# Benchmarking ionization potentials from pCCD tailored coupled cluster models

Marta Gałyńska  Institute of Physics, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University in Toruń, Grudziadzka 5, 87-100
Toruń, Poland.  Katharina Boguslawski<EMAIL_ADDRESS>Institute
of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus
University in Toruń, Grudziadzka 5, 87-100 Toruń, Poland.

###### Abstract

The ionization potential (IP) is an important parameter providing essential
insights into the reactivity of chemical systems. IPs are also crucial for
designing, optimizing, and understanding the functionality of modern
technological devices. We recently showed that limiting the CC ansatz to the
seniority-zero sector proves insufficient in predicting reliable and accurate
ionization potentials within an IP equation-of-motion coupled-cluster
formalism. Specifically, the absence of dynamic correlation in the seniority-
zero pair coupled cluster doubles (pCCD) model led to unacceptably significant
errors of approximately 1.5 eV. In this work, we aim to explore the impact of
dynamical correlation and the choice of the molecular orbital basis (canonical
vs. localized) in CC-type methods targeting 201 ionized states in 41
molecules. We focus on pCCD-based approaches as well as the conventional IP-
EOM-CCD and IP-EOM-CCSD. Their performance is compared to the CCSDT equivalent
and experimental reference data. Our statistical analysis reveals that all
investigated frozen-pair coupled cluster methods exhibit similar performance,
with differences in errors typically within chemical accuracy (1 kcal/mol or
0.05 eV). Notably, the effect of the molecular orbital basis, such as
canonical Hartree-Fock or natural pCCD-optimized orbitals, on the IPs is
marginal if dynamical correlation is accounted for. Our study suggests that
triple excitations are crucial in achieving chemical accuracy in IPs when
modeling electron detachment processes with pCCD-based methods.

††preprint: AIP/123-QED

The reliable determination of ionization potentials (IP) is crucial for the
theoretical modeling of molecular electronic structures and molecular
properties. The IP provides information about the system’s reactivity as it
facilitates measuring the strength of one electron being attached to the
molecular bulk and quantifying the molecule’s ability to form a more
positively charged ion. This information can be further utilized to design,
optimize, and comprehensively understand the functionality of modern
technological devices such as photovoltaic (PV) cells, light-emitting diodes,
or sensors.Yuan _et al._ (2019); Zhu _et al._ (2022) For instance, a critical
factor in designing novel organic-based donor and acceptor molecules in
organic PV devices Yuan _et al._ (2019) is the knowledge of the energies of
the highest occupied molecular orbital (HOMO) and the lowest unoccupied
molecular orbital (LUMO) and the corresponding HOMO–LUMO gap. From a
theoretical perspective, one of the simplest approximations to deduce orbital
energies exploits the diagonal elements of the Fock matrix and the electron
repulsion energy. Limacher (2015) More reliable orbital energies can be, for
instance, obtained from the ionization potential (IP) Nooijen and Snijders
(1992, 1993); Stanton and Gauss (1994, 1999) and electron affinity (EA) Musiał
_et al._ (2014) variants of equation-of-motion (EOM) Rowe (1968); Stanton and
Bartlett (1993); Bartlett (2012) applied on top of a coupled cluster reference
wave function. The resulting IPs and EAs are then exploited to predict the so-
called charge gap.

Apart from CC approaches, Paldus (1992); Bartlett and Musiał (2007); Lyakh _et
al._ (2012) various calculation protocols have been proposed to investigate
the electron detachment process, including density functional theory and its
time-dependent formulation, Parr and Yang (1989); Cohen _et al._ (2012)
configuration interaction models, Foster and Boys (1960); Szalay _et al._
(2012) perturbation theory, Shavitt and Bartlett (2009) algebraic-diagrammatic
construction (ADC) schemes, Schirmer _et al._ (1998); Dempwolff _et al._
(2019); Banerjee and Sokolov (2019, 2021) and Monte Carlo methods. Zhang
(2013); Hochkeppel _et al._ (2009) These methods feature a diverse spectrum
related to their accuracy and computational complexity, cost, and resource
requirements. Furthermore, they can be applied to a wide range of chemical
compounds, varying in complexity and size. Among these methods, different
variants of IP-EOM-CC Nooijen and Snijders (1992, 1993); Stanton and Gauss
(1994, 1999); Shee _et al._ (2018) have become well-established correlation-
based methods, mainly employed for simulating photoelectron spectroscopy.
Geertsen _et al._ (1989); Stanton and Bartlett (1993); Watts and Bartlett
(1994); Van Voorhis and Head-Gordon (2000); Musiał _et al._ (2003); Krylov
(2008); Manohar and Krylov (2008); Cooper and Knowles (2010); Evangelista
(2011); Kuś and Krylov (2011); Musiał _et al._ (2014); Lischka _et al._
(2018); Gulania _et al._ (2021); Marie _et al._ (2021)

The most common CC ansatz is constrained to single and double excitations
(CCSD) but can be rather easily extended to perturbatively account for triple
substitutions (CCSD(T)), which is commonly known as the gold standard of
quantum chemistry. Those, as well as further extensions of the CC ansatz,
including full triples (CCSDT), quadruples, and higher excitations, can be
combined with an IP-EOM formalism to describe ionized states. Musiał _et al._
(2003) Although those EOM-CC methods Krylov (2008); Sneskov and Christiansen
(2012); Bartlett (2012) are highly reliable in terms of accuracy, they are
remarkably expensive and hence limited to relatively small system sizes. Thus,
a significant effort has been made to devise alternatives of similar accuracy
but more reasonable computational complexity. The simplified IP-EOM pair
coupled cluster doubles (IP-EOM-pCCD) variant Boguslawski (2021) proved an
inexpensive alternative to model open-shell electronic structures within the
pCCD Limacher _et al._ (2013); Boguslawski _et al._ (2014); Stein _et al._
(2014) model. pCCD was originally introduced as a geminal-based wavefunction
Tecmer and Boguslawski (2022) ansatz using two-electron functions as building
blocks for the electronic wave function. Hurley _et al._ (1953); Parr _et al._
(1956); Bardeen _et al._ (1957); Parks and Parr (1958); Coleman (1965); Miller
and Ruedenberg (1968); Surján (1999); Surján _et al._ (2012); Tecmer _et al._
(2014); Johnson _et al._ (2013, 2017); Fecteau _et al._ (2020); Johnson _et
al._ (2020, 2022); Faribault _et al._ (2022); Fecteau _et al._ (2022); Moisset
_et al._ (2022); Tecmer and Boguslawski (2022) Other examples are strictly
localized geminals, Surján (1984); Poirier and Surján (1987); Surján (1994)
the antisymmetrized product of strongly orthogonal geminals, Hurley _et al._
(1953); Parks and Parr (1958); Kutzelnigg (1964); Surján (1999) and the
generalized valence bond perfect pairing Bobrowicz and Goddard (1977); Cullen
(1996) models, to name a few. Such models are a promising alternative to
conventional electronic structure approaches, which are typically constructed
from one-electron functions.

Initial numerical studies Boguslawski (2021) demonstrated that the accuracy of
IP-EOM-pCCD approaches closely matches the accuracy of CCSD(T) or the density
matrix renormalization group White (1992); Legeza _et al._ (2008); Marti and
Reiher (2010); Chan and Sharma (2011) (DMRG) algorithm in open-shell
electronic structures. We recently presented a benchmark study to assess the
accuracy of the IP-EOM-pCCD method in predicting ionization energies. Mamache
_et al._ (2023) In Ref. 74, we compared the vertical ionization energies
obtained in the space of one-hole (1h) and two-hole-one-particle (2h1p) states
for three types of molecular orbitals (canonical Hartree-Fock, Pipek-Mezey
localized, and natural pCCD orbitals). Our study suggests that the orbital-
optimized IP-EOM-pCCD method, restricted to the 2h1p operator, demonstrated
the highest accuracy among the investigated methods. However, due to the
absence of dynamic correlation, we observed unacceptably large errors in IPs
of approximately 1.5 eV.

As demonstrated previously, Tecmer _et al._ (2023); Leszczyk _et al._ (2021)
natural pCCD orbitals present a promising alternative to canonical Hartree-
Fock orbitals, serving as a reference wave function for more sophisticated
calculations. Consequently, the question arises whether it is possible to
achieve an accuracy comparable to more elaborate approaches (like CCSDT or
higher) employing natural pCCD-optimized orbitals in combination with
simplified CC ansätze, which account for dynamical correlation. Thus, in the
current work, we explore, for the first time, both the effect of dynamical
correlation and the choice of the molecular orbital basis (canonical vs.
localized) in CC-type methods including up to double excitations.
Specifically, we focus on various pCCD-tailored CC flavours Henderson _et al._
(2014); Boguslawski and Ayers (2015); Leszczyk _et al._ (2021) and compare
their performance to the conventional CCSD method exploiting a canonical and
pCCD-optimized localized molecular orbital basis. Specifically, we investigate
the influence of dynamical correlation to the IP values determined by six
different approaches using the natural pCCD-optimized orbitals, namely frozen-
pair (fp)CC Henderson _et al._ (2014) methods (IP-EOM-fpCCD and IP-EOM-
fpCCSD), their linearized (fpLCC) Boguslawski and Ayers (2015) variants (IP-
EOM-fpLCCD and IP-EOM-fpLCCSD), and conventional IP-EOM-CCD and IP-EOM-CCSD
Nooijen and Snijders (1992, 1993); Stanton and Gauss (1994, 1999), and compare
their performance to the CCSDT equivalent and experimental reference data.

This work is structured as follows: In section I, we briefly review the
investigated theoretical models. Section II provides an overview of the
computational methodology. Section III presents the numerical results,
including a statistical analysis. Finally, we conclude in Section IV.

##  I THEORY

The pCCD Limacher _et al._ (2013); Boguslawski _et al._ (2014); Stein _et al._
(2014); Tecmer and Boguslawski (2022) ansatz is a simple reduction of the
single-reference CCD approach, where the cluster operator only contains
electron-pair excitations $\hat{T}_{\textrm{pCCD}}$,

$\ket{\textrm{pCCD}}=e^{\hat{T}_{\textrm{pCCD}}}\ket{\Phi_{0}},$ (1)

and

$\hat{T}_{\textrm{pCCD}}=\sum_{i}^{n_{\rm occ}}\sum_{a}^{n_{\rm
virt}}c_{i}^{a}{a_{a}^{\dagger}}{a_{\bar{a}}^{\dagger}}{a_{\bar{i}}}{a_{i}},$
(2)

where $\big{|}\Phi_{0}\rangle$ is some reference determinant, $\hat{a}_{p}$
($\hat{a}_{p}^{\dagger}$) are the elementary annihilation (creation) operators
for $\alpha$ ($p$) and $\beta$ $(\overline{p})$ electrons, and $c^{a}_{i}$ are
the pCCD cluster amplitudes. The above sum runs over all occupied $i$ and
virtual $a$ orbitals. Typically, the pCCD molecular orbitals are optimized,
Boguslawski _et al._ (2014, 2014, 2014); Stein _et al._ (2014) which re-
establishes size consistency and yields localized and symmetry-broken orbitals
that allow us to simulate quantum states with (quasi-)degeneracies.
Boguslawski _et al._ (2016) Numerical examples comprise bond-breaking
processes in small molecules Tecmer _et al._ (2014); Limacher (2015); Tecmer
_et al._ (2015); Brzęk _et al._ (2019); Henderson and Scuseria (2019); Nowak
_et al._ (2021); Leszczyk _et al._ (2021, 2022), heavy-element-containing
compounds featuring lanthanide Tecmer _et al._ (2019) or actinide Tecmer _et
al._ (2015); Garza _et al._ (2015); Boguslawski (2016, 2017); Nowak _et al._
(2019); Leszczyk _et al._ (2022); Nowak and Boguslawski (2023); Chakraborty
_et al._ (2023) atoms, organic electronicsJahani _et al._ (2023); Tecmer _et
al._ (2023), and electronically excited states. Boguslawski (2016, 2017,
2018); Nowak _et al._ (2019); Kossoski _et al._ (2021); Boguslawski (2021);
Ravi _et al._ (2023) Although these numerical studies support pCCD to be a
promising alternative to capture static/nondynamic electron correlation
effects, Sinanoğlu and Tuan (1963); Bartlett and Stanton (1994); Boguslawski
_et al._ (2012) a large fraction of the correlation energy cannot be captured
by electron-pair states alone. This missing correlation energy is commonly
attributed to so-called broken-pair states. These correlation effects are
commonly included a posteriori using various state-of-the-art techniques.
Tecmer and Boguslawski (2022)

One possibility to account for dynamical correlation is to exploit a coupled
cluster correction with a pCCD reference function. Henderson _et al._ (2014);
Boguslawski and Ayers (2015); Leszczyk _et al._ (2021) For instance, in the
frozen-pair coupled cluster (fpCC) ansatz, Henderson _et al._ (2014); Leszczyk
_et al._ (2021)

$\ket{{\rm fpCC}}=e^{\hat{T}^{\rm ext}}\ket{{\rm pCCD}}=e^{\hat{T}^{\rm
ext}}e^{\hat{T}_{\rm pCCD}}\ket{\Phi_{0}},$ (3)

the pCCD wave function is taken as the fixed reference function and the
$\hat{T}^{\rm ext}$ cluster operator contains electron excitations beyond
electron-pair excitations. Thus, the cluster operator of fpCCD is defined as
$\hat{T}^{\rm ext}=\hat{T}_{2}^{\prime}=\hat{T}_{2}-\hat{T}_{\rm pCCD}$, while
the cluster operator of fpCCSD includes also single excitations, $\hat{T}^{\rm
ext}=\hat{T}_{1}+\hat{T}_{2}^{\prime}$. We should stress that fpCC theory can
be considered as a conventional tailored coupled cluster approach. Kinoshita
_et al._ (2005); Hino _et al._ (2006); Lyakh _et al._ (2011); Melnichuk and
Bartlett (2012) The fpCC ansatz can be further simplified by truncating the
Baker–Campbell–Hausdorff expansion after the second term (concerning all non-
pair excitations) and hence including only linear terms in $\hat{T}^{\rm
ext}$. Boguslawski and Ayers (2015); Boguslawski and Tecmer (2017) Strictly
speaking, the frozen-pair Linearized CC correction (fpLCC) does not fall into
the category of tailored CC methods. Nonetheless, we will use the acronym
fpLCC due to its simplicity (originally, fpLCC was introduced as pCCD-LCC).
The wave function ansatz of fpLCC is approximated as

$\ket{{\rm fpLCC}}\approx(1+\hat{T}^{\rm ext})\ket{{\rm pCCD}}=(1+\hat{T}^{\rm
ext})e^{\hat{T}_{\rm pCCD}}\ket{\Phi_{0}}.$ (4)

We should note that all disconnected terms containing $\hat{T}_{\rm pCCD}$
still appear in the fpLCC amplitude equations as the exponential ansatz of
pCCD is not linearized. For instance, terms associated with $\hat{T}^{\rm
ext}\hat{T}_{\rm pCCD}$ and $\hat{T}_{\rm pCCD}^{2}$ have to be considered in
fpLCCD-type methods. Thus, in fpLCC, the coupled cluster equations are linear
concerning non-pair amplitudes $\hat{T}^{\rm ext}$ but the coupling between
all pair- and non-pair amplitudes is included.

Since we formally work in a single-reference CC picture, we can
straightforwardly employ single-reference CC techniques to target, for
instance, electronically excited, ionized, and electron-attached states.
Musiał _et al._ (2014); Perera _et al._ (2014); Gulania _et al._ (2021)
Another possible extension are spin-flip EOM-CC methods. Casanova and Krylov
(2020) Specifically, for ionized states, we can employ the IP-EOM formalism,
where we use a linear ansatz on top of the closed-shell fp(L)CC reference
function to parametrize the k-th (ionized) state

$\ket{\Psi_{k}}=\hat{R}(k)\ket{\textrm{fpCC}}$ (5)

where the operator $\hat{R}(k)$ generates the targeted ionized state $k$ from
the initial fp(L)CC reference state. In the single IP-EOM formalism,
$\hat{R}(k)$ reads

$\hat{R}^{\rm
IP}=\sum_{i}r_{i}\hat{a}_{i}+\frac{1}{2}\sum_{ij{a}}{r}^{a}_{ij}\hat{a}_{a}^{\dagger}\hat{a}_{j}\hat{a}_{i}+\dots~{}=\hat{R}_{1h}+\hat{R}_{2h1p}+\dots~{}$
(6)

where we introduced the hole (h, encoding $\hat{a}_{i}$) and particle (p,
encoding $\hat{a}^{\dagger}_{a}$) labels and dropped the k-dependence for
better readability. The ionized states are then obtained by solving the
corresponding EOM equations

${[\hat{H}_{N},\hat{R}]}\ket{\textrm{fpCC}}=\omega\hat{R}\ket{\textrm{fpCC}},$
(7)

where $\omega=\Delta E-\Delta E_{0}$ is the energy corresponding to the
ionization process concerning the fpCC ground state, while
$\hat{H}_{N}=\hat{H}-\langle\Phi_{0}\big{|}\hat{H}|\Phi_{0}\rangle$ is the
normal product form of the Hamiltonian. We can rewrite the above equation in
the well-known form

${\cal
H}^{\textrm{fpCC}}_{N}\hat{R}\ket{\Phi_{0}}=\omega\hat{R}\ket{\Phi_{0}}$ (8)

where ${\cal H}^{\textrm{fpCC}}_{N}$ is the similarity transformed Hamiltonian
of the used fpCC flavour in its normal-product form

${\cal
H}^{\textrm{fpCC}}_{N}=e^{-\hat{T}_{\textrm{pCCD}}}e^{-\hat{T}^{\textrm{ext}}}\hat{H}_{N}e^{\hat{T}^{\textrm{ext}}}e^{\hat{T}_{\textrm{pCCD}}}.$
(9)

The ionization energies are obtained by iteratively diagonalizing ${\cal
H}^{\textrm{fpCC}}_{N}$ of the chosen pCCD-based CC correction. Note that
depending on the selected CC model, the similarity transformed Hamiltonian
either contains all non-vanishing non-linear terms (here, ${\cal
H}^{\textrm{fpCCD}}_{N}$ and ${\cal H}^{\textrm{fpCCSD}}_{N}$) or only non-
linear terms associated with the electron-pair excitation operator (here,
${\cal H}^{\textrm{fpLCCD}}_{N}$ and ${\cal H}^{\textrm{fpLCCSD}}_{N}$. The
diagrammatic representation of the IP-EOM-fpCC equations is shown in Fig. 1
(see also, for instance, Ref. 32 for the diagrammatic form and its algebraic
expressions). We should note that the IP-EOM-fpCC(S)D equations are similar in
form to the conventional IP-EOM-CCSD formalism. However, in the IP-EOM-fpLCCSD
method, the effective Hamiltonian diagrams (a1), (b1), (b2), (b3), (b4), (b5)
do not contain the disconnected $\hat{T}_{1}\hat{T}_{1}$ terms, while, in
addition, diagram (b1) lacks the $\hat{T}_{1}\hat{T}_{1}\hat{T}_{1}$ part.
Furthermore, the $\hat{T}_{1}\hat{T}_{2}$ term contained in (b1) is replaced
by the simpler $\hat{T}_{1}\hat{T}_{p}$ counterpart due to the truncation of
the BCH expansion of the LCC correction (see also eq. (4)). Finally, we should
note that we focused on the $S_{z}=-0.5$ case Musiał _et al._ (2014);
Boguslawski (2021) in its spin-dependent and spin-summed versions. The spin-
summed working equations can be obtained by either diagrammatic or algebraic
spin summation (see also Ref. 32) of the IP-EOM-fp(L)CC equations shown in
Fig. 1. For the latter case, the targeted ionized (open-shell) states are the
doublet states of the $S_{z}=-0.5$ spin block.

Figure 1: Diagrammatic representation of the IP-EOM-fp(L)CCSD equations
(antisymmetrized formalism). Figure 2: The benchmark set containing 41
molecules relaxed at the CCSD(T)/aug-cc-pVTZ level provided in Ref. 109. Table
1: Statistical error measures in electronvolts (eV), such as mean error (ME),
mean absolute error (MAE), and root-mean-square error (RMSE), were assessed
based on the ionization potentials (IP) calculated using various methods: IP-
EOM-fpCCD, IP-EOM-fpCCSD, IP-EOM-fpLCCD, IP-EOM-fpLCCSD, IP-EOM-CCD(pCCD), and
IP-EOM-CCSD(pCCD), listed in the upper part of each block, are derived in this
work. All these CC approaches exploit natural pCCD-optimized orbitals (or the
orbital-optimized pCCD reference determinant) and are conducted in the space
of two-hole-one-particle (2h1p) states. The corresponding errors in IPs for
CCSD done with Hartree-Fock orbitals, unitary coupled cluster (UCC),
algebraic-diagrammatic construction (ADC) methods, and pCCD, using natural
pCCD-optimized orbitals, are presented in the lower part of each block. The
errors in IPs were calculated concerning IP-EOM-CCSDT Ranasinghe _et al._
(2019) (top) and experimental data Ranasinghe _et al._ (2019) (bottom). The
definitions for ME, MAE, and RMSE are printed in the table footnote.

| Errors w.r.t. IP-EOM-CCSDT
---|---
Method | ME | MAE | RMSE
IP-fpCCD |  $-$0.295 | 0.346 | 0.461
IP-fpLCCD |  $-$0.335 | 0.382 | 0.489
IP-CCD(pCCD) |  $-$0.241 | 0.297 | 0.418
IP-fpCCSD |  $-$0.239 | 0.293 | 0.399
IP-fpLCCSD |  $-$0.293 | 0.341 | 0.432
IP-CCSD(pCCD) |  $-$0.183 | 0.238 | 0.350
IP-pCCDMamache _et al._ (2023) |  $-$1.535 | 1.535 | 1.633
IP-CCSD(HF)Ranasinghe _et al._ (2019) |  $-$0.186 | 0.241 | 0.341
IP-UCC2Dempwolff _et al._ (2022) |  $-$0.488 | 0.579 | 0.728
IP-UCC3Dempwolff _et al._ (2022) |  $-$0.260 | 0.306 | 0.377
IP-ADC(2)Dempwolff _et al._ (2022) |  $-$0.545 | 0.607 | 0.737
IP-ADC(3(3))Dempwolff _et al._ (2022) |  $-$0.269 | 0.351 | 0.442
IP-ADC(3(4+))Dempwolff _et al._ (2022) |  $-$0.306 | 0.339 | 0.418
IP-ADC(3(DEM))Dempwolff _et al._ (2022) |  $-$0.292 | 0.334 | 0.411
| Errors w.r.t. experiment
Method | ME | MAE | RMSE
IP-fpCCD |  $-$0.259 | 0.336 | 0.457
IP-fpLCCD |  $-$0.299 | 0.361 | 0.481
IP-CCD(pCCD) |  $-$0.206 | 0.303 | 0.422
IP-fpCCSD |  $-$0.203 | 0.283 | 0.391
IP-fpLCCSD |  $-$0.256 | 0.311 | 0.415
IP-CCSD(pCCD) |  $-$0.147 | 0.253 | 0.253
IP-pCCDMamache _et al._ (2023) |  $-$1.570 | 1.570 | 1.697
IP-CCSD(HF)Ranasinghe _et al._ (2019) |  $-$0.150 | 0.252 | 0.351
IP-CCSDTRanasinghe _et al._ (2019) |  $-$0.035 | 0.197 | 0.262
IP-UCC2Dempwolff _et al._ (2022) |  $-$0.523 | 0.684 | 0.841
IP-UCC3Dempwolff _et al._ (2022) |  $-$0.225 | 0.312 | 0.394
IP-ADC(2)Dempwolff _et al._ (2022) |  $-$0.580 | 0.693 | 0.844
IP-ADC(3(3))Dempwolff _et al._ (2022) |  $-$0.233 | 0.356 | 0.419
IP-ADC(3(4+))Dempwolff _et al._ (2022) |  $-$0.271 | 0.342 | 0.423
IP-ADC(3(DEM))Dempwolff _et al._ (2022) |  $-$0.256 | 0.339 | 0.419

  * *

ME = $\sum_{i}^{N}\frac{E_{i}^{\rm method}-E_{i}^{\rm ref}}{N}$

  * *

MAE = $\sum_{i}^{N}\frac{|E_{i}^{\rm method}-E_{i}^{\rm ref}|}{N}$

  * *

RMSE = $\sqrt{\sum_{i}^{N}\frac{(E_{i}^{\rm method}-E_{i}^{\rm ref})^{2}}{N}}$

##  II COMPUTATIONAL DETAILS

The vertical ionization potentials (IP) were calculated using different CCD-
and CCSD-type flavours as implemented in a developer version of the PyBEST
v1.4.0-dev0 software package. Boguslawski _et al._ (2021, 2024); Brzęk _et
al._ These included IP-EOM-fpCCD, IP-EOM-fpCCSD, IP-EOM-fpLCCD, IP-EOM-
fpLCCSD, IP-EOM-CCD, and IP-EOM-CCSD. In all CC calculations (electronic
ground states and IP-EOM), the pCCD-optimized orbitals (labeled as “(pCCD)”)
were used to construct the reference determinant $\ket{\Phi_{0}}$, which were
obtained through a variational orbital-optimization protocol of the pCCD
reference calculation. Boguslawski _et al._ (2014); Stein _et al._ (2014);
Boguslawski _et al._ (2014, 2014) The optimization protocol used in pCCD
calculations automatically selects the reference determinant according to the
pCCD natural occupation numbers.

The ionization energies were computed in the space of two-hole-one-particle
(2h1p) states as previous studies indicate that errors in IPs are
significantly reduced compared to the use of only one-hole (1h) states.
Mamache _et al._ (2023) No symmetry constraints were applied to allow the
algorithm to freely relax the orbitals resulting in a symmetry-broken,
localized molecular orbital basis.

A frozen core was used in all CC calculations, keeping the 1s orbitals for C,
N, O, and F and 1s, 2s, and 2p orbitals for Si, P, S, and Cl frozen. We should
note that preliminary tests indicate minimal impact on the IP values when
freezing core orbitals. All calculations employed the cc-pVTZ basis set by
Dunning, Dunning Jr (1989) facilitating a direct comparison with previously
published vertical IPs obtained using the IP-EOM-CCSDT model, Ranasinghe _et
al._ (2019) various forms of the unitary coupled-cluster (IP-UCC) approach,
and algebraic-diagrammatic construction (IP-ADC) methods.Dempwolff _et al._
(2022)

Furthermore, our test set contains 41 molecules shown in Fig. 2, whose
molecular geometries were relaxed using the CCSD(T)/aug-cc-pVTZ method Kendall
_et al._ (1992) and are available in the supplementary material of Ref. 109.
In total, we optimized 201 IP states and compared the performance of our IP-
EOM-CC methdos to theoretical and experimental reference data.

##  III RESULTS AND DISCUSSION

Table 1 summarizes our statistical analysis, including mean errors (ME), mean
absolute errors (MAE), and root-mean-square errors (RMSE) calculated using IP-
EOM-fpCCD, IP-EOM-fpCCSD, IP-EOM-fpLCCD, IP-EOM-fpLCCSD, conventional IP-EOM-
CCD(pCCD), and IP-EOM-CCSD(pCCD) of 201 ionized states in 41 molecules shown
in Fig 2. The footnote in Table 1 defines the error measures used in this
work. We should note that we use the labels CCD(pCCD) and CCSD(pCCD) to
indicate that the corresponding CCD and CCSD calculations were done employing
pCCD-optimized natural orbitals (or equivalently the orbital-optimized pCCD
reference determinant in the CC ansatz). The upper section of Table 1 presents
error values concerning IP-EOM-CCSDT reference data, Ranasinghe _et al._
(2019) while the lower section reports the corresponding errors relative to
experimental results.Ranasinghe _et al._ (2019) For a direct comparison, the
table includes data from IP-EOM-CCSD calculated with a canonical Hartree-Fock
reference determinant (or molecular orbital basis) indicated as IP-CCSD(HF),
two variations of the unitary coupled cluster ansatz (IP-UCC2 and IP-UCC3),
and four variants of non-Dyson algebraic diagrammatic construction schemes
(ADC(2), ADC(3(3)), ADC(3(4+)), and ADC(3(DEM)). Dempwolff _et al._ (2022)
Furthermore, Fig. 3(a) visualizes, using box and violin plots, the locality,
spread, skewness, and distribution of errors of all targeted ionization
potentials concerning IP-EOM-CCSDT. Fig. 3(b) displays an equivalent analysis
for experimental data. The individual ionization energies obtained by all
methods investigated in this work are accessible in the Electronic
Supplementary Information (ESI)†.

Figure 3: Box plots presented at the top and violin plots at the bottom,
illustrating errors [eV] derived from selected methods (refer to Table 1 for
numerical values). All errors are reported relative to either (a) IP-EOM-CCSDT
or (b) experimental reference data. For brevity, we have omitted the EOM
prefix in IP-EOM-CC-type methods. A star in each box plot denotes the mean
value, while a white dot in each violin plot represents the median value.

The effect of adding dynamical correlation, that is including the seniority
two and seniority four sectors in the fpCC reference function of the IP-EOM-
fpCCD method, provides a considerable improvement of the ionization energies
in comparison to the seniority zero orbital-optimized pCCD method.
Specifically, the MAE and RMSE errors are reduced by approximately 1.2 eV on
average if we go beyond the seniority-zero sector in the CC reference
function. This supports the original finding Mamache _et al._ (2023) that
dynamical correlation is needed to correctly describe the electron detachment
process within pCCD-based methods. Using a simplified version of the frozen-
pair methods, IP-EOM-fpLCCD increases slightly the ME, MAE, and RSME errors
compared to IP-EOM-fpCCD by 0.040 (0.040) eV, 0.036 (0.025) eV, 0.028 (0.024)
eV, respectively, with respect to IP-EOM-CCSDT reference data (experiment). On
the other hand, IP-EOM-CCD(pCCD) (conventional IP-EOM-CCD with an orbital-
optimized pCCD reference determinant) decreases the errors of IP-EOM-fpCCD by
0.054 (0.053) eV for ME, 0.049 (0.033) eV for MEA, and 0.043 (0.035) eV for
RMSE with respect to IP-EOM-CCSDT (experimental) results.

A similar trend is observed for CCSD-based approaches. Specifically, IP-EOM-
fpLCCSD yields the largest errors, while IP-EOM-CCSD(pCCD) (conventional IP-
EOM-CCSD with an orbital-optimized pCCD reference determinant) exhibits the
smallest errors compared to theoretical (experimental) reference values.
Nonetheless, their differences in errors are acceptable, amounting up to
around 0.11 eV (or 2.5 kcal/mol). Most importantly, including single
excitations slightly reduces the ME, MAE, and RMSE values by 0.041 to 0.061 eV
between the frozen-pair CCD and CCSD methods, respectively. Regarding
statistical errors, IP-EOM-fpCCSD is the most accurate among all investigated
pCCD-based approaches investigated in this work. Specifically, it yields very
similar errors to the IP-EOM-UCC3 method identified as the best among recently
investigated approximations. Ranasinghe _et al._ (2019) Concerning IP-EOM-
CCSDT (experiment), the relative error measures between IP-EOM-fpCCSD and IP-
EOM-UCC3 are $\Delta\Delta$ME = 0.021 (0.022) eV, $\Delta\Delta$MAE = 0.013
(0.029) eV, and $\Delta\Delta$RMSE = $-$0.022 (0.003) eV, where $\Delta\Delta$
indicates the difference between the IP-EOM-fpCCSD and IP-EOM-UCC3 error
measures.

Our box plots (see Fig. 3 top) illustrate that the differences in errors among
all fpCC methods are very similar, displaying an almost identical dispersion
of 50% of errors (highlighted in yellow and blue boxes). The total range of
scope (indicated by the whiskers) diminishes slightly with the addition of
single excitations. However, the differences are minimal. The violin plots
(see Fig. 3 bottom) highlight interquartile ranges distributed closely around
the median. The skewness of errors is left-shifted in all cases. Although the
dispersion of results is slightly smaller when using the IP-EOM-CCSDT method
as a reference, a similar trend of error dispersion can be seen for both
references (theoretical and experimental). All frozen-pair variants exhibited
an accuracy range similar to IP-EOM-CCSD conducted with canonical HF molecular
orbitals, demonstrating a comparable dispersion and skewness of errors.

The IP-EOM-CCSD results reported by Ranasinghe et al.Ranasinghe _et al._
(2019) allow us to directly assess the effect of the choice of the molecular
orbital basis on molecular properties, that is if the performance is
significantly different between canonical HF and natural pCCD-optimized
orbitals. Surprisingly, the errors are almost identical and exhibit resilience
to the choice of the reference wave function. Furthermore, the overall
appearance of error distribution, skewness, and even the positioning of
outliers presented in the box and violin plots in Fig. 3 is almost identical
in both cases (differences lie within chemical accuracy). This suggests that
including triple excitations in the theoretical model will be crucial for
further improving the accuracy of pCCD-based approaches in predicting IPs.

##  IV CONCLUSIONS

As recently shown, Mamache _et al._ (2023) restricting the CC ansatz to the
seniority-zero sector is insufficient in predicting reliable and accurate IPs.
Although the seniority-zero pCCD model can capture static correlation
reliably, it is inadequate to describe electron detachment with sufficient
accuracy. This deficiency was attributed to the missing broken-pair states,
that is the exclusion of the seniory-two, seniority-four, etc. sectors. In
this work, we investigated the impact of dynamical correlation and the choice
of the molecular orbital basis (canonical vs. localized) on vertical
ionization potentials using various pCCD-based and conventional CC approaches.
Specifically, we studied six CC variants: IP-EOM-fpCCD, IP-EOM-fpLCCD, IP-EOM-
CCD(pCCD), IP-EOM-fpCCSD, IP-EOM-fpLCCSD, and IP-EOM-CCSD(pCCD). Throughout
this work, we included (up to) 2p1h operators in the IP-EOM formalism as the
resulting pCCD-based model turned out to be superior to the corresponding IP-
EOM approach restricted to 1h operators. Mamache _et al._ (2023) Our analysis
encompasses a set of 41 molecules, targeting 201 ionized states. These
ionization energies are compared to IP-EOM-CCSDT and experimental reference
data. Furthermore, our results are juxtaposed with those obtained using
various conventional CC methods, UCC flavors, and non-Dyson ADC second and
third-order schemes.

Our statistical analysis (mean errors, mean absolute errors, root mean square
errors) highlights that all investigated frozen-pair coupled cluster methods
feature similar performance. Specifically, the differences in errors are
typically within chemical accuracy (1 kcal/mol or 0.05 eV). Adding single
excitations slightly reduces error measures with respect to the corresponding
CCD model. Yet, these changes approach chemical accuracy, constituting
approximately 0.06 eV. Our benchmark data renders IP-EOM-fpCCSD the best-
performing method among all tested frozen-pair variants. We should stress,
however, that differences between the investigated frozen-pair methods are
nearly invisible on box and violin plots. Furthermore, the scattering of
errors and their distribution around the median make them comparable to the
conventional IP-EOM-CCSD method. On the other hand, the error measures of IP-
EOM-fpCCSD are comparable to the accuracy of IP-EOM-UCC(3), identified as the
best among recently investigated approximations. Ranasinghe _et al._ (2019)
Finally, the influence of the molecular orbital basis or CC reference
determinant (that is canonical vs. localized) is marginal as the conventional
IP-EOM-CCSD and IP-EOM-CCSD(pCCD) result in almost identical errors in
ionization potentials. This observation suggests that triple excitations are
crucial for further improving IPs and approaching chemical accuracy for
modeling electron detachment processes with pCCD-based methods.

## Acknowledgements

The research leading to these results has received funding from the Norway
Grants 2014–2021 via the National Centre for Research and Development. M.G.
acknowledges financial support from a Ulam NAWA – Seal of Excellence research
grant (no. BPN/SEL/2021/1/00005). We acknowledge that the results of this
research have been achieved using the DECI resource Bem (Grant No. 412) based
in Poland at Wroclaw Centre for Networking and Supercomputing (WCSS,
http://wcss.pl) with support from the PRACE aisbl.

## References

  * Yuan _et al._ (2019) J. Yuan, Y. Zhang, L. Zhou, G. Zhang, H.-L. Yip, T.-K. Lau, X. Lu, C. Zhu, H. Peng, P. A. Johnson _et al._ , _Joule_ , 2019, 3, 1140–1151.
  * Zhu _et al._ (2022) L. Zhu, M. Zhang, J. Xu, C. Li, J. Yan, G. Zhou, W. Zhong, T. Hao, J. Song, X. Xue _et al._ , _Nat. Material_ , 2022, 21, 656–663.
  * Limacher (2015) P. A. Limacher, _J. Chem. Theory Comput._ , 2015, 11, 3629–3635.
  * Nooijen and Snijders (1992) M. Nooijen and J. G. Snijders, _Int. J. Quantum Chem._ , 1992, 44, 55–83.
  * Nooijen and Snijders (1993) M. Nooijen and J. G. Snijders, _Int. J. Quantum Chem._ , 1993, 48, 15–48.
  * Stanton and Gauss (1994) J. F. Stanton and J. Gauss, _J. Chem. Phys._ , 1994, 101, 8938–8944.
  * Stanton and Gauss (1999) J. F. Stanton and J. Gauss, _J. Chem. Phys._ , 1999, 111, 8785–8788.
  * Musiał _et al._ (2014) M. Musiał, Ł. Lupa and S. A. Kucharski, _J. Chem. Phys._ , 2014, 140, 114107.
  * Rowe (1968) D. Rowe, _Rev. Mod. Phys._ , 1968, 40, 153.
  * Stanton and Bartlett (1993) J. F. Stanton and R. J. Bartlett, _J. Chem. Phys._ , 1993, 98, 7029–7039.
  * Bartlett (2012) R. J. Bartlett, _WIREs Comput. Mol. Sci._ , 2012, 2, 126–138.
  * Paldus (1992) J. Paldus, in _Methods in computational molecular physics_ , Springer, 1992, pp. 99–194.
  * Bartlett and Musiał (2007) R. J. Bartlett and M. Musiał, _Rev. Mod. Phys._ , 2007, 79, 291\.
  * Lyakh _et al._ (2012) D. I. Lyakh, M. Musiał, V. F. Lotrich and . J. Bartlett, _Chem. Rev._ , 2012, 112, 182–243.
  * Parr and Yang (1989) R. G. Parr and W. Yang, _Density Functional Theory of atoms and molecules_ , Oxford, New York, 1989.
  * Cohen _et al._ (2012) A. J. Cohen, P. Mori-Sánchez and W. Yang, _Chem. Rev._ , 2012, 112, 289.
  * Foster and Boys (1960) J. Foster and S. Boys, _Rev. Mod. Phys._ , 1960, 32, 300.
  * Szalay _et al._ (2012) P. G. Szalay, T. Müller, G. Gidofalvi, H. Lischka and R. Shepard, _Chem. Rev._ , 2012, 112, 108–181.
  * Shavitt and Bartlett (2009) I. Shavitt and R. J. Bartlett, _Many-body methods in chemistry and physics_ , Cambridge University Press, New York, 2009.
  * Schirmer _et al._ (1998) J. Schirmer, A. B. Trofimov and G. Stelter, _J. Chem. Phys._ , 1998, 109, 4734–4744.
  * Dempwolff _et al._ (2019) A. L. Dempwolff, M. Schneider, M. Hodecker and A. Dreuw, _J. Chem. Phys._ , 2019, 150, 064108.
  * Banerjee and Sokolov (2019) S. Banerjee and A. Y. Sokolov, _J. Chem. Phys._ , 2019, 151, 224112\.
  * Banerjee and Sokolov (2021) S. Banerjee and A. Y. Sokolov, _J. Chem. Phys._ , 2021, 154, 074105\.
  * Zhang (2013) S. Zhang, _Emergent Phenomena in Correlated Matter: Autumn School Organized by the Forschungszentrum Jülich and the German Research School for Simulation Sciences at Forschungszentrum Jülich 23-27 September 2013; Lecture Notes of the Autumn School Correlated Electrons 2013_ , 2013, 3, 15.1–15.30.
  * Hochkeppel _et al._ (2009) S. Hochkeppel, T. C. Lang, C. Brünger, F. F. Assaad and W. Hanke, in _High Performance Computing in Science and Engineering, Garching/Munich 2007: Transactions of the Third Joint HLRB and KONWIHR Status and Result Workshop, Dec. 3–4, 2007, Leibniz Supercomputing Centre, Garching/Munich, Germany_ , ed. S. Wagner, M. Steinmetz, A. Bode and M. Brehm, Springer Berlin Heidelberg, Berlin, Heidelberg, 2009, ch. Quantum Monte Carlo Studies of Strongly Correlated Electron Systems, pp. 669–686.
  * Nooijen and Snijders (1992) M. Nooijen and J. G. Snijders, _Int. J. Quantum Chem._ , 1992, 44, 55–83.
  * Nooijen and Snijders (1993) M. Nooijen and J. G. Snijders, _Int. J. Quantum Chem._ , 1993, 48, 15–48.
  * Shee _et al._ (2018) A. Shee, T. Saue, L. Visscher and A. Severo Pereira Gomes, _J. Chem. Phys._ , 2018, 149, 174113.
  * Geertsen _et al._ (1989) J. Geertsen, M. Rittby and R. J. Bartlett, _Chem. Phys. Lett._ , 1989, 164, 57–62.
  * Watts and Bartlett (1994) J. D. Watts and R. J. Bartlett, _J. Chem. Phys._ , 1994, 101, 3073–3078.
  * Van Voorhis and Head-Gordon (2000) T. Van Voorhis and M. Head-Gordon, _J. Chem. Phys._ , 2000, 113, 8873–8879.
  * Musiał _et al._ (2003) M. Musiał, S. A. Kucharski and R. J. Bartlett, _J. Chem. Phys._ , 2003, 118, 1128–1136.
  * Krylov (2008) A. I. Krylov, _Annu. Rev. Phys. Chem._ , 2008, 59, 433–462.
  * Manohar and Krylov (2008) P. U. Manohar and A. I. Krylov, _J. Chem. Phys._ , 2008, 129, 194105\.
  * Cooper and Knowles (2010) B. Cooper and P. J. Knowles, _J. Chem. Phys._ , 2010, 133, 234102\.
  * Evangelista (2011) F. A. Evangelista, _J. Chem. Phys._ , 2011, 134, 224102.
  * Kuś and Krylov (2011) T. Kuś and A. I. Krylov, _J. Chem. Phys._ , 2011, 135, 084109\.
  * Lischka _et al._ (2018) H. Lischka, D. Nachtigallova, A. J. Aquino, P. G. Szalay, F. Plasser, F. B. Machado and M. Barbatti, _Chem. Rev._ , 2018, 118, 7293–7361.
  * Gulania _et al._ (2021) S. Gulania, E. F. Kjønstad, J. F. Stanton, H. Koch and A. I. Krylov, _J. Chem. Phys._ , 2021, 154, 114115.
  * Marie _et al._ (2021) A. Marie, F. Kossoski and P.-F. Loos, _J. Chem. Phys._ , 2021, 155, 104105.
  * Sneskov and Christiansen (2012) K. Sneskov and O. Christiansen, _WIREs Comput. Mol. Sci._ , 2012, 566–584.
  * Boguslawski (2021) K. Boguslawski, _Chem. Commun._ , 2021, 57, 12277–12280.
  * Limacher _et al._ (2013) P. A. Limacher, P. W. Ayers, P. A. Johnson, S. De Baerdemacker, D. Van Neck and P. Bultinck, _J. Chem. Theory Comput._ , 2013, 9, 1394–1401.
  * Boguslawski _et al._ (2014) K. Boguslawski, P. Tecmer, P. W. Ayers, P. Bultinck, S. De Baerdemacker and D. Van Neck, _Phys. Rev. B_ , 2014, 89, 201106.
  * Stein _et al._ (2014) T. Stein, T. M. Henderson and G. E. Scuseria, _J. Chem. Phys._ , 2014, 140, 214113.
  * Tecmer and Boguslawski (2022) P. Tecmer and K. Boguslawski, _Phys. Chem. Chem. Phys._ , 2022, 24, 23026–23048.
  * Hurley _et al._ (1953) A. C. Hurley, J. E. Lennard-Jones and J. A. Pople, _Proc. R. Soc. A_ , 1953, 220, 446–455.
  * Parr _et al._ (1956) R. G. Parr, F. O. Ellison and P. G. Lykos, _J. Chem. Phys._ , 1956, 24, 1106–1106.
  * Bardeen _et al._ (1957) J. Bardeen, L. N. Cooper and J. R. Schrieffer, _Phys. Rev._ , 1957, 108, 1175.
  * Parks and Parr (1958) J. M. Parks and R. G. Parr, _J. Chem. Phys._ , 1958, 28, 335–345.
  * Coleman (1965) A. Coleman, _J. Math. Phys._ , 1965, 6, 1425–1431.
  * Miller and Ruedenberg (1968) K. J. Miller and K. Ruedenberg, _J. Chem. Phys._ , 1968, 48, 3444–3449.
  * Surján (1999) P. R. Surján, in _Correlation and Localization_ , ed. P. R. Surján, R. J. Bartlett, F. Bogár, D. L. Cooper, B. Kirtman, W. Klopper, W. Kutzelnigg, N. H. March, P. G. Mezey, H. Müller, J. Noga, J. Paldus, J. Pipek, M. Raimondi, I. Røeggen, J. Q. Sun, P. R. Surján, C. Valdemoro and S. Vogtner, Springer Berlin Heidelberg, Berlin, Heidelberg, 1999, pp. 63–88.
  * Surján _et al._ (2012) P. R. Surján, Á. Szabados, P. Jeszenszki and T. Zoboki, _J. Math. Chem._ , 2012, 50, 534–551.
  * Tecmer _et al._ (2014) P. Tecmer, K. Boguslawski, P. A. Johnson, P. A. Limacher, M. Chan, T. Verstraelen and P. W. Ayers, _J. Phys. Chem. A_ , 2014, 118, 9058–9068.
  * Johnson _et al._ (2013) P. A. Johnson, P. W. Ayers, P. A. Limacher, S. De Baerdemacker, D. Van Neck and P. Bultinck, _Comput. Theor. Chem._ , 2013, 1003, 101–113.
  * Johnson _et al._ (2017) P. A. Johnson, P. A. Limacher, T. D. Kim, M. Richer, R. A. Miranda-Quintana, F. Heidar-Zadeh, P. W. Ayers, P. Bultinck, S. De Baerdemacker and D. Van Neck, _Comput. Theor. Chem._ , 2017, 1116, 207–219.
  * Fecteau _et al._ (2020) C.-É. Fecteau, H. Fortin, S. Cloutier and P. A. Johnson, _J. Chem. Phys._ , 2020, 153, 164117.
  * Johnson _et al._ (2020) P. A. Johnson, C.-É. Fecteau, F. Berthiaume, S. Cloutier, L. Carrier, M. Gratton, P. Bultinck, S. De Baerdemacker, D. Van Neck, P. Limacher _et al._ , _J. Chem. Phys._ , 2020, 153, 104110.
  * Johnson _et al._ (2022) P. A. Johnson, P. W. Ayers, S. De Baerdemacker, P. A. Limacher and D. Van Neck, _Comput. Theor. Chem._ , 2022, 1212, 113718.
  * Faribault _et al._ (2022) A. Faribault, C. Dimo, J.-D. Moisset and P. A. Johnson, _J. Chem. Phys._ , 2022, 157, 214104.
  * Fecteau _et al._ (2022) C.-É. Fecteau, S. Cloutier, J.-D. Moisset, J. Boulay, P. Bultinck, A. Faribault and P. A. Johnson, _J. Chem. Phys._ , 2022, 156, 194103\.
  * Moisset _et al._ (2022) J.-D. Moisset, C.-É. Fecteau and P. A. Johnson, _J. Chem. Phys._ , 2022, 156, 214110.
  * Surján (1984) P. R. Surján, _Phys. Rev. A_ , 1984, 30, 43.
  * Poirier and Surján (1987) R. A. Poirier and P. R. Surján, _J. Comput. Chem._ , 1987, 8, 436–441.
  * Surján (1994) P. R. Surján, _Int. J. Quantum Chem._ , 1994, 52, 563–574.
  * Kutzelnigg (1964) W. Kutzelnigg, _J. Chem. Phys._ , 1964, 40, 3640–3647.
  * Bobrowicz and Goddard (1977) F. W. Bobrowicz and W. A. Goddard, in _Methods of electronic structure theory_ , Springer, 1977, pp. 79–127.
  * Cullen (1996) J. Cullen, _Chem. Phys._ , 1996, 202, 217–229.
  * White (1992) S. R. White, _Phys. Rev. Lett._ , 1992, 69, 2863–2866.
  * Legeza _et al._ (2008) O. Legeza, R. M. Noack, J. Sólyom and L. Tincani, in _Computational Many-Particle Physics_ , ed. H. Fehske, R. Schneider and A. Weiße, Springer, Berlin/Heidelerg, 2008, vol. 739 of Lect. Notes Phys., pp. 653–664.
  * Marti and Reiher (2010) K. H. Marti and M. Reiher, _Z. Phys. Chem._ , 2010, 224, 583–599.
  * Chan and Sharma (2011) G. K.-L. Chan and S. Sharma, _Annu. Rev. Phys. Chem._ , 2011, 62, 465–481.
  * Mamache _et al._ (2023) S. Mamache, M. Gałyńska and K. Boguslawski, _Phys. Chem. Chem. Phys._ , 2023, 25, 18023–18029.
  * Tecmer _et al._ (2023) P. Tecmer, M. Gałyńska, L. Szczuczko and K. Boguslawski, _J. Phys. Chem. Lett._ , 2023, 14, 9909–9917.
  * Leszczyk _et al._ (2021) A. Leszczyk, M. Máté, O. Legeza and K. Boguslawski, _J. Chem. Theory Comput._ , 2021, 18, 96–117.
  * Henderson _et al._ (2014) T. M. Henderson, I. W. Bulik, T. Stein and G. E. Scuseria, _J. Chem. Phys._ , 2014, 141, 244104.
  * Boguslawski and Ayers (2015) K. Boguslawski and P. W. Ayers, _J. Chem. Theory Comput._ , 2015, 11, 5252–5261.
  * Boguslawski _et al._ (2014) K. Boguslawski, P. Tecmer, P. Bultinck, S. De Baerdemacker, D. Van Neck and P. W. Ayers, _J. Chem. Theory Comput._ , 2014, 10, 4873–4882.
  * Boguslawski _et al._ (2014) K. Boguslawski, P. Tecmer, P. A. Limacher, P. A. Johnson, P. W. Ayers, P. Bultinck, S. De Baerdemacker and D. Van Neck, _J. Chem. Phys._ , 2014, 140, 214114.
  * Boguslawski _et al._ (2016) K. Boguslawski, P. Tecmer and Ö. Legeza, _Phys. Rev. B_ , 2016, 94, 155126.
  * Limacher (2015) P. A. Limacher, _J. Chem. Theory Comput._ , 2015, 11, 3629–3635.
  * Tecmer _et al._ (2015) P. Tecmer, K. Boguslawski and P. W. Ayers, _Phys. Chem. Chem. Phys._ , 2015, 17, 14427–14436.
  * Brzęk _et al._ (2019) F. Brzęk, K. Boguslawski, P. Tecmer and P. S. Żuchowski, _J. Chem. Theory Comput._ , 2019, 15, 4021–4035.
  * Henderson and Scuseria (2019) T. M. Henderson and G. E. Scuseria, _J. Chem. Phys._ , 2019, 151, 051101\.
  * Nowak _et al._ (2021) A. Nowak, Ö. Legeza and K. Boguslawski, _J. Chem. Phys._ , 2021, 154, 084111.
  * Leszczyk _et al._ (2022) A. Leszczyk, T. Dome, P. Tecmer, D. Kedziera and K. Boguslawski, _Phys. Chem. Chem. Phys._ , 2022, 24, 21296–21307.
  * Tecmer _et al._ (2019) P. Tecmer, K. Boguslawski, M. Borkowski, P. S. Żuchowski and D. Kędziera, _Int. J. Quantum Chem._ , 2019, 119, e25983.
  * Garza _et al._ (2015) A. J. Garza, A. G. Sousa Alencar and G. E. Scuseria, _J. Chem. Phys._ , 2015, 143, 244106.
  * Boguslawski (2016) K. Boguslawski, _J. Chem. Phys._ , 2016, 145, 234105.
  * Boguslawski (2017) K. Boguslawski, _J. Chem. Phys._ , 2017, 147, 139901.
  * Nowak _et al._ (2019) A. Nowak, P. Tecmer and K. Boguslawski, _Phys. Chem. Chem. Phys._ , 2019, 21, 19039–19053.
  * Nowak and Boguslawski (2023) A. Nowak and K. Boguslawski, _Phys. Chem. Chem. Phys._ , 2023, 25, 7289–7301.
  * Chakraborty _et al._ (2023) R. Chakraborty, K. Boguslawski and P. B. Tecmer, _Phys. Chem. Chem. Phys._ , 2023, 25, 25377–25388.
  * Jahani _et al._ (2023) S. Jahani, K. Boguslawski and P. Tecmer, _RSC advances_ , 2023, 13, 27898–27911.
  * Boguslawski (2018) K. Boguslawski, _J. Chem. Theory Comput._ , 2018, 15, 18–24.
  * Kossoski _et al._ (2021) F. Kossoski, A. Marie, A. Scemama, M. Caffarel and P.-F. Loos, _J. Chem. Theory Comput._ , 2021, 17, 4756–4768.
  * Ravi _et al._ (2023) M. Ravi, A. Perera, Y. C. Park and R. J. Bartlett, _J. Chem. Phys._ , 2023, 159, 094101.
  * Sinanoğlu and Tuan (1963) O. Sinanoğlu and D. F. Tuan, _J. Chem. Phys._ , 1963, 38, 1740–1748.
  * Bartlett and Stanton (1994) R. J. Bartlett and J. F. Stanton, _Rev. Comput. Chem._ , 1994, 5, 165–169.
  * Boguslawski _et al._ (2012) K. Boguslawski, P. Tecmer, O. Legeza and M. Reiher, _J. Phys. Chem. Lett._ , 2012, 3, 3129–3135.
  * Kinoshita _et al._ (2005) T. Kinoshita, O. Hino and R. J. Bartlett, _J. Chem. Phys._ , 2005, 123, 074106.
  * Hino _et al._ (2006) O. Hino, T. Kinoshita, G. K.-L. Chan and R. J. Bartlett, _J. Chem. Phys._ , 2006, 124, 114311.
  * Lyakh _et al._ (2011) D. I. Lyakh, V. F. Lotrich and R. J. Bartlett, _Chem. Phys. Lett._ , 2011, 501, 166–171.
  * Melnichuk and Bartlett (2012) A. Melnichuk and R. J. Bartlett, _J. Chem. Phys._ , 2012, 137, 214103\.
  * Boguslawski and Tecmer (2017) K. Boguslawski and P. Tecmer, _J. Chem. Theory Comput._ , 2017, 13, 5966–5983.
  * Perera _et al._ (2014) A. Perera, R. W. Molt Jr., V. F. Lotrich and R. J. Bartlett, _Theor. Chem. Acc._ , 2014, 133, 1514.
  * Casanova and Krylov (2020) D. Casanova and A. I. Krylov, _Phys. Chem. Chem. Phys._ , 2020, 22, 4326–4342.
  * Ranasinghe _et al._ (2019) D. S. Ranasinghe, J. T. Margraf, A. Perera and R. J. Bartlett, _J. Chem. Phys._ , 2019, 150, 074108.
  * Dempwolff _et al._ (2022) A. L. Dempwolff, M. Hodecker and A. Dreuw, _J. Chem. Phys._ , 2022, 156, 054114.
  * Boguslawski _et al._ (2021) K. Boguslawski, A. Leszczyk, A. Nowak, F. Brzęk, P. S. Żuchowski, D. Kędziera and P. Tecmer, _Comput. Phys. Commun._ , 2021, 264, 107933.
  * Boguslawski _et al._ (2024) K. Boguslawski, F. Brzęk, R. Chakraborty, K. Cieślak, S. Jahani, A. Leszczyk, A. Nowak, E. Sujkowski, J. Świerczyński, S. Ahmadkhani, D. Kędziera, M. H. Kriebel, P. S. Żuchowski and P. Tecmer, _Comput. Phys. Commun._ , 2024, 297, 109049.
  * (113) F. Brzęk, R. Chakraborty, K. Cieślak, S. Jahani, A. Leszczyk, A. Nowak, E. Sujkowski, J. Świerczyński, S. Ahmadkhani, K. Boguslawski, D. Kędziera, M. Kriebel, P. Tecmer and P. S. Żuchowski, _PyBEST_ , 10.5281/zenodo.3925277.
  * Dunning Jr (1989) T. H. Dunning Jr, _J. Chem. Phys._ , 1989, 90, 1007–1023.
  * Kendall _et al._ (1992) R. A. Kendall, T. H. Dunning Jr and R. J. Harrison, _J. Chem. Phys._ , 1992, 96, 6796–6806.