####  6.1.2 On the definition of the photonic character of excited states

From the QED-Casida equations, it is possible to define a relative
electronic/photonic contribution to the polaritonic excitation.95, 99 Given a
normalized eigenvector of Eq. (235), we define the "photonic character"
$\chi_{n}$ as the sum over the cavity modes $\alpha$ of the squares of the
photonic response parameters $\gamma_{\alpha}$95, 99

$\chi_{n}=\sum_{\alpha}|\gamma_{\alpha}|^{2}.$ (243)

The electronic character of the excitation $\rho_{n}$ is defined as the sum of
the orbital rotation $\kappa_{ai}$, such that $\chi_{n}+\rho_{n}=1$

$\rho_{n}=\sum_{ai}|k_{ai}|^{2}=1-\chi.$ (244)

However, such a definition of the excitation character is not
straightforwardly connected to photon number nor to electronic excitations in
a molecular sense. In fact, for the QED-HF equations, we used the dipole
Hamiltonian in the length form. As a consequence of this choice, as pointed
out at the end of section (4.2), the photon creation and annihilation
operators cannot be simply identified with $b^{\dagger}$ and $b$. These
operators refer instead to the displacement field and therefore include matter
contributions from the polarization in Eq. (38). Moreover, as the coupling
strength increases, the ground state might gain significant contributions from
photon states. The "photon character" of the excitation becomes, therefore, a
slippery concept that requires careful examination also of the ground state. A
clarification of this photon character ambiguity could be obtained by
implementing the quadratic response equations, from which we obtain the
expectation values of observables for excited states. Accordingly, we may
compute the following expectation values

$\displaystyle\braket{n}{b_{\alpha}^{\dagger}b_{\alpha}}{n}$ (245)
$\displaystyle\braket{n}{\big{(}b_{\alpha}^{\dagger}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\bm{d}\big{)}\big{(}b_{\alpha}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\bm{d}\big{)}}{n}.$
(246)

In the length representation, Eq. (245) refers to the mean occupation number
of the displacement field number states, while Eq. (246) to the mean electric
field number states. We can then compare these calculations to the QED-HF
ground state expectation values

$\displaystyle\braket{\mathrm{R}}{b_{\alpha}^{\dagger}b_{\alpha}}{\mathrm{R}}=\frac{1}{2\omega_{\alpha}}\big{(}\bm{\lambda}_{\alpha}\cdot\braket{\bm{d}}_{\text{QED-
HF}}\big{)}^{2}$ (247)
$\displaystyle\braket{\mathrm{R}}{\big{(}b_{\alpha}^{\dagger}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\bm{d}\big{)}\big{(}b_{\alpha}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\bm{d}\big{)}}{\mathrm{R}}=0$
(248)

to obtain an estimate of the cavity-field contribution to the excitations.

Employing these less straightforward but more precise definitions might
clarify some results presented in the literature. When using the definition in
Eq. (243), Yang et al.95 found that the photon contribution of the lower
polariton increases with the coupling strength, while the opposite is found
for the upper polariton.95 At the same time, the intensity of the lower
(upper) polariton increases (decreases) with the coupling strength, which
seems in contradiction with the zero oscillator strength associated with the
photon states. It is argued that this apparent discrepancy results from the
intrusion of higher-energy electronic states, which strengthen the intensity
of the lower polariton, overcompensating for the increased photon character of
the excitation.95 Nevertheless, in this representation, $b^{\dagger}$ involves
both the photons and the matter polarization, which suggests that this
interpretation might be revised.

We also note that since QED-HF introduces a coherent state transformation, it
would be interesting to investigate the following expectation values

$\displaystyle\braket{n}{b_{\alpha}^{\dagger}b_{\alpha}}{n}$ (249)
$\displaystyle\braket{n}{\big{(}b_{\alpha}^{\dagger}-\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\braket{\bm{d}}_{\text{QED-
HF}}\big{)}\big{(}b_{\alpha}-\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\braket{\bm{d}}_{\text{QED-
HF}}\big{)}}{n}$ (250)
$\displaystyle\braket{n}{\big{(}b_{\alpha}^{\dagger}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot(\bm{d}-\braket{\bm{d}}_{\text{QED-
HF}})\big{)}\big{(}b_{\alpha}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot(\bm{d}-\braket{\bm{d}}_{\text{QED-
HF}})\big{)}}{n},$ (251)

where Eq. (249) refers to the occupation of the displacement field coherent
states, Eq. (250) refers to the displacement field number states, and Eq.
(251) refers to the occupation of the electric field number states.

####  6.1.3 Equivalent transition moments in TD-QED-HF

Following the same procedure as for standard HF response theory,227, 84 we can
show that Eqs. (102), (110), and (113), derived for exact wave functions, also
hold in the QED-HF response framework. In Tab. (1) we report the TD-QED-HF
transition moments for the lower polariton of an ethylene molecule using
different basis sets. We report the oscillator strengths in the length $f_{L}$
and velocity $f_{V}$ forms

$\displaystyle
f_{L}^{0n}=\frac{2}{3}\;\omega_{n}\sum_{i=x,y,z}|\braket{0}{d_{i}}{n}|^{2}$
(252) $\displaystyle
f_{V}^{0n}=\frac{2}{3\omega_{n}}\sum_{i=x,y,z}|\braket{0}{p_{i}}{n}|^{2},$
(253)

and the transition photon displacement coordinate and momentum

$\displaystyle\braket{0}{p_{\alpha}}{n}$
$\displaystyle=i\sqrt{\frac{\omega_{\alpha}}{2}}\braket{0}{(b_{\alpha}^{\dagger}-b_{\alpha})}{n}$
(254) $\displaystyle\braket{0}{q_{\alpha}}{n}$
$\displaystyle=\frac{1}{\sqrt{2\omega_{\alpha}}}\braket{0}{(b_{\alpha}^{\dagger}+b_{\alpha})}{n}.$
(255)

We numerically demonstrate the validity of Eqs. (102), (110), and (113).
Notice that since Eq. (102) is fulfilled only in the complete basis set limit,
Eqs. (252) and (253) will converge only for large basis sets, as seen from the
table. At the same time, Eq. (113) holds independently of the basis set size
and the photon space truncation. Notice that although the equivalence
expressed by Eqs. (109) and (110) do not depend on the quality of the basis
set, the computed values of the transition moments change with the basis, as
expected from the connection between matter and photon moments expressed by
(110). In particular, the use of diffuse functions strongly affects the
computed results. Equivalence relations such as Eqs. (102) and (113) are well
known in molecular response theory. However, they have not been explicitly
investigated for QED systems. Furthermore, these relations do not hold for the
Tamm-Dancoff approximation. The formal equivalence between the transition
dipole and velocity momenta can be exploited in the computation of electronic
circular dichroism (ECD) and optical rotation. From standard molecular
response theory, the ECD spectrum is known to be proportional to the
rotational strength

$R_{n0}=\Im\\{\braket{0}{\bm{\mu}}{n}\cdot\braket{n}{\bm{m}}{0}\\}.$ (256)

The dipole formulation in Eq. (256) is origin dependent for a finite basis
set, which can lead to unphysical results. If we are interested in the
chirality effects promoted by a chiral cavity, such a formulation might be
misleading. However, the equivalent velocity form

$R_{n0}=\frac{1}{\omega_{n}}\Re\\{\braket{0}{\bm{p}}{n}\cdot\braket{n}{\bm{m}}{0}\\}$
(257)

is always origin independent, and Eqs. (256) and (257) are equivalent in the
limit of a complete basis.

Basis set |  $f_{L}$ |  $f_{V}$ |  $\braket{0}{q_{\alpha}}{n}$ [a.u.] |  $i\braket{0}{p_{\alpha}}{n}$ [a.u.] |  $\omega_{n}\braket{0}{q_{\alpha}}{n}$ [a.u.] |  $\frac{\braket{0}{\bm{d}\cdot\bm{\lambda}_{\alpha}}{n}\omega_{n}\omega_{\alpha}}{\omega_{n}^{2}-\omega_{\alpha}^{2}}$ [a.u.]
---|---|---|---|---|---|---
sto-3g |  0.00234 |  0.00096 |  1.36138 |  0.36641 |  0.36641 |  0.36641
6-31g |  0.02540 |  0.01902 |  1.34118 |  0.35965 |  0.35965 |  0.35965
6-311g |  0.04362 |  0.03158 |  1.32054 |  0.35349 |  0.35349 |  0.35349
6-311g* |  0.04978 |  0.04891 |  1.30958 |  0.35036 |  0.35036 |  0.35036
6-311g** |  0.05454 |  0.05192 |  1.30320 |  0.34852 |  0.34852 |  0.34852
6-311+g** |  0.20610 |  0.19318 |  1.03720 |  0.27413 |  0.27413 |  0.27413
6-311++g** |  0.20631 |  0.19327 |  1.03677 |  0.27401 |  0.27401 |  0.27401
cc-pVDZ |  0.04267 |  0.04121 |  1.31895 |  0.35309 |  0.35309 |  0.35309
aug-cc-pVDZ |  0.21694 |  0.21309 |  0.99843 |  0.26353 |  0.26353 |  0.26353
aug-cc-pV5Z |  0.22561 |  0.22564 |  0.96941 |  0.25559 |  0.25559 |  0.25559
Table 1: Oscillator strength in dipole $f_{L}$ and velocity $f_{V}$ form and
transition photon displacement coordinate $\braket{0}{q_{\alpha}}{n}$ and
momentum $i\braket{0}{p_{\alpha}}{n}$ for the lower polariton of an ethylene
molecule, for different basis sets at TD-QED-HF level. The cavity frequency is
set to $0.2695$ a.u., and the coupling strength to $0.01$ a.u. with field
polarization along the transition dipole moment of the first non-dark
excitation of ethylene. Eq. (102) holds only in the complete basis limit, and
convergence between the velocity and length form of the oscillator strength
strongly depends on the quality of the basis set. On the other hand, Eqs.
(109) and (110) hold independently of the truncation of the electronic or
photonic space.

####  6.1.4 QED-HF static response equations

For static perturbations, the response equations for the optimized QED-HF
state $\ket{\mathrm{R}}$ can be derived using the same parametrization shown
in Eq. (204), which now is time-independent

$\ket{\text{QED-HF}}=\text{exp}\big{(}i\Lambda\big{)}\ket{\mathrm{R}}.$ (258)

Here $\Lambda$ has the same form showed in Eq. (6.1.1) and may be written
compactly as

$\displaystyle\Lambda=\bm{\xi}^{\dagger}\cdot\bm{\Theta},$ (259)

where $\bm{\Theta}$ collects both photon and electron parameters and
$\bm{\xi}$ the respective operators:

$\bm{\Theta}=\begin{pmatrix}\gamma_{\alpha}\\\ \kappa_{ai}\\\
\gamma_{\alpha}^{*}\\\
\kappa_{ai}^{*}\end{pmatrix};\quad\bm{\xi}=\begin{pmatrix}b_{\alpha}\\\
\frac{1}{\sqrt{2}}E_{ia}\\\ b^{\dagger}_{\alpha}\\\
\frac{1}{\sqrt{2}}E_{ai}\end{pmatrix}.$

Following the perturbation expansion of the energy in Sec. (5), the first-
order energy derivative is

$\displaystyle E^{(1)}_{\text{QED-
HF}}=\braket{\mathrm{R}}{H^{(1)}}{\mathrm{R}},$ (260)

which expresses the Hellmann-Feynman theorem for the QED-HF state. In Eq.
(260), $H^{(1)}$ refers to the first-order interactions. The second-order
energy derivative is

$\displaystyle E_{\text{QED-
HF}}^{(2)}=\braket{\mathrm{R}}{H^{(2)}}{\mathrm{R}}+\braket{\mathrm{R}}{[\Lambda^{(1)},H^{(1)}]}{\mathrm{R}},$
(261)

where $H^{(2)}$ is the second-order interaction and $\Lambda^{(1)}$ is the
first-order correction to the parameterization. Since QED-HF state is a
variational theory,60 the optimization conditions are equivalent to those in
standard HF87

$\displaystyle\braket{\mathrm{R}}{[\bm{\xi},H^{(0)}]}{\mathrm{R}}$
$\displaystyle=0$ (262)
$\displaystyle\sum_{j}\braket{\mathrm{R}}{[{\xi}^{\dagger}_{i},[{H^{(0)},{\xi}}_{j}]]}{\mathrm{R}}{\Theta}_{j}^{(1)}$
$\displaystyle=i\braket{\mathrm{R}}{[{\xi}_{i},H^{(1)}]}{\mathrm{R}},$ (263)

where Eq. (262) corresponds to the QED-HF Brillouin’s theorem ,60 and Eq.
(263) to the first-order static response equations. The left-hand side of Eq.
(263) includes the generalized Hessian matrix, whose blocks are given in Eq.
(227)-Eq. (228), and the first-order parameters $\bm{\Theta}^{(1)}$. The
right-hand side contains the first-order Hamiltonian, which includes the
first-order interaction terms. We note that Eq. (263) is equivalent to the
time-dependent response equation (222) with $\omega=0$.

###  6.2 QED-CC

The reference wave function used in QED-CC is QED-HF, where we employ the
coherent-state transformed Hamiltonian in Eq. (6.1).60 The QED-CC state is
defined as

$\displaystyle\ket{\text{QED-CC}}$ $\displaystyle=\exp(T)\ket{\text{HF},0},$
(264)

where $T$ is the cluster operator

$\displaystyle T$ $\displaystyle=T_{e}+T_{p}+T_{int}.$ (265)

The electronic cluster $T_{e}$ is the standard electronic excitation
operator185

$\displaystyle T_{e}=\sum_{\mu}t_{\mu}\tau_{\mu}$ (266)
$\displaystyle\tau_{\mu}\ket{\text{HF}}=\ket{\mu},$ (267)

where $\ket{\mu}$ is an excited HF determinant. The cluster operator $T_{p}$
includes pure photonic excitations

$\displaystyle
T_{p}=\sum_{\bm{n}}\Gamma_{\bm{n}}=\sum_{\bm{n}}\gamma_{\bm{n}}\prod_{\alpha}(b^{\dagger}_{\alpha})^{n_{\alpha}},$
(268)

where ${\bm{n}}$ is a vector of integers $n_{\alpha}$ referring to the
$\alpha$-mode. Finally, the interaction operator $T_{int}$ includes
simultaneous excitations of matter and cavity modes

$\displaystyle
T_{int}=\sum_{\bm{n}}(S_{1}^{\bm{n}}+S_{2}^{\bm{n}}+\dots+S^{\bm{n}}_{N_{e}}),$
(269)

where fir instance

$\displaystyle S_{1}^{\bm{n}}$
$\displaystyle=\sum_{ai}s_{ai}^{\bm{n}}E_{ai}\prod_{\alpha}(b^{\dagger})^{n_{\alpha}}$
(270) $\displaystyle S_{2}^{\bm{n}}$
$\displaystyle=\frac{1}{2}\sum_{aibj}s_{aibj}^{\bm{n}}E_{ai}E_{bj}\prod_{\alpha}(b^{\dagger})^{n_{\alpha}}.$
(271)

The Schrödinger equation is solved by projection onto the space ${S}$ spanned
by the electronic determinants and the number states60

${S}=\text{Span}\big{\\{}\ket{\text{HF},0},\ket{\mu,0},\ket{\text{HF},{\bm{n}}},\ket{\mu,{\bm{n}}}\big{\\}}.$
(272)

In Eq. (272), the state $\ket{\mu,{\bm{n}}}$ is a simultaneous excitation to
the $\mu$ electronic excited determinant and the ${\bm{n}}$ photon state

$\ket{\mu,\bm{n}}=\ket{\mu}\otimes\ket{\bm{n}}=\tau_{\mu}\prod_{\alpha}\frac{(b_{\alpha}^{\dagger})^{n_{\alpha}}}{\sqrt{n_{\alpha}!}}\ket{\text{HF},0}.$
(273)

The ground state QED-CC equations are therefore

$\displaystyle\braket{\mu,\bm{n}}{e^{-T}He^{T}}{\text{HF},0}=0$ (274)
$\displaystyle\braket{\text{HF},0}{H}{\text{QED-CC}}=E_{\text{QED-CC}}.$ (275)

The QED-CC dual state is defined as in standard CC theory

$\bra{\Lambda}=\bra{\text{HF},0}+\sum\bar{t}_{\mu{\bm{n}}}\bra{\mu,{\bm{n}}}e^{-T},$
(276)

where $\bar{t}_{\mu\bm{n}}$ are the Lagrangian multipliers.60, 185 A
collective index ${t}_{\mu\bm{n}}$ can also be defined for the cluster
operator such that we can write $T$ more compactly as

$T=\sum_{\mu,\bm{n}}{t}_{\mu\bm{n}}\tau_{\mu}\prod_{\alpha}(b^{\dagger}_{\alpha})^{n_{\alpha}},$
(277)

where

$\displaystyle t_{\mu 0}$ $\displaystyle=t_{\mu}$ (278) $\displaystyle
t_{\mu\bm{n}}$ $\displaystyle=s^{\bm{n}}_{\mu}$ (279) $\displaystyle
t_{\text{HF}\bm{n}}$ $\displaystyle=\gamma_{\bm{n}}.$ (280)

As in standard CC theory, QED-CC is based on a hierarchy of approximations
where the cluster operators and the projection space are truncated.185, 60
Since the space ${S}$ is larger than in standard CC theory, as it also
includes the photonic excitations, the QED-CC Jacobian $\mathbf{A}$ has
additional dimensions

$A_{\mu{\bm{n}},\nu\bm{m}}=\braket{\mu,{\bm{n}}}{[e^{-T}{H}e^{T},\tau_{\nu}\prod_{\alpha}(b^{\dagger}_{\alpha})^{m_{\alpha}}]}{\text{HF},0}.$
(281)

In addition to the electronic block $\mathbf{A}_{e,e}$ similar to electronic
CC, there is also a photonic block $\mathbf{A}_{p,p}$ and blocks involving the
electronic-photonic parameters

$\mathbf{A}=\begin{pmatrix}\mathbf{A}_{e,e}&\mathbf{A}_{e,ep}&\mathbf{A}_{e,p}\\\
\mathbf{A}_{ep,e}&\mathbf{A}_{ep,ep}&\mathbf{A}_{ep,p}\\\
\mathbf{A}_{p,e}&\mathbf{A}_{p,ep}&\mathbf{A}_{p,p}\end{pmatrix}.$ (282)

From the Jacobian $\mathbf{A}$ and the $\bm{\eta}$ vector

$\eta_{\mu{\bm{n}}}=\braket{\text{HF},0}{e^{-T}{H}e^{T}}{\mu,{\bm{n}}},$ (283)

we obtain the Lagrangian multipliers and the equation of motion (EOM)-QED-CC
formalism for properties and excited states.60

####  6.2.1 Time-dependent QED-CC

Following the response theory for electronic CC,234, 235, 236 we parametrize
the time-evolution of the QED-CC state by a time-dependent cluster operator
$T(t)$

$\ket{\text{QED-CC}}(t)=e^{T(t)}\ket{\text{HF},0}e^{i\epsilon(t)}.$ (284)

The time-dependent parameters are obtained by projection of the time dependent
Schrödinger equation

$i\frac{d}{dt}\big{(}e^{T(t)}\ket{\text{HF},0}e^{i\epsilon(t)}\big{)}=(H+V^{t})e^{T(t)}\ket{\text{HF},0}e^{i\epsilon(t)}$
(285)

onto the space ${S}$. We obtain the following response equations for the
cluster parameters

$\displaystyle\frac{d\epsilon}{dt}$
$\displaystyle=-\braket{\text{HF},0}{(H+V^{t})e^{T(t)}}{\text{HF},0}$ (286)
$\displaystyle\frac{dt_{\mu}}{dt}$
$\displaystyle=-i\braket{\mu,0}{e^{-T(t)}(H+V^{t})e^{T(t)}}{\text{HF},0}$
(287) $\displaystyle\frac{d\gamma_{\bm{n}}}{dt}$
$\displaystyle=-i\braket{\text{HF},{\bm{n}}}{e^{-T(t)}(H+V^{t})e^{T(t)}}{\text{HF},0}$
(288) $\displaystyle\frac{ds_{\mu}^{\bm{n}}}{dt}$
$\displaystyle=-i\braket{\mu,{\bm{n}}}{e^{-T(t)}(H+V^{t})e^{T(t)}}{\text{HF},0},$
(289)

where Eqs. (286) and (287) are analogous to standard CC response theory,235,
234 while Eqs. (288) and (289) are additional equations for the photon and
electron-photon parameters. We then perform a perturbative expansion of the
amplitudes and a Fourier decomposition

$\displaystyle t_{\mu n}=$ $\displaystyle t_{\mu n}^{(0)}+t_{\mu
n}^{(1)}+t_{\mu n}^{(2)}+\dots$ $\displaystyle=$ $\displaystyle t_{\mu
n}^{(0)}+\int d\omega_{1}X_{\mu n}^{(1)}(\omega_{1})e^{-i\omega_{1}t}+\int
d\omega_{1}\int d\omega_{2}X_{\mu
n}^{(2)}(\omega_{1},\omega_{2})e^{-i\omega_{1}t-i\omega_{2}t}+\dots$ (290)

Focusing now on a single cavity mode, although the generalization to a
multimode system is straightforward, we obtain the following set of equations

$\displaystyle\braket{{\mu,n}}{e^{-T^{(0)}}He^{T^{(0)}}}{\text{HF},0}=0$ (291)
$\displaystyle\sum_{\nu m}\left(\omega_{1}\bm{I}-\mathbf{A}\right)_{\mu n,\nu
m}X^{(1)}_{\nu m}=\xi^{(1)}_{\mu n}$ (292) $\displaystyle\sum_{\nu
m}\left((\omega_{1}+\omega_{2})\bm{I}-\mathbf{A}\right)_{\mu n,\nu
m}X^{(2)}_{\nu m}=\xi_{\mu n}^{(2)},$ (293)

where

$\displaystyle\xi_{\mu n}^{(1)}(\omega_{1})$
$\displaystyle=\braket{\overline{\mu,n}}{V^{\omega_{1}}}{\text{QED-CC}}$ (294)
$\displaystyle\xi_{\mu n}^{(2)}(\omega_{1},\omega_{2})$
$\displaystyle=\frac{1}{2}\hat{P}(\omega_{1}\,\omega_{2})\left(\braket{\overline{\mu,n}}{[V^{\omega_{1}},T^{(1)}(\omega_{2})]}{\text{QED-
CC}}+\frac{1}{2}\braket{\overline{\mu,n}}{[[H,T^{(1)}(\omega_{1})],T^{(1)}(\omega_{2})]}{\text{QED-
CC}}\right),$ (295)

and we defined the QED-CC-transformed states

$\displaystyle\bra{\overline{\mu,n}}=\bra{\mu,n}e^{-T^{(0)}}$ (296)
$\displaystyle\ket{\text{QED-CC}}=e^{T^{(0)}}\ket{\text{HF},0}.$ (297)

The zero, first and second-order cluster operators read

$\displaystyle T^{(0)}$ $\displaystyle=\sum_{\mu n}t_{\mu n}\tau_{\mu n}$
(298) $\displaystyle T^{(1)}(\omega_{1})$ $\displaystyle=\sum_{\mu
n}X^{(1)}_{\mu n}(\omega_{1})\tau_{\mu n}$ (299) $\displaystyle
T^{(2)}(\omega_{1},\omega_{2})$ $\displaystyle=\sum_{\mu n}X^{(2)}_{\mu
n}(\omega_{1},\omega_{2})\tau_{\mu n},$ (300)

and we note that the zero-order equation in Eq (291) is the ground state QED-
CC optimization condition. In an analogous way, we can parametrize the time
dependence of the dual QED-CC state

$\bra{\Lambda(t)}=\left(\bra{\text{HF}}+\sum_{\mu n}\bar{t}_{\mu
n}(t)\bra{\mu,n}e^{-T(t)}\right)e^{-i\epsilon(t)},$ (301)

where we have the standard normalization condition of CC at all times185

$\braket{\Lambda(t)}{\text{QED-CC}(t)}=1.$ (302)

The equations for the Lagrange multipliers are obtained by projection onto
${S}$

$\displaystyle\frac{d\bar{t}_{\mu n}}{dt}$
$\displaystyle=i\braket{\Lambda(t)}{[H+V_{t},\tau_{\mu}(b^{\dagger})^{n}]}{\text{QED-
CC}(t)}$ (303)

and using the same expansion as Eq. (6.2.1) we can write

$\displaystyle\bar{t}_{\mu n}=$ $\displaystyle\bar{t}_{\mu
n}^{(0)}+\bar{t}_{\mu n}^{(1)}+\bar{t}_{\mu n}^{(2)}+\dots$ $\displaystyle=$
$\displaystyle\bar{t}_{\mu n}^{(0)}+\int d\omega_{1}Y_{\mu
n}^{(1)}(\omega_{1})e^{-i\omega_{1}t}+\int d\omega_{1}\int d\omega_{2}Y_{\mu
n}^{(2)}(\omega_{1},\omega_{2})e^{-i\omega_{1}t-i\omega_{2}t}+\dots$ (304)

We then obtain the equations

$\displaystyle\sum_{\nu n}\bar{t}_{\nu n}^{(0)}A_{\nu n,\mu
m}=-\braket{R}{[H,\tau_{\mu}(b^{\dagger})^{n}]}{\text{QED-CC}}$ (305)
$\displaystyle\sum_{\nu n}Y^{(1)}_{\nu n}(\omega_{1}\bm{I}+\mathbf{A})_{\nu
n,\mu m}=-\eta_{\mu m}^{(1)}-\sum_{\gamma k}F_{\mu m,\gamma k}X^{(1)}_{\gamma
k}$ (306) $\displaystyle\sum_{\nu n}Y^{(2)}_{\nu
n}((\omega_{1}+\omega_{2})\bm{I}+\mathbf{A})_{\nu n,\mu m}=-\eta_{\mu
m}^{(2)}-\sum_{\gamma k}F_{\mu m,\gamma k}X^{(2)}_{\gamma k}.$ (307)

The zero-order equation in Eq. (305) is the ground state multiplier equation,
and the vectors $\eta$ are defined as

$\displaystyle\eta_{\mu n}^{(1)}(\omega_{1})=$
$\displaystyle\braket{\Lambda}{[V^{\omega_{1}},\tau_{\mu}(b^{\dagger})^{n}]}{\text{QED-
CC}}$ (308) $\displaystyle\eta_{\mu n}^{(2)}(\omega_{1},\omega_{2})=$
$\displaystyle\frac{1}{2}\hat{P}(\omega_{1},\omega_{2})\bigg{(}\braket{\Lambda}{[[V^{\omega_{1}},\tau_{\mu}(b^{\dagger})^{n}],T^{(1)}(\omega_{2})]}{\text{QED-
CC}}+\frac{1}{2}\braket{\Lambda}{[[[H,\tau_{\mu}(b^{\dagger})^{n}],T^{(1)}(\omega_{1})],T^{(1)}(\omega_{2})]}{\text{QED-
CC}}$ $\displaystyle+$ $\displaystyle\sum_{\nu
m}Y^{(1)}_{\nu,n}(\omega_{1})\braket{\overline{\nu,m}}{[[H,\tau_{\mu}(b^{\dagger})^{n}],T^{(1)}(\omega_{2})]+[V^{\omega_{2}},\tau_{\mu}(b^{\dagger})^{n}]}{\text{QED-
CC}}\bigg{)},$ (309)

where the matrix $\bm{F}$ as

$F_{\mu m,\nu
k}=\braket{\Lambda}{[[H,\tau_{\mu}(b^{\dagger})^{m}],\tau_{\nu}(b^{\dagger})^{k}]}{\text{QED-
CC}}.$ (311)

As for QED-HF, the response matrices and vectors have additional dimensions
connected to the electromagnetic degrees of freedom. The $X_{\mu n}$ and
$Y_{\nu m}$ vectors include, together with the standard CC electronic
excitation response parameters, the purely photonic and the simultaneous
electronic-photonic response parameters of Eqs. (268) and (269). As for the
Jacobian matrix $\mathbf{A}$, the $\bm{F}$ matrix has additional photon and
electron-photon blocks

$\bm{F}=\begin{pmatrix}\bm{F}_{e,e}&\bm{F}_{e,ep}&\bm{F}_{e,p}\\\
\bm{F}_{ep,e}&\bm{F}_{ep,ep}&\bm{F}_{ep,p}\\\
\bm{F}_{p,e}&\bm{F}_{p,ep}&\bm{F}_{p,p}\end{pmatrix}.$ (312)

Moreover, the perturbation $V^{\omega}$ can act on both the electronic and the
photon degrees of freedom, as in Eq. (147).

The CC expectation value is defined as234

$\braket{A}_{\text{QED-
CC}}(t)=\frac{1}{2}\big{(}\braket{\Lambda(t)}{A}{\text{QED-
CC}(t)}+\braket{\Lambda(t)}{A}{\text{QED-CC}(t)}^{*}\big{)}.$ (313)

From the perturbative expansion

$\displaystyle\braket{A}_{\text{QED-CC}}(t)$
$\displaystyle=\braket{A}_{\text{QED-CC}}+\int
d\omega_{1}\braket{\braket{A,V^{\omega_{1}}}}_{\omega_{1}+i\eta}e^{-i\omega_{1}t+i\eta
t}+\dots$

we can identify the QED-CC response functions

$\displaystyle\braket{\braket{A,V^{\omega_{1}}}}_{\omega_{1}}$
$\displaystyle=\frac{1}{2}\left(\mathcal{F}^{A,V^{\omega_{1}}}_{\omega_{1}}+\left(\mathcal{F}^{A,V^{-\omega_{1}}}_{-\omega_{1}}\right)^{*}\right)$
(314)
$\displaystyle\braket{\braket{A,V^{\omega_{1}},V^{\omega_{2}}}}_{\omega_{1},\omega_{2}}$
$\displaystyle=\frac{1}{2}\left(\mathcal{F}^{A,V^{\omega_{1}},V^{\omega_{2}}}_{\omega_{1},\omega_{2}}+\left(\mathcal{F}^{A,V^{-\omega_{1}},V^{\omega_{2}}}_{-\omega_{1},-\omega_{2}}\right)^{*}\right)$
(315)

where

$\displaystyle\mathcal{F}^{A,V^{\omega_{1}}}_{\omega_{1}}=$
$\displaystyle\sum_{\mu n}\left(Y^{(1)}_{\mu
n}(\omega_{1})\braket{\overline{\mu,n}}{A}{\text{QED-CC}}+X_{\mu
n}^{(1)}(\omega_{1})\braket{\Lambda}{[A,\tau_{\mu n}]}{\text{QED-CC}}\right)$
(316)
$\displaystyle\mathcal{F}^{A,V^{\omega_{1}},V^{\omega_{2}}}_{\omega_{1},\omega_{2}}=$
$\displaystyle 2\sum_{\mu n}\left(Y_{\mu
n}^{(2)}(\omega_{1},\omega_{2})\braket{\overline{\mu,n}}{A}{\text{QED-
CC}}+X_{\mu
n}^{(2)}(\omega_{1},\omega_{2})\braket{\Lambda}{[A,\tau_{\mu}(b^{\dagger})^{n}]}{\text{QED-
CC}}\right)$ $\displaystyle+$ $\displaystyle\sum_{\mu n,\nu
m}\left(X^{(1)}_{\mu n}(\omega_{1})F^{A}_{\mu n,\nu m}X^{(1)}_{\nu
m}(\omega_{2})+\hat{P}(\omega_{1},\omega_{2})Y^{(1)}_{\mu
n}(\omega_{1})\braket{\overline{\mu,m}}{[A,\tau_{\nu}(b^{\dagger})^{n}]}{\text{QED-
CC}}X^{(1)}_{\nu n}(\omega_{2})\right).$ (317)

Here we have used the notation

$F^{A}_{\mu m,\nu
k}=\braket{\Lambda}{[[A,\tau_{\mu}(b^{\dagger})^{m}],\tau_{\nu}(b^{\dagger})^{k}]}{\text{QED-
CC}}.$ (318)

As for QED-HF, the static response equations can be obtained by setting the
frequency of the external to zero. The poles and the residues of the response
functions can be obtained assuming that $\mathbf{A}$ can be diagonalized

$(\bm{S}^{-1}\mathbf{A}\bm{S})_{mn}=\delta_{mn}\omega_{n},$ (319)

where $\omega_{n}$ is the n-th (real) excitation energy. We further introduce
the notation

$\displaystyle\tilde{\tau}_{n}=\sum_{\mu,m}S_{\mu
m,n}\tau_{\mu}(b^{\dagger})^{m}$ (320)
$\displaystyle\tilde{\tau}_{n}^{\dagger}=\sum_{\mu,m}S^{-1}_{n,\mu
m}\tau_{\mu}^{\dagger}b^{m}$ (321)

to describe the diagonal representation in Eq. (319). From the poles of the
linear response function, we identify the residues

$\displaystyle\lim_{\omega\to\omega_{k}}(\omega-\omega_{k})\braket{\braket{A;B}}_{\omega}=$
$\displaystyle\frac{1}{2}\bigg{(}\Gamma^{A}_{k}\Theta^{B}_{k}+\big{(}\Gamma^{B}_{k}\Theta^{A}_{k}\big{)}^{*}\bigg{)}\equiv\braket{0}{A}{k}\braket{k}{B}{0}$
(322)
$\displaystyle\lim_{\omega\to-\omega_{k}}(\omega+\omega_{k})\braket{\braket{A;B}}_{\omega}=$
$\displaystyle-\frac{1}{2}\bigg{(}\big{(}\Gamma^{A}_{k}\Theta^{B}_{k}\big{)}^{*}+\Gamma^{B}_{k}\Theta^{A}_{k}\bigg{)}\equiv-\big{(}\braket{0}{A}{k}\braket{k}{B}{0}\big{)}^{*}$
(323)

where

$\displaystyle\Theta^{A}_{k}$ $\displaystyle=\braket{{\text{QED-
HF}}}{\tilde{\tau}_{k}^{\dagger}e^{-T^{(0)}}A}{\text{QED-CC}}$ (324)
$\displaystyle\Gamma^{A}_{k}$
$\displaystyle=\braket{\Lambda}{[A,\tilde{\tau}_{k}]}{\text{QED-CC}}$
$\displaystyle-\sum_{n}\frac{\braket{{\text{QED-
HF}}}{\tilde{\tau}_{k}^{\dagger}e^{-T^{(0)}}A}{\text{QED-
CC}}F_{nk}}{\omega_{n}+\omega_{k}}.$ (325)

From these equations, we can compute the polaritonic properties.234 Since we
employ the QED-HF coherent-state transformed Hamiltonian in Eq. (6.1), the
operators and the perturbations in the response functions must be described in
the same representation.

####  6.2.2 QED-CC electronic and photonic excitation character

Since QED-CC is a highly correlated method, defined for the QED-HF coherent-
state transformed Hamiltonian in Eq. (6.1), the definition of the electronic
or photonic character of the excitation is not trivial. In the EOM framework,
Haugland et. al60 defined the electronic weight $w_{el}$ of the state by means
of projection operators

$w^{k}_{el}=\sqrt{\frac{\braket{\Lambda_{k}}{P_{el}}{R_{k}}}{\braket{\Lambda_{k}}{P}{R_{k}}}}$
(326)

where $\Lambda_{k}$ and $R_{k}$ are the $k$-th left and right state, $P$ is
the projection operator onto the space $S$ of Eq. (272) and $P_{el}$ is the
projector onto the states of $S$ with zero photons. In the response framework,
a similar definition can be obtained by considering the electronic components
of the $X_{\mu n}$ and $Y_{\mu n}$ vectors, for instance

$\displaystyle w_{el}$ $\displaystyle=\sqrt{\sqrt{\frac{\sum_{\mu}|X_{\mu
0}|^{2}}{\sum_{\mu n}|X_{\mu n}|^{2}}}\times\sqrt{\frac{\sum_{\mu}|Y_{\mu
0}|^{2}}{\sum_{\mu n}|Y_{\mu n}|^{2}}}}.$ (327)

Notice, however, that such a definition does not consider the contribution of
the simultaneous electron-photon excitations. A more consistent definition of
the photonic character could be obtained by considering the mean values of the
field number operators from the quadratic response function, and comparing
them to the ground state values, as discussed for QED-HF

$\displaystyle\braket{\Lambda_{k}}{b^{\dagger}b}{R_{k}}$ (328)
$\displaystyle\braket{\Lambda_{k}}{\big{(}b_{\alpha}^{\dagger}-\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\braket{\bm{d}}_{HF}\big{)}\big{(}b_{\alpha}-\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot\braket{\bm{d}}_{HF}\big{)}}{R_{k}}$
(329)
$\displaystyle\braket{\Lambda_{k}}{\big{(}b_{\alpha}^{\dagger}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot(\bm{d}-\braket{\bm{d}}_{HF})\big{)}\big{(}b_{\alpha}+\frac{1}{\sqrt{2\omega_{\alpha}}}\bm{\lambda}_{\alpha}\cdot(\bm{d}-\braket{\bm{d}}_{HF})\big{)}}{R_{k}}.$
(330)

##  7 Concluding remarks

In this paper, we proposed a systematic discussion of polaritonic response
theory based on the well-established molecular response theory routinely
employed in quantum chemistry. The fundamental definitions and features of the
response functions are still valid, but the explicit treatment of the
electromagnetic degrees of freedom allows for novel perspectives. Additional
equivalence relations between matter and photonic observables are introduced,
and novel ways to probe the system are discussed. Particular care is needed
when using different mathematical representations of the operators, as this
can lead to misinterpretations of the computed results. We also provided QED-
HF and QED-CC response equations that resemble the standard electronic
response theory, providing the reader with a general framework for ab inito
QED response theory. While significant progress in the theoretical description
of polaritonic systems has been made, we emphasize several challenges that
future research will have to address. These include a description of light-
matter interaction beyond the electric dipole approximation, the issue of
disorder in optical devices, the role of collective effects on polaritonic
properties and chemical reactions, and chiral polaritonics.

##  8 Acknowledgments

We acknowledge Tor S. Haugland for insightful discussions.

##  9 Funding information

M.C., A.B., and H.K. acknowledge funding from the European Research Council
(ERC) under the European Union’s Horizon 2020 Research and Innovation
Programme (grant agreement No. 101020016). R.R.R and H.K. acknowledge funding
from the Research Council of Norway through FRINATEK Project No. 275506. E.R
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon Europe Research and Innovation Programme (Grant n.
ERC-StG-2021-101040197 - QED-SPIN). We acknowledge computing resources through
UNINETT Sigma2—the National Infrastructure for High Performance Computing and
Data Storage in Norway, through Project No. NN2962k.

##  10 Conflict of interest

The authors declare no conflict of interest for this paper.

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