# A Comparison of Relativistic Coupled Cluster and Equation of Motion Coupled Cluster Quadratic Response Theory

Xiang Yuan, Loïc Halbert, Lucas Visscher, André Severo Pereira Gomes

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Xiang Yuan, Loïc Halbert, Lucas Visscher, André Severo Pereira Gomes. A Comparison of Relativistic Coupled Cluster and Equation of Motion Coupled Cluster Quadratic Response Theory. 2025. hal-05137563

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# A Comparison of Relativistic Coupled Cluster and Equation of Motion Coupled Cluster Quadratic Response Theory

Xiang Yuan,†,‡ Loïc Halbert,† Lucas Visscher,¶ and André Severo Pereira

Gomes*,†

†Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France

‡Department of Chemistry and Pharmaceutical Sciences, Faculty of Science, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands

¶Department of Chemistry and Pharmaceutical Sciences, Faculty of Science, Vrije Universiteit Amsterdam, 1081 HZ Amsterdam, The Netherlands

E-mail: andre.gomes@univ-lille.fr

## Abstract

We present the implementation of relativistic coupled cluster quadratic response theory (QR-CC), following our development of relativistic equation of motion coupled cluster quadratic response theory (QR-EOMCC) [X. Yuan et al., J. Chem. Theory Comput. 2023, 19, 9248]. These codes, which can be used in combination with relativistic (2- and 4-component based) as well as non-relativistic Hamiltonians, are capable of treating both static and dynamic perturbations for electric and magnetic operators. We have employed this new implementation to revisit the calculation of static and frequency-dependent first hyperpolarizabilities of hydrogen halides (HX, X=F-Ts) and
the Verdet constant of heavy noble gas atoms (Xe, Rn, Og), in order to investigate the
differences and similarities of QR-CC and the more approximate QR-EOMCC. Fur-
thermore, we have determined the relative importance of scalar relativistic effects and
spin-orbit coupling to these properties, through a comparison of different Hamiltonians,
and extended our calculations to superheavy element species (HTs for hyperpolariz-
abilities, Og for the Verdet constant). Our results show that as one moves towards the
bottom of the periodic table, QR-EOMCC can yield rather different results (hyperpo-
larizabilities) or perform rather similarly (Verdet constant) to QR-CC. These results
underscore the importance of further characterizing the performance of QR-EOMCC
for heavy element systems.

Introduction

External perturbations such as those induced by electric or magnetic fields allow us to
probe the electronic structure of atoms, molecules and materials, with different perturbations
providing us with complementary information about the system.<sup>1,2</sup> The first-order (linear)
responses to such a perturbation are particularly sought-after quantities, since from them one
can obtain information about electronically or vibrationally<sup>3</sup> excited states though processes
involving a single photon, or magnetic properties such as NMR shieldings or indirect spin-
spin coupling constants<sup>4-6</sup> to name but a few.

Properties arising from the interaction with two or more photons<sup>7,8</sup> at a time are also
of great interest be it for technological applications in non-linear optics<sup>9,10</sup> or for speciation
through two-photon absorption,<sup>11</sup> as the differences in selection rules compared to one-
photon processes allow for probing different states.

Theoretical models are indispensable to interpret experimental results, and the last fifty
years have seen the development<sup>12,13</sup> of approaches based on *ab initio* molecular electronic
structure theory gain popularity in applications, thanks in no small part to the development
of response theory<sup>2,14,15</sup> and its application to different models such as multi-configurational
self-consistent field theory,16–18 density functional theory,19 and coupled cluster theory.20–22
The use of the time-averaged quasienergy formalism21,23–35 has provided not only a uni-
fied framework, with which to derive response for both variational and non-variational ap-
proaches, but also a way to enable the development of embedding approaches in which
variational and non-variational electronic structure methods are combined36–43 to calculate
molecular properties in large systems within a fully quantum mechanical description in a
cost-effective manner.

In the case of coupled cluster theory, one can identify two main approaches, one which employs an exponential parametrization for the time evolution of the cluster operators (CC response), and another based on the equation of motion formalism (EOMCC response)44–53 for which a linear parametrization is employed to obtain molecular properties.

From the outset, key differences between the two approaches have been identified with
respect to separability,²¹,⁵⁴,⁵⁵ with as consequence that CC transition moments are size-
extensive whereas EOMCC ones are not. More recently the interconnections between these
two approaches were further investigated from a formal point of view.⁵⁶,⁵⁷

In spite of such differences, the calculation of response properties within the EOMCC
formalism remains appealing from both the implementation and the application points of
view,⁵⁸⁻⁶⁰ as simplifications in the equations also translate into a somewhat more modest
computational cost. This calls for an understanding of how the two approaches differ in
practical calculations. In the literature comparisons between linear response CC (LR-CC)
and EOMCC (LR-EOMCC) properties⁴⁹,⁵⁰,⁵⁷,⁶¹ so far suggest that, in spite of the lack of
size-extensivity of EOM-CC transition moments, in practice the two approaches yield quite
similar results.

Similar conclusions have been reached when comparing quadratic response CC (QR-CC)
and EOMCC (QR-EOMCC) results for dipole hyperpolarizabilities,⁴⁶,⁴⁸,⁵⁷ though compar-
isons were initially limited to a few systems. More recently, an extensive benchmark study
on excited state transition moments,⁶² which also probe quadratic response functions, seem
to confirm the tendencies observed previously. These previous evaluations were carried out
for valence properties of molecules containing light elements, that is, occupying the first two
rows of the periodic table. For properly representing the molecular properties of heavier
systems it is necessary to account for relativistic effects⁶³⁻⁶⁹ and to the best of our knowl-
edge there is currently no data in the literature assessing the differences between the two
approaches beyond the second row of the periodic table.

Response theory can be used in combination with relativistic Hamiltonians,⁷⁰ and im-
plementations for mean-field approaches (Hartree-Fock and DFT) are for example available
for the calculation of frequency-dependent linear and non-linear response properties for 4-
component-based Hamiltonians.⁷¹⁻⁸¹ To go beyond mean-field, relativistic coupled cluster
theory⁸²⁻⁸⁷ offers a flexible framework for obtaining accurate energies and properties which
treats electron correlation and relativistic effects on the same footing. It can thereby be used
to investigate systems of interest across the periodic table, and benchmark more approximate
methods.

In recent years there have been extensive efforts by different groups to address states
beyond closed-shell ground states through EOMCC theory for excitation and ionization en-
ergies, and electron affinities.⁸⁸⁻¹⁰³,¹⁰³⁻¹¹² While some of these implementations allow for the
calculation of transition moments or static properties (through the application of finite fields
or via approximate expectation values), full-fledged implementations of CC or EOMCC re-
sponse theory are much more scarce. Yuan et al.¹¹³ have reported an implementation capable
of carrying out frequency-dependent LR-CC and LR-EOMCC calculations for different (from
1- to 4-component based) Hamiltonians for electric, magnetic and mixed electric-magnetic
properties. A comparison of CC and EOMCC showed that for dipole polarizabilities the two
approaches do yield similar results as relativistic effects become more pronounced, but for in-
direct spin-spin coupling constants there is growing difference between the two. Chakraborty
et al.¹¹⁴ reported an efficient implementation of LR-CC for the spin-free eXact 2-Component
Hamiltonian (x2C) and applied it to investigate the static and dynamic dipole polarizabilities
of an extensive set of molecular systems.

Beyond linear response, the analytical treatment of quadratic response coupled cluster
theory would be particularly useful in applications in which finite field calculations are not
straightforward, easy to converge or inapplicable, such for frequency-dependent calculations
of non-linear optical properties or in the study of birefringences. However, general implemen-
tations beyond linear response are scarcer still, with our report of relativistic QR-EOMCC¹¹⁵
being, to the best of our knowledge, the first to appear in the literature. This implementa-
tion allowed us to gauge the performance of more approximate approaches such as TDDFT
and TDHF in the calculation of hyperpolarizabilities, two-photon absorption cross-sections
and Verdet constants. However, as discussed above QR-CC and QR-EOMCC are not equiv-
alent, and it is desirable to further characterize the behavior of the latter with respect to the
former across the periodic table, before carrying out more extensive applications with the
more approximate QR-EOMCC scheme.

The aim of the current work is therefore to report our implementation of quadratic
response coupled cluster for relativistic Hamiltonians, capable of handling for both static and
frequency-dependent perturbations involving electric or magnetic operators. We showcase
it by providing a first comparison of how QR-CC and QR-EOMCC behave as one goes
towards the bottom of the periodic table. We thereby chose to re-investigate the dipole first
hyperpolarizabilities of hydrogen halides, and the Verdet constants of heavy noble gas atoms
as they show significant relativistic effects. ¹¹⁵ Going further than this previous work, we
carried out a more extensive exploration on the effect of the Hamiltonian on these properties,
and investigated the properties of systems made up of superheavy elements, such as the
Tennessine hydride molecule and the Oganesson atom.

Tennessine and Oganesson, whose existence has been recognized by IUPAC and IUPAC
in 2015, are the two heaviest elements known. They have been synthesized at the Joint
Institute for Nuclear Research in Dubna by nuclear fusion upon bombardment of Berkelium
and Californium targets by between atoms of Calcium-48 ions, respectively. ¹¹⁶,¹¹⁷ While they
are expected to be members of groups 17 and 18, the difficulties in their production-only a few atoms are generated at a time, and their lifetimes are in the order of milliseconds- present formidable challenges to the experimental characterization of their physico-chemical properties,¹¹⁸ making theoretical work an indispensable tool to learn about their electronic structure.⁶⁹ In view of their large polarizabilities and more pronounced spin-orbit coupling than that in 6p, 5d and 5f block elements, they are also of potential interest as probes for any subtle difference between CC and EOMCC in the calculation of response properties.

# Theory and Implementation

## Quadratic Response Theory

As was the case in our previous work on coupled cluster linear response¹¹³ (LR-CC) and EOM-CC quadratic response¹¹⁵ (QR-EOMCC), our work is carried out within the framework of the quasienergy formulation of response theory.³¹ For the sake of brevity, we only review the formalism for CC quadratic response (QR-CC), and refer to the supplementary information for the working equations of QR-EOMCC as presented in.¹¹⁵ Throughout the text, the CC acronym will imply the use of coupled cluster singles and doubles (CCSD) wavefunctions.

We shall remain within the no-pair approximation,⁸³,¹¹⁹,¹²⁰ with the electronic part of the unperturbed molecular Hamiltonian given by

$$ \hat{H}_0 = \sum_{pq} h_q^p \{ \hat{a}_p^\dagger \hat{a}_q \} + \frac{1}{4} \sum_{pqrs} V_{rs}^{pq} \{ \hat{a}_p^\dagger \hat{a}_q^\dagger \hat{a}_s \hat{a}_r \} \quad (1) $$

where and p, q, r, s denote general (occupied or positive energy virtual) indexes, $h_{pq} = \langle p | \hat{h} | q \rangle$ denotes matrix elements of the one-body operator $\hat{h}$, and $V_{rs}^{pq} = \langle pq | \hat{g} | rs \rangle - \langle pq | \hat{g} | sr \rangle$ the anti-symmetrized matrix elements of the two-body operator $\hat{g}$. In the calculations reported in this work, $\hat{g}$ corresponds to Coulomb operator. In our current implementation, Gaunt or
Breit interactions are assumed to be treated in an effective manner.¹²¹⁻¹²⁴ External time-(in)dependent perturbations will be represented by one-body operators

$$ \hat{Y} = \sum_{pq} Y_q^p \{ \hat{a}_p^\dagger \hat{a}_q \} \qquad (2) $$

where $Y_q^p = \langle p | \hat{Y} | q \rangle$.

Due to the fact that in CC response theory the time-dependent quasienergy Lagrangian³¹ is made stationary with respect to the wavefunction parametrization (CC amplitudes and multipliers), we can use the (2n + 1) and (2n + 2) rules in perturbation theory.¹³,³¹ The calculation of a third-order property such as quadratic response, thus only requires the determination of the unperturbed and first-order perturbed CC amplitudes and Lagrange multipliers.

The unperturbed CC amplitudes $T$ and Lagrange multipliers $\bar{T}$ are defined as

$$ T = T_1 + T_2 = \sum_{ai} t_i^a \{ \hat{a}_a^\dagger \hat{a}_i \} + \frac{1}{4} \sum_{abij} t_{ij}^{ab} \{ \hat{a}_a^\dagger \hat{a}_b^\dagger \hat{a}_j \hat{a}_i \} = \sum_{ai} t_i^a \hat{\tau}_1 + \frac{1}{4} \sum_{abij} t_{ij}^{ab} \hat{\tau}_2 \qquad (3) $$

$$ \bar{T} = \bar{T}_1 + \bar{T}_2 = \sum_{ia} \bar{t}_a^i \{ \hat{a}_i^\dagger \hat{a}_a \} + \frac{1}{4} \sum_{ijab} \bar{t}_{ab}^{ij} \{ \hat{a}_i^\dagger \hat{a}_j^\dagger \hat{a}_b \hat{a}_a \} = \sum_{ia} \bar{t}_a^i \hat{\tau}_1 + \frac{1}{4} \sum_{ijab} \bar{t}_{ab}^{ij} \hat{\tau}_2 \qquad (4) $$

with $a, b$ for particle lines and $i, j$ for hole lines. As a shorthand notation, in the following we shall use greek letters {$\mu, \nu, \sigma$} to represent the strings of second quantization operators, with subscripts to represent single or double excitation; for instance, $\hat{\tau}_{\mu_1} \equiv \hat{\tau}_1 = \hat{a}_a^\dagger \hat{a}_i$ whereas $\hat{\tau}_\mu$ will represent a generic (either single or double) excitation operator. Conversely, $\hat{\tau}_\mu^\dagger$ will denote a de-excitation operator, which is biorthogonal to operator $\hat{\tau}_\mu$, that is, $\langle R | \hat{\tau}_\mu^\dagger \hat{\tau}_\nu | R \rangle = \delta_{\mu\nu}$, with $|R\rangle$ representing the reference state, here a single Slater determinant made up of molecular orbitals-or molecular spinors whenever the Hamiltonian accounts for spin-orbit coupling. Also, $|\mu\rangle$ denotes the ensemble of excited determinants with respect to the reference.
From the exponential ansatz for the CC wavefunction¹²⁵,¹²⁶

$$|CC\rangle = \exp(T)|R\rangle, \qquad (5)$$

the time-independent Dirac (Schrödinger) wave equation

$$\hat{H}_0|CC\rangle = E|CC\rangle \qquad (6)$$

and the Λ operator

$$\langle\Lambda| = \langle R| + \sum_{\mu} \bar{t}_{\mu} \langle\mu| \exp(-T), \qquad (7)$$

we have the unperturbed amplitudes and multipliers obtained from the equations

$$\langle\mu|\bar{H}|R\rangle = 0 \qquad (8)$$

$$\bar{t}\bar{H} + \eta = 0 \qquad (9)$$

where $\bar{H}$ is matrix representation of the normal-ordered similarity transformed Hamiltonian. The definitions of $\bar{H}$ and $\eta$ are given in table 1. Further details on the evaluation of these equations in our implementation can be found in.⁸⁴

After solving for the unperturbed wave function parameters, the first-order parameters are next obtained by solving the response equations³¹,¹¹³,¹¹⁵

$$(\bar{H} - \omega_Y \mathbf{I}) \dot{t}(\omega_Y) + \xi^Y = 0 \qquad (10a)$$

$$\dot{t}(\omega_Y)(\bar{H} + \omega_Y \mathbf{I}) + \eta^Y + \mathbf{F}\dot{t}(\omega_Y) = 0, \qquad (10b)$$

where $\mathbf{I}$ is the identity matrix. We employ the shorthand notation $\dot{t}^Y, \bar{t}^Y$ to describe the first-order perturbed amplitudes and multipliers, obtained for perturbation $\hat{Y}$ at frequency $\omega_Y$, respectively. We note that, in contrast to the first-order response equations in QR-EOMCC, the equations for the first-order perturbed multipliers depend on the first-order perturbed
amplitudes. The definitions of terms **F**, $\xi^Y$, $\eta^Y$ are also given in table 1.

The frequency-(in)dependent quadratic response function is given by the expression

$$ \langle\langle Z; X, Y \rangle\rangle_{\omega_X, \omega_Y} = \frac{1}{2} C^{\pm\omega} P(X(\omega_X), Y(\omega_Y), Z(\omega_Z)) \\ \times \left\{ \left[ \frac{1}{2} \mathbf{F}^X + \frac{1}{6} \mathbf{G}^x t(\omega_X) \right]^y t(\omega_Y) + \frac{x}{t}(\omega_X) \left[ \mathbf{A}^Y + \frac{1}{2} \mathbf{B}^y t(\omega_Y) \right] \right\}^z t(\omega_Z) , \quad (11) $$

where the perturbing frequencies satisfy the relationship $\omega_Z = -(\omega_X + \omega_Y)$. The operator $P(X(\omega_X), Y(\omega_Y), Z(\omega_Z))$ takes care of the permutation of the perturbing operators, and

$$ C^{\pm\omega} f^{XYZ}(\omega_X, \omega_Y, \omega_Z) = f^{XYZ}(\omega_X, \omega_Y, \omega_Z) + [f^{XYZ}(-\omega_X, -\omega_Y, -\omega_Z)]^* \quad (12) $$

is used for the symmetrization of the response functions,³¹ which requires the solution of response $f^{XYZ}(\omega_X, \omega_Y, \omega_Z)$ also for the negative of the frequencies (inverting the frequencies followed by complex conjugation). Eq. 11 can also be rewritten as

$$ \langle\langle X, Y, Z \rangle\rangle_{\omega_X, \omega_Y, \omega_Z} = \frac{1}{2} C^{\pm\omega} P(X(\omega_X), Y(\omega_Y), Z(\omega_Z)) \times \left\{ [O^{X,Y}] + [O^{\bar{X},Y}] \right\}^z t(\omega_Z) , \quad (13) $$

where $O^{X,Y}$ and $O^{\bar{X},Y}$ group the terms involving contractions only with perturbed amplitudes or with both perturbed amplitudes and multipliers, respectively:

$$ O^{X;Y} = \left[ \frac{1}{2} \mathbf{F}^X + \frac{1}{6} \mathbf{G}^x t(\omega_X) \right]^y t(\omega_Y) = \frac{1}{2} \mathbf{F}^{X,Y} t(\omega_Y) + \frac{1}{6} \mathbf{G}^x t(\omega_X)^y t(\omega_Y) \quad (14a) $$

$$ O^{\bar{X};Y} = \frac{x}{t}(\omega_X) \left[ \mathbf{A}^Y + \frac{1}{2} \mathbf{B}^y t(\omega_Y) \right] = \frac{x}{t}(\omega_X) \mathbf{A}^Y + \frac{1}{2} \frac{x}{t}(\omega_X) \mathbf{B}^y t(\omega_Y) \quad (14b) $$

and that for all combinations allowed by the $C^{\pm\omega}P$ product.

As is the case for $\bar{\mathbf{H}}$ and $\mathbf{F}$,¹¹³,¹¹⁵ the $\mathbf{F}^X$, $\mathbf{A}^Y$, $\mathbf{G}$ and $\mathbf{B}$ tensors, given by the expression in the table 1, are never constructed explicitly but rather as contractions with the perturbed amplitudes and/or multipliers. The programmable expressions required to assemble the $\mathbf{G}^{XYZ}$ term are presented in the supplementary information. The programmable expressions
<table><caption>Table 1: Expressions for the tensors appearing in the coupled cluster quadratic response functions as well as in the linear response equations determining perturbed amplitudes and Lagrange multipliers. Here $|R\rangle$ denotes the reference state in the definition of the coupled cluster wave function $|CC\rangle = \exp(T)|R\rangle$ and for the $\langle\Lambda| = \langle R| + \sum_{\mu} \bar{t}_{\mu} \langle\bar{\mu}|$ operator, with $\langle\bar{\mu}| = \langle R|\hat{\tau}_{\mu}^{\dagger} \exp(-T)$. Finally, $\hat{X}$ and $\hat{Y}$ denote one-body perturbation operators. We note that $\bar{\mathbf{H}}$ is strictly equivalent to the $\mathbf{A}$, the CC Jacobian defined by Christiansen et al.³¹.</caption><tr><td>η<sub>ν</sub></td><td>⟨R|[H̃<sub>0</sub>, τ̃<sub>μ</sub>]|CC⟩</td></tr><tr><td>H̃<sub>μν</sub></td><td>⟨μ|[H̃<sub>0</sub>, τ̃<sub>μ</sub>]|CC⟩</td></tr><tr><td>F<sub>μν</sub></td><td>⟨Λ|[[H̃<sub>0</sub>, τ̃<sub>μ</sub>], τ̃<sub>ν</sub>]|CC⟩</td></tr><tr><td>η<sup>Y</sup><sub>μ</sub></td><td>⟨Λ|[Ỹ, τ̃<sub>μ</sub>]|CC⟩</td></tr><tr><td>ξ<sup>Y</sup><sub>μ</sub></td><td>⟨μ|[Ỹ|CC⟩</td></tr><tr><td>F<sup>X</sup><sub>μν</sub></td><td>⟨Λ|[[X̃, τ<sub>μ</sub>], τ<sub>ν</sub>]|CC⟩</td></tr><tr><td>G<sub>μνσ</sub></td><td>⟨Λ|[[[H̃<sub>0</sub>, τ<sub>μ</sub>], τ<sub>ν</sub>], τ<sub>σ</sub>]|CC⟩</td></tr><tr><td>A<sup>Y</sup><sub>μν</sub></td><td>⟨μ|[Ỹ, τ<sub>ν</sub>]|CC⟩</td></tr><tr><td>B<sub>μνσ</sub></td><td>⟨μ|[H̃<sub>0</sub>, τ<sub>ν</sub>], τ<sub>σ</sub>]|CC⟩</td></tr></table>

for $F^X$, $A^Y$ and $B$ can be obtained from minor modifications of those already presented in our previous work.¹¹³,¹¹⁵

Implementation details

Our QR-CC implementation has been carried out in the ExaCorr module⁸⁴ of the relativistic quantum chemistry program DIRAC.¹²⁷ Below we provide details on the implementation of the QR-CC response function calculation, and refer the reader to prior works concerning (a) the solution of ground-state amplitudes and multipliers;⁸⁴ and (b) the solution of first-order response equations,¹¹³ which leveraged our prior implementation of a generalized Davidson eigenvalue solver¹⁰²,¹²⁸ to implement the solution of systems of linear equations.¹²⁹ We recall that through the use of the TAL-SH tensor library,¹³⁰ a modular component of the ExaTEN-SOR library,¹³¹ all tensor operations in our code are parallelized via OpenMP and can be offloaded to NVIDIA or AMD GPUs whenever these are available.

In our implementation, we divide the calculation into different levels of subroutines: (a) a top-level driver to organize calculations for $\omega$ and $-\omega$ and to apply the symmetrization operator as in Eq. 12; (b) a driver that calculates $\xi^Y$, $\eta^Y$ for all operators under consideration,
solves all corresponding linear response equations for either QR-CC and QR-EOMCC, and
calls the driver to calculate the appropriate response function.

In the case of QR-CC, the response function driver carries out the six permutations
between the triplet of X, Y, Z operators; it calls a subroutine that calculates the terms
within the curly braces in Eq.13 for a given pair or operators X, Y and stores it on set of
2-/4-dimension tensors (singles and doubles manifolds) and performs the contraction with
the $t^z$ perturbed amplitudes. This allows us to, if desired, extract the contributions from
each of the four terms (for each triplet of operators) and further break these down in terms
of the contributions from the singles and doubles manifolds.

## Computational details

All QR-CC, QR-EOMCC calculations were carried out with a development version (revision number 8ef365b7) of the DIRAC code.<sup>127,132</sup> All coupled cluster calculations have been carried out in GPU-accelerated nodes.

As done in our prior EOM-QR calculations,<sup>115</sup> we have used some of the functionality connected to the Cholesky-decomposition approach in ExaCorr<sup>84</sup> to avoid the explicit construction of the 2-electron repulsion integral tensor in AO basis in the integral transformation step and with that reduce the memory footprint of our calculations. We employed a thresholds of 10<sup>-5</sup> for all systems except HAt and HTs, for which the threshold was set to 10<sup>-4</sup>. The outcome of response calculations is not significantly affected by these approximations.

The Dunning basis set<sup>133-135</sup> augmented by three diffuse functions (denoted by t-aug-cc-pVTZ) was used for the hydrogen atom. For all other atoms, the Dyall triple zeta sets (denoted by t-aug-dyall.v3z) were employed in general, though in some cases and for elements up to bromide the t-aug-cc-pVTZ set was used to make a direct comparison to our prior EOM-QR calculations<sup>115</sup> possible. The diffuse functions in question were automatically generated by DIRAC as an even-tempered sequence based on the most diffuse exponents for
all angular momenta primitives in the original basis.

In addition to the non-relativistic<sup>136,137</sup> (as activated by the `.Levy-Leblond` keyword) and eXact 2-Component (X2C) relativistic,<sup>123</sup> in the coupled cluster calculations we also investigated the molecular mean-field X2C method based on the Dirac-Coulomb (<sup>2</sup>DC<sup>M</sup>) and Dirac-Coulomb-Gaunt (<sup>2</sup>DCG<sup>M</sup>) Hamiltonians, as well as the spin-free (SF) variants of the X2C and <sup>2</sup>DCG<sup>M</sup> Hamiltonians. In what follows, we shall use the term orbital as shorthand for both spinors and spin-orbitals, depending on the Hamiltonian used in the calculation. In X2C calculations including spin-orbit coupling, we make use of atomic-mean field SOC integrals, obtained from a calculation employing the Douglas-Kroll-Hess Hamiltonian.

The relativistic and non-relativistic calculations have been carried out with the Gaussian type<sup>138</sup> and point charge nucleus model, respectively.

The molecular structures employed in our calculations have been taken from Huber and Herzberg<sup>139</sup> for HX (X=F, Cl, Br, I), from Gomes and Visscher<sup>140</sup> for HAt, and from de Farias<sup>141</sup> for HTs, so respectively: H-F (0.91680 Å), H-Cl (1.27455 Å), H-Br (1.41443 Å), H-I (1.60916 Å), H-At (1.722 Å) and H-Ts (1.9676 Å). As the choice of orientation in these systems may affect the signs of the electric dipole and first hyperpolarizability (but not their absolute value), we note our results have been tabulated following the convention that the heavy atoms are placed at the origin of the coordinate system, with the molecular axis aligned with the z axis of the coordinate system.

For the most part we retained the correlation spaces defined by Yuan et al.<sup>115</sup>, which include valence electrons and virtual orbitals with energies up to 5 a.u.; However, in some cases (Xe, Rn, Og) we also explored the correlation of the (n-1)d shell and in selected cases, larger virtual cutoffs (100 a.u.), and in others (HF, HCl, HBr) we also carried out calculations correlating all electrons and all virtual orbitals.

EOM-CC excitation energy calculations exploiting point-group symmetry were carried out for the noble gas atoms using the RELCCSD module of DIRAC with the <sup>2</sup>DC<sup>M</sup> and the same basis sets and correlating spaces as employed for the QR-CC calculations.
All inputs, outputs and structures used in this paper are available in the zenodo repository as supplemental information.

## Results and discussion

In the sections below we discuss our results for the first hyperpolarizability and Verdet constant for systems across the periodic table. We note that the choice of computational setup concerning the basis sets and correlation space, as well as a comparison to the B3LYP functional, have been discussed in detail in our work discussing the implementation of QR-EOMCC.¹¹⁵ Therefore, in the following we will mostly focus on a comparison between QR-CC and QR-EOMCC for different Hamiltonians.

### First hyperpolarizability of hydrogen halides (HX, X=F-Ts)

Due to the symmetry of the systems under consideration, we restrict ourselves to the calculation of the parallel component of the static first hyperpolarizability ($\beta_{||}$)¹⁴²

$$ \beta_{||} = \frac{1}{5} \sum_{i=x,y,z} (\beta_{iiz} + \beta_{izi} + \beta_{zii}) \quad (15) $$

where the $\beta_{ijk}$ element is given by the sum-over-states expression:¹

$$ \beta_{ijk}(-\omega_{\sigma}; \omega_{1}, \omega_{2}) = \sum P_{-\sigma,1,2} \sum_{n,m} \frac{\langle 0 | \hat{\mu}_{i} | n \rangle \langle n | \hat{\mu}_{j} | m \rangle \langle m | \hat{\mu}_{k} | 0 \rangle}{(\omega_{n0} - \omega_{\sigma})(\omega_{m0} - \omega_{2})} \quad (16) $$

with $\hat{\mu}_i$ the Cartesian components of the electric dipole operators, $\sum P_{-\sigma,1,2}$ indicating the sum of six terms by permuting the pairs $(i, -\omega_\sigma)$, $(j, \omega_1)$, $(k, \omega_2)$, and $\omega_{n0}$ denoting the excitation energy between the ground state and excited state $n$. In the following we only
calculated two unique components, $\beta_{zzz}$ and $\beta_{zxx}$, so that $\beta_{||}$ and $\beta_{aniso}$ are calculated from

$$ \beta_{||} = \frac{3}{5}(\beta_{zzz} + 2\beta_{zxx}) \qquad (17) $$

$$ \beta_{aniso} = \beta_{zzz} - 3\beta_{zxx} \qquad (18) $$

respectively. Our results for the frequency-independent case are summarized in tables 2 and 3 for the $\beta_{||}$ and $\beta_{aniso}$. To illustrate the trends for the individual components, in figure 1 we present results for the SF-$^2$DC$^M$ and $^2$DC$^M$ Hamiltonians, while in tables 1 and 2 in the SI we report the values for the $\beta_{zzz}$ and $\beta_{zxx}$ components respectively for all Hamiltonians under consideration.

We recall that our use of a Gaussian nuclear model and Dyall bases in all calculations makes for a slightly different setup than that of Yuan et al.¹¹⁵. For ease of comparison, we also provide results obtained with the Dunning basis set on the halogens, and note small discrepancies for calculations with the Levy-Leblond Hamiltonian arising from the difference in nuclear model.

Focusing first on the QR-EOMCC and QR-CC results for the different Hamiltonians for both $\beta_{||}$ and $\beta_{aniso}$, we observe that for HF and HCl scalar relativistic effects are dominant, with spin-orbit effects representing less than 1% of the total response, something to be expected for these molecules which contain only light elements. For HBr we see that spin-orbit coupling is of some importance–in absolute terms, the hyperpolarizabilities increase with respect to spin-free calculations by about 5% (20%) for $\beta_{||}$ QR-CC (QR-EOMCC) and reach about 8% for HI and 10% for HAt. For HTs, on the other hand, spin-orbit coupling is the dominant effect, with nearly a twofold increase in value compared to spin-free calculations. For HF to HI the absolute values for $\beta_{aniso}$ are typically rather small, and become larger for HAt and HTs.
<table><caption>Table 2: QR-CCSD (CC) and QR-EOMCCSD (EOM) values for the $β_{||}$ component (in a.u.) of the static first hyperpolarizability of hydrogen halides, as well as their difference ($Δβ_{||}$). All calculations correlate eight (valence) electrons up to 5 a.u., except those marked by “*”, in which all electrons and virtuals are correlated; $^†$calculated with Dunning basis sets for the halogens; nc: not calculated.</caption><thead><tr><th>Method</th><th>Hamiltonian</th><th>HF</th><th>HCl</th><th>HBr</th><th>HI</th><th>HAt</th><th>HTs</th></tr></thead><tbody><tr><td rowspan="8">CC</td><td>LL</td><td>-8.0441</td><td>-8.2236</td><td>-5.5096</td><td>1.5590</td><td>9.9337</td><td>46.4544</td></tr><tr><td>LL<sup>†</sup></td><td>-8.0370</td><td>-8.7902</td><td>-5.8960</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-8.0483</td><td>-8.0824</td><td>-4.1250</td><td>6.3168</td><td>35.6445</td><td>173.6980</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-8.0483</td><td>-8.0821</td><td>-4.1253</td><td>6.3164</td><td>35.6457</td><td>173.9446</td></tr><tr><td>X2C</td><td>-8.0537</td><td>-8.1031</td><td>-4.3473</td><td>5.7901</td><td>39.5212</td><td>476.3573</td></tr><tr><td>X2C<sup>†</sup></td><td>-8.0407</td><td>-8.6627</td><td>-4.7106</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-8.0498</td><td>-8.0942</td><td>-4.3343</td><td>5.7341</td><td>39.7956</td><td>495.1556</td></tr><tr><td><sup>2</sup>DC<sup>M,*</sup></td><td>-7.9142</td><td>-8.0537</td><td>-4.7243</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>-8.0494</td><td>-8.0918</td><td>-4.3212</td><td>5.7574</td><td>39.8091</td><td>493.6418</td></tr><tr><td rowspan="8">EOM</td><td>LL</td><td>-7.4972</td><td>-5.6034</td><td>-0.2535</td><td>11.1413</td><td>23.5288</td><td>66.9526</td></tr><tr><td>LL<sup>†</sup></td><td>-7.5315</td><td>-6.2493</td><td>-0.6946</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-7.4993</td><td>-5.4499</td><td>1.1741</td><td>15.6862</td><td>46.4367</td><td>164.0866</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-7.4994</td><td>-5.4502</td><td>1.1731</td><td>15.6839</td><td>46.4288</td><td>164.1359</td></tr><tr><td>X2C</td><td>-7.5046</td><td>-5.4712</td><td>0.9285</td><td>14.8533</td><td>45.5480</td><td>405.8769</td></tr><tr><td>X2C<sup>†</sup></td><td>-7.5333</td><td>-6.1101</td><td>0.5181</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-7.5009</td><td>-5.4627</td><td>0.9445</td><td>14.8124</td><td>45.8277</td><td>423.0797</td></tr><tr><td><sup>2</sup>DC<sup>M,*</sup></td><td>-7.6050</td><td>-5.6405</td><td>-0.4133</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>-7.5016</td><td>-5.4658</td><td>0.9341</td><td>14.7813</td><td>45.8455</td><td>422.7453</td></tr><tr><td rowspan="8">Δβ<sub>||</sub></td><td>LL</td><td>-0.5469</td><td>-2.6203</td><td>-5.2561</td><td>-9.5822</td><td>-13.5951</td><td>-20.4982</td></tr><tr><td>LL<sup>†</sup></td><td>-0.5055</td><td>-2.5409</td><td>-5.2014</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-0.5490</td><td>-2.6324</td><td>-5.2991</td><td>-9.3694</td><td>-10.7921</td><td>9.6114</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-0.5490</td><td>-2.6319</td><td>-5.2985</td><td>-9.3675</td><td>-10.7831</td><td>9.8088</td></tr><tr><td>X2C</td><td>-0.5491</td><td>-2.6319</td><td>-5.2758</td><td>-9.0631</td><td>-6.0268</td><td>70.4803</td></tr><tr><td>X2C<sup>†</sup></td><td>-0.5074</td><td>-2.5526</td><td>-5.2287</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-0.5490</td><td>-2.6316</td><td>-5.2787</td><td>-9.0782</td><td>-6.0321</td><td>72.0758</td></tr><tr><td><sup>2</sup>DC<sup>M,*</sup></td><td>-0.3092</td><td>-2.4132</td><td>-4.3110</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>-0.5478</td><td>-2.6261</td><td>-5.2553</td><td>-9.0239</td><td>-6.0364</td><td>70.8965</td></tr></tbody></table>

In all cases, there are no significant differences between the (SF-)X2C and (SF-)²DCM Hamiltonians, in spite of the more approximate starting point of the X2C calculations, and the effect of the Gaunt interaction is only non-negligible for the HAt and HTs molecules (though it only contributes to about 1%).

We have noticed small differences between our results and those previously reported,¹¹⁵
the latter showing larger (in absolute value) hyperpolarizabilities. We have traced this difference to the use of Dyall basis sets for the current calculations, as opposed to the Dunning basis sets. This difference can be understood as a consequence of the bias towards somewhat less diffuse exponents as part of the process of obtaining relativistic energy-optimized basis sets.

<figure><img src="image_2.png" /><figcaption>Figure 1: QR-CC and QR-EOMCC dipole hyperpolarizabilities components ($\beta_{zxx}$ and $\beta_{zzz}$, in a.u.) and the respective differences ($\Delta\beta_{zxx}$ and $\Delta\beta_{zzz}$, in a.u.) calculated with the $^2$DC$^M$ Hamiltonians for the HX series, upon correlating the nsnp shells (and with virtual threshold at 5 a.u.).</figcaption></figure>

Focusing now first on QR-CC values for $\beta_{||}$, for HF to HBr we observe negative values (similar in magnitude for HF and HCl, but less negative for HBr), and positive for HI, HAt, and HTs, with rather large values for the latter molecule. These trends result from the fact that the clearly negative $\beta_{zzz}$ component for HF initially becomes less negative as the halogen becomes heavier, changes sign for HAt, and becomes strongly positive for HTs. The $\beta_{zxx}$ component shows a similar trend, with positive values from HBr onwards, but is smaller in magnitude than $\beta_{zzz}$ for most systems. These trends are qualitatively the same for QR-EOMCC across the series, with as important difference that the sign change in $\beta_{||}$
upon going down the series already occurs for HBr.

Quantifying the differences between QR-CC and QR-EOMCC by computing $(\Delta\beta_\parallel)$, we see that differences are small for HF but become non-negligible already for HCl and remain sizable for the other species. In relative terms, the differences for $\beta_\parallel$ are about 6% for HF, but grow to about 30% for HCl, 120% for HBr and 150% for HI. For the very large positive hyperpolarizability of HTs, the difference is in absolute sense the largest, but amounts to only 15% of the $^2\text{DCG}^M$ value. For the more accessible HAt molecule, the effect is similar but less pronounced. Finally, we see essentially the same trends when correlating more electrons and virtuals in the lighter systems (HF to HBr), though we do observe a somewhat smaller difference between QR-CC and QR-EOMCC for HBr than when only the valence is correlated. This is driven by a larger increase in the $\beta_{zzz}$ for QR-EOMCC than for QR-CC.

We note the situation is a bit different for $\beta_{\text{aniso}}$, where we observe that apart from HTs, the difference between QR-CC and QR-EOMCC is fairly modest across the series.

Going beyond the static case, in figure 2 we present the results for second harmonic generation (SHG) calculations on HI, which corresponds to the case in which $\omega_B = \omega_C = \omega_A/2$. As this property has been discussed in detail in our work presenting QR-EOMCC,$^{115}$ we refer the reader to it for further details on the behavior of the dispersion curves. We nevertheless highlight the fact that in the region around $\omega = \omega_B = \omega_C = 0.1$ a.u., corresponding to half the excitation energy of the spin-orbit allowed transition towards the $a^3\Pi_{0+}$ state, we see the expected pole for the quadratic response functions. Beside that, we observe that across the range of frequencies sampled, the difference between QR-CC and QR-EOMCC results amounts to a constant shift.
<table><caption>Table 3: QR-CCSD (CC) and QR-EOMCCSD (EOM) values for the $\beta_{aniso}$ component (in a.u.) of the static first hyperpolarizability of hydrogen halides, as well as their difference ($\Delta\beta_{aniso}$). All calculations correlate eight (valence) electrons virtuals up to 5 a.u., except those marked by '*' in which all electrons and virtuals are correlated; $^1$calculated with Dunning basis sets for the halogens; nc: not calculated.</caption><thead><tr><th>Method</th><th>Hamiltonian</th><th>HF</th><th>HCl</th><th>HBr</th><th>HI</th><th>HAt</th><th>HTs</th></tr></thead><tbody><tr><td rowspan="7">CC</td><td>LL</td><td>-4.9459</td><td>-10.3349</td><td>-14.5008</td><td>-16.3776</td><td>-18.5536</td><td>-50.1339</td></tr><tr><td>LL<sup>†</sup></td><td>-4.6309</td><td>-10.0969</td><td>-14.3693</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-4.9455</td><td>-10.4108</td><td>-15.4252</td><td>-20.1042</td><td>-41.2161</td><td>-191.5107</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-4.9454</td><td>-10.4091</td><td>-15.4184</td><td>-20.0927</td><td>-41.1858</td><td>-191.6326</td></tr><tr><td>X2C</td><td>-4.9536</td><td>-10.4424</td><td>-16.0964</td><td>-24.8101</td><td>-83.2791</td><td>-599.5770</td></tr><tr><td>X2C<sup>†</sup></td><td>-4.6278</td><td>-10.1948</td><td>-15.9223</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-4.9467</td><td>-10.4348</td><td>-16.0811</td><td>-25.0659</td><td>-84.7735</td><td>-618.4277</td></tr><tr><td rowspan="3">EOM</td><td><sup>2</sup>DC<sup>M,*</sup></td><td>-5.1300</td><td>-10.6165</td><td>-16.5149</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>-4.9478</td><td>-10.4450</td><td>-16.1219</td><td>-25.1406</td><td>-84.9810</td><td>-617.7646</td></tr><tr><td>LL</td><td>-5.2227</td><td>-10.2724</td><td>-14.6625</td><td>-15.0853</td><td>-16.4412</td><td>-43.4775</td></tr><tr><td rowspan="7">Δβ<sub>aniso</sub></td><td>LL<sup>†</sup></td><td>-4.9146</td><td>-9.9864</td><td>-14.4596</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-5.2236</td><td>-10.3699</td><td>-15.8135</td><td>-19.9561</td><td>-43.5784</td><td>-195.9931</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-5.2234</td><td>-10.3687</td><td>-15.8100</td><td>-19.9571</td><td>-43.5978</td><td>-196.3284</td></tr><tr><td>X2C</td><td>-5.2315</td><td>-10.4017</td><td>-16.4911</td><td>-24.6949</td><td>-85.2353</td><td>-553.0503</td></tr><tr><td>X2C<sup>†</sup></td><td>-4.9126</td><td>-10.1052</td><td>-16.2428</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-5.2248</td><td>-10.3948</td><td>-16.4807</td><td>-24.9707</td><td>-86.8113</td><td>-570.8098</td></tr><tr><td><sup>2</sup>DC<sup>M,*</sup></td><td>-5.3591</td><td>-10.5605</td><td>-16.7843</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>-5.2252</td><td>-10.4008</td><td>-16.5012</td><td>-24.9851</td><td>-86.8673</td><td>-570.0000</td></tr><tr><td rowspan="7"></td><td>LL</td><td>0.2768</td><td>-0.0624</td><td>0.1618</td><td>-1.2923</td><td>-2.1124</td><td>-6.6564</td></tr><tr><td>LL<sup>†</sup></td><td>0.2838</td><td>-0.1105</td><td>0.0903</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>0.2780</td><td>-0.0409</td><td>0.3883</td><td>-0.1482</td><td>2.3622</td><td>4.4824</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>0.2781</td><td>-0.0404</td><td>0.3915</td><td>-0.1357</td><td>2.4120</td><td>4.6959</td></tr><tr><td>X2C</td><td>0.2780</td><td>-0.0407</td><td>0.3947</td><td>-0.1151</td><td>1.9562</td><td>-46.5267</td></tr><tr><td>X2C<sup>†</sup></td><td>0.2848</td><td>-0.0896</td><td>0.3205</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>0.2781</td><td>-0.0400</td><td>0.3997</td><td>-0.0952</td><td>2.0379</td><td>-47.6178</td></tr><tr><td></td><td><sup>2</sup>DC<sup>M,*</sup></td><td>0.2291</td><td>-0.0560</td><td>0.2695</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>0.2774</td><td>-0.0442</td><td>0.3793</td><td>-0.1555</td><td>1.8864</td><td>-47.7646</td></tr></tbody></table>

In the following we look more closely at the differences between QR-CC and QR-EOMCC by analyzing the contributions from the different terms to the response functions, found in tables 4 and 3 for QR-CC and QR-EOMCC respectively, as we go across the series from HF to HI.

For this analysis, it is interesting to also consider the results for the static dipole polarizability.
<figure><img src="image_3.png" /><figcaption>Figure 2: QR-CC and QR-EOMCC $^2$DC$^M$ dispersion curves for the $\beta_{zzz}$, $\beta_{zxx}$ components (top: general view; bottom: detailed view of the low-frequency region) of the hyperpolarizability of the HI molecule for second harmonic generation (SHG) calculations ($\omega = \omega_B = \omega_C$, in a.u.). We note the pole at around 0.10 a.u. corresponding to half of the excitation energy towards the $a^3\Pi_{0+}$ state.$^{115}$</figcaption></figure>
larizabilties for the same molecules, since the difference between LR-CC and LR-EOMCC calculations is connected to the presence of the $Ft^Y$ term in the LR-CC response function (and therefore the right-hand side transition moment), whereas LR-EOMCC employs a simpler expression (for both methods perturbed amplitudes $t^Y$ are obtained from eq. 10a).

These dipole polarizabilities, shown in figure 3 (values for dipole moments and polarizabilities are reported in the SI) increase across the series, which is consistent with the increase in size of the halides accompanied by their decreasing electronegativity. The differences between CC and EOMCC do increase across the series for the $\alpha_{zz}$ and $\alpha_{xx}$ components and consequently, for the mean polarizability $\bar{\alpha}$ as well. However, in contrast to the hyperpolarizabilities, in relative terms these differences are roughly the same and of about 2% for $\alpha_{zz}$ and 1.5% for $\alpha_{xx}$.

<figure><img src="image_4.png" /><figcaption>Figure 3: Mean dipole polarizabilities (left, $\bar{\alpha} = 1/3(\alpha_{zz} + 2\alpha_{xx})$ in a.u.) and anisotropies (right, $\Delta\alpha = \alpha_{zz} - \alpha_{xx}$ in a.u.) calculated with $^2DC^M$ Hamiltonian for the HX series with LR-CC and LR-EOMCC, upon correlating the $nsnp$ shells (and with virtual threshold at 5 a.u.). The absolute differences between the two approaches ($\Delta$, in a.u.) are also presented.</figcaption></figure>

The difference between CC and EOMCC is more strongly felt for the anisotropy, which
is about 3% for HF, but 8-12% for HCl to HAt, and reaches about 30% for HTs, though
in absolute terms the differences are still modest. We note that our CCSD dipole moments
compare well with available experimental¹⁴³,¹⁴⁴ or theoretical work (for HTs),¹⁴⁵ though for
reaching benchmark accuracy we would need to account for higher excitations, something
which is beyond the scope of this work. These findings add to the body of data supporting
the argument that differences between LR-CC and LR-EOMCC properties are in practice
small.

Considering now the individual contributors to the value of the QR-CC functions, we
see that dominant contributions typically arise from the terms from eq. 14b, involving both
the perturbed amplitudes and multipliers ($\bar{t}^x A^Y \bar{t}$ and $\bar{t}^x B \bar{t} \bar{t}$). Furthermore, we observe that
these two contributions to the final value in general tend to counteract each other, and
that the magnitude of the “doubles” contributions increases as the halide becomes heavier.
Terms from eq. 14a, involving only perturbed amplitudes and ground-state multipliers ($F^x \bar{t} \bar{t}$
and $G \bar{t} \bar{t} \bar{t}$) show smaller (but non-negligible) net contributions than those above. They also
increase in absolute value as the halide becomes heavier.

The relative importance of contribution of the terms in eq. 14b also changes depending
on which response function we look at. For $\langle\langle Z; Z, Z \rangle\rangle_0$, the term $\bar{t}^x A^Y \bar{t}$ is the one which
contributes the most, but for $\langle\langle Z; X, X \rangle\rangle_0$ there are more important cancellations between
the “singles” and “doubles” contributions from it so that the $\bar{t}^x B \bar{t}^y$ term becomes a more
important net contributor, being the dominant one for the HI molecule followed by $F^x \bar{t} \bar{t} \bar{t}$,
whereas $\bar{t}^x A^Y \bar{t}^z$ and $G \bar{t} \bar{t} \bar{t} \bar{t}$ show similar contributions.

For QR-EOMCC, we see some similarities to the QR-CC picture, in that contributions
beyond “singles” become increasingly important as the halogen becomes heavier, while the
term $^{EOM}\eta^X A^Y \bar{t}$ is typically a key contribution and dominates for the lightest elements. The
value of the $^{EOM}\eta^X A^Y \bar{t}$ contribution also appears to follow closely that of $\bar{t}^x A^Y \bar{t} + \bar{t}^x B \bar{t} \bar{t}$ in QR-
CC. The terms $-\bar{t} \bar{t} \bar{t} \bar{\xi}^Z$ and $-\bar{t} \bar{t} \bar{t} \bar{\xi}^X$, on the other hand, become increasingly more important
for heavier systems, and for $\langle\langle Z; Z, Z \rangle\rangle_0$ of HBr and HI, they offset the other contribution
and are largely responsible for the sign changes between QR-CC and QR-EOMCC for these systems.

## Verdet constant of Xe, Rn and Og

In table 4 we present our results for the calculation of the Verdet constant for Xe, Rn and Og. In line with our prior work, we considered three laser wavelengths (589 nm, 694.3 nm, and 1064 nm) to compare QR-EOMCC and QR-CC for different Hamiltonians. The Verdet constants are evaluated as quadratic response functions:³⁰,⁷³

$$V(\omega) = \omega \frac{eN\epsilon_{xyz}}{24c_0\epsilon_0m_e} \mathrm{Im}\langle\langle\hat{\mu}_x; \hat{\mu}_y, \hat{m}_z\rangle\rangle_{\omega,0} \quad (19)$$

with N the number density of the gas, e the elementary charge, $m_e$ the electron mass, $c_0$ the speed of light in vacuo, and $\hat{m}_z$ is magnetic dipole moment operator. Given the atomic symmetry, we only needed to calculate one unique component and could obtain $V(\omega)$ through the expression

$$V(\omega) = \omega \frac{eN}{4c_0\epsilon_0m_e} \mathrm{Im}\langle\langle\hat{Y}; \hat{X}, \hat{m}_z\rangle\rangle_{\omega,0}. \quad (20)$$

Comparing first the differences between QR-CC and QR-EOMCC, we observe that these become larger as the wavelength decreases, and increase as well as the atoms become heavier. However, if we consider the magnitude of these differences, we see that the differences are rather small overall, between 1.1% and 1.3%, and are very similar for the different wavelengths. These additional calculations with respect to those already reported by,¹¹⁵ allow us to more clearly separate contributions from different relativistic effects. With respect to the Hamiltonian and as discussed previously, the Verdet constant for Xe is still well described by non-relativistic Hamiltonians,¹¹⁵,¹⁴⁶ and scalar relativity is the dominant relativistic effect. Furthermore, we observe no difference between (SF-)X2C and (SF-)²DCM results, and find only rather small differences X2C and ²DCM. As this is an atomic system, these differences are primarily due to the difference in Hamiltonian used to compute the spin-orbit screening
in the mean-field approximation used in the X2C approach. Adding the Gaunt interaction to this mean-field screening acts to slightly decrease the $^2$DC$^M$ value, in the direction of the spinfree and X2C (Douglas-Kroll-Hess based screening) result. We also observe that in spite of the fact that QR-CC values are smaller than QR-EOMCC ones, they still overestimate the experimental results. For Rn relativistic effects are important and spin-orbit coupling effects significantly increase the Verdet constant. We again see some small differences between X2C and $^2$DC$^M$. For Og, spin-free results with X2C and $^2$DC$^M$ are roughly the same, indicating that the bare nucleus and screened X2C transformation matrices are very similar (this is the only difference between the two spin-free approaches, see$^{122}$ for a more extensive discussion). With spin-orbit coupling differences between X2C and $^2$DC$^M$ are more noticeable at about 3%. For this superheavy element, the spin-orbit effect is very significant and even dominates over scalar relativity, with $^2$DC$^M$ values being roughly three times that of SF-$^2$DC$^M$. While much larger in magnitude, the Gaunt interaction behaves similarly in Og as in Xe, reducing the Verdet constant at all frequencies.

In figure 4 we present the frequency dependence of the Verdet constant obtained with the $^2$DC$^M$ Hamiltonian (see SI for the values for each atom at different frequencies). From these data, we observe that no significant discrepancy arises from the use of QR-EOMCC instead of QR-CC. Across the range of studied wave lengths, the differences are only of the order of 1.1-1.3 %. The magnitude of the Verdet constant for Rn is found to be roughly twice of that for Xe across the range of studied wavelengths, while the one of Og is much larger still, with its magnitude being 4.5 times larger for the longest to about 6.5 times larger for the shortest wavelengths. This is in stark contrast to what is observed for the spin-free calculation presented in table 4, where we see changes of only a factor of roughly 1.5 going from Xe to Rn, and of 1.8 going from Rn to Og. This is consistent with prior theoretical investigations of Og, that predict this element to be much more polarizable due to the extremely large spin-orbit effects in this superheavy element.$^{147,148}$ While the first excited state is lowered due to relativity, this strongly increased polarizability due to spin-
orbit effects manifests itself already far away from this resonance. To document this, we include in the SI calculations for the lowest excited states of these atoms. These show that while the first excited state is progressively lowered by spin-orbit coupling as the atom gets heavier, the wavelengths probed are still far away from the resonances.

<table><caption>Table 4: Verdet Constant V(ω) [in 10⁻³ rad/(T m)] for Gaseous Xe, Rn and Og at Different Laser Wavelengths, calculated with QR-CCSD (denoted by CC) and QR-EOMCCSD (denoted by EOM) upon correlating the *nsnp* shells (and with virtual threshold at 5 a.u.), as well as the difference in Verdet constant between the two approaches (ΔV = V<sup>CC</sup> - V<sup>EOM</sup>). For Xe, experimental values at λ(589 nm) and λ(1064 nm) are respectively 12.30 10⁻³ rad/(T m)<sup>149</sup> and 3.56±0.10 10⁻³ rad/(T m).<sup>146</sup></caption><thead><tr><th rowspan="2">Atom</th><th rowspan="2">Hamiltonian</th><th colspan="3">λ(589 nm)</th><th colspan="3">λ(694.3 nm)</th><th colspan="3">λ(1064 nm)</th></tr><tr><th>CC</th><th>EOM</th><th>ΔV</th><th>CC</th><th>EOM</th><th>ΔV</th><th>CC</th><th>EOM</th><th>ΔV</th></tr></thead><tbody><tr><td rowspan="5">Xe</td><td>SF-X2C</td><td>12.39</td><td>12.54</td><td>-0.15</td><td>8.72</td><td>8.82</td><td>-0.11</td><td>3.60</td><td>3.64</td><td>-0.04</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>12.40</td><td>12.55</td><td>-0.15</td><td>8.72</td><td>8.83</td><td>-0.11</td><td>3.60</td><td>3.64</td><td>-0.04</td></tr><tr><td>X2C</td><td>12.95</td><td>13.10</td><td>-0.15</td><td>9.10</td><td>9.20</td><td>-0.11</td><td>3.74</td><td>3.79</td><td>-0.05</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>13.00</td><td>13.16</td><td>-0.15</td><td>9.14</td><td>9.24</td><td>-0.11</td><td>3.76</td><td>3.81</td><td>-0.05</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>12.97</td><td>13.12</td><td>-0.15</td><td>9.11</td><td>9.22</td><td>-0.11</td><td>3.75</td><td>3.80</td><td>-0.05</td></tr><tr><td rowspan="5">Rn</td><td>SF-X2C</td><td>18.79</td><td>19.01</td><td>-0.22</td><td>13.14</td><td>13.30</td><td>-0.15</td><td>5.38</td><td>5.44</td><td>-0.06</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>18.82</td><td>19.04</td><td>-0.22</td><td>13.16</td><td>13.31</td><td>-0.15</td><td>5.38</td><td>5.45</td><td>-0.06</td></tr><tr><td>X2C</td><td>25.62</td><td>25.90</td><td>-0.28</td><td>17.73</td><td>17.92</td><td>-0.19</td><td>7.14</td><td>7.22</td><td>-0.08</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>26.07</td><td>26.35</td><td>-0.28</td><td>18.03</td><td>18.22</td><td>-0.20</td><td>7.26</td><td>7.34</td><td>-0.08</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>25.95</td><td>26.24</td><td>-0.28</td><td>17.95</td><td>18.15</td><td>-0.20</td><td>7.23</td><td>7.31</td><td>-0.08</td></tr><tr><td rowspan="5">Og</td><td>SF-X2C</td><td>35.45</td><td>35.83</td><td>-0.38</td><td>24.44</td><td>24.71</td><td>-0.27</td><td>9.81</td><td>9.91</td><td>-0.11</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>35.58</td><td>35.95</td><td>-0.38</td><td>24.53</td><td>24.79</td><td>-0.26</td><td>9.84</td><td>9.95</td><td>-0.11</td></tr><tr><td>X2C</td><td>144.66</td><td>146.52</td><td>-1.85</td><td>92.02</td><td>93.20</td><td>-1.18</td><td>33.34</td><td>33.77</td><td>-0.43</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>151.00</td><td>152.95</td><td>-1.95</td><td>95.74</td><td>96.98</td><td>-1.24</td><td>34.55</td><td>35.00</td><td>-0.45</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>149.96</td><td>151.92</td><td>-1.96</td><td>95.12</td><td>96.36</td><td>-1.24</td><td>34.34</td><td>34.79</td><td>-0.45</td></tr></tbody></table>

To further complete the picture, we computed the $^2DC^M$ dipole polarizabilities for the noble gases with LR-CC and LR-EOMCC for the same range of frequencies as for the Verdet constant. The results, presented in the SI, are in line with the findings for the static polarizabilities for the HX systems and our work¹¹³ in that there is no appreciable difference between the two CC variants for this property. Our results are in good agreement with the experimental static dipole polarizabilities for Xe and Rn (27.3 and 35.8 a.u. respectively),¹⁵⁰,¹⁵¹ and likewise for the theoretical predictions for Og (57.98 a.u.) of Smits and coworkers.
ers.⁶⁹ Though we did not calculated values for atomic Ts, our results from HTs capture well the trend of decreasing polarizability from Ts to Og (76.3 to 57.98 a.u.)⁶⁹ predicted in that work. We attribute the observed discrepancies to the lack of higher excitations in CCSD.

<figure><img src="image_5.png" /><figcaption>Figure 4: Left: frequency dependence of the QR-CC Verdet constants calculated with $^2$DC$^M$ Hamiltonian for Xe, Rn and Og, upon correlating the $(n-1)nsnp$ shells (and with virtual threshold at 5 a.u.). † Experimental data from Ingersoll and Liebenberg¹⁴⁹ and Cadène et al.¹⁴⁶. Left: frequency dependence of the difference between QR-CC and QR-EOMCC $(\Delta V = (V_{CC} - V_{EOM})/V_{CC}$, in %).</figcaption></figure>

To check for the contribution of subvalence electrons, we also considered additional calculations in which we retained the virtual space truncation at 5 a.u. but included also the $(n-1)d$ electrons. In our prior QR-EOMCC calculations¹¹⁵ we found, based on non-relativistic calculations (the difference of calculations with the same correlating space, and by correlating all occupied and virtual orbitals), that the effect of correlating electrons beyond the valence shell could result in a decrease of the Verdet constant by about -0.08 [$10^{-3}$ rad/(T m)] for Xe. Our current results, shown in the SI, confirm that this trend is indeed followed by all atoms, though with the current virtual space truncation we only recover about half of this value. This change is consistent with our earlier finding, that upon cor-
relating the (n − 1) d electrons, the excitation energies for all systems are (slightly) shifted
upward. Another change seen upon correlating the (n − 1) d electrons is that the difference
between QR-CC and QR-EOMCC values increases somewhat, though in relative terms these
differences remain close to 1% overall.

To conclude this section we note that, unlike the hyperpolarizability case, our results
indicate that QR-CC and QR-EOMCC perform very much alike for the Verdet constant,
and notably better than common density functional approximations such as B3LYP.¹¹⁵ To
keep computational costs manageable compared to such more approximate, but also much
more efficient approaches, it may indeed be advantageous to resort to QR-EOMCC given its
somewhat smaller memory footprint and cost as compared to QR-CC.

Conclusions

In this work we have presented an implementation in the GPU-accelerated ExaCorr module
of coupled cluster quadratic response (QR-CC) theory which can be used in combination
with both (2- and 4-component based) relativistic and non-relativistic Hamiltonians, thus
extending our prior implementation quadratic response based on the equation of motion
(QR-EOMCC) formalism .<sup>115</sup>

We have showcased this implementation by carrying out a detailed comparison between
QR-CC and QR-EOMCC in the calculation of both static and dynamic first hyperpolariz-
abilities for the hydrogen halide series (for HF to HTs), as well as the calculation of the
Verdet constant for noble gases (Xe, Rn and Og). In addition to the comparison of non-
relativistic and X2C Hamiltonians as done previously, we have carried out calculations with
the X2C molecular mean field approach, and gauged the contribution of spin-orbit coupled
and scalar relativistic effects.

For the first hyperpolarizabilities, we find that for the HF molecule the difference between
QR-CC and EOM-CC is relatively small (around 4%), in line with the studies reported in
the literature employing non-relativistic approaches that do not see significant differences between the two approaches. The differences between QR-CC and QR-EOMCC do however increase as one goes down the periodic table, with differences of about 20% already for HCl and a sign change for $\beta_{zzz}$ in HI. The overall shape of the dispersion curves remains unchanged, indicating that the EOM approximation yields qualitatively but not always quantitatively correct results. The contribution of spin-orbit coupling to the hyperpolarizability is rather small up to HBr, but becomes significant from HI onwards, also for the static case. Truncation of the correlating space appears to affect QR-EOMCC more than QR-CC for the $\beta_{zzz}$ component for HBr, but the additional electron correlation recovered does not modify the trends upon going down the series from HF to HBr.

In contrast to the hyperpolarizabilities, for the Verdet constant the difference between QR-CC and EOM-CC is minor and roughly constant percentage-wise (about 1%) for all three systems studied. Spin-orbit coupling accounts for a relatively small part of the response (about 4%) in Xe, becomes important for Rn (about 35-40%) and absolutely dominates for Og. Correlating the $(n-1)d$ electrons in addition to the valence reduces the magnitude of the Verdet constants by about 10%.

The calculations presented here, albeit restricted to a relatively small set of atomic and molecular systems serve to highlight similarities, but also sometimes important differences, in behavior between the QR-CC and QR-EOMCC approaches across the periodic table and for different molecular properties. While QR-EOMCC is advantageous in terms of computational cost and implementation complexity with respect to QR-CC, it is thus important to further characterize differences between the two approaches in the calculation of nonlinear properties for heavier element systems.
## Acknowledgement

This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725 (allocations CHM160, CHM191 and CHP109). XY, LH and ASPG acknowledge funding from projects CPER WaveTech, Labex CaPPA (Grant No. ANR-11-LABX-0005-01), ANR CompRIXS (Grant Nos. ANR-19-CE29-0019 and DFG JA 2329/6-1), ANR SCREECHES (Grant Nos. ANR-24-CE29-0904) the I-SITE ULNE project OVERSEE and MESONM International Associated Laboratory (LAI) (Grant No. ANR-16-IDEX-0004), as well support from the French national supercomputing facilities (Grant Nos. DARI A0150801859, A0170801859) and the HPC center at the University of Lille. For the purpose of Open Access, a CC-BY public copyright license has been applied by the authors to the present document and will be applied to all subsequent versions up to the Author Accepted Manuscript arising from this submission.

## Supporting Information Available

The data (input/output) corresponding to the calculations of this paper are available at the Zenodo repository under DOI: 10.5281/zenodo.15738821.

The Supporting Information is available free of charge on the ACS Publications website at DOI: XXX.

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Supplementary information: A Comparison of
Relativistic Coupled Cluster and Equation of
Motion Coupled Cluster Quadratic Response
Theory

Xiang Yuan,†,‡ Loïc Halbert,† Lucas Visscher,¶ and André Severo Pereira

Gomes*,†

†Univ. Lille, CNRS, UMR 8523 - PhLAM - Physique des Lasers Atomes et Molécules, F-59000 Lille, France

‡Department of Chemistry and Pharmaceutical Sciences, Faculty of Science, Vrije Universiteit Amsterdam, 1081 HZ Amsterdam, The Netherlands

¶Department of Chemistry and Pharmaceutical Sciences, Faculty of Science, Vrije Universiteit Amsterdam, 1081 Hz Amsterdam, The Netherlands

E-mail: andre.gomes@univ-lille.fr
Working equations for CCSD quadratic response

In the quadratic response function,¹

$$ \langle\langle X; Y, Z \rangle\rangle_{\omega_Z; \omega_X, \omega_Y} = \frac{1}{2} C^{\pm\omega} P'(X(\omega_X), Y(\omega_Y), Z(\omega_Z)) \\ \times \left\{ \left[ \frac{1}{2} \mathbf{F}^X + \frac{1}{6} \mathbf{G}^X \mathbf{t}(\omega_X) \right]^\gamma \mathbf{t}(\omega_Y) + \bar{\mathbf{t}}^\gamma(\omega_X) \left[ \mathbf{A}^Y + \frac{1}{2} \mathbf{B}^Y \mathbf{t}(\omega_Y) \right] \right\}^z \mathbf{t}(\omega_Z) \quad (1) $$

we have that, with the exception of $G^{XYZ}$, all other terms can be obtained by adapting the working equations reported previously² and the respective computer implementations. Employing Einstein notation, for $G^{XYZ}$ the programmable expressions for are:

$$ \begin{aligned} \left( G^{XY} \right)_e^m = & - \bar{t}_c^i (V_{ae}^{km}) \cdot t_i^a \cdot t_k^Y - \bar{t}_a^k (V_{ce}^{im}) \cdot t_i^a \cdot t_k^Y - \bar{t}_e^k (V_{ca}^{mi}) \cdot t_i^a \cdot t_k^Y \\ & - \bar{t}_e^i (V_{ac}^{mk}) \cdot t_i^a \cdot t_k^Y - \bar{t}_c^m (V_{ea}^{ki}) \cdot t_i^a \cdot t_k^Y - \bar{t}_a^m (V_{ec}^{ik}) \cdot t_i^a \cdot t_k^Y \\ & + \bar{t}_{ac}^{nk} (V_{nc}^{im}) \cdot t_i^a \cdot t_k^Y - \bar{t}_{fa}^{mk} (V_{ec}^{fi}) \cdot t_i^a \cdot t_k^Y - \bar{t}_{fc}^{im} (V_{ae}^{fk}) \cdot t_i^a \cdot t_k^Y \\ & + \bar{t}_{ec}^{ni} (V_{na}^{mk}) \cdot t_i^a \cdot t_k^Y - \bar{t}_{fe}^{ik} (V_{ac}^{fm}) \cdot t_i^a \cdot t_k^Y + \bar{t}_{ac}^{nm} (V_{ne}^{ik}) \cdot t_i^a \cdot t_k^Y \\ & + \bar{t}_{ae}^{ko} (V_{cg}^{im} t_o^g) \cdot t_i^a \cdot t_k^Y + \bar{t}_{cf}^{im} (V_{ae}^{ko} t_o^g) \cdot t_i^a \cdot t_k^Y \\ & + \bar{t}_{eg}^{ki} (V_{ca}^{mo} t_o^g) \cdot t_i^a \cdot t_k^Y + \bar{t}_{ca}^{mo} (V_{eg}^{ki} t_o^g) \cdot t_i^a \cdot t_k^Y \\ & + \bar{t}_{ce}^{io} (V_{ag}^{km} t_o^g) \cdot t_i^a \cdot t_k^Y + \bar{t}_{ag}^{km} (V_{ce}^{io} t_o^g) \cdot t_i^a \cdot t_k^Y \\ & - \frac{1}{2} \bar{t}_{bc}^{ji} (V_{ae}^{km}) \cdot t_{ij}^{ab} \cdot t_k^Y - \frac{1}{2} \bar{t}_{ba}^{jk} (V_{ce}^{im}) \cdot t_{ij}^{ab} \cdot t_k^Y - \frac{1}{2} \bar{t}_{eb}^{ij} (V_{ac}^{mk}) \cdot t_{ij}^{ab} \cdot t_k^Y \\ & - \frac{1}{2} \bar{t}_{ba}^{jm} (V_{ec}^{ik}) \cdot t_{ij}^{ab} \cdot t_k^Y + \frac{1}{4} \bar{t}_{ce}^{ij} (V_{ab}^{km}) \cdot t_{ij}^{ab} \cdot t_k^Y + \frac{1}{4} \bar{t}_{ab}^{km} (V_{ce}^{ij}) \cdot t_{ij}^{ab} \cdot t_k^Y \\ & - \bar{t}_{be}^{jk} (V_{ac}^{im}) \cdot t_{ij}^{ab} \cdot t_k^Y - \bar{t}_{bc}^{jm} (V_{ae}^{ik}) \cdot t_{ij}^{ab} \cdot t_k^Y \\ & - \frac{1}{2} \bar{t}_{ad}^{kl} (V_{ce}^{im}) \cdot t_i^a \cdot t_{kl}^{cd} - \frac{1}{2} \bar{t}_{cd}^{il} (V_{ae}^{km}) \cdot t_i^a \cdot t_{kl}^{cd} - \frac{1}{2} \bar{t}_{ed}^{kl} (V_{ca}^{mi}) \cdot t_i^a \cdot t_{kl}^{cd} \\ & - \frac{1}{2} \bar{t}_{cd}^{ml} (V_{ea}^{ki}) \cdot t_i^a \cdot t_{kl}^{cd} + \frac{1}{4} \bar{t}_{ae}^{lk} (V_{dc}^{im}) \cdot t_i^a \cdot t_{kl}^{cd} + \frac{1}{4} \bar{t}_{cd}^{mi} (V_{ae}^{lk}) \cdot t_i^a \cdot t_{kl}^{cd} \\ & - \bar{t}_{ed}^{il} (V_{ca}^{km}) \cdot t_i^a \cdot t_{kl}^{cd} - \bar{t}_{ad}^{ml} (V_{ce}^{ki}) \cdot t_i^a \cdot t_{kl}^{cd} \end{aligned} $$
$$ \left( G \bar{t} t \right)_{ef}^{\mathrm{XY}} = - P_{-ef} \bar{t}_{ec}^{mn} (V_{fa}^{ki}) \cdot t_i^a \cdot t_k^c - P_{-mn} \bar{t}_{ef}^{mk} (V_{ca}^{ni}) \cdot t_i^a \cdot t_k^c \\ - P_{-ef} \bar{t}_{ea}^{mn} (V_{fc}^{ik}) \cdot t_i^a \cdot t_k^c - P_{-mn} \bar{t}_{ef}^{mi} (V_{ac}^{nk}) \cdot t_i^a \cdot t_k^c \\ + \bar{t}_{ef}^{ki} (V_{ca}^{mn}) \cdot t_i^a \cdot t_k^c + \bar{t}_{ca}^{mn} (V_{ef}^{ki}) \cdot t_i^a \cdot t_k^c \\ - P_{-mn} P_{-ef} \bar{t}_{fa}^{nk} (V_{ec}^{mi}) \cdot t_i^a \cdot t_k^c - P_{-mn} P_{-ef} \bar{t}_{ec}^{mk} (V_{fa}^{nk}) \cdot t_i^a \cdot t_k^c \quad (2) $$

where a, b, c, .. indicate particle lines, i, j, k, ... hole lines, and p, q, r, s, ... general indexes,³ and $P_{pq}$ indicates a permutation operator, with: $P_{-pq}f(...pq...)=f(...pq...)-f(...qp...)$.
Furthermore, $\bar{t}_q^p, \bar{t}_{rs}^{pq}$ denote the single and double ground-state Lagrange multipliers, and $V_{rs}^{pq}$ an antisymmetrized two-electron integral.

## QR-EOMCC equations

Within the EOM-CC approximation, the quadratic response function is expressed below:⁴⁻⁶

$$ \mathrm{EOM} \langle\langle X; Y, Z \rangle\rangle_{\omega_Y, \omega_Z} = \frac{1}{2} C^{\pm \omega} P^{X,Y,Z} \left[ \begin{aligned} & - \mathrm{EOM} \bar{t}^X(\omega_X) \bar{t}^Y(\omega_Y) \bar{t}^Z \xi^Z \\ & + \mathrm{EOM} \bar{t}^X(\omega_X) \mathrm{EOM} \mathbf{A}^Y \bar{t}^Z(\omega_Z) \\ & - \bar{t}^Y(\omega_Y) \mathrm{EOM} \bar{t}^Z(\omega_Z) \xi^X \end{aligned} \right] \quad (3) $$

where $\bar{t}$ indicates the zeroth-order multipliers and $\mathrm{EOM} \mathbf{A}^X$ is the EOM-CC property
Jacobian matrix

$$ \mathrm{EOM} \mathbf{A}_{\mu\nu}^{Y} = \langle \mu | \left[ \bar{Y}, |\nu\rangle \langle \mathrm{HF} | \right] | \mathrm{HF} \rangle \quad (4) $$

$$ \bar{Y} = e^{-\hat{T}} \hat{Y} e^{\hat{T}} \quad (5) $$

The EOM-CC right-hand side first order response equation is the same as the one pre-
sented in the manuscript, whereas the left-hand side first order response equation the $Ft^Y$ term is replaced by $\bar{t}_D\xi_S^Y + (\bar{t}\xi^Y)\bar{t}$, yielding

$$ \text{EOM}^Y \dot{t}(\bar{\mathbf{H}} + \omega_Y \mathbf{I}) = -\eta^X - \bar{t}_D \xi_S^Y + (\bar{t}\xi^Y)\bar{t}. \quad (6) $$

The subscripts "S" and "D" denote singles and doubles blocks, respectively.
First hyperpolarizabilities of HX (X = F-Ts)

<table><caption>Table 1: QR-CCSD (CC) and QR-EOMCCSD (EOM) values for the $\beta_{zxx}$ component (in a.u.) of the first hyperpolarizability of hydrogen halides, as well as their difference ($\Delta\beta_{zxx}$). All calculations correlate eight (valence) electrons virtuals up to 5 a.u., except those marked by ‘*’, in which all electrons and virtuals are correlated; $^\dagger$calculated with Dunning basis sets for the halogens; nc: not calculated.</caption><thead><tr><th>Method</th><th>Hamiltonian</th><th>HF</th><th>HCl</th><th>HBr</th><th>HI</th><th>HAt</th><th>HTs</th></tr></thead><tbody><tr><td rowspan="7">CC</td><td>LL</td><td>-1.6922</td><td>-0.6742</td><td>1.0636</td><td>3.7952</td><td>7.0220</td><td>25.5116</td></tr><tr><td>LL<sup>†</sup></td><td>-1.7528</td><td>-0.9107</td><td>0.9085</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-1.6937</td><td>-0.6120</td><td>1.7100</td><td>6.1264</td><td>20.1247</td><td>96.2015</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-1.6937</td><td>-0.6122</td><td>1.7086</td><td>6.1240</td><td>20.1191</td><td>96.3081</td></tr><tr><td>X2C</td><td>-1.6939</td><td>-0.6125</td><td>1.7702</td><td>6.8921</td><td>29.8295</td><td>278.7012</td></tr><tr><td>X2C<sup>†</sup></td><td>-1.7547</td><td>-0.8486</td><td>1.6143</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-1.6939</td><td>-0.6111</td><td>1.7714</td><td>6.9246</td><td>30.2199</td><td>288.7374</td></tr><tr><td rowspan="3">EOM</td><td><sup>2</sup>DC<sup>M,*</sup></td><td>-1.6121</td><td>-0.5613</td><td>1.7282</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>-1.6936</td><td>-0.6083</td><td>1.7840</td><td>6.9473</td><td>30.2659</td><td>288.1002</td></tr><tr><td>LL</td><td>-1.4545</td><td>0.1867</td><td>2.8480</td><td>6.7308</td><td>11.1312</td><td>31.0130</td></tr><tr><td rowspan="7">Δβ<sub>zxx</sub></td><td>LL<sup>†</sup></td><td>-1.5276</td><td>-0.0858</td><td>2.6604</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-1.4550</td><td>0.2573</td><td>3.5540</td><td>9.2200</td><td>24.1946</td><td>93.8942</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-1.4551</td><td>0.2570</td><td>3.5530</td><td>9.2194</td><td>24.1958</td><td>93.9776</td></tr><tr><td>X2C</td><td>-1.4552</td><td>0.2566</td><td>3.6077</td><td>9.8901</td><td>32.2297</td><td>245.9024</td></tr><tr><td>X2C<sup>†</sup></td><td>-1.5286</td><td>-0.0157</td><td>3.4213</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-1.4553</td><td>0.2581</td><td>3.6110</td><td>9.9316</td><td>32.6382</td><td>255.1885</td></tr><tr><td><sup>2</sup>DC<sup>M,*</sup></td><td>-1.4632</td><td>0.2319</td><td>3.2191</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>-1.4555</td><td>0.2582</td><td>3.6116</td><td>9.9241</td><td>32.6553</td><td>254.9151</td></tr><tr><td rowspan="7"></td><td>LL</td><td>-0.2377</td><td>-0.8609</td><td>-1.7844</td><td>-2.9356</td><td>-4.1092</td><td>-5.5015</td></tr><tr><td>LL<sup>†</sup></td><td>-0.2253</td><td>-0.8249</td><td>-1.7519</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-0.2386</td><td>-0.8693</td><td>-1.8440</td><td>-3.0935</td><td>-4.0698</td><td>2.3073</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-0.2386</td><td>-0.8692</td><td>-1.8445</td><td>-3.0954</td><td>-4.0768</td><td>2.3304</td></tr><tr><td>X2C</td><td>-0.2386</td><td>-0.8691</td><td>-1.8375</td><td>-2.9980</td><td>-2.4002</td><td>32.7988</td></tr><tr><td>X2C<sup>†</sup></td><td>-0.2261</td><td>-0.8329</td><td>-1.8070</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-0.2386</td><td>-0.8692</td><td>-1.8395</td><td>-3.0070</td><td>-2.4183</td><td>33.5489</td></tr><tr><td></td><td><sup>2</sup>DC<sup>M,*</sup></td><td>-0.1489</td><td>-0.7932</td><td>-1.4909</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>-0.2381</td><td>-0.8665</td><td>-1.8276</td><td>-2.9769</td><td>-2.3894</td><td>33.1851</td></tr></tbody></table>
<table><caption>Table 2: QR-CCSD (CC) and QR-EOMCCSD (EOM) values for the $\beta_{zzz}$ component (in a.u.) of the first hyperpolarizability of hydrogen halides, as well as their difference ($\Delta\beta_{zzz}$). All calculations correlate eight (valence) electrons virtuals up to 5 a.u., except those marked by ‘*’, in which all electrons and virtuals are correlated; $^†$calculated with Dunning basis sets for the halogens; nc: not calculated.</caption><thead><tr><th>Method</th><th>Hamiltonian</th><th>HF</th><th>HCl</th><th>HBr</th><th>HI</th><th>HAt</th><th>HTs</th></tr></thead><tbody><tr><td rowspan="7">CC</td><td>LL</td><td>-10.0225</td><td>-12.3576</td><td>-11.3099</td><td>-4.9920</td><td>2.5122</td><td>26.4008</td></tr><tr><td>LL<sup>†</sup></td><td>-9.8894</td><td>-12.8290</td><td>-11.6437</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-10.0265</td><td>-12.2467</td><td>-10.2951</td><td>-1.7249</td><td>19.1581</td><td>97.0937</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-10.0265</td><td>-12.2457</td><td>-10.2927</td><td>-1.7207</td><td>19.1714</td><td>97.2916</td></tr><tr><td>X2C</td><td>-10.0351</td><td>-12.2800</td><td>-10.7858</td><td>-4.1339</td><td>6.2096</td><td>236.5265</td></tr><tr><td>X2C<sup>†</sup></td><td>-9.8918</td><td>-12.7406</td><td>-11.0795</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-10.0285</td><td>-12.2682</td><td>-10.7667</td><td>-4.2923</td><td>5.8862</td><td>247.7845</td></tr><tr><td></td><td><sup>2</sup>DC<sup>M,*</sup></td><td>-9.9662</td><td>-12.3004</td><td>-11.3303</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>-10.0285</td><td>-12.2698</td><td>-10.7700</td><td>-4.2988</td><td>5.8167</td><td>246.5360</td></tr><tr><td rowspan="7">EOM</td><td>LL</td><td>-9.5863</td><td>-9.7123</td><td>-6.1185</td><td>5.1071</td><td>16.9523</td><td>49.5616</td></tr><tr><td>LL<sup>†</sup></td><td>-9.4973</td><td>-10.2439</td><td>-6.4784</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-9.5887</td><td>-9.5979</td><td>-5.1513</td><td>7.7038</td><td>29.0053</td><td>85.6894</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-9.5887</td><td>-9.5976</td><td>-5.1508</td><td>7.7011</td><td>28.9897</td><td>85.6045</td></tr><tr><td>X2C</td><td>-9.5972</td><td>-9.6318</td><td>-5.6680</td><td>4.9753</td><td>11.4539</td><td>184.6568</td></tr><tr><td>X2C<sup>†</sup></td><td>-9.4983</td><td>-10.1522</td><td>-5.9790</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-9.5908</td><td>-9.6206</td><td>-5.6478</td><td>4.8241</td><td>11.1031</td><td>194.7558</td></tr><tr><td></td><td><sup>2</sup>DC<sup>M,*</sup></td><td>-9.7486</td><td>-9.8647</td><td>-7.1271</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>-9.5917</td><td>-9.6261</td><td>-5.6664</td><td>4.7873</td><td>11.0986</td><td>194.7453</td></tr><tr><td rowspan="7">Δβ<sub>zzz</sub></td><td>LL</td><td>-0.4362</td><td>-2.6452</td><td>-5.1914</td><td>-10.0991</td><td>-14.4401</td><td>-23.1608</td></tr><tr><td>LL<sup>†</sup></td><td>-0.3920</td><td>-2.5851</td><td>-5.1653</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td>SF-X2C</td><td>-0.4378</td><td>-2.6488</td><td>-5.1438</td><td>-9.4287</td><td>-9.8472</td><td>11.4044</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>-0.4377</td><td>-2.6481</td><td>-5.1419</td><td>-9.4218</td><td>-9.8182</td><td>11.6871</td></tr><tr><td>X2C</td><td>-0.4379</td><td>-2.6482</td><td>-5.1179</td><td>-9.1092</td><td>-5.2443</td><td>51.8697</td></tr><tr><td>X2C<sup>†</sup></td><td>-0.3935</td><td>-2.5884</td><td>-5.1005</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>-0.4377</td><td>-2.6476</td><td>-5.1189</td><td>-9.1163</td><td>-5.2169</td><td>53.0287</td></tr><tr><td></td><td><sup>2</sup>DC<sup>M,*</sup></td><td>-0.2176</td><td>-2.4356</td><td>-4.2032</td><td>nc</td><td>nc</td><td>nc</td></tr><tr><td></td><td><sup>2</sup>DCG<sup>M</sup></td><td>-0.4368</td><td>-2.6437</td><td>-5.1036</td><td>-9.0861</td><td>-5.2818</td><td>51.7907</td></tr></tbody></table>
<table><caption>Table 3: Breakdown of the contributions (in a.u.) from each of the terms in eq. 1 to the static QR-CCSD response functions $\langle\langle X; Y, Z \rangle\rangle_0$, (after all permutations) for the calculations with the $^2$DCM Hamiltonian and correlating eight (valence) electrons up to 5 a.u. The column “singles” (“doubles”) denote the contributions from the contraction between the singles(doubles) block of the terms in square brackets and the $^Z$ tensor. The column “total” correspond to the sum of the “singles” and “doubles” contributions.</caption><thead><tr><th rowspan="2">Sytem</th><th rowspan="2">Term</th><th colspan="3">⟨⟨Z; Z, Z⟩⟩<sub>0</sub></th><th colspan="3">⟨⟨Z; X, X⟩⟩<sub>0</sub></th></tr><tr><th>total</th><th>singles</th><th>doubles</th><th>total</th><th>singles</th><th>doubles</th></tr></thead><tbody><tr><td rowspan="5">HF</td><td>F<sup>X</sup><sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-0.5139</td><td>-0.2901</td><td>-0.2238</td><td>-0.0748</td><td>-0.0458</td><td>-0.0290</td></tr><tr><td>G<sup>X</sup> t(ω<sub>X</sub>)<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-0.0513</td><td>-0.0698</td><td>0.0185</td><td>0.0011</td><td>-0.0017</td><td>0.0028</td></tr><tr><td>t(ω<sub>X</sub>) B<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-3.1113</td><td>-3.0875</td><td>-0.0238</td><td>-0.9085</td><td>-0.9718</td><td>0.0634</td></tr><tr><td>t(ω<sub>X</sub>) A<sup>Y</sup> t(ω<sub>Z</sub>)</td><td>13.7050</td><td>15.3507</td><td>-1.6457</td><td>2.6761</td><td>3.2016</td><td>-0.5255</td></tr><tr><td>sum</td><td>10.0285</td><td>11.9034</td><td>-1.8749</td><td>1.6939</td><td>2.1823</td><td>-0.4884</td></tr><tr><td rowspan="5">HCl</td><td>F<sup>X</sup><sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-1.2492</td><td>-0.7032</td><td>-0.5460</td><td>-0.3202</td><td>-0.2411</td><td>-0.0791</td></tr><tr><td>G<sup>X</sup> t(ω<sub>X</sub>)<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-0.0845</td><td>-0.1599</td><td>0.0754</td><td>0.0442</td><td>0.0353</td><td>0.0088</td></tr><tr><td>t(ω<sub>X</sub>) B<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-6.3625</td><td>-6.7142</td><td>0.3517</td><td>-2.6790</td><td>-2.9266</td><td>0.2476</td></tr><tr><td>t(ω<sub>X</sub>) A<sup>Y</sup> t(ω<sub>Z</sub>)</td><td>19.9643</td><td>25.0924</td><td>-5.1281</td><td>3.5662</td><td>5.4898</td><td>-1.9235</td></tr><tr><td>sum</td><td>12.2682</td><td>17.5151</td><td>-5.2469</td><td>0.6111</td><td>2.3573</td><td>-1.7462</td></tr><tr><td rowspan="5">HBr</td><td>F<sup>X</sup><sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-1.9007</td><td>-1.0708</td><td>-0.8299</td><td>-0.5289</td><td>-0.4014</td><td>-0.1275</td></tr><tr><td>G<sup>X</sup> t(ω<sub>X</sub>)<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-0.0834</td><td>-0.2054</td><td>0.1219</td><td>0.0779</td><td>0.0622</td><td>0.0157</td></tr><tr><td>t(ω<sub>X</sub>) B<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-7.7564</td><td>-7.9359</td><td>0.1794</td><td>-4.1734</td><td>-4.5494</td><td>0.3759</td></tr><tr><td>t(ω<sub>X</sub>) A<sup>Y</sup> t(ω<sub>Z</sub>)</td><td>20.5073</td><td>26.9694</td><td>-6.4621</td><td>2.8530</td><td>6.1112</td><td>-3.2582</td></tr><tr><td>sum</td><td>10.7667</td><td>17.7573</td><td>-6.9906</td><td>-1.7714</td><td>1.2225</td><td>-2.9940</td></tr><tr><td rowspan="5">HI</td><td>F<sup>X</sup><sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-1.9820</td><td>-1.1233</td><td>-0.8587</td><td>-0.9588</td><td>-0.7776</td><td>-0.1812</td></tr><tr><td>G<sup>X</sup> t(ω<sub>X</sub>)<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-0.0214</td><td>-0.1844</td><td>0.1630</td><td>0.1575</td><td>0.1370</td><td>0.0205</td></tr><tr><td>t(ω<sub>X</sub>) B<sup>Y</sup> t(ω<sub>Y</sub>)<sup>Z</sup> t(ω<sub>Z</sub>)</td><td>-7.0690</td><td>-6.8321</td><td>-0.2369</td><td>-6.3643</td><td>-6.8736</td><td>0.5093</td></tr><tr><td>t(ω<sub>X</sub>) A<sup>Y</sup> t(ω<sub>Z</sub>)</td><td>13.3647</td><td>19.4919</td><td>-6.1272</td><td>0.2410</td><td>5.1993</td><td>-4.9583</td></tr><tr><td>sum</td><td>4.2923</td><td>11.3520</td><td>-7.0597</td><td>-6.9246</td><td>-2.3150</td><td>-4.6096</td></tr></tbody></table>
<table><caption>Table 4: Breakdown of the contributions (in a.u.) from each of the terms in eq. 3 to the static QR-EOMCCSD response functions $\langle\langle X; Y, Z \rangle\rangle_0$, (after all permutations) for the calculations with the $^2$DC$^M$ Hamiltonian and correlating eight (valence) electrons up to 5 a.u.. The column “singles” (“doubles”) denote the contributions from the contraction between the singles(doubles) block of the $[^{\text{EOM}}\eta^X A^Y]$ and $\bar{t}$ tensors. The column “total” correspond to the sum of the “singles” and “doubles” contributions, plus the ”sd+ds” column, which corresponds to the cross-terms arising from the 4-tensor products $([\bar{t}\bar{t}][\bar{t}\xi^Z]$ and $-[\bar{t}\bar{t}][\bar{t}\xi^X]$).</caption><thead><tr><th rowspan="2">Sytem</th><th rowspan="2">Term</th><th colspan="4">⟨⟨Z; Z, Z⟩⟩<sub>0</sub></th><th colspan="4">⟨⟨Z; X, X⟩⟩<sub>0</sub></th></tr><tr><th>total</th><th>singles</th><th>doubles</th><th>sd+ds</th><th>total</th><th>singles</th><th>doubles</th><th>sd+sd</th></tr></thead><tbody><tr><td rowspan="3">HF</td><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>Z</sup></td><td>-0.7616</td><td>-0.2959</td><td>-0.0249</td><td>-0.4407</td><td>-0.2216</td><td>-0.0860</td><td>-0.0074</td><td>-0.1282</td></tr><tr><td><sup>XY</sup><sub>EOM</sub>η<sup>X</sup><span style="text-decoration: overline;">A</span><sup>Y</sup><sup>Z</sup><span style="text-decoration: overline;">t</span></td><td>11.0952</td><td>13.1353</td><td>-2.0401</td><td>0.0000</td><td>1.8815</td><td>2.4852</td><td>-0.6037</td><td>0.0000</td></tr><tr><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>X</sup></td><td>-0.7429</td><td>-0.2827</td><td>-0.0218</td><td>-0.4384</td><td>-0.2046</td><td>-0.0780</td><td>-0.0058</td><td>-0.1208</td></tr><tr><td>sum</td><td>9.5908</td><td>12.5567</td><td>-2.0868</td><td>-0.8791</td><td>1.4553</td><td>2.3212</td><td>-0.6168</td><td>-0.2490</td></tr><tr><td rowspan="3">HCl</td><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>Z</sup></td><td>-2.3768</td><td>-1.3300</td><td>-0.0495</td><td>-0.9962</td><td>-0.7690</td><td>-0.4307</td><td>-0.0160</td><td>-0.3223</td></tr><tr><td><sup>XY</sup><sub>EOM</sub>η<sup>X</sup><span style="text-decoration: overline;">A</span><sup>Y</sup><sup>Z</sup><span style="text-decoration: overline;">t</span></td><td>14.6500</td><td>21.2000</td><td>-6.5375</td><td>0.0000</td><td>1.3102</td><td>3.4600</td><td>-2.1498</td><td>0.0000</td></tr><tr><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>X</sup></td><td>-2.6526</td><td>-1.3700</td><td>-0.0607</td><td>-1.2181</td><td>-0.7992</td><td>-0.4150</td><td>-0.0174</td><td>-0.3668</td></tr><tr><td>sum</td><td>9.6206</td><td>18.5000</td><td>-6.6477</td><td>-2.2143</td><td>-0.2581</td><td>2.6144</td><td>-2.1833</td><td>-0.6892</td></tr><tr><td rowspan="3">HBr</td><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>Z</sup></td><td>-4.0194</td><td>-2.1189</td><td>-0.0887</td><td>-1.8118</td><td>-1.3421</td><td>-0.7077</td><td>-0.0295</td><td>-0.6049</td></tr><tr><td><sup>XY</sup><sub>EOM</sub>η<sup>X</sup><span style="text-decoration: overline;">A</span><sup>Y</sup><sup>Z</sup><span style="text-decoration: overline;">t</span></td><td>13.9372</td><td>22.4470</td><td>-8.5098</td><td>0.0000</td><td>-0.9642</td><td>2.6929</td><td>-3.4997</td><td>0.0000</td></tr><tr><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>X</sup></td><td>-4.2700</td><td>-2.1914</td><td>-0.0980</td><td>-1.9806</td><td>-1.3046</td><td>-0.6717</td><td>-0.0281</td><td>-0.6048</td></tr><tr><td>sum</td><td>5.6478</td><td>18.1367</td><td>-8.6965</td><td>-3.7923</td><td>-3.6110</td><td>1.3135</td><td>-3.5573</td><td>-1.2097</td></tr><tr><td rowspan="3">HI</td><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>Z</sup></td><td>-6.6787</td><td>-3.5975</td><td>-0.1503</td><td>-2.9309</td><td>-2.2608</td><td>-1.2200</td><td>-0.0492</td><td>-0.9916</td></tr><tr><td><sup>XY</sup><sub>EOM</sub>η<sup>X</sup><span style="text-decoration: overline;">A</span><sup>Y</sup><sup>Z</sup><span style="text-decoration: overline;">t</span></td><td>8.4536</td><td>17.1299</td><td>-8.6763</td><td>0.0000</td><td>-5.6300</td><td>-0.1012</td><td>-5.5288</td><td>-0.0000</td></tr><tr><td>-<span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">t</span><span style="text-decoration: overline;">ξ</span><sup>X</sup></td><td>-6.5990</td><td>-3.8031</td><td>-0.1373</td><td>-2.6586</td><td>-2.0407</td><td>-1.1815</td><td>-0.0390</td><td>-0.8203</td></tr><tr><td>sum</td><td>-4.8241</td><td>9.7293</td><td>-8.9639</td><td>-5.5895</td><td>-9.9316</td><td>-2.5026</td><td>-5.6171</td><td>-1.8119</td></tr></tbody></table>
Polarizabilities of HX (X = F-Ts) and noble gas atoms

<table><caption>Table 5: CCSD equilibrium dipole moment ($\mu_z$) and static dipole polarizabilities for hydrogen halides obtained with the symmetric (CC) and asymmetric (aCC) formulations of LR-CCSD, and LR-EOMCCSD (EOM). Apart from components ($\alpha_{zz}$, $\alpha_{xx}$) we report the mean values and the anisotropy ($\bar{\alpha} = (\alpha_{zz} + 2\alpha_{xx})/3$). $\Delta = \alpha^{CC} - \alpha^{EOM}$ denotes the difference between properties calculated with CC and EOM. All calculations correlate eight (valence) electrons virtuals up to 5 a.u., and all values are given in a.u.; The experimental dipole moment values$^7$ correspond to $\mu_0$ instead of the equilibrium values obtained from the calculations.</caption><thead><tr><th>Property</th><th>Method</th><th>HF</th><th>HCl</th><th>HBr</th><th>HI</th><th>HAt</th><th>HTs</th></tr></thead><tbody><tr><td rowspan="2">μ<sub>z</sub></td><td>CC</td><td>0.6916</td><td>0.4229</td><td>0.3011</td><td>0.1499</td><td>-0.0880</td><td>-0.7989</td></tr><tr><td>Exp.<sup>7</sup><br>CCSD(T)<sup>8</sup></td><td>0.7186</td><td>0.4362</td><td>0.3254</td><td>0.1762</td><td></td><td>-0.7644</td></tr><tr><td rowspan="4">α<sub>zz</sub></td><td>CC</td><td>6.5763</td><td>18.6242</td><td>25.5904</td><td>38.0011</td><td>46.5183</td><td>70.1581</td></tr><tr><td>CC asym</td><td>6.5763</td><td>18.6242</td><td>25.5904</td><td>38.0011</td><td>46.5183</td><td>70.1581</td></tr><tr><td>EOM</td><td>6.6476</td><td>18.9614</td><td>26.0809</td><td>38.8239</td><td>47.4742</td><td>71.5697</td></tr><tr><td>Δ</td><td>-0.0713</td><td>-0.3372</td><td>-0.4905</td><td>-0.8228</td><td>-0.9559</td><td>-1.4116</td></tr><tr><td rowspan="4">α<sub>xx</sub></td><td>CC</td><td>5.4560</td><td>16.9257</td><td>23.5976</td><td>35.4944</td><td>43.5843</td><td>71.5091</td></tr><tr><td>CC asym</td><td>5.4560</td><td>16.9257</td><td>23.5976</td><td>35.4944</td><td>43.5843</td><td>71.5091</td></tr><tr><td>EOM</td><td>5.4924</td><td>17.1394</td><td>23.9058</td><td>36.0189</td><td>44.1797</td><td>72.4536</td></tr><tr><td>Δ</td><td>-0.0364</td><td>-0.2137</td><td>-0.3082</td><td>-0.5244</td><td>-0.5954</td><td>-0.9445</td></tr><tr><td rowspan="4">ᾱ</td><td>CC</td><td>5.8295</td><td>17.4919</td><td>24.2619</td><td>36.3300</td><td>44.5623</td><td>71.0588</td></tr><tr><td>CC asym</td><td>5.8295</td><td>17.4919</td><td>24.2619</td><td>36.3300</td><td>44.5623</td><td>71.0588</td></tr><tr><td>EOM</td><td>5.8775</td><td>17.7467</td><td>24.6308</td><td>36.9539</td><td>45.2779</td><td>72.1590</td></tr><tr><td>Δ</td><td>-0.0480</td><td>-0.2548</td><td>-0.3690</td><td>-0.6239</td><td>-0.7156</td><td>-1.1002</td></tr><tr><td rowspan="4">Δα</td><td>Exp.<sup>9</sup></td><td>5.6±0.1</td><td>17.4±0.01</td><td>23.79±0.5</td><td>37.1±0.5</td><td></td><td></td></tr><tr><td>CC</td><td>1.1203</td><td>1.6985</td><td>1.9928</td><td>2.5067</td><td>2.9340</td><td>-1.3510</td></tr><tr><td>CC asym</td><td>1.1203</td><td>1.6984</td><td>1.9929</td><td>2.5066</td><td>2.9340</td><td>-1.3510</td></tr><tr><td>EOM</td><td>1.1551</td><td>1.8220</td><td>2.1751</td><td>2.8051</td><td>3.2946</td><td>-0.8839</td></tr><tr><td></td><td>Δ</td><td>-0.0349</td><td>-0.1235</td><td>-0.1823</td><td>-0.2984</td><td>-0.3605</td><td>-0.4671</td></tr></tbody></table>
<table><caption>Table 6: Static and dynamic dipole polarizabilities ($\alpha(\omega)$), in a.u.) for noble gases obtained with the symmetric (CC) formulation of LR-CCSD and LR-EOMCCSD (EOM). $\Delta = \alpha^{\text{CC}} - \alpha^{\text{EOM}}$ denotes the difference between properties calculated with CC and EOM. All calculations correlate 8 ($nsnp$) or 18 (($n-1$)d$nsnp$) electrons up to 5 a.u.</caption><thead><tr><th rowspan="2">Atom</th><th rowspan="2">ω</th><th colspan="3">nsnp</th><th colspan="3">(n − 1)dnsnp</th></tr><tr><th>CC</th><th>EOM</th><th>Δ</th><th>CC</th><th>EOM</th><th>Δ</th></tr></thead><tbody><tr><td rowspan="8">Xe</td><td>0.0000</td><td>28.1736</td><td>28.5905</td><td>-0.4169</td><td>27.9533</td><td>28.3785</td><td>-0.4253</td></tr><tr><td>0.0100</td><td>28.1876</td><td>28.6047</td><td>-0.4171</td><td>27.9670</td><td>28.3924</td><td>-0.4254</td></tr><tr><td>0.0200</td><td>28.2297</td><td>28.6473</td><td>-0.4176</td><td>28.0083</td><td>28.4342</td><td>-0.4260</td></tr><tr><td>0.0300</td><td>28.3003</td><td>28.7187</td><td>-0.4185</td><td>28.0774</td><td>28.5043</td><td>-0.4268</td></tr><tr><td>0.0428</td><td>28.4333</td><td>28.8534</td><td>-0.4201</td><td>28.2078</td><td>28.6363</td><td>-0.4285</td></tr><tr><td>0.0500</td><td>28.5293</td><td>28.9505</td><td>-0.4212</td><td>28.3019</td><td>28.7315</td><td>-0.4296</td></tr><tr><td>0.0656</td><td>28.7939</td><td>29.2184</td><td>-0.4244</td><td>28.5612</td><td>28.9941</td><td>-0.4329</td></tr><tr><td>0.0774</td><td>29.0457</td><td>29.4732</td><td>-0.4274</td><td>28.8079</td><td>29.2439</td><td>-0.4359</td></tr><tr><td>0.0900</td><td>29.3720</td><td>29.8033</td><td>-0.4313</td><td>29.1275</td><td>29.5674</td><td>-0.4399</td></tr><tr><td rowspan="8">Rn</td><td>0.0000</td><td>36.4840</td><td>37.0015</td><td>-0.5175</td><td>36.1210</td><td>36.6936</td><td>-0.5725</td></tr><tr><td>0.0100</td><td>36.5100</td><td>37.0278</td><td>-0.5178</td><td>36.1461</td><td>36.7190</td><td>-0.5729</td></tr><tr><td>0.0200</td><td>36.5883</td><td>37.1070</td><td>-0.5187</td><td>36.2217</td><td>36.7956</td><td>-0.5739</td></tr><tr><td>0.0300</td><td>36.7199</td><td>37.2401</td><td>-0.5202</td><td>36.3487</td><td>36.9243</td><td>-0.5755</td></tr><tr><td>0.0428</td><td>36.9695</td><td>37.4925</td><td>-0.5230</td><td>36.5896</td><td>37.1682</td><td>-0.5787</td></tr><tr><td>0.0500</td><td>37.1506</td><td>37.6756</td><td>-0.5250</td><td>36.7644</td><td>37.3453</td><td>-0.5809</td></tr><tr><td>0.0656</td><td>37.6553</td><td>38.1860</td><td>-0.5307</td><td>37.2512</td><td>37.8384</td><td>-0.5872</td></tr><tr><td>0.0774</td><td>38.1426</td><td>38.6787</td><td>-0.5360</td><td>37.7211</td><td>38.3144</td><td>-0.5933</td></tr><tr><td>0.0900</td><td>38.7851</td><td>39.3282</td><td>-0.5431</td><td>38.3403</td><td>38.9415</td><td>-0.6012</td></tr><tr><td rowspan="8">Og</td><td>0.0000</td><td>62.3732</td><td>64.9005</td><td>-2.5273</td><td>60.4439</td><td>61.4848</td><td>-1.0409</td></tr><tr><td>0.0100</td><td>62.4796</td><td>65.0665</td><td>-2.5869</td><td>60.5419</td><td>61.5844</td><td>-1.0424</td></tr><tr><td>0.0200</td><td>62.8024</td><td>65.4570</td><td>-2.6546</td><td>60.8392</td><td>61.8861</td><td>-1.0469</td></tr><tr><td>0.0300</td><td>63.3526</td><td>66.0841</td><td>-2.7315</td><td>61.3456</td><td>62.4002</td><td>-1.0546</td></tr><tr><td>0.0428</td><td>64.4241</td><td>67.2700</td><td>-2.8459</td><td>62.3302</td><td>63.3996</td><td>-1.0695</td></tr><tr><td>0.0500</td><td>65.2263</td><td>68.1455</td><td>-2.9192</td><td>63.0659</td><td>64.1465</td><td>-1.0805</td></tr><tr><td>0.0656</td><td>67.5821</td><td>70.6899</td><td>-3.1079</td><td>65.2194</td><td>66.3321</td><td>-1.1127</td></tr><tr><td>0.0774</td><td>70.0500</td><td>73.3341</td><td>-3.2841</td><td>67.4630</td><td>68.6090</td><td>-1.1460</td></tr><tr><td>0.0900</td><td>73.6588</td><td>77.1808</td><td>-3.5221</td><td>70.7189</td><td>71.9126</td><td>-1.1937</td></tr></tbody></table>
Verdet constants for Xe, Rn and Og

<table><caption>Table 7: Verdet Constant V(ω) [in 10⁻³ rad/(T m)] for Gaseous Xe, Rn and Og at Different Laser Wavelengths, calculated with QR-CCSD (denoted by CC) and QR-EOMCCSD (denoted by EOM) upon correlating the (n-1)dnsp shells (and with virtual threshold at 5 a.u.), as well as the difference in Verdet constant between the two approaches (ΔV = V<sup>CC</sup> - V<sup>EOM</sup>).</caption><thead><tr><th rowspan="2">Atom</th><th rowspan="2">Hamiltonian</th><th colspan="3">λ(589 nm)</th><th colspan="3">λ(694.3 nm)</th><th colspan="3">λ(1064 nm)</th></tr><tr><th>CC</th><th>EOM</th><th>ΔV</th><th>CC</th><th>EOM</th><th>ΔV</th><th>CC</th><th>EOM</th><th>ΔV</th></tr></thead><tbody><tr><td rowspan="5">Xe</td><td>SF-X2C</td><td>12.39</td><td>12.54</td><td>-0.15</td><td>8.72</td><td>8.82</td><td>-0.11</td><td>3.60</td><td>3.64</td><td>-0.04</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>12.40</td><td>12.55</td><td>-0.15</td><td>8.72</td><td>8.83</td><td>-0.11</td><td>3.60</td><td>3.64</td><td>-0.04</td></tr><tr><td>X2C</td><td>12.95</td><td>13.10</td><td>-0.15</td><td>9.10</td><td>9.20</td><td>-0.11</td><td>3.74</td><td>3.79</td><td>-0.05</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>13.00</td><td>13.16</td><td>-0.15</td><td>9.14</td><td>9.24</td><td>-0.11</td><td>3.76</td><td>3.81</td><td>-0.05</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>12.97</td><td>13.12</td><td>-0.15</td><td>9.11</td><td>9.22</td><td>-0.11</td><td>3.75</td><td>3.80</td><td>-0.05</td></tr><tr><td rowspan="5">Rn</td><td>SF-X2C</td><td>18.79</td><td>19.01</td><td>-0.22</td><td>13.14</td><td>13.30</td><td>-0.15</td><td>5.38</td><td>5.44</td><td>-0.06</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>18.82</td><td>19.04</td><td>-0.22</td><td>13.16</td><td>13.31</td><td>-0.15</td><td>5.38</td><td>5.45</td><td>-0.06</td></tr><tr><td>X2C</td><td>25.62</td><td>25.90</td><td>-0.28</td><td>17.73</td><td>17.92</td><td>-0.19</td><td>7.14</td><td>7.22</td><td>-0.08</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>26.07</td><td>26.35</td><td>-0.28</td><td>18.03</td><td>18.22</td><td>-0.20</td><td>7.26</td><td>7.34</td><td>-0.08</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>25.95</td><td>26.24</td><td>-0.28</td><td>17.95</td><td>18.15</td><td>-0.20</td><td>7.23</td><td>7.31</td><td>-0.08</td></tr><tr><td rowspan="5">Og</td><td>SF-X2C</td><td>35.45</td><td>35.83</td><td>-0.38</td><td>24.44</td><td>24.71</td><td>-0.27</td><td>9.81</td><td>9.91</td><td>-0.11</td></tr><tr><td>SF-<sup>2</sup>DC<sup>M</sup></td><td>35.58</td><td>35.95</td><td>-0.38</td><td>24.53</td><td>24.79</td><td>-0.26</td><td>9.84</td><td>9.95</td><td>-0.11</td></tr><tr><td>X2C</td><td>144.66</td><td>146.52</td><td>-1.85</td><td>92.02</td><td>93.20</td><td>-1.18</td><td>33.34</td><td>33.77</td><td>-0.43</td></tr><tr><td><sup>2</sup>DC<sup>M</sup></td><td>151.00</td><td>152.95</td><td>-1.95</td><td>95.74</td><td>96.98</td><td>-1.24</td><td>34.55</td><td>35.00</td><td>-0.45</td></tr><tr><td><sup>2</sup>DCG<sup>M</sup></td><td>149.96</td><td>151.92</td><td>-1.96</td><td>95.12</td><td>96.36</td><td>-1.24</td><td>34.34</td><td>34.79</td><td>-0.45</td></tr></tbody></table>
Sample timings for quadratic response evaluation

<table><caption>Table 9: Comparison of timings for the quadratic response evaluation step for the $\beta_{zzz}$ component of the dipole hyperpolarizability (wall time, in hh:mm:ss format) between for QR-CC and QR-EOMCC, and the ratio between them. Calculations with the $^2$DC$^M$ Hamiltonian. O: number of correlated electrons, V: number of virtual spinors. QRF: quadratic response function evaluation step; total: all steps (QR, solution of linear systems etc.)</caption><thead><tr><th>system</th><th>O</th><th>V</th><th>Method</th><th>total walltime</th><th>ratio</th><th>QRF walltime<sup>†</sup></th></tr></thead><tbody><tr><td colspan="7" style="text-align: center;">valence electrons correlated</td></tr><tr><td rowspan="2">HF</td><td rowspan="2">8</td><td rowspan="2">196</td><td>CC</td><td>0:34:07</td><td></td><td>0:13:51</td></tr><tr><td>EOMCC</td><td>0:18:51</td><td>0.6</td><td>0:00:12</td></tr><tr><td rowspan="2">HCl</td><td rowspan="2">8</td><td rowspan="2">226</td><td>CC</td><td>0:37:37</td><td></td><td>0:21:36</td></tr><tr><td>EOMCC</td><td>0:16:18</td><td>0.4</td><td>0:00:06</td></tr><tr><td rowspan="2">HBr</td><td rowspan="2">8</td><td rowspan="2">226</td><td>CC</td><td>1:05:55</td><td></td><td>0:26:31</td></tr><tr><td>EOMCC</td><td>0:39:35</td><td>0.6</td><td>0:00:08</td></tr><tr><td colspan="7" style="text-align: center;">all electrons correlated</td></tr><tr><td rowspan="2">HF</td><td rowspan="2">10</td><td rowspan="2">282</td><td>CC</td><td>0:57:44</td><td></td><td>0:17:31</td></tr><tr><td>EOMCC</td><td>0:41:06</td><td>0.7</td><td>0:00:11</td></tr><tr><td rowspan="2">HCl</td><td rowspan="2">18</td><td rowspan="2">300</td><td>CC</td><td>1:56:59</td><td></td><td>0:28:44</td></tr><tr><td>EOMCC</td><td>1:32:24</td><td>0.8</td><td>0:00:33</td></tr><tr><td rowspan="2">HBr</td><td rowspan="2">36</td><td rowspan="2">402</td><td>CC</td><td>8:36:03</td><td></td><td>2:24:50</td></tr><tr><td>EOMCC</td><td>6:14:54</td><td>0.7</td><td>0:02:36</td></tr></tbody></table>
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