviews Chemistry (2018), 2 (3), 0118CODEN: NRCAF7; ISSN:2397-3358. (Nature Research)Insights from spectroscopic expts. led to the development of quantum mechanics as the common theor. framework for describing the phys. and chem. properties of atoms, mols. and materials. Later, a full quantum description of charged particles, electromagnetic radiation and special relativity was developed, leading to quantum electrodynamics (QED). This is, to our current understanding, the most complete theory describing photon-matter interactions in correlated many-body systems. In the low-energy regime, simplified models of QED have been developed to describe and analyze spectra over a wide spatiotemporal range as well as phys. systems. In this Review, we highlight the interrelations and limitations of such theor. models, thereby showing that they arise from low-energy simplifications of the full QED formalism, in which antiparticles and the internal structure of the nuclei are neglected. Taking mol. systems as an example, we discuss how the breakdown of some simplifications of low-energy QED challenges our conventional understanding of light-matter interactions. In addn. to high-precision at. measurements and simulations of particle physics problems in solid-state systems, new theor. features that account for collective QED effects in complex interacting many-particle systems could become a material-based route to further advance our current understanding of light-matter interactions.**36**Groenhof, G.; Toppari, J. J. Coherent light harvesting through strong coupling to confined light.*J. Phys. Chem. Lett.*2018,*9*, 4848– 4851, DOI: 10.1021/acs.jpclett.8b02032[ACS Full Text ], [CAS], Google Scholar36https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsVKjtbfE&md5=f1f63c555637939a5fc7b8524a597ff8Coherent Light Harvesting through Strong Coupling to Confined LightGroenhof, Gerrit; Toppari, J. JussiJournal of Physical Chemistry Letters (2018), 9 (17), 4848-4851CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)When photoactive mols. interact strongly with confined light modes, new hybrid light-matter states may form: the polaritons. These polaritons are coherent superpositions of excitations of the mols. and of the cavity photon. Recently, polaritons were shown to mediate energy transfer between chromophores at distances beyond the Forster limit. Here we explore the potential of strong coupling for light-harvesting applications by means of atomistic mol. dynamics simulations of mixts. of photoreactive and non-photo-reactive mols. strongly coupled to a single confined light mode. These mols. are spatially sepd. and present at different concns. Our simulations suggest that while the excitation is initially fully delocalized over all mols. and the confined light mode, it very rapidly localizes onto one of the photoreactive mols., which then undergoes the reaction.**37**Vendrell, O. Collective Jahn-Teller interactions through light-matter coupling in a cavity.*Phys. Rev. Lett.*2018,*121*, 253001, DOI: 10.1103/PhysRevLett.121.253001[Crossref], [PubMed], [CAS], Google Scholar37https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXltFWhu78%253D&md5=2588a03813dba311ef4680461d4eaa4fCollective Jahn-Teller Interactions through Light-Matter Coupling in a CavityVendrell, OriolPhysical Review Letters (2018), 121 (25), 253001CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)The ultrafast nonradiative relaxation of a mol. ensemble coupled to a cavity mode is considered theor. and by real-time quantum dynamics. For equal coupling strength of single mols. to the cavity mode, the nonradiative relaxation rate from the upper to the lower polariton states is found to strongly depend on the no. of coupled mols. The coupling of both bright and dark polaritonic states among each other constitutes a special case of (pseudo-)Jahn-Teller interactions involving collective displacements the internal coordinates of the mols. in the ensemble, and the strength of the first order vibronic coupling depends exclusively on the gradient of the energy gaps between mol. electronic states. For N>2 mols., the N-1 dark light-matter states between the two optically active polaritons feature true collective conical intersection crossings, whose location depends on the internal at. coordinates of each mol. in the ensemble, and which contribute to the ultrafast nonradiative decay from the upper polariton.**38**Reitz, M.; Sommer, C.; Genes, C. Langevin approach to quantum optics with molecules.*Phys. Rev. Lett.*2019,*122*, 203602, DOI: 10.1103/PhysRevLett.122.203602[Crossref], [PubMed], [CAS], Google Scholar38https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXhtFKlsbfE&md5=dd16d30dffec617059d2846562fed4c4Langevin Approach to Quantum Optics with MoleculesReitz, Michael; Sommer, Christian; Genes, ClaudiuPhysical Review Letters (2019), 122 (20), 203602CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)We investigate the interaction between light and mol. systems modeled as quantum emitters coupled to a multitude of vibrational modes via a Holstein-type interaction. We follow a quantum Langevin equations approach that allows for anal. derivations of absorption and fluorescence profiles of mols. driven by classical fields or coupled to quantized optical modes. We retrieve anal. expressions for the modification of the radiative emission branching ratio in the Purcell regime and for the asym. cavity transmission assocd. with dissipative cross talk between upper and lower polaritons in the strong coupling regime. We also characterize the F.ovrddot.orster resonance energy transfer process between donor-acceptor mols. mediated by the vacuum or by a cavity mode.**39**Triana, J. F.; Sanz-Vicario, J. L. Revealing the Presence of Potential Crossings in Diatomics Induced by Quantum Cavity Radiation.*Phys. Rev. Lett.*2019,*122*, 063603, DOI: 10.1103/PhysRevLett.122.063603[Crossref], [PubMed], [CAS], Google Scholar39https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1MXnvFSrsLs%253D&md5=53316bbfd4a8311f93286fe1ca9ce117Revealing the Presence of Potential Crossings in Diatomics Induced by Quantum Cavity RadiationTriana, Johan F.; Sanz-Vicario, Jose LuisPhysical Review Letters (2019), 122 (6), 063603CODEN: PRLTAO; ISSN:1079-7114. (American Physical Society)We propose an expt. to find evidence of the formation of light-induced crossings provoked by cavity quantum radiation on simple mols. by using state-of-the-art optical cavities, mol. beams, pump-probe laser schemes, and velocity mapping detectors for fragmentation. The procedure is based on prompt excitation and subsequent dissocn. in a three-state scheme of a polar diat. mol., with two Σ1 states (ground and first excited) coupled first by the UV pump laser and then by the cavity radiation, and a third fully dissociative state Π1 coupled through the delayed UV/V probe laser. The obsd. enhancement of photodissocn. yields in the Π1 channel at given time delays between the pump and probe lasers unambiguously indicates the formation of a light-induced crossing between the two Σ1 field-dressed potential energy curves of the mol. Also, the prodn. of cavity photons out of the vacuum field state via nonadiabatic effects represents a showcase of a mol. dynamical Casimir effect. To simulate the expt. outcome, we perform ab initio coherent quantum dynamics of the mol. LiF subject to external lasers and quantum cavity interactions in the strong coupling regime, using a product grid representation of the total polaritonic wave function for both vibrational and photon degrees of freedom.**40**Galego, J.; Climent, C.; Garcia-Vidal, F. J.; Feist, J. Cavity Casimir-Polder forces and their effects in ground state chemical reactivity.*Phys. Rev. X*2019,*9*, 1– 22, DOI: 10.1103/PhysRevX.9.021057**41**Csehi, A.; Kowalewski, M.; Halász, G. J.; Vibók, Á. Ultrafast dynamics in the vicinity of quantum light-induced conical intersections.*New J. Phys.*2019, na, DOI: 10.1088/1367-2630/ab3fcc**42**Martínez-Martínez, L. A.; Ribeiro, R. F.; Campos-González-Angulo, J.; Yuen-Zhou, J. Can Ultrastrong Coupling Change Ground-State Chemical Reactions?.*ACS Photonics*2018,*5*, 167– 176, DOI: 10.1021/acsphotonics.7b00610[ACS Full Text ], [CAS], Google Scholar42https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhs1aksL%252FK&md5=3c3aa349f79c593e175d976bc588a60dCan Ultrastrong Coupling Change Ground-State Chemical Reactions?Martinez-Martinez, Luis A.; Ribeiro, Raphael F.; Campos-Gonzalez-Angulo, Jorge; Yuen-Zhou, JoelACS Photonics (2018), 5 (1), 167-176CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)Recent advancements on the fabrication of org. micro- and nanostructures have permitted the strong collective light-matter coupling regime to be reached with mol. materials. Pioneering works in this direction have shown the effects of this regime in the excited state reactivity of mol. systems and at the same time have opened up the question of whether it is possible to introduce any modifications in the electronic ground energy landscape which could affect chem. thermodn. and/or kinetics. In this work, we use a model system of many mols. coupled to a surface-plasmon field to gain insight on the key parameters which govern the modifications of the ground-state potential energy surface. Our findings confirm that the energetic changes per mol. are detd. by effects that are essentially on the order of single-mol. light-matter couplings, in contrast with those of the electronically excited states, for which energetic corrections are of a collective nature. Hence the prospects of ultrastrong coupling to change ground-state chem. reactions for the parameters studied in this model are limited. Still, we reveal some intriguing quantum-coherent effects assocd. with pathways of concerted reactions, where two or more mols. undergo reactions simultaneously and which can be of relevance in low-barrier reactions. Finally, we also explore modifications to nonadiabatic dynamics and conclude that, for our particular model, the presence of a large no. of dark states yields negligible effects. Our study reveals new possibilities as well as limitations for the emerging field of polariton chem.**43**Viehmann, O.; Von Delft, J.; Marquardt, F. Superradiant phase transitions and the standard description of circuit QED.*Phys. Rev. Lett.*2011,*107*, 1– 5, DOI: 10.1103/PhysRevLett.107.113602**44**Dreizler, E. G.*Density Functional Theory - An Approach to the Quantum Many-Body Problem*; Springer: Berlin Heidelberg, 1990.Google ScholarThere is no corresponding record for this reference.**45**Dreizler, R. M.; Engel, E.*Density Functional Theory: An Advanced Course*; Springer, 2011.Google ScholarThere is no corresponding record for this reference.**46**Fetter, A.; Walecka, J.*Quantum Theory of Many-Particle Systems*; Dover: Mineola, NY, 2003.Google ScholarThere is no corresponding record for this reference.**47**Stefanucci, G.; van Leeuwen, R.*Nonequilibrium many-body theory of quantum systems: a modern introduction*; Cambridge University Press, 2013.**48**Onida, G.; Reining, L.; Rubio, A. Electronic excitations: Density-functional versus many-body Green’s-function approaches.*Rev. Mod. Phys.*2002,*74*, 601– 659, DOI: 10.1103/RevModPhys.74.601[Crossref], [CAS], Google Scholar48https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38Xlt1ymsL0%253D&md5=904c22dc306e014cab96b27d8d971951Electronic excitations: density-functional versus many-body Green's-function approachesOnida, Giovanni; Reining, Lucia; Rubio, AngelReviews of Modern Physics (2002), 74 (2), 601-659CODEN: RMPHAT; ISSN:0034-6861. (American Physical Society)A review. Electronic excitations lie at the origin of most of the commonly measured spectra. However, the 1st-principles computation of excited states requires a larger effort than ground-state calcns., which can be very efficiently carried out within d.-functional theory. However, two theor. and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green's-function equations, starting with a 1-electron propagator and considering the electron-hole Green's function for the response. Key ingredients are the electron's self-energy Σ and the electron-hole interaction. A good approxn. for Σ was obtained with Hedin's GW approach, using d.-functional theory as a zero-order soln. First-principles GW calcns. for real systems were successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional deriv. of Σ. An alternative approach to calcg. electronic excitations is the time-dependent d.-functional theory (TDDFT), which offers the important practical advantage of a dependence on d. rather than on multivariable Green's functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-d. approxn. gave promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green's functions and the TDDFT approaches profit from mutual insight. This review compares the theor. and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.**49**Theophilou, I.; Buchholz, F.; Eich, F. G.; Ruggenthaler, M.; Rubio, A. Kinetic-Energy Density-Functional Theory on a Lattice.*J. Chem. Theory Comput.*2018,*14*, 4072– 4087, DOI: 10.1021/acs.jctc.8b00292[ACS Full Text ], [CAS], Google Scholar49https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXht1Kgu73L&md5=cb188271a155f26e3b03de3813850e75Kinetic-Energy Density-Functional Theory on a LatticeTheophilou, Iris; Buchholz, Florian; Eich, F. G.; Ruggenthaler, Michael; Rubio, AngelJournal of Chemical Theory and Computation (2018), 14 (8), 4072-4087CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)We present a kinetic-energy d.-functional theory and the corresponding kinetic-energy Kohn-Sham (keKS) scheme on a lattice and show that, by including more observables explicitly in a d.-functional approach, already simple approxn. strategies lead to very accurate results. Here, we promote the kinetic-energy d. to a fundamental variable alongside the d. and show for specific cases (anal. and numerically) that there is a one-to-one correspondence between the external pair of on-site potential and site-dependent hopping and the internal pair of d. and kinetic-energy d. On the basis of this mapping, we establish two unknown effective fields, the mean-field exchange-correlation potential and the mean-field exchange-correlation hopping, which force the keKS system to generate the same kinetic-energy d. and d. as the fully interacting one. We show, by a decompn. based on the equations of motions for the d. and the kinetic-energy d., that we can construct simple orbital-dependent functionals that outperform the corresponding exact-exchange Kohn-Sham (KS) approxn. of std. d.-functional theory. We do so by considering the exact KS and keKS systems and comparing the unknown correlation contributions as well as by comparing self-consistent calcns. based on the mean-field exchange (for the effective potential) and a uniform (for the effective hopping) approxn. for the keKS and the exact-exchange approxn. for the KS system, resp.**50**Ruggenthaler, M.; Mackenroth, F.; Bauer, D. Time-dependent Kohn-Sham approach to quantum electrodynamics.*Phys. Rev. A: At., Mol., Opt. Phys.*2011,*84*, 042107, DOI: 10.1103/PhysRevA.84.042107[Crossref], [CAS], Google Scholar50https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhsVWmt7fF&md5=6afdf8c78c3ecb60dfb2f6c712472dcaTime-dependent Kohn-Sham approach to quantum electrodynamicsRuggenthaler, M.; Mackenroth, F.; Bauer, D.Physical Review A: Atomic, Molecular, and Optical Physics (2011), 84 (4, Pt. A), 042107/1-042107/7CODEN: PLRAAN; ISSN:1050-2947. (American Physical Society)We prove a generalization of the van Leeuwen theorem toward quantum electrodynamics, providing the formal foundations of a time-dependent Kohn-Sham construction for coupled quantized matter and electromagnetic fields. We circumvent the symmetry-causality problems assocd. with the action-functional approach to Kohn-Sham systems. We show that the effective external four-potential and four-current of the Kohn-Sham system are uniquely defined and that the effective four-current takes a very simple form. Further we rederive the Runge-Gross theorem for quantum electrodynamics.**51**Tokatly, I. V. Time-dependent density functional theory for many-electron systems interacting with cavity photons.*Phys. Rev. Lett.*2013,*110*, 1– 5, DOI: 10.1103/PhysRevLett.110.233001**52**Ruggenthaler, M.; Flick, J.; Pellegrini, C.; Appel, H.; Tokatly, I. V.; Rubio, A. Quantum-electrodynamical density-functional theory: Bridging quantum optics and electronic-structure theory.*Phys. Rev. A: At., Mol., Opt. Phys.*2014,*90*, 1– 26, DOI: 10.1103/PhysRevA.90.012508**53**Ruggenthaler, M. Ground-State Quantum-Electrodynamical Density-Functional Theory.*arXiv:1509.01417 [quant-ph]*2015, 1– 6Google ScholarThere is no corresponding record for this reference.**54**Flick, J.; Ruggenthaler, M.; Appel, H.; Rubio, A. Kohn–Sham approach to quantum electrodynamical density-functional theory: Exact time-dependent effective potentials in real space.*Proc. Natl. Acad. Sci. U. S. A.*2015,*112*, 15285– 15290, DOI: 10.1073/pnas.1518224112[Crossref], [PubMed], [CAS], Google Scholar54https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXhvFalt7zL&md5=ab2f497f36a34a13df4f6731ea546c60Kohn-Sham approach to quantum electrodynamical density-functional theory: Exact time-dependent effective potentials in real spaceFlick, Johannes; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelProceedings of the National Academy of Sciences of the United States of America (2015), 112 (50), 15285-15290CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)The d.-functional approach to quantum electrodynamics extends traditional d.-functional theory and opens the possibility to describe electron-photon interactions in terms of effective Kohn-Sham potentials. In this work, we numerically construct the exact electron-photon Kohn-Sham potentials for a prototype system that consists of a trapped electron coupled to a quantized electromagnetic mode in an optical high-Q cavity. Although the effective current that acts on the photons is known explicitly, the exact effective potential that describes the forces exerted by the photons on the electrons is obtained from a fixed-point inversion scheme. This procedure allows us to uncover important beyond-mean-field features of the effective potential that mark the breakdown of classical light-matter interactions. We observe peak and step structures in the effective potentials, which can be attributed solely to the quantum nature of light; i.e., they are real-space signatures of the photons. Our findings show how the ubiquitous dipole interaction with a classical electromagnetic field has to be modified in real space to take the quantum nature of the electromagnetic field fully into account.**55**Flick, J.; Ruggenthaler, M.; Appel, H.; Rubio, A. Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistry.*Proc. Natl. Acad. Sci. U. S. A.*2017,*114*, 3026– 3034, DOI: 10.1073/pnas.1615509114[Crossref], [PubMed], [CAS], Google Scholar55https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXjvF2htLs%253D&md5=c469fe9a59a87cc6422835a1b0824ea4Atoms and molecules in cavities, from weak to strong coupling in quantum-electrodynamics (QED) chemistryFlick, Johannes; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelProceedings of the National Academy of Sciences of the United States of America (2017), 114 (12), 3026-3034CODEN: PNASA6; ISSN:0027-8424. (National Academy of Sciences)A review is provided of how well-established concepts in the fields of quantum chem. and material sciences have to be adapted when the quantum nature of light becomes important in correlated matter-photon problems. Model systems in optical cavities were analyzed, where the matter-photon interaction is considered from the weak- to the strong-coupling limit and for individual photon modes as well as for the multimode case. The authors identify fundamental changes in Born-Oppenheimer surfaces, spectroscopic quantities, conical intersections, and efficiency for quantum control. The authors conclude by applying the recently developed quantum-electrodynamical d.-functional theory to spontaneous emission and show how a straightforward approxn. accurately describes the correlated electron-photon dynamics. This work paves the way to describe matter-photon interactions from 1st principles and addresses the emergence of new states of matter in chem. and material science.**56**Flick, J.; Schäfer, C.; Ruggenthaler, M.; Appel, H.; Rubio, A. Ab Initio Optimized Effective Potentials for Real Molecules in Optical Cavities: Photon Contributions to the Molecular Ground State.*ACS Photonics*2018,*5*, 992– 1005, DOI: 10.1021/acsphotonics.7b01279[ACS Full Text ], [CAS], Google Scholar56https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXkvFanug%253D%253D&md5=fc695d87a81e7fa21abf2b6797a5e419Ab Initio Optimized Effective Potentials for Real Molecules in Optical Cavities: Photon Contributions to the Molecular Ground StateFlick, Johannes; Schaefer, Christian; Ruggenthaler, Michael; Appel, Heiko; Rubio, AngelACS Photonics (2018), 5 (3), 992-1005CODEN: APCHD5; ISSN:2330-4022. (American Chemical Society)A simple scheme to efficiently compute photon exchange-correlation contributions due to the coupling to transversal photons as formulated in the newly developed quantum-electrodynamical d. functional theory (QEDFT) is introduced. The construction employs the optimized-effective potential (OEP) approach by the Sternheimer equation to avoid the explicit calcn. of unoccupied states. The efficiency of the scheme was demonstrated by applying it to an exactly solvable GaAs quantum ring model system, a single azulene mol., and chains of Na2, all located in optical cavities and described in full real space. While the 1st example is a 2-dimensional system and allows to benchmark the employed approxns., the latter 2 examples demonstrate that the correlated electron-photon interaction appreciably distorts the ground-state electronic structure of a real mol. By using this scheme, the authors not only construct typical electronic observables, such as the electronic ground state d., but also illustrate how photon observables, such as the photon no., and mixed electron-photon observables, e.g. electron-photon correlation functions, become accessible in a DFT framework. This work constitutes the 1st 3-dimensional ab initio calcn. within the new QEDFT formalism and thus opens up a new computational route for the ab initio study of correlated electron-photon systems in quantum cavities.**57**Flick, J.; Welakuh, D. M.; Ruggenthaler, M.; Appel, H.; Rubio, A. Light-Matter Response Functions in Quantum-Electrodynamical Density-Functional Theory: Modifications of Spectra and of the Maxwell Equations.*arXiv Preprint arXiv:1803.02519*2018, 1– 27Google ScholarThere is no corresponding record for this reference.**58**Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Challenges for Density Functional Theory.*Chem. Rev.*2012,*112*, 289– 320, DOI: 10.1021/cr200107z[ACS Full Text ], [CAS], Google Scholar58https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhs1GltrbE&md5=51a7564af74b194a423868c40e5bc3caChallenges for Density Functional TheoryCohen, Aron J.; Mori-Sanchez, Paula; Yang, WeitaoChemical Reviews (Washington, DC, United States) (2012), 112 (1), 289-320CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review includes the following topics: the entrance of DFT into chem., constructing approx. functionals and minimizing the total energy, insight into large systematic errors of functionals, and strong correlation.**59**Niemczyk, T.; Deppe, F.; Huebl, H.; Menzel, E. P.; Hocke, F.; Schwarz, M. J.; Zueco, D.; Hümmer, T.; Solano, E.; Marx, A.; Gross, R. Circuit quantum electrodynamics in the ultrastrong-coupling regime.*Nat. Phys.*2010,*6*, 772– 776, DOI: 10.1038/nphys1730[Crossref], [CAS], Google Scholar59https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3cXht1ehtbvM&md5=48ded8663f7ae7736c09d0937be3a636Circuit quantum electrodynamics in the ultrastrong-coupling regimeNiemczyk, T.; Deppe, F.; Huebl, H.; Menzel, E. P.; Hocke, F.; Schwarz, M. J.; Garcia-Ripoll, J. J.; Zueco, D.; Huemmer, T.; Solano, E.; Marx, A.; Gross, R.Nature Physics (2010), 6 (10), 772-776CODEN: NPAHAX; ISSN:1745-2473. (Nature Publishing Group)In circuit quantum electrodynamics (QED), where superconducting artificial atoms are coupled to on-chip cavities, the exploration of fundamental quantum physics in the strong-coupling regime has greatly evolved. In this regime, an atom and a cavity can exchange a photon frequently before coherence is lost. Nevertheless, all expts. so far are well described by the renowned Jaynes-Cummings model. Here, we report on the first exptl. realization of a circuit QED system operating in the ultrastrong-coupling limit, where the atom-cavity coupling rate reaches a considerable fraction of the cavity transition frequency . Furthermore, we present direct evidence for the breakdown of the Jaynes-Cummings model. We reach remarkable normalized coupling rates of up to 12% by enhancing the inductive coupling of a flux qubit to a transmission line resonator. Our circuit extends the toolbox of quantum optics on a chip towards exciting explorations of ultrastrong light-matter interaction.**60**Bayer, A.; Pozimski, M.; Schambeck, S.; Schuh, D.; Huber, R.; Bougeard, D.; Lange, C. Terahertz Light-Matter Interaction beyond Unity Coupling Strength.*Nano Lett.*2017,*17*, 6340– 6344, DOI: 10.1021/acs.nanolett.7b03103[ACS Full Text ], [CAS], Google Scholar60https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2sXhsFelsbvK&md5=13d1e3958a3ae5177905d5ddce178da9Terahertz Light-Matter Interaction beyond Unity Coupling StrengthBayer, Andreas; Pozimski, Marcel; Schambeck, Simon; Schuh, Dieter; Huber, Rupert; Bougeard, Dominique; Lange, ChristophNano Letters (2017), 17 (10), 6340-6344CODEN: NALEFD; ISSN:1530-6984. (American Chemical Society)Achieving control over light-matter interaction in custom-tailored nanostructures is at the core of modern quantum electrodynamics. In strongly and ultrastrongly coupled systems, the excitation is repeatedly exchanged between a resonator and an electronic transition at a rate known as the vacuum Rabi frequency ΩR. For ΩR approaching the resonance frequency ωc, novel quantum phenomena including squeezed states, Dicke superradiant phase transitions, the collapse of the Purcell effect, and a population of the ground state with virtual photon pairs are predicted. Yet, the exptl. realization of optical systems with ΩR/ωc ≥ 1 has remained elusive. Here, we introduce a paradigm change in the design of light-matter coupling by treating the electronic and the photonic components of the system as an entity instead of optimizing them sep. Using the electronic excitation to not only boost the electronic polarization but furthermore tailor the shape of the vacuum mode, we push ΩR/ωc of cyclotron resonances ultrastrongly coupled to metamaterials far beyond unity. As one prominent illustration of the unfolding possibilities, we calc. a ground state population of 0.37 virtual photons for our best structure with ΩR/ωc = 1.43 and suggest a realistic exptl. scenario for measuring vacuum radiation by cutting-edge terahertz quantum detection.**61**Gilbert, T. L. Hohenberg-Kohn theorem for nonlocal external potentials.*Phys. Rev. B*1975,*12*, 2111– 2120, DOI: 10.1103/PhysRevB.12.2111**62**Müller, A. M. K. Explicit approximate relation between reduced two- and one-particel density matrices.*Phys. Lett. A*1984,*105*, 446, DOI: 10.1016/0375-9601(84)91034-X**63**Goedecker, S.; Umrigar, C. J. Natural orbital functional for the many-electron problem.*Phys. Rev. Lett.*1998,*81*, 866– 869, DOI: 10.1103/PhysRevLett.81.866[Crossref], [CAS], Google Scholar63https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXkslertr8%253D&md5=241786c71cc5bc0ab33616c403155db4Natural Orbital Functional for the Many-Electron ProblemGoedecker, S.; Umrigar, C. J.Physical Review Letters (1998), 81 (4), 866-869CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)The exchange-correlation energy in Kohn-Sham d. functional theory is expressed as a functional of the electronic d. and the Kohn-Sham orbitals. An alternative to Kohn-Sham theory is to express the energy as a functional of the reduced first-order d. matrix or equivalently the natural orbitals. We present an approx., simple, and parameter-free functional of the natural orbitals, based solely on scaling arguments and the near satisfaction of a sum rule. Our tests on atoms show that it yields on av. more accurate energies and charge densities than the Hartree-Fock method, the local d. approxn., and the generalized gradient approxns.**64**Sharma, S.; Dewhurst, J. K.; Shallcross, S.; Gross, E. K. U. Spectral density and metal-insulator phase transition in mott insulators within reduced density matrix functional theory.*Phys. Rev. Lett.*2013,*110*, 1– 5, DOI: 10.1103/PhysRevLett.110.116403**65**Coleman, A. J. Structure of Fermion Density Matrices.*Rev. Mod. Phys.*1963,*35*, 668– 686, DOI: 10.1103/RevModPhys.35.668**66**Klyachko, A. A. Quantum marginal problem and N-representability.*J. Phys.: Conf. Ser.*2006,*36*, 72, DOI: 10.1088/1742-6596/36/1/014**67**Mazziotti, D. A. Structure of Fermionic Density Matrices: Complete*N*-Representability Conditions.*Phys. Rev. Lett.*2012,*108*, 263002, DOI: 10.1103/PhysRevLett.108.263002[Crossref], [PubMed], [CAS], Google Scholar67https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC38XhtFartLjN&md5=944b437a01adf56f2bb39287d8636988Structure of fermionic density matrices: complete N-representability conditionsMazziotti, David A.Physical Review Letters (2012), 108 (26), 263002/1-263002/5CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)We present a constructive soln. to the N-representability problem: a full characterization of the conditions for constraining the two-electron reduced d. matrix to represent an N-electron d. matrix. Previously known conditions, while rigorous, were incomplete. Here, we derive a hierarchy of constraints built upon (i) the bipolar theorem and (ii) tensor decompns. of model Hamiltonians. Existing conditions D, Q, G, T1, and T2, known classical conditions, and new conditions appear naturally. Subsets of the conditions are amenable to polynomial-time computations of strongly correlated systems.**68**Nielsen, S. E. B.; Schäfer, C.; Ruggenthaler, M.; Rubio, A. Dressed-Orbital Approach to Cavity Quantum Electrodynamics and Beyond.*arXiv preprint arXiv:1812.00388*2018, naGoogle ScholarThere is no corresponding record for this reference.**69**Grynberg, G.; Aspect, A.; Fabre, C.*Introduction to quantum optics: from the semi-classical approach to quantized light*; Cambridge University Press, 2010.**70**Rokaj, V.; Welakuh, D. M.; Ruggenthaler, M.; Rubio, A. Light–matter interaction in the long-wavelength limit: no ground-state without dipole self-energy.*J. Phys. B: At., Mol. Opt. Phys.*2018,*51*, 034005, DOI: 10.1088/1361-6455/aa9c99[Crossref], [CAS], Google Scholar70https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC1cXhsFynu7bL&md5=406a602d898e5ab2e1f6a85f8383ec7eLight-matter interaction in the long- wavelength limit: no ground-state without dipole self-energyRokaj, Vasil; Welakuh, Davis M.; Ruggenthaler, Michael; Rubio, AngelJournal of Physics B: Atomic, Molecular and Optical Physics (2018), 51 (3), 034005/1-034005/13CODEN: JPAPEH; ISSN:0953-4075. (IOP Publishing Ltd.)Most theor. studies for correlated light-matter systems are performed within the long- wavelength limit, i.e., the electromagnetic field is assumed to be spatially uniform. In this limit the so-called length-gauge transformation for a fully quantized light-matter system gives rise to a dipole self-energy term in the Hamiltonian, i.e., a harmonic potential of the total dipole matter moment. In practice this term is often discarded as it is assumed to be subsumed in the kinetic energy term. In this work we show the necessity of the dipole self-energy term. First and foremost, without it the light-matter system in the long-wavelength limit does not have a ground-state, i.e., the combined light-matter system is unstable. Further, the mixing of matter and photon degrees of freedom due to the length-gauge transformation, which also changes the representation of the translation operator for matter, gives rise to the Maxwell equations in matter and the omittance of the dipole self-energy leads to a violation of these equations. Specifically we show that without the dipole self-energy the so-called 'depolarization shift' is not properly described. Finally we show that this term also arises if we perform the semi-classical limit after the length-gauge transformation. In contrast to the std. approach where the semi-classical limit is performed before the length-gauge transformation, the resulting Hamiltonian is bounded from below and thus supports ground-states. This is very important for practical calcns. and for d.-functional variational implementations of the non-relativistic QED formalism. For example, the existence of a combined light-matter ground-state allows one to calc. the Stark shift non-perturbatively.**71**Spohn, H.*Dynamics of charged particles and their radiation field*; Cambridge university press, 2004.**72**Power, E. A.; Thirunamachandran, T. Quantum electrodynamics in a cavity.*Phys. Rev. A: At., Mol., Opt. Phys.*1982,*25*, 2473– 2484, DOI: 10.1103/PhysRevA.25.2473[Crossref], [CAS], Google Scholar72https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL38Xit1Gnu7c%253D&md5=d61053ce33b923c88c1a790685834d50Quantum electrodynamics in a cavityPower, E. A.; Thirunamachandran, T.Physical Review A: Atomic, Molecular, and Optical Physics (1982), 25 (5), 2473-84CODEN: PLRAAN; ISSN:0556-2791.The theory of interaction of atoms and mols. with radiation confined in a cavity is presented. The effect of the confinement on the field commutation relations is examd. and the modified commutation relations are derived for an arbitrary cavity. A canonical transformation on the minimal-coupling Hamiltonian for such systems leads to a new Hamiltonian contg. explicitly neither the interat. nor the atom image potentials. The new multipolar Hamiltonian is used to calc. energy shifts of an atom and a pair of atoms near a conducting wall. Changes in the rate of spontaneous emission from an excited atom are also evaluated.**73**Shahbazyan, T. V.; Stockman, M. I.*Plasmonics: theory and applications*; Springer, 2013.**74**Giesbertz, K. J. H.; Ruggenthaler, M. One-body reduced density-matrix functional theory in finite basis sets at elevated temperatures.*Phys. Rep.*2019,*806*, 1– 47, DOI: 10.1016/j.physrep.2019.01.010**75**Bonitz, M.*Quantum Kinetic Theory*, 2nd ed.; Springer: Berlin, 2016.**76**van Leeuwen, R.; Stefanucci, G.*Nonequilibrium Many-Body Theory of Quantum Systems*; Cambridge University Press, 2013.Google ScholarThere is no corresponding record for this reference.**77**Coleman, A. J.; Yukalov, V. I.*Reduced density matrices: Coulson’s challenge*; Springer Science & Business Media, 2000; Vol. 72.**78**Coulson, C. Present State of Molecular Structure Calculations.*Rev. Mod. Phys.*1960,*32*, 170– 177, DOI: 10.1103/RevModPhys.32.170[Crossref], [CAS], Google Scholar78https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF3cXovF2gtA%253D%253D&md5=5f84ea370cca7ff42889b1d5a03f3a39Present state of molecular structure calculationsCoulson, C. A.Reviews of Modern Physics (1960), 32 (), 170-7CODEN: RMPHAT; ISSN:0034-6861.There is no expanded citation for this reference.**79**Rokaj, V.; Penz, M.; Sentef, M. A.; Ruggenthaler, M.; Rubio, A. Quantum Electrodynamical Bloch Theory with Homogeneous Magnetic Fields.*Phys. Rev. Lett.*2019,*123*, 1– 6, DOI: 10.1103/PhysRevLett.123.047202**80**Watson, J. K. Simplification of the molecular vibration-rotation Hamiltonian.*Mol. Phys.*1968,*15*, 479– 490, DOI: 10.1080/00268976800101381[Crossref], [CAS], Google Scholar80https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaF1MXlt1Kjug%253D%253D&md5=220a477da0af4bbb6b19fbbf53e08e50Simplification of the molecular vibration-rotation hamiltonianWatson, James K. G.Molecular Physics (1968), 15 (5), 479-90CODEN: MOPHAM; ISSN:0026-8976.By use of the commutation relations and sum rules, the Darling-Dennison vibration-rotation Hamiltonian for a nonlinear mol. is rearranged. A simple expansion is given for the μαβ tensor in terms of the normal coordinates.**81**Piris, M. Global Method for Electron Correlation.*Phys. Rev. Lett.*2017,*119*, 1– 5, DOI: 10.1103/PhysRevLett.119.063002**82**Mazziotti, D. A. Two-electron reduced density matrix as the basic variable in many-electron quantum chemistry and physics.*Chem. Rev.*2012,*112*, 244– 262, DOI: 10.1021/cr2000493[ACS Full Text ], [CAS], Google Scholar82https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXhtVOns7zN&md5=fb6dcf6b6470ff39479a446778190941Two-Electron Reduced Density Matrix as the Basic Variable in Many-Electron Quantum Chemistry and PhysicsMazziotti, David A.Chemical Reviews (Washington, DC, United States) (2012), 112 (1), 244-262CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)A review including the following topics: contracted Schrodinger theory, variational 2-RDM method, parametric 2-RDM method, and parametric 2-RDM method.**83**Hohenberg, P.; Kohn, W. Inhomogeneous electron gas.*Phys. Rev.*1964,*136*, B864, DOI: 10.1103/PhysRev.136.B864**84**Theophilou, I.; Lathiotakis, N. N.; Marques, M. A.; Helbig, N. Generalized Pauli constraints in reduced density matrix functional theory.*J. Chem. Phys.*2015,*142*, 154108, DOI: 10.1063/1.4918346[Crossref], [PubMed], [CAS], Google Scholar84https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXms1Oqu74%253D&md5=833ab0d65a3f8cb49ff27845f5070783Generalized Pauli constraints in reduced density matrix functional theoryTheophilou, Iris; Lathiotakis, Nektarios N.; Marques, Miguel A. L.; Helbig, NicoleJournal of Chemical Physics (2015), 142 (15), 154108/1-154108/7CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)Functionals of the one-body reduced d. matrix (1-RDM) are routinely minimized under Coleman's ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any no. of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced d.-matrix functional theory calcns. In particular, we examine whether the std. minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addn. to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation nos. with those obtained by the enforcement of the ensemble conditions alone. (c) 2015 American Institute of Physics.**85**Lieb, E. H. Variational principle for many-fermion systems.*Phys. Rev. Lett.*1981,*46*, 457, DOI: 10.1103/PhysRevLett.46.457[Crossref], [CAS], Google Scholar85https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL3MXptVKruw%253D%253D&md5=445e4a36b3d06cc10d1a1ef959a82789Variational principle for many-fermion systemsLieb, Elliot H.Physical Review Letters (1981), 46 (7), 457-9CODEN: PRLTAO; ISSN:0031-9007.If ψ is a determinantal eigenfunction for the N-fermion Hamiltonian, H, with one- and two-body terms, the e0 ≤ 〈ψ,Hψ〉 = E(K), where e0 is the ground-state energy, K is the one-body reduced d. matrix of ψ, and E(K) is the well-known expression in terms of direct and exchange energies. If an arbitrary one-body K is given, which does not come from a determinantal ψ, then E ≥ e0 does not necessarily hold. If the two-body of H is pos., then in fact e0 ≤ eHF ≤ E(K), where eHF is the Hartree-Fock ground-state energy.**86**Buijse, M. A.; Baerends, E. J. An approximate exchange-correlation hole density as a functional of the natural orbitals.*Mol. Phys.*2002,*100*, 401– 421, DOI: 10.1080/00268970110070243[Crossref], [CAS], Google Scholar86https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD38XitFSitro%253D&md5=afe235742d04e5fe1951437f0faccde0An approximate exchange-correlation hole density as a functional of the natural orbitalsBuijse, M. A.; Baerends, E. J.Molecular Physics (2002), 100 (4), 401-421CODEN: MOPHAM; ISSN:0026-8976. (Taylor & Francis Ltd.)The Fermi and Coulomb holes that can be used to describe the physics of electron correlation are calcd. and analyzed for a no. of typical cases, ranging from prototype dynamical correlation to purely nondynamical correlation. Their behavior as a function of the position of the ref. electron and of the nuclear positions is exhibited. The notion that the hole can be written as the square of a hole amplitude, which is exactly true for the exchange hole, is generalized to the total holes, including the correlation part. An Ansatz is made for an approx. yet accurate expression for the hole amplitude in terms of the natural orbitals, employing the local (at the ref. position) values of the natural orbitals and the d. This expression for the hole amplitude leads to an approx. two-electron d. matrix that: (a) obeys correct permutation symmetry in the electron coordinates; (b) integrates to the exact one-matrix; and (c) yields exact correlation energies in the limiting cases of predominant dynamical correlation (high Z two-electron ions) and pure nondynamical correlation (dissocd. H2).**87**Frank, R. L.; Lieb, E. H.; Seiringer, R.; Siedentop, H. Müller’s exchange-correlation energy in density-matrix-functional theory.*Phys. Rev. A: At., Mol., Opt. Phys.*2007,*76*, 1– 16, DOI: 10.1103/PhysRevA.76.052517**88**Andrade, X. Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems.*Phys. Chem. Chem. Phys.*2015,*17*, 31371– 31396, DOI: 10.1039/C5CP00351B[Crossref], [PubMed], [CAS], Google Scholar88https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC2MXjtVyktLw%253D&md5=2457aa1e92d75da4f45ac2d66297c5caReal-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systemsAndrade, Xavier; Strubbe, David; De Giovannini, Umberto; Larsen, Ask Hjorth; Oliveira, Micael J. T.; Alberdi-Rodriguez, Joseba; Varas, Alejandro; Theophilou, Iris; Helbig, Nicole; Verstraete, Matthieu J.; Stella, Lorenzo; Nogueira, Fernando; Aspuru-Guzik, Alan; Castro, Alberto; Marques, Miguel A. L.; Rubio, AngelPhysical Chemistry Chemical Physics (2015), 17 (47), 31371-31396CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new phys. models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calc. response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact soln. of the Schr.ovrddot.odinger equation for low-dimensionality systems.**89**Piris, M.; Ugalde, J. M. Iterative Diagonalization for Orbital Optimization in Natural Orbital Functional Theory.*J. Comput. Chem.*2009,*30*, 2078– 2086, DOI: 10.1002/jcc.21225[Crossref], [PubMed], [CAS], Google Scholar89https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD1MXptleqsbw%253D&md5=cd6c06c44b6d1b4861af90c14e21c39dIterative diagonalization for orbital optimization in natural orbital functional theoryPiris, M.; Ugalde, J. M.Journal of Computational Chemistry (2009), 30 (13), 2078-2086CODEN: JCCHDD; ISSN:0192-8651. (John Wiley & Sons, Inc.)A challenging task in natural orbital functional theory is to find an efficient procedure for doing orbital optimization. Procedures based on diagonalization techniques have confirmed its practical value since the resulting orbitals are automatically orthogonal. In this work, a new procedure is introduced, which yields the natural orbitals by iterative diagonalization of a Hermitian matrix F. The off-diagonal elements of the latter are detd. explicitly from the hermiticity of the matrix of the Lagrange multipliers. An expression for diagonal elements is absent so a generalized Fockian is undefined in the conventional sense, nevertheless, they may be detd. from an aufbau principle. Thus, the diagonal elements are obtained iteratively considering as starting values those coming from a single diagonalization of the matrix of the Lagrange multipliers calcd. with the Hartree-Fock orbitals after the occupation nos. have been optimized. The method has been tested on the G2/97 set of mols. for the Piris natural orbital functional. To help the convergence, we have implemented a variable scaling factor which avoids large values of the off-diagonal elements of F. The elapsed times of the computations required by the proposed procedure are compared with a full sequential quadratic programming optimization, so that the efficiency of the method presented here is demonstrated. © 2009 Wiley Periodicals, Inc. J Comput Chem, 2009.**90**Ruggenthaler, M.; Bauer, D. Rabi oscillations and few-level approximations in time-dependent density functional theory.*Phys. Rev. Lett.*2009,*102*, 2– 5, DOI: 10.1103/PhysRevLett.102.233001**91**Fuks, J. I.; Helbig, N.; Tokatly, I. V.; Rubio, A. Nonlinear phenomena in time-dependent density-functional theory: What Rabi oscillations can teach us.*Phys. Rev. B: Condens. Matter Mater. Phys.*2011,*84*, na, DOI: 10.1103/PhysRevB.84.075107

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## Supporting Information

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Survey on the bosonic symmetry of the photon wave function. Details about the convergence study of the numerical examples shown in the paper. Protocol for the convergence of a dressed HF/RDMFT calculation (PDF)

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