# Diatomic molecules of alkali-metal and alkaline-earth-metal atoms:
interaction potentials,
dipole moments, and polarizabilities

Hela Ladjimi  Faculty of Physics, University of Warsaw, Pasteura 5, 02-093
Warsaw, Poland  Michał Tomza<EMAIL_ADDRESS>Faculty of Physics,
University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

###### Abstract

Ultracold diatomic molecules find application in quantum studies ranging from
controlled chemistry and precision measurement physics to quantum many-body
simulation and potentially quantum computing. Accurate knowledge of molecular
properties is required to guide and explain ongoing experiments. Here, in an
extensive and comparative study, we theoretically investigate the electronic
properties of the ground-state diatomic molecules composed of alkali-metal
(Li, Na, K, Rb, Cs, Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr, Ba, Ra)
atoms. We study 78 hetero- and homonuclear diatomic combinations, including 21
alkali-metal molecules in the $X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$
electronic states, 36 alkali-metal–alkaline-earth-metal molecules in the
$X^{2}\Sigma^{+}$ electronic state, and 21 alkaline-earth-metal molecules in
the $X^{1}\Sigma^{+}$ electronic state. We calculate potential energy curves,
permanent electric dipole moments, and polarizabilities using the hierarchy of
coupled cluster methods upto CCSDTQ with large Gaussian basis sets and small-
core relativistic energy-consistent pseudopotentials. We collect and analyze
corresponding spectroscopic constants. We estimate computational uncertainties
and compare the present values with previous experimental and theoretical data
to establish a new theoretical benchmark. The presented results should be
useful for further application of the studied molecules in modern ultracold
physics and chemistry experiments.

##  I Introduction

Cold and ultracold molecules offer exciting prospects for fundamental quantum
physics and physical chemistry studies as well as for new quantum technology
developments Carr  _et al._ (2009). Within the past two decades, ultracold
polar molecules have been established as great candidates for a plethora of
applications ranging from ultracold controlled chemistry Quemener and Julienne
(2012); Bohn  _et al._ (2017) and precision measurements of fundamental
constants Safronova  _et al._ (2018); DeMille  _et al._ (2017) to quantum
simulation of many-body systems Micheli  _et al._ (2006); Lahaye  _et al._
(2009) and quantum computation DeMille (2002); Ni  _et al._ (2018). One of the
unique features of ultracold polar molecules is the possibility of controlling
their states and intermolecular dipolar interactions with an external electric
field Gadway and Yan (2016). The achieved exquisite control of both molecular
internal quantum states and external motion, enabled by ultralow temperatures
Quemener and Julienne (2012), propels further intensive theoretical and
experimental investigations.

Ultracold molecules can be produced either directly using laser Shuman  _et
al._ (2010); Barry  _et al._ (2014); Truppe  _et al._ (2017), evaporative
Stuhl  _et al._ (2012); Valtolina  _et al._ (2020); Schindewolf  _et al._
(2022); Bigagli  _et al._ (2023), or sympathetic Son  _et al._ (2020) cooling
or indirectly by associating pre-cooled atoms Köhler  _et al._ (2006); Jones
_et al._ (2006). However, ultracold diatomic molecules of alkali-metal and
alkaline-earth-metal atoms have only been produced using indirect methods.
Fortunately, alkali-metal and alkaline-earth-metal atoms can be favorably
laser cooled to ultralow temperatures. Ultracold gases of alkali-metal
molecules such as KRb Ni  _et al._ (2008); Hu  _et al._ (2019), RbCs Takekoshi
_et al._ (2014); Molony  _et al._ (2014), NaK Park  _et al._ (2015); Liu  _et
al._ (2019); Voges  _et al._ (2020); Bause  _et al._ (2021a), NaRb Guo  _et
al._ (2016), and NaCs Stevenson  _et al._ (2023) in the ground rovibrational
level of the singlet electronic state were produced from ultracold atoms by
magnetoassociation using magnetic Feshbach resonances Köhler  _et al._ (2006);
Chin  _et al._ (2010) followed by an optical stabilization using the Stimulate
Adiabatic Raman Passage (STIRAP) Jones  _et al._ (2006); Vitanov  _et al._
(2017). A similar scheme was used to obtain ultracold gases of Rb2 Lang  _et
al._ (2008a), Cs2 Lang  _et al._ (2008b); Danzl  _et al._ (2008), NaLi Rvachov
_et al._ (2017), and Li2 Polovy  _et al._ (2020) molecules in the ground
rovibrational level of the lowest triplet electronic state. Alternatively,
molecules such as RbCs Sage  _et al._ (2005), Cs2 Viteau  _et al._ (2008),
LiCs Deiglmayr  _et al._ (2008a), and recently Sr2 Leung  _et al._ (2021) were
produced in the rovibrational ground state using all-optical photoassociation
schemes. Degenerate Fermi gases of polar KRb De Marco  _et al._ (2019) and NaK
Duda  _et al._ (2023) molecules were formed. The Bose-Einstein condensation of
weakly bound Li2, K2, and Cs2 Feshbach molecules was realized Jochim  _et al._
(2003); Greiner  _et al._ (2003); Zhang  _et al._ (2021), while the
condensation of deeply-bound ground-state molecules remains one of the most
important goals in the field. First steps toward producing ultracold SrRb
Barbé  _et al._ (2018) and SrLi Ye  _et al._ (2020) molecules using narrow
magnetic Feshbach resonances were achieved. Finally, the formation of
ultracold NaCs Liu  _et al._ (2018); Zhang  _et al._ (2020); Yu  _et al._
(2021); Cairncross  _et al._ (2021), Rb2 He  _et al._ (2020), and RbCs Ruttley
_et al._ (2023) molecules was possible at the single-molecule level using
optical tweezers Kaufman and Ni (2021).

Ultracold quantum-controlled chemical reactions were studied in pioneering
experiments with KRb molecules Ospelkaus  _et al._ (2010); Ni  _et al._
(2010); De Miranda  _et al._ (2011); Hu  _et al._ (2019); Liu  _et al._ (2020,
2021); Hu  _et al._ (2021); Tobias  _et al._ (2022), followed by
investigations involving Rb2 Drews  _et al._ (2017), NaRb Guo  _et al._
(2018); Ye  _et al._ (2018); Gersema  _et al._ (2021), NaCs Gregory  _et al._
(2019, 2020), and NaK Bause  _et al._ (2021b); Gersema  _et al._ (2021)
species. Suppressing chemical reactivity and short-range losses was realized
by shielding with electric and microwave fields for KRb Matsuda  _et al._
(2020) and NaK Schindewolf  _et al._ (2022) molecules, respectively. Feshbach
resonances controlled with a magnetic field were demonstrated in ultracold
NaK+K Yang  _et al._ (2019); Su  _et al._ (2022), NaLi+Na Son  _et al._
(2022); Park  _et al._ (2023a), and NaLi+NaLi Park  _et al._ (2023b)
collisions and used to create ultracold weakly-bound NaK2 triatomic molecules
from a NaK+K mixture Yang  _et al._ (2022a, b). Ultracold KRb molecules were
trapped in a three-dimensional optical lattice, and dipolar spin-exchange
interactions between lattice-confined molecules were observed Yan  _et al._
(2013). The first realization of a molecular quantum gas microscope with NaRb
molecules Rosenberg  _et al._ (2022); Christakis  _et al._ (2022) was also
reported. Long-lived coherence of molecular qubits based on ultracold NaK Park
_et al._ (2017) and RbCs Gregory  _et al._ (2021) molecules was shown.
Ultracold KRb molecules were employed for precision measurements of the
variation of the electron-to-proton mass ratio Kobayashi  _et al._ (2019),
while ultracold Sr2 molecules in an optical lattice were established as a
molecular clock for metrology and probing the fundamental laws of nature
McGuyer  _et al._ (2015); McDonald  _et al._ (2016); Kondov  _et al._ (2019).

Molecular formation and application described above would not be possible
without preceding detailed experimental spectroscopic studies and extensive
theoretical ab initio electronic structure calculations of underlying
molecular properties. Different applications require data at different levels
of accuracy. Accurate measurements can ultimately, in most cases, provide more
accurate results than theoretical computations. Nevertheless, ab initio
quantum-chemical calculations of potential energy curves, permanent and
transition electric dipole moments, and fine and hyperfine couplings, used
next in rovibrational and scattering calculations, are often essential to
guide and explain experimental efforts.

Table 1: References to previous experimental and theoretical works on the potential energy curves of the alkali-metal diatomic molecules in the $X^{1}\Sigma^{+}$ or $a^{3}\Sigma^{+}$ electronic state. Molecule | Experiment | Theory
---|---|---
Li2 | Hessel and Vidal (1979); Martin  _et al._ (1988); Abraham  _et al._ (1995); Le Roy  _et al._ (2009); Dattani and Le Roy (2011); Barakat  _et al._ (1986); Xie and Field (1985); Linton  _et al._ (1989, 1999); Semczuk  _et al._ (2013) | Igel-Mann  _et al._ (1986); Boldyrev  _et al._ (1993); Noro  _et al._ (2008); Musiał  _et al._ (2013); Qi  _et al._ (2021); Nasiri and Zahedi (2015); Musiał and Kucharski (2014); Barysz (2016); Jasik and Sienkiewicz (2006); Lesiuk  _et al._ (2020); Nakatsuji and Nakashima (2022); Sheng  _et al._ (2022)
LiNa | Engelke  _et al._ (1982); Fellows  _et al._ (1988); Steinke  _et al._ (2012); Fellows (1991); Steinke  _et al._ (2012); Rvachov  _et al._ (2018) | Igel-Mann  _et al._ (1986); Schmidt-Mink  _et al._ (1984); Boldyrev  _et al._ (1993); Aymar and Dulieu (2005); Fedorov  _et al._ (2014); Mabrouk and Berriche (2008); Bellayouni  _et al._ (2014); Mieszczanin  _et al._ (2014); Mabrouk  _et al._ (2020); Gronowski  _et al._ (2020); Musiał  _et al._ (2021)
LiK | Bednarska  _et al._ (1998); Martin  _et al._ (2001); Salami  _et al._ (2007); Tiemann  _et al._ (2009) | Igel-Mann  _et al._ (1986); Rousseau  _et al._ (1999); Aymar and Dulieu (2005); Fedorov  _et al._ (2014); Bellayouni  _et al._ (2014); Xiao  _et al._ (2013a); Musiał  _et al._ (2016a)
LiRb | Ivanova  _et al._ (2011a) | Igel-Mann  _et al._ (1986); Korek  _et al._ (2000a); Aymar and Dulieu (2005); Jendoubi  _et al._ (2012); Dardouri  _et al._ (2012); Fedorov  _et al._ (2014); Bellayouni  _et al._ (2014)
LiCs | Staanum  _et al._ (2007) | Igel-Mann  _et al._ (1986); Korek  _et al._ (2000b); Aymar and Dulieu (2005); Dardouri  _et al._ (2012); Bellayouni  _et al._ (2014); Fedorov  _et al._ (2014); Mabrouk  _et al._ (2010)
LiFr | - | Bellayouni  _et al._ (2014); Jendoubi (2022); Shundalau and Lamberti (2023)
Na2 | Demtröder  _et al._ (1969); Barrow  _et al._ (1984); Luh  _et al._ (1985); Li  _et al._ (1985); Jones  _et al._ (1996); Samuelis  _et al._ (2000); Ho  _et al._ (2000); Ivanov  _et al._ (2003); Lau  _et al._ (2016); Knoop  _et al._ (2011); Bauer and Toennies (2019) | Partridge  _et al._ (1983); Jeung  _et al._ (1983); Magnier  _et al._ (1993); Igel-Mann  _et al._ (1986); Neogrády  _et al._ (2005); Noro  _et al._ (2008); Musiał  _et al._ (2013); Barysz (2016); Zhang  _et al._ (2016); Soorkia  _et al._ (2007); Musiał  _et al._ (2022)
NaK | Wormsbecher  _et al._ (1981); Ross  _et al._ (1985); Kowalczyk (1989); Ishikawa  _et al._ (1994); Ferber  _et al._ (2000); Russier-Antoine  _et al._ (2000); Gerdes  _et al._ (2008, 2011); Schwarzer and Toennies (2021); Temelkov  _et al._ (2015); Hartmann  _et al._ (2019) | Igel-Mann  _et al._ (1986); Aymar and Dulieu (2005, 2007); Allouche and Aubert-Frécon (2011); Fedorov  _et al._ (2014); Musiał  _et al._ (2016b); Zhang  _et al._ (2019); Segovia and Ventura (2019); Baczek  _et al._ (2022)
NaRb | Kasahara  _et al._ (1996); Docenko  _et al._ (2004a, 2002); Pashov  _et al._ (2005); Wang  _et al._ (1991); Takahashi and Katô (1981); Zemke and Stwalley (2001) | Igel-Mann  _et al._ (1986); Korek  _et al._ (2000a); Zaitsevskii  _et al._ (2001); Aymar and Dulieu (2005, 2007); Korek and Fawwaz (2009); Dardouri  _et al._ (2012); Wiatr  _et al._ (2018); Fedorov  _et al._ (2014); Chaieb  _et al._ (2014); Wiatr  _et al._ (2015)
NaCs | Docenko  _et al._ (2004b, 2006) | Igel-Mann  _et al._ (1986); Korek  _et al._ (2000b); Aymar and Dulieu (2005); Dardouri  _et al._ (2012); Mabrouk and Berriche (2014); Fedorov  _et al._ (2014); Aymar and Dulieu (2007); Schwarzer and Toennies (2021); Korek  _et al._ (2007)
NaFr | - | Jellali  _et al._ (2018)
K2 | Luh  _et al._ (1985); Ross  _et al._ (1986); Heinze  _et al._ (1987); Li  _et al._ (1990); Amiot (1991); Zemke  _et al._ (1994); Amiot  _et al._ (1995); Zhao  _et al._ (1996); Zavitsas (2006); Falke  _et al._ (2008); Pashov  _et al._ (2008); Lau  _et al._ (2016); Tiemann  _et al._ (2020) | Partridge  _et al._ (1983); Engelke  _et al._ ; Igel-Mann  _et al._ (1986); Krauss and Stevens (1990); Lim  _et al._ (2005); Noro  _et al._ (2008); Jraij  _et al._ (2009); Pershina  _et al._ (2012)
KRb | Ross  _et al._ (1990); Amiot and Vergés (2000); Pashov  _et al._ (2007); Stwalley (2005); Wang  _et al._ (2007); Schwarzer and Toennies (2020) | Igel-Mann  _et al._ (1986); Rousseau  _et al._ (2000); Park  _et al._ (2000); Kotochigova  _et al._ (2003); Aymar and Dulieu (2005); Soldán and Spirko (2007); Jellali and Habli (2021); Guo  _et al._ (2013); Fedorov  _et al._ (2014); Stwalley  _et al._ (2010); Chen  _et al._ (2012); Shundalau  _et al._ (2016)
KCs | Ferber  _et al._ (2008, 2009, 2013); Schwarzer and Toennies (2021); Krumins  _et al._ (2022a, b) | Igel-Mann  _et al._ (1986); Korek  _et al._ (2000b); Aymar and Dulieu (2005); Kotochigova and Tiesinga (2005); Korek  _et al._ (2006); Kim  _et al._ (2009); Habli  _et al._ (2020)
KFr | - | -
Rb2 | Caldwell  _et al._ (1980); Amiot  _et al._ (1985); Amiot (1990); Seto  _et al._ (2000); Beser  _et al._ (2009); Strauss  _et al._ (2010); Guan  _et al._ (2013); Lau  _et al._ (2016) | Partridge  _et al._ (1983); Spiegelmann  _et al._ (1989); Igel-Mann  _et al._ (1986); Krauss and Stevens (1990); Foucrault  _et al._ (1992); Park  _et al._ (2001); Edvardsson  _et al._ (2003); Lim  _et al._ (2005); Aymar  _et al._ (2006); Noro  _et al._ (2008); Pershina  _et al._ (2012); Allouche and Aubert-Frécon (2012); Tomza  _et al._ (2012, 2013a); Hill and Peterson (2017)
RbCs | Kato and Kobayashi (1983); Gustavsson  _et al._ (1988); Fellows  _et al._ (1999); Docenko  _et al._ (2011); Takekoshi  _et al._ (2012); Schwarzer and Toennies (2021) | Igel-Mann  _et al._ (1986); Pavolini  _et al._ (1989); Allouche  _et al._ (2000); Fahs  _et al._ (2002); Aymar and Dulieu (2005); Aymar  _et al._ (2006); Fedorov  _et al._ (2014)
RbFr | - | Aymar  _et al._ (2006); Jendoubi  _et al._ (2016)
Cs2 | Weickenmeier  _et al._ (1985); Amiot  _et al._ (1988); Amiot and Dulieu (2002); Xie  _et al._ (2009); Coxon and Hajigeorgiou (2010); Xie  _et al._ (2011); Sainis  _et al._ (2012); Sovkov  _et al._ (2013); Lau  _et al._ (2016); Sovkov  _et al._ (2017) | Walch  _et al._ (1982); Jeung  _et al._ (1983); Laskowski and Langhoff (1982); Igel-Mann  _et al._ (1986); Moullet  _et al._ (1989); Krauss and Stevens (1990); Foucrault  _et al._ (1992)
|  | Lim  _et al._ (2005); Aymar  _et al._ (2006); Noro  _et al._ (2008); Allouche and Aubert-Frécon (2012); Pershina  _et al._ (2012); Hill and Peterson (2017)
CsFr | - | Aymar  _et al._ (2006); Alyousef  _et al._ (2020)
Fr2 | - | Aymar  _et al._ (2006); Roos  _et al._ (2004); Lim  _et al._ (2005); Noro  _et al._ (2008); Pershina  _et al._ (2012); Campos  _et al._ (2017); Hill and Peterson (2017); Jellali and Habli (2022); Neto  _et al._ (2022)

In this work, we use state-of-the-art ab initio electronic structure methods
to calculate the potential energy curves, permanent electric dipole moments,
and static electric dipole polarizabilities for all the hetero- and
homonuclear diatomic molecules composed of alkali-metal (Li, Na, K, Rb, Cs,
Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr, Ba, Ra) atoms. We employ the
hierarchy of coupled cluster methods upto CCSDTQ with large Gaussian basis
sets and small-core relativistic energy-consistent pseudopotentials. We study
a large number of 78 hetero- and homonuclear diatomic combinations, including
21 alkali-metal molecules in the $X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$
electronic states, 36 alkali-metal–alkaline-earth-metal molecules in the
$X^{2}\Sigma^{+}$ electronic state, and 21 alkaline-earth-metal molecules in
the $X^{1}\Sigma^{+}$ electronic state. We also analyze the convergence and
accuracy of our calculations with the size of the orbital basis sets and the
quality of the wave functions. In this way, we establish a new theoretical
benchmark.

Table 2: References to previous experimental and theoretical works on the potential energy curves of the alkali-metal–alkaline-earth-metal diatomic molecules in the $X^{2}\Sigma^{+}$ electronic state. Molecule | Experiment | Theory
---|---|---
LiBe | Persinger  _et al._ (2021a) | Jones (1980); Safonov  _et al._ (1983); Schlachta  _et al._ (1990); Fischer  _et al._ (1991); Pak  _et al._ (1991); Marino  _et al._ (1992); Bauschlicher Jr  _et al._ (1992); Boldyrev  _et al._ (1993); Kotochigova  _et al._ (2011); You  _et al._ (2015a); Pototschnig  _et al._ (2016); Koput (2022)
NaBe | - | Bauschlicher Jr  _et al._ (1992); Pototschnig  _et al._ (2016); Chmaisani  _et al._ (2017)
KBe | - | Xiao  _et al._ (2013b); Pototschnig  _et al._ (2016); Wan  _et al._ (2017); Awad  _et al._ (2020)
RbBe | - | Pototschnig  _et al._ (2016); You  _et al._ (2015b); Awad  _et al._ (2020)
CsBe | - | You  _et al._ (2015b); Awad  _et al._ (2020)
FrBe | - | -
LiMg | Berry and Duncan (1997); Persinger  _et al._ (2021b) | Jones (1980); Fantucci  _et al._ (1989); Bauschlicher Jr  _et al._ (1992); Boldyrev  _et al._ (1993); Kotochigova  _et al._ (2011); Gopakumar  _et al._ (2011); Augustovičová and Soldán (2012); Gao and Gao (2014); Pototschnig  _et al._ (2016); Houalla  _et al._ (2019)
NaMg | - | Benard and Michels (1982); Fantucci  _et al._ (1989); Bauschlicher Jr  _et al._ (1992); Augustovičová and Soldán (2012); Pototschnig  _et al._ (2016); Li  _et al._ (2018); Houalla  _et al._ (2019)
KMg | - | Augustovičová and Soldán (2012); Houalla  _et al._ (2017); Pototschnig  _et al._ (2016)
RbMg | - | Augustovičová and Soldán (2012); Houalla  _et al._ (2017); Pototschnig  _et al._ (2016)
CsMg | - | Augustovičová and Soldán (2012); Houalla  _et al._ (2017)
FrMg | - | -
LiCa | Wu  _et al._ (1983); Russon  _et al._ (1998); Ivanova  _et al._ (2011b); Krois  _et al._ (2013) | Neumann  _et al._ (1980); Jones (1980); Allouche and Aubert-Frécon (1994a); Russon  _et al._ (1998); Kotochigova  _et al._ (2011); Gopakumar  _et al._ (2011, 2013, 2014); Pototschnig  _et al._ (2016, 2017)
NaCa | - | Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2016, 2017); Moussa  _et al._ (2022)
KCa | Gerschmann  _et al._ (2017) | Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2016, 2017); Moussa  _et al._ (2021)
RbCa | - | Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2016, 2015, 2017)
CsCa | - | Moussa  _et al._ (2022)
FrCa | - | -
LiSr | Schwanke  _et al._ (2017) | Guérout  _et al._ (2010); Gopakumar  _et al._ (2011); Kotochigova  _et al._ (2011); Gopakumar  _et al._ (2013); Guo  _et al._ (2013); Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2016, 2017); Zeid  _et al._ (2018)
NaSr | - | Guérout  _et al._ (2010); Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2016, 2017); Zeid  _et al._ (2018)
KSr | Szczepkowski  _et al._ (2018) | Guérout  _et al._ (2010); Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2016, 2017); Szczepkowski  _et al._ (2018); Zeid  _et al._ (2018)
RbSr | Pototschnig  _et al._ (2014); Krois  _et al._ (2014); Lackner  _et al._ (2014); Ciamei  _et al._ (2018) | Żuchowski  _et al._ (2010); Guérout  _et al._ (2010); Chen  _et al._ (2014); Pototschnig  _et al._ (2014); Żuchowski  _et al._ (2014); Pototschnig  _et al._ (2016); Gopakumar  _et al._ (2014); Pototschnig  _et al._ (2017); Devolder  _et al._ (2018)
CsSr | - | Guérout  _et al._ (2010)
FrSr | - | -
LiBa | Stringat  _et al._ (1994) | Allouche and Aubert-Frécon (1994b); Gopakumar  _et al._ (2011); Gou  _et al._ (2015)
NaBa | - | Boutassetta  _et al._ (1994); Gou  _et al._ (2015)
KBa | - | Boutassetta  _et al._ (1995); Gou  _et al._ (2015)
RbBa | - | Gou  _et al._ (2015); Ladjimi  _et al._ (2020)
CsBa | - | Gou  _et al._ (2015); Ladjimi  _et al._ (2019)
FrBa | - | -
LiRa | - | Fleig and DeMille (2021)
NaRa | - | Fleig and DeMille (2021)
KRa | - | Fleig and DeMille (2021)
RbRa | - | Fleig and DeMille (2021)
CsRa | - | Fleig and DeMille (2021)
FrRa | - | Fleig and DeMille (2021)
Table 3: References to previous experimental and theoretical works on the potential energy curves of the alkaline-earth-metal diatomic molecules in the $X^{1}\Sigma^{+}$ electronic state. Molecule | Experiment | Theory
---|---|---
Be2 | Kaledin  _et al._ (1999); Merritt  _et al._ (2009); Špirko (2006); Meshkov  _et al._ (2014); Bondybey (1984) | Liu and McLean (1980); Chiles and Dykstra (1981); Kaledin  _et al._ (1999); Røeggen and Veseth (2005); Zhao and Truhlar (2006); Patkowski  _et al._ (2007, 2009); Schmidt  _et al._ (2010); Heaven  _et al._ (2011); Koput (2011); Li  _et al._ (2011); Jeung (2012); Helal  _et al._ (2013); Sheng  _et al._ (2013); Sharma  _et al._ (2014); El Khatib  _et al._ (2014); Lesiuk  _et al._ (2015); Deible  _et al._ (2015); Kalemos (2016); Magoulas  _et al._ (2018); Lesiuk  _et al._ (2019); Bao  _et al._ (2019); Cheng  _et al._ (2019); Xu and Dunning Jr (2020); Derbov  _et al._ (2021); Guther  _et al._ (2021)
BeMg | - | Chiles and Dykstra (1982); Borin and Rodrigues (2003); Ruette  _et al._ (2005); Heaven  _et al._ (2011); Kerkines and Nicolaides (2012); Mahapatra  _et al._ (2015)
BeCa | - | Heaven  _et al._ (2011)
BeSr | - | Heaven  _et al._ (2011)
BeBa | - | Heaven  _et al._ (2011)
BeRa | - | -
Mg2 | Balfour and Douglas (1970); Vidal and Scheingraber (1977); Wu  _et al._ (1983); Knöckel  _et al._ (2013) | Chiles and Dykstra (1981); Partridge  _et al._ (1990); Dyall and McLean (1992); Czuchaj  _et al._ (2001); Tiesinga  _et al._ (2002); Zhao and Truhlar (2006); Patkowski  _et al._ (2007); Li  _et al._ (2010); Heaven  _et al._ (2011); Li  _et al._ (2011); Amaran  _et al._ (2013); Yuwono  _et al._ (2019); Bao  _et al._ (2019); Cheng  _et al._ (2019); Yuwono  _et al._ (2020); Dyall  _et al._ (2023)
MgCa | Atmanspacher  _et al._ (1985) | Heaven  _et al._ (2011); Wei  _et al._ (2015); Rizkallah  _et al._ (2021)
MgSr | - | Heaven  _et al._ (2011); Wei  _et al._ (2015)
MgBa | - | Heaven  _et al._ (2011); Wei  _et al._ (2015)
MgRa | - | -
Ca2 | Balfour and Whitlock (1975); Wyss (1979); Vidal (1980); Wu  _et al._ (1983); Allard  _et al._ (2002, 2003) | Dyall and McLean (1992); Czuchaj  _et al._ (2003a); Bussery-Honvault  _et al._ (2003); Zhao and Truhlar (2006); Patkowski  _et al._ (2007); Yang  _et al._ (2009); Bouissou  _et al._ (2010); Heaven  _et al._ (2011); Li  _et al._ (2011); Yang and Wang (2012); Mosyagin  _et al._ (2013); Bao  _et al._ (2019); Cheng  _et al._ (2019)
CaSr | - | Heaven  _et al._ (2011); Wei  _et al._ (2015)
CaBa | - | Heaven  _et al._ (2011); Wei  _et al._ (2015)
CaRa | - | -
Sr2 | Bergeman and Liao (1980); Gerber  _et al._ (1984); Stein  _et al._ (2008, 2010) | Boutassetta  _et al._ (1996); Wang  _et al._ (2000); Czuchaj  _et al._ (2003b); Kotochigova (2008); Mitin (2009); Yin  _et al._ (2010); Li  _et al._ (2011); Heaven  _et al._ (2011); Skomorowski  _et al._ (2012); Yang and Wang (2012); Bao  _et al._ (2019); Cheng  _et al._ (2019); Szczepkowski  _et al._ (2023); Dyall  _et al._ (2023)
SrBa | - | Heaven  _et al._ (2011); Wei  _et al._ (2015)
SrRa | - | -
Ba2 | Lebeault  _et al._ (1998) | Allouche  _et al._ (1995); Mitin (2009); Li  _et al._ (2011); Heaven  _et al._ (2011); Yang and Wang (2012); Bao  _et al._ (2019); Cheng  _et al._ (2019)
BaRa | - | -
Ra2 | - | Roos  _et al._ (2004); Teodoro  _et al._ (2015); Campos  _et al._ (2017); Bao  _et al._ (2019); Neto  _et al._ (2022)

A significant portion of the molecules under investigation has already been
studied experimentally or theoretically. Notably, alkali-metal dimers are one
of the most extensively studied classes of molecules. However, alkali-
metal–alkaline-earth-metal and alkaline-earth-metal molecules have received
less attention, particularly in experimental studies, with a large number of
such combinations without any experimental data. In the following sections, we
will compare our results with the most recent and accurate experimental and
theoretical values. Because of the large number of studied molecules,
refereeing and comparing to all previous theoretical results is not feasible.
Therefore, we have collected references to previous experimental and
theoretical works on the potential energy curves of the alkali-metal–alkali-
metal, alkali-metal–alkaline-earth-metal, and alkaline-earth-metal–alkaline-
earth-metal diatomic molecules in their ground electronic states in Tables 1,
2, and 3, respectively.

Experimental works in this field can be categorized into several classes,
including: 1) laser photoionization spectroscopy, 2) laser-induced
fluorescence spectroscopy, 3) polarisation labeling spectroscopy in hot vapors
or beams, 4) spectroscopy of molecules on helium nanodroplets, and 5) highly
accurate measurements in ultracold atomic or molecular gases in traps.
Previous theoretical works can also be classified into several groups,
including: 1) the oldest results, often without including the electron
correlation, 2) the results with large-core pseudopotentials and the full
conﬁguration interaction method for valence electrons, 3) the results with
small-core (scalar-relativistic) pseudopotentials and truncated coupled
cluster or multireference configuration interactions methods, and 4) all-
electron calculations often with relativistic Hamiltonians.

The main advancements of the present work lie in the following:

  * •

the coherent calculation and comparison of three classes of experimentally
relevant molecules at the consistent level of theory,

  * •

the computation of electronic properties of some molecules, such as those
containing francium and radium, for the first time,

  * •

the investigation of permanent electric dipole moments of heteronuclear
alkaline-earth molecules, for the first time,

  * •

the use of large recently developed Gaussian basis sets Hill and Peterson
(2017),

  * •

the inclusion of full triple and quadruple excitations in the coupled cluster
method to obtain potential energy curves, resulting in the description of
valence electrons at the full configuration interaction level for all
molecules,

  * •

the publication of all results in a numerical form in the Supplemental
Material sup .

For completeness, it is worth mentioning the existence of the studies of
isoelectronic diatomic molecules containing: 1) alkaline-earth-metal-like
ytterbium atom Buchachenko  _et al._ (2007); Kitagawa  _et al._ (2008); Tecmer
_et al._ (2019), including its combinations with alkali-metal Sorensen  _et
al._ (2009); Zhang  _et al._ (2010); Gopakumar  _et al._ (2010); Münchow  _et
al._ (2011); Tohme and Korek (2013); Borkowski  _et al._ (2013); Tohme  _et
al._ (2015); Tohme and Korek (2015, 2016); Shao  _et al._ (2017); Meniailava
and Shundalau (2017); Shundalau and Minko (2017); Guttridge  _et al._ (2018);
Green  _et al._ (2019) or alkaline-earth-metal Tomza  _et al._ (2011); Shao
_et al._ (2017) atom, 2) alkaline-earth-metal-like zinc, cadmium, or mercury
atom with alkali-metal or alkaline-earth-metal atom Borkowski  _et al._
(2017); Zaremba-Kopczyk and Tomza (2021), and 3) alkali-metal-like copper,
silver, or gold atom with alkali-metal or alkaline-earth-metal atom
Śmiałkowski and Tomza (2021); Kłos  _et al._ (2022); Xiao  _et al._ (2022).
Finally, diatomic molecular ions of alkali-metal and alkaline-earth-metal
atoms have also been investigated extensively (see, e.g., Ref. Śmiałkowski and
Tomza (2020) and references therein). Such systems are, however, out of the
scope of this investigation.

The structure of the paper is the following. In Section II, we describe the
employed computational methods. In Section III, we present and discuss the
obtained results. In section IV, we provide a summary and outlook.

##  II COMPUTATIONAL METHODS

We adopt and employ the computational scheme based on the composite approach
to calculate potential energy curves in the Born-Oppenheimer approximation for
the relevant molecular electronic states, which we established and tested in
several previous studies on different classes of molecules containing alkali-
metal and alkaline-earth-metal atoms Gronowski  _et al._ (2020); Śmiałkowski
and Tomza (2020, 2021); Zaremba-Kopczyk and Tomza (2021). The final
interaction energies, $V_{\text{int}}(R)$, as a function of the internuclear
distance $R$, are calculated as a sum of different contributions

$\begin{split}V_{\text{int}}(R)=&V_{\text{CCSD(T)}}^{\text{apwCV5Z+bf}}(R)+\delta
V_{\text{CCSDT}}^{\text{apwCVTZ}}(R)\\\ &+\delta
V_{\text{CCSDTQ,val}}^{\text{apVTZ}}(R)\,,\end{split}$ (1)

where the leading part, $V_{\text{CCSD(T)}}^{\text{apwCV5Z+bf}}(R)$, is
obtained with the closed-shell (for the $X^{1}\Sigma^{+}$ states) or spin-
restricted open-shell (for the $X^{2}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ states)
coupled cluster method restricted to single, double, and noniterative triple
excitations [CCSD(T)] Bartlett and Musial (2007) and the augmented
correlation-consistent polarized weighted core-valence quintuple-$\zeta$
quality basis sets (aug-cc-pwCV5Z) Prascher  _et al._ (2010); Hill and
Peterson (2017). The atomic basis sets are additionally augmented in these
calculations by the set of the $[4s4p3d3f1g]$ bond functions (bf) 111Bond
function exponents, $s$: 0.6, 0.2, 0.067, 0.02, $p$: 0.6, 0.2, 0.067, 0.02,
$d$: 0.4, 0.13, 0.04, $f$: 0.4, 0.13, 0.04, $g$: 0.2. to accelerate the
convergence toward the complete basis set limit Tao and Pan (1992).

The next leading electron-correlation correction to the interaction energy, to
account for the contribution of the iterative full triple excitations in the
coupled cluster method, $\delta V_{\text{CCSDT}}^{\text{apwCVTZ}}(R)$, is
obtained as

$\delta
V_{\text{CCSDT}}^{\text{apwCVTZ}}(R)=V_{\text{CCSDT}}^{\text{apwCVTZ}}(R)-V_{\text{CCSD(T)}}^{\text{apwCVTZ}}(R)\,,$
(2)

where $V_{\text{CCSDT}}^{\text{apwCVTZ}}(R)$ is the interaction energy
calculated with the coupled cluster method restricted to single, double, and
triple excitations (CCSDT) and $V_{\text{CCSD(T)}}^{\text{apwCVTZ}}(R)$ – with
the CCSD(T) method, both with the same augmented correlation-consistent
polarized weighted core-valence triple-$\zeta$ quality basis sets (aug-cc-
pwCVTZ) Prascher  _et al._ (2010); Hill and Peterson (2017).

Finally, the valence electron-correlation correction to the interaction energy
to account for the contribution of the quadruple excitations in the coupled
cluster method, $\delta V_{\text{CCSDTQ,val}}^{\text{apVTZ}}(R)$, is obtained
as

$\delta
V_{\text{CCSDTQ,val}}^{\text{apVTZ}}(R)=V_{\text{CCSDTQ,val}}^{\text{apVTZ}}(R)-V_{\text{CCSDT,val}}^{\text{apCVZ}}(R)\,,$
(3)

where $V_{\text{CCSDTQ,val}}^{\text{apVTZ}}(R)$ is the interaction energy
calculated with the coupled cluster method restricted to single, double,
triple, and quadruple excitations (CCSDTQ) and
$V_{\text{CCSDT,val}}^{\text{apCVZ}}(R)$ – with the CCSDT method, both with
the same augmented correlation-consistent polarized valence triple-$\zeta$
quality basis sets (aug-cc-pVTZ) Prascher  _et al._ (2010); Hill and Peterson
(2017). In these calculations, only valence electrons are correlated,
therefore $\delta V_{\text{CCSDTQ,val}}^{\text{apVTZ}}(R)$ is identically
equal to zero for alkali-metal and alkali-metal–alkaline-earth-metal molecules
with two and three valence electrons, respectively.

The interaction energies, $V_{method}^{basis}(R)$, in Eqs. (1)-(3) are
obtained using the super-molecule approach with the basis set superposition
error (BSSE) corrected by using the Boys-Bernardi counterpoise correction Boys
and Bernardi (1970),

$V_{method}^{basis}(R)=E_{AB}(R)-E_{A}(R)-E_{B}(R)\,,$ (4)

where $E_{AB}(R)$ is the total energy of the molecule $AB$, and $E_{A}(R)$ and
$E_{B}(R)$ are the total energies of the atoms $A$ and $B$, all computed with
the given $method$ and diatom $basis$ set at a distance $R$.

The employed computational scheme given by Eq. (1) works very well at all
internuclear distances for all molecules in the considered doublet
$X^{2}\Sigma^{+}$ and triplet $a^{3}\Sigma^{+}$ molecular electronic states
and alkaline-earth-metal molecules in the $X^{1}\Sigma^{+}$ states because
they are well described at all internuclear distances by single-reference
methods. On the other hand, the singlet $X^{1}\Sigma^{+}$ molecular electronic
states of the alkali-metal molecules have single-reference nature at small and
intermediate internuclear distances and multireference nature at larger
distances, which originates from the open-shell character of the interacting
alkali-metal atoms. At larger distances, the single-reference CCSD(T) method
gives incorrect results, and the CCSDT correction only partially improves
them. Therefore, we compute the $X^{1}\Sigma^{+}$ states of alkali-metal
molecules with the coupled cluster methods in the vicinity of the interaction
potential well at short and intermediate distances and smoothly merge them
with results obtained with the internally contracted multireference
configuration interaction method restricted to single and double excitations
with the active space of all valence orbitals and including the Davidson
corrections (MRCISD+Q) Knowles and Werner (1992) at larger distances. The
MRCISD+Q results are shifted to impose correct asymptotic energies. Depending
on the system, the switching distance is around 11-14 bohr.

All electrons in the Li, Be, Na, and Mg atoms and outer-shell electrons in
other atoms are explicitly described by the selected large atomic augmented
correlation-consistent one-electron Gaussian basis sets, while the inner-shell
electrons in heavier atoms are replaced by the small-core relativistic energy-
consistent pseudopotentials (ECP) Dolg and Cao (2012) to include the scalar
relativistic effects. The ECP10MDF, ECP10MDF, ECP28MDF, ECP28MDF, ECP46MDF,
ECP46MDF, ECP78MDF, and ECP78MDF pseudopotentials Lim  _et al._ (2005, 2006)
are used for the K, Ca, Rb, Sr, Cs, Ba, Fr, and Ra atoms, respectively. The
electrons of two outermost shells, i.e., $(n-1)s^{2}(n-1)p^{6}{n}s^{1}$ from
alkali-metal and $(n-1)s^{2}(n-1)p^{6}{n}s^{2}$ from alkaline-earth-metal
atoms, are correlated in calculations with the aug-cc-pwCV$n$Z basis sets,
while only valence electrons, i.e., ${n}s^{1}$ from alkali-metal and
${n}s^{2}$ from alkaline-earth-metal atoms – with the aug-cc-pV$n$Z basis
sets.

The composite approach of Eq. (1) rely on error cancellation observed in high-
level molecular electronic structure calculations Lesiuk  _et al._ (2019);
Gronowski  _et al._ (2020); Lesiuk  _et al._ (2020). Effectively, for three
classes of molecules, we describe their valence electrons at the full
configuration interaction (FCI) level because CCSD, CCSDT, and CCSDTQ are
equivalent to FCI for two-, three-, and four-electron systems, respectively.
In order to evaluate the accuracy of the used approach, additional test
calculations are carried out for exemplary KRb, RbSr, and CaSr molecules.
Different basis sets from the aug-cc-pwCV$n$Z and aug-cc-pV$n$Z families with
$n$=D, T, Q, 5 are employed and extrapolate the complete basis set limit (CBS)
using the two-point formula Helgaker  _et al._ (1997). Additionally, potential
energy curves are obtained with a series of different less accurate methods
Helgaker  _et al._ (2000): spin-restricted Hartree-Fock (RHF), second-order
Møller-Plesset perturbation theory (MP2), coupled cluster method restricted to
single and double excitations (CCSD), configuration interaction method
restricted to single and double excitations (CISD) and its variant including
the Davidson correction (CISD+Q), as well as multireference variants of the
CISD and CISD+Q methods (MRCISD and MRCISD+Q), respectively.

Figure 1: Potential energy curves of all the alkali-metal diatomic molecules
in the ground $X^{1}\Sigma^{+}$ electronic state.

We interpolate the potential energy curves, $V_{\text{int}}(R)$, using the
cubic spline method to obtain spectroscopic parameters. The equilibrium
distance, $R_{e}$, is defined by

$\left.\frac{dV_{\text{int}}(R)}{dR}\right|_{R_{e}}=0\,,$ (5)

and the corresponding potential energy well depth, $D_{e}$, is given by

$D_{e}=-V_{\text{int}}(R_{e})\,.$ (6)

The harmonic constant, $\omega_{e}$, of the interaction potential is
calculated at the equilibrium distance as

$\omega_{e}=\sqrt{\frac{1}{\mu}\left.\frac{d^{2}V_{\text{int}}(R)}{dR^{2}}\right|_{R_{e}}}\,,$
(7)

where $\mu$ represents the reduced mass. The equilibrium rotational constant,
$B_{e}$, is given as

$B_{e}=\frac{\hbar^{2}}{2\mu R_{e}}\,.$ (8)

Finally, the first anharmonicity constant, $\omega_{e}x_{e}\approx-Y_{20}$, is
obtained by fitting the Dunham expansion to the lowest vibrational energy
levels obtained with the discrete variable representation (DVR).

The permanent electric dipole moments $d(R)$ and static electric dipole
polarizabilities $\alpha(R)$ are calculated with the finite field approach
using the CCSD(T) method and the aug-cc-pwCV5Z+bf basis sets. The $z$ axis is
chosen along the internuclear axis and oriented from more electronegative
atoms to less electronegative ones. The value of the external field
perturbation equal to $\pm$0.0005 a.u. is used.

All electronic structure calculations are performed with the Molpro Werner
_et al._ (2012a, b) and MRCC Kállay  _et al._ (2020) packages of ab initio
programs. Atomic masses of the most abundant isotopes are assumed.

##  III RESULTS AND DISCUSSION

Figure 2: Potential energy curves of all the alkali-metal diatomic molecules
in the $a^{3}\Sigma^{+}$ electronic state. Figure 3: Potential energy curves
of all the alkali-metal–alkaline-earth-metal diatomic molecules in the ground
$X^{2}\Sigma^{+}$ electronic state. Figure 4: Potential energy curves of all
the alkaline-earth-metal diatomic molecules in the ground $X^{1}\Sigma^{+}$
electronic state.

The interaction of two open-shell alkali-metal atoms in their ground doublet
${}^{2}S$ electronic state results in the ground molecular electronic state of
the singlet $X^{1}\Sigma^{+}$ symmetry and the first excited electronic state
of the triplet $a^{3}\Sigma^{+}$ symmetry. The interaction of two closed-shell
alkaline-earth-metal atoms in their ground single ${}^{1}S$ electronic state
results in the ground molecular electronic state of the singlet
$X^{1}\Sigma^{+}$ symmetry. Finally, the interaction of an alkali-metal atom
with an alkaline-earth-metal atom results in the ground molecular electronic
state of the doublet $X^{2}\Sigma^{+}$ symmetry. Additionally, homonuclear
dimers have the gerade and ungerade symmetries in the singlet
($X^{1}\Sigma^{+}_{g}$) and triplet ($a^{3}\Sigma^{+}_{u}$) states,
respectively.

###  III.1 Potential energy curves

The computed potential energy curves for the alkali-metal diatomic molecules
in the $X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ electronic states are presented
in Fig. 1 and Fig. 2, respectively, for the alkali-metal–alkaline-earth-metal
diatomic molecules in the $X^{2}\Sigma^{+}$ electronic state – in Fig. 3, and
for the alkaline-earth-metal diatomic molecules in the $X^{1}\Sigma^{+}$ state
– in Fig. 4. Calculations are performed for all the combinations of the
alkali-metal (Li, Na, K, Rb, Cs, Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr,
Ba, Ra) atoms. All potential energy curves show a smooth behavior with well-
defined minima. The corresponding spectroscopic characteristics such as the
equilibrium interatomic distance $R_{e}$, well depth $D_{e}$, harmonic
constant $\omega_{e}$, first anharmonicity constant $\omega_{e}x_{e}$, and
rotational constant $B_{e}$ are collected in Tables 4-7, along with available
experimental spectroscopic data.

####  III.1.1 Alkali-metal diatomic molecules

The alkali-metal diatomic molecules in their ground $X^{1}\Sigma^{+}$
electronic state are relatively strongly chemically bound. Formally, they are
covalently bound with a bond order of one. The calculated well depths
systematically decrease with increasing the atomic number of the alkali-metal
atoms and range from 3333 cm-1 for Fr2 to 8509 cm-1 for Li2 (with an average
of 3892 cm-1 calculated over a set of all molecules of their type). The
corresponding equilibrium distances systematically increase with increasing
the atomic number of the alkali-metal atoms and take values between 5.05 bohr
for Li2 and 8.80 bohr for Cs2 (with an average of 7.31 bohr).

The alkali-metal diatomic molecules in their first excited $a^{3}\Sigma^{+}$
electronic state with two valence electrons spin-polarized are weakly bound
van der Waals complexes. Formally, they are not chemically bound with a bond
order of zero and are stabilized by the dispersion interaction. The calculated
well depths range from 198 cm-1 for Fr2 to 334 cm-1 for Li2 (with an average
of 239 cm-1), displaying less variation and regularity than in their ground
$X^{1}\Sigma^{+}$ states. The corresponding equilibrium distances
systematically increase with increasing the atomic number of the alkali-metal
atoms and take values between 7.88 bohr for Li2 and 12.51 bohr for Fr2 (with
an average of 10.78 bohr). Thus, the alkali-metal diatomic molecules in the
$a^{3}\Sigma^{+}$ state have well depths more than an order of magnitude
smaller and equilibrium distances almost twice as large as those in their
$X^{1}\Sigma^{+}$ states.

All the alkali-metal diatomic molecules in the $X^{1}\Sigma^{+}$ and
$a^{3}\Sigma^{+}$ electronic states consisting of stable isotopes (15 atomic
combinations) have already been studied experimentally (see Table 1).
Experimental high-resolution spectroscopic studies typically provide potential
energy curves with accuracy better than 1 cm-1. To evaluate the accuracy of
our results, we compare our spectroscopic constants with the available
experimental values in Tables 4 and 5 and calculate the root-mean-square error
(RMSE) between accurate experimental and present well depths and equilibrium
distances.

Table 4: Spectroscopic constants of the alkali-metal diatomic molecules in the ground $X^{1}\Sigma^{+}$ electronic state: equilibrium interatomic distance $R_{e}$, well depth $D_{e}$, harmonic constant $\omega_{e}$, first anharmonicity constant $\omega_{e}x_{e}$, rotational constant $B_{e}$, permanent electric dipole moments $d_{e}$, and parallel $\alpha_{e}^{\parallel}$ and perpendicular $\alpha_{e}^{\perp}$ components of the static electric dipole polarizability. Available experimental values are also collected. Molecule |  $R_{e}\,$(bohr) |  $D_{e}\,$(cm-1) |  $\omega_{e}\,$(cm-1) |  $\omega_{e}\chi_{e}\,$(cm-1) |  $B_{e}\,$(cm-1) |  $d_{e}\,$(D) |  $\alpha_{e}^{\parallel}\,$(a.u.) |  $\alpha_{e}^{\perp}\,$(a.u.) | Ref.
---|---|---|---|---|---|---|---|---|---
Li2 | 5.053 | 8509 | 351.2 | 2.59 | 0.6722 | 0 | 296.2 | 165.7 | This work
| 5.051225(4) | 8516.774(4) | 351.4 | 2.58 | 0.6726 | 0 |  |  | Le Roy  _et al._ (2009); Hessel and Vidal (1979)
LiNa | 5.464 | 7085 | 256.4 | 1.69 | 0.3751 | 0.47 | 342.3 | 185.6 | This work
| 5.45859 | 7103.5(1.0) | 256.46 | 1.58 | 0.3758 | 0.47(3) |  |  | Dagdigian  _et al._ (1971); Fellows (1991); Steinke  _et al._ (2012)
LiK | 6.269 | 6197 | 211.9 | 1.21 | 0.2576 | 3.36 | 480.3 | 245\. 5 | This work
| 6.2722(1) | 6216.9(1) | 211.91(2) | 1.224(6) | 0.2576(2) | 3.45(1) |  |  | Tiemann  _et al._ (2009); Engelke  _et al._ (1984); Dagdigian and Wharton (1972)
LiRb | 6.558 | 5902 | 195.0 | 1.14 | 0.2160 | 4.00 | 531.5 | 261.6 | This work
| 6.5501(1) | 5928(4) | 195.18 | 1.096 | 0.2165 | 4.00(1) |  |  | Ivanova  _et al._ (2011a); Dagdigian and Wharton (1972)
LiCs | 6.943 | 5817 | 184.4 | 1.05 | 0.1874 | 5.28 | 605.7 | 284.6 | This work
| 6.9317 | 5875.5(1) | 184.70 | 1.000 | 0.1880 | 5.5(2) |  |  | Staanum  _et al._ (2007); Deiglmayr  _et al._ (2008a)
LiFr | 6.976 | 5460 | 180.7 | 1.79 | 0.1819 | 4.24 | 605.1 | 276.5 | This work
Na2 | 5.827 | 5993 | 159.5 | 0.90 | 0.1542 | 0 | 378.8 | 204.7 | This work
| 5.8191 | 6022.029(5) | 159.18 | 0.760 | 0.1547 | 0 |  |  | Barrow  _et al._ (1984); Jones  _et al._ (1996)
NaK | 6.629 | 5248 | 120.2 | 0.20 | 0.0947 | 2.71 | 521.8 | 274.7 | This work
| 6.6122 | 5273.6(1) | 124.03 | 0.496 | 0.0952 | 2.72(6) |  |  | Dagdigian and Wharton (1972); Gerdes  _et al._ (2008, 2011)
NaRb | 6.902 | 4993 | 105.7 | 0.16 | 0.0700 | 3.30 | 574.3 | 293.4 | This work
| 6.8850 | 5030.50(5) | 106.85 | 0.380 | 0.0702 | 3.2(1) |  |  | Docenko  _et al._ (2004a); Pashov  _et al._ (2005); Guo  _et al._ (2016)
NaCs | 7.295 | 4897 | 100.0 | 0.39 | 0.0577 | 4.52 | 665.9 | 323.7 | This work
| 7.2766 | 4954.24(1) | 98.88 | 0.321 | 0.0580 | 4.8(2) |  |  | Docenko  _et al._ (2004b, 2006); Dagdigian and Wharton (1972)
NaFr | 7.301 | 4599 | 95.8 | 0.20 | 0.0542 | 3.51 | 646.7 | 309.7 | This work
K2 | 7.405 | 4415 | 92.6 | 0.32 | 0.0564 | 0 | 705.7 | 372.0 | This work
| 7.415(2) | 4450.67(7) | 92.40 | 0.324 | 0.0562 | 0 |  |  | Amiot  _et al._ (1995); Zhao  _et al._ (1996)
KRb | 7.687 | 4171 | 76.2 | 0.28 | 0.0382 | 0.63 | 762.4 | 398.6 | This work
| 7.6868(1) | 4217.82(1) | 75.85 | 0.230 | 0.0381 | 0.57(2) |  |  | Amiot and Vergés (2000); Pashov  _et al._ (2007); Ni  _et al._ (2008)
KCs | 8.097 | 3996 | 68.5 | 0.20 | 0.0305 | 1.86 | 879.0 | 445.2 | This work
| 8.096(13) | 4069.3(1.5) | 68.394(3) | 0.193(1) | 0.03048(5) |  |  |  | Ferber  _et al._ (2008)
KFr | 8.095 | 3838 | 65.2 | 0.16 | 0.0277 | 0.91 | 833.3 | 421.6 | This work
Rb2 | 7.962 | 3933 | 57.5 | 0.01 | 0.0223 | 0 | 821.6 | 428.1 | This work
| 7.9556(1) | 3993.593(3) | 57.75 | 0.125 | 0.0224 | 0 |  |  | Amiot (1990); Strauss  _et al._ (2010)
RbCs | 8.379 | 3750 | 50.1 | 0.11 | 0.0166 | 1.21 | 939.7 | 479.2 | This Work
| 8.3660 | 3836.37(0) | 50.01 | 0.110 | 0.0166 | 1.23(1) |  |  | Fellows  _et al._ (1999); Docenko  _et al._ (2011); Molony  _et al._ (2014)
RbFr | 8.370 | 3619 | 46.0 | 0.14 | 0.0140 | 0.29 | 891.8 | 452.8 | This work
Cs2 | 8.801 | 3538 | 42.0 | 0.081 | 0.0117 | 0 | 1061.0 | 538.4 | This work
| 8.7781 | 3649.9(5) | 42.021 | 0.082 | 0.0117 | 0 |  |  | Amiot and Dulieu (2002)
CsFr | 8.779 | 3440 | 37.7 | 0.077 | 0.0094 | 0.88 | 1008.5 | 507.8 | This work
Fr2 | 8.768 | 3333 | 32.8 | 0.04 | 0.0070 | 0 | 953.9 | 477.6 | This work
Table 5: Spectroscopic constants of the alkali-metal diatomic molecules in the first triplet $a^{3}\Sigma^{+}$ electronic state: equilibrium interatomic distance $R_{e}$, well depth $D_{e}$, transition energies $T_{e}$, harmonic constant $\omega_{e}$, first anharmonicity constant $\omega_{e}x_{e}$, rotational constant $B_{e}$, permanent electric dipole moments $d_{e}$, and parallel $\alpha_{e}^{\parallel}$ and perpendicular $\alpha_{e}^{\perp}$ components of the static electric dipole polarizability. Available experimental values are also collected. Molecule |  $R_{e}\,$(bohr) |  $D_{e}\,$(cm-1) |  $T_{e}\,$(cm-1) |  $\omega_{e}\,$(cm-1) |  $\omega_{e}\chi_{e}\,$(cm-1) |  $B_{e}\,$(cm-1) |  $d_{e}\,$(D) |  $\alpha_{e}^{\parallel}\,$(a.u.) |  $\alpha_{e}^{\perp}\,$(a.u.) | Ref
---|---|---|---|---|---|---|---|---|---|---
Li2 | 7.880 | 333.6 | 8175 | 65.1 | 3.31 | 0.2768 | 0 | 696.0 | 252.4 | This work
| 7.88017(6) | 333.780(6) | 8183.0 | 65.2(2) | 3.19(4) | 0.2764 | 0 |  |  | Martin  _et al._ (1988); Semczuk  _et al._ (2013)
LiNa | 8.910 | 238 | 6847 | 42.12 | 1.80 | 0.1413 | 0.18 | 560.7 | 270.8 | This work
|  | 229.753 | 6874 |  |  |  |  |  |  | Rvachov  _et al._ (2018)
LiK | 9.402 | 283 | 5914 | 43.5 | 1.70 | 0.1146 | 0.32 | 817.8 | 369.9 | This work
| 9.43(17) | 287(4) | 5930(4) | 44.2(1.5) |  |  |  |  |  | Salami  _et al._ (2007)
LiRb | 9.718 | 276 | 5626 | 41.0 | 1.60 | 0.0984 | 0.37 | 845.2 | 399.1 | This work
| 9.71(2) | 278(4) | 5650(8) |  |  |  |  |  |  | Ivanova  _et al._ (2011a)
LiCs | 9.991 | 295 | 5522 | 40.7 | 1.30 | 0.0905 | 0.45 | 985.2 | 461.5 | This work
| 9.916 | 309(10) | 5566(10) | 44.51 | 1.25 |  |  |  |  | Staanum  _et al._ (2007)
LiFr | 10.239 | 250 | 5209 | 37.7 | 1.5 | 0.0844 | 0.21 | 813.0 | 415.7 | This work
Na2 | 9.736 | 185 | 5808 | 25.2 | 0.88 | 0.0552 | 0 | 491.6 | 282.0 | This work
| 9.719(2) | 172.91(4) | 5849.1 | 23.79 | 0.643 | 0.0565 | 0 |  |  | Li  _et al._ (1985); Knoop  _et al._ (2011)
NaK | 10.301 | 206 | 5042 | 23.5 | 0.97 | 0.0392 | 0.03 | 697.9 | 387.3 | This work
| 10.280 | 207.86(2) | 5065.8 | 23.01 | 0.622 | 0.0394 |  |  |  | Gerdes  _et al._ (2008)
NaRb | 10.615 | 201 | 4792 | 19.9 | 0.47 | 0.0295 | 0.06 | 728.0 | 416.0 | This work
| 10.583 | 203.36(5) | 4827.1 | 18.86 |  | 0.0282 |  |  |  | Wang  _et al._ (1991); Pashov  _et al._ (2005)
NaCs | 10.902 | 212 | 4686 | 19.30 | 0.44 | 0.0259 | 0.09 | 837.71 | 482.0 | This work
| 10.856 | 217.17(1) | 4737.1 | 19.57 | 0.44 | 0.0261 |  |  |  | Docenko  _et al._ (2006)
NaFr | 11.092 | 186 | 4413 | 17.75 | 0.40 | 0.0235 | 0.10 | 710.4 | 430.6 | This work
K2 | 10.853 | 248 | 4167 | 21.0 | 0.44 | 0.0262 | 0 | 946.3 | 483.8 | This work
| 10.8364(2) | 255.02(5) | 4195.7 | 21.63(4) | 0.470 | 0.0260 | 0 |  |  | Li  _et al._ (1990); Pashov  _et al._ (2008)
KRb | 11.211 | 242 | 3928 | 19.32 | 0.39 | 0.0180 | 0.06 | 972.9 | 512.5 | This work
| 11.1549(2) | 249.03(1) | 3968.8 | 18.79 | 0.98 | 0.0181 |  |  |  | Pashov  _et al._ (2007)
KCs | 11.475 | 256 | 3740 | 16.5 | 0.24 | 0.0152 | 0.10 | 1096.1 | 576.0 | This work
| 11.4355(2) | 267.14(2) | 3803.0 | 17.52 | 0.306 | 0.0158 |  |  |  | Ferber  _et al._ (2008, 2009)
KFr | 11.685 | 220 | 3617 | 14.8 | 0.26 | 0.0133 | 0.13 | 935.5 | 530.4 | This work
Rb2 | 11.491 | 235 | 3698 | 13.4 | 0.16 | 0.0107 | 0 | 999.0 | 541.3 | This work
| 11.5160(2) | 241.503(3) | 3752.1 | 13.50 | 0.171 | 0.0108 | 0 |  |  | Beser  _et al._ (2009); Strauss  _et al._ (2010)
RbCs | 11.810 | 249 | 3501 | 12.29 | 0.15 | 0.0083 | 0.03 | 1118.2 | 604.9 | This work
| 11.7528(2) | 259.34(3) | 3577.0 | 12.55 | 0.15 | 0.00841 |  |  |  | Docenko  _et al._ (2011)
RbFr | 12.010 | 215 | 3535 | 10.3 | 0.03 | 0.0068 | 0.07 | 961.3 | 558.8 | This work
Cs2 | 12.143 | 263 | 3275 | 10.9 | 0.082 | 0.0062 | 0 | 1237.4 | 668.2 | This work
| 11.915(2) | 278.58(4) | 3370.7 | 11.37 | 0.184 | 0.00652 | 0 |  |  | Xie  _et al._ (2009); Lau  _et al._ (2016)
CsFr | 12.344 | 226 | 3215 | 9.20 | 0.11 | 0.0048 | 0.07 | 1073.6 | 626.8 | This work
Fr2 | 12.502 | 198 | 3135 | 7.37 | 0.04 | 0.0035 | 0 | 929.3 | 576.0 | This work
Table 6: Spectroscopic constants of the alkali-metal–alkaline-earth-metal diatomic molecules in the ground $X^{2}\Sigma^{+}$ electronic state: equilibrium interatomic distance $R_{e}$, well depth $D_{e}$, harmonic constant $\omega_{e}$, first anharmonicity constant $\omega_{e}x_{e}$, rotational constant $B_{e}$, permanent electric dipole moments $d_{e}$, and parallel $\alpha_{e}^{\parallel}$ and perpendicular $\alpha_{e}^{\perp}$ components of the static electric dipole polarizability. Available experimental values are also collected. Molecule |  $R_{e}\,$(bohr) |  $D_{e}\,$(cm-1) |  $\omega_{e}\,$(cm-1) |  $\omega_{e}\chi_{e}\,$(cm-1) |  $B_{e}\,$(cm-1) |  $d_{e}\,$(D) |  $\alpha_{e}^{\parallel}\,$(a.u.) |  $\alpha_{e}^{\perp}\,$(a.u.) | Ref
---|---|---|---|---|---|---|---|---|---
LiBe | 4.86 | 2545 | 311.8 | 5.2 | 0.6466 | 3.60 | 371.9 | 111.6 | This work
|  | 2794(143) | 326(4) | 9.5(7) |  |  |  |  | Persinger  _et al._ (2021a)
NaBe | 5.62 | 1293 | 166.2 | 3.6 | 0.2943 | 2.21 | 397.6 | 136.5 | This work
KBe | 6.63 | 898 | 120.3 | 4.9 | 0.1871 | 1.99 | 629.7 | 246.4 | This work
RbBe | 7.02 | 778 | 99.1 | 3.1 | 0.1501 | 1.74 | 657.2 | 277.5 | This work
CsBe | 7.45 | 720 | 91.4 | 3.4 | 0.1287 | 1.72 | 784.3 | 336.5 | This work
FrBe | 7.63 | 620 | 83.1 | 3.3 | 0.1194 | 1.27 | 621.3 | 297.3 | This work
LiMg | 5.83 | 1561 | 185.1 | 6.1 | 0.3262 | 1.04 | 484.7 | 169.1 | This work
| 5.86 | 1424(21) | 187.3 | 6.16 | 0.320 |  |  |  | Persinger  _et al._ (2021b); Berry and Duncan (1997)
NaMg | 6.61 | 910 | 87.3 | 2.5 | 0.1173 | 0.73 | 369.7 | 183.8 | This work
KMg | 7.63 | 722 | 61.5 | 1.61 | 0.0696 | 0.86 | 631.2 | 292.0 | This work
RbMg | 8.01 | 658 | 50.3 | 1.04 | 0.0502 | 0.83 | 659.2 | 321.0 | This work
CsMg | 8.43 | 631 | 46.1 | 1.01 | 0.0417 | 0.88 | 772.1 | 383.5 | This work
FrMg | 8.57 | 569 | 42.3 | 0.91 | 0.0379 | 0.70 | 635.9 | 335.4 | This work
LiCa | 6.34 | 2553 | 203.1 | 4.3 | 0.2510 | 1.15 | 582.5 | 228.1 | This work
| 6.3419(2) | 2605(10) | 202.239 | 3.405 | 0.25071 |  |  |  | Ivanova  _et al._ (2011b)
NaCa | 6.97 | 1636 | 100.0 | 1.53 | 0.0849 | 1.15 | 582.2 | 240.2 | This work
KCa | 8.03 | 1250 | 66.7 | 0.90 | 0.0474 | 1.90 | 879.3 | 328.0 | This work
| 8.011 |  | 67.983 | 0.0940 | 0.04754 |  |  |  | Gerschmann  _et al._ (2017)
RbCa | 8.40 | 1131 | 52.8 | 0.69 | 0.0314 | 1.89 | 935.7 | 353.4 | This work
CsCa | 8.84 | 1085 | 47.0 | 0.59 | 0.0251 | 2.18 | 1113.6 | 403.5 | This work
FrCa | 8.95 | 965 | 42.1 | 0.46 | 0.0222 | 1.71 | 927.0 | 340.9 | This work
LiSr | 6.69 | 2473 | 181.5 | 2.9 | 0.0207 | 0.39 | 620.3 | 269.7 | This work
| 6.699 | - | 182.93 | 3.026 | 0.02072 | - | - | - | Schwanke  _et al._ (2017)
NaSr | 7.32 | 1612 | 91.0 | 4.3 | 0.0617 | 0.57 | 630.8 | 281.5 | This work
KSr | 8.40 | 1222 | 54.1 | 0.67 | 0.0316 | 1.40 | 927.6 | 368.4 | This work
|  |  | 54.6(1.0) | 0.56(2) | 0.0322 |  |  |  | Szczepkowski  _et al._ (2018)
RbSr | 8.78 | 1108 | 39.4 | 0.35 | 0.0181 | 1.50 | 980.7 | 393.9 | This work
| 8.683 | 1158(20) | 40.3(7) | 0.4(1) | 0.0185 |  |  |  | Ciamei  _et al._ (2018)
CsSr | 9.24 | 1060 | 33.6 | 0.22 | 0.0133 | 1.79 | 1151.2 | 445.3 | This work
FrSr | 9.34 | 952 | 29.6 | 0.31 | 0.0110 | 1.41 | 971.2 | 411.0 | This work
LiBa | 6.87 | 3335 | 183.3 | 2.6 | 0.1908 | 0.51 | 596.4 | 347.5 | This work
| 6.8395 |  | 196.282 | 2.313 | 0.19277 |  |  |  | Stringat  _et al._ (1994)
NaBa | 7.39 | 2198 | 94.1 | 1.21 | 0.0560 | 0.08 | 637.3 | 363.0 | This work
KBa | 8.56 | 1539 | 56.22 | 0.52 | 0.0270 | 1.58 | 1014.2 | 423.0 | This work
RbBa | 8.96 | 1387 | 43.5 | 0.84 | 0.0143 | 1.76 | 1092.2 | 452.3 | This work
CsBa | 9.45 | 1292 | 32.0 | 0.12 | 0.0100 | 2.19 | 1288.2 | 497.8 | This work
FrBa | 9.52 | 1165 | 27.3 | 0.06 | 0.0078 | 1.71 | 1106.3 | 468.7 | This work
LiRa | 7.23 | 2152 | 157.6 | 2.5 | 0.1692 | 0.89 | 686.6 | 328.2 | This work
NaRa | 7.87 | 1408 | 70.4 | 0.98 | 0.0466 | 0.35 | 696.5 | 336.9 | This work
KRa | 9.03 | 1053 | 42.40 | 0.41 | 0.0222 | 0.50 | 954.6 | 428.2 | This work
RbRa | 9.42 | 957 | 28.9 | 0.24 | 0.0110 | 0.65 | 997.7 | 453.9 | This work
CsRa | 9.90 | 910 | 23.4 | 0.15 | 0.0074 | 0.87 | 1141.2 | 510.2 | This work
FrRa | 9.98 | 831 | 19.3 | 0.037 | 0.0054 | 0.68 | 980.4 | 470.2 | This work
Table 7: Spectroscopic constants of the alkaline-earth-metal diatomic molecules in the ground $X^{1}\Sigma^{+}$ electronic state: equilibrium interatomic distance $R_{e}$, well depth $D_{e}$, harmonic constant $\omega_{e}$, first anharmonicity constant $\omega_{e}x_{e}$, rotational constant $B_{e}$, permanent electric dipole moments $d_{e}$, and parallel $\alpha_{e}^{\parallel}$ and perpendicular $\alpha_{e}^{\perp}$ components of the static electric dipole polarizability. Available experimental values are also collected. Molecule |  $R_{e}\,$(bohr) |  $D_{e}\,$(cm-1) |  $\omega_{e}\,$(cm-1) |  $\omega_{e}\chi_{e}\,$(cm-1) |  $B_{e}\,$(cm-1) |  $d_{e}\,$(D) |  $\alpha_{e}^{\parallel}\,$(a.u.) |  $\alpha_{e}^{\perp}\,$(a.u.) | Ref
---|---|---|---|---|---|---|---|---|---
Be2 | 4.62 | 919 | 270.3 | 25.2 | 0.6255 | 0 | 136.2 | 61.1 | This work
| 4.620(1) | 934.8(3) | 270.7 | 26.0 | 0.623 | 0 |  |  | Meshkov  _et al._ (2014); Bondybey (1984)
BeMg | 6.20 | 435 | 80.2 | 5.73 | 0.2390 | 0.17 | 173.3 | 94.9 | This work
BeCa | 6.53 | 783 | 113.0 | 5.0 | 0.1918 | 0.39 | 320.7 | 169.2 | This work
BeSr | 6.93 | 756 | 101.0 | 4.5 | 0.1535 | 0.44 | 374.4 | 208.6 | This work
BeBa | 7.16 | 868 | 105.4 | 4.1 | 0.1387 | 0.60 | 489.8 | 279.0 | This work
BeRa | 7.62 | 632 | 81.6 | 3.4 | 0.1198 | 0.46 | 427.7 | 259.2 | This work
Mg2 | 7.35 | 434 | 51.9 | 1.8 | 0.0930 | 0 | 214.6 | 124.9 | This work
| 7.352 | 430.3(5) | 51.12(4) | 1.64(1) | 0.0929(1) | 0 |  |  | Knöckel  _et al._ (2013); Balfour and Douglas (1970)
MgCa | 7.67 | 673 | 60.3 | 1.87 | 0.0683 | 0.07 | 375.5 | 195.7 | This work
| 7.632 | 691.5(5) | 60.26(2) | 1.652(6) | 0.06896(2) |  |  |  | Atmanspacher  _et al._ (1985)
MgSr | 8.01 | 676 | 50.2 | 0.57 | 0.0498 | 0.009 | 426.3 | 234.9 | This work
MgBa | 8.25 | 758 | 51.4 | 0.87 | 0.0433 | 0.002 | 547.0 | 303.9 | This work
MgRa | 8.63 | 614 | 43.8 | 0.88 | 0.0373 | 0.13 | 482.6 | 284.7 | This work
Ca2 | 8.13 | 1050 | 63.4 | 1.16 | 0.0456 | 0 | 574.1 | 260.0 | This work
| 8.083 | 1102.08(9) | 64.93(1) | 1.07(3) | 0.0461(1) | 0 |  |  | Allard  _et al._ (2002); Balfour and Whitlock (1975)
CaSr | 8.50 | 1046 | 50.8 | 0.24 | 0.0303 | 0.15 | 645.4 | 295.5 | This work
CaBa | 8.76 | 1196 | 51.0 | 0.53 | 0.0253 | 0.21 | 804.1 | 359.6 | This work
CaRa | 9.12 | 943 | 42.0 | 0.47 | 0.0213 | 0.43 | 711.1 | 343.1 | This work
Sr2 | 8.88 | 1046 | 39.6 | 0.43 | 0.0174 | 0 | 718.6 | 330.2 | This work
| 8.8288(2) | 1081.64(2) | 40.328 | 0.399 | 0.01758 | 0 |  |  | Stein  _et al._ (2008, 2010)
SrBa | 9.15 | 1191 | 37.4 | 0.34 | 0.0134 | 0.007 | 886.1 | 392.9 | This work
SrRa | 9.50 | 945 | 28.4 | 0.20 | 0.0106 | 0.31 | 783.5 | 377.3 | This work
Ba2 | 9.43 | 1366 | 34.3 | 0.20 | 0.0098 | 0 | 1080.4 | 453.5 | This work
|  |  | 33.2(2) | 0.5(2) |  | 0 |  |  | Lebeault  _et al._ (1998)
BaRa | 9.78 | 1068 | 26.7 | 0.21 | 0.0074 | 0.43 | 952.3 | 439.2 | This work
Ra2 | 10.13 | 858 | 20.5 | 0.14 | 0.0052 | 0 | 842.3 | 424.6 | This work

For the $X^{1}\Sigma^{+}$ state, the RMSE of the well depths is 54 cm-1 (1.3
%). The smallest error of 8 cm-1 (0.1 %) is for Li2 and the largest error of
112 cm-1 (3 %) is for Cs2. Only for the heaviest Cs2 dimer, the error is
larger than 100 cm-1, and only for RbCs and Cs2, the error is larger than 2 %.
For most alkali-metal molecules in the $X^{1}\Sigma^{+}$ state, the error is
smaller than 1 %. The RMSE of the equilibrium distances is 0.012 bohr (0.16
%). The smallest error of 0.001 bohr (0.03 %) is for Li2 and the largest error
of 0.02 bohr (0.25 %) is for NaRb and Rb2. The theoretical and experimental
harmonic and anharmonicity constants agree with each other very well, mostly
within 0.1-0.2 cm-1.

For the $a^{3}\Sigma^{+}$ state, the RMSE of the well depths is 8.5 cm-1 (3.5
%). The smallest error of $<1\,$cm-1 ($<0.3\,$%) is for Li2 and the largest
error of 16 cm-1 (5.7 %) is for Cs2. The RMSE of the equilibrium distances is
0.07 bohr (0.6 %). The smallest error of $<0.001\,$bohr ($<0.01\,$%) is for
Li2 and the largest error of 0.22 bohr (1.9 %) is for Cs2. Molecules
containing Cs have the largest errors, but the error is larger than 0.1 bohr
for the heaviest Cs2 dimer only. The agreement for other spectroscopic
constants is also good.

All the alkali-metal diatomic molecules have already been studied
theoretically (see Table 1). The results for the $X^{1}\Sigma^{+}$ states,
reported within the last two decades, typically agree with the experimental
values within 200 cm-1 (5 %), with some of them having errors smaller than 100
cm-1 (3 %). For example, the calculations for 10 heteronuclear molecules in
the $X^{1}\Sigma^{+}$ state reported in Ref. Fedorov  _et al._ (2014) reached
the RMSE of the well depths of 84 cm-1 and the RMSE of the equilibrium
distances of 0.014 bohr, which are by 55 % and 15 % larger, respectively, than
the RMSEs reported in this study. The results for the $a^{3}\Sigma^{+}$
states, reported within the last two decades, typically agree with the
experimental values within 50 cm-1 (20 %), with some of them having errors
smaller than 10 cm-1 (5 %). For example, the highly-accurate calculations for
the lightest Li2 Lesiuk  _et al._ (2020) and NaLi Gronowski  _et al._ (2020)
molecules in the $a^{3}\Sigma^{+}$ state reached recently the spectroscopic
accuracy ($<1$cm-1). The observed overall agreement of our results with the
experimental values for the alkali-metal molecules confirms the accuracy and
sufficiency of the employed methodology for describing the electronic
structure of such molecules, is better on average than in previous theoretical
studies but cross-validates the accuracy of different approaches, and allows
to establish the present results as a benchmark for future more accurate
calculations.

####  III.1.2 Alkali-metal–alkaline-earth-metal diatomic molecules

The alkali-metal–alkaline-earth-metal diatomic molecules in their ground
$X^{2}\Sigma^{+}$ electronic state are weakly to moderately bound and have the
van der Waals character. Formally, they are chemically bound with a bond order
of half. They are radicals because of their single unpaired valence electron.
The calculated well depths range from 569 cm-1 for FrMg to 3335 cm-1 for LiBa
(with an average of 1311 cm-1). They systematically decrease with increasing
the atomic number of the alkali-metal atoms for a given alkaline-earth-metal
atom, as clearly visible in Fig. 3, but the trend is less regular with
changing alkaline-earth-metal atoms. The molecules containing Li are
significantly stronger bound than others. The corresponding equilibrium
distances systematically increase with increasing the atomic numbers of both
the alkali-metal and alkaline-earth-metal atoms and take moderate values
between 4.86 bohr for LiBe and 9.98 bohr for RaFr (with an average of 7.94
bohr). The alkali-metal–alkaline-earth-metal molecules have two-to-four times
smaller (larger) well depth than the alkali-metal molecules in the
$X^{1}\Sigma^{+}$ ($a^{3}\Sigma^{+}$) electronic state, but their equilibrium
distances are only slightly larger than those of the ground-state alkali-metal
dimers.

Spectroscopic measurements of the rovibrational levels of the ground
electronic state were recorded for the LiBe, LiMg, LiCa, KCa, LiSr, KSr, RbSr,
and LiBa molecules (see Table 2). Our spectroscopic constants are compared
with the available experimental values in Table 6. However, the experimental
data did not allow for accurate determination of well depths of corresponding
potential energy curves. Nevertheless, our well depths agree within 10 % with
the rough experimental estimates for LiBa Stringat  _et al._ (1994) and LiMg
Persinger  _et al._ (2021b) obtained from the extrapolations of the Morse
vibrational constants, and within 2 % and 5 % with the experimental estimates
guided by previous theoretical calculations for LiCa Ivanova  _et al._ (2011b)
and RbSr Ciamei  _et al._ (2018), respectively. The theoretical and
experimental vibrational and rotational constants agree even better, mostly
with differences below 3 %. Therefore, we estimate the uncertainty of the
calculated potential energy curves for the alkali-metal–alkaline-earth-metal
diatomic molecules to be around 3-6 %.

Most of the alkali-metal–alkaline-earth-metal diatomic molecules have already
been studied theoretically (see Table 2). Our results agree well with previous
accurate calculations. For example, for the family of the LiSr, NaSr, KSr,
RbSr, and CsSr molecules studied in Ref. Guérout  _et al._ (2010), the average
absolute differences between the present and previous well depths is 49 cm-1
(3.1 %) and equilibrium distances – 0.11 bohr (1.4 %), with all equilibrium
distances systematically shorter in Ref. Guérout  _et al._ (2010). Slightly
worse agreement with the average absolute differences of 90 cm-1 (6.7 %) for
well depths and 0.07 bohr (0.9 %) for equilibrium distances is found when our
results are compared with another systematic study of 16 alkali-
metal–alkaline-earth-metal combinations presented in Ref. Pototschnig  _et
al._ (2016). It is worth mentioning that different methods, basis sets, and
pseudopotentials were used in the present work and Refs. Guérout  _et al._
(2010); Pototschnig  _et al._ (2016), thus the observed overall agreement
additionally cross-validates the accuracy of different approaches. The
differences are a bit larger between the present and some older results, which
used smaller basis sets and less accurate wave functions.

Figure 5: Potential energy curves of (a) the KRb molecule in the
$X^{1}\Sigma^{+}$ electronic state, (b) the KRb molecule in the
$a^{3}\Sigma^{+}$ electronic state, (c) the RbSr molecule in the
$X^{2}\Sigma^{+}$ electronic state, and (d) the CaSr molecule in the
$X^{1}\Sigma^{+}$ electronic state computed using different basis sets and the
CCSD(T) method. See the text for details.

####  III.1.3 Alkaline-earth-metal diatomic molecules

The alkaline-earth-metal diatomic molecules in their ground $X^{1}\Sigma^{+}$
electronic state are weakly bound van der Waals complexes. Formally, they are
not chemically bound with a bond order of zero and are stabilized by the
dispersion interaction. The calculated well depths range from 434 cm-1 for Mg2
to 1366 cm-1 for Ba2 (with an average of 869 cm-1). The trends are not regular
with changing the atomic number of the alkaline-earth-metal atoms (see Fig.
4), but generally heavier atoms are stronger bound due to their larger
polarizability, as expected for van der Waals systems, with an exception for
the unusually strongly bound Be2 dimer Merritt  _et al._ (2009). The
corresponding equilibrium distances systematically increase with increasing
the atomic number of the alkaline-earth-metal atoms and take moderate values
between 4.62 bohr for Be2 and 10.13 bohr for Ra2 (with an average of 8.11
bohr). Thus, the alkaline-earth-metal molecules have slightly smaller well
depths and slightly larger equilibrium distances than the corresponding
alkali-metal–alkaline-earth-metal molecules.

Spectroscopic measurements of the rovibrational levels of the ground
electronic state were recorded for the Be2, Mg2, MgCa, Ca2, Sr2, and Ba2
molecules (see Table 3), and the accurate dissociation energies were
determined for the listed molecules except for Ba2. Our spectroscopic
constants are compared with the available experimental values in Table 7. The
RMSE of the well depths is 30 cm-1 (3.0%). The smallest error of 4 cm-1 (0.9
%) is for Mg2 and the largest error of 52 cm-1 (4.7 %) is for Ca2. The RMSE of
the equilibrium distances is 0.03 bohr (0.4 %). The theoretical and
experimental vibrational and rotational constants also mostly agree within 3%.
Therefore, we estimate the uncertainty of the calculated potential energy
curves for the alkaline-earth-metal diatomic molecules to be around 3-6% with
possibly larger values for heavier molecules.

Most of the alkaline-earth-metal diatomic molecules have already been studied
theoretically (see Table 3), but the detailed or accurate calculations have
been reported for homonuclear Be2, Mg2, Ca2, Sr2, and Ba2 dimers, only, while
heteronuclear combinations were studied using small basis sets and low-level
methods. The highly-accurate calculations reaching the spectroscopic accuracy
($<1$cm-1) were presented for the lightest Be2 Patkowski  _et al._ (2009);
Lesiuk  _et al._ (2019), and recently Mg2 Yuwono  _et al._ (2020). Our results
agree within 1 % with these more accurate approaches and establish the most
accurate reference for all other alkaline-earth-metal molecules.

###  III.2 Convergence and accuracy

Figure 6: Potential energy curves of (a) the KRb molecule in the
$X^{1}\Sigma^{+}$ electronic state, (b) the KRb molecule in the
$a^{3}\Sigma^{+}$ electronic state, (c) the RbSr molecule in the
$X^{2}\Sigma^{+}$ electronic state, and (d) the CaSr molecule in the
$X^{1}\Sigma^{+}$ electronic state computed with the different electronic-
structure methods and the aug-cc-pCV5Z basis sets. See the text for details.

Estimating the theoretical uncertainties of molecular electronic structure
calculations is challenging. A prerequisite is an accurate description of the
involved atoms. The performance of the CCSD(T) method with the aug-cc-pwCV5Z
basis sets for obtaining properties of the alkali-metal and alkaline-earth-
metal atoms was carefully tested in Refs. Hill and Peterson (2017);
Śmiałkowski and Tomza (2021) and demonstrated to reproduce well the most
accurate available theoretical or experimental data. For example, the
calculated atomic static electric dipole polarizabilities coincide with
previous accurate values with the RMSE of 3$\,e^{2}a_{0}^{2}/E_{\text{h}}$
(1%), and the atomic ionization potentials and the lowest $S$–$P$ excitation
energies agree with experimental data with the RMSE of 172 cm-1 (0.5 %) and
109 cm-1 (0.8 %), respectively.

In this subsection, we analyze the convergence of the interatomic interaction
energy calculations with the size of the employed atomic basis sets and the
quality of the employed wave-function representation for the selected
exemplary KRb, RbSr, and CaSr molecules.

Figure 5 presents the potential energy curves of the KRb molecule in the
$X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ electronic states, the RbSr molecule
in the $X^{2}\Sigma^{+}$ electronic state, and the CaSr molecule in the
$X^{1}\Sigma^{+}$ electronic state calculated with several different basis
sets and the CCSD(T) method. The family of the aug-cc-pwCV$n$Z basis sets is
employed, including the largest aug-cc-pwCV5Z basis sets augmented by bond
functions (aug-cc-pwCV5Z+bf). We also show the interaction energies obtained
with the two largest basis sets, with $n$=Q and 5 extrapolated to the complete
basis set limit (CBS) using the $1/n^{3}$ two-point formula Helgaker  _et al._
(1997). Additionally, the aug-cc-pV5Z basis sets with all electrons correlated
(aug-cc-pV5Z/all) and with only valence electrons correlated (aug-cc-pV5Z/val)
are used.

The smooth convergence toward the complete basis set limit can be observed,
with the aug-pwCVTZ basis sets performing reasonably well and the aug-pwCVQZ
basis sets being less than 5 % from the CBS limit. To reach an accuracy better
than 5 %, the largest basis sets have to be used. The CBS limit can be
approached using either the extrapolation scheme or bond functions. In our
test calculations, the extrapolated potential energy curves have slightly
larger (by 0.5-2 %) well depths than the curves obtained with the basis sets
augmented by bond functions, with bond functions performing best for the
alkaline-earth-metal dimer. Additionally, we studied the convergence of the
CCSDT correction of Eq. (2) with the basis set size and found the opposite
trend. In the final calculations, we decided to use the bond functions to
accelerate the convergence toward the CBS limit because this approach is
simpler and works very well for weakly-bound van der Waals complexes Tao
(2001), including other metal-atom molecules Zaremba-Kopczyk and Tomza (2021).

The importance of including the inner-shell electron correlation is evident
when the potential energy curves obtained using the aug-cc-pV5Z basis sets
with and without correlating $(n-1)s^{2}(n-1)p^{6}$ electrons are compared.
The equilibrium distances are larger by around 0.5 bohr for all the studied
molecules, and the well depths are significantly larger for all the molecules
except ground-state alkali-metal dimers when only valence electrons are
correlated. Thus, including the inner-shell electron correlation (core-core
and core-valence contributions) is crucial for accurately describing the
interatomic interactions in the studied systems. This correlation can be
included directly, as in the present work, or effectively using core
polarization potentials in calculations with large-core pseudopotentials,
which provides results in good agreement with the present values Aymar and
Dulieu (2005).

Figure 6 presents the potential energy curves of the KRb molecule in the
$X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ electronic states, the RbSr molecule
in the $X^{2}\Sigma^{+}$ electronic state, and the CaSr molecule in the
$X^{1}\Sigma^{+}$ electronic state calculated at several different levels of
theory: RHF, MP2, CISD, CISD+Q, MRCISD, MRCISD+Q, MRCISD+$p$, MRCISD+Q+$p$,
CCSD, CCSD(T), CCSD(T)+$\Delta$T, and CCSD(T)+$\Delta$T+$\Delta$Q, as
introduced in Sec. II. The aug-cc-pwCV5Z+bf basis sets are used in all
calculations except for the CCSD(T)+$\Delta$T and CCSD(T)+$\Delta$T+$\Delta$Q
curves, which are obtained with Eq. (1), including corrections given by Eqs.
(2) and (3).

The mean-field spin-restricted Hartree-Fock method (RHF) does not properly
describe the studied molecules because the correlation energy is crucial for
binding in all the systems. Low-level methods such as the second-order
perturbation theory (MP2) or the configuration interaction method, including
single and double excitations (CISD) are presented for completeness and poorly
reproduce the correlation energy, although the curve at the MP2 level is
accidentally close to the final result for the alkaline-earth-metal dimer. The
multireference version of the configuration interaction method with the active
space of valence orbitals ($ns+ns$ in MRCISD) describes better the ground-
state alkali-metal and alkali-metal–alkaline-earth-metal molecules but does
not improve the CISD results for other molecules, where only single spin
configuration can be constructed from the valence orbitals. Increasing the
size of the active space in the configuration interaction approach by
including the lowest unoccupied $p$ orbitals ($nsnp+nsnp$ in MRCISD/$p$)
noticeably improves its accuracy. The inclusion of the Davidson correction to
the CISD and MRCISD results (CISD+Q and MRCISD+Q, respectively) brings them
closer to the most accurate coupled cluster values. However, the MRCI approach
is size-inconsistent, and its convergence toward the full configuration
interaction results by increasing the CI active space is hard to control and
limited by available computing power.

The coupled cluster (CC) method provides the most accurate interaction
energies with a well-controlled convergence toward the full configuration
interaction results by including higher and higher excitations in the CC wave
function in a systematic way. It is also size-consistent. The inclusion of
triple excitations is important for all the studied molecules. The so-called
gold standard of quantum chemistry, the CCSD(T) method, which provides a good
estimate of connected triple excitations perturbatively, performs very well
for the alkali-metal molecules. For these molecules, the inclusion of full
triple excitations beyond the CCSD(T) results (CCSD(T)+$\Delta$T) changes the
potential well depth by less than $0.5\,\%$ and is more important at larger
distances. In contrast, the CCSDT corrections increase the well depths for the
alkali-metal–alkaline-earth-metal and alkaline-earth-metal molecules by more
than $5\,\%$. The importance of full triple excitations for the molecules
containing alkaline-earth-metal atoms is not surprising since the CCSDT and
CCSDTQ methods are needed to describe the alkali-metal–alkaline-earth-metal
and alkaline-earth-metal molecules at the valence full configuration level,
respectively. Therefore, the CCSDTQ corrections increase the well depths for
the alkaline-earth-metal molecules by an additional $5\,\%$.

Figure 7: Permanent electric dipole moment curves of the alkali-metal diatomic
molecules in the $X^{1}\Sigma^{+}$ electronic state. The points indicate
values for equilibrium distances. Figure 8: Permanent electric dipole moment
curves of the alkali-metal diatomic molecules in the $a^{3}\Sigma^{+}$
electronic state. The points indicate values for equilibrium distances.

We can assess the importance of higher excitations in the coupled cluster
calculations based on statistics for all the studied molecules. The inclusion
of the CCSDT corrections of Eq. (2) increases on average the well depths for
the alkali-metal molecules in the $X^{1}\Sigma^{+}$ state by 10 cm-1 (0.3 %)
(except Cs2, CsFr, Fr2 with the opposite effect of the same magnitude), for
the alkali-metal–alkaline-earth-metal molecules in the $X^{2}\Sigma^{+}$ state
by 88 cm-1 (7.5 %), and for the alkaline-earth-metal molecules in the
$X^{1}\Sigma^{+}$ state by 52 cm-1 (8.1 %). For the alkali-metal molecules in
the $a^{3}\Sigma^{+}$ state, the well depths change on average by 1.1 cm-1
(0.4 %) with an increase (decrease) of binding energy for lighter (heavier)
molecules. The inclusion of the CCSDT corrections also increases, on average,
the equilibrium distances for the alkali-metal molecules in the
$X^{1}\Sigma^{+}$ state by 0.017 bohr (0.2 %) and decreases on average the
equilibrium distances for the alkali-metal molecules in the $a^{3}\Sigma^{+}$
state by $<0.01\,$bohr ($<0.1\,$%), for the alkali-metal–alkaline-earth-metal
molecules in the $X^{2}\Sigma^{+}$ by 0.04 bohr (0.5 %), and for the alkaline-
earth-metal molecules in the $X^{1}\Sigma^{+}$ state by 0.1 bohr (2.0 %).
Additionally, the CCSDTQ corrections of Eq. (3) increase on average the well
depths for the alkaline-earth-metal molecules in the $X^{1}\Sigma^{+}$ state
by 44 cm-1 (5.1 %) and decrease on average their equilibrium distances by 0.03
bohr (0.4 %).

Based on the convergence analysis in this subsection and the comparison with
available experimental data in the previous subsection, where a good agreement
was observed, we can conclude that the employed electronic structure methods,
basis sets, and energy-consistent pseudopotentials properly treat the
relativistic effects and reproduce the correlation energy while being close to
being converged in the size of the basis function set. We can also estimate
the uncertainties of the employed computational approach of Eq. (1) for
calculating potential energy curves are around:

  * •

25-100 cm-1 (0.5-2 %) of $D_{e}$ and 0.005-0.02 bohr (0.05-0.2 %) of $R_{e}$
for the alkali-metal molecules in the $X^{1}\Sigma^{+}$ state,

  * •

5-15 cm-1 (2-6 %) of $D_{e}$ and 0.05-0.2 bohr (0.5-2 %) of $R_{e}$ for the
alkali-metal molecules in the $a^{3}\Sigma^{+}$ state,

  * •

50-100 cm-1 (3-6 %) of $D_{e}$ and 0.01-0.04 bohr (0.1-0.5 %) of $R_{e}$ for
the alkali-metal–alkaline-earth-metal molecules in the $X^{2}\Sigma^{+}$
state,

  * •

50-100 cm-1 (3-6 %) of $D_{e}$ and 0.01-0.05 bohr (0.1-0.6 %) of $R_{e}$ for
the alkaline-earth-metal molecules in the $X^{1}\Sigma^{+}$ state.

Figure 9: Permanent electric dipole moment curves of the alkali-
metal–alkaline-earth-metal diatomic molecules in the $X^{2}\Sigma^{+}$
electronic state. The points indicate values for equilibrium distances. Figure
10: Permanent electric dipole moment curves of the alkaline-earth-metal
diatomic molecules in the $X^{1}\Sigma^{+}$ electronic state. The points
indicate values for equilibrium distances.

###  III.3 Permanent electric dipole moments

The computed permanent electric dipole moments as functions of the
internuclear distance for the alkali-metal diatomic molecules in the
$X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ electronic states are presented in
Fig. 7 and Fig. 8, respectively, for the alkali-metal–alkaline-earth-metal
diatomic molecules in the $X^{2}\Sigma^{+}$ electronic state – in Fig. 9, and
for the alkaline-earth-metal diatomic molecules in the $X^{1}\Sigma^{+}$ state
– in Fig. 10. Calculations are performed for all the combinations of the
alkali-metal (Li, Na, K, Rb, Cs, Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr,
Ba, Ra) atoms. All curves exhibit smooth behavior, and different classes of
molecules share similar characteristics. The corresponding values for the
equilibrium distances, $d_{e}\equiv d(R_{e})$, are indicated in the plots and
collected in Tables 4-7, along with available experimental data.

The permanent electric dipole moments originate from the uneven distribution
of charge in heteronuclear molecules. They describe the response of polar
molecules to external static electric fields, resulting in molecular
polarization Lemeshko  _et al._ (2013). The dipole-dipole interactions also
dominate long-range intermolecular interactions and ultracold collisions
between polar molecules and are crucial for their applications Gadway and Yan
(2016).

The alkali-metal diatomic molecules in their ground $X^{1}\Sigma^{+}$
electronic state possess the largest permanent electric dipole moments ranging
from 0.3 D for RbFr to 5.3 D for LiCs (with an average of 2.5 D) at their
equilibrium distances. Molecules containing Li are the most polar, followed by
those containing Na. In contrast, the alkali-metal diatomic molecules in the
$a^{3}\Sigma^{+}$ electronic state have the smallest permanent electric dipole
moments ranging from 0.03 D for NaK and RbCs to 0.45 D for LiCs (with an
average of 0.15 D) at their equilibrium distances. Triplet-state molecules
containing Li are also the most polar.

The alkali-metal–alkaline-earth-metal diatomic molecules in their ground
$X^{2}\Sigma^{+}$ electronic state exhibit intermediate permanent electric
dipole moments ranging from 0.08 D for NaBa to 3.6 D for LiBe (with an average
of 1.3 D) at their equilibrium distances. Among them, molecules containing Be
or Cs are the most polar. Finally, the alkaline-earth-metal diatomic molecules
in their ground $X^{2}\Sigma^{+}$ electronic state have small permanent
electric dipole moments ranging from almost zero for combinations containing
Mg to 0.6 D for BeBa (with an average of 0.25 D) at their equilibrium
distances. In this group, molecules containing Be or Ra are the most polar.

The magnitude and orientation of the permanent electric dipole moments at
equilibrium and large interatomic distances correlate with the difference in
atomic electronegativities for all the molecules, except for singlet-state
CsFr. Electronegativity is a measure of an atom’ ability to attract shared
electrons to itself. Thus, the permanent electric dipole moments are oriented
from more electronegative atoms to less electronegative ones.

The permanent electric dipole moments have been measured for almost all the
alkali-metal diatomic molecules consisting of stable isotopes (except KCs) in
their ground vibrational level of the ground $X^{1}\Sigma^{+}$ electronic
state Dagdigian  _et al._ (1971); Dagdigian and Wharton (1972); Deiglmayr  _et
al._ (2008a); Ni  _et al._ (2008); Deiglmayr  _et al._ (2010); Molony  _et
al._ (2014); Guo  _et al._ (2016). The RMSE of our calculated values is 0.13 D
(4.5 %). No experimental measurements of permanent electric dipole moments
have been reported for other classes of molecules studied in this work.

A part of theoretical studies collected in Tables 1-3 reported calculations of
permanent electric dipole moments alongside potential energy curves at
different levels of theory, generally in agreement with the present results.
Here, we compare our values with the previous most accurate and systematic
studies. For the 10 ground-state alkali-metal molecules consisting of stable
isotopes, the average absolute differences between the present and previous
values are 0.1 D (4.8 %) Aymar and Dulieu (2005), 0.04 D (2.5 %) Fedorov  _et
al._ (2014), and 0.09 D (4.7 %) Mitra  _et al._ (2020). For the 16 lightest
ground-state alkali-metal–alkaline-earth molecules, the average absolute
differences between the present and previous values are 0.15 D (13 %)
Pototschnig  _et al._ (2016) and 0.08 D (6.6 %) Mitra  _et al._ (2022). The
average absolute differences between the present results and values for LiSr,
NaSr, KSr, RbSr, and CsSr reported in Ref. Guérout  _et al._ (2010) are 0.08 D
(7.9 %). It is worth mentioning that different methods, basis sets, and
pseudopotentials were used in the present work and Refs. Aymar and Dulieu
(2005); Guérout  _et al._ (2010); Pototschnig  _et al._ (2016). Furthermore,
the authors of Refs. Mitra  _et al._ (2020, 2022) included the relativistic
effects directly in all-electron calculations in contrast to our scalar
relativistic pseudopotentials. Thus the observed overall agreement
additionally cross-validates the accuracy of different approaches. To the best
of our knowledge, the calculations of permanent electric dipole moments have
not been previously reported for alkaline-earth molecules. We estimate the
uncertainty of our permanent electric dipole moments to be around 5 %.

###  III.4 Static electric dipole polarizabilities

The polarizability tensor of $\Sigma$-state molecules has two independent
components: parallel $\alpha^{\parallel}(R)\equiv\alpha^{zz}(R)$ and
perpendicular $\alpha^{\perp}(R)\equiv\alpha^{xx}(R)=\alpha^{yy}(R)$ ones with
$z$ axis chosen along the internuclear axis in a molecule-fixed reference
frame. The calculated static electric dipole polarizabilities as functions of
the internuclear distance for the alkali-metal diatomic molecules in the
$X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ electronic states, the alkali-
metal–alkaline-earth-metal diatomic molecules in the $X^{2}\Sigma^{+}$
electronic state, and the alkaline-earth-metal diatomic molecules in the
$X^{1}\Sigma^{+}$ state are provided in the Supplemental Material sup .
Calculations are performed for all the combinations of the alkali-metal (Li,
Na, K, Rb, Cs, Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr, Ba, Ra) atoms.
The corresponding values for the equilibrium distances,
$\alpha^{\parallel}_{e}\equiv\alpha^{\parallel}(R_{e})$ and
$\alpha^{\perp}_{e}\equiv\alpha^{\perp}(R_{e})$, are collected in Tables 4-7.
The polarizabilities are reported in atomic units of
$e^{2}a_{0}^{2}/E_{\text{h}}$ throughout this paper.

At large internuclear distances, the polarizabilities approach their
asymptotic behavior given by the atomic polarizabilities $\alpha_{A}$ and
$\alpha_{B}$ Jensen  _et al._ (2002)

$\begin{split}\alpha^{\parallel}(R)&\approx\alpha_{A}+\alpha_{B}+\frac{4\alpha_{A}\alpha_{B}}{R^{3}}+\frac{4(\alpha_{A}+\alpha_{B})\alpha_{A}\alpha_{B}}{R^{6}}\,,\\\
\alpha^{\perp}(R)&\approx\alpha_{A}+\alpha_{B}-\frac{2\alpha_{A}\alpha_{B}}{R^{3}}+\frac{(\alpha_{A}+\alpha_{B})\alpha_{A}\alpha_{B}}{R^{6}}\,.\end{split}$
(9)

Two independent polarizability tensor components can also be transformed to
the isotropic $\bar{\alpha}(R)$ and anisotropic $\Delta\alpha(R)$ ones using

$\begin{split}\bar{\alpha}(R)&=\frac{2\alpha^{\perp}(R)+\alpha^{\parallel}(R)}{3}\,,\\\
\Delta\alpha(R)&=\alpha^{\parallel}(R)-\alpha^{\perp}(R)\,.\end{split}$ (10)

The polarizability describes the molecular response to an electric field in
the second order of perturbation theory. For example, optical dipole trapping
is governed by the isotropic polarizability $\bar{\alpha}$, while the laser-
induced molecular alignment is controlled by its anisotropic component
$\Delta\alpha$ Lemeshko  _et al._ (2013). Molecular polarizabilities are also
useful in evaluating long-range intermolecular interactions Jeziorski  _et
al._ (1994).

The calculated polarizabilities for all the studied molecules are well
described by the formulas in Eq. (9) at intermediate and large internuclear
distances. For the most weakly-bound alkali-metal dimers in the
$a^{3}\Sigma^{+}$ state, these formulas also apply around their equilibrium
distances, while the more strongly-bound alkali-metal dimers in the
$X^{1}\Sigma^{+}$ state have smaller polarizabilities at equilibrium
distances, influenced by stronger chemical bonding of a polarized-covalent
nature. In general, the magnitude of the molecular polarizabilities correlates
with the atomic polarizabilities, which increase with increasing the atomic
number of the alkali-metal and alkaline-earth-metal atoms, with an exception
for Fr and Ra due to the contraction of electronic orbitals by the
relativistic effects. The parallel components at equilibrium distances are
around twice larger than the perpendicular ones for all atomic combinations.

The polarizabilities have been measured only for a few ground-state alkali-
metal molecules Dagdigian  _et al._ (1971); Dagdigian and Wharton (1972);
Molof  _et al._ (1974); Knight  _et al._ (1985); Tarnovsky  _et al._ (1993);
Antoine  _et al._ (1999); Bowlan  _et al._ (2011); Hohm (2013). Our results
agree with the experimental values within their uncertainties. For example,
for LiNa, the calculated isotropic and anisotropic polarizabilities of
$\bar{\alpha}_{e}=237.8\,$ and $\Delta\alpha_{e}=156.6\,$ can be compared with
the experimental ones of $\bar{\alpha}_{0}=270(34)$ and
$\Delta\alpha_{0}=162(13)$ from Ref. . Our isotropic values $\bar{\alpha}_{e}$
of 209.2, 262.7, 483.2, 559.2, 712.6, 357.1, and 589.8 for Li2, Na2, K2, Rb2,
Cs2, NaK, and KCs, respectively, agree with the corresponding experimental
estimates of 216(20), 256(20), 499(40), 533(40), 702(54), 344(20), and 601(34)
from Ref. Tarnovsky  _et al._ (1993). The experimental values for the
molecules containing the alkaline-earth-metal atoms are even more scarce. For
the Ba2 dimer, the ratio of the polarizabilities of the ground state molecules
and separate atoms was measured to be 1.30(13) Schäfer  _et al._ (2007), which
agrees with our value of 1.21.

The present polarizabilities agree well within a few percent with previous
systematic computational studies employing different methods, basis sets, and
pseudopotentials presented for the alkali-metal molecules in Refs. Deiglmayr
_et al._ (2008b); Vexiau  _et al._ (2017); Mitra  _et al._ (2020) and the
alkali-metal–alkaline-earth-metal molecules in Ref. Gopakumar  _et al._
(2014); Pototschnig  _et al._ (2016); Mitra  _et al._ (2022). The agreement is
also satisfactory with other calculations reported in some older works, which
used smaller basis sets and less accurate wave functions Müller and Meyer
(1986); Urban and Sadlej (1995); RéRAT and BUSSERY-HONVAULT (2003); Dyall  _et
al._ (2023).

##  IV Summary and conclusions

Motivated by the growing experimental interest in producing and using
ultracold gases of molecules that contain different alkali-metal or alkaline-
earth-metal atoms, we have conducted a comprehensive theoretical study on the
ground-state electronic properties of such molecules. We have calculated the
interactions energies, permanent electric dipole moments, and static electric
dipole polarizabilities as functions of the internuclear distance for all 78
possible homonuclear and homonuclear diatomic combinations of alkali-metal
(Li, Na, K, Rb, Cs, Fr) and alkaline-earth-metal (Be, Mg, Ca, Sr, Ba, Ra)
atoms. We have employed the hierarchy of coupled cluster methods up to CCSDTQ
with large Gaussian basis sets and small-core relativistic energy-consistent
pseudopotentials. The inclusion of full triple and quadruple excitations in
the coupled cluster method to obtain potential energy curves has allowed for
the description of valence electrons at the full configuration interaction
level for all molecules. Thus, results for three classes of experimentally
relevant molecules have been presented at a consistent level of theory.
Corresponding spectroscopic constants have been collected. We have computed
the electronic properties of some molecules, such as those containing francium
and radium, for the first time. The permanent electric dipole moments of
heteronuclear alkaline-earth molecules have also been presented for the first
time. We have analyzed the convergence, estimated computational uncertainties,
and compared previous experimental and theoretical data with the present
values.

The presented results can serve as a reference and theoretical benchmark for
future potentially more accurate electronic-structure computations Gronowski
_et al._ (2020), including both ground and excited electronic states. The
potential energy curves can be used to obtain rovibrational levels to guide
and explain the formation and new spectroscopic measurements for
experimentally unexplored atomic combinations. Our interaction potentials can
also serve as a starting point for fitting curves to future accurate
experimental rovibrational energy levels for molecules containing alkaline-
earth-metal atoms Ciamei  _et al._ (2018).

The calculated permanent electric dipole moments and static electric dipole
polarizabilities can be employed to construct long-range intermolecular
interaction potentials Christianen  _et al._ (2019). They can also be used to
describe a molecular response to static electric and off-resonant laser fields
Lemeshko  _et al._ (2013), respectively, and, in general, to design and guide
new formation, control, and manipulation schemes Tomza  _et al._ (2014);
Devolder  _et al._ (2021).

Finally, the determined well depths and related dissociation energies can
allow for assessing the energetics of chemical reactions between ground-state
molecules and in atom-molecule collisions. The chemical reactivity of the
alkali-metal diatomic molecules is well understood Żuchowski and Hutson
(2010); Byrd  _et al._ (2010); Tomza  _et al._ (2013b). On the other hand, the
reactivity of the alkali-metal–alkaline-earth-metal (except RbSr Man  _et al._
(2022)) and alkaline-earth-metal molecules have not been theoretically
studied, while fast chemical reactions between ground-state Sr2 were recently
obsreved Leung  _et al._ (2021). The presented data can directly provide the
energy changes for atom-exchange chemical reactions in molecular gases and
atom-molecule mixtures. They can also be used to evaluate the energy changes
for trimer formation chemical reactions if binding energies of triatomic
molecules are calculated.

Full potential energy curves, permanent electric dipole moments, and static
electric dipole polarizabilities as functions of the interatomic distance in a
numerical form are collected in the Supplemental Material sup .

###### Acknowledgements.

We gratefully acknowledge the National Science Centre Poland (grant no.
2020/38/E/ST2/00564) for financial support and Poland’s high-performance
computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing
computer facilities and support within computational grant no.
PLG/2021/015237.

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