PHYSICAL REVIEW X 10, 041043 (2020) Featured in Physics
Coupled Cluster Theory for Molecular Polaritons: Changing Ground and Excited States
Tor S. Haugland ,1 Enrico Ronca ,2,3 Eirik F. Kjønstad ,1 Angel Rubio ,3,4,5 and Henrik Koch 6,1,* 1Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway 2Istituto per i Processi Chimico Fisici del CNR (IPCF-CNR), Via G. Moruzzi, 1, 56124, Pisa, Italy 3Max Planck Institute for the Structure and Dynamics of Matter and Center Free-Electron Laser Science, Luruper Chaussee 149, 22761 Hamburg, Germany 4Center for Computational Quantum Physics (CCQ), The Flatiron Institute, 162 Fifth Avenue, New York, New York 10010, USA 5Nano-Bio Spectroscopy Group, Departamento de Física de Materiales, Universidad del País Vasco, 20018 San Sebastian, Spain 6Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56124 Pisa, Italy
(Received 9 May 2020; revised 6 September 2020; accepted 14 October 2020; published 2 December 2020)
We present an ab initio correlated approach to study molecules that interact strongly with quantum fields in an optical cavity. Quantum electrodynamics coupled cluster theory provides a nonperturbative description of cavity-induced effects in ground and excited states. Using this theory, we show how quantum fields can be used to manipulate charge transfer and photochemical properties of molecules. We propose a strategy to lift electronic degeneracies and induce modifications in the ground-state potential energy surface close to a conical intersection.
DOI: 10.1103/PhysRevX.10.041043 Subject Areas: Atomic and Molecular Physics, Chemical Physics, Photonics
I. INTRODUCTION quantized electromagnetic field interacts with the molecu- lar system, producing new hybrid light-matter states called Manipulation by strong electron-photon coupling [1] polaritons [19]. These states exhibit new and interesting and laser fields [2–4] is becoming a popular technique to properties, leading to unexpected phenomena. In the past design and explore new states of matter. Recent advances in few years, improvement of optical cavities [20–23] has experimental and theoretical research include new ways to resulted in devices that can reach the strong and ultrastrong generate exciton-polariton condensates [5], induce phase coupling limits, which has made polaritonic states acces- transitions [6–8], tune exciton energies in monolayers of sible at room temperature, also for a small number of 2D materials and interfaces [5,9,10], and even enhance the electron-phonon coupling, with possible effects on super- molecules [21,24]. Theoretical modeling is an essential tool to provide conductivity [11,12]. Nevertheless, general techniques for manipulating molecules via strong coupling have not yet fundamental understanding and outline new strategies for reached maturity. applications in polaritonic chemistry. The challenge is to Chemistry is one field that has witnessed the most progress in strong light-matter coupling applications. In particular, Ebbesen and co-workers have found that strong coupling to vibrational excited states in molecules can inhibit [13–15], catalyze [16,17], and induce selective change in the reactive path of a chemical reaction [18]. These experiments use an optical cavity, the simplest device in which entanglement between matter and light can be observed. In an optical cavity (see Fig. 1), the
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, FIG. 1. Illustration of an optical cavity interacting with a and DOI. molecule.
2160-3308=20=10(4)=041043(17) 041043-1 Published by the American Physical Society HAUGLAND, RONCA, KJØNSTAD, RUBIO, and KOCH PHYS. REV. X 10, 041043 (2020) develop an accurate theoretical description of entangled molecules, commonly used as prototypes for photovoltaic light-matter systems. In the quantum optics community, applications, can be modified by the quantized electro- several groups have developed model Hamiltonians to magnetic field. Furthermore, we show that the presence of reproduce the main features of polaritonic physics the cavity can break molecular symmetry and change [15,25–29]. The objective of the current work is to formulate relaxation mechanisms. Suitably defined fields can induce and implement a quantitative ab initio method for polari- significant changes in both ground- and excited-state tonic chemistry. properties. These results pave the way for novel strategies Presently, the only available ab initio theory is quantum to control photochemical reaction paths. electrodynamical density functional theory (QEDFT) – [1,30 32], which can describe interacting electrons and II. COUPLED CLUSTER THEORY FOR photons on an equal footing. This method is a natural ELECTRONS extension of density functional theory (DFT) [33] to quantum electrodynamics (QED). In QEDFT, the Kohn- In this section, we introduce the notation and important Sham formalism treats electrons and photons as indepen- concepts needed to develop the electron-photon interaction dent particles interacting through an exchange correlation model. For a complete outline of coupled cluster theory, we potential. The QEDFT method is computationally cheap refer to Ref. [37]. In standard coupled cluster theory for and reproduces the main polaritonic features for large singlet states, the many-body wave function is expressed systems, even at the mean-field level. However, its accuracy using the exponential parametrization, is limited due to the unknown form of the exchange correlation functional for the electron-photon interaction. jCCi¼expðTÞjHFi; ð1Þ In particular, in a mean-field treatment, no explicit electron- photon correlation is accounted for in the ground state. The where jHFi is a reference wave function that is usually problem can be overcome with a properly designed chosen to be the closed-shell Hartree-Fock (HF) determi- exchange-correlation functional, but current functionals nant. The cluster operator T generates electronic excitations are still not sufficiently accurate. Recently, Rubio and when operating on the reference, and in this way, the co-workers [34,35] proposed an extension of optimized exponential produces a superposition of Slater determi- effective potentials (OEPs). However, the accuracy of this nants. In the case of purely electronic many-body states, the functional still needs to be assessed. cluster operator is defined as Coupled cluster theory is one of the most successful T ¼ T1 þ T2 þ þT ; ð2Þ methods for treating electron correlation in both ground and Ne excited states of molecular systems [36,37]. Nowadays, it is where N is the number of electrons. Each term corre- routinely applied to compelling chemical problems due to e sponds to excitations (single, double, triple, and so on), i.e., major advances in computational resources and several decades of algorithmic developments. Coupled cluster X methods are available in several programs; we use the T1 ¼ taiEai; ð3Þ electronic structure program eT [38] for our developments. ai In this paper, we extend coupled cluster theory to treat 1 X strongly interacting electron-photon systems in a non- T2 ¼ 2 taibjEaiEbj; ð4Þ relativistic QED framework. We refer to the resulting aibj method as quantum electrodynamics coupled cluster † † † (QED-CC) theory. To the best of our knowledge, this where Epq ¼ apαaqα þ apβaqβ [here, a and a denote theory is the first coupled cluster formalism that incorpo- fermionic operators, and (α, β) denote spin projections] rates many-body electron-photon operators for an ab initio are singlet one-electron operators and the parameters tai Hamiltonian. Recently, a different coupled cluster formal- and taibj are called cluster amplitudes. Furthermore, we let ism was proposed by Mordovina et al. [39]. Their study indices ði; j; k; lÞ and ða; b; c; dÞ label occupied and virtual was limited to model Hamiltonians, and they used state- HF orbitals, respectively. General orbitals are labeled transfer operators instead of many-body operators to ðp; q; r; sÞ. The cluster operator can be expressed as describe the photonic part of the wave function. We should X also mention that studies using other electronic structure T ¼ tμτμ; ð5Þ methods and model or semiempirical Hamiltonians have μ been presented by other authors [40–42]. In addition to presenting the complete formulation and where the excitation operators τμ generate an orthonormal implementation of QED-CC, we consider some interesting set of excited configurations: applications in photochemistry. In particular, we demon- strate how light-induced charge transfer in small dye jμi¼τμjHFi: ð6Þ
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Together with jHFi, these configurations define a subspace not change with the total system size [43]. The excitation of the Hilbert space in which the Schrödinger equation is energies are the eigenvalues of the Jacobian matrix solved. Inserting the coupled cluster wave function in ∂Ω Eq. (1) into the time-independent Schrödinger equation, μ ¯ Aμν ¼ ¼hμj½He; τν jHFi: ð13Þ we obtain ∂tν
HejCCi¼jCCiECC; ð7Þ Since this matrix is non-Hermitian, special attention is required at electronic degeneracies; at such points, the matrix where He is the electronic Born-Oppenheimer Hamiltonian can become defective or nondiagonalizable. Therefore, defects are expected close to conical intersections, as dis- X X 1 cussed in more detail in Sec. III C. H h E g e h : e ¼ pq pq þ 2 pqrs pqrs þ nuc ð8Þ In coupled cluster theory, there are two prevailing pq pqrs approaches to electronic excited states. One is coupled The quantities h and g are one- and two-electron cluster response theory (CCRT), which is based on a time- pq pqrs dependent formalism [43]. This theory provides both size- integrals, respectively; for convenience, we have intro- extensive and size-intensive molecular properties, such as duced the operator excitation energies and transition moments. The other is X − δ † † based on a time-independent formalism and is referred to as epqrs ¼ EpqErs qrEps ¼ apσarτasτaqσ: ð9Þ equation-of-motion coupled cluster (EOM-CC) theory σ τ α β ; ¼ ; [44]. In EOM-CC theory, the excitation energies are the same as in CCRT. However, some molecular properties In coupled cluster theory, Eq. (7) is projected onto the set are not guaranteed to scale correctly with system size. For fjHFi; jμig. Consequently, the coupled cluster energy is instance, transition moments are not necessarily size given by intensive [45]. For the purpose of the present developments, which mainly relates to ground- and excited-state energies, E ¼hHFjH¯ jHFi; ð10Þ CC e the EOM-CC formalism is sufficient. In EOM-CC, the similarity transformed Hamiltonian is expressed in the and the cluster amplitudes are determined from the basis fjHFi; jμig, equations ¯ ¯ ν ¯ ¯ hHFjHejHFihHFjHej i Ωμ ¼hμjHejHFi¼0: ð11Þ H ¼ hμjH¯ jHFihμjH¯ jνi e e η Here, we have introduced the similarity transformed ECC ν : 14 Hamiltonian, ¼ 0 δ ð Þ Aμν þ μνECC ¯ − ¯ He ¼ expð TÞHe expðTÞ: ð12Þ The left and right eigenvectors of H define the excited-state ¯ vectors, hLkj and jRki, and the eigenvalues of H are the Coupled cluster theory is equivalent to exact diagonal- energies of the states. We have used Eqs. (10), (11), and — ization of the Hamiltonian in Eq. (8) also called full (13) in the last equality. To extend coupled cluster theory to — configuration interaction (FCI) when the cluster operator electron-photon systems, we need to introduce a new in Eq. (2) is untruncated and contains all possible excita- parametrization of the cluster operator. tions. When the excitation space is truncated, we obtain different levels of approximation and, needless to say, III. COUPLED CLUSTER THEORY FOR reduced computational cost. For example, T ¼ T1 defines ELECTRON-PHOTON SYSTEMS the coupled cluster singles model (CCS) and T ¼ T1 þ T2 the coupled cluster singles and doubles model (CCSD). To describe the interaction of the electromagnetic field Coupled cluster theory is manifestly size extensive (also in with atoms, molecules, and condensed matter systems, the its truncated forms), a property that ensures that the total low-energy limit of QED is usually sufficient [46,47].In energy of noninteracting subsystems is the sum of the particular, the nonrelativistic Pauli-Fierz Hamiltonian in the subsystem energies. This is unlike similar truncation in dipole approximation [1,46,48,49], configuration interaction theory, where extensivity errors X 1 ω † λ d 2 can become arbitrarily large for an increasing number of HPF ¼ He þ αbαbα þ 2 ð α · Þ subsystems. α rffiffiffiffiffiffi ! Another important feature of coupled cluster theory is ω − α λ d † the size intensivity of excitation energies: for noninteract- 2 ð α · Þðbα þ bαÞ ; ð15Þ ing subsystems, excitation energies in each subsystem do
041043-3 HAUGLAND, RONCA, KJØNSTAD, RUBIO, and KOCH PHYS. REV. X 10, 041043 (2020) is usually an accurate starting point for describing elec- cn, where n ¼ðn1;n2; …Þ, corresponds to the state with nα tronic systems in optical cavities. The Hamiltonian in photons in mode α. Starting from Eq. (18), and assuming Eq. (15) is expressed in Coulomb and length gauge that jRi is normalized, the energy [46–48]. We use the Born-Oppenheimer approximation and keep the nuclear positions fixed. The second term in the EQED−HF ¼hRjHjRið20Þ Hamiltonian is the purely photonic part, represented by a sum of harmonic oscillators, one for each frequency. We is minimized with respect to Hartree-Fock orbitals and † have neglected the zero-point energies. The operators bα photon coefficients cn. Note that QED-HF can be consid- and bα are bosonic creation and annihilation operators, ered a special case of QEDFT with an appropriately chosen respectively. The third term is the dipole self-energy term, exchange-correlation functional. which ensures that the Hamiltonian is bounded from below For a given Hartree-Fock state, the energy can be [49] and independent of origin. The last term, the bilinear minimized with respect to the photon coefficients. This coupling, couples the electronic and photonic degrees of can be achieved by diagonalizing the photonic freedom. In the length gauge, the light-matter coupling is Hamiltonian, via the dipole operator X 1 † 2 X ω α λ d d hHPFi¼EHF þ αb bα þ 2 hð α · Þ i d ¼ d E ; d ¼ pjd þ nuc jq ; ð16Þ α pq pq pq e N rffiffiffiffiffiffi pq e ω − α λ d † 2 ð α · h iÞðbα þ bαÞ ; ð21Þ which consists of an electronic and a constant nuclear contribution, d and d . The elements d denotes one- e nuc pq where the mean value is with respect to jHFi. This electron dipole integrals. The coupling is described through e Hamiltonian can be diagonalized by the unitary coher- the transversal polarization vector multiplied by the ent-state transformation [50] coupling strength λα: Y sffiffiffiffiffiffiffiffiffiffiffiffiffiffi † UðzÞ¼ expðzαbα − zαbαÞð22Þ 1 α λα ¼ λαe; λα ¼ : ð17Þ ε0εrVα with a suitable choice of z. In the transformed basis, the photonic Hamiltonian is Here, ε0 and εr are the permittivities of the vacuum and the dielectric materials separating the cavity mirrors, respec- X † 1 2 tively [9]. The α-mode quantization volume is denoted Vα. ω λ d hHPFiz ¼ EHF þ αðbα þ zαÞðbα þ zαÞþ2hð α · Þ i The dipole approximation is usually valid when the α rffiffiffiffiffiffi wavelength of the electromagnetic field is significantly ω − α λ d † larger than the size of the electronic system. However, there 2 ð α · h iÞðbα þ bα þ zα þ zαÞ : ð23Þ are cases where the dipole approximation is not sufficiently accurate—for instance, when the size of the system is If we choose comparable to the cavity wavelength or when matter interacts with a circularly or elliptically polarized field. λα · hdi These aspects will be discussed in a forthcoming paper. ðz0Þα ¼ − pffiffiffiffiffiffiffiffi ; ð24Þ 2ωα
A. QED-HF method the Hamiltonian reduces to In order to formulate QED-CC, we need to define a X X 1 2 † suitable reference wave function. We formulate an exten- hH iz ¼ E þ hðλα · ðd − hdiÞÞ iþ ωαbαbα: PF 0 HF 2 sion of the Hartree-Fock method to QED, hereafter referred α α to as QED-HF. The noncorrelated electrons and photons in ð25Þ QED-HF are described by
⊗ The eigenvectors of this operator are the photon number jRi¼jHFi jPi; ð18Þ states, and the lowest eigenvalue corresponds to the X Y — † vacuum state. In the untransformed basis that is, for nα 0 jPi¼ ðbαÞ j icn; ð19Þ the Hamiltonian in Eq. (21)—the eigenstates are the n α generalized coherent states where j0i is the photon vacuum and cn are expansion † − coefficients for the photon number states. The coefficient jzα;nαi¼expðzαbα zαbαÞjnαi; ð26Þ
041043-4 COUPLED CLUSTER THEORY FOR MOLECULAR POLARITONS: … PHYS. REV. X 10, 041043 (2020) where zα is given in Eq. (24) and where the correction to the electronic energy can be understood as the variance in the dipole interacting with † ðbαÞnα the photon field. For an infinite number of modes, we jnαi¼ pffiffiffiffiffiffiffi j0ið27Þ nα! expect, as is standard in QED [52], to encounter diver- gencies due to the second term of Eq. (31). are the normalized photon number states for mode α. We should point out that the eigenvalues of F, normally Applying the unitary transformation to the original interpreted as orbital energies, are origin dependent for Pauli-Fierz Hamiltonian, Eq. (15),givesusthefinal charged systems. As a consequence, applying concepts that expression for the Hamiltonian in the coherent-state basis: depend on the orbital energies, like Koopmans’ theorem [53], will require a different choice of the occupied- X 1 F ω † λ d − d 2 occupied and virtual-virtual blocks of . For the same H ¼ He þ αbαbα þ 2 ð α · ð h iÞÞ F α reason, cannot be used as a zeroth-order Hamiltonian in rffiffiffiffiffiffi perturbation theories such as CC2 [54] and CC3 [55,56]. ω α † An origin-independent F can be obtained by performing an − ðλα · ðd − hdiÞÞðbα þ bαÞ : ð28Þ 2 appropriate unitary transformation [57] that mixes the electronic and photonic degrees of freedom. In this way, This Hamiltonian will be used in QED-CC. Note that the dressed electrons also have well-defined orbital ener- the operator is manifestly origin invariant, which is gies for charged systems. This case will be considered in a different from Eq. (15), where the invariance is obtained future publication. However, note that the origin depend- through a gauge transformation. This is also true for ence of the eigenvalues of F does not imply a loss of origin charged systems, where the dipole moment operator invariance in QED-HF (see the Appendix A). depends on the choice of origin. In the coherent-state basis, the bilinear coupling and self-energy terms depend on fluctuations of the dipole moment away from the mean B. QED-CC method value. We point out that the nuclear dipole gives no Extending the exponential parametrization in Eq. (1) to contributions to the Hamiltonian. However, note that QED requires that the cluster operator generates excitations the dependence on the nuclear dipole now resides in both in the purely photonic and in the electron-photon the wave function through the coherent-state transforma- coupling spaces [39]. The cluster operator can therefore be tion [see Eq. (24)]. partitioned as As we have shown, the QED-HF reference state is now given by T ¼ Te þ Tp þ Tint; ð32Þ
jRi¼jHFi ⊗ j0i: ð29Þ where Te is the standard cluster operator for the electrons, and Tp and Tint consist of photon and electron-photon The Hartree-Fock equations are solved using standard operators, respectively. techniques [37,51]. In every iteration, the Hartree- The purely photonic operator is defined as Fock orbitals are updated and used to evaluate z0 [see X X Y Eq. (24)]. The Fock matrix used in the optimization is n n † nα Tp ¼ Γ ¼ γ ðbαÞ : ð33Þ given by n n α X X 1 n e λ d λ d In this equation, γ are photon amplitudes. The form of the Fpq ¼ Fpq þ 2 ð α · paÞð α · aqÞ α photonic operator was chosen to expand the photonic part a X of the Hilbert space and to give commuting cluster − λ d λ d ð α · piÞð α · iqÞ ; ð30Þ operators (as in the electronic case). Since expðTpÞ does i not terminate, the parametrization is able to incorporate many-body effects within the limitation imposed by the where Fe is the standard closed-shell electronic Fock projection space. In contrast, Mordovina et al. [39] used a matrix [37] and the stationary condition is equivalent to nilpotent photonic operator that only enters linearly in the Fia ¼ 0. The QED-HF ground-state energy can now be written as expansion of the coupled cluster state. The excitations in the electron-photon interaction oper- 1 X ator T are defined as direct products of electronic and λ d − d 2 int EQED−HF ¼ EHF þ 2 hð α · ð h iÞÞ i photonic excitations. Thus, the operator can be expressed as α X X λ d 2 n n n ¼ EHF þ ð α · aiÞ ; ð31Þ T ¼ S þ S þ þS ; ð34Þ int 1 2 Ne α;ai n
041043-5 HAUGLAND, RONCA, KJØNSTAD, RUBIO, and KOCH PHYS. REV. X 10, 041043 (2020) where, for instance, The projection space used in Eq. (11) is defined by X Y the excitations included in the cluster operator. With the n n † nα S1 ¼ saiEai ðbαÞ ; ð35Þ notation ai α HF;n HF ⊗ n ; 42 X Y j i¼j i j i ð Þ n 1 n † S s E E bα nα : 36 2 ¼ 2 aibj ai bj ð Þ ð Þ μ μ ⊗ aibj α j ;ni¼j i jni; ð43Þ
γn n n the projection basis is The new cluster amplitudes, ;sai, saibj, etc., are param- eters that will be determined from a set of projection HF; 0 ; μ; 0 ; μ; 1 ; HF; 1 ; 44 equations. fj i j i j i j ig ð Þ Hierarchies of approximations are formulated by trun- where HF; 0 R and μ is restricted to single and double cating the cluster operator and the associated projection j i¼j i excitations. The derivation of the amplitude equations space. Here, we implement CCSD with the special case of a follows the same procedure as in the electronic case, single photon mode, where we only include one photon in and the truncation of the equations is determined by the the cluster operator. The coupled cluster wave function is projection space and the commutator expansion of the then given by similarity transformed Hamiltonian. Explicit formulas are presented in Appendix B. jCCi¼expðTÞjRi; ð37Þ The formation of polaritons usually appears in the optical spectrum as a Rabi splitting (proportional to the coupling where jRi is the QED-HF reference given in Eq. (29), and strength λ) of the electronic states due to the coupling to the the cluster operator is quantum field. Hence, we must also describe the excited 1 1 1 states of the coupled system. In coupled cluster theory, T T1 T2 Γ S S : 38 ¼ þ þ þ 1 þ 2 ð Þ electronic excitation energies may be determined using EOM-CC theory, as described in Sec. II. The projection Electronic excitations are described at the singles and space in QED-CCSD is extended, relative to the electronic doubles level, and the photon mode is coupled to these case, giving rise to additional blocks in the Jacobian matrix: excitations through S1 and S1, respectively. The electronic 1 2 0 1 operators T1 and T2 are given in Eqs. (3) and (4). The A A A photon and electron-photon operators are defined as B e;e e;ep e;p C B A A A C A ¼ @ ep;e ep;ep ep;p A: ð45Þ Γ1 γ † ¼ b ; ð39Þ Ap;e Ap;ep Ap;p X 1 † S1 ¼ saiEaib ; ð40Þ In addition to the electronic Jacobian Ae;e, there are ai coupling blocks between electronic (e), electronic-photonic ep p 1 X ( ), and photonic ( ) configurations; see Eqs. (38) and 1 † (44). Explicit formulas for the sub-blocks of the Jacobian, S2 ¼ 2 saibjEaiEbjb : ð41Þ aibj ¯ † m Aμn;νm ¼hμ;nj½H; τνðb Þ jRi; ð46Þ This model is referred to as QED-CCSD-1, with one photon mode, where “1” refers to the photonic excitation order. are presented in Appendix C, along with the corresponding More involved terminology will be required to describe the sub-blocks of the η vector. The ground- and excited-state full hierarchy. In the notation used by Mordovina et al. [39], QED-CCSD equations are solved using standard methods this corresponds to a QED-CC-SD-S-DT model. However, [37,58,59]. because of the difference in photonic excitation operators To ensure a balanced description of the electron-photon and the coherent-state basis, the model described here is not states, it is important to use all product operators between directly comparable to the one in Ref. [39]. Even if only one the included electronic and photonic excitations in the † n photon creation operator is included, the exponential will cluster operator, τμðb Þ . For instance, suppose that the partially incorporate two photon contributions into the wave coupling is zero, and consider the two states je; 1i and 0 ω function. Thus, we expect the convergence with respect to je; i.As cav tends to zero, these states will only become 1 photons to be faster than CI-like diagonalization in photon degenerate if Γ is coupled to both T1 and T2. Neglecting 1 number states. Furthermore, notice that the generalized S2 would lead to unphysical results. coherent-state basis incorporates higher photonic excita- Additional properties of the electron-photon system can tions as well. be calculated from the left and right eigenvectors of H¯ .
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For instance, we can evaluate the ground- and excited-state In all calculations, we have chosen a coupling strength electronic EOM density matrices as λ ¼ 0.05. The coupling strength is usually calibrated using the Jaynes-Cummings model, where λ ¼ 0.05 corresponds k Dpq ¼hLkj expð−TÞEpq expðTÞjRki: ð47Þ to a value well within the strong-coupling regime. Although this coupling leads to relatively large Rabi splittings For a description of other molecular properties, we refer the (∼1 eV), comparable but smaller Rabi splittings have also reader to Refs. [43,44]. been observed experimentally (∼0.3 eV) [21,69].
C. Some technical aspects A. Diatomic molecules The Pauli-Fierz Hamiltonian in Eq. (15) is defined on the Interesting QED effects can be observed for small diatomic molecules, such as H2 and HF. We also use these direct product Hilbert space H ¼ He ⊗ Hp. In the trun- molecules to benchmark the coupled cluster model against cated description, where He and Hp are finite dimen- sional, expðTÞjRi has an effectively finite expansion due to the more accurate QED-FCI approach. The comparison the finite projection basis. For instance, in QED-CCSD-1, shows an excellent agreement; see Appendix E for a the terms in expðTÞjRi that give nonzero contributions are detailed discussion. For the calculations here, we use a up to quadruple electronic excitations and double photonic Gaussian basis set, in particular, cc-pVDZ [70]. excitations. However, in the limit of an infinite-dimensional The potential energy curves for the ground and excited states of these systems are shown in Fig. 2. The conical Hp, special care might be required to define the exponen- tial operator [60]. intersections and avoided crossings in the UV range define Because of the non-Hermiticity of coupled cluster the photochemical properties of these molecules. An theory, it is known to give nonphysical complex energies optical cavity set in resonance with one of the excited close to conical intersections between excited states of the states can completely restructure the excitation landscape same symmetry [61–63]. The same issue can arise in QED- and redefine the photochemistry of the system. The color map in Fig. 2 indicates the electronic or photonic con- CC and is mentioned by Mordovina et al. [39]. The tributions to the states. The electronic states are highlighted problem can be traced to defects in the Jacobian matrix in blue, while the photonic states are transparent white. For for a truncated projection basis. In the untruncated case, the more details, see Appendix D. states satisfy generalized orthogonality relations, ensuring a Considering first H2 set in resonance with the first singlet correct description of conical intersections [63]. To obtain a excited state (at the ground-state equilibrium geometry), we physically correct description with a truncated excitation are able to induce significant changes of the potential space, one can enforce the orthogonality conditions energy curves. Here, we focus on the first Rabi splitting, † where, in particular, the upper (UP) polariton is more bound hR j expðT ÞP expðTÞjR i¼δ ; ð48Þ k l kl than the bare electronic state. Hence, it should be possible to trap the molecule in the UP state. In contrast, the bare where P is some projection operator [64,65]. This electronic state has, to a larger extent, a dissociative approach, unlike a posteriori corrections [39,62], can be character. extended to analytical energy gradients and nonadiabatic Similar conclusions can be drawn for the HF molecule. coupling elements using well-established Lagrangian tech- Differently from H2, it has a permanent dipole moment. niques [66,67]. In passing, we note that no defects or However, this seems to have minor effects on the general complex eigenvalues were encountered in the results landscape, which is consistent with the fact that the total reported in this work. energy depends only on the fluctuations in the dipole moment (see Sec. III A). IV. MOLECULAR POLARITONS From the above analysis, we see that the application of a In this section, the QED-CCSD-1 model is used to quantum field inside an optical cavity can be used to fine- investigate cavity-induced effects on the chemistry of tune the excited-state properties of molecules. This opens molecules. All calculations are performed using a develop- the way towards new decay paths, the possibility of ment version of eT [38]. Molecular geometries are provided trapping systems in excited dark polaritons, and many in the Supplemental Material [68]. A single cavity mode is other effects redesigning the molecular photochemistry. used throughout; this approximation typically breaks down ω B. Charge transfer molecules for small values of cav, when several replica states overlap energetically with electronic states. This can occur in large Recently, some research groups have suggested that ω cavities, where cav may be smaller than the electronic quantum fields can have a significant impact on the charge spectral range. In the following calculations, we assume transfer [71,72] and energy transfer [71,73] properties in that the single-mode approximation is valid for the inves- molecules. These preliminary studies are based on model tigated properties. Hamiltonians; thus, only qualitative interpretations of the
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FIG. 2. Potential energy curves calculated with (blue solid lines) and without (red dashed lines) an optical cavity for (a) H2 and (b) HF. The polarization is along the main axis of the molecules, and the field is in resonance with the first bright excitation of the system at its equilibrium geometry. The coupling strength is set to λ ¼ 0.05 in both cases. The blue color map indicates the electronic or photonic character of the states. phenomena are provided. Here, we present a quantitative density difference with and without the cavity, Δρ ¼ ρcav − ρno cav analysis of cavity-induced effects on a charge transfer gs gs . The cavity is set in resonance with the process. most intense, low-lying, charge transfer excited state, ω 4 84 We investigate p-nitroaniline (PNA), a simple amine often cav ¼ . eV. Although the charge displacement is used as a prototype dye for solar energy applications— small, a clear cavity-induced charge reorganization is for instance, in dye-sensitized solar cells [74,75]. This observed in the ground state. Specifically, we have a charge − molecule has an intense, low-lying, charge transfer transfer of about 0.005 e going from the acceptor (NO2)to excitation (at about 3–4 eV) that can potentially be used the nitrogen atom of the donor (NH2) group, reducing the to inject charge into a semiconductor and produce a current. Developing strategies to control the charge transfer process is of fundamental importance to increase photovoltaic efficiencies. In our calculations on PNA, we have used the cc-pVDZ basis and oriented the polarization (λ ¼ 0.05) along the principal axis of the molecule. The molecular structure was optimized with DFT/B3LYP using the 6-31+G basis set [76]. Initially, we investigate the effects of the cavity on the electronic ground state by analyzing the electron density. This can conveniently be carried out using the charge displacement analysis [77–79]. The charge displacement function is defined as Z Z Z þ∞ þ∞ z ΔqðzÞ¼ Δρðx; y; z0Þdxdydz0; ð49Þ −∞ −∞ −∞ where we integrate over the electron density difference Δρ. This function measures the amount of charge that has been Δ moved along the z coordinate. In particular, if qðzÞ is FIG. 3. QED-CCSD-1 ground-state density difference induced positive, charge is transferred from right to left, and if λ 0 05 ω 4 84 in PNA by an optical cavity ( ¼ . and cav ¼ . eV) and negative, charge is transferred in the opposite direction. the corresponding charge displacement analysis. The blue and In Fig. 3, we show the QED-CCSD-1 charge displace- red regions represent charge accumulations and depletions. The ment function and isosurface for the ground-state electron isosurface value is 5 × 10−5 e− a:u:3.
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FIG. 4. QED-CCSD-1 dispersion with respect to cavity fre- FIG. 5. QED-CCSD-1 excited-state charge displacement analy- ω sis of PNA with and without an optical cavity (λ ¼ 0.05 and quency cav of the excitation energies in PNA. The blue color ω 4 84 map indicates the electronic or photonic character of the states. cav ¼ . eV). The charge displacement functions with the cavity for lower (LP, blue) and upper (UP, red) polaritons are shown. In green, we show the function for the charge transfer state without the cavity. The dashed black line represents the sum magnitude of the dipole moment from 6.87 D in a vacuum of the curves for LP and UP. to 6.77 D in the cavity. This counterintuitive effect is in agreement with previous QEDFT/OEP studies from Flick et al. [35]. The cavity field accumulates more charge in the charge transfer character. We note that the cavity field high-density regions. In this way, the variance of the dipole slightly reduces the total charge transfer (see the black operator is reduced [see Eq. (31)]. dashed line in Fig. 5) because the charge transfer state also In the excited states of PNA, the cavity-induced effects contributes to the other excited states, not just the are more evident. In Fig. 4, we show the dispersion of the polaritons. low-lying excitation energies of PNA with respect to In Fig. 6, we show how fine-tuning the cavity frequency ω the cavity frequency cav. Because of the large transition can be used to change the degree of charge transfer in the dipole moment for the charge transfer excitation, a large lower and upper polaritons. This provides another pos- Rabi splitting is observed when the cavity is resonant with sibility for charge transfer control, with the potential for this state. As discussed in the previous section, the cavity technological applications. can induce significant changes in the excited states. Specifically, we have a state inversion between the lower C. Photochemical processes polariton and the first two excited states. This effect in PNA and other dye molecules could be important for photo- We now turn our attention to photochemical processes voltaic applications, where a proper alignment of charge and the possibility of changing the ground-state potential transfer states with the states in a semiconductor is essential energy surface using an optical cavity. For this purpose, we to optimize solar-cell efficiency. choose the pyrrole molecule that exhibits conical inter- In Fig. 5, we show a charge displacement analysis of the sections between the ground state and two low-lying charge transfer state, with a resonant quantum field. Here, excited states. A detailed analysis of these conical inter- we use the density difference between the ground and sections, and of the involved relaxation mechanism, can be Δρ ρ − ρ excited states, ¼ es gs, with and without the cavity. found in Refs. [80,81].InC2v symmetry, the ground state is 1 1 1 Both the lower and upper polaritons have charge transfer A1, while the first two excited states are A2 and B1. The character and are shown separately. equilibrium geometry is calculated with CCSD and a Differently from the ground state, a sizable charge cc-pVDZ basis set. transfer of almost 0.4 e− is moved from the donor to the We investigate the behavior of the potential energy acceptor group. The cavity divides the total charge transfer curves when the NH bond distance R is varied, preserving between the polaritons; thus, a compromise must be made the C2v symmetry (Fig. 7). The polarization of the cavity is 1 1 1 e pffiffi pffiffi pffiffi between energetically aligning states and maintaining the chosen as ¼ð 3 ; 3 ; 3Þ, such that the point group
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FIG. 8. Potential energy curves in QED-CCSD-1 for pyrrole FIG. 6. QED-CCSD-1 excited-state charge displacement analy- calculated without (a,c) and with (b,d) the cavity. Panels (c) and sis of PNA with and without an optical cavity (λ ¼ 0.05 and (d) are zoom-in views of (a) and (b), respectively, in the conical ωres ¼ 4.84 eV). The charge displacement functions with the cav intersections’ region. The energies are relative to equilibrium cavity for lower (LP, blue) and upper (UP, red) polaritons are energy. In panels (b) and (d), the blue color map indicates the shown in panels (a)–(d). In green, we show the function for the electronic or photonic character of the states. charge transfer state without the cavity. The dashed black line represents the sum of the curves for LP and UP. 1 B1 states close to the conical intersection. The correct ordering can be recovered by including triple excitations in symmetry of the Hamiltonian reduces to C1. The the model, as we have confirmed with CC3 [56] calcu- coupling strength is set to λ ¼ 0.05, and the cavity 1 lations (see Appendix F). Since the ordering of the states frequency is set in resonance with the B1 state at R ¼ does not change the conclusions, we perform the analysis at 2 0 ω 1 06 . Å( cav ¼ . eV). the QED-CCSD-1 level. In Fig. 8, we show the potential energy curves along We first observe the lifting of the degeneracy as shown in the coordinate R with and without the cavity. Without the Fig. 8(d). In the coupled system, now in C1 symmetry, all cavity (λ ¼ 0), the CCSD model mostly reproduces the states can interact, as they have the same symmetry. This accurate potential energy curves calculated in Refs. [80,81]. unequivocally demonstrates that the cavity can also sig- 1 The main difference is an inverted ordering of the A2 and nificantly impact the potential energy surface of the electronic ground state, which is the first time this phe- nomenon has been demonstrated in a molecule using an ab initio Hamiltonian. Previously, a similar observation was made with a model Hamiltonian for graphene [6]. Interestingly, a coupling strength (λ ¼ 0.05) that pro- duces a relatively small Rabi splitting of 0.03 eV around 2 Å is able to open a gap almost twice this size, 0.05 eV, at the conical intersection. We can rationalize this observation in the following way. The Rabi splitting is mainly due to the bilinear term in the Hamiltonian, whereas the lifting of the degeneracy is mainly due to the self-energy term. In Appendix F, we show that the symmetry breaking is insensitive to basis sets and cavity frequencies. A consequence of the above observation is the possibility FIG. 7. Orientation of pyrrole with indication of the to change relaxation pathways in chemical reactions. The coordinates. electron-photon coupling may cause intersection seams to
041043-10 COUPLED CLUSTER THEORY FOR MOLECULAR POLARITONS: … PHYS. REV. X 10, 041043 (2020) move or vanish. Note that the avoided crossing in ACKNOWLEDGMENTS Fig. 8(d) does not exclude the possibility that the seam We acknowledge Laura Grazioli, Rosario Riso, and has been moved to a distorted molecular geometry. In any Christian Schäfer for insightful discussions. We acknowl- case, the changes to the energy surfaces may, for example, edge computing resources through UNINETT Sigma2 lead to reduced relaxation through the conical intersection (National Infrastructure for High Performance Computing to the ground state, such that the relaxation can be and Data Storage in Norway) through Project No. NN2962k. dominated by radiative processes (which are generally We acknowledge funding from the Marie Skłodowska-Curie slower). Note also that the majority of molecular orienta- European Training Network COSINE (Computational tions allow for this type of symmetry breaking, which Spectroscopy in Natural Sciences and Engineering) should make the gap opening experimentally observable Grant Agreement No. 765739, and the Research Council in standard temperature conditions where the molecules of Norway through FRINATEK Projects No. 263110 and can rotate freely. No. 275506. A. R. was supported by the European Research Council (ERC-2015-AdG694097), the Cluster V. CONCLUDING REMARKS of Excellence Advanced Imaging of Matter (AIM), and We have developed a coupled cluster theory that Grupos Consolidados Grants No. IT1249-19 and explicitly incorporates quantized electromagnetic fields, No. SFB925. The Flatiron Institute is a division of the denoted as QED-CC. This nonperturbative theory can Simons Foundation. describe molecular photochemistry inside an optical T. S. H. and E. R. contributed equally to this work. cavity. The QED-CC model is a natural extension of the well-established coupled cluster model used in elec- tronic structure theory. The method provides a highly APPENDIX A: DERIVATION OF accurate description of electron-electron and electron- QED-HF THEORY photon correlation, at least in regions where the electronic Consider the Pauli-Fierz Hamiltonian in the coherent- ground state is dominated by a single determinant. These state basis, Eq. (28), with a single photon mode, correlations are not accounted for in commonly used 1 model Hamiltonians and mean-field methods. The accu- ω † λ d − d 2 H ¼ He þ b b þ 2 ð · ð h iÞÞ racy is demonstrated by comparison with exact diagonal- rffiffiffiffi ization (within an orbital basis) in a truncated photon ω − λ d − d † space (QED-FCI). Unlike QED-FCI, the QED-CC hier- 2ð · ð h iÞÞðb þ bÞ: ðA1Þ archy is computationally feasible for larger molecules. Extension of QED-CC to approximately include the When averaging over the photon vacuum state j0i and environment, such as a solvent, is a natural next step in using Eq. (9), we obtain further developments. However, one must then carefully X X consider how the quantum field should be incorporated 1 hHi0 ¼ h þ ðλ · d Þðλ · d Þ into the environment. pq 2 pr rq pq r Initially, we investigated the restructuring of potential energy curves in diatomic molecules. In particular, we − ðλ · hdiÞðλ · dpqÞ Epq found that the interaction with the cavity creates polaritons that are more bound than the corresponding bare electronic 1 X λ d λ d excited states. Clearly, polaritons have crucial implications þ 2 ðgpqrs þð · pqÞð · rsÞÞepqrs on the photochemistry. For instance, they can alter the pqrs 1 relaxation pathways and trap molecules in dark excited 2 þ ðλ · hdiÞ þ h : ðA2Þ states. A further study of these phenomena would be very 2 nuc interesting. This operator has the same form as the electronic Cavity-induced effects on charge transfer processes were HamiltonianinEq.(8), with modified integrals and con- also investigated quantitatively for PNA. We explained how the cavity restructures the charge inside the molecule and stants. These modified integrals can be inserted directly into the expression for the inactive electronic Fock matrix, how these effects could be applied in photovoltaics. X Finally, we demonstrated how the cavity field can be Fe ¼ h þ ð2g − g Þ; ðA3Þ used to manipulate conical intersections in molecules. We pq pq pqii piiq i showed that the quantum field is able to lift degeneracies between ground and excited states. This analysis suggests and we obtain Eq. (30) for a single mode. This procedure is that new experimental strategies can be developed to easily generalized to the multimode case. manipulate ultrafast molecular relaxation mechanisms We now consider the origin invariance in QED-HF by through conical intersections. shifting the dipole by a constant, d → d þ Δd. For neutral
041043-11 HAUGLAND, RONCA, KJØNSTAD, RUBIO, and KOCH PHYS. REV. X 10, 041043 (2020) molecules, the dipole moment operator is origin invariant, energy is also invariant. On the other hand, the inactive Δd ¼ 0, but for charged molecules, this is not the case. Fock matrix in Eq. (30) is not invariant. As seen from the Nevertheless, since the Hamiltonian (A1) is invariant, the shifted Fock matrix,
! F F F − ðλ · ΔdÞðλ · d Þ − 1 ðλ · ΔdÞ2δ F ij ib → ij ij 2 ij ib 1 2 ; ðA4Þ Faj Fab Faj Fab þðλ · ΔdÞðλ · dabÞþ2 ðλ · ΔdÞ δab
the occupied-virtual blocks of the Fock matrix, Fib and Faj, APPENDIX C: QED-CCSD-1 EXCITED-STATE are unchanged, whereas the purely occupied and virtual EQUATIONS blocks, F and F , are dependent on the origin. Explicitly, ij ab The expressions for the Jacobian in Eq. (45) are derived for the occupied-virtual block of the Fock matrix, we have using the commutator expansion. We obtain X 1 ¯ 1 → δ λ Δd λ d hμ; 0j½H; τν jRi¼hμ; 0j½H þ½H;T2 þ½H;S1 Fjb Fjb þ 2 abð · Þð · jaÞ a 1 þ½H;S þ½H; Γ1 ; τν jRi; ðC1Þ 1 X 2 − δ ðλ · ΔdÞðλ · d Þ¼F : ðA5Þ 2 ij ib jb μ 0 ¯ † μ 0 H H † i h ; j½H; b jRi¼h ; j½ þ½ ;T2 ;b jRi; ðC2Þ
¯ † † hμ; 0j½H; τνb jRi¼hμ; 0j½H þ½H;T2 ; τνb jRi; ðC3Þ APPENDIX B: QED-CCSD-1 GROUND-STATE ¯ 1 EQUATIONS hHF; 1j½H; τν jRi¼hHF; 1j½H þ½H;S1 H 1 H Γ1 τ Using the QED-CCSD-1 cluster operator defined in þ½ ;S2 þ½ ; ; ν jRi; ðC4Þ Eq. (38), the coupled cluster ground-state equations [see ¯ † 1 † Eqs. (10) and (11)] take the form hHF; 1j½H; b jRi¼hHF; 1j½H þ½H;S1 ;b jRi; ðC5Þ
¯ † † ¯ 1 1 hHF; 1j½H; τνb jRi¼hHF; 1j½H; τνb jRi; ðC6Þ hRjHjRi¼hRjH þ½H;T2 þ½H;S1 þ½H; Γ jRi; ðB1Þ ¯ 1 hμ; 1j½H; τν jRi¼hμ; 1j½H þ½H;T2 þ½H;S 1 1 ¯ 1 1 1 1 μ; 0 H R μ; 0 H H;T2 H;T2 ;T2 H H Γ H Γ h j j i¼h j þ½ þ2 ½½ þ½ ;S2 þ½ ; þ½½ ; ;S1 1 1 1 H Γ1 1 τ þ½H;S1 þ½½H;S1 ;T2 þ½H;S2 þ½½ ; ;S2 ; ν jRi; ðC7Þ 1 1 þ½H; Γ þ½½H; Γ ;T2 jRi; ðB2Þ ¯ † 1 1 † hμ; 1j½H; b jRi¼hμ; 1j½½H;S1 þ½H;S2 ;b jRi; ðC8Þ
¯ 1 1 μ 1 ¯ τ † μ 1 H H H 1 hμ; 1jHjRi¼hμ; 1jH þ½H;T2 þ½½H;S2 ;T2 þ½H;S1 h ; j½H; νb jRi¼h ; j½ þ½ ;T2 þ½ ;S1 1 H 1 H Γ1 τ † H 1 H 1 1 H 1 1 þ½ ;S2 þ½ ; ; νb jRi: ðC9Þ þ½ ;S2 þ2 ½½ ;S1 ;S1 þ½½ ;S1 ;S2 1 1 1 1 For completeness, we also give the expressions for the ην þ½H; Γ þ½½H; Γ ;T2 þ½½H; Γ ;S1 block of Eq. (14): 1 1 1 þ½½H; Γ ;S2 þ½½H;S1 ;T2 jRi; ðB3Þ ¯ 1 hRjHjν; 0i¼hRjH þ½H;T2 þ½H;S1 jν; 1i; ðC10Þ ¯ 1 1 1 hHF; 1jHjRi¼hHF; 1jH þ½H; Γ þ½H;S1 þ½H;S2 jRi: hRjH¯ jHF; 1i¼hRjHjHF; 1i; ðC11Þ ðB4Þ hRjH¯ jν; 1i¼hRjHjν; 1i: ðC12Þ Here, we have introduced the notation
H ¼ expð−T1ÞH expðT1Þ: ðB5Þ APPENDIX D: ELECTRONIC WEIGHTS e The ground-state electronic weights wgs in QED-CCSD-1 This operator can be expressed as H with transformed one- are calculated by projecting the ground-state wave function and two-electron integrals [37]. jCCi on the elementary electronic basis, that is,
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FIG. 9. Comparison between QED-CCSD-1 (red solid lines) and QED-FCI (contour map in the background) energy dispersion plots for (a) H2 and (b) HF. The intensity of the red lines indicates the electronic or photonic contribution to the states, λ ¼ 0.05 and η ¼ 0.01 a:u:. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hCCjP jCCi Hamiltonian in Eq. (15) with one photonic excitation in we ¼ el : ðD1Þ order to obtain a consistent comparison. gs hCCjPjCCi We use a 3-21G basis [82] for H2 and a STO-3G [83] basis for HF, with internuclear distances R ¼ 1.0 Å and The electronic and total projection operators are defined H2 0 917 λ 0 05 here as RHF ¼ . Å. In both cases, the coupling value ¼ . X is used. P R R μ; 0 μ; 0 ; D2 In Fig. 9, we show the energy dispersion with respect to el ¼j ih jþ j ih j ð Þ ω μ the cavity frequency cav. In this figure, we use the QED- FCI linear response spectral function, P ¼jRihRjþjHF; 1ihHF; 1j X P P þ ðjμ; 0ihμ; 0jþjμ; 1ihμ; 1jÞ: ðD3Þ X † † 1 hΨ0j a a jΨ ihΨ j a a jΨ0i μ ω − ij i j n n ij i j Að Þ¼ π Im ω − − η ; n≠0 ðEn E0Þþi Excited-state weights we are calculated in an approximate es ðE1Þ way using the norm of the electronic part of the excitation vector R, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP Ψ 2 where j ni are the eigenfunctions of the QED Hamiltonian. μðRμ;0Þ we ¼ : ðD4Þ In this case, we calculate the density-density spectral es jjRjj2 function instead of the more appropriate optical spectrum (with transition dipole moments) in order to compare the In principle, an equation equivalent to Eq. (D1) should also states independently from the selection rules. Coupled be used in this case, substituting jCCi with jRi. Considering cluster results are displayed in Fig. 9 as red lines with that we are only interested in a qualitative estimate of the electronic weights. weights, the approximate form of Eq. (D4) is adequate. For both molecules, we observe excellent agreement. The only noticeable difference is the absence of a few electronically excited states in the FCI spectrum. These APPENDIX E: COMPARISON OF states have large double-excitation character and, thus, QED-CCSD-1 AND QED-FCI small contributions to the spectral function. Notice here that Here, we compare QED-CCSD-1 and QED-FCI. For the QED-CCSD-1 is also very accurate for hydrogen fluoride, latter method, we perform an exact diagonalization of the where the electronic structure is not exact in CCSD.
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FIG. 10. Potential energy curves in QED-CCSD-1 for pyrrole FIG. 12. Potential energy curves in QED-CCSD-1 for pyrrole ω calculated with the cavity for λ ¼ 0.05 using the aug-cc-pVDZ with cavities at different frequencies cav, coupling strength basis set. The blue color map indicates the electronic or photonic λ ¼ 0.05, and with cc-pVDZ. The energies are relative to character of the states. equilibrium energy.
APPENDIX F: ADDITIONAL RESULTS FOR electromagnetic fields may require more diffuse basis PYRROLE functions. In Fig. 10, potential energy curves calculated using an aug-cc-pVDZ [70] basis are shown. The inclusion We also provide some additional results that are of diffuse functions does not change the general qualitative important to validate the accuracy of QED-CCSD-1 for picture described in Sec. IV C, confirming the accuracy pyrrole. First, we address the basis set appropriateness, as of our predictions. In particular, the position of the intersections is nearly unaffected by the size of the basis, and the qualitative shape of the potential energy curves is unchanged. In Sec. IV C, we noted that CCSD gives an incorrect 1 1 ordering of the A2 and B1 excited states. This can be rectified by including triple excitations in the electronic treatment, as shown with CC3 in Fig. 11. The opening of the conical intersection is quite robust ω with respect to changes in the cavity frequency, cav,as seen in Fig. 12. This supports the claim that the dipole self- energy term is mainly responsible for lifting the degen- eracy. The position of the Rabi splitting is, as expected, highly sensitive to the cavity frequency.
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