# A configuration interaction correction on top of pair coupled cluster
doubles

Artur Nowak  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

Numerous numerical studies have shown that geminal-based methods are a
promising direction to model strongly correlated systems with low
computational costs. Several strategies have been introduced to capture the
missing dynamical correlation effects, which typically exploit a posteriori
corrections to account for correlation effects associated with broken-pair
states or inter-geminal correlations. In this article, we scrutinize the
accuracy of the pair coupled cluster doubles (pCCD) method extended by
configuration interaction (CI) theory. Specifically, we benchmark various CI
models, including, at most double excitations against selected CC corrections
as well as conventional single-reference CC methods. A simple Davidson
correction is also tested. The accuracy of the proposed pCCD-CI approaches is
assessed for challenging small model systems such as the N2 and F2 dimers and
various di- and triatomic actinide-containing compounds. In general, the
proposed CI methods considerably improve spectroscopic constants compared to
the conventional CCSD approach, provided a Davidson correction is included in
the theoretical model. At the same time, their accuracy lies between the
linearized frozen pCCD and frozen pCCD variants.

††preprint: AIP/123-QED

##  I Introduction

The reliable and efficient description of electron correlation effects is an
open problem in quantum chemistry. Although the correlation energy is
quantitatively small compared to the total electronic energy, the proper
description of the correlated motion of the electrons is crucial to understand
physical and chemical phenomena like bond-breaking processes. Typically,
electron correlation effects are divided into a dynamical and a non-
dynamical/static part. To capture both contributions simultaneously, we
require advanced methods that exploit a multireference description. Examples
thereof are multireference coupled cluster approaches Jeziorski (2010); Lyakh
_et al._ (2012); Ivanov, Lyakh, and Adamowicz (2009), the density matrix
renormalization group (DMRG) algorithm White (1992, 1993); White and Martin
(1999); Legeza  _et al._ (2008); Marti and Reiher (2010); Chan and Sharma
(2011); Szalay  _et al._ (2015); Hachmann, Cardoen, and Chan (2006); Kurashige
and Yanai (2009); Yanai  _et al._ (2015); Baiardi and Reiher (2020); Gunst
_et al._ (2018); Ma  _et al._ (2017), or quantum Monte Carlo methods Zhang
(2013); Hochkeppel  _et al._ (2009). Despite their high accuracy, these
methods are computationally rather expensive.

An alternative strategy to account for strong electron correlation effects is
to exploit geminal-based methods. Tecmer and Boguslawski (2022) The key idea
of geminal-based electronic structure approaches is to use non-interacting
electron pairs as the fundamental building blocks of the electronic wave
function. Various numerical studies using geminal-based methods, such as the
antisymmetric product of interacting geminals Coleman (1965); Bratoz and
Durand (1965); Silver (1969, 1970); Náray-Szabó (1973, 1975), the
antisymmetric product of strongly orthogonal geminals (APSG) Hurley, Lennard-
Jones, and Pople (1953); Parks and Parr (1958); Kapuy (1966); Kutzelnigg
(1964); Surján (1984, 1985, 1994, 1995); Surjan (1999); Rosta and Surján
(2000); Surján  _et al._ (2012), singlet-type strongly orthogonal geminals,
Rassolov (2002); Rassolov, Xu, and Garashchuk (2004); Cagg and Rassolov (2014)
geminals constructed from Richardson–Gaudin states, Johnson  _et al._ (2013,
2017, 2020); Fecteau  _et al._ (2021, 2022) and the antisymmetric product of
1-reference orbital geminals Limacher  _et al._ (2013); Boguslawski  _et al._
(2014a), also known as the pair-coupled cluster doubles (pCCD) ansatz Stein,
Henderson, and Scuseria (2014), yield promising results. Specifically, the
pCCD approach is a product of geminal creation operators $\phi_{i}^{\dagger}$
acting on some vacuum state

$|\textrm{pCCD}\rangle=\prod_{i=1}^{P}\phi_{i}^{\dagger}|0\rangle,$ (1)

with $P$ being the number of electron pairs and

$\phi_{i}^{\dagger}=a^{\dagger}_{i}a^{\dagger}_{\bar{i}}+\sum_{a}^{\rm
virt}c_{i}^{a}a^{\dagger}_{a}a^{\dagger}_{\bar{a}},$ (2)

where the sum runs over all virtual orbitals $a$ and $\\{c_{i}^{a}\\}$ are the
geminal expansion coefficients. In eq. (2), $i$ ($\bar{i}$) indicates spin-up
(spin-down) electrons. Using an exponential ansatz Limacher  _et al._ (2013),
the pCCD wave function can be rewritten as

$\displaystyle|\textrm{pCCD}\rangle$ $\displaystyle=\exp(\sum_{i}^{\rm
occ}\sum_{a}^{\rm
virt}c_{i}^{a}a^{\dagger}_{a}a^{\dagger}_{\bar{a}}a_{\bar{i}}a_{i})|\Phi_{0}\rangle$
$\displaystyle=\exp(\hat{T}_{\textrm{p}})\ket{\Phi_{0}},$ (3)

where $|\Phi_{0}\rangle$ is some independent-particle wave function, for
instance, the Hartree–Fock (HF) determinant, and $\hat{T}_{\textrm{p}}$ is the
electron-pair cluster operator. One benefit in using the exponential form is
the proper (linear) scaling of the method with the number of electrons (size-
extensivity). To recover size-consistency it is necessary to apply an orbital-
optimization protocol Boguslawski  _et al._ (2014a); Stein, Henderson, and
Scuseria (2014); Boguslawski  _et al._ (2014b, c) which also leads to
localized orbitals, which are symmetry-broken Boguslawski  _et al._ (2014a);
Stein, Henderson, and Scuseria (2014); Boguslawski  _et al._ (2014b, c).
Investigations on the one-dimensional Hubbard model Boguslawski  _et al._
(2014a); Boguslawski, Tecmer, and Legeza (2016) or molecules with stretched
bonds Tecmer  _et al._ (2014a); Stein, Henderson, and Scuseria (2014);
Henderson  _et al._ (2014); Boguslawski and Tecmer (2015, 2017a); Tecmer,
Boguslawski, and Ayers (2015); Garza, Alencar, and Scuseria (2015);
Boguslawski and Ayers (2015); Boguslawski and Tecmer (2017b); Brzęk  _et al._
(2019); Kossoski  _et al._ (2021), even those containing lanthanide Tecmer
_et al._ (2019) or actinide atoms Tecmer, Boguslawski, and Ayers (2015);
Garza, Alencar, and Scuseria (2015); Nowak, Tecmer, and Boguslawski (2019)
demonstrate that geminal-based approaches can capture an important part of
static/nondynamic electron correlation effects. Since pCCD requires a
reasonable amount of computational resources (with a scaling of $N^{4}$ or
$N^{3}$ if proper intermediates are defined), heavy-element-containing
compounds are a promising scope of applications. Leszczyk, Tecmer, and
Boguslawski (2019) Nevertheless, by restricting the wave function ansatz to
electron-pair states, a large fraction of the correlation energy is not
accounted for. This missing dynamical electron correlation energy that cannot
be described using electron-pair states, is commonly included a posteriori by
means of, for instance, perturbation theory Rosta and Surján (2002);
Christiansen, Koch, and Jørgensen (1996); Limacher  _et al._ (2014);
Boguslawski and Tecmer (2017b), a coupled cluster ansatzHenderson  _et al._
(2014); Zoboki, Szabados, and Surjan (2013); Boguslawski and Ayers (2015);
Leszczyk  _et al._ (2022), or density functional approximations Garza  _et
al._ (2015a, b). In this work, we develop a different a posteriori correction
for a pCCD reference function using configuration interaction (CI) theory.
Specifically, we will focus on a CI correction, which is restricted to hole-
particle excitations with respect to a reference determinant. Similar CI
models have been already combined with, for instance, an APSG Kállay and
Surján (1999), generalized valence bond Røeggen (1987); Carter and Goddard III
(1988); Faglioni and Goddard III (1999); Røeggen (1999), or an antisymmetrized
geminal power Henderson and Scuseria (2019) reference function.

This work is organized as follows. In section II, we briefly summarize the
theory of the CI corrections on top of pCCD, which are introduced using a
spin-free formalism. The computational details are summarized in section III,
while the numerical results are presented in sections IV–V. Finally, we
conclude in section VI.

##  II CI Corrections with a pCCD Reference Function

Conventional (truncated) CI methods define the electronic wave function as a
linear combination of all possible “excited” configurations with respect to
some reference configuration $\ket{\Phi_{0}}$,

$\displaystyle\ket{\textrm{CI}}=\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}\ket{\Phi_{0}}=\sum_{\mu}c_{\mu}\ket{{\Phi_{\mu}}},$
(4)

where in this compact representation $c_{\mu}$ are the CI coefficients and
$\hat{\tau}_{\mu}$ is an excitation operator including occupied–virtual (or
hole–particle) excitations up to a predefined level. To arrive at a CI-type
correction, we replace the reference determinant with a pCCD reference
function eq. (I),

$\displaystyle|\textrm{pCCD-
CI}\rangle=\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}e^{\hat{T}_{\rm
p}}|{\rm\Phi_{0}}\rangle.$ (5)

Substituting this ansatz for the electronic wave function $\ket{\Psi}$ in the
Schrödinger equation $\hat{H}\ket{\Psi}=E\ket{\Psi}$, we obtain

$\displaystyle\sum_{\mu}\hat{H}c_{\mu}\hat{\tau}_{\mu}e^{\hat{T}_{\rm
p}}|{\rm\Phi_{0}}\rangle$
$\displaystyle=E\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}e^{\hat{T}_{\rm
p}}|{\rm\Phi_{0}}\rangle$ $\displaystyle\sum_{\mu}\hat{H}e^{\hat{T}_{\rm
p}}c_{\mu}\hat{\tau}_{\mu}|{\rm\Phi_{0}}\rangle$
$\displaystyle=E\sum_{\mu}e^{\hat{T}_{\rm
p}}c_{\mu}\hat{\tau}_{\mu}|{\rm\Phi_{0}}\rangle$ (6)

where $\hat{H}$ is the electronic Hamiltonian comprising one- and two-electron
integrals and we utilized that $\hat{\tau}_{\mu}$ and $\hat{T}_{\rm p}$
commute as they contain only hole–particle excitation operators,
$[\hat{\tau}_{\mu},\hat{T}_{\rm p}]=0$. Unfortunately, the eigenvalue problem
of eq. (II) cannot be solved in its symmetric representation as the
corresponding series expansion, that enters the working equation, does not
truncate. In order to solve for the eigenvalues and eigenvectors efficiently,
we can project out the components of interests, which will lead to a
diagonalization problem of a non-Hermitian matrix. There are several choices
for the projection space, which is defined in terms of the de-excitation
operator $\hat{\tau}^{\dagger}_{\mu}$. We can either project by
$\bra{\Phi_{0}}\hat{\tau}^{\dagger}_{\mu}$ or
$\bra{\Phi_{0}}\hat{\tau}^{\dagger}_{\mu}e^{-\hat{T_{\rm p}}}$. We should note
that if we restrict the CI operator to comprise at most single and double
excitations, both projection schemes will result in the same solutions.Kállay
and Surján (1999) In the following, we will focus on the latter approach.
Multiplying eq. (II) from left with $e^{-\hat{T_{\rm p}}}$ and using the
property that $\hat{T_{\rm p}}$ and $\hat{\tau}_{\mu}$ commute, results in

$\displaystyle e^{-\hat{T_{\rm p}}}\hat{H}e^{\hat{T_{\rm
p}}}\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}|{\rm\Phi_{0}}\rangle$
$\displaystyle=E\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}e^{-\hat{T_{\rm
p}}}e^{\hat{T_{\rm p}}}|{\rm\Phi_{0}}\rangle$
$\displaystyle\mathcal{{H}}^{\rm(p)}\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}|{\rm\Phi_{0}}\rangle$
$\displaystyle=E\sum_{\mu}c_{\mu}\hat{\tau}_{\mu}|{\rm\Phi_{0}}\rangle.$ (7)

Thus, the complex CI problem has now the form of a conventional single-
reference (truncated) CI model, where the Hamiltonian is substituted by the
similarity-transformed Hamiltonian of pCCD $\mathcal{{H}}^{\rm(p)}$. The
diagonalization problem becomes computationally feasible and can be solved in
efficient time as the commutator expansion of $e^{-\hat{T_{\rm
p}}}\hat{H}e^{\hat{T_{\rm p}}}$ naturally truncates. The corresponding working
equations bear similarities to the EOM formalism Rowe (1968); Geersten,
Rittby, and Bartlett (1989); Kowalski  _et al._ (2011); Krylov (2008) as EOM-
CC diagonalizes the similarity-transformed Hamiltonian of a given CC model.
The main difference between conventional EOM-CC and the proposed pCCD-CI model
is that the similarity-transformed Hamiltonian to be diagonalized has a
particular simple form (as it only accounts for electron-pair excitation),
while the CI ansatz includes any choice for the excitation operator. In
standard EOM-CC theory, both the CC cluster operator and the CI ansatz are
restricted to the same order of excitations.

Furthermore, we will consider only singlet excitations. The corresponding
projection manifold will result in a spin-free CI problem, where all
eigenstates of $\mathcal{{H}}^{\rm(p)}$ will be singlet states. Moreover, from
all working equations, we have subtracted the energy of the reference
function, that is, the pCCD total energy (similar to the conventional CI
equations, where the energy of the reference determinant is subtracted). Thus,
the eigenenergies correspond to excitation energies with respect to the pCCD
reference function. In this spin-free formulation, the excitation operator
that contains single excitations can be expressed as follows

$\displaystyle\hat{\tau}=\hat{\tau}_{0}+\hat{\tau}_{\rm S}$
$\displaystyle=c_{0}+\sum_{ia}c_{i}^{a}(a_{a}^{\dagger}a_{i}+a_{\bar{a}}^{\dagger}a_{\bar{i}})$
$\displaystyle=c_{0}+\sum_{ia}c_{i}^{a}E_{i}^{a},$ (8)

where we introduced the singlet one-electron excitation operator
$E_{i}^{a}=a_{a}^{\dagger}a_{i}+a_{\bar{a}}^{\dagger}a_{\bar{i}}$. To account
for double excitations, the above operator is modified as follows

$\displaystyle\hat{\tau}$ $\displaystyle=\hat{\tau}_{0}+\hat{\tau}_{\rm D}$
$\displaystyle=c_{0}+\frac{1}{2}\sum_{ijab}c_{ij}^{ab}(a_{a}^{\dagger}a_{b}^{\dagger}a_{j}a_{i}+a_{a}^{\dagger}a_{\bar{b}}^{\dagger}a_{\bar{j}}a_{i}+a_{\bar{a}}^{\dagger}a_{b}^{\dagger}a_{j}a_{\bar{i}}^{\dagger}+a_{\bar{a}}^{\dagger}a_{\bar{b}}^{\dagger}a_{\bar{j}}a_{\bar{i}})$
$\displaystyle=c_{0}+\frac{1}{2}\sum_{ijab}c_{ij}^{ab}E_{i}^{a}E_{j}^{b}.$ (9)

To arrive at a CISD correction, the excitation operators for singles and
doubles are combined, that is, $\hat{\tau}=\hat{\tau}_{0}+\hat{\tau}_{\rm
S}+\hat{\tau}_{\rm D}$. We should note that we tested two different
$\hat{\tau}_{\rm D}$ excitation operators. In the pCCD-CID and pCCD-CISD
model, all electron pair excitations are excluded from the CI ansatz as they
are already accounted for in the pCCD reference calculations. If we are
interested in targeting several roots, that is, the electronic ground state
and some lowest-lying electronically excited states, this approximation will
neglect any double excitations of electron-pair character and, hence, bi-
excited states might miss important contributions to the wave function
expansion. Thus, in the pCCD-CID′ and pCCD-CISD′ models, the electron-pair
sector is added to the general excitation operator $\hat{\tau}_{\rm D}$.
However, including electron-pair excitations in the CI ansatz might lead to
double-counting problems of electron correlation effects associated with
electron-pair states. In the following, we assess both CI ansätze (with and
without electron-pair excitations) in describing ground and electronically
excited states.

In order to solve eq. (II), that is, to find the eigenvalues and eigenvectors
of $\mathcal{{H}}^{\rm(p)}$, its matrix representation has to be diagonalized.
In practical calculations, iterative techniques are used to compute only a few
of the lowest eigenvalues and corresponding eigenvectors, for instance non-
Hermitian extensions of the Davidson algorithm (the corresponding working
equations are collected in the SI). We should note that the projection
manifold has a special form Helgaker, T., Jørgensen, P., Olsen, J (2000)
(similar to spin-free CC theory) to allow for an algebraic spin summation,
which results in the spin-free pCCD-CI working equations,

$\displaystyle\bra{\overline{\Phi_{I}^{A}}}$
$\displaystyle=\frac{1}{2}\bra{\Phi_{0}}E_{a}^{i}$ (10)
$\displaystyle\bra{\overline{\Phi_{IJ}^{AB}}}$
$\displaystyle=\frac{1}{3}\bra{\Phi_{0}}E^{j}_{b}E^{i}_{a}+\frac{1}{6}\bra{\Phi_{0}}E^{j}_{a}E^{i}_{b}.$
(11)

Finally, we include a Davidson correction on top of the proposed pCCD-CI
models to minimize the size-consistency error intrinsic to (truncated) CI
methods. Szalay (2005); Szalay  _et al._ (2012) Specifically, we combine a
renormalized Davidson correction introduced in Ref. 84 exploiting the pCCD-CI
correlation energy,

$E_{\rm{RDC}}=\Big{(}\frac{1-c_{0}^{2}}{c_{0}^{2}}\Big{)}\Big{(}{E_{\rm pCCD-
CI}-{E_{\rm RF}}}\Big{)},$ (12)

where ${E_{\rm pCCD-CI}}$ indicates the total energy of the pCCD-CI method,
${E_{\rm RF}}$ is the energy of the reference method (here pCCD), and $c_{0}$
is the contribution of the reference determinant ($\ket{\Phi_{0}}$) of the
reference method.

##  III Computational details

###  III.1 Basis sets, relativistic effects, and frozen core

For the light diatomic molecules $\ce{H_{2}}$ and $\ce{N_{2}}$, we used the
triple-$\zeta$ correlation consistent basis sets of Dunning Dunning Jr. (1989)
(cc-pVTZ), while for $\ce{F_{2}}$ the augmented version (aug-cc-pVTZ) was
employed. For all actinide compounds, we used the double-$\zeta$ correlation
consistent basis sets of Peterson Peterson (2015) (cc-pVDZ-DK3) for all heavy
elements, optimized specifically for the DKH3 Hamiltonian, Reiher and Wolf
(2004a, b, 2009); Tecmer, P., Boguslawski, K., Kȩdziera, D. (2017) and
Dunning’s aug-cc-pVDZ basis set for all the remaining light elements. Scalar
relativistic effects were accounted for by the DKH3 Hamiltonian. All
calculations for systems containing heavy-elements were performed with a
frozen core to ensure a compromise between computational efficiency, the
reliability of the results for heavy-element-containing systems, and the
quality of the atomic basis sets Réal  _et al._ (2009); Tecmer  _et al._
(2014b); Tecmer, Boguslawski, and Ayers (2015); Tecmer and González-Espinoza
(2018). Specifically, the atomic 1s orbitals of the O and N atom, the 1s, 2s,
and 2p orbitals of the S atom, and up to the 5d orbitals of the Th, Pa, and U
center were frozen. All post-pCCD methods as well as all CCD and CCSD
calculations were performed using a developer version of the PyBEST v1.2.0
software package Boguslawski  _et al._ (2021); Brzęk  _et al._ (2022); pyb
(2022). All pCCD-based calculations exploited the natural orbitals obtained
from orbital-optimized pCCD, while all conventional CC calculations featured
canonical Hartree–Fock orbitals. The CCSD(T) reference calculations were
performed in the Molpro2020 Werner  _et al._ (2009) software suite.

###  III.2 Evaluations of spectroscopic constants

The potential energy curves were obtained from a polynomial fit of 8-th order
using the fitting scripts available in the PyBEST Brzęk  _et al._ (2022)
software package. From these fitted potential energy curves, we derived the
corresponding spectroscopic constants such as equilibrium bond lengths
($r_{\rm e}$) and harmonic vibrational frequencies ($\omega_{\rm e}$) for all
actinide compounds and light diatomic molecules. Additionally, the potential
energy well depths ($D_{e}$) were calculated from fitting a generalized Morse
function. To determine the harmonic vibrational frequencies ($\omega_{\rm
e}$), we performed numerical calculations using the five-point finite
difference stencil Abramowitz and Stegun (1970) with the following averaged
masses: uranium: 238.0508, thorium: 232.0381, protactinium: 231.0359, oxygen:
15.9949, sulfur: 31.9721, and nitrogen: 14.0031. Sansonetti and Martin (2005)

Figure 1: Potential energy surfaces of the ground states of $N_{2}$ determined
for various post-pCCD with (a) doubles excitation and (b) singles and doubles
excitations. The superscript ′ indicates that the CI operator also includes
the electron pair sector. RDC: renormalized Davidson correction.

##  IV Assessing the accuracy of the CI corrections for light diatomic
molecules

Our first test set to assess the performance of the CI corrections contains
the N2 and F2 diatomic molecules. In these systems, the different flavors of
electron correlation effects are difficult to describe with various pCCD
corrections. For instance, a remarkable challenging problem for computational
chemistry is to accurately model the electronic structure of the N_2 molecule
Chan, Kállay, and Gauss (2004); Boguslawski  _et al._ (2012); Henderson  _et
al._ (2014); Boguslawski  _et al._ (2017); Nowak, Legeza, and Boguslawski
(2021); Leszczyk  _et al._ (2022) along its dissociation pathway. This problem
is related to dynamic changes in the contributions of electron correlation
effects for an increasing interatomic distance. In general, around the
equilibrium distances, dynamical electron correlation effects are prevalent,
while the contribution of nondynamic/static electron correlation significantly
increases when the N–N bond is stretched. Furthermore, all CC corrections on
top of pCCD restricted to at most double excitations fail and do not yield a
physically meaningful dissociation limit. Unlike in the nitrogen dimer,
dynamical correlation effects are dominant in the F_2 molecule. Conventional
methods require even triple excitations to model the dissociation pathway of
F_2 reliablyKowalski and Piecuch (2001); Boguslawski  _et al._ (2017); Nowak,
Legeza, and Boguslawski (2021); Leszczyk  _et al._ (2022). Thus, the F_2 dimer
represents a good test system to assess if novel methods can accurately
describe a bond-breaking process without increasing the computational cost
related to the inclusion of triple (or higher) excitations. Since pCCD-based
methods do not provide sufficiently accurate results in these systems, they
represent a good testing ground to assess the accuracy and robustness of the
CI corrections.

Table 1: Spectroscopic parameters for the ground state of the N_2 molecule obtained from various post-pCCD methods and conventional CC models. $r_{\mathrm{e}}$ is the equilibrium bond length, $\omega_{\mathrm{e}}$ the harmonic vibrational frequency, and $D_{\mathrm{e}}$ the dissociation energy, respectively. The differences with respect to experimental result $\Delta_{\rm{exp}}$ or MRCI reference data $\Delta_{\rm{MRCI}}$ are given in parentheses. The total electronic energies are summarized in the SI. RDC: renormalized Davidson correction.  Method |  $r_{\mathrm{e}}$ [Å] | $\Delta_{\rm{MRCI}}$ | $\Delta_{\rm{expr}}$ |  $\omega_{\mathrm{e}}$ [cm-1] | $\Delta_{\rm{MRCI}}$ | $\Delta_{\rm{exp}}$ |  $D_{\mathrm{e}}$ [$\frac{kcal}{mol}$] | $\Delta_{\rm{MRCI}}$ | $\Delta_{\rm{exp}}$
---|---|---|---|---|---|---|---|---|---
oo-pCCD | 1.085 |  ($-0.019$) |  ($-0.013$) | 2517 |  ($+176$) |  ($+158$) | 244.8 |  ($+26.9$) |  ($+19.7$)
pCCD-CID′ | 1.091 |  ($-0.013$) |  ($-0.007$) | 2454 |  ($+113$) |  ($+95$) | 304.3 |  ($+86.4$) |  ($+79.2$)
pCCD-CID′+RDC | 1.095 |  ($-0.009$) |  ($-0.003$) | 2407 |  ($+66$) |  ($+48$) | 263.3 |  ($+45.4$) |  ($+38.2$)
pCCD-CID | 1.096 |  ($-0.008$) |  ($-0.002$) | 2399 |  ($+58$) |  ($+40$) | 266.1 |  ($+48.2$) |  ($+41.0$)
pCCD-CID+RDC | 1.099 |  ($-0.005$) |  ($+0.001$) | 2369 |  ($+28$) |  ($+10$) | 231.9 |  ($+14.0$) |  ($+6.8$)
fpLCCD | 1.101 |  ($-0.003$) |  ($+0.003$) | 2347 |  ($+6$) |  ($-12$) |  |  |
fpCCD | 1.100 |  ($-0.004$) |  ($+0.002$) | 2365 |  ($+24$) |  ($+6$) |  |  |
CCD | 1.093 |  ($-0.011$) |  ($-0.005$) | 2447 |  ($+106$) |  ($+88$) |  |  |
pCCD-CISD′ | 1.088 |  ($-0.016$) |  ($-0.013$) | 2447 |  ($+106$) |  ($+88$) | 305.4 |  ($+87.5$) |  ($+80.3$)
pCCD-CISD′+RDC | 1.095 |  ($-0.009$) |  ($-0.003$) | 2396 |  ($+55$) |  ($+37$) | 263.3 |  ($+45.4$) |  ($+19.7$)
pCCD-CISD | 1.097 |  ($-0.007$) |  ($-0.001)$ | 2394 |  ($+53$) |  ($+35$) | 270.4 |  ($+52.5$) |  ($+45.3$)
pCCD-CISD+RDC | 1.099 |  ($-0.005$) |  ($+0.001$) | 2360 |  ($+19$) |  ($+1$) | 234.4 |  ($+16.5$) |  ($+9.3$)
fpLCCSD | 1.102 |  ($-0.002$) |  ($+0.004$) | 2340 |  ($-1$) |  ($-19$) |  |  |
fpCCSD | 1.100 |  ($-0.011$) |  ($+0.002$) | 2365 |  ($+24$) |  ($+6$) |  |  |
CCSD | 1.093 |  ($-0.011$) |  ($-0.005$) | 2444 |  ($+103$) |  ($+85$) |  |  |
MRCI Peterson, Kendall, and Dunning (1993) | 1.104 |  |  ($+0.006$) | 2341 |  |  ($-18$) | 217.9 |  |  ($-7.2$)
exp Huber and Herzberg (1979); Shimanouchi  _et al._ (1978) | 1.098 |  ($-0.006$) |  | 2359 |  ($+18$) |  | 225.1 |  ($+7.2$) |

###  IV.1 The Nitrogen Dimer

Figure 1 shows the PESs obtained from various pCCD-based methods and
conventional single-reference CC models restricted to double and single and
double excitations. We performed pCCD-CID and pCCD-CISD calculations with and
without a renormalized Davidson correction (RDC). The corresponding results
are indicated by including “+RDC” in the label. Furthermore, the superscripts
′ indicates that the electron pair sector is included in CI calculations.

All investigated corrections generally predict a similar behavior around the
equilibrium distances, while the shape of the PES features only small changes
up to a bond distance of 1.6 Å. From this point onward, the PESs obtained from
various pCCD-CI methods differ from those predicted by fp(L)CC as well as
conventional CC approaches. This observations is true for both variants,
including only double (Figure 1-a) as well as single and double (Figure 1-b)
excitations in the CI ansatz. Specifically, for interatomic distances of $\geq
2$ Å, conventional methods as well as pCCD-based CC corrections diverge. On
the other hand, the potential energy curves predicted by pCCD-CI are smooth
and do not suffer from divergencies along the reaction coordinate. In the
vicinity of dissociation (around 3 Å), the simple RD correction, however,
breaks down and yields undershooting PESs. Nonetheless, a single-reference CI
correction restricted to at most double excitations is able to provide a sound
dissociation pathway for the N_2 molecule, while approaches based on CC
corrections break down for stretched N–N bond lengths.

For a quantitative analysis, we determined the spectroscopic constants from
the fitted PESs, which are summarized in Table 1 and compared to MRCI and
experimental reference data, respectively. In general, all investigated CI and
CC methods underestimate $r_{\mathrm{e}}$ and overestimate
$\omega_{\mathrm{e}}$ and $D_{\mathrm{e}}$, respectively. Both the CID and
CISD correction provide similar spectroscopic constants. Most importantly,
including the electron pair sector in the CI ansatz considerably worsens
vibrational frequencies and potential energy well depths, where the errors are
approximately doubled and approach the CCD/CCSD level of accuracy. Including a
Davidson correction on top of pCCD-CID or pCCD-CISD significantly lowers
errors in spectroscopic constants by more than a factor of 2. In general,
pCCD-CISD+RDC yields values for $r_{\mathrm{e}}$ and $\omega_{\mathrm{e}}$
that lie between fpLCCSD and fpCCD accuracy, while fpLCCSD predicts
spectroscopic constants closest to MRCI reference data. Conventional CC
methods (CCD and CCSD) feature the largest deviations from MRCI and
experimental reference data. We should note that a CI correction excluding the
electron pair sector, but including a Davidson correction, yields results
closest to experiment (with differences up to 0.001 Å, 1 cm-1, or 9 kcal mol-1
for $r_{\mathrm{e}}$, $\omega_{\mathrm{e}}$, and $D_{\mathrm{e}}$,
respectively). A significant advantage of pCCD-CI methods over CC-type
corrections is the ability to dissociate the N2 molecule without encountering
divergencies. Similar to $r_{\mathrm{e}}$ and $\omega_{\mathrm{e}}$, the most
reliable results for $D_{\mathrm{e}}$ are obtained from pCCD-CISD+RDC (that
is, excluding the electron pair sector). Although the fitted value of
$D_{\mathrm{e}}$ deviates by about 4% from the experimental value, the
absolute difference of 9 kcal mol-1 is dissatisfying. Compared to MRCI
reference data, this error further increases to 16 kcal mol-1. In summary,
pCCD-CI methods excluding the electron pair sector are more accurate than the
corresponding CI flavours explicitly including electron pair excitations.
Moreover, an RDC correction noticeably improves the performance of the
proposed pCCD-CI models, significantly lowering the $\Delta_{\rm{MRCI}}$ and
$\Delta_{\rm{exp}}$ errors. In general, pCCD-CI outperforms conventional CC
methods of similar cost, provides spectroscopic constants between the fpLCC
and fpCC level of accuracy, and can efficiently model the static/nondynamical
and dynamical correlation along the reaction coordinate.

Figure 2: Potential energy surfaces for the ground states of F_2 determined for various post-pCCD methods with (a) doubles excitation and (b) singles and doubles excitations. The superscript ′ indicates that the CI operator also includes the electron pair sector. RDC: renormalized Davidson correction.  Table 2: Spectroscopic parameters for the ground state of the F_2 molecule obtained from various post-pCCD methods and conventional CC models. $r_{\mathrm{e}}$ is the equilibrium bond length, $\omega_{\mathrm{e}}$ the harmonic vibrational frequency, and $D_{\mathrm{e}}$ the dissociation energy, respectively. The differences with respect to experimental result $\Delta_{\rm{exp}}$ or MRCI reference data $\Delta_{\rm{MRCI}}$ are given in parentheses. The total electronic energies are summarized in the SI. RDC: renormalized Davidson correction.  Method |  $r_{\mathrm{e}}$ [Å] | $\Delta_{\rm{MRCI}}$ | $\Delta_{\rm{expr}}$ |  $\omega_{\mathrm{e}}$ [cm-1] | $\Delta_{\rm{MRCI}}$ | $\Delta_{\rm{exp}}$ |  $D_{\mathrm{e}}$ [$\frac{kcal}{mol}$] | $\Delta_{\rm{MRCI}}$ | $\Delta_{\rm{exp}}$
---|---|---|---|---|---|---|---|---|---
oo-pCCD | 1.458 |  ($+0.038$) |  ($+0.046$) | 733 |  ($-159$) |  ($-184$) | 18.6 |  ($-15.3$) |  ($-18.7$)
pCCD-CID′ | 1.397 |  ($-0.023$) |  ($-0.015)$ | 975 |  ($+83$) |  ($+58$) | 60.5 |  ($+26.6$) |  ($+22.8$)
pCCD-CID′+RDC | 1.403 |  ($-0.017$) |  ($-0.009$) | 954 |  ($+62$) |  ($+37$) | 59.0 |  ($+25.1$) |  ($+21.3$)
pCCD-CID | 1.431 |  ($+0.011$) |  ($+0.019$) | 857 |  ($-35$) |  ($-60$) | 42.2 |  ($+8.3$) |  ($+4.5$)
pCCD-CID+RDC | 1.430 |  ($+0.010$) |  ($+0.018$) | 857 |  ($-35$) |  ($-60$) | 42.4 |  ($+8.5$) |  ($+4.7$)
fpLCCD | 1.434 |  ($+0.014$) |  ($+0.022$) | 866 |  ($-26$) |  ($-51$) | 44.3 |  ($+10.4$) |  ($+6.6$)
fpCCD | 1.433 |  ($+0.013$) |  ($+0.021$) | 863 |  ($-29$) |  ($-48$) | 43.3 |  ($+9.4$) |  ($+5.6$)
CCD | 1.383 |  ($-0.037$) |  ($-0.029$) | 1053 |  ($+161$) |  ($+136$) | 82.2 |  ($+48.3$) |  ($+44.5$)
pCCD-CISD′ | 1.397 |  ($-0.023$) |  ($-0.015$) | 978 |  ($+86$) |  ($+71$) | 60.9 |  ($+27.0$) |  ($+23.2$)
pCCD-CISD′+RDC | 1.403 |  ($-0.017$) |  ($-0.009$) | 957 |  ($+65$) |  ($+40$) | 59.3 |  ($+25.4$) |  ($+21.6$)
pCCD-CISD | 1.429 |  ($+0.009$) |  ($+0.017$) | 864 |  ($-28$) |  ($-53$) | 42.7 |  ($+8.8$ ) |  ($+5.0$)
pCCD-CISD+RDC | 1.429 |  ($+0.009$) |  ($+0.017$) | 867 |  ($-25$) |  ($-50$) | 43.1 |  ($+9.2$) |  ($+5.4$)
fpLCCSD | 1.431 |  ($+0.011$) |  ($+0.019$) | 879 |  ($-13$) |  ($-38$) | 45.2 |  ($+11.3$) |  ($+7.5$)
fpCCSD | 1.431 |  ($+0.011$) |  ($+0.019$) | 873 |  ($-19$) |  ($-44$) | 44.2 |  ($+10.3$) |  ($+6.5$)
CCSD | 1.392 |  ($-0.028$) |  ($-0.020$) | 1018 |  ($+126$) |  ($+101$) | 70.1 |  ($+36.2$) |  ($+32.4$)
MRCI Peterson, Kendall, and Dunning (1993) | 1.420 |  |  ($+0.008$) | 892 |  |  ($-25$) | 33.9 |  |  ($-3.8$)
exp Huber and Herzberg (1979); Irikura (2007) | 1.412 |  ($-0.008$) |  | 917 |  ($+25$) |  | 37.7 |  ($+3.8$) |

###  IV.2 The Fluorine Dimer

In contrast to the N_2 molecule, we do not observe any significant differences
in the shape of the PESs predicted by all investigated pCCD corrections for
the F2 dimer (see Figure 2). The similarities in the shape of the potential
energy curves translate into similar values of spectroscopic constants, which
are summarized in Table 2. As expected, major differences
($\Delta_{\rm{MRCI}}$) are obtained if the electron pair sector is included in
the CI ansatz. We should note that, in contrast to the N2 dimer,
$\Delta_{\rm{exp}}$ for $r_{\mathrm{e}}$ and $\omega_{\mathrm{e}}$
significantly decreases in CI′ corrections, while errors in $D_{\mathrm{e}}$
considerably increase. Most importantly, the changes in spectroscopic
constants are only minor if single excitations are included in the CI or CC
correction. Moreover, the accuracy in spectroscopic constants is similar for
pCCD-CI methods (without electron pair excitations) and fp(L)CC models. While
pCCD-CI provides smaller errors in $r_{\mathrm{e}}$ and $D_{\mathrm{e}}$,
fp(L)CC yields values of $\omega_{\mathrm{e}}$ that are closer to MRCI
reference data. Specifically, both types of corrections overestimate
$r_{\mathrm{e}}$ by around 0.02 (0.01) Å compared to experimental (MRCI)
reference data, while the differences in $D_{\mathrm{e}}$ amount to
approximately 10 kcal mol-1. Furthermore, the proposed pCCD-CI methods
increase the error in harmonic vibrational frequencies $\omega_{\mathrm{e}}$
by about 10 cm-1 with respect to fp(L)CC. Including a RD correction on top of
the pCCD-CI models has only a minor effect on spectroscopic constants. While
$r_{\mathrm{e}}$ remains unchanged, $\omega_{\mathrm{e}}$ and $D_{\mathrm{e}}$
only marginally worsen by 3 cm-1 and 0.4 kcal mol-1. Finally, all pCCD
corrections outperform the conventional CCD or CCSD models, which provide
large errors in $r_{\mathrm{e}}$ (about 0.03 Å), $\omega_{\mathrm{e}}$ (126
cm-1), and $D_{\mathrm{e}}$ (about 36 kcal mol-1).

To sum up, the quality and performance of the investigated CI corrections on
top of a pCCD reference function is comparable to fp(L)CC methods featuring a
similar excitation order. Nonetheless, for problems which feature a
substantial amount of static electron correlation effects (like the
dissociation process of N2), the proposed pCCD-CI models seem more robust and
less dependent on the quality of the pCCD reference. Thus, pCCD-CI might be
less prone to unexpected failures encountered in CC corrections like
divergencies.

##  V Assessing the accuracy of CI corrections in actinide-containing
compounds

In the following, we extend our test set of light diatomic molecules with
small di- and triatomic actinide compounds. Specifically, we investigate the
simplest actinide compounds, ThO Marian  _et al._ (1988); Paulovic  _et al._
(2002); Meyer and Bohn (2008); Fleig and Nayak (2014); Skripnikov and Titov
(2015); Skripnikov (2016); Denis and Fleig (2016); Tecmer and González-
Espinoza (2018) and ThS Liang and Andrews (2002); Tecmer and González-Espinoza
(2018), which are important systems in studies on the electron electric dipole
moment (eEDM) Heaven, Barker, and Antonov (2014); Wang  _et al._ (2011); Baron
_et al._ (2014); Le, Heaven, and Steimie (2014); Tecmer and González-Espinoza
(2018). Furthermore, we scrutinize the performance of the CI corrections for
the the UO2^2+ isoelectronic series Wadt (1981); Dyall (1999); de Jong,
Visscher, and Nieuwpoort (1999); Zhang and Pitzer (1999); Kaltsoyannis (2000);
Matsika  _et al._ (2001); Réal  _et al._ (2007, 2009); Tecmer  _et al._
(2011); Wei  _et al._ (2011); Tecmer  _et al._ (2012a, b, 2013, 2014b); Gomes
_et al._ (2015); Gomes, Jacob, and Visscher (2008); Tecmer, Hong, and
Boguslawski (2016), containing UO2^2+, UN2, PaO2^+ and ThO_2 Denning (2007);
Baker (2012); Clavaguéra-Sarrio  _et al._ (2004); Pierloot and van Besien
(2005); Réal  _et al._ (2009); Hunt, Yustein, and Andrews (1993); A. Kovács,
R.J.M. Konings (2011); Tecmer  _et al._ (2012a); Kovács  _et al._ (2015);
Tecmer, Hong, and Boguslawski (2016). Finally, we augment our numerical
analysis to target also the lowest-lying excited states in the UO2^2+ cation.

Table 3: Spectroscopic parameters of the ground states of the investigated actinide-containing compounds obtained from various post-pCCD methods and conventional CC methods. $r_{\mathrm{e}}$ is the equilibrium bond length, $\omega_{\mathrm{e}}$ the harmonic vibrational frequency, respectively. The differences with respect to CCSD(T) are given in parentheses. The total electronic energies, as also PES curves are summarized in the ESI†. RDC: renormalized Davidson correction. Method |  $r_{\mathrm{e}}$ [Å] |  $\omega_{\mathrm{e}}$ [cm-1] |  $r_{\mathrm{e}}$ [Å] |  $\omega_{\mathrm{e}}$ [cm-1] |  $r_{\mathrm{e}}$ [Å] |  $\omega_{\mathrm{e}}$ [cm-1]
---|---|---|---|---|---|---
| ThO | ThS | $\ce{UO_{2}}^{2+}$
oo-pCCD |  1.854 ($-0.012$) |  894 ($+56$) |  2.387 ($+0.011$) |  484 ($+12$) |  1.670 ($-0.038$) |  1139 ($+72$)
pCCD-CID′ |  1.850 ($-0.016$) |  898 ($+60$) |  2.365 ($-0.011$) |  494 ($+22$) |  1.681 ($-0.027$) |  1164 ($+97$)
pCCD-CID′+RDC |  1.853 ($-0.013$) |  890 ($+52$) |  2.366 ($-0.010$) |  492 ($+20$) |  1.687 ($-0.021$) |  1154 ($+87$)
pCCD-CID |  1.854 ($-0.012$) |  888 ($+50$) |  2.366 ($-0.010$) |  494 ($+22$) |  1.687 ($-0.021$) |  1129 ($+62$)
pCCD-CID+RDC |  1.855 ($-0.011$) |  886 ($+48$) |  2.365 ($-0.011$) |  493 ($+21$) |  1.690 ($-0.018$) |  1114 ($+47$)
fpLCCD |  1.858 ($-0.008$) |  876 ($+38$) |  2.367 ($-0.009$) |  488 ($+16$) |  1.701 ($-0.007$) |  1067 ($\pm 0$)
fpCCD |  1.856 ($-0.010$) |  869 ($+31$) |  2.365 ($-0.011$) |  491 ($+19$) |  1.698 ($-0.010$) |  1087 ($+20$)
CCD |  1.851 ($-0.015$) |  895 ($+57$) |  2.362 ($-0.014$) |  492 ($+20$) |  1.683 ($-0.025$) |  1174 ($+107$)
pCCD-CISD′ |  1.852 ($-0.014$) |  894 ($+56$) |  2.368 ($-0.008$) |  494 ($+22$) |  1.683 ($-0.025$) |  1158 ($+91$)
pCCD-CISD′+RDC |  1.856 ($-0.010$) |  889 ($+51$) |  2.370 ($-0.006$) |  487 ($+15$) |  1.689 ($-0.019$) |  1129 ($+62$)
pCCD-CISD |  1.855 ($-0.011$) |  889 ($+51$) |  2.367 ($-0.009$) |  492 ($+20$) |  1.689 ($-0.019$) |  1125 ($+58$)
pCCD-CISD+RDC |  1.857 ($-0.009$) |  871 ($+33$) |  2.369 ($-0.007$) |  487 ($+15$) |  1.694 ($-0.014$) |  1102 ($+35$)
fpLCCSD |  1.863 ($-0.003$) |  840 ($+2$) |  2.384 ($+0.008$) |  457 ($-15$) |  1.717 ($+0.009$) |  976 ($-91$)
fpCCSD |  1.857 ($-0.009$) |  874 ($+36$) |  2.367 ($-0.009$) |  489 ($+17$) |  1.698 ($-0.010$) |  1115 ($+48$)
CCSD |  1.855 ($-0.011$) |  885 ($+47$) |  2.366 ($-0.010$) |  487 ($+15$) |  1.690 ($-0.018$) |  1142 ($+75$)
CCSD(T) | 1.866 | 838 | 2.376 |  472 | 1.708 |  1067
| $\ce{ThO_{2}}$ | $\ce{PaO_{2}^{+}}$ | $\ce{UN_{2}}$
oo-pCCD |  1.917 ($-0.012$) |  805 ($-13$) |  1.763 ($-0.021$) |  972 ($\pm 0$) |  1.711 ($-0.030$) |  1176 ($+74$)
pCCD-CID′ |  1.913 ($-0.016$) |  858 ($+40$) |  1.763 ($-0.021$) |  1024 ($+52$) |  1.717 ($-0.024$) |  1171 ($+69$)
pCCD-CID′+RDC |  1.917 ($-0.012$) |  841 ($+23$) |  1.767 ($-0.017$) |  1011 ($+39$) |  1.722 ($-0.019$) |  1164 ($+62$)
pCCD-CID |  1.917 ($-0.012$) |  852 ($+34$) |  1.768 ($-0.016$) |  1000 ($+28$) |  1.720 ($-0.021$) |  1124 ($+22$)
pCCD-CID+RDC |  1.918 ($-0.011$) |  850 ($+32$) |  1.771 ($-0.013$) |  987 ($+15$) |  1.725 ($-0.016$) |  1122 ($+20$)
fpLCCD |  1.925 ($-0.004$) |  829 ($+11$) |  1.778 ($-0.006$) |  995 ($+23$) |  1.736 ($-0.005$) |  1077 ($-25$)
fpCCD |  1.924 ($-0.005$) |  830 ($+12$) |  1.776 ($-0.008$) |  985 ($+13$) |  1.731 ($-0.011$) |  1102 ($\pm 0$)
CCD |  1.914 ($-0.015$) |  850 ($+32$) |  1.765 ($-0.019$) |  1027 ($+55$) |  1.722 ($-0.019$) |  1184 ($+82$)
pCCD-CISD′ |  1.914 ($-0.015$) |  851 ($+33$) |  1.765 ($-0.019$) |  1014 ($+42$) |  1.719 ($-0.022$) |  1162 ($+40$)
pCCD-CISD′+RDC |  1.918 ($-0.011$) |  842 ($+24$) |  1.770 ($-0.014$) |  1004 ($+32$) |  1.725 ($-0.016$) |  1152 ($+50$)
pCCD-CISD |  1.918 ($-0.011$) |  837 ($+19$) |  1.770 ($-0.014$) |  999 ($+27$) |  1.722 ($-0.019$) |  1118 ($+16$)
pCCD-CISD+RDC |  1.920 ($-0.009$) |  836 ($+18$) |  1.774 ($-0.010$) |  982 ($+10$) |  1.727 ($-0.014$) |  1106 ($+4$)
fpLCCSD |  1.934 ($+0.005$) |  773 ($-45$) |  1.789 ($+0.005$) |  895 ($-77$) |  1.748 ($+0.007$) |  960 ($-142$)
fpCCSD |  1.923 ($-0.006$) |  829 ($+11$) |  1.776 ($-0.008$) |  989 ($+17$) |  1.733 ($-0.008$) |  1113 ($+11$)
CCSD |  1.918 ($-0.011$) |  804 ($-14$) |  1.770 ($-0.014$) |  965 ($-75$) |  1.727 ($-0.014$) |  1158 ($+56$)
CCSD(T) | 1.929 | 818 | 1.784 | 972 | 1.741 | 1102

###  V.1 Targeting the ground states of actinide compounds

The spectroscopic constants for the electronic ground states of the
investigated actinide compounds are summarized in Table 3 and are obtained
from the corresponding PESs, which are collected in the SI. For ThO, the
inclusion of single excitations on top of doubles in the correction
significantly improves the equilibrium distance $r_{\mathrm{e}}$. This is a
common feature for all investigated pCCD corrections, where fpLCCSD yields the
most accurate values, which differ only about 0.003 Å from the CCSD(T)
reference results. fpCCSD and pCCD-CISD+RDC (that is, including a RD
correction) perform similarly, both underestimating $r_{\mathrm{e}}$ by about
0.009 Å. Similar observation can be made for the harmonic vibrational
frequencies. Specifically, the LCCSD correction approaches spectroscopic
accuracy (deviating by 2 cm-1 from the reference), while fpCCSD and pCCD-
CISD+RDC feature larger differences of about 36 and 33 cm-1, respectively.

In case of the ThS molecule, all investigated pCCD corrections yield
spectroscopic constants of similar accuracy. While all CI corrections and the
conventional CCSD model underestimate $r_{\mathrm{e}}$, the LCCSD correction
overestimates equilibrium bond lengths. In absolute value, all corrections
provide similar error measures amounting from 0.007 to 0.011 Å with respect to
CCSD(T) reference values. The fitted values for $\omega_{\mathrm{e}}$ are very
consistent, where all corrections result in an error of approximately 20 cm-1
with respect to the CCSD(T) harmonic vibrational frequency. As observed above,
adding a RD correction generally improves the performance of all pCCD-CI-type
approaches. However, we should note that the pCCD approximation already
delivers a well description of spectroscopic constants for both the ThO and
ThS molecules. All applied corrections do not significantly improve the pCCD
results.

Quantum chemistry studies on the UO2^2+ isoelectronic series are still an
active field of research. These small triatomic molecules have a linear
geometry beyond the ThO_2 species, which features a bent geometry and a
peculiar electronic structure. In contrast to its linear analogues, the
Thorium 5f orbitals of ThO_2 participate neither in bonding nor in electronic
excitations. This is caused by differences in the relative energetic ordering
of the 5f and 6d orbitals in the thorium and uranium atoms, respectively. By
scrutinizing the spectroscopic constants summarized in Table 3, we are able to
assess whether the investigated post-pCCD methods can reliably model electron
correlation effects in these compounds along the PESs. Starting from the
uranyl cation, pCCD considerably underestimates the equilibrium distance
$r_{\mathrm{e}}$. All CI corrections significantly improve this value
decreasing the errors by a factor of 2. Specifically, pCCD-CISD+RDC
underestimate $r_{\mathrm{e}}$ by $-0.014$ Å. A better performance can be
observed for fp(L)CC methods, which differ by only $0.007-0.009$ Å in absolute
value. Predicting accurate $\omega_{\mathrm{e}}$ is more challenging. In order
to reduce the pCCD errors by a factor of 2, a CISD+RDC correction is required,
while the remaining CI flavours feature large differences with respect to
CCSD(T) reference values (more than 60 cm-1). The performance of the fp(L)CC
methods is more erratic. While fpLCCD predicts a $\omega_{\mathrm{e}}$ that
perfectly agrees with the CCSD(T) reference, inclusion of single excitations
leads to large errors (up to 90 cm-1). Moving to a fpCCSD formalism improves
spectroscopic constants, where the errors in $\omega_{\mathrm{e}}$ amount to
48 cm-1. We should highlight that the conventional CCSD method provides larger
errors in spectroscopic constants than the alternatives fp(L)CCSD or pCCD-
CISD+RDC.

Analogous to the examples above, the CI corrections significantly improves
pCCD results for the UN2 molecule, where pCCD-CISD+RDC provides the smallest
errors with respect to CCSD(T) reference data. Furthermore, we observe the
same tendency for all investigated corrections to underestimate the
equilibrium bond length and to overestimate the harmonic vibrational
frequency, except for fpLCCSD, which shows the opposite behaviour.
Specifically, the (L)CCSD corrections provide more accurate equilibrium bond
lengths, while the CISD flavours reduce the errors in $\omega_{\mathrm{e}}$ to
4 cm-1. Most importantly, fpCCSD and pCCD-CISD+RDC outperform the conventional
CCSD approach and predict spectroscopic constants that deviate the least from
CCSD(T) reference values.

For the protactinium dioxygen cation, we observe similar trends, where the
CI/CC corrections are able to cure some of the deficiencies originated from
the pCCD model. Specifically, the errors in $r_{\mathrm{e}}$ are reduced by a
factor of 2 in CI-type corrections (pCCD-CISD+RDC features an error of 0.01 Å)
and further lowered to 0.005 Å in the LCC corrections. However, all
investigated pCCD corrections deteriorate harmonic vibrational frequencies,
where pCCD already provides the proper shape of the PESs around the
equilibrium distance. While pCCD-CISD+RDC yields the smallest errors of 10
cm-1, fpLCCSD increases the discrepancy to 77 cm-1. Nonetheless, both CID/CISD
and (L)CCD/(L)CCSD corrections again outperform the conventional CCD/CCSD
approaches, which feature one of the largest errors in spectroscopic
constants.

A particular difficult case is the ThO2 molecule, where the CI corrections do
not significantly change or improve the spectroscopic constants derived from
the pCCD PES. The best performance is obtained from the fp(L)CC methods, where
the errors in $r_{\mathrm{e}}$ are reduced to 0.005 Å, while the absolute
errors in $\omega_{\mathrm{e}}$ remain unchanged (except for fpLCCSD, which
provides the largest errors). The most accurate CI flavor is again pCCD-
CISD+RDC, which underestimates $r_{\mathrm{e}}$ by around 0.009 Å, while
$\omega_{\mathrm{e}}$ deviates by 18 cm-1. The performance of the CI
corrections lies between fp(L)CC and conventional CC theory.

###  V.2 Dissecting the accuracy of pCCD corrections based on an error
analysis.

To scrutinize the accuracy of the investigated corrections in predicting
accurate values for $r_{\mathrm{e}}$ and $\omega_{\mathrm{e}}$, we perform a
statistical analysis. Specifically, we evaluate the mean error (ME) of the
fitted spectroscopic constants with respect to CCSD(T) reference data,

$\mathrm{ME}=\sum_{i}^{N}\Delta x_{i}/N,$ (13)

where $\Delta x_{i}$ is the difference in $r_{\mathrm{e}}$ or
$\omega_{\mathrm{e}}$ between a selected method and the reference $\Delta
x_{i}=x_{i}^{\rm method}-x_{i}^{\rm CCSD(T)}$. The sum runs over the all
investigated actinide-containing compounds ($N=6$). Furthermore, we also
determine the root-mean-square deviation (RMSD) in $r_{\mathrm{e}}$ and
$\omega_{\mathrm{e}}$, according to

$\mathrm{RMSD}=\sqrt{\sum_{i}^{N}\Delta x_{i}^{2}/N}.$ (14)

The graphical representation of the ME and RMSD of $r_{\mathrm{e}}$ is shown
in Figure 3 for CI and CC methods including (a) only double and (b) single and
double excitations. On average, all investigated methods underestimates the
equilibrium bond distance. The smallest deviations from CCSD(T) data are
obtained from fpLCCD/fpLCCSD, followed by fpCCD/fpCCSD and then by pCCD-
CID+RDC/pCCD-CISD+RDC. The largest discrepancies are deduced from the pCCD-
CID′/pCCD-CISD′ PESs. These large errors might be related to the double-
counting problem of electron correlation effects associated with the electron
pairs, which are included in both the pCCD reference function and the a
posteriori CI correction. Excluding pair excitations in the CI operator
significantly reduces errors. Furthermore, including single excitations in the
pCCD correction only slightly affects equilibrium bond lengths, where the ME
and RMSE are reduced by 0.002 Å in CI corrections and by 0.001 Å in the (L)CC
flavors. However, the performance of fpLCC methods is more erratic; while
fpCCD underestimates $r_{\mathrm{e}}$, fpCCSD overestimates equilibrium
distances. Finally, we should note that all pCCD corrections (excluding the
pair sector and including a RD correction) provide smaller ME and RMSE
compared to the conventional CC methods featuring similar excitation
operators.

Figure 4 highlights the ME and RMSD for $\omega_{\mathrm{e}}$ for all CI and
CC methods restricted to (a) double excitations and (b) single and double
excitations. A similar behaviour in ME and RMSD can be be observed as found
for $r_{\mathrm{e}}$. Considering only double excitations, fpLCCD yields the
smallest ME of 10 cm-1, followed by fpCCD with a ME up to 20 cm-1, while pCCD-
CID+RDC features an increased ME of about 30 cm-1. All investigated approaches
tend to overestimate the harmonic vibrational frequency. Note that the largest
errors are obtained for the conventional CCD model. Including both single and
double excitations significantly changes the distribution of ME and RMSE. The
smallest ME is obtained for pCCD-CISD+RDC (19 cm-1), followed by fpCCSD with a
ME of about 23 cm-1. Furthermore, the inclusion of single excitations in the
correction significantly increases the accuracy of all CI-type corrections,
while the ME of fpCC methods grows by about 70%. Similarly, the performance of
fpLCC considerably deteriorates raising ME and RMSE by a factor of 6 and 4,
respectively. To summarize, fpLCCSD provides the smallest errors in
$r_{\mathrm{e}}$, while pCCD-CISD+RDC properly describes the shape of the PESs
yielding the most accurate harmonic vibrational frequencies.

Figure 3: Mean errors in $r_{\mathrm{e}}$ obtained for various post-pCCD and conventional CC methods restricted to (a) double excitations and (b) single and double excitations with respect to CCSD(T) reference data calculated from eq. 13 for all investigate actinide-containing compounds. The standard deviation (RMSE) is indicates as black lines. The corresponding numerical values are summarized in Table S19 in the SI. RDC: renormalized Davidson correction.  Figure 4: Mean errors in $\omega_{\mathrm{e}}$ obtained for various post-pCCD and conventional CC methods restricted to (a) double excitations and (b) single and double excitations with respect to CCSD(T) reference data calculated from eq. 13 for all investigate actinide-containing compounds. The standard deviation (RMSE) is indicates as black lines. The corresponding numerical values are summarized in Table S20 in the SI. RDC: renormalized Davidson correction.  Table 4: Spectroscopic constants of some low-lying singlet excited states of UO2^2+ obtained from various post-pCCD methods and conventional EOM-CC theory. $r_{\mathrm{e}}$ is the equilibrium bond length, $\omega_{\mathrm{e}}$ the harmonic vibrational frequency, $T_{\mathrm{e}}$ the adiabatic excitation energy, and $T_{\mathrm{v}}$ the vertical excitation energy, respectively. $T_{\mathrm{v}}$ has been calculated with respect to the equilibrium bond distance $r_{\mathrm{e}}$ of the corresponding ground state obtained for each method. The differences with respect to CR-EOM-CCSD(T) are given in parentheses. The superscripts ′ indicates that the pCCD-CISD method includes the electron pair sector in the excitation operator. EOM-fpLCCSD was originally introduces as EOM-pCCD-LCCSD Boguslawski (2019).  |  UO_2^2+
---|---
Method |  $r_{\mathrm{e}}$ [Å] |  $\omega_{\mathrm{e}}$ [cm-1] |  $T_{\mathrm{e}}$ [eV] |  $T_{\mathrm{v}}$[eV]
| $1^{1}\Phi_{g}(\sigma_{u}\rightarrow\phi_{u})$
pCCD-CISD′ |  1.773($+$0.018) |  888($-$49) |  14.50($+$10.74) |  14.92($+$10.93)
pCCD-CISD |  1.773($+$0.018) |  888($-$49) |  14.50($+$10.74) |  14.92($+$10.93)
EOM-fpLCCSD |  1.786($+$0.031) |  837($-$100) |  3.66($-$0.10) |  3.87($-$0.12)
EOM-CCSD |  1.762($+$0.007) |  924($-$13) |  3.57($-$0.19) |  3.86($-$0.13)
CR-EOM-CCSD | 1.755 | 937 | 3.76 | 3.99
| $1^{1}\Delta_{g}(\sigma_{u}\rightarrow\delta_{u})$
pCCD-CISD′ |  1.791($+$0.039) |  764($-169$) |  14.72($+$10.56) |  15.17($+$10.80)
pCCD-CISD |  1.791($+$0.039) |  764($-169$) |  14.72($+$10.56) |  15.17($+$10.80)
EOM-fpLCCSD |  1.781($+$0.029) |  840($-$93) |  4.07($-$0.09) |  4.25($-$0.12)
EOM-CCSD |  1.760($+$0.008) |  910($-$23) |  3.96($-$0.20) |  4.22($-$0.15)
CR-EOM-CCSD | 1.752 | 933 | 4.16 | 4.37
| $2^{1}\Phi_{g}(\pi_{u}\rightarrow\delta_{u})$
pCCD-CISD′ |  1.803($+$0.013) |  928($+22$) |  14.75($+$9.85) |  15.34($+$9.89)
pCCD-CISD |  1.801($+$0.012) |  975($+69$) |  14.74($+$9.84) |  15.34($+$9.89)
EOM-fpLCCSD |  1.804($+$0.014) |  941($+$35) |  4.87($-$0.03) |  5.29($-$0.16)
EOM-CCSD |  1.797($+$0.007) |  902($-$4) |  4.46($-$0.44) |  5.08($-$0.37)
CR-EOM-CCSD(T) | 1.790 | 906 | 4.90 | 5.45
| $1^{1}\Pi_{g}(\pi_{u}\rightarrow\delta_{u})$
pCCD-CISD′ |  1.784($-$0.006) |  1023($+118$) |  14.83($+$9.84) |  15.53($+$10.00)
pCCD-CISD |  1.785($-$0.005) |  1042($+137$) |  14.82($+$9.83) |  15.52($+$9.99)
EOM-fpLCCSD |  1.804($+$0.014) |  965($+$60) |  4.98($-$0.01) |  5.38($-$0.15)
EOM-CCSD |  1.796($-$0.006) |  904($-$1) |  4.55($-$0.44) |  5.16($-$0.37)
CR-EOM-CCSD(T) | 1.790 | 905 | 4.99 | 5.53

###  V.3 Performance of pCCD-CI methods for excited states.

Finally, we will turn our discussion toward electronically excited states and
scrutinize the performance of the investigated pCCD-CI methods to model the
lowest-lying electronic states in the UO_2^2+ molecule. Specifically, we
target the following electronically excited states:
$\sigma_{u}\rightarrow\delta_{u}$, $\sigma_{u}\rightarrow\phi_{u}$ and two
$\pi_{u}\rightarrow\delta_{u}$ transitions. Or analysis is based on our
previous study Nowak, Tecmer, and Boguslawski (2019), where we focused on
dissecting the accuracy of the LCC corrections in targeting electronically
excited states in selected actinide compounds. As a reference method, we chose
the completely reonormalized equation of motion coupled cluster singles
doubles and perturbative triples (CR-EOM-CCSD(T)) method. Kowalski and Piecuch
(2004)

Table 4 contains the fitted spectroscopic constants for the four lowest-lying
targeted excited states. pCCD-CISD with and without electron pair excitations
yield equivalent results (for $r_{\mathrm{e}}$ and excitation energies) due to
the purely singly-excited character of the targeted states. However, the
excitation energies predicted by the CI corrections strongly deviate from all
EOM-CC results. While the ground state energies are corrected towards lower
energies (by accounting for the missing dynamical correlation energy), all
electronically excited states miss this shift in energy. Although pCCD-CISD
provides reasonable excitation energies with respect to the pCCD reference
energy, it misses a large fraction of the dynamical correlation energy of the
excited states. The adiabatic and vertical excitation energies collected in
Table 4 demonstrate that this shift in energy between pCCD-CISD and the
CCSD/CR-EOM-CCSD(T) reference amounts to over 10 eV. This disproportion in
excitation energies is significant, especially compared to EOM-fpLCCSD, which
differs only around from 0.1 to 0.5 eV from reference values. Although these
quantitative deficiencies of pCCD-CISD methods in capturing electron
correlation effects in excited states forbid us to precisely target excitation
energies, the CI corrections predict equilibrium bond lengths with reasonable
accuracy providing smaller differences from CR-EOM-CCSD(T) reference data than
EOM-fpLCCSD. The prediction of harmonic vibrational frequencies is, however,
more erratic, where errors lie between 12 and 140 cm-1. Specifically, for the
lowest excited state $1^{1}\Phi_{g}$, pCCD-CISD predicts spectroscopic
constants that are closer to reference values than the corresponding EOM-
fpLCCSD results. Thus, pCCD-CISD generally fails in describing electronically
excited states and provides spectroscopic constants of irregular accuracy.
Similar disadvantages of CI corrections in predicting excitation energies were
noticed in previous numerical studies Kállay and Surján (1999).

##  VI Conclusions

In this work, we developed various CI corrections with a pCCD references
function that are restricted to at most double (hole-particle) excitations.
Furthermore, we tested CI models, where the linear CI operator excludes and
includes electron pair excitations, where the former case might be important
to properly describe doubly-excited states. The accuracy of the proposed pCCD-
CI approaches was assessed for the ground states of light diatomic molecules,
namely F_2 and N_2, as well as for selected di- and tri-atomic f0 actinide
species, such as the ThO, ThS, UO^2+_2, NUN, PaO^+_2, and ThO_2 molecules. We
also scrutinized the performance of the CI corrections on the lowest-lying
electronically excited states in the UO^2+_2 cation. The CI corrections were
compared to fp(L)CC methods as well as to conventional CC variants. Finally,
we introduced a simple renormalized Davidson correction to minimize size-
consistency errors in the ground state electronic structures.

Our study demonstrates that the proposed pCCD-CI methods provide a reliable
and promising alternative to conventional CC methods as well as to the other
unconventional approaches such as the fp(L)CC model. A major advantage of the
CI-type corrections is their ability to dissociate the N2 molecule, where the
contributions to electron correlation effects change along the reaction
coordinate. The accuracy of the pCCD-CI models can be improved by including an
(approximate) renormalized Davidson correction, where the errors in
spectroscopic constants are reduced by a factor of 2. In general, pCCD-
CISD+RDC allows us to obtain smooth PESs and predicts spectroscopic constants
that agree well with experimental or MRCI reference data. Specifically, pCCD-
CISD+RDC outperforms conventional CCSD, while the spectroscopic constants lie
between fpLCCSD and fpCCSD accuracy. The good performance of pCCD-CISD+RDC is
also observed for heavy-element-containing systems, where it features the
smallest ME and RMSE in $\omega_{\mathrm{e}}$, while the corresponding error
measures for $r_{\mathrm{e}}$ are slightly smaller than the CCSD values. The
inclusion of electron-pair excitations in the CI operator, however,
deteriorates the performance of the various CI corrections, which might
originate from a double-counting problem of correlation effects associated
with electron pair states as they are included in both the pCCD reference
function and the a posteriori CI correction.

Finally, the pCCD-CISD model fails for electronically excited states, where
large errors in excitation energies are observed. Although pCCD-CISD provides
more accurate values for $r_{\mathrm{e}}$ in electronic excited states than
the EOM-fpLCCSD model, the prediction of $\omega_{\mathrm{e}}$ and (adiabatic
and vertical) excitation energies is rather erratic. To sum up, a CI
correction (including a Davidson correction) on top of pCCD is a promising
alternative to conventional (like CCSD) or unconventional (like fp(L)CC) CC
methods to model electronic ground states. However, in order to reach chemical
or spectroscopic accuracy, further improvements are required, like approximate
protocols to account for higher-order excitations. Furthermore, in this work,
the CI operators are restricted to singlet excitations, while the projection
manifold has been defined to allow for an algebraic spin-summation. By
eliminating this constraint, the CI corrections would allow us to target
different spin sectors. If we restrict the excitation operators to be of SD-
type, both singlet and triplet states will be accessible in the CI
corrections. The corresponding diagonalization problem has to be performed in
the Slater determinant basis containing all singly- and doubly-excited
determinants with respect to $\ket{\Phi_{0}}$ instead of the spin-summed
states defined in eqs. (10) and (11). This formulation, however, increases the
number of degrees of freedom of the diagonalization problem.

##  VII Data Availability Statement

The data that supports the findings of this study are available within the
article and its supplementary material.

###### Acknowledgements.

A.N. and K.B. acknowledge financial support from a SONATA BIS grant of the
National Science Centre, Poland (no. 2015/18/E/ST4/00584). A.N. also received
financial support from a PRELUDIUM 17 grant of the National Science Centre,
Poland (no. 2019/33/N/ST4/01880).

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