# Assessing the accuracy of tailored coupled cluster methods corrected by
electronic wave functions of polynomial cost.

Aleksandra Leszczyk  Institute of Physics, Nicolaus Copernicus University in
Toruń, Torun, Poland  Mihály Máté  Strongly Correlated Systems "Lendület"
Research Group, Wigner Research Center for Physics, H-1525 Budapest, Hungary
Örs Legeza  Strongly Correlated Systems "Lendület" Research Group, Wigner
Research Center for Physics, H-1525 Budapest, Hungary  Katharina Boguslawski
<EMAIL_ADDRESS>Institute of Physics, Nicolaus Copernicus
University in Toruń, Torun, Poland

###### Abstract

Tailored coupled cluster theory represents a computationally inexpensive way
to describe static and dynamical electron correlation effects. In this work,
we scrutinize the performance of various tailored coupled cluster methods
externally corrected by electronic wave functions of polynomial cost.
Specifically, we focus on frozen-pair coupled-cluster (fpCC) methods, which
are externally corrected by pair-coupled cluster doubles (pCCD), and coupled
cluster theory tailored by matrix product state wave functions optimized by
the density matrix renormalization group (DMRG) algorithm. As test system, we
selected a set of various small- and medium-sized molecules containing
diatomics (N2, F2, C2, CN+, BN, BO+, and Cr2) and molecules (ammonia,
ethylene, cyclobutadiene, benzene) for which conventional single-reference
coupled cluster singles and doubles (CCSD) is not able to produce accurate
results for spectroscopic constants, potential energy surfaces, and barrier
heights. Most importantly, DMRG-tailored and pCCD-tailored approaches yield
similar errors in spectroscopic constants and potential energy surfaces
compared to multireference and/or experimental reference data and generally
outrank the conventional single-reference CCSD approach. Although fpCC methods
provide a reliable description for the dissociation pathway of molecules
featuring single and quadruple bonds, they fail in the description of triple
or hextuple bond-breaking processes or avoided crossing regions.

###### keywords:

Coupled Cluster, DMRG, pCCD, tailored Coupled Cluster

DMRG,pCCD,tCC

##  1 Introduction

Conventional single-reference coupled cluster theory provides a robust and
systematically improvable treatment of electron correlation effects.1, 2, 3,
4, 5, 6, 7, 8 The wave function ansatz features an exponential cluster
operator that ensures size-extensivity, while the straightforward truncation
scheme of the cluster operator based on the excitation level yields an ordered
hierarchy of approximations that converges toward the full configuration
interaction limit. The coupled cluster model in its standard single-reference
formulation is one of the most accurate tools in describing dynamical electron
correlation but it fails when the electronic systems under study has
multireference character. In such cases, the hierarchy of approximations
breaks down and the truncation of the cluster operator provides incorrect
approximations to the exact electronic wave function with unphysical coupling
between cluster amplitudes.9, 10, 11

One possible remedy dedicated to capture strong electron correlation effects
are externally corrected or tailored coupled cluster methods12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22. In this methodology, a subset of cluster
amplitudes is extracted from an external model that guarantees the proper
description of the multireference nature of the molecular system under study.
Popular wave function approaches to except some external coupled cluster
amplitudes include the multireference configuration interaction (MRCI) or
complete active space self-consistent field (CASSCF) methods. However, these
approaches are computationally infeasible for large molecules and force us to
resign from black-box computational setups. As an alternative to conventional
electronic structure methods, the density matrix renormalization group (DMRG)
algorithm23, 24, 25, 26, 27, 28, 29 and various geminal-based approaches30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49,
50, 51, 52, 53 offer a computationally less complex way to model strongly-
correlated electrons.

The DMRG algorithm represents a computationally efficient method to optimize
matrix product state (MPS) wave functions. The evaluation of the electronic
energy scales only polynomially with system size. Therefore, DMRG allows us to
efficiently handle large active orbital spaces. Due to its favorable
computational scaling, the DMRG algorithm found numerous applications to
investigate strongly-correlated systems including transition metal-54, 55, 56,
57, 58, 59, 60, 61 or actinide-containing molecules. 62, 63, 64 Although being
computationally more efficient, DMRG still requires us to select active space
orbitals. This can be done efficiently exploiting, for instance, tools based
on quantum information theory65, 66, 67, 68, 69, 70, 71, 60, 61, 72, 73, 74,
75, 76, 77, 78, 63, 79, 80, fractional occupancies of unrestricted natural
orbitals81, 82, or high-spin-state unrestricted Hartre–Fock (UHF) natural
orbitals. 83, 84

In this work, we benchmark various coupled cluster singles and doubles models
tailored by unconventional electronic structure approaches. Specifically, we
focus on orbital-optimized pair coupled cluster doubles (pCCD) 39, 44 and DMRG
wave functions. The optimization of these wave functions scales only
polynomially with system size. Hence, these approaches provide an efficient
way to capture strong electron correlation effects (within an active orbital
space in the case of DMRG). Most importantly, both methods allow for an
accurate description of static/nondynamic electron correlation effects 68, 73
and thus represent a promising choice for reference coupled cluster amplitudes
in externally corrected coupled cluster methods. Furthermore, the quality of
DMRG calculations is affected by the type of molecular orbitals used in the
active orbital space. Recent studies report that local pair natural orbitals
and domain-based local pair natural orbitals perform better than canonical RHF
orbitals in DMRG-tCCSD.20, 85, 86, 87 Here, we benchmark another type of
orbitals of localized nature, namely pCCD-optimized orbitals as they have not
yet been combined with coupled cluster theory tailored by DMRG wave functions.

Furthermore, the optimal active orbital space used in DMRG calculations that
are then employed in the tailored coupled cluster flavour can be chosen
according to the selection protocol presented by some of us 21. This active
space selection protocol provides accurate values for the correlation energy.
In this work, however, we focus on a different approach. We aim at
constructing a minimal but optimal active orbital space that captures the
dominant part of static/nondynamic electron correlation using a one-step
procedure exploiting tools based on quantum information theory. Such an
active-space selection scheme will be beneficial for large-scale modelling or
the accurate and efficient prediction of potential energy surfaces as it
reduces the number of DMRG calculations to be performed on the daily basis.
Besides, active orbital spaces can change along the reaction pathway. Thus,
ensuring the same composition of active orbitals comprised in large active
space calculation might be difficult to achieve along the reaction coordinate.

This work is organized as follows. In Section 2, we briefly review the main
ideas of tailored-coupled-cluster methods, followed by the pCCD- and DMRG-
tailored flavours. Computational details are presented in Section 3. We
discuss the results and performance of these methods in Section 4. Finally, we
conclude in section 5.

##  2 Tailored coupled cluster theory

The core of coupled cluster theory is the exponential parametrization of the
wave function. Tailored coupled cluster approaches take advantage of this
ansatz as the exponential form allows us to utilize the Baker-Campbell-
Hausdorff (BCH) formula and to factorize any operator of the form
${e}^{\hat{T}}={e}^{\hat{T}_{\mathrm{a}}+\hat{T}_{b}}$ to
$\mathrm{e}^{\hat{T}_{\mathrm{a}}}\mathrm{e}^{\hat{T}_{\mathrm{b}}}$ if and
only if the operators $\hat{T}_{\mathrm{a}}$ and $\hat{T}_{\mathrm{b}}$
commute. Furthermore, the particular partitioning scheme of the cluster
operator (here in $\hat{T}_{\mathrm{a}}$ and $\hat{T}_{\mathrm{b}}$) depends
on the external model for strong correlation. The tailored coupled cluster
wave function is thus expressed as

$\ket{\Phi_{\rm
tCC}}=e^{\hat{T}}\ket{\Phi_{0}}=e^{\hat{T}_{\mathrm{a}}}e^{\hat{T}_{\mathrm{b}}}\ket{\Phi_{0}},$
(1)

where $\ket{\Phi_{0}}$ is a reference Slater determinant and $\hat{T}$ is the
cluster operator that is partitioned into a sum of two cluster operators,
$\hat{T}_{\mathrm{a}}$ and $\hat{T}_{\mathrm{b}}$. Note that we assume that
$\hat{T}_{\mathrm{a}}$ and $\hat{T}_{\mathrm{b}}$ do commute. The cluster
amplitudes of one part of the composite cluster operators $\hat{T}$, here say
$\hat{T}_{\mathrm{b}}$, are then derived from some external calculation that
provides a proper treatment of strong correlation. With the
$\hat{T}_{\mathrm{b}}$ amplitudes being frozen, the remaining cluster
amplitudes of $\hat{T}_{\mathrm{a}}$ can be obtained using optimization
algorithms that are analogous to single-reference coupled cluster methods.
That is, the cluster amplitudes of $\hat{T}_{\mathrm{a}}$ can be obtained
using projection techniques, where the Schrödinger equation for this
particular wave function ansatz reads

$\displaystyle
e^{-\hat{T}_{\mathrm{b}}}e^{-\hat{T}_{\mathrm{a}}}\hat{H}e^{\hat{T}_{\mathrm{a}}}e^{\hat{T}_{\mathrm{b}}}\ket{\Phi_{0}}$
$\displaystyle=E\ket{\Phi_{0}}.$ (2)

In the above equation, the $\hat{T}_{\mathrm{b}}$ amplitudes are kept fixed
during the optimization. Thus, the projection manifold of the tailored coupled
cluster wave function contains only the set of determinants
$\\{\hat{T}_{\mathrm{a}}\ket{\Phi_{0}}\\}$.

###  2.1 Frozen-pair coupled cluster theory

Frozen-pair coupled cluster theory originates from the idea of seniority-based
coupled cluster approaches where the components that differ in the number of
unpaired electrons are treated separately. The most significant part of the
wave function is the seniority-zero sector, which includes only those
amplitudes where the number of unpaired electrons is zero 40. The pCCD wave
function is an example for such an ansatz. That is, the pCCD wave function is
constructed from two-electron functions, also called geminals, using an
exponential cluster operator of the form,

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

where the cluster operator $\hat{T}_{\mathrm{p}}$ contains only electron pair-
excitations (geminals),

$\hat{T}_{\mathrm{p}}=\sum_{i}^{\rm
occ}\sum_{a}^{virt}t_{ii}^{aa}a_{a}^{\dagger}a_{\bar{a}}^{\dagger}a_{\bar{i}}a_{i}$
(4)

with $a_{p}$ ($a_{p}^{\dagger}$) and $a_{\bar{p}}$ ($a_{\bar{p}}^{\dagger}$)
being electron annihilation (creation) operators for $\alpha$\- and
$\beta$-spin electrons, respectively. If combined with an orbital optimization
protocol, the pCCD method is size-consistent and provides a proper qualitative
description of the (exact) electronic wave function for strongly-correlated
electronic systems 37, 39, 40, 42, 43, 44. However, energetics and other
properties cannot be described precisely (for instance, satisfying chemical
accuracy) by restricting the wave function to the seniority-zero sector alone.
To reach quantitative accuracy, we need to go beyond the electron-pair
approximation and to extend the electronic wave function by components that
account for unpaired electrons, so-called broken-pairs 45, 88, 48, 89, 90.

Frozen-pair coupled cluster (fpCC) theory chooses the pCCD wave function as
the fixed reference function 45. That is, the cluster operator
$\hat{T}_{\mathrm{b}}$ in eq. (1) is equivalent to the electron-pair cluster
operator of eq. (4) and the cluster amplitudes are thus divided to pair-
amplitudes ($\hat{T}_{\mathrm{b}}$) and non-pair amplitudes
($\hat{T}_{\mathrm{a}}$). The fpCC ansatz therefore reads

$\ket{\Phi_{\rm fpCC}}=e^{\hat{T}^{\prime}}\ket{\Phi_{\rm
pCCD}}=e^{\hat{T}^{\prime}}e^{\hat{T_{\mathrm{p}}}}\ket{\Phi_{0}},$ (5)

where $\hat{T}^{\prime}$ is a cluster operator that is restricted to contain
electron excitations (singles, broken-pair doubles, etc.) beyond electron-pair
excitations. In the fpCCD method, the cluster operator is defined as
$\hat{T}^{\prime}=\hat{T}_{2}-\hat{T}_{\rm p}$, while for fpCCSD, the cluster
operator includes also single excitations,
$\hat{T}^{\prime}=\hat{T}_{1}+\hat{T}_{2}-\hat{T}_{\rm p}$. The geminal
amplitudes $\\{c_{i}^{a}\\}$ account for strong electron correlation effects,
while the remaining amplitudes introduce broken-pair components to complement
the wave function.

The difficulties in coupled cluster theory arise from the non-linearity of the
amplitude equations that have to be solved to obtain the cluster amplitudes.
This technical obstacle can be bypassed by truncating/neglecting the non-
linear terms in the BCH expansion. Although linearized coupled cluster (LCC)
theory did not gain popularity due to its poor performance, the linearized
version of pCCD-tailored coupled cluster approaches allows us to reach
chemical accuracy for many challenging systems. 48, 90, 91, 52, 92
Specifically, it has been shown that the pCCD-tailored LCC method is an
efficient and reliable alternative to conventional electronic structure
methods for both ground- 48, 90, 91 and excited states 52, 92. The ansatz is
given by

$\ket{\Phi_{\rm fpLCC}}\approx(1+\hat{T}^{\prime})\ket{\Phi_{\rm
pCCD}}=(1+\hat{T}^{\prime})e^{\hat{T_{\mathrm{p}}}}\ket{\Phi_{0}},$ (6)

where $\hat{T}^{\prime}$ is again some cluster operator that is restricted to
contain non-pair electron excitations. The coupled cluster equations are
linear with respect to non-pair amplitudes but the coupling between all pair-
and non-pair amplitudes is included (in addition to all non-linear terms
originating from the pCCD reference function). In this work, we use truncated
coupled cluster models that include either only double excitations (fpLCCD) or
single and double excitations (fpLCCSD).

###  2.2 The DMRG-tailored coupled cluster method

The DMRG algorithm provides qualitatively correct solutions within some active
orbital space, while the wave function components that include external or
inactive orbitals can be included a posteriori using, for example, DMRG-
tailored CC approaches. 20 In these externally corrected CC flavours, the
matrix product state ansatz, which is optimized by the DMRG algorithm, is
translated to a CI-type wave function 93 with some specific reference
determinant. Spin-dependent coupled cluster amplitudes can then be evaluated
from the reconstructed CI coeffcients using standard equations,

$\displaystyle t_{i}^{a}=c_{i}^{a}/c_{0}$ (7) $\displaystyle
t_{ij}^{ab}=c_{ij}^{ab}/c_{0}-(c_{i}^{a}c_{j}^{b}-c_{i}^{b}c_{j}^{a})/c_{0}^{2},$
(8)

where $c_{0}$ is the CI coefficient of the chosen reference determinant and
the indices indicate spin orbitals. The spin-free amplitudes optimized by
solving the spin-summed CC amplitude equations can be deduced form the spin-
dependent ones as $t_{I}^{A}=t_{I_{\alpha}}^{A_{\alpha}}$ and
$t_{IJ}^{AB}=t_{I_{\alpha}J_{\beta}}^{A_{\alpha}B_{\beta}}$, where the spin
degree of freedom is labeled as a subscript and capital letters indicate
spatial orbitals. Within the DMRG-tCCSD formalism, the wave function is
optimized in its split-amplitude form,

$\ket{\Phi_{\rm DMRG-tCCSD}}=e^{\hat{T}_{\rm CAS}}e^{\hat{T}_{\rm
ext}}\ket{\Phi_{0}},$ (9)

where the $\hat{T}_{\rm CAS}$ cluster operator includes amplitudes comprising
excitations within the active space orbitals, while the operator $\hat{T}_{\rm
ext}$ incorporates all excitations beyond the active space 94. The
$\hat{T}_{\rm CAS}$ amplitudes are extracted from DMRG calculations and kept
frozen during the optimization of the remaining amplitudes. That is, the
$\hat{T}_{\rm ext}$ amplitudes are obtained from the solution of the
conventional CCSD equations where the $\hat{T}_{\rm CAS}$ amplitudes are
fixed. Preventing the relaxation of the $\hat{T}_{\rm CAS}$ amplitudes allows
us to capture strong electron correlation effects within the CCSD framework,
while the relaxed $\hat{T}_{\rm ext}$ amplitudes are optimized to supplement
the wave function by the missing dynamical electron correlation effects.

##  3 Computational Details

###  3.1 Electronic structure calculations

All pCCD and (tailored) coupled cluster calculations (using the spin-summed
equations) were performed in a developer version of the PyBEST software
package95, 96. We used the aug-cc-pVDZ and aug-cc-pVTZ basis sets for the F
atom, the cc-pCVDZ basis set for the benzene molecule, and Dunning’s cc-pVDZ
and cc-pVTZ basis sets for all other atoms. For the N2 dimer, we performed
additional calculations for the cc-pVQZ basis set. 97, 98, 99, 100 The depths
of the potential energy well were obtained from a generalized Morse
function101 fit. The vibrational frequencies and equilibrium bond lengths were
calculated numerically from a polynomial fit of sixth order around the
equilibrium distance, where we used $M_{\textrm{B}}=11.0093\,u$ for the B
atom, $M_{\textrm{C}}=12\,u$ for the C atom, $M_{\textrm{N}}=14.0031\,u$ for
the N atom, $M_{\textrm{O}}=15.9949\,u$ for the O atom, and
$M_{\textrm{F}}=18.9984\,u$ for the F atom. The CCSD(T) and CCSDT calculations
for ammonia, ethylene, and cyclobutadiene have been performed with the Molpro
2012.1.12 software package.102, 103

The spin-free DMRG calculations were performed using the Budapest QC-DMRG
program. 104 Two different sets of molecular orbitals were investigated,
namely canonical restricted Hartree-Fock (RHF) orbitals and pCCD-optimized
orbitals. Furthermore, we aimed at constructing small but chemically
reasonable active orbital spaces as they allow us to maintain the same level
of approximation for orbitals of similar weight and correlation strength.
Specifically, it allows us to avoid errors driven by an unbalanced composition
of the active space. The active space selection for RHF orbitals was based on
quantum information measures obtained from $m={64,128}$ DMRG calculations
since DMRG calculations exploiting even small bond dimensions provide robust
single-orbital entropies and orbital-pair mutual information profiles.

In all calculations involving active orbital spaces, we performed calculations
with different values of the bond dimension $m$ to ensure that DMRG energy
converged with respect to $m$, that is, we used $m={32,64,128}$ for all first-
row atom dimers, $m={128,256,512}$ for Cr2, ammonia, ethylene, and the benzene
molecules, and $m={256,512,1024}$ for cyclobutadiene molecule. The DMRG
energies of all first-row atom dimers, ammonia, and benzene were converged up
to $\Delta E=10^{-8}$ with respect to $m$, while for the Cr2, ethylene and
cyclobutadiene molecules the convergence threshold was relaxed to $\Delta
E=10^{-5}$ and $\Delta E=10^{-4}$, respectively. The converged DMRG wave
functions were first used to reconstruct the CI and then CC coefficients.

The active space of the N2 and F2 molecules consists of one $3\sigma_{g}$, two
$\pi_{u}$, two $\pi_{g}$, and one $3\sigma_{u}$ orbital. In the case of the
carbon dimer, the active orbital space was extended by the $2\sigma_{g}$ and
$2\sigma_{u}$ orbitals as the single orbital entropy and orbital-pair mutual
information in the pCCD-optimized orbital basis suggested that these orbitals
might have non-negligible impact on the balanced description of electron
correlation effects in the dissociation region. The orbital interactions are
similar for the C2 isoelectronic analogues — BN, BO+, and CN+ — and,
therefore, their active orbital spaces were composed of two $\sigma$, two
$\sigma^{*}$, two $\pi$, and two $\pi^{*}$ orbitals occupied by eight
electrons. For the chromium dimer, we used all twelve valence orbitals (4s and
3d) following the recommendations of Refs. 105, 20.

We studied two active spaces in the case of the ammonia compound, which were
selected solely on the values of the single orbital entropies and the orbital-
pair mutual information. Specifically, we looked for pronounced gaps in the
single-orbital entropy and orbital-pair mutual information distributions to
find reasonable cutoff values. This procedure resulted in a small CAS(6,6)
containing orbitals with $s_{i}$ > 0.04 and $I_{ij}$ > 0.07, while a slightly
bigger CAS(8,8) was derived from a decreased cutoff value of $s_{i}$ > 0.03
and $I_{ij}$ > 0.01. A similar active space selection protocol was used to
obtain a CAS(12,12) for the ethylene molecule and a CAS(20,20) for the
cyclobutadiene complex.

For benzene, the thresholds was tightened up ($I_{ij}$ > 0.1) resulting in a
CAS(6,6). The diagrams for the single-orbital entropy and orbital-pair mutual
information of selected systems (molecules and bond lengths) are summarized in
the Supporting Information (SI). The coupled cluster amplitudes were
reconstructed from matrix product state wave functions using the method
described by some of us.20 All tailored coupled cluster calculations were
performed using the CC amplitude equations in the spin-summed form.

###  3.2 Abbreviations of method names

In all (conventional and tailored) coupled cluster calculations, we used two
different reference wave functions and molecular orbitals: (a) canonical RHF
and (b) (variationally) orbital-optimized pCCD. In this work, all coupled
cluster methods were truncated at the doubles (CCD) and singles and doubles
(CCSD) levels. Thus, CCDa and CCSDa represent the traditional coupled cluster
methods with a canonical RHF reference function, while the abbreviations CCDb
and CCSDb indicate that the reference determinant (and hence the molecular
orbitals) of the orbital-optimized pCCD wave function was selected as the
reference determinant in conventional, that is untailored, coupled cluster
calculations. In these flavours, all cluster amplitudes are thus optimized and
all information about the electron-pair amplitudes is lost. We should note
that the linearized CC corrections with a pCCD reference function are labeled
as fpLCCD and fpLCCSD, respectively, while they were originally introduced
using the acronyms pCCD-LCCD and pCCD-LCCSD. All fpCC calculations, including
the linearized variants, have been performed in the pCCD-optimized orbital
basis only.

##  4 Results

Table 1: Spectroscopic constants for the dissociation of homonuclear main-group diatomic molecules for different quantum chemistry methods and basis sets. Errors with respect to MRCI, FCIQMC, or DMRG are given in parentheses. The reference data has no error given in parentheses. The superscript $a$ denotes that calculations have been performed in the canonical RHF orbital basis, while the superscript $b$ stands for pCCD-optimized orbitals. The “$*$” denotes the lowest-lying singlet excited state.  |  |  re [Å] |  De [$\rm\frac{kcal}{mol}$] |  $\omega_{\mathrm{e}}$ [cm-1] |  re [Å] |  De [$\rm\frac{kcal}{mol}$] |  $\omega_{\mathrm{e}}$ [cm-1]
---|---|---|---|---|---|---|---
|  | aug-cc-pVDZ | aug-cc-pVTZ |  |
F2 |  RHF |  1.338 | ( 0.115 ) |  181.0 | ( -152.5 ) |  1216 | ( -415 ) |  1.328 | ( 0.092 ) |  |  |  1271 | ( -379 )
|  CCDa |  1.414 | ( 0.039 ) |  69.1 | ( -40.6 ) |  964 | ( -163 ) |  1.383 | ( 0.037 ) |  82.2 | ( -48.3 ) |  1053 | ( -161 )
|  CCSDa |  1.425 | ( 0.028 ) |  57.0 | ( -28.5 ) |  922 | ( -121 ) |  1.392 | ( 0.028 ) |  70.1 | ( -36.2 ) |  1018 | ( -126 )
|  pCCD |  1.502 | ( -0.049 ) |  12.6 | ( 15.9 ) |  622 | ( 179 ) |  1.466 | ( -0.046 ) |  16.2 | ( 17.7 ) |  708 | ( 184 )
|  CCDb |  1.425 | ( 0.028 ) |  55.5 | ( -27.0 ) |  924 | ( -123 ) |  1.391 | ( 0.029 ) |  68.8 | ( -34.9 ) |  1021 | ( -129 )
|  CCSDb |  1.424 | ( 0.029 ) |  55.9 | ( -27.4 ) |  926 | ( -125 ) |  1.391 | ( 0.029 ) |  69.3 | ( -35.4 ) |  1022 | ( -130 )
|  fpLCCD |  1.473 | ( -0.020 ) |  37.5 | ( -9.0 ) |  776 | ( 25 ) |  1.434 | ( -0.014 ) |  44.3 | ( -10.4 ) |  866 | ( 26 )
|  fpLCCSD |  1.469 | ( -0.016 ) |  38.1 | ( -9.6 ) |  788 | ( 13 ) |  1.431 | ( -0.011 ) |  45.2 | ( -11.3 ) |  879 | ( 13 )
|  fpCCD |  1.473 | ( -0.020 ) |  36.7 | ( -8.2 ) |  773 | ( 28 ) |  1.433 | ( -0.013 ) |  43.3 | ( -9.4 ) |  863 | ( 29 )
|  fpCCSD |  1.469 | ( -0.016 ) |  37.1 | ( -8.6 ) |  782 | ( 19 ) |  1.431 | ( -0.011 ) |  44.2 | ( -10.3 ) |  873 | ( 19 )
|  DMRG(8,8)-tCCSDa |  1.493 | ( -0.040 ) |  37.7 | ( -9.2 ) |  728 | ( 73 ) |  1.454 | ( -0.034 ) |  43.7 | ( -9.8 ) |  758 | ( 134 )
|  DMRG(8,8)-tCCSDb |  1.468 | ( -0.015 ) |  38.4 | ( -9.9 ) |  728 | ( 73 ) |  1.427 | ( -0.007 ) |  45.9 | ( -12.0 ) |  875 | ( 17 )
|  MRCI126 |  1.453 |  |  28.5 |  |  801 |  |  1.420 |  |  33.9 |  |  892 |
|  exp.127, 128 |  1.412 | ( 0.041 ) |  37.7 | ( -9.2 ) |  917 | ( -116 ) |  1.412 | ( 0.008 ) |  37.7 | ( -3.8 ) |  917 | ( -25 )
|  | cc-pVDZ | cc-pVTZ |  |
|  pCCD |  1.471 | ( -0.006 ) |  15.5 | ( 10.8 ) |  693 | ( 64 ) |  1.471 | ( -0.052 ) |  15.5 | ( 17.0 ) |  693 | ( 198 )
|  CCDb |  1.392 | ( 0.074 ) |  66.1 | ( -39.8 ) |  1017 | ( -261 ) |  1.392 | ( 0.028 ) |  66.2 | ( -33.7 ) |  1016 | ( -126 )
|  CCSDb |  1.391 | ( 0.074 ) |  66.5 | ( -40.2 ) |  1019 | ( -263 ) |  1.391 | ( 0.028 ) |  66.6 | ( -34.1 ) |  1018 | ( -128 )
|  fpLCCD |  1.432 | ( 0.034 ) |  42.0 | ( -15.7 ) |  868 | ( -112 ) |  1.432 | ( -0.013 ) |  42.1 | ( -9.6 ) |  867 | ( 23 )
|  fpLCCSD |  1.429 | ( 0.036 ) |  43.1 | ( -16.8 ) |  879 | ( -123 ) |  1.429 | ( -0.010 ) |  43.2 | ( -10.7 ) |  878 | ( 13 )
|  fpCCD |  1.432 | ( 0.034 ) |  41.0 | ( -14.7 ) |  864 | ( -108 ) |  1.432 | ( -0.012 ) |  41.2 | ( -8.7 ) |  864 | ( 27 )
|  fpCCSD |  1.429 | ( 0.036 ) |  41.9 | ( -15.6 ) |  874 | ( -117 ) |  1.429 | ( -0.010 ) |  42.1 | ( -9.6 ) |  872 | ( 18 )
|  DMRG(8,8)-tCCSDa |  1.471 | ( -0.006 ) |  35.0 | ( -8.7 ) |  765 | ( -9 ) |  1.426 | ( -0.007 ) |  43.8 | ( -11.3 ) |  878 | ( 12 )
|  DMRG(8,8)-tCCSDb |  1.474 | ( -0.006 ) |  35.8 | ( -9.5 ) |  742 | ( 15 ) |  1.426 | ( -0.007 ) |  43.4 | ( -10.9 ) |  872 | ( 19 )
|  MRCI126 |  1.465 |  |  26.3 |  |  756 |  |  1.419 |  |  32.5 |  |  891 |
|  exp.127, 128 |  1.412 | ( 0.053 ) |  |  |  917 | ( -161 ) |  1.412 | ( 0.007 ) |  |  |  917 | ( -27 )
|  | cc-pVDZ | cc-pVTZ |  |
N2 |  pCCD |  1.099 | ( 0.020 ) |  239.6 | ( -37.6 ) |  2482 | ( -153 ) |  1.085 | ( 0.019 ) |  244.8 | ( -26.9 ) |  2517 | ( -176 )
|  CCDb |  1.111 | ( 0.008 ) |  |  |  2415 | ( -86 ) |  1.093 | ( 0.011 ) |  |  |  2447 | ( -106 )
|  CCSDb |  1.112 | ( 0.008 ) |  |  |  2412 | ( -84 ) |  1.093 | ( 0.011 ) |  |  |  2444 | ( -104 )
|  fpLCCD |  1.119 | ( 0.001 ) |  |  |  2320 | ( 8 ) |  1.101 | ( 0.003 ) |  |  |  2347 | ( -7 )
|  fpLCCSD |  1.120 | ( 0.000 ) |  |  |  2315 | ( 14 ) |  1.102 | ( 0.002 ) |  |  |  2340 | ( 0 )
|  fpCCD |  1.118 | ( 0.002 ) |  |  |  2335 | ( -7 ) |  1.100 | ( 0.004 ) |  |  |  2365 | ( -24 )
|  fpCCSD |  1.118 | ( 0.002 ) |  |  |  2336 | ( -8 ) |  1.100 | ( 0.004 ) |  |  |  2365 | ( -24 )
|  DMRG(6,6)-tCCSDa |  1.118 | ( 0.002 ) |  208.1 | ( -6.1 ) |  2327 | ( 2 ) |  1.100 | ( 0.004 ) |  227.4 | ( -9.5 ) |  2356 | ( -15 )
|  DMRG(6,6)-tCCSDb |  1.122 | ( -0.002 ) |  219.0 | ( -17.0 ) |  2383 | ( -55 ) |  1.102 | ( 0.002 ) |  239.3 | ( -21.4 ) |  2387 | ( -46 )
|  MRCI 126 |  1.120 |  |  202.0 |  |  2329 |  |  1.104 |  |  217.9 |  |  2341 |
|  | cc-pVQZ | aug-cc-pVTZ |  |
|  pCCD |  1.077 | ( 0.023 ) |  252.8 | ( -28.6 ) |  2723 | ( -371 ) |  1.087 | ( 0.011 ) |  256.1 | ( -31.0 ) |  2450 | ( -91 )
|  CCDb |  1.090 | ( 0.010 ) |  |  |  2447 | ( -95 ) |  1.092 | ( 0.006 ) |  |  |  2455 | ( -96 )
|  CCSDb |  1.090 | ( 0.010 ) |  |  |  2450 | ( -98 ) |  1.092 | ( 0.006 ) |  |  |  2448 | ( -89 )
|  fpLCCD |  1.098 | ( 0.003 ) |  |  |  2370 | ( -18 ) |  1.100 | ( -0.002 ) |  |  |  2357 | ( 2 )
|  fpLCCSD |  1.098 | ( 0.002 ) |  |  |  2373 | ( -21 ) |  1.101 | ( -0.003 ) |  |  |  2344 | ( 15 )
|  fpCCD |  1.096 | ( 0.005 ) |  |  |  2394 | ( -43 ) |  1.099 | ( -0.001 ) |  |  |  2374 | ( -15 )
|  fpCCSD |  1.096 | ( 0.005 ) |  |  |  2402 | ( -51 ) |  1.099 | ( -0.001 ) |  |  |  2369 | ( -10 )
|  DMRG(6,6)-tCCSDa |  1.097 | ( 0.004 ) |  230.9 | ( -6.7 ) |  2353 | ( -2 ) |  1.094 | ( 0.004 ) |  222.1 | ( 3.0 ) |  2280 | ( 79 )
|  DMRG(6,6)-tCCSDb |  1.099 | ( 0.001 ) |  244.4 | ( -20.2 ) |  2392 | ( -40 ) |  1.117 | ( -0.019 ) |  237.0 | ( -11.9 ) |  2232 | ( 127 )
|  MRCI126 |  1.101 |  |  224.2 |  |  2352 |  |  |  |  |  |  |
|  exp.127, 129 |  1.098 | ( 0.002 ) |  225.1 | ( -0.9 ) |  2359 | ( -7 ) |  1.098 |  |  225.1 |  |  2359 |
|  | cc-pVDZ | cc-pVTZ |  |
C2 |  CCDb |  1.263 | ( 0.009 ) |  136.0 | ( -9.2 ) |  1873 | ( -62 ) |  1.240 | ( 0.013 ) |  156.1 | ( -20.8 ) |  1911 | ( -76 )
|  CCSDb |  1.263 | ( 0.009 ) |  136.5 | ( -9.7 ) |  1873 | ( -62 ) |  1.240 | ( 0.013 ) |  156.6 | ( -21.2 ) |  1915 | ( -80 )
|  fpLCCD |  1.264 | ( 0.008 ) |  131.3 | ( -4.5 ) |  1901 | ( -90 ) |  1.241 | ( 0.011 ) |  136.7 | ( -1.3 ) |  1871 | ( -36 )
|  fpLCCSD |  1.265 | ( 0.008 ) |  134.2 | ( -7.4 ) |  1890 | ( -79 ) |  1.241 | ( 0.011 ) |  140.0 | ( -4.6 ) |  1862 | ( -27 )
|  fpCCD |  1.261 | ( 0.012 ) |  130.4 | ( -3.6 ) |  1905 | ( -94 ) |  1.237 | ( 0.015 ) |  143.4 | ( -8.0 ) |  1901 | ( -67 )
|  fpCCSD |  1.260 | ( 0.013 ) |  131.7 | ( -4.9 ) |  1906 | ( -95 ) |  1.236 | ( 0.017 ) |  145.1 | ( -9.7 ) |  1905 | ( -70 )
|  DMRG(8,8)-tCCSDa |  1.266 | ( 0.007 ) |  134.3 | ( -7.5 ) |  1850 | ( -39 ) |  1.242 | ( 0.011 ) |  147.5 | ( -12.1 ) |  1893 | ( -58 )
|  DMRG(8,8)-tCCSDb |  1.265 | ( 0.008 ) |  136.0 | ( -9.2 ) |  1899 | ( -88 ) |  1.249 | ( 0.004 ) |  146.3 | ( -11.0 ) |  1885 | ( -51 )
|  MRCI116 |  1.251 | ( 0.021 ) |  135.4 | ( -8.6 ) |  1873 | ( -62 ) |  1.252 | ( 0.001 ) |  140.4 | ( -5.0 ) |  1840 | ( -5 )
|  DMRG(12,28)123 |  1.272 | ( 0.001 ) |  130.1 | ( -3.3 ) |  1816 | ( -5 ) |  |  |  |  |  |
|  FCIQMC122 |  1.273 |  |  126.8 |  |  1811 |  |  1.253 |  |  135.4 |  |  1835 |
|  exp.127 |  1.243 | ( 0.030 ) |  147.8 | ( -21.0 ) |  1855 | ( -44 ) |  1.243 | ( 0.010 ) |  147.8 | ( -12.4 ) |  1855 | ( -20 )
|  | cc-pVDZ | cc-pVTZ |  |
C${}_{2}^{*}$ |  CCDb |  1.415 | ( -0.005 ) |  111.3 | ( -23.8 ) |  1641 | ( -274 ) |  1.393 | ( -0.016 ) |  133.4 | ( -32.2 ) |  1970 | ( -546 )
|  CCSDb |  1.416 | ( -0.006 ) |  111.3 | ( -23.8 ) |  1648 | ( -281 ) |  1.320 | ( 0.057 ) |  131.9 | ( -30.7 ) |  1387 | ( 37 )
|  fpLCCD |  1.466 | ( -0.056 ) |  87.5 | ( 0.0 ) |  1421 | ( -54 ) |  1.404 | ( -0.026 ) |  94.9 | ( 6.3 ) |  1389 | ( 35 )
|  fpLCCSD |  1.449 | ( -0.039 ) |  91.1 | ( -3.6 ) |  1317 | ( 50 ) |  1.406 | ( -0.028 ) |  98.4 | ( 2.8 ) |  1348 | ( 76 )
|  fpCCD |  1.425 | ( -0.015 ) |  91.9 | ( -4.4 ) |  1286 | ( 80 ) |  1.394 | ( -0.016 ) |  106.7 | ( -5.5 ) |  1379 | ( 45 )
|  fpCCSD |  1.426 | ( -0.016 ) |  94.4 | ( -6.9 ) |  1301 | ( 66 ) |  1.395 | ( -0.018 ) |  109.4 | ( -8.2 ) |  1370 | ( 54 )
|  DMRG(12,28)123 |  1.410 |  |  87.6 |  |  1367 |  |  |  |  |  |  |
|  exp.130 |  1.377 | ( 0.033 ) |  101.2 | ( -13.6 ) |  1424 | ( -57 ) |  1.377 |  |  101.2 |  |  1424 |
Table 2: Spectroscopic constants for the dissociation of main-group heteronuclear diatomic molecules for different quantum chemistry methods and basis sets. Errors with respect to MRCI, CMRCI+Q, FCIQMC, or experiment are given in parentheses. The superscript $a$ denotes that calculations have been performed in the canonical RHF orbital basis, while the superscript $b$ stands for pCCD-optimized orbitals. The “$*$” denotes the lowest-lying singlet excited state that could be optimized within pCCD.  |  |  re [Å] |  De [$\rm\frac{kcal}{mol}$] |  $\omega_{\mathrm{e}}$ [cm-1] |  re [Å] |  De [$\rm\frac{kcal}{mol}$] |  $\omega_{\mathrm{e}}$ [cm-1]
---|---|---|---|---|---|---|---
|  | cc-pVDZ | cc-pVTZ |  |
BO+ |  CCDa |  1.199 | ( 0.027 ) |  114.8 | ( 3.2 ) |  1901 | ( -168 ) |  1.186 | ( 0.030 ) |  126.5 | ( -0.5 ) |  1966 | ( -182 )
|  CCSDa |  1.205 | ( 0.021 ) |  119.6 | ( -1.6 ) |  1883 | ( -150 ) |  1.191 | ( 0.025 ) |  130.6 | ( -4.6 ) |  1949 | ( -165 )
|  CCDb |  1.198 | ( 0.028 ) |  108.5 | ( 9.6 ) |  1903 | ( -169 ) |  1.184 | ( 0.031 ) |  122.8 | ( 3.2 ) |  2005 | ( -221 )
|  CCSDb |  1.205 | ( 0.021 ) |  111.5 | ( 6.6 ) |  1894 | ( -160 ) |  1.191 | ( 0.025 ) |  124.3 | ( 1.7 ) |  1952 | ( -168 )
|  fpLCCD |  1.205 | ( 0.021 ) |  107.5 | ( 10.6 ) |  1854 | ( -121 ) |  1.191 | ( 0.025 ) |  116.3 | ( 9.7 ) |  1942 | ( -158 )
|  fpLCCSD |  1.232 | ( -0.006 ) |  120.5 | ( -2.5 ) |  2264 | ( -530 ) |  1.245 | ( -0.029 ) |  132.0 | ( -6.0 ) |  1865 | ( -81 )
|  fpCCD |  1.203 | ( 0.023 ) |  106.1 | ( 12.0 ) |  1874 | ( -140 ) |  1.189 | ( 0.026 ) |  114.8 | ( 11.3 ) |  1958 | ( -174 )
|  fpCCSD |  1.205 | ( 0.021 ) |  107.0 | ( 11.0 ) |  1925 | ( -191 ) |  1.193 | ( 0.023 ) |  114.4 | ( 11.6 ) |  1936 | ( -152 )
|  DMRG(8,8)-tCCSDa |  1.230 | ( -0.004 ) |  130.8 | ( -12.8 ) |  1823 | ( -89 ) |  1.218 | ( -0.002 ) |  141.7 | ( -15.7 ) |  1868 | ( -84 )
|  DMRG(8,8)-tCCSDb |  1.188 | ( 0.038 ) |  102.0 | ( 16.0 ) |  1676 | ( 58 ) |  1.166 | ( 0.049 ) |  93.3 | ( 32.7 ) |  1718 | ( 66 )
|  CASSCF116 |  1.211 | ( 0.016 ) |  133.7 | ( -15.7 ) |  1815 | ( -82 ) |  1.205 | ( 0.011 ) |  138.5 | ( -12.4 ) |  1835 | ( -52 )
|  CMRCI116 |  1.225 | ( 0.001 ) |  118.6 | ( -0.5 ) |  1741 | ( -7 ) |  1.214 | ( 0.001 ) |  126.7 | ( -0.7 ) |  1792 | ( -8 )
|  CMRCI+Q116 |  1.226 |  |  118.0 |  |  1734 |  |  1.216 |  |  126.0 |  |  1784 |
|  | cc-pVDZ | cc-pVTZ |  |
BN |  pCCD |  1.269 | ( 0.030 ) |  100.3 | ( 48.2 ) |  1776 | ( -125 ) |  | ( 1.285 ) |  | ( 154.4 ) |  | ( 1682 )
|  CCDb |  1.293 | ( 0.005 ) |  150.8 | ( -2.2 ) |  1689 | ( -39 ) |  1.279 | ( 0.006 ) |  164.4 | ( -10.1 ) |  1683 | ( -1 )
|  CCSDb |  1.288 | ( 0.010 ) |  153.3 | ( -4.7 ) |  1714 | ( -63 ) |  1.271 | ( 0.014 ) |  167.7 | ( -13.3 ) |  1759 | ( -77 )
|  fpLCCD |  1.294 | ( 0.004 ) |  148.5 | ( 0.1 ) |  1683 | ( -33 ) |  1.281 | ( 0.004 ) |  187.7 | ( -33.3 ) |  1461 | ( 221 )
|  fpLCCSD |  1.293 | ( 0.005 ) |  154.3 | ( -5.8 ) |  1695 | ( -45 ) |  1.297 | ( -0.012 ) |  193.9 | ( -39.5 ) |  1595 | ( 87 )
|  fpCCD |  1.291 | ( 0.008 ) |  146.2 | ( 2.4 ) |  1700 | ( -49 ) |  1.282 | ( 0.003 ) |  184.0 | ( -29.7 ) |  1589 | ( 93 )
|  fpCCSD |  1.287 | ( 0.012 ) |  147.8 | ( 0.8 ) |  1724 | ( -74 ) |  1.276 | ( 0.009 ) |  185.7 | ( -31.4 ) |  1606 | ( 76 )
|  DMRG(8,8)-tCCSDa |  1.293 | ( 0.005 ) |  159.6 | ( -11.0 ) |  1787 | ( -136 ) |  1.282 | ( 0.003 ) |  169.7 | ( -15.4 ) |  1836 | ( -154 )
|  DMRG(8,8)-tCCSDb |  1.297 | ( 0.001 ) |  138.7 | ( 9.9 ) |  1742 | ( -91 ) |  1.298 | ( -0.013 ) |  159.2 | ( -4.8 ) |  1852 | ( -170 )
|  CASSCF116 |  1.294 | ( 0.004 ) |  158.8 | ( -10.2 ) |  1681 | ( -30 ) |  1.288 | ( -0.003 ) |  160.8 | ( -6.4 ) |  1686 | ( -4 )
|  CMRCI116 |  1.298 | ( 0.001 ) |  149.8 | ( -1.2 ) |  1655 | ( -4 ) |  1.284 | ( 0.001 ) |  155.9 | ( -1.5 ) |  1687 | ( -5 )
|  CMRCI+Q116 |  1.298 |  |  148.6 |  |  1651 |  |  1.285 |  |  154.4 |  |  1682 |
|  | cc-pVDZ | cc-pVTZ |  |
CN+ |  CCDb |  1.186 | ( 0.012 ) |  173.6 | ( -9.8 ) |  |  |  1.186 | ( 0.012 ) |  173.6 | ( -9.8 ) |  2106 | ( -127 )
|  CCSDb |  1.184 | ( 0.014 ) |  173.0 | ( -9.2 ) |  |  |  1.184 | ( 0.014 ) |  173.0 | ( -9.2 ) |  2121 | ( -141 )
|  fpLCCD |  1.174 | ( 0.024 ) |  179.5 | ( -15.7 ) |  2068 | ( -88 ) |  1.174 | ( 0.024 ) |  179.5 | ( -15.7 ) |  2068 | ( -88 )
|  fpLCCSD |  1.173 | ( 0.025 ) |  176.3 | ( -12.5 ) |  2080 | ( -100 ) |  1.173 | ( 0.025 ) |  176.3 | ( -12.5 ) |  2080 | ( -100 )
|  fpCCD |  1.169 | ( 0.029 ) |  167.5 | ( -3.7 ) |  2075 | ( -95 ) |  1.169 | ( 0.029 ) |  167.5 | ( -3.7 ) |  2070 | ( -91 )
|  fpCCSD |  1.181 | ( 0.017 ) |  176.8 | ( -13.0 ) |  2081 | ( -101 ) |  1.160 | ( 0.038 ) |  176.8 | ( -13.0 ) |  2174 | ( -194 )
|  DMRG(8,8)-tCCSDa |  1.191 | ( 0.007 ) |  171.2 | ( -7.4 ) |  2013 | ( -34 ) |  1.179 | ( 0.019 ) |  155.3 | ( 8.5 ) |  2156 | ( -177 )
|  DMRG(8,8)-tCCSDb |  1.190 | ( 0.008 ) |  178.5 | ( -14.7 ) |  1986 | ( -7 ) |  1.170 | ( 0.028 ) |  170.9 | ( -7.0 ) |  2042 | ( -62 )
|  CASSCF116 |  1.191 | ( 0.007 ) |  178.8 | ( -14.9 ) |  2030 | ( -51 ) |  1.182 | ( 0.016 ) |  181.0 | ( -17.2 ) |  2018 | ( -39 )
|  CMRCI116 |  1.197 | ( 0.001 ) |  165.2 | ( -1.4 ) |  1985 | ( -6 ) |  1.182 | ( 0.016 ) |  170.6 | ( -6.8 ) |  2006 | ( -26 )
|  CMRCI+Q116 |  1.198 |  |  163.8 |  |  1979 |  |  1.198 |  |  163.8 |  |  1979 |
|  exp.127 |  1.173 | ( 0.025 ) |  |  |  2033 | ( -54 ) |  1.173 | ( 0.025 ) |  |  |  2033 | ( -54 )
|  | cc-pVDZ | cc-pVTZ |  |
(CN+)∗ |  CCDb |  1.335 |  |  160.7 |  |  1962 |  |  1.335 |  |  160.7 |  |  1962 |
|  CCSDb |  1.339 |  |  162.8 |  |  2084 |  |  1.339 |  |  162.8 |  |  2084 |
|  fpLCCD |  1.389 |  |  123.0 |  |  1099 |  |  1.389 |  |  123.0 |  |  1099 |
|  fpLCCSD |  1.391 |  |  142.8 |  |  1407 |  |  1.391 |  |  142.8 |  |  1407 |
|  fpCCD |  1.375 |  |  128.0 |  |  1264 |  |  1.375 |  |  128.0 |  |  1363 |
|  fpCCSD |  1.373 |  |  144.7 |  |  1515 |  |  1.373 |  |  144.7 |  |  1510 |
|  | cc-pVDZ | cc-pVTZ |  |
CO |  CCDa |  1.135 | ( 0.010 ) |  326.7 | ( -85.2 ) |  2248 | ( -104 ) |  1.104 | ( 0.031 ) |  307.0 | ( -55.1 ) |  2419 | ( -265 )
|  CCSDa |  1.138 | ( 0.007 ) |  329.0 | ( -87.4 ) |  2212 | ( -68 ) |  1.123 | ( 0.013 ) |  334.0 | ( -82.1 ) |  2264 | ( -110 )
|  CCSD(T)a |  1.145 |  |  241.5 |  |  2144 |  |  1.136 |  |  251.9 |  |  2154 |
|  pCCD |  1.117 | ( 0.027 ) |  224.8 | ( 16.7 ) |  2295 | ( -151 ) |  1.116 | ( 0.020 ) |  237.9 | ( 14.1 ) |  2317 | ( -163 )
|  CCDb |  1.132 | ( 0.013 ) |  262.8 | ( -21.3 ) |  2246 | ( -102 ) |  1.125 | ( 0.011 ) |  288.8 | ( -36.9 ) |  2257 | ( -103 )
|  CCSDb |  1.137 | ( 0.007 ) |  264.0 | ( -22.4 ) |  2219 | ( -75 ) |  1.125 | ( 0.011 ) |  289.7 | ( -37.8 ) |  2238 | ( -84 )
|  fpLCCD |  1.134 | ( 0.011 ) |  229.6 | ( 11.9 ) |  2171 | ( -26 ) |  1.129 | ( 0.006 ) |  244.3 | ( 7.6 ) |  2218 | ( -64 )
|  fpLCCSD |  1.143 | ( 0.001 ) |  236.2 | ( 5.3 ) |  2105 | ( 39 ) |  1.130 | ( 0.006 ) |  253.4 | ( -1.5 ) |  2155 | ( -2 )
|  fpCCD |  1.133 | ( 0.011 ) |  235.1 | ( 6.5 ) |  2178 | ( -33 ) |  1.128 | ( 0.008 ) |  254.3 | ( -2.4 ) |  2228 | ( -74 )
|  fpCCSD |  1.137 | ( 0.007 ) |  240.3 | ( 1.2 ) |  2178 | ( -34 ) |  1.128 | ( 0.008 ) |  260.7 | ( -8.8 ) |  2218 | ( -64 )
|  exp.131, 132, 128 |  1.128 | ( 0.016 ) |  255.8 | ( -14.3 ) |  2170 | ( -26 ) |  1.128 | ( 0.007 ) |  255.8 | ( -3.9 ) |  2170 | ( -16 )

###  4.1 Diatomic molecules

As a first test case, we selected seven diatomic molecules containing only
main-group elements (namely B, C, N, O, and F), which feature complex
electronic structures driven by quasi-degenerate 2p orbitals. The dissociation
process of these main-group dimers highlights the disparate interplay of
nondynamic/static and dynamic electron correlation. Specifically, the
dissociation of the fluorine dimer, despite its single bond, cannot be
reliably modeled with the gold standard of quantum chemistry, that is CCSD(T),
as it produces an unphysical shape of the potential energy surface (PES).106,
107, 108, 109, 48 A similar outcome is observed for the CCSD model in the case
of the nitrogen dimer. This molecule is known as one of the most challenging
(diatomic) systems due to the triple bonding mechanism, which requires higher
excitation operators in the theoretical model (as well as high angular momenta
in the basis set) to reach spectroscopic accuracy. 110, 111, 112, 113, 114,
45, 47, 90 The PESs of the carbon dimer, cyano cation, and boron nitride
feature electron configurations of energetic proximity and avoided crossing.
115, 116, 117, 118, 119, 120, 121, 122, 113, 123, 40, 48, 124, 125 We also
included the carbon monoxide molecule and the boron monoxide cation in our
test set, which are less affected by electron correlation effects.116, 48 The
spectroscopic constants obtained for our diatomic test set by various coupled-
cluster models (and basis sets) are summarized in Tables 1 and 2.

For the F2 molecule, coupled cluster methods with an RHF reference result in
noticeably deeper potential energy wells. Specifically, CCD(RHF) considerably
overestimates the dissociation energy De with an error of 44.5 kcal/mol. The
inclusion of single excitations only slightly improves De. Changing the
reference determinant (and hence the molecular orbital basis) to the pCCD-
optimized solution results in similar shapes of the PES as obtained in
conventional CCSD calculations. Thus, switching to the pCCD reference
determinant allows us to obtain CCSD accuracy by only solving the CCD
equations. The DMRG-tailored and pCCD-tailored CCD and CCSD flavours
substantial improve the accuracy of the predicted potential energy well depths
and vibrational frequencies. While single-reference CCSD overestimates the
potential energy well depth by 28–40 kcal/mol, this error drops down to 8–17
kcal/mol if the ansatz is tailored by multireference wave functions. Similar
gain is observed for vibrational frequencies, where tailoring decreases the
difference with respect to MRCI results below the basis set error threshold.
The equilibrium bond lengths are estimated with the highest accuracy by the
DMRG-tCCSD method in the pCCD-optimized orbital basis and differ from
reference MRCI results by 0.007–0.015 Å, which is less than the difference
between the MRCI result and the experimental value. DMRG-tCCSD in the RHF
orbital basis provides similar results for correlation-consistent basis sets
(cc-pVDZ, cc-pVTZ) but the errors increase when augmented basis sets (aug-cc-
pVDZ, aug-cc-pVTZ) are used.

For the singlet ground-state N2 molecule, all tested coupled cluster flavours
achieve good accuracy in the near-equilibrium region and predict accurate bond
lengths and vibrational frequencies. However, most coupled-cluster methods
fail to accurately describe the region with a stretched N–N bond and the
vicinity of dissociation. This also holds for pCCD-tailored coupled cluster
methods, where the linearized models feature particularly large divergencies
in the dissociation limit. This problem is cured by both DMRG-tCCSD models in
the RHF and pCCD orbital basis, which are able to (indirectly) include triple
excitations in the active space spanned by the 2p orbitals.

Figure 1: The potential energy curves for the two singlet states of the carbon
dimer using the cc-pVDZ basis set compared to DMRG123 reference data. The left
upper panel corresponds to the ground state pCCD orbitals, while the left
bottom panel was obtained for the excited-state pCCD orbitals. The superscript
$a$ denotes that calculations have been performed in the canonical RHF orbital
basis, while the superscript $b$ stands for pCCD-optimized orbitals. The right
panels shows two sets of (valence) pCCD orbitals for the C2 molecules with
occupations numbers in parentheses.

The carbon dimer, cyano cation, boron nitride, and boron monoxide cation are
isoelectronic diatomic species where a balanced description of electron
correlation effects is required to obtain accurate energetics and
spectroscopic properties.115, 116, 40, 48, 125 The challenge in modelling
their PESs arises from the near-degeneracies of the valence orbitals and the
energetic proximity of electronic configurations.117, 118, 119, 120, 121, 122,
113, 123, 124, 48 For example, the two lowest ${}^{1}\Sigma_{g}^{+}$ states of
C2 are affected by an avoided crossing since the $\ket{1\pi_{u}^{4}}$
configuration is favoured around the equilibrium bond length while the
$\ket{3\sigma_{g}^{2}1\pi_{u}^{2}}$ configuration dominates in the ground
state wave function in the dissociation limit. The CN+ and BN molecules
exhibit an analogous bonding pattern to the carbon dimer 133, 117, 118, while
BO+ features a more single-reference character.118

For the carbon dimer and cyano cation, we obtained two sets of pCCD-optimized
orbitals corresponding to pCCD solutions dominated by either the
$\ket{1\pi^{4}}$ or $\ket{3\sigma^{2}1\pi_{u}^{2}}$ determinant. The latter
causes symmetry breaking of the orbitals since the pCCD model does not
describe two equivalent $\ket{3\sigma_{g}^{2}1\pi_{u}^{2}}$ determinants on an
equal footing. The adiabatic excitation energies of these two states are
presented in Table 3, while the PESs of the C2 molecule obtained with the two
different sets of pCCD-optimized orbitals are presented in Figure 1. Only for
DMRG-tCCSD, we were not able to optimize a different PES using the second set
of pCCD orbitals as we obtained the same total energies for both orbital sets.
Our results are consistent with DMRG(12,28)123 and FCIQMC122 reference data,
but the PESs around the avoiding crossing region is not smooth in the case of
fpCCD, fpCCSD, fpLCCD, and fpLCCSD.

Table 3: Adiabatic excitation energies [eV] between the singlet ground- and first excited state of the C2 and CN+ molecules. The acronym in parentheses indicates the molecular orbital basis employed in calculations. The superscript $b$ denotes that calculations have been performed in the pCCD-optimized orbital basis.  | C2 | CN+
---|---|---
| cc-pVDZ | cc-pVTZ | cc-pVDZ | cc-pVTZ
CCDb | 1.064 | 0.997 | 1.567 | 1.567
CCSDb | 1.080 | 1.031 | 1.578 | 1.578
fpLCCD | 1.811 | 1.792 | 2.643 | 2.643
fpLCCSD | 1.819 | 1.774 | 2.669 | 2.669
fpCCD | 1.652 | 1.591 | 2.426 | 2.425
fpCCSD | 1.601 | 1.549 | 2.533 | 2.533
DMRG(12,28)123 | 1.913 |  |  |

The electron-pair cluster (geminal) amplitudes $t_{ii}^{aa}$ of the C2
molecule obtained with DMRG-tCCSD, pCCD, and (conventional) CCSD are presented
in Figure 3. The pair-amplitudes for other diatomic molecules are presented in
the SI.§ For the near-equilibrium geometry, the largest cluster amplitudes (in
absolute value) are found for the space spanned by six 2p-type orbitals. Two
additional orbitals participate in the bond-breaking process. In the canonical
RHF basis, the electron-pair amplitudes of the conventional CCD and CCSD model
agree well with the DMRG-tCCSD reference amplitudes in the equilibrium region,
but disagree in the vicinity of dissociation. For the pCCD-optimized orbital
basis, the differences between pCCD and DMRG electron-pair amplitudes are
small and rather of quantitative nature, while the general structure of the
wave function is similar in both cases. The CCD and CCSD electron-pair
amplitudes (in both the canonical RHF and pCCD orbital basis) tend to
substantially differ from the pCCD and DMRG-tailored amplitudes, which is
particularly pronounced for the BO+ molecule in the dissociation limit.

Figure 2: Mean errors with respect to accurate theoretical reference data
(MRCISD+Q for BN, BO+, CN+, MRCI for C2, CN+, N2, F2, and CCSD(T) for CO
molecule) including the standard deviation (black lines) determined for all
fitted spectroscopic constants for our test set of main-group diatomics (BN,
BO+, C2, CN+, N2, F2, CO). The superscript $a$ denotes that calculations have
been performed in the canonical RHF orbital basis, while the superscript $b$
indicates pCCD-optimized orbitals. See also Table 4 for the definition of the
corresponding error measures. Table 4: Error measures determined for all
fitted spectroscopic parameters (equilibrium bond lengths, potential energy
well depths, and harmonic vibrational frequencies) of our test set containing
main-group diatomics (BN, BO+, C2, CN+, CO, N2, F2) with respect to accurate
multireference methods (MRCISD+Q for BN, BO+, CN+, MRCI for C2, CN+, N2, F2,
and CCSD(T) for the CO molecule). ME: mean error, MAE: mean absolute error,
RMSD: root mean square deviation. The superscript $a$ denotes that
calculations have been performed in the canonical RHF orbital basis, while the
superscript $b$ indicates pCCD-optimized orbitals.

|  |  $\delta$re [Å] |  $\delta$De [$\rm\frac{kcal}{mol}$] |  $\delta\omega_{\mathrm{e}}$ [cm-1]
---|---|---|---|---
|  | ME | MAE | RMSD | ME | MAE | RMSD | ME | MAE | RMSD
cc-pVDZ |  CCDb | -0.021 | 0.021 | 0.031 | 2.3 | 9.1 | 11.3 | 112 | 112 | 141
CCSDb | -0.021 | 0.021 | 0.030 | 11.8 | 14.0 | 19.3 | 130 | 130 | 159
fpLCCD | -0.015 | 0.015 | 0.018 | 4.9 | 13.8 | 18.9 | 68 | 68 | 79
fpLCCSD | -0.010 | 0.012 | 0.017 | 5.0 | 7.1 | 8.9 | 129 | 142 | 227
fpCCD | -0.017 | 0.017 | 0.020 | -0.9 | 7.7 | 9.2 | 76 | 76 | 87
fpCCSD | -0.015 | 0.015 | 0.018 | 1.8 | 7.4 | 9.3 | 92 | 92 | 107
DMRG-tCCSDa | -0.002 | 0.005 | 0.005 | 8.6 | 9.0 | 10.2 | 40 | 49 | 69
DMRG-tCCSDb | -0.007 | 0.010 | 0.016 | 2.5 | 11.2 | 12.5 | 11 | 40 | 50
cc-pVTZ |  CCDb | -0.016 | 0.016 | 0.018 | 17.2 | 18.2 | 22.2 | 108 | 108 | 127
CCSDb | -0.017 | 0.017 | 0.018 | 18.1 | 18.7 | 22.8 | 112 | 112 | 118
fpLCCD | -0.009 | 0.012 | 0.015 | 6.3 | 13.3 | 16.4 | 16 | 98 | 121
fpLCCSD | 0.001 | 0.014 | 0.016 | 11.6 | 11.8 | 17.7 | 18 | 51 | 64
fpCCD | -0.011 | 0.014 | 0.017 | 6 | 9.8 | 13.6 | 47 | 87 | 98
fpCCSD | -0.013 | 0.016 | 0.019 | 9.3 | 13.2 | 15.7 | 64 | 95 | 112
DMRG-tCCSDa | -0.005 | 0.008 | 0.010 | 8.4 | 11.3 | 11.7 | 78 | 82 | 104
DMRG-tCCSDb | -0.011 | 0.017 | 0.024 | 2.9 | 13.8 | 17.1 | 40 | 68 | 83

ME (mean error)
$=\frac{1}{N}\sum_{i}^{N}(x_{i}^{\mathrm{method}}-x_{i}^{\mathrm{reference}})$
MAE (mean absolute error)
$=\frac{1}{N}\sum_{i}^{N}|x_{i}^{\mathrm{method}}-x_{i}^{\mathrm{reference}}|$
RMSD (root mean square deviation)
$=\sqrt{\frac{1}{N}\sum_{i}^{N}(x_{i}^{\mathrm{method}}-x_{i}^{\mathrm{reference}})^{2}}$

Figure 3: Electron-pair amplitudes represented as a matrix for the C2 molecule
obtained by CCD, CCSD, pCCD, and DMRG-tCCSD. The horizontal axis denotes
occupied orbitals, while the vertical axis stands for virtual orbitals. The
value of each amplitude is color-coded. The superscript $a$ denotes that
calculations have been performed in the canonical RHF orbital basis, while the
superscript $b$ indicates pCCD-optimized orbitals. Figure 4: The potential
energy curves for the CO molecule (cc-pVTZ basis set). The superscript $b$
indicates pCCD-optimized orbitals.

Figure 4 summarizes the dissociation path of the CO molecule obtained by
various CC methods. Although the reference wave functions (RHF and pCCD)
provide smooth Morse potential-shaped plots, fpLCCSD diverges in the region
between 1.5–2.3 Å, while CCSD in the RHF orbital basis fails in predicting a
smooth PES. Inspecting the nature and composition of the pCCD wave function,
we observe that the pCCD solution becomes multireference for those bond
lengths. When the C–O distance reaches 1.48 Å, the geminal coefficients for
two dominant excited determinants are about $-0.1$. They gradually increase in
absolute value when the molecule is stretched reaching values of $-0.95$ and
$-0.8$ for a distance of 3.17 Å. The failure of fpLCCSD may be further
attributed to the linearized nature of the coupled-cluster amplitude
equations, which may feature divergencies and poles in their solutions. 134

Figure 2 and Table 4 show the mean errors including the standard deviation of
the fitted spectroscopic constants (equilibrium distance, dissociation energy,
and harmonic vibrational frequencies) for our test set of main-group diatomic
molecules with respect to reference data. All studied coupled cluster methods,
including CCD and CCSD with pCCD-optimized orbitals, tailored CCSD approaches
with both RHF and pCCD-optimized orbitals, fpCCD, and fpCCSD, overestimate
equilibrium bond lengths. In general, tailoring selected coupled-cluster
amplitudes reduces the errors in all spectroscopic constants. DMRG-tCC is the
most accurate method in predicting dissociation energies (in the cc-pVTZ
basis), but lies in between fpCC and fpLCC quality for equilibrium distances
and vibrational frequencies. Furthermore, pCCD orbitals are slightly better in
the dissociation region and thus allow us to predict more precise De and
$\omega$ values. In general, DMRG-tCCSD(RHF) provides the smallest errors only
for equilibrium bond lengths, while its performance strongly deteriorates for
dissociation energies and harmonic vibrational frequencies. Moreover, the
addition of single excitations in the fpCCD model does not improve the
accuracy compared to fpCCD. Comparing the mean errors and mean absolute errors
suggests that the error measures provided by tailored coupled cluster
approaches are not as systematic as those obtained by single-reference coupled
cluster theory, where equilibrium bond lengths are systematically too large
and vibrational frequencies are systematically underestimated. Finally, our
statistical analysis on the bond-breaking process of selected main-group
diatomics suggest that the results of the linearized pCCD-tCC models are of
similar quality as its non-linear version. However, pCCD-tailored wave
functions (restricted to at most double excitations) are insufficient to
accurately describe the dissociation pathway of molecules featuring a triple
bond. As expected, the fpLCCSD method may, however, show unphysical features
in the PESs like divergencies or poles. These divergencies can be cured by
including non-linear terms in the CC amplitude equations resulting in the
fpCCSD framework.

###  4.2 The dissociation pathway of the Cr2 dimer

The chromium dimer and its dissociation process are widely used as a benchmark
problem in quantum chemistry primarily because its complicated electronic
structure and formal hextuple bond pose a remarkable challenge for present-day
quantum chemical methods.135, 136, 137, 55, 105, 138 Even excitations of
fourth order are insufficient to accurately capture electron correlation
effects within a single-reference framework.139 Multi-reference methods
represent the most robust and trustworthy approach to study the electronic
structure of Cr2. For instance, Veis et al. report that DMRG-tCCSD effectively
describes the Cr2 energy around the equilibrium geometry and outperforms the
conventional CCSDTQ method in terms of total energies.20

Figure 5 summarizes the PESs obtained with different flavours of conventional
and tailored coupled-cluster theory. pCCD provides a smooth curve, albeit
overestimating the equilibrium bond length (r${}_{e}=1.881$ Å). Note that pCCD
does not converge for bond lengths r${}_{\rm Cr-Cr}>2.5$ Å. CCDb and CCSDb
yield potential energy curves with too short bond lengths (r${}_{e}=1.533$ Å)
and too large slopes. fpCCD reproduces the proper shape of the PES around the
equilibrium and predicts a bond length of 1.641 Å, which is closest to the
experimental value of 1.6788 Å. However, pCCD, and thus, pCCD-tailored CC
theory fails in the description of the dissociation path and dissociation
limit. We should note that we encountered convergence problems for stretched
Cr–Cr distances in all frozen-pair coupled-cluster calculations, while the
linearized pCCD-tailored coupled cluster flavours (both fpLCCD and fpLCCSD)
fail due to divergencies near the equilibrium. Thus, those points are not
shown in Figure 5. DMRG(12,12) calculations in the RHF orbital basis do not
yield a bonded PES and diverge for r${}_{\rm Cr-Cr}>1.8$ Å, while DMRG(12,12)
in the pCCD orbital basis provides an unphysical PES similar to
CASSCF(12,12).55, 105, 138 Note that the poor performance of minimal active
space calculations for Cr2, that is 12 electrons in 12 orbitals, is a well
known problem in computational chemistry 136, 137, 105, 55, 138. All
investigated DMRG-tailored coupled cluster methods perform well in the near-
equilibrium region. However, for $r_{\rm Cr-Cr}>1.9$ Å, we observe
overcorrelation in the RHF orbital basis, while for pCCD-optimized orbital the
DMRG-tCCSD equations do not converge. Thus, DMRG-tCCSD exploiting the minimal
active space in DMRG calculations cannot be used to model the dissociation of
the sextuple bond of the chromium dimer. Further investigations are required
to determine whether extending the active space can cure the problems related
to overcorrelation, divergencies, and convergence difficulties in the CC
amplitude equations.

Table 5: Computed spectroscopic constants of the chromium dimer for various coupled cluster methods and atomic basis sets. The superscript $a$ denotes that calculations have been performed in the canonical RHF orbital basis, while the superscript $b$ indicates pCCD-optimized orbitals. | re [Å] |  $\omega_{\mathrm{e}}$ [cm-1] | re [Å] |  $\omega_{\mathrm{e}}$ [cm-1]
---|---|---|---|---
| cc-pVDZ | cc-pVTZ
pCCD | 1.884 | 264 | 1.881 | 244
CCSDb | 1.542 | 950 | 1.535 | 943
fpCCD | 1.676 | 397 | 1.637 | 425
fpCCSD | 1.651 | 525 | 1.623 | 567
DMRG(12,12)-tCCSDa | 1.584 | 785 | 1.576 | 790
DMRG(12,12)-tCCSDb | 1.626 | 635 | 1.627 | 602
exp. | 1.6788 140 | 481 141 | 1.679 140 | 481 141
Figure 5: The potential energy curves of Cr2 (cc-pVTZ basis set). The
superscript $a$ denotes that calculations have been performed in the canonical
RHF orbital basis, while the superscript $b$ stands for pCCD orbitals. Note
that DMRG(12,12)b yields an unbound PES and is hence not shown here.

###  4.3 Umbrella inversion of ammonia

Theoretical models have been struggling for many years to obtain spectroscopic
accuracy for the six vibrational modes of the NH3 molecule. The main reason
for this struggle is an inversion mode that is characterized by a high
amplitude but a low frequency. Since this system does not feature strong
electron correlation and the energy converges fast with respect to the order
of the cluster operator, single-reference coupled cluster methods are
sufficient to approach spectroscopic accuracy.142 To assess the performance of
various tailored coupled cluster flavours, we modeled the path of the
conversion of the ground-state equilibrium pyramidal-shaped molecule to the
planar complex. Furthermore, the accuracy of tCC theory is benchmarked against
theoretical results rather than experimental ones due to the non-monotonic
behavior concerning the basis set for structural and spectroscopic
properties.143

Figure 6: Ammonia umbrella inversion – energy difference compared to CCSDT
data (cc-pVTZ basis set). The type of orbitals used in DMRG calculations is
denoted by upper index with $a$ standing for RHF orbitals and $b$ standing for
pCCD orbitals.

Figure 6 shows the potential energy curves for the umbrella inversion of the
NH3 complex, while Table 6 presents the equilibrium angles, barrier hights,
and non-parallelity errors. Both DMRG featuring a small active space and
orbital-optimized pCCD (that is models capturing mostly static/nondynamic
electron correlation) greatly overestimate the barrier height of the umbrella
inversion process. Specifically, the small DMRG active space comprises six
valence electrons distributed in six hybridized bonding and antibonding sp3
orbitals. We also studied a CAS that was extended by two additional sp3
orbitals that feature significant values of the single orbital entropy and
orbital-pair mutual information. This overestimation originates from the fact
that pCCD, DMRG(6,6), and DMRG(8,8) are insufficient to capture dynamical
electron correlation effects. Note that the shape of the PES obtained by
DMRG(8,8)b at $\alpha=90^{\circ}$ features an unphysical shape and diverges.
Thus, the corresponding value for De is not shown in Table 6. For DMRG-tCCSD,
we observe that augmenting the active space deteriorates the accuracy of the
results compared to CCSDT reference energies. Nonetheless, a large fraction of
the missing dynamical correlation energy can be recovered by various tailored
coupled cluster flavours. Most importantly, the differences in the shape of
the PESs, the equilibrium angles, and the barrier height are small with
respect to CCSDT reference data for all investigated tailored CC approaches
and lie within chemical accuracy as long as dynamical correlation effects have
been accounted for in the theoretical model.

Table 6: Equilibrium angles and barrier heights for the umbrella inversion of
NH3. The difference with respect to CCSDT results is given in parentheses. The
superscript $a$ denotes that calculations have been performed in the canonical
RHF orbital basis, while the superscript $b$ indicates pCCD-optimized
orbitals.

| cc-pVDZ | cc-pVTZ
---|---|---
|  $\alpha_{\rm e}$ |  De [$\rm\frac{kcal}{mol}$] |  NPEa [$\rm\frac{kcal}{mol}$] |  $\alpha_{\rm e}$ |  De [$\rm\frac{kcal}{mol}$] |  NPEa [$\rm\frac{kcal}{mol}$]
RHF | 67.1 | ( 1.1 ) | 7.9 | ( -0.7 ) | 1.9 | 68.1 | ( 1.0 ) | 6.4 | ( -0.4 ) | 1.7
CCD | 66.1 | ( 0.2 ) | 8.4 | ( -0.2 ) | 0.4 | 67.4 | ( 0.4 ) | 6.5 | ( -0.4 ) | 0.9
CCSD | 66.1 | ( 0.2 ) | 8.4 | ( -0.2 ) | 0.4 | 67.4 | ( 0.4 ) | 6.5 | ( -0.4 ) | 0.9
CCSD(T) | 65.9 | ( 0.0 ) | 8.6 | ( 0.0 ) | 0.0 | 67.1 | ( 0.0 ) | 6.9 | ( 0.0 ) | 0.0
CCSDT | 65.9 | ( 0.0 ) | 8.6 | ( 0.0 ) |  | 67.1 | ( 0.0 ) | 6.9 | ( 0.0 ) |
pCCD | 66.7 | ( 0.8 ) | 10.0 | ( 1.4 ) | 1.6 | 67.9 | ( 0.8 ) | 8.9 | ( 2.0 ) | 2.1
fpCCD | 66.5 | ( 0.6 ) | 8.6 | ( 0.0 ) | 0.5 | 67.6 | ( 0.5 ) | 6.6 | ( -0.3 ) | 1.3
fpCCSD | 66.3 | ( 0.3 ) | 8.8 | ( 0.2 ) | 0.6 | 67.5 | ( 0.4 ) | 6.9 | ( 0.0 ) | 0.9
fpLCCD | 66.3 | ( 0.4 ) | 8.6 | ( 0.0 ) | 0.7 | 67.6 | ( 0.5 ) | 6.5 | ( -0.3 ) | 1.2
fpLCCSD | 66.2 | ( 0.3 ) | 8.8 | ( 0.2 ) | 0.5 | 67.4 | ( 0.3 ) | 6.9 | ( 0.1 ) | 0.7
DMRG(6,6)a | 68.1 | ( 2.2 ) | 8.7 | ( 0.1 ) | 2.2 | 66.8 | ( -0.3 ) | 6.8 | ( 0.0 ) | 1.1
DMRG(6,6)-tCCSDa | 67.2 | ( 1.3 ) | 8.4 | ( -0.2 ) | 0.3 | 67.2 | ( 0.1 ) | 6.6 | ( -0.3 ) | 0.7
DMRG(6,6)b | 65.8 | ( -0.1 ) | 10.9 | ( 2.2 ) | 0.6 | 65.7 | ( -1.4 ) | 9.0 | ( 2.2 ) | 2.3
DMRG(6,6)-tCCSDb | 65.8 | ( -0.1 ) | 8.5 | ( -0.1 ) | 0.2 | 66.1 | ( -1.0 ) | 6.4 | ( -0.5 ) | 0.8
DMRG(8,8)a | 65.4 | ( -0.6 ) | 9.7 | ( 1.0 ) | 1.0 | 67.2 | ( 0.1 ) | 7.2 | ( 0.4 ) | 0.6
DMRG(8,8)-tCCSDa | 65.4 | ( -0.5 ) | 8.7 | ( 0.0 ) | 0.0 | 67.1 | ( 0.0 ) | 6.6 | ( -0.3 ) | 0.6
DMRG(8,8)b | 66.5 | ( 0.5 ) | - | ( - ) | 2.5 | 68.0 | ( 0.9 ) | - | ( - ) | 2.8
DMRG(8,8)-tCCSDb | 65.6 | ( -0.3 ) | 8.4 | ( -0.2 ) | 0.4 | 67.4 | ( 0.3 ) | 6.1 | ( -0.8 ) | 1.4

a NPE (non-parallelity error) $=\max\limits_{\alpha}(|\Delta
E_{\alpha}|)-\min\limits_{\alpha}(|\Delta E_{\alpha}|)$ and $\Delta
E_{\alpha}=E^{\rm CC}_{\alpha}-E^{\rm CCSDT}_{\alpha}$

###  4.4 Ethylene twist

By twisting the dihedral angle in the ethylene molecule, we can scrutinize the
flexibility of various tCC models to describe varying degrees of strong and
weak electron correlation effects. The ground state molecule in its
equilibrium geometry features D2h symmetry and its electronic wave function is
dominated by a single Slater determinant. The orbital interaction picture
changes when one CH2 group is rotated by ninety degrees and the molecular
point group is hence reduced to D2d symmetry. For this twisted geometry, the
$\pi$ and $\pi^{*}$ orbitals become degenerate and multi-reference approaches
are required to capture static/non-dynamical electron correlation effects.

Figure 7: Potential energy surfaces for the ethylene twist (cc-pVTZ basis set)
obtained by various coupled cluster methods. The superscript $a$ denotes that
calculations have been performed in the canonical RHF orbital basis, while the
superscript $b$ stands for pCCD orbitals.

Figure 7 shows the potential energy curves of the ethylene torsion for various
electronic structure methods. Specifically, RHF as well as single-reference
CCD and CCSD (with canonical RHF orbitals) feature an unphysical cusp in the
PES for a dihedral angle of 90∘. This cusp is a common problem encountered in
the transition state when RHF orbitals are used together with some post-HF
treatment that is not sufficiently accurate to account for (static/nondynamic
and dynamic) electron correlation effects in a balanced way.144 A smooth
potential energy curve can be obtained either with the inclusion of triple
excitations, complete active space calculations containing at least two
electrons and two orbitals in the active space, or with pCCD-tailored coupled
cluster approaches. Although being limited to pair excitations, the pCCD model
correctly predicts that both configurations with (occupied) $\pi$ and
$\pi^{*}$ orbitals are degenerate. Therefore, the orbital-optimized pCCD model
is sufficient to provide a smooth reaction pathway, that is, without a cusp. A
qualitatively correct PES is also provided by DMRG(12,12) calculations for
both a canonical RHF and orbital-optimized pCCD molecular orbital basis. In
general, all tCC flavours improve the shape of the PES and the barrier height
for the twist without introducing unphysical effects as observed in standard
single-reference approaches.

###  4.5 Automerization of cyclobutadiene

The cyclobutadiene molecule features a significant multi-reference character
even in its equilibrium rectangle-shaped geometry. The self-automerization of
cyclobutadiene is a process where the carbon atoms are rearranged so that the
final geometry is similar to the initial geometry, albeit rotated by 90∘.
During the self-automerization process, cyclobutadiene passes through a
transition state where all C–C bonds have equal lengths and the HOMO and LUMO
orbitals are exactly degenerate. For the square geometry, the symmetry
breaking of the RHF wave function affects post-HF treatments.145 Specifically,
single-reference coupled cluster methods tend to underestimate the weight of
one of the two equivalent $\ket{\pi^{1}\pi^{*1}}$ determinants associated with
the C–C double bonds. Previous works demonstrate 146, 145, 147 that
multireference approaches, like state-specific MRCCSD, outperform the so-
called “gold standard” of quantum chemistry CCSD(T). The selection of the
reference wave function is, however, crucial since increasing the size of the
active orbital space can exacerbate the quality of the results.42

Table 7 summarizes the automerization barrier heights obtained from various
conventional and unconventional electronic structure methods. The considerable
difference in the barrier heights predicted by a perturbative treatment of
triple excitations in CCSD(T) compared to full-T calculations indicates that
triple excitations are important and a perturbative treatment is not
sufficient. Thus, in order to achieve reliable results using single-reference
coupled cluster methods, triple excitations are to be included in the cluster
operator. Although it remains difficult to assess whether the CCSDT results
are already converged with respect to the truncation order of the cluster
operator as the difference between CCSD, CCSD(T), and CCSDT energies are
significant, CCSDT provides a reliable description of the automerization
process yielding barrier heights that agree well with experimental results and
MkCCSD reference calculations. In general, all tailored CC flavours restricted
to at most double excitations overestimate the barrier height significantly,
which lies up to 15 kcal/mol beyond the experimental range of 1.6–10
kcal/mol.148, 149 The best performance is observed for DMRG-tCCSD, which
predicts a barrier height between CCSD(T) and CCSDT accuracy. Nonetheless,
DMRG and DMRG-tCCSD feature a strong dependence on the choice of both the
atomic basis set size and the molecular orbitals (differences amount to
approximately 5–9 kcal/mol). The method- and basis-set-dependence is smallest
for (localized) pCCD-optimized orbitals (less than 1 kcal/mol). Note that all
pCCD-tCC methods provide similar barrier heights of approximately 23 kcal/mol.

Table 7: Barrier heights in kcal/mol for the automerization of cyclobutadiene obtained for various quantum chemistry methods and basis sets. The superscript $a$ denotes that calculations have been performed in the canonical RHF orbital basis, while the superscript $b$ stands for pCCD orbitals. | cc-pVDZ | aug-cc-pVDZ
---|---|---
RHF | 28.5 | 27.2
CCSDa | 20.3 | 20.0
CCSD(T)a | 15.4 | 15.7
CCSDTa | 7.3 | 8.1
pCCD | 23.8 | 23.1
fpLCCD | 24.4 | 23.6
fpLCCSD | 24.5 | 23.8
fpCCD | 22.2 | 21.4
fpCCSD | 22.4 | 21.6
DMRG(20,20)a | 10.3 | 19.0
DMRG(20,20)-tCCSDa | 11.1 | 14.9
DMRG(20,20)b | 15.7 | 16.2
DMRG(20,20)-tCCSDb | 16.8 | 17.2
NEVPT2/CAS(20,16)42 | 41.2 |
MkCCSD150 | 7.8 |
exp.148 | 1.6 – 10

###  4.6 Benzene distortion

The equilibrium structure of the benzene molecule features D6h point group
symmetry where all carbon–carbon bond lengths are equivalent. Some approximate
wave function models (including orbital-optimized pCCD 42) tend to distort
this symmetry and thus break aromaticity due to the formation of three partial
double bonds caused by the localization of the three $\pi$\- and
$\pi^{*}$-orbitals. Here, we will assess whether frozen-pair coupled cluster
methods can restore the proper symmetry and hence aromaticity when imposed on
top of the pCCD ansatz. For this purpose, we scrutinize the deformation
pathway of benzene as depicted in Ref. 151 where the distortion is determined
by the difference between the equilibrium and distorted angle defined through
some carbon atom, the center of the molecule, and the neighboring C atom.

Figure 8: Potential energy surfaces for the distortion of benzene using
different tailored and untailored coupled cluster methods (cc-pVTZ basis set).
The superscript $a$ denotes that calculations have been performed in the
canonical RHF orbital basis, while the superscript $b$ stands for pCCD
orbitals.

Figure 8 shows the PESs of the deformation process obtained from pCCD, pCCD-
tLCC, pCCD-tCC, DMRG-tCC, and CCSD exploiting pCCD-optimized orbitals. Similar
to conventional CCSD (with a canonical RHF reference), which does not break
the symmetry of benzene, CCSD, frozen-pair linearized CC, and DMRG-tCCSD
calculations performed on pCCD-optimized orbitals correctly predict the
aromatic structure (D6h point group symmetry) as the minimum geometry.
Although the DMRG(6,6) potential energy curves differ, we obtained almost
exactly the same energies in DMRG(6,6)-tCCSD calculations exploiting both the
pCCD and canonical RHF orbital basis. On the contrary, the fpCCD and fpCCSD
methods anticipate the minimum energy to be associated with some slightly
deformed molecular structure, thus breaking the aromaticity of benzene.
Specifically, fpCCD and fpCCSD predict that the most stable molecular
structure is distorted by approximately 0.2–0.3∘ and lies about 0.09–0.17
kcal/mol lower in energy than the D6h structure. Most importantly, the
predicted symmetry breaking is reduced compared to the pure pCCD ansatz where
the distortion angle reached 1.8∘ and the energy difference between the
aromatic and distorted structures amount to 3.01 kcal/mol.

##  5 Conclusions

In this work, we scrutinized the performance of various coupled cluster
methods tailored by pCCD and DMRG wave functions for various small- and
medium-sized molecules (F2, C2, CN+, BN, BO+, CO, Cr2, ammonia, ethylene,
cyclobutadiene, and benzene) and analyzed the limitations of this
computationally cheap and conceptually simple approaches. Specifically, all
conventional and DMRG-tailored CC calculations were performed for two
different molecular orbital basis sets and thus reference determinants: (i)
delocalized canonical RHF orbitals and (ii) localized pCCD-optimized molecular
orbitals. The active spaces in all DMRG calculations were further selected
using a black-box orbital-selection protocol based on the single-orbital
entropy and orbital-pair mutual information. If pCCD-optimized orbitals are
chosen as molecular orbital basis in DMRG calculations, an entropy- and/or
correlation-based active space can be furthermore straightforwardly and
cheaply selected from the complete set of orbitals as these measures are
readily available at the end of an orbital-optimized pCCD calculation 73. This
feature greatly facilitates and speeds up DMRG calculations with respect to
selecting proper active orbital spaces, DMRG convergence, and computational
cost (that is, the maximum number of bond dimensions $m$ required for energy
convergence).

Tailored CC approaches noticeably improve the quality of spectroscopic
properties, that is equilibrium bond lengths, harmonic vibrational
frequencies, and dissociation energies, in comparison to their single-
reference counterparts CCD and CCSD. Although restricting the cluster operator
to at most double excitations is insufficient to reach the accuracy of the
more expensive CCSDT, MRCC, or MRCI methods, the mean errors in spectroscopic
constants can be reduced to 0.005 Å for bond lengths, 2.5 kcal/mol for
dissociation energies, and 40 cm-1 for harmonic vibrational frequencies. Thus,
tailored coupled cluster theory constitutes a promising alternative to the
conventional CCSD approach if the inclusion of triple and higher excitation
operators is computationally too demanding. Furthermore, for F2, C2, CN+, BN,
BO+, CO, ammonia, and ethylene, we obtained congruent coupled cluster
electron-pair amplitudes and spectroscopic constants for both DMRG- and pCCD-
tailored approached. This indicates that electron-pair amplitudes are
qualitatively well described by the rather simple and cheap pCCD approach.
Statistically, the performance of DMRG-tCCSD in predicting reliable
spectroscopic constants lies between fpCCSD and its linearized version
fpLCCSD, while fpCC(S)D and fpLCC(S)D typically provide results of similar
accuracy (like relative energy differences and spectroscopic constants). The
major drawback of linearized fpCC methods is that divergencies and poles can
be encountered when solving the set of the (linearized) amplitude equations.
These divergencies and poles disappear if non-linear terms are added in the
amplitude equations thus resulting in fpCC-type methods. Finally, the matrix
product state ansatz optimized by the DMRG algorithm is a more flexible
reference wave function for externally-corrected CC methods than pCCD as it
correctly describes the dissociation of the triple bond of the nitrogen dimer
and does not break in the region of avoided crossings. The performance of
DMRG-tCC is, however, dependent on the choice of the molecular orbital basis:
while for some cases (like equilibrium bond lengths, automerization of
cyclobutadiene, benzene distortion) delocalized canonical RHF orbitals yield
results that agree better with reference data (both from experiment and
multireference calculations), DMRG-tCC exploiting localized pCCD-optimized
orbitals performs better for others (like vibrational frequencies,
dissociation energies, Cr2). However, none of the studied tailored CC methods
was able to predict the correct barrier height for the automerization of
cyclobutadiene or to reliably describe the complete potential energy surface
of the chromium dimer. Most likely, full triple excitations are required to
reach chemical accuracy for such challenging systems.

A. L. and K. B. thank for financial support from the National Science Centre,
Poland (SONATA BIS 5 Grant No. 2015/18/E/ST4/00584). A. L. acknowledges
funding from Interdisciplinary Doctoral School Academia Copernicana. M. M. and
Ö. L. acknowledges financial support from the Hungarian National Research,
Development and Innovation Office (Grant Nos. K120569 and K134983), the
Hungarian Quantum Technology National Excellence Program (Grant No.
2017-1.2.1-NKP-2017-00001) and the Hungarian Quantum Information National
Laboratory (QNL). Ö. L. acknowledges financial support from the Alexander von
Humboldt foundation. Calculations have been carried out using resources
provided by Wroclaw Centre for Networking and Supercomputing (http://wcss.pl),
Grant No. 412. The development of the DMRG libraries was supported by the
Center for Scalable and Predictive methods for Excitation and Correlated
phenomena (SPEC), which is funded from the Computational Chemical Sciences
Program by the U.S. Department of Energy (DOE), at Pacific Northwest National
Laboratory.

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