# Electric Dipole Moments and Static Dipole Polarizabilities of
Alkali–Alkaline-Earth Molecules: Non-relativistic versus relativistic coupled-
cluster theory analyses

R. Mitra<EMAIL_ADDRESS>Physical Research Laboratory, Atomic,
Molecular and Optical Physics Division, Navrangpura, Ahmedabad-380009, India
V. S. Prasannaa<EMAIL_ADDRESS>Centre for Quantum Engineering,
Research and Education, TCG CREST, Salt Lake, Kolkata 700091, India  B. K.
Sahoo<EMAIL_ADDRESS>Physical Research Laboratory, Atomic, Molecular and
Optical Physics Division, Navrangpura, Ahmedabad-380009, India

###### Abstract

We analyze the electric dipole moments (PDMs) and static electric dipole
polarizabilities of the alkali–alkaline-earth (Alk-AlkE) dimers by employing
finite-field coupled-cluster methods, both in the frameworks of non-
relativistic and four-component spinfree relativistic theory. In order to
carry out comparative analyses rigorously, we consider those Alk-AlkE
molecules made out of the lightest to the medium-heavy constituent atoms (Alk:
Li to Rb and AlkE: Be through Sr). We present behaviour of electron
correlation effects as well as relativistic effects with the size of the
molecules. Uncertainties to the above quantities of the investigated Alk-AlkE
molecules are inferred by analyzing our results from different form of
Hamiltonian, basis set, and perturbative parameter in a few representative
molecules. We have also provided empirical relations by connecting average
polarizabilities of the Alk-AlkE molecules with their PDMs, and atomic numbers
and polarizabilities of the corresponding Alk and AlKE atoms, which can be
used to roughly estimate the average polarizabilities of other heavier Alk-
AlkE molecules. We finally give our recommended results, and compare them with
the literature values.

##  I Introduction

The physics of ultracold diatomic molecules has received enormous attention
over the last couple of decades for their exclusive properties [1, 2, 3, 4]
and a wide array of potential applications [5, 6]. Several ultracold diatomic
molecules have been successfully produced recently via different mechanisms
including Feshbach resonance [7, 8], photoassociation [9], deceleration of
molecular beams [10], and buffer gas cooling [11]. The very low temperature of
ultracold molecules helps to reduce decoherence and other systematic effects,
which enables one to conduct high precision measurements for investigating
fundamental constants like proton to electron mass ratio [12, 13, 14],
electric dipole moment of electron [15, 16] et cetera. Ultracold polar
molecules are also being considered as useful tools to produce qubits in the
booming field of quantum computation [17].

Historically, homonuclear alkali dimers were the first set of molecules that
were cooled to ultracold temperatures [18, 19, 20]. However, the focus
gradually shifted from non-polar homonuclear alkali-dimers to heteronuclear
alkali-dimers after the latter were successfully cooled to temperatures in the
ultracold regime [21, 22, 23, 24]. By the virtue of having permanent electric
dipole moments (PDMs), the heteronuclear alkali-dimers offer the opportunity
to properly interrogate and control the systems in the presence of external
electric fields. Molecules possessing sufficiently large values of PDM ($\mu$)
would enhance long range and anisotropic dipole-dipole interactions, which can
be controlled by external electric fields [25]. A large $\mu$ for a given
molecule also means that we require a sufficiently low electric field to align
the molecular beam for understanding dipole-dipole interactions. For a given
density of trapped molecules, a prior knowledge of $\mu$ would help to
understand better the required dipole interaction strengths [26]. The electric
dipole-dipole interactions have useful applications in the physics of quantum
phase transitions [27]. These dipole-dipole interactions could couple
molecular electric dipoles parallel or anti-parallel to an external electric
field. Each of the molecules can act as qubits and coupled molecular dipoles
can be used to realize entanglements for quantum computation [17, 28, 29].
Information of molecular PDM is important in understanding the chaining of
molecules. The interaction strength for the chaining process of molecules in a
one-dimensional optical lattice is proportional to the square of the PDM [30].

Another important property that finds crucial applications in ultracold
physics is the static dipole polarizability ($\alpha$) of molecules. Its
magnitude plays a significant role in trapping a molecule, as the restoring
force of a trapping laser beam in an optical tweezer is directly proportional
to $\alpha$. Therefore, molecules with large values of $\alpha$ are more
suitable for trapping and cooling [31]. Moreover, for molecules trapped inside
a far-off resonance optical trap, the value of $\alpha$ determines the depth
of the trap depending upon the laser field intensity [26]. In femtosecond
physics, the information on $\alpha$ plays an important role in laser-induced
impulsive alignment of molecules [32]. Hence, knowledge of both $\mu$ and
$\alpha$ has significant implications in unraveling the physics of ultracold
molecules.

In an earlier work [33], we conducted a comparative study of PDMs and static
dipole polarizabilities of alkali dimers between non-relativistic (NR) and
relativistic (Rel) theories. As alkali dimers are closed-shell systems, our
next logical step is to extend the analysis to open-shell systems of interest
to ultracold physics, that is, those molecules having non-zero magnetic dipole
moments (or non-zero spin). A non-zero magnetic moment would enable one to
control and manipulate the systems with an external magnetic field. All the
alkali (Alk) and alkaline-earth (AlkE) atoms have been successfully cooled to
ultracold temperatures, thus opening up the opportunity to produce
alkali–alkaline-earth (Alk-AlkE) molecules. For this reason, we chose Alk-AlkE
molecules to study their $\mu$ and $\alpha$ using both NR and Rel methods. One
of the first theoretical studies on Alk-AlkE molecular bond-lengths and their
PDMs were carried out by Bauschlicher et al. [34]. Coupled-cluster (CC) [35,
36] calculations to investigate the electronic properties of Li-AlkE molecules
were performed by Kotochigova et al. [37]. In Ref. [38], the authors examined
the ground state properties of eight Alk-AlkE molecules (Alk: Li, Na, K, Rb,
and AlkE: Ca, Sr) using singles-doubles and partial triples CC (CCSD(T))
method, but in a NR theory framework. In another recent work [39], electronic
and spectroscopic properties of sixteen Alk-AlkE molecules (Alk: Li, Na, K, Rb
and AlkE: Be, Mg, Ca, Sr) were examined using multi-reference configuration
interaction (MRCI) method. In this work, we consider Alk-AlkE molecules made
from the combination of four Alk atoms (Li, Na, K, Rb) and four AlkE atoms
(Be, Mg, Ca, Sr). We compare results from both the NR and Rel calculations of
$\mu$ and $\alpha$ of these sixteen molecules in their ground state
(${}^{2}\Sigma^{+}$). This is accompanied by an analysis of correlation
effects. These discussions are followed by studying the precision of our
results, finding empirical relationship between PDMs and polarizabilities, and
finally comparing our results with available literature values.

This paper is organised in the following manner: in Sec. II, we expound on the
theory of $\mu$ and $\alpha$, noting that we employ the finite-field (FF)
approach to calculate molecular properties in the presence of a static
electric field perturbation of certain strength. In Sec. III, we discuss CC
theory. We also briefly describe the basis sets used in our calculations, cut-
offs imposed on high-lying virtual orbitals, and other important parameters
crucial to our calculations. In Sec. IV, we present our results and analyses.
We use atomic units (a.u.) throughout, unless stated otherwise explicitly.

##  II Theory

The PDM (the intrinsic electric dipole moment) of a molecule can be inferred
from the first-order energy shift of the ground electronic state of the
molecule in the presence of a weak static electric field. Whereas, the static
dipole polarizability is defined as a property that causes a second-order
shift under the application of a weak static electric field. If the static
electric field perturbation strength is $\epsilon$, the molecular ground state
energy can be expressed as

$\displaystyle E_{0}=E_{0}^{(0)}+\epsilon
E_{0}^{(1)}+\epsilon^{2}E_{0}^{(2)}+\cdots,$ (1)

where $E_{0}^{(i)}$ denotes the $i^{th}$ order energy shift, and $E_{0}^{(0)}$
is the unperturbed molecular ground state energy. Eq. (1) can be conveniently
written as

$\displaystyle
E_{0}=E_{0}^{(0)}-\mu_{i}\epsilon_{i}-\frac{1}{2}\alpha_{ij}\epsilon_{i}\epsilon_{j}+\cdots,$
(2)

where indices $i$, $j$ denote for the direction of the applied electric field
perturbation and they assume integer values from 1 to 3 (with 1, 2, and 3
denoting $x$, $y$, and $z$-directions, respectively), while $\mu_{i}$ and
$\alpha_{ij}$ are the components of PDM and second-rank static dipole
polarizability tensor, respectively. Using the Taylor series expansion, Eq.
(1) can be rewritten as

$\displaystyle E_{0}=E_{0}^{(0)}+\epsilon_{i}\frac{\partial
E_{0}}{\partial\epsilon_{i}}\Bigg{|}_{\epsilon_{i}\to
0}+\frac{1}{2!}\epsilon_{i}\epsilon_{j}\frac{\partial^{2}E_{0}}{\partial\epsilon_{i}\partial\epsilon_{j}}\Bigg{|}_{\epsilon_{i},\epsilon_{j}\to
0}+\cdots.\ \ \ $ (3)

Now, comparing Eq. (2) and Eq. (3), we can get the expressions for $\mu_{i}$
and $\alpha_{ij}$ as

$\displaystyle\mu_{i}=-\frac{\partial
E_{0}}{\partial\epsilon_{i}}\Bigg{|}_{\epsilon_{i}\to 0}$ (4)

and

$\displaystyle\alpha_{ij}=-\frac{\partial^{2}E_{0}}{\partial\epsilon_{i}\epsilon_{j}}\Bigg{|}_{\epsilon_{i},\epsilon_{j}\to
0}.$ (5)

Therefore, from Eq. (4) and Eq. (5) we see that the PDM and the components of
the static dipole polarizability tensor can be expressed as first-order and
second-order energy derivatives, respectively. This approach of computing
energy derivatives to evaluate properties is known as the FF method. We employ
the two point central difference scheme for calculating the PDMs and the
corresponding three point stencil to evaluate the static dipole
polarizabilities numerically. For our calculations, we choose a static
electric field perturbation $\epsilon\sim 10^{-4}$ a.u.. The average static
dipole polarizability ($\bar{\alpha}$) for a diatomic molecule is given as

$\displaystyle\bar{\alpha}=\frac{1}{3}(\alpha_{xx}+\alpha_{yy}+\alpha_{zz}).$
(6)

If the chosen diatomic molecule lies on the $z$-axis, Eq. (6) can be expressed
as

$\displaystyle\bar{\alpha}=\frac{1}{3}(\alpha_{zz}+2\alpha_{xx}),$ (7)

as here $\alpha_{xx}=\alpha_{yy}$. $\alpha_{zz}$ is the parallel component of
polarizability ($\alpha_{zz}=\alpha_{\parallel}$), and $\alpha_{xx}$ and
$\alpha_{yy}$ constitute the perpendicular components of polarizability
($\alpha_{xx}=\alpha_{yy}=\alpha_{\perp}$). Therefore, Eq. (7) can be
expressed as

$\displaystyle\bar{\alpha}=\frac{1}{3}(\alpha_{\parallel}+2\alpha_{\perp}).$
(8)

Hereafter, we shall follow this notation. Another interesting property that we
report in this work, which is crucial in understanding the alignment of
molecules along the applied external electric field is the polarizability
anisotropy ($\Delta\alpha$) defined as the difference between the parallel and
perpendicular components of polarizability

$\displaystyle\Delta\alpha=(\alpha_{\parallel}-\alpha_{\perp}).$ (9)

Table 1: Values of PDM (in a.u.) of the considered Alk-AlkE molecules from the HF, DF, CCSD, RCCSD, CCSD(T), and RCCSD(T) methods. Molecule | Method | Non-relativistic | Relativistic
---|---|---|---
LiBe | HF/DF | 2.04 | 2.04
| (R)CCSD | 1.39 | 1.39
| (R)CCSD(T) | 1.33 | 1.33
LiMg | HF/DF | 0.33 | 0.34
| (R)CCSD | 0.36 | 0.35
| (R)CCSD(T) | 0.41 | 0.40
LiCa | HF/DF | 0.85 | 0.84
| (R)CCSD | 0.47 | 0.44
| (R)CCSD(T) | 0.43 | 0.40
LiSr | HF/DF | 0.30 | 0.23
| (R)CCSD | 0.18 | 0.13
| (R)CCSD(T) | 0.16 | 0.11
NaBe | HF/DF | 0.09 | 0.12
| (R)CCSD | 0.70 | 0.76
| (R)CCSD(T) | 0.85 | 0.86
NaMg | HF/DF | 0.25 | 0.25
| (R)CCSD | 0.25 | 0.24
| (R)CCSD(T) | 0.31 | 0.30
NaCa | HF/DF | 0.09 | 0.06
| (R)CCSD | 0.45 | 0.42
| (R)CCSD(T) | 0.46 | 0.43
NaSr | HF/DF | 0.09 | 0.08
| (R)CCSD | 0.28 | 0.21
| (R)CCSD(T) | 0.26 | 0.20
KBe | HF/DF | 0.33 | 0.32
| (R)CCSD | 0.58 | 0.57
| (R)CCSD(T) | 0.76 | 0.75
KMg | HF/DF | 0.47 | 0.46
| (R)CCSD | 0.25 | 0.24
| (R)CCSD(T) | 0.37 | 0.35
KCa | HF/DF | 0.05 | 0.08
| (R)CCSD | 0.66 | 0.61
| (R)CCSD(T) | 0.76 | 0.70
KSr | HF/DF | 0.16 | 0.13
| (R)CCSD | 0.57 | 0.44
| (R)CCSD(T) | 0.64 | 0.51
RbBe | HF/DF | 0.45 | 0.40
| (R)CCSD | 0.49 | 0.46
| (R)CCSD(T) | 0.69 | 0.64
RbMg | HF/DF | 0.54 | 0.49
| (R)CCSD | 0.23 | 0.21
| (R)CCSD(T) | 0.37 | 0.33
RbCa | HF/DF | 0.09 | 0.09
| (R)CCSD | 0.69 | 0.60
| (R)CCSD(T) | 0.81 | 0.70
RbSr | HF/DF | 0.04 | 0.14
| (R)CCSD | 0.60 | 0.50
| (R)CCSD(T) | 0.72 | 0.58

##  III Methodology

###  III.1 Approximated CC methods

As have been explained in the previous section, in order to evaluate $\mu$,
$\bar{\alpha}$ and $\Delta\alpha$ using the FF approach, one needs to first
calculate the molecular energies, which in turn require calculating the wave
function of the molecule. As the Schrödinger (the Dirac equation in the
relativistic framework) cannot be solved exactly for many-body systems, we opt
for an approximate method to estimate the wave function. The starting point of
our calculation is the mean-field wave function ($\arrowvert\Phi_{0}\rangle$),
obtained by solving the Hartree-Fock (HF) equations (or the Dirac-Fock (DF)
equations for the relativistic case). To rigorously include the missing
electron correlation effects of the mean-field calculation in the
determination of wave function and energy, we employ the CC method for both NR
and Rel (RCC) framework with the reference wave function
$\arrowvert\Phi_{0}\rangle$ chosen to be the respective HF and DF wave
function. In the (R)CC method, the wave function is expressed as

$\displaystyle\arrowvert\Psi\rangle=e^{T}\arrowvert\Phi_{0}\rangle,$ (10)

where $T=T_{1}+T_{2}+T_{3}+\cdots+T_{N_{e}}$ is the cluster operator, where
$T_{n}$ would denote the $n$-tuple excitation operator, that is, $n$ occupied
orbitals are virtually excited to $n$ unoccupied ones. $N_{e}$ is the total
number of electrons in the molecule. In the singles and doubles excitations
approximation ((R)CCSD method), we retain only $T_{1}$ and $T_{2}$ operators.
In the second-quantization formalism, they are expressed as

$\displaystyle T_{1}$ $\displaystyle=$
$\displaystyle\sum_{i,a}t_{i}^{a}a^{\dagger}_{a}a_{i}$ (11)

and

$\displaystyle T_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\sum_{i,j,a,b}t_{i,j}^{a,b}a^{\dagger}_{a}b^{\dagger}_{b}a_{j}a_{i},$
(12)

where occupied orbitals are denoted by the subscripts $i,$ $j\cdots$ and
virtual orbitals by $a,$ $b\cdots$, and $a_{i}$ defines an annihilation
operator acting on the $i^{th}$ occupied orbital, while $a_{a}^{\dagger}$
refers to a creation operator acting on the $a^{th}$ virtual orbital.
$t_{i}^{a}$ is the amplitude corresponding to a single excitation from
$i^{th}$ occupied to $a^{th}$ virtual orbital while $t_{i,j}^{a,b}$ is the
amplitude corresponding to a doubles excitation from the $i^{th}$ and $j^{th}$
occupied orbitals to $a^{th}$ and $b^{th}$ virtual orbitals, respectively. We
also account for contributions from the dominant triples excitations in the
perturbative approach using the (R)CCSD operators (denoted as (R)CCSD(T)
method) to improve the results.

|
---|---
(a) | (b)
Figure 1: Plots demonstrating relative percentage changes in the values of $\mu$ in the Alk-AlkE molecules due to (a) the electron correlation effects ($\delta^{corr}_{\mu}$) and (b) the relativistic effects ($\delta_{\mu}^{rel}$) through the four-component spinfree Hamiltonian in the RCCSD(T) method. The x-axis shows atomic number of the alkaline-earth atoms.  Table 2: Values of different components of the electric dipole polarizabilities ($\alpha_{\parallel}$ and $\alpha_{\perp}$) as well as average polarizability ($\bar{\alpha}$) and polarizability anisotropy ($\Delta\alpha$) of the considered Alk-AlkE molecules from the HF, DF, CCSD, RCCSD, CCSD(T), and RCCSD(T) methods. All units are in a.u.. |  | Non-relativistic |  | Relativistic
---|---|---|---|---
Molecule |  Method |  $\alpha_{\parallel}$ |  $\alpha_{\perp}$ |  $\bar{\alpha}$ |  $\Delta\alpha$ |  |  $\alpha_{\parallel}$ |  $\alpha_{\perp}$ |  $\bar{\alpha}$ |  $\Delta\alpha$
LiBe |  HF/DF |  211.36 |  88.52 |  129.46 |  122.84 |  |  212.06 |  88.56 |  129.73 |  123.5
|  (R)CCSD |  373.58 |  111.54 |  198.87 |  262.04 |  |  374.13 |  111.6 |  199.11 |  262.53
|  (R)CCSD(T) |  376.55 |  114.42 |  201.78 |  262.13 |  |  376.9 |  114.12 |  201.91 |  262.48
LiMg |  HF/DF |  557.42 |  197.55 |  317.51 |  359.87 |  |  553.14 |  197.66 |  316.15 |  355.48
|  (R)CCSD |  497.42 |  169.19 |  278.6 |  328.23 |  |  495.95 |  169.37 |  278.23 |  326.58
|  (R)CCSD(T) |  481.64 |  166.64 |  271.64 |  315 |  |  480.63 |  166.84 |  271.44 |  313.79
LiCa |  HF/DF |  445.36 |  252.82 |  317 |  192.54 |  |  455.26 |  250.32 |  318.63 |  204.94
|  (R)CCSD |  559.03 |  231.68 |  340.8 |  327.35 |  |  563.01 |  229.97 |  340.98 |  333.04
|  (R)CCSD(T) |  580.16 |  229.88 |  346.64 |  350.28 |  |  584.28 |  228.08 |  346.81 |  356.2
LiSr |  HF/DF |  504.27 |  332.4 |  389.69 |  171.87 |  |  464.91 |  296.6 |  352.7 |  168.31
|  (R)CCSD |  597.14 |  283.13 |  387.8 |  314.01 |  |  560.36 |  268.63 |  365.78 |  291.73
|  (R)CCSD(T) |  622.61 |  276.08 |  391.59 |  346.53 |  |  620.03 |  268.5 |  385.68 |  351.53
NaBe |  HF/DF |  532.87 |  153.96 |  280.26 |  378.91 |  |  536.96 |  147.47 |  277.3 |  389.49
|  (R)CCSD |  396.7 |  144.08 |  228.27 |  252.62 |  |  402.3 |  140.48 |  227.75 |  261.82
|  (R)CCSD(T) |  390.48 |  137.79 |  222.02 |  252.69 |  |  392.99 |  140.2 |  224.46 |  252.79
NaMg |  HF/DF |  490.11 |  220.49 |  310.36 |  269.62 |  |  485.25 |  219.52 |  308.1 |  265.73
|  (R)CCSD |  446.77 |  186.06 |  272.96 |  260.71 |  |  442.72 |  185.37 |  271.15 |  257.35
|  (R)CCSD(T) |  441.11 |  183.04 |  269.06 |  258.07 |  |  437.29 |  182.39 |  267.36 |  254.9
NaCa |  HF/DF |  743.08 |  276.5 |  432.03 |  466.58 |  |  730.78 |  274.73 |  426.75 |  456.05
|  (R)CCSD |  605.78 |  246.26 |  366.1 |  359.52 |  |  600.19 |  245.06 |  363.44 |  355.13
|  (R)CCSD(T) |  590.94 |  243.6 |  359.38 |  347.34 |  |  585.93 |  242.25 |  356.81 |  343.68
NaSr |  HF/DF |  816.5 |  325.57 |  489.21 |  490.93 |  |  784.82 |  316.82 |  472.82 |  468
|  (R)CCSD |  666.07 |  299.65 |  421.79 |  366.42 |  |  648.63 |  284.99 |  406.2 |  363.64
|  (R)CCSD(T) |  652.97 |  296.04 |  415.02 |  356.93 |  |  636.1 |  280.9 |  399.3 |  355.2
KBe |  HF/DF |  763.69 |  433.35 |  543.46 |  330.34 |  |  745.42 |  424.52 |  531.49 |  329.9
|  (R)CCSD |  646.62 |  276.35 |  399.77 |  370.27 |  |  628.81 |  271.91 |  390.88 |  356.9
|  (R)CCSD(T) |  638.91 |  250.30 |  379.84 |  388.61 |  |  621.5 |  246.63 |  371.59 |  374.87
KMg |  HF/DF |  764.42 |  451.98 |  555.86 |  312.84 |  |  749.44 |  443.96 |  545.79 |  305.48
|  (R)CCSD |  671.65 |  315.14 |  433.98 |  356.51 |  |  657.19 |  310.20 |  425.86 |  346.99
|  (R)CCSD(T) |  658.97 |  295.79 |  416.85 |  363.18 |  |  644.47 |  291.31 |  409.03 |  353.16
KCa |  HF/DF |  1209.95 |  462.82 |  711.86 |  747.13 |  |  1171.9 |  468.66 |  703.07 |  703.24
|  (R)CCSD |  956.7 |  347.2 |  550.37 |  609.5 |  |  931.75 |  350.76 |  544.42 |  580.99
|  (R)CCSD(T) |  909.43 |  330.71 |  523.62 |  578.72 |  |  888.38 |  334.4 |  519.06 |  553.98
KSr |  HF/DF |  1339.52 |  505.45 |  783.47 |  834.07 |  |  1249.65 |  498.21 |  748.69 |  751.44
|  (R)CCSD |  1026.44 |  388.06 |  600.85 |  638.38 |  |  975.42 |  387.44 |  583.43 |  587.98
|  (R)CCSD(T) |  971.74 |  372.68 |  572.37 |  599.06 |  |  928.23 |  372.13 |  557.5 |  556.1
RbBe |  HF/DF |  819.56 |  554.86 |  643.09 |  264.7 |  |  773.14 |  507.91 |  576.32 |  265.23
|  (R)CCSD |  697.67 |  333.56 |  454.93 |  364.11 |  |  648.61 |  312.05 |  424.24 |  336.56
|  (R)CCSD(T) |  692.69 |  295.05 |  427.6 |  397.64 |  |  644.08 |  278.14 |  400.12 |  365.94
RbMg |  HF/DF |  862.3 |  567.18 |  665.55 |  295.12 |  |  806.75 |  523.44 |  617.88 |  283.31
|  (R)CCSD |  740.42 |  366.43 |  491.09 |  373.99 |  |  684.47 |  343.39 |  457.08 |  341.08
|  (R)CCSD(T) |  720.48 |  335.83 |  464.05 |  384.65 |  |  667.39 |  316.44 |  433.42 |  350.95
RbCa |  HF/DF |  1372.88 |  567.82 |  836.17 |  805.06 |  |  1260.82 |  533.44 |  775.9 |  727.38
|  (R)CCSD |  1072.62 |  393.68 |  619.99 |  678.94 |  |  986.42 |  380.38 |  582.39 |  606.04
|  (R)CCSD(T) |  1015.59 |  368.26 |  584.04 |  647.33 |  |  939.92 |  357.71 |  551.78 |  582.21
RbSr |  HF/DF |  1526.59 |  599.47 |  908.51 |  927.12 |  |  1349.35 |  562.62 |  824.86 |  786.73
|  (R)CCSD |  1150.27 |  438.2 |  675.56 |  712.07 |  |  1027.13 |  407.66 |  614.25 |  619.77
|  (R)CCSD(T) |  1080.95 |  414.26 |  636.49 |  666.69 |  |  970.25 |  385.59 |  580.48 |  584.66

###  III.2 Bond-lengths and basis functions

We have used the equilibrium bond-lengths of the Alk-AlkE molecules from Ref.
[39], where the authors carried out MRCI calculations to obtain the ground
state energies of the chosen molecules, and obtained the equilibrium bond-
lengths from the minima of the corresponding potential energy curves. For
lighter elements (Li, Be, Na, Mg), we used correlation-consistent polarized
core-valance quadruple zeta (cc-pCVQZ) basis sets [40, 41], and for the
heavier ones (K, Ca, Rb, Sr), we used Dyall’s quadruple zeta (QZ) [42] basis
functions. For the reduction of computational cost, at the (R)CC level, we
have cut-off virtual orbitals having energies greater than 1000 a.u. for the
relatively heavier molecules (NaSr, KSr, RbBe, RbMg, RbCa and RbSr). All the
computations were carried out using the DIRAC program [43].

##  IV Results and discussion

We investigate the trends in relativistic as well as electron correlation
effects to the undertaken properties in the Alk-atom family-wise. For example,
Li-family refers to LiX molecules, where X could be Be, Mg, Ca, or Sr atom.
The rationale behind categorizing the chosen molecules as families rather than
looking for trends based on electron number is due to the fact that two
molecules that are next to each other in the number of electrons can display
very dissimilar trends, since one molecule may contain a combination of light-
heavy nuclei, while the other moderate-moderate nuclei. There is no
particularly strong reason to provide our results based on Alk-atom family
rather than AlkE-atom family, and therefore when certain trends are easier to
see with the latter, we have appropriately discussed them in the main text.
For the purpose of analyzing the trends, we use two quantities in the
subsequent sub-sections – one for correlation effects and the other for
relativistic ones. For the former, we define
$\delta^{corr}_{P}=\left(\frac{P_{(R)CC}-P_{DF/HF}}{P_{DF/HF}}\times
100\right)$ that signifies the relative percentage of correlation
contributions to the property, $P$. For the latter, we define
$\delta^{rel}_{P}=\left(\frac{P_{Rel}-P_{NR}}{P_{NR}}\times 100\right)$.

###  IV.1 Results for PDM

In Table 1, we present the values of PDM for all the considered systems from
both the NR and Rel calculations. To demonstrate the roles of electron
correlation effects explicitly, we give values from the HF/DF, (R)CCSD, and
(R)CCSD(T) methods for all the molecules. We observe from the table that
electron correlation plays a significant role in determining the quality of
the final values. These effects are largest in the RCCSD(T) values for NaCa
(where the percentage fraction correlation is about 620), KCa (775), and RbCa
(680). In the LiX family, the magnitudes of the PDM decrease gradually from
the HF method to the CCSD(T) method, except in the case of LiMg, which shows
the opposite behaviour. However, this trend changes for NaX family, where the
values increase from the HF method to the CCSD(T) method, with two exceptions
– NaSr, where the PDM increases from HF to CCSD, but decreases slightly
(within error margins) in the CCSD(T) method, and NaMg, where the relativistic
results decrease ever so slightly (again within error margins) from HF to
CCSD, and then show a clear increase in the magnitude of the PDM. For both the
KX and RbX families, the values increase from the HF method to the CCSD(T)
method gradually, except for KMg and RbMg. In summary, for non-Mg containing
molecules, we observe from our results for PDM that except for the lightest Li
family, where correlation effects decrease the PDM, the general tendency of
electron correlation is to increase the value of the property. For those
chosen molecules that contain Mg, correlation effects increase the PDM in the
lighter Li and Na families, whereas for the heavier K and Rb families,
correlation decreases the PDM. Fig. 1(a), which plots the related quantity,
$\delta^{corr}_{\mu}$ for the relativistic case for the four families, shows
these observed trends. An interesting observation that arises from the
distribution of $\delta^{corr}_{\mu}$ for the Mg family in the figure is that
$|\mu_{corr}/\mu_{HF}|$ is almost a constant, especially relative to the same
quantity for the other families.

|
---|---
(a) | (b)
Figure 2: Schematic figures showing relative percentage changes in (a) the ${\bar{\alpha}}$ values and (b) the ${\Delta{\alpha}}$ values of the Alk-AlkE molecules due to the electron correlation effects at the RCCSD(T) method. The values in y-axis are given in % while x-axis shows atomic number of the alkaline-earth atom for a given alkali atom family. |
---|---
(a) | (b)
Figure 3: Schematic figures showing relative percentage changes in (a)
${\delta_{\bar{\alpha}}^{rel}}$, and (b) ${\delta_{\Delta{\alpha}}^{rel}}$ in
the Alk-AlkE molecules using the RCCSD(T) method. The x-axis provides atomic
number of the alkaline-earth atom, as in the previous figure.

Having discussed correlation effects, we now compare the results for PDM
between the non-relativistic and relativistic methods, from the data presented
in Table 1 and Fig. 1(b), with the figure plotting percentage fraction
difference due to relativity at the CCSD(T) level of theory. The first
observation that we can draw from our results is that the effect of relativity
is to lower the value of PDM in most cases. The only exceptions are the
lightest LiBe, where there is practically no difference between NR and Rel
values, and NaBe, where relativistic effects are around 1 percent. This is
reflected in Fig. 1(b), where for each of the families, the slope is negative.
The figure also shows that in each family, the importance of relativity
increases with the atomic number of the AlkE atom. However, the rate of change
is non-trivial, with crossings observed between families. We also note that
the importance of relativistic effects is most pronounced in the heaviest
molecules considered of each family, with the PDM of LiSr changing the most
with the inclusion of relativistic effects (31 percent). In summary, while the
PDM is lowered when one switches to a relativistic framework from a non-
relativistic one, the degree to which the quantity decreases increases with
the atomic number of a given AlkE atom. Note that this is not directly obvious
visually from Fig. 1(b), since the x-axis is not linear.

Relativistic effects impact the results both at mean field and the correlation
levels of theory. In the former, we see that relativity accounts for as much
as 250 percent for RbSr, but also note that this is an exception; they are at
most about 33 percent in the the rest of the molecules. In the correlation
sector, inclusion of relativity can decrease the correlation value by as large
as 35 percent for RbSr.

###  IV.2 Results for polarizability

In Table 2, we present values of $\alpha_{\parallel}$, $\alpha_{\perp}$,
$\bar{\alpha}$, and $\Delta\alpha$, both from NR and Rel calculations, and in
each of these cases, we provide results at mean field, CCSD, and CCSD(T)
levels of theory. We note that the individual values as well as the trends for
$\bar{\alpha}$ and $\Delta\alpha$ are obviously influenced by those in
$\alpha_{\parallel}$ and $\alpha_{\perp}$. Therefore, in view of the former
two quantities being the ones that are obtained in experiment, we focus on the
individual values and trends of these properties. Otherwise, the discussion of
the flow of results in this sub-section will be very similar to that from the
previous one.

We begin with a discussion on correlation effects, based on data from Table 2
and Fig. 2. We expect the trends in polarizabilities to be at least somewhat
different from those in PDMs not only because the quantities are intrinsically
different, but also because the stress would not be on $\alpha_{\parallel}$
and $\alpha_{\perp}$, but rather on $\bar{\alpha}$ and $\Delta\alpha$. We
observe that electron correlation influences the results significantly, with
the largest percentage fraction difference between the RCCSD(T) and DF results
occurring for the lightest LiBe molecule (about 60 percent for $\bar{\alpha}$
and about 110 percent for $\Delta\alpha$). However, it is not as striking as
in the case of PDMs, where we encountered changes as large as about 775
percent. Trend-wise, the plots in sub-figures 2(a) and 2(b) display an almost
monotonic trend for each of the families, with the molecules containing Be in
each family being an exception, unlike in the case of PDMs. Note that in the
non-relativistic case, the correlation trends are similar for $\Delta\alpha$,
but are slightly different in the case of $\bar{\alpha}$. One can also observe
from sub-figures 2(a) and 2(b) that with the exception of the Li family, the
correlation trends vary relatively mildly for the rest of the families for
$\bar{\alpha}$, whereas for $\Delta\alpha$, we see more variation. This
indicates that accurate determination of the values of $\Delta\alpha$ for the
Alk-AlkE systems would be more sensitive to proper inclusion of the electron
correlation effects. We observe similar signs of the peculiarity seen in the
Mg family of Fig. 1 in our relativistic results, and in this case, we find
that $|{\Delta\alpha}_{corr}/{\Delta\alpha}_{HF}|$ varies within around 30
percent, which is not small but still significantly smaller relative to other
families. The same quantity constructed for $\bar{\alpha}$ also varies within
30 percent across the Mg family, which is again much lesser than the maximum
variation seen in each of the other families. In summary, we find that
correlation effects play a very important role in determining $\bar{\alpha}$
and $\Delta\alpha$, but not as extreme as in the case of PDMs. We also observe
that the trends for both $\bar{\alpha}$ and $\Delta\alpha$ are smoother than
the trend found for the PDMs. Lastly, we see a similar behaviour in
$|P_{corr}/P_{HF}|$ for the Mg family as it was for PDMs.

Table 3: The values of PDM and parallel component of polarizabilities of the LiBe, KBe, and RbBe molecules using different Hamiltonians in the RCCSD(T) method. All units are in a.u..  Molecule | $\mu$ | $\alpha_{\parallel}$
---|---|---
| Dirac-Coulomb
LiBe | 1.33 | 376.96
KBe | 0.75 | 621.64
RbBe | 0.64 | 644.01
| 4-component spinfree
LiBe | 1.33 | 376.9
KBe | 0.75 | 621.5
RbBe | 0.64 | 644.08
| Spinfree-X2C
LiBe | 1.32 | 376.86
KBe | 0.75 | 621.76
RbBe | 0.64 | 644.58
| DKH2
LiBe | 1.32 | 376.99
KBe | 0.72 | 620.31
RbBe | 0.64 | 644.87
| Non-relativistic
LiBe | 1.32 | 376.55
KBe | 0.76 | 638.91
RbBe | 0.69 | 692.69

We will now turn our attention to the importance of relativistic effects on
polarizabilities. From Fig. 3(a), we immediately notice that the effect of
relativity is to decrease the value of $\bar{\alpha}$ in each of the families,
with the Li family being an exception. On the other hand, relativity increases
the value of $\Delta\alpha$ for six out of the sixteen systems considered. The
trends are a reflection of those in $\alpha_{\parallel}$ and $\alpha_{\perp}$,
as $\bar{\alpha}$ involves a sum of these quantities, while $\Delta\alpha$ is
constructed from the difference between them. When we compare the importance
of relativistic and correlation effects, we find that while the former can be
as large as 10 percent and therefore not negligible in their own right, the
latter is more pronounced, and is as large as 120 percent for LiBe. In
summary, the relativistic trends in $\bar{\alpha}$ and $\Delta\alpha$ are not
as clear as in the case of PDM, and relativistic effects are not as dominant
as correlation effects, while being non-negligible on their own.

Table 4: The values of PDM and polarizability of four representative molecules (LiBe, KBe, RbBe, and RbSr) with basis sets of increasing cardinal number (DZ to QZ) in the RCCSD(T) method, and with the four-component spinfree relativistic Hamiltonian. We also give results from a CBS extrapolation scheme. All units are in a.u.. Molecule | Basis | $\mu$ | $\alpha_{\parallel}$ | $\alpha_{\perp}$ | $\bar{\alpha}$ | $\Delta\alpha$
---|---|---|---|---|---|---
LiBe | DZ | 1.1 | 387.33 | 114.73 | 205.6 | 272.6
| TZ | 1.27 | 378.12 | 114.06 | 202.08 | 264.06
| QZ | 1.33 | 376.9 | 114.12 | 201.91 | 262.48
| CBS | 1.37 | 376.01 | 114.69 | 201.8 | 261.32
KBe | DZ | 0.35 | 555.73 | 257.82 | 357.12 | 297.91
| TZ | 0.65 | 608.5 | 248.04 | 368.19 | 360.46
| QZ | 0.75 | 621.5 | 246.63 | 371.59 | 374.87
| CBS | 0.81 | 630.99 | 245.6 | 374.06 | 385.39
RbBe | DZ | 0.27 | 577.9 | 289.86 | 385.87 | 288.04
| TZ | 0.55 | 627.78 | 279.22 | 395.41 | 348.56
| QZ | 0.64 | 644.08 | 278.14 | 400.12 | 365.94
| CBS | 0.71 | 655.97 | 277.34 | 403.55 | 378.63
RbSr | DZ | 0.26 | 915.82 | 387.95 | 563.91 | 527.87
| TZ | 0.52 | 957.48 | 384.35 | 575.39 | 573.13
| QZ | 0.58 | 970.25 | 385.59 | 580.48 | 584.66
| CBS | 0.62 | 979.56 | 386.49 | 584.18 | 593.07

###  IV.3 Reliability tests

In order to present our recommended values along with their uncertainties from
our calculations, we first assess the reliability of our results in the form
of multiple precision checks. The precision of a calculation would depend on
the choice of Hamiltonian and the wave function. The choice of wave function
entails selecting the single particle basis as well as the quantum many-body
theory to employ. Further, since we adopt the FF approach, we also need to
check the dependence of the precision in our results on the choice of
perturbing parameter. We will address each of these aspects very briefly in
the next few paragraphs, in order to arrive at a reasonable estimate for the
recommended values of the properties for the chosen molecules. Note that
choosing a better stencil than our simplest central difference scheme would
have little effect on results given that the perturbing parameter, $\epsilon$,
is already small. For carrying out these reliability tests, we choose three
representative systems, which include the lightest LiBe, the slightly heavier
KBe, and the moderately heavy RbBe. In view of the steep computational cost,
we did not choose the heavier systems for reliability tests.

The first of our precision checks is the influence of the choice of
Hamiltonian. For this analysis, we assume that our RCCSD(T) values are the
more accurate results compared to those from the RCCSD method due to the fact
that the former takes into account more physical effects. We consider a
hierarchy of Hamiltonians in terms of physical effects included as well as the
resources that they consume. They include the computationally very expensive
four-component Dirac-Coulomb Hamiltonian, the four-component spinfree
Hamiltonian that we used for all our main results, an exact two-component
Hamiltonian (spinfree X2C) [44], an approximate two-component Hamiltonian (we
choose second-order Douglas-Kroll-Hess [45, 46, 47] Hamiltonian for this
purpose), and finally the non-relativistic Hamiltonian. Note that the four-
component Dirac-Coulomb Hamiltonian (DCH) does not include the Coulomb
integrals of the type SSSS (where S stands for small component), but rather
replaced by interatomic corrections [48, 49]. This approximation is known to
be very accurate, and save lots of computational time. In spite of this
approximation, the DCH is very expensive. In view of this situation, we only
test the dependence of results on the choice of Hamiltonian for $\mu$ and
$\alpha_{\parallel}$, and assume that the difference in results between any
two chosen Hamiltonians would be comparable for $\alpha_{\perp}$ too. The
results in Table 3 show that the PDM of the considered systems are practically
unchanged between the DCH and our four-component spinfree calculations. We
also observe a similar behaviour for $\alpha_{\parallel}$. This indicates that
the effect of spin-orbit coupling is negligible in these systems, and we
expect a similar behaviour to hold for the other Alk-AlkE molecules. We also
see that the results agree very well with the spinfree X2C Hamiltonian. Note
that spinfree X2C Hamiltonian, which is a two-component Hamiltonian, still
contains in it relativistic effects (excluding spin-orbit coupling). We also
see no noteworthy deviations when we compare our results with DKH2
Hamiltonian. We take this a step further by carrying out calculations of the
PDM and $\alpha_{\parallel}$ for the heaviest RbSr molecule with spinfree X2C
Hamiltonian, and find that the PDM remains unchanged while
$\alpha_{\parallel}$ changes well within 1 percent.

Table 5: The PDM and dipole polarizabilities (all in atomic units) of the LiBe, KBe, and RbBe molecules with different choices of the perturbing parameter, $\epsilon$. Molecule | $\epsilon$ | $\mu$ | $\bar{\alpha}$ | $\Delta\alpha$
---|---|---|---|---
LiBe | 0.00005 | 1.33 | 203.77 | 259.68
| 0.0001 | 1.33 | 201.91 | 262.48
| 0.0005 | 1.32 | 201.29 | 263.92
KBe | 0.00005 | 0.75 | 370.42 | 376.64
| 0.0001 | 0.75 | 371.59 | 374.87
| 0.0005 | 0.75 | 372.08 | 372.98
RbBe | 0.00005 | 0.64 | 404.22 | 359.91
| 0.0001 | 0.64 | 400.12 | 365.94
| 0.0005 | 0.64 | 399.18 | 366.73

Another important test of reliability of our calculations is to analyze the
basis set dependence of the results. To that end, we have performed
calculations using the four-component spinfree relativistic Hamiltonian in the
RCCSD(T) method with Dyall’s triple-zeta (TZ) and double-zeta (DZ) basis
functions. Again, we have considered LiBe, KBe and RbBe as representative
molecules. Further, we carry out a two-point complete basis set (CBS) [50, 51]
extrapolation with our TZ and QZ results upto $\zeta=50$. We enlist the
RCCSD(T) values of $\mu$, $\bar{\alpha}$ and $\Delta\alpha$ using the
aforementioned basis sets in Table 4. The table shows that $\mu$ increases and
converges systematically from cardinal numbers 1 through 50. However, we see a
relatively strong dependence of PDMs on basis, as the results change from QZ
to the CBS limit by about $4\%$ for LiBe, 8$\%$ for KBe, and 11$\%$ for RbBe.
We extended our analysis to the heaviest RbSr molecule in view of this
dependence being important, and found that the percentage fraction difference
between results using QZ basis and with CBS extrapolation is 7.5$\%$. We carry
out the same analysis for polarizabilities too, and we find that the QZ basis
is reasonably good, with the percentage fraction differences between QZ and
CBS extrapolation being utmost $1\%$ for $\bar{\alpha}$ and $4\%$ for
$\Delta\alpha$.

The other possible source of uncertainty in our calculation could stem from
the contributions from high-lying orbitals that are not considered to account
for electron correlation effects in the RCCSD(T) method. To carry out the
calculations with the available computational resources, we have imposed a
cut-off to virtual orbitals having more than energies at 1000 a.u. for
relatively heavier systems. To get an estimate of error arising from the
neglected higher virtual orbitals, we calculate PDMs and polarizabilities for
RbBe, by considering all the generated virtual orbitals in the QZ basis. We
found that the changes in the values are insignificant and within the
precision of the present interest. This further assured us about the precision
in the calculations.

We now consider the error due to the missing triple excitations in the CC
method. Since CCSD(T) is widely regarded as a gold standard for molecular
property calculations, we expect that the error due to missing triple
excitations from RCCSD(T) would be much lesser than the percentage fraction
difference between the RCCSD and the RCCSD(T) values, and assume that it would
be roughly half its magnitude or lower. We find that the percentage fraction
difference between the RCCSD and the RCCSD(T) methods is at most about 6
percent for $\bar{\alpha}$, and is within 7 percent for $\Delta\alpha$, except
in the case of LiSr, where it is significant at about 17 percent. However, for
the PDM, we find that the percentage fraction difference is well over 15
percent for some of the chosen systems, and is as large as 57 percent in the
case of RbMg. We therefore expect that while the error due to missing higher
order excitations are usually reasonably small for polarizabilities, the PDM
is much more sensitive to this factor. Hence, we defer detailed analysis to a
future study and conservatively set an error estimates in the PDM as mentioned
in the beginning of this paragraph.

Another factor that needs to be taken into account in determining the
precision of our results is the dependence on the choice of $\epsilon$. We
again choose our representative molecules LiBe, KBe and RbBe. We check the
change in our results in the neighborhood of $\epsilon=10^{-4}$ a.u., as shown
in Table 5, and find that the results are stable in that range, with
negligible variation.

Table 6: The estimated $\bar{\alpha}$ values (in a.u.) of the considered Alk-AlkE molecules using the empirical relation given by Eq. (13). In the table, the entry in the first row and third column of the results, for example, corresponds to LiCa.  Alk AlkE |  Be |  Mg |  Ca |  Sr
---|---|---|---|---
Li |  218.41 |  250.69 |  348.71 |  374.54
Na |  216.81 |  250.82 |  353.17 |  386.97
K |  344.42 |  381.07 |  495.26 |  544.84
Rb |  375.47 |  412.58 |  527.57 |  583.29
---
(a)
(b)
(c)
Figure 4: (a) Plot showing the agreement between average polarizabilities calculated using RCCSD(T) method (in blue) and our empirical relation (in red). The dominant part of the empirical relation is shown in green. We find that the red and the blue curves agree to within 10 percent for each of the points, whereas the green curve deviates from the blue one for heavier systems. The shaded background regions have been given to distinguish between families. Sub-figures (b) and (c) show the same plots, but for alkali-alkali molecules and alkaline-earth–monofluorides, respectively. The RCCSD(T) values of PDM and $\bar{\alpha}$ given in sub-figure (b) were taken from our previous work [33], whereas the ab initio values of PDM and $\bar{\alpha}$ in sub-figure (c) were taken from Ref. [52] and Ref. [53], respectively.  Table 7: Our final recommended results for $\mu$, $\alpha_{\parallel}$, $\alpha_{\perp}$, $\bar{\alpha}$, and $\Delta\alpha$ from RCCSD(T) calculations, along with the estimated uncertainties that are quoted in the parentheses. We have also compared our results with the previously reported values using MRCI and CCSD(T) methods. $\bar{\alpha}$ and $\Delta\alpha$ are rounded-off to the nearest whole number, given their large values. All the results are given in a.u.. Molecule |  $\mu$ |  $\alpha_{\parallel}$ |  $\alpha_{\perp}$ |  $\bar{\alpha}$ |  $\Delta\alpha$ |  Method |  Reference
---|---|---|---|---|---|---|---
LiBe |  1.33(18) |  376.9 |  114.12 |  202(3) |  263(11) |  RCCSD(T) |  This work
|  1.32 |  376.55 |  114.42 |  201.78 |  262.13 |  CCSD(T) |  This work
|  1.36 |  365 |  |  |  |  MRCI |  [39]
LiMg |  0.40(7) |  480.63 |  166.84 |  271(6) |  314(19) |  RCCSD(T) |  This work
|  0.41 |  481.64 |  166.64 |  271.64 |  315 |  CCSD(T) |  This work
|  0.46 |  470 |  |  |  |  MRCI |  [39]
LiCa |  0.40(6) |  584.28 |  228.08 |  347(6) |  356(26) |  RCCSD(T) |  This work
|  0.43 |  580.16 |  229.88 |  346.64 |  350.28 |  CCSD(T) |  This work
|  0.43 |  594 |  230 |  352 |  364 |  CCSD(T) |  [38]
|  0.47 |  588 |  |  |  |  MRCI |  [39]
LiSr |  0.11(2) |  620.03 |  268.5 |  386(14) |  352(44) |  RCCSD(T) |  This work
|  0.16 |  622.61 |  276.08 |  391.59 |  346.53 |  CCSD(T) |  This work
|  0.12 |  621 |  271 |  395 |  372 |  CCSD(T) |  [38]
|  0.11 |  653 |  |  |  |  MRCI |  [39]
NaBe |  0.86(15) |  392.99 |  140.2 |  224(4) |  253(15) |  RCCSD(T) |  This work
|  0.85 |  390.48 |  137.79 |  222.02 |  252.69 |  CCSD(T) |  This work
|  0.92 |  397 |  |  |  |  MRCI |  [39]
NaMg |  0.30(7) |  437.29 |  182.39 |  267(5) |  255(11) |  RCCSD(T) |  This work
|  0.31 |  441.11 |  183.04 |  269.06 |  258.07 |  CCSD(T) |  This work
|  0.34 |  432 |  |  |  |  MRCI |  [39]
NaCa |  0.43(5) |  585.93 |  242.25 |  357(5) |  344(27) |  RCCSD(T) |  This work
|  0.46 |  590.94 |  243.6 |  359.38 |  347.34 |  CCSD(T) |  This work
|  0.39 |  581 |  240 |  354 |  361 |  CCSD(T) |  [38]
|  0.46 |  577 |  |  |  |  MRCI |  [39]
NaSr |  0.20(3) |  636.1 |  280.9 |  399(7) |  355(18) |  RCCSD(T) |  This work
|  0.26 |  652.97 |  296.04 |  415.02 |  356.93 |  CCSD(T) |  This work
|  0.19 |  633 |  281 |  398 |  352 |  CCSD(T) |  [38]
|  0.20 |  636 |  |  |  |  MRCI |  [39]
KBe |  0.75(20) |  621.5 |  246.63 |  372(13) |  375(24) |  RCCSD(T) |  This work
|  0.76 |  638.91 |  250.30 |  379.84 |  388.61 |  CCSD(T) |  This work
|  0.87 |  628 |  |  |  |  MRCI |  [39]
KMg |  0.35(12) |  644.47 |  291.31 |  409(12) |  353(17) |  RCCSD(T) |  This work
|  0.37 |  658.97 |  295.79 |  416.85 |  363.18 |  CCSD(T) |  This work
|  0.42 |  656 |  |  |  |  MRCI |  [39]
KCa |  0.70(13) |  888.38 |  334.4 |  519(17) |  554(36) |  RCCSD(T) |  This work
|  0.76 |  909.43 |  330.71 |  523.62 |  578.72 |  CCSD(T) |  This work
|  0.64 |  892 |  326 |  515 |  566 |  CCSD(T) |  [38]
|  0.83 |  869 |  |  |  |  MRCI |  [39]
KSr |  0.51(10) |  928.23 |  372.13 |  558(18) |  556(38) |  RCCSD(T) |  This work
|  0.64 |  971.74 |  372.68 |  572.37 |  599.06 |  CCSD(T) |  This work
|  0.50 |  942 |  367 |  559 |  574 |  CCSD(T) |  [38]
|  0.60 |  925 |  |  |  |  MRCI |  [39]
RbBe |  0.64(20) |  644.08 |  278.14 |  400(15) |  366(29) |  RCCSD(T) |  This work
|  0.69 |  692.69 |  295.05 |  427.6 |  397.64 |  CCSD(T) |  This work
|  0.78 |  631 |  |  |  |  MRCI |  [39]
RbMg |  0.33(13) |  667.39 |  316.44 |  433(16) |  351(19) |  RCCSD(T) |  This work
|  0.37 |  720.48 |  335.83 |  464.05 |  384.65 |  CCSD(T) |  This work
|  0.41 |  664 |  |  |  |  MRCI |  [39]
RbCa |  0.70(14) |  939.92 |  357.71 |  552(20) |  582(35) |  RCCSD(T) |  This work
|  0.81 |  1015.59 |  368.26 |  584.04 |  647.33 |  CCSD(T) |  This work
|  0.68 |  961 |  357 |  558 |  604 |  CCSD(T) |  [38]
|  0.86 |  922 |  |  |  |  MRCI |  [39]
RbSr |  0.58(11) |  970.25 |  385.59 |  581(22) |  585(41) |  RCCSD(T) |  This work
|  0.72 |  1080.95 |  414.26 |  636.49 |  666.69 |  CCSD(T) |  This work
|  0.55 |  1009 |  394 |  599 |  615 |  CCSD(T) |  [38]
|  0.64 |  972 |  |  |  |  MRCI |  [39]

###  IV.4 A simple empirical relation between PDM and average polarizability

In this subsection, we seek to explore interesting connections between $\mu$
and $\bar{\alpha}$. Previous works have sought such empirical relationships,
but they usually give complicated functions; for example, the authors in Ref.
[39] find such a function connecting PDM and average atomic polarizabilities.
We intend to find a relatively simple relation,
$\bar{\alpha}=f(\mu,Z_{Alk},Z_{AlkE},\bar{\alpha}_{at})$, whose predictions
for $\bar{\alpha}$ should agree reasonably well with our RCCSD(T) results.
Here, $Z_{Alk}$ and $Z_{AlkE}$ are the atomic numbers of the alkali and
alkaline earth atoms, and $\bar{\alpha}_{at}$ is the average atomic
polarizability, given by
${\bar{\alpha}}_{at}=(\alpha_{Z_{Alk}}+\alpha_{Z_{AlkE}})/2$. We find that
$\bar{\alpha}$ predicted by the empirical relation

$\displaystyle\bar{\alpha}=[a{\bar{\alpha}}_{at}+\pi ln(Z_{Alk})+b\mu
Z_{AlkE}]/c,$ (13)

with $a=2$, $b=2.5$, and $c=1$. This agrees reasonably well with our RCCSD(T)
results of $\bar{\alpha}$. We note that the dominant contribution comes from
the first term, that is, the term containing average atomic polarizability.
However, as Fig. 4(a) shows, using just the first term does not reproduce the
RCCSD(T) results for $\bar{\alpha}$ well for heavier systems in each family.
The next two terms play an important role in improving the predictability of
the empirical relation. Within those two terms, the result is not strongly
influenced by the term containing the atomic number of the alkali atom, owing
to the $log$ dependence. An interesting occurrence in the expression is the
presence of $\pi$. We have used the ${\bar{\alpha}}_{at}$ values from Ref.
[54] in our analyses. In Table 6, we have listed $\bar{\alpha}$ estimated
using the above relation. By comparing these results with the corresponding
values from Table 2, it is clear that the relation predicts $\bar{\alpha}$
well within 10 percent of the RCCSD(T) values. It is worth commenting at this
point that we have made a conscious choice to exclude the dependence of the
function on electronegativity differences, as the PDMs that we obtained do not
follow the straightforward and simplistic dependence on electronegativity
differences.

We take the applicability of the empirical relation a step further, and check
if we can reproduce the correct value of molecular polarizability of LiBa,
given its PDM. For this purpose, we use the RCCSD(T) results for the PDM, and
find that the average polarizability thus obtained using our empirical
relation (459.09 a.u.) matches remarkably well with the RCCSD(T) result for
the same quantity (443.69 a.u.), to within 4 percent.

Lastly, we check if the function from Eq. (13) can faithfully predict
$\bar{\alpha}$s for at least a few other diatomic molecules. For this purpose,
we choose two other systems that find important applications, heteronuclear
Alk-Alk and AlkE-F molecules. For the former, the empirical relation that
agrees reasonably well (within 15 percent) with results from RCCSD
calculations (both PDM and $\bar{\alpha}$ have been taken from Ref. [33]) is
with $a=1.5$, $b=2.5$, and $c=1$. i.e.

$\displaystyle\bar{\alpha}=1.5{\bar{\alpha}}_{at}+\pi ln(Z_{Alk1})+2.5\mu
Z_{Alk2}.$ (14)

Here, Alk2 can be Li, Na, K, or Rb. For a given Alk2, Alk1 is either Li, Na,
K, or Rb. We also test the quality of the results obtained from the above
mentioned equation for homonuclear alkali-alkali molecules. We borrow the
values of average polarizabilities from Deiglmayr et al [55] for comparison,
and find that Eq. (14) gives results that are in reasonable agreement (within
10 percent, except in the case of Li2, which gives 20 percent).

A similar empirical equation is found to give reasonably good agreement
(within 15 percent) with ab initio theory for AlkE-F molecules when $a=2$,
$b=2.5$, and $c=2$. i.e.

$\displaystyle\bar{\alpha}=[2{\bar{\alpha}}_{at}+\pi ln(Z_{F})+2.5\mu
Z_{AlkE}]/2.$ (15)

For this purpose, we chose the PDMs from our previous work [52], and
$\bar{\alpha}$s from Ref. [53]. The former work uses RCCSD method, while the
latter employs Kramers Restricted Configuration Interaction (KRCI) method. It
is worth noting that in spite of the values of $\mu$ and $\bar{\alpha}$ being
taken from different relativistic many-body methods, the empirical data
generated from Eq. (15) agrees well with theory. This is not very surprising,
as the KRCI results for the PDMs agree well with the corresponding RCCSD
results for these systems. The causes behind the seemingly unreasonable
effectiveness of the simple functional form based on Eq. (13) are unclear, and
future studies on these aspects with more systems may shed light on the
rationale behind these observations. Nonetheless, we assume that
$\bar{\alpha}$ of other heavier Alk-AlkE molecules, which are not considered
here, will also obey our above suggested empirical relation.

###  IV.5 Recommended values

After analysing the reliability of our results and possible uncertainties in
their evaluation using the RCCSD(T) method, we would like to quantify now
their total uncertainties to the investigated properties. From the above
discussions, it is clear that the uncertainties to our calculated values of
$\mu$, $\bar{\alpha}$ and $\Delta\alpha$ come mainly from the dependence on
basis functions and missing higher order coupled-cluster excitations. From our
CBS extrapolation results in an earlier section, we conservatively assign a
maximum uncertainty in the PDMs of about 11$\%$, about 1$\%$ uncertainties to
$\bar{\alpha}$, and about 4$\%$ to $\Delta\alpha$. The uncertainties due to
the neglected electron correlation effects are calculated for each of the
molecules and are added linearly to the error arising from incompleteness of
basis, and are quoted in Table 7 in brackets next to our recommended values,
which are our RCCSD(T) results. In the same table, we also give the previously
calculated values of some of the above quantities that are obtained using the
MRCI method [39] and non-relativistic CCSD(T) method [38]. Our RCCSD(T) values
and the MRCI values for the PDMs mostly agree within our quoted uncertainties,
thus reinforcing our error estimates. It can be noted that the values of $\mu$
from the CCSD(T) calculations in Ref. [38] differ from the present CCSD(T)
results listed in Table 1. This may be due to the fact that different basis
functions (their ANO-RCC contracted bases versus our cc-pCVQZ and Dyall bases)
are used in both the works. Moreover, we carry out all-electron calculations,
and we expect that to have a non-negligible bearing on the precision of our
results, whereas the authors in Ref. [38] freeze inner orbitals. We did not
find any other calculations of the $\bar{\alpha}$ and $\Delta\alpha$ values
apart from the CCSD(T) calculations in Ref. [38].

##  V Conclusion

We have investigated the roles of relativistic and electron correlation
effects in the determination of the PDMs and static electric dipole
polarizabilities of the alkali-alkaline earth heteronuclear dimers. For this
purpose, we have employed finite-field coupled-cluster approach in the
CCSD(T)) method in both non-relativistic and relativistic frameworks. We find
that electron correlation effects can have a significant impact on the final
values of the PDMs of some of the considered molecules. Trend-wise, we observe
that while correlation effects decrease the PDMs in the Li family, they
increase it for the rest of the families, with Mg-containing molecules being
exceptions. We also find that for molecules containing Mg, the ratio of the
correlation contribution to the PDM to the mean field value of PDM is almost a
constant. We find that relativistic effects decrease the PDMs in most cases.
The importance of relativistic effects increase from the lighter to the
heavier molecules in each family, as expected. Correlation effects play an
important role and are pronounced in polarizabilities too, but not as much as
in the case of PDMs. Moreover, the observed correlation trends in
polarizability anisotropy indicate that the property is likely more sensitive
to correlation effects than the average polarizability. Relativistic effects
in polarizabilities are found to be non-negligible but are much less prominent
than in the case of PDMs. The analysis of relativistic and correlation effects
is followed by tests of precision in our results, where we find that the
errors due to spin-orbit coupling is negligible. We also test the stability of
our numerical results by varying the perturbing parameter in the neighborhood
of the chosen value for this work. We observe that the uncertainties due to
choice of basis sets and missing coupled-cluster excitations are the dominant
error sources. We find a simple and interesting empirical functional form that
connects PDM and average polarizability reasonably well and does so
consistently, not only for the considered alkali-alkaline earth molecules, but
also for homonuclear as well as heteronuclear alkali-alkali systems and for
alkaline earth-fluorine molecules. We finally provide recommended relativistic
CCSD(T) values of PDMs and polarizabilities of the alkali-alkaline earth
systems and compare them with available literature values as well as our non-
relativistic calculations.

## Acknowledgments

All the computations were performed on the VIKRAM-100 cluster at Physical
Research Laboratory, Ahmedabad. We thank Prof. Trond Saue for providing useful
insights on the symmetry aspects of the Dirac program.

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