# Ab initio spectroscopic studies of AlF and AlCl molecules

R. Bala<EMAIL_ADDRESS>V. S. Prasannaa  D. Chakravarti  D. Mukherjee  B.
P. Das  Centre for Quantum Engineering, Research and Education, TCG CREST,
Salt Lake, Kolkata 700091, India

###### Abstract

In this work, we report results from our extensive spectroscopic study on AlF
and AlCl molecules, keeping in mind potential laboratory as well as
astrophysical applications. We carry out detailed electronic structure
calculations in both the molecules, including obtaining the potential energy
surfaces of the $X^{1}\Sigma$ ground electronic state and some of the relevant
low-lying excited electronic states belonging to $\Sigma$ and $\Pi$
symmetries. This is followed by evaluating spectroscopic constants and
molecular properties such as electric dipole moments and electric quadrupole
moments. Throughout, we employ the multi-reference configuration interaction
method and work with high-quality quadruple zeta basis sets, keeping in mind
the need for precise results. Further, transition dipole moments between the
ground electronic state and singlet excited states are also studied. The
relevant vibrational parameters are computed by solving the vibrational
Schrödinger equation. Subsequently, the vibrational energy spacings and
transition dipole moments between the vibrational levels belonging to the same
electronic states are used to evaluate the spontaneous and black-body
radiation induced transition rates, followed by computing lifetimes. Finally,
the energy differences between rotational levels belonging to different
vibrational levels and within an electronic state as well as Einstein
coefficients are reported.

Keywords

ab initio calculations, spectroscopic constants, molecular properties,
potential energy curves, dipole moment, quadrupole moment, transition dipole
moment, transition rate, lifetime, vibrational and rotational transitions.

††preprint: APS/123-QED

##  I INTRODUCTION

The field of cold and ultracold molecules is one of the fastest growing areas
of research due to its scope in a wide range of fields, including quantum
chemistry [1], fundamental physics [2], quantum computing [3, 4], quantum
simulation [5], controlled chemical reactions [6, 7], anisotropic- and long-
range dipole-dipole interactions [8, 9], etc. An important feature that makes
a molecule particularly attractive for many applications is its laser
coolability. Further, extensive knowledge and careful analysis of the
properties and spectroscopy of such systems could open avenues for a wider
range of applications in numerous interrelated areas of research.

AlF and AlCl molecules have recently been proposed as promising candidates for
laser cooling and trapping experiments, and efforts towards laser cooling
these systems are currently underway [10, 11, 12, 13, 14]. The rate
coefficients for the formation of AlF and AlCl molecules from Al, F and Cl
atoms via radiative association have been predicted by Andreazza et al. [15,
16]. These molecules can be produced in gas phase by heating Al with AlF3 or
AlCl3 to high temperatures [17].

These molecules are also of interest to astrophysics due to their occurrence
in astrophysical environments [18, 19]. In particular, the rotational energy
levels of these molecules have been identified in the circumsteller envelope
of a C-type asymptotic giant branch (AGB) star, IRC +10216 [20]. Quite
recently, AlF emission lines have been detected in five M-type AGB stars [21].

Table 1:  Spectroscopic constants for the ground electronic state of the AlF
molecule. The results computed in our work are shown in bold fonts. The term
Exp. is used to represent experimental results.

Method |  $R_{e}$ (Å) |  $D_{e}$ (eV) |  $B_{e}$ (cm-1) |  $\alpha_{e}$ (cm-1) |  $\omega_{e}$ (cm-1) |  $\omega_{e}x_{e}$ (cm-1) |  $\omega_{e}y_{e}$ (cm-1)
---|---|---|---|---|---|---|---
SCF | 1.644 | 9.343 | 0.5609 | 0.00509 | 845.22 | 5.51 | 0.0243
CASSCF | 1.669 | 9.666 | 0.5447 | 0.00519 | 802.26 | 5.09 | 0.0231
MRCI | 1.666 | 7.206 | 0.5459 | 0.00500 | 799.49 | 4.54 | 0.0161
MRCI + Q | 1.667 | 7.022 | 0.5433 | 0.00416 | 798.02 | 5.25 | 0.0314
Published works |  |  |  |  |  |  |
CASSCF [22] | 1.681 | 5.710 | $-$ | $-$ | 773.00 | $-$ | $-$
MRCI [22] | 1.675 | 6.760 | $-$ | $-$ | 788.00 | $-$ | $-$
MRCI + Q [22] | 1.679 | 6.870 | $-$ | $-$ | 777.00 | $-$ | $-$
MRCI + Q [23] | 1.664 | 7.036 | 0.5459 | $-$ | 796.11 | 4.45 | $-$
CCSD(T) [24] | 1.680 | $-$ | 0.5360 | $-$ | $-$ | $-$ | $-$
MRCI [25] | 1.638 | $-$ | $-$ | $-$ | 777.00 | $-$ | $-$
CAM-B3LYP [26] | 1.659 | $-$ | $-$ | $-$ | 792.87 | $-$ | $-$
ACPF [27] | 1.654 | $-$ | 0.5530 | 0.00494 | 806.35 | 4.72 | $-$
CCSD(T) [28] | 1.660 | 7.010 | 0.5490 | $-$ | 803.00 | $-$ | $-$
Exp.[17] | 1.654 | 6.943 | $-$ | $-$ | $-$ | $-$ | $-$
Exp. [29] | 1.65437 $\pm$ 0.00001 | $-$ | 0.55246 | 0.00495 | $-$ | $-$ | $-$
Exp. [30, 31] | 1.654369 | $-$ | 0.5524798 | 0.0049841 | 802.26 | 4.77 | $-$

† All the results from literature have been rounded off (Except experimental
data) up to the same decimal place to that reported in the current work.

Using a jet-cooled, pulsed molecular beam, spectroscopic characterization of
the lowest rotational levels in the $a^{3}\Pi$ state of AlF for $v=0$ to $8$
have been reported by Walter et al. [26]. Recently, spectroscopic measurements
of AlF have been performed for energy level structures in $X^{1}\Sigma$,
$A^{1}\Pi$ and $a^{3}\Pi$ states by Truppe et al. [32]. For the molecule, a
high-resolution emission spectrum lying in the infrared region has been
observed by Bernath et al. [33], using Fourier transform infrared
spectrometer. The emission and absorption band spectra between singlet
($A^{1}\Pi\,-\,X^{1}\Sigma,\,C^{1}\Sigma\,-\,A^{1}\Pi,\,D^{1}\Delta\,-\,A^{1}\Pi,\,F^{1}\Pi\,-\,A^{1}\Pi,\,G^{1}\Sigma\,-\,A^{1}\Pi,\,F^{1}\Pi\,-\,B^{1}\Sigma$),
triplet (${}^{3}\Sigma\,-\,^{3}\Pi$) and intercombination electronic states
($a^{3}\Pi\,-\,X^{1}\Sigma$) fall in different spectral ranges, and have been
observed and reported in literature [34, 35, 36, 37, 38].

In addition to experimental data for spectroscopic parameters of AlF molecule,
there are several theoretical results reported in literature. For instance,
Langhoff et al. [22] have reported the spectroscopic parameters (equilibrium
bond length ($R_{e}$), harmonic frequency ($\omega_{e}$), vertical transition
energy ($T_{e}$), and dissociation energy ($D_{e}$)) for the ground and low-
lying excited states. However, they have plotted the potential energy curves
(PECs) for only $X^{1}\Sigma$ and $A^{1}\Pi$ states. The feasibility to obtain
a lower Doppler temperature of around $3\,\mu K$ using spin forbidden
transition $a^{3}\Pi\,\leftarrow\,X^{1}\Sigma$ has been studied in Ref. [25].
Gutsev et al. [28] have studied diatomic constants ($R_{e}$, $\omega_{e}$, and
rotational constant ($B_{e}$)) and first-order molecular properties (dipole
moment ($\mu$) and $z$-component of quadrupole moment ($\Theta_{zz}$)) using
coupled cluster singles and doubles with partial triples (CCSD(T)) method and
atomic natural orbital (ANO) basis sets. The equilibrium bond lengths,
rotational constants and components of dipole polarizabilities for the ground
states of both the molecules have been reported by Woon and Herbst [24] at
CCSD(T)/augmented correlation-consistent polarized valence triple zeta (aug-
cc-pVTZ) level. Zhang et al. [23] have studied the $X^{1}\Sigma$ and
$A^{1}\Pi$ states at multi-reference configuration interaction (MRCI) level of
theory using AWCV5Z basis set. The ro-vibrational transitions in the
electronic ground states of AlF and AlCl molecular systems have been studied
in Ref. [27].

For AlCl, the chemical mechanisms of laser ablation sources for the production
of AlCl molecules for laser cooling have been studied by Lewis et al. [39].
The ground and excited states of AlCl molecule have been observed by David et
al. [40] using resonance-enhanced multiphoton ionization spectroscopy. High
resolution emission spectra of AlCl at $20\mu m$ has been analysed by
Hedderich et al. [17]. The emission spectrum of $A^{1}\Pi-X^{1}\Sigma$
transition has been analyzed by Mahieu et al. [41] to obtain the vibrational
and rotational constants for the involved electronic states. Ram et al. [42]
have studied the rotational bands of $A^{1}\Pi-X^{1}\Sigma$,
$a^{3}\Pi-X^{1}\Sigma$, and $b^{3}\Sigma-a^{3}\Pi$ transitions to obtain the
molecular constants for the electronic states of AlCl.

In theory, a three-electronic level scheme for laser cooling of AlCl molecule
has been discussed in Ref. [12]. Xu et al. [43] have studied the line
intensities of $X^{1}\Sigma\rightarrow A^{1}\Pi$ transition along with the
spectroscopic parameters of $X^{1}\Sigma$, $a^{3}\Pi$, and $A^{1}\Pi$
electronic states. AlCl and its cationic species, AlCl+ and AlCl2+, have been
studied and their ground and excited state parameters obtained by Brites et
al. [44]. The work by Yang et al. [11] has reported on the suitability of AlCl
and AlBr molecules for laser cooling using ab initio methods.

Most of the theoretical works discussed above do not report all the
spectroscopic parameters ($R_{e}$, $D_{e}$, $B_{e}$, $\omega_{e}$, rotational-
vibrational coupling constant ($\alpha_{e}$), anharmonic constants
($\omega_{e}x_{e}$ and $\omega_{e}y_{e}$)) for both ground and excited states.
For example, the dissociation energies are not available for the excited
states of AlF molecule except for $A^{1}\Pi$ state in Ref. [23]. The value of
$\omega_{e}y_{e}$, for the $X^{1}\Sigma$ state of AlF is not reported by any
work, while for AlCl, $\omega_{e}y_{e}$ is reported only in Ref. [44] to the
best of our knowledge. The $\alpha_{e}$ and $\omega_{e}x_{e}$ results for
excited states are reported only in Ref. [30]. Further, the rotational
constants, $B_{e}$, for $A^{1}\Pi$, $B^{1}\Sigma$ and $b^{3}\Sigma$ electronic
states are communicated in Ref. [30].

|
---|---
(a) | (b)
|
(c) | (d)
Figure 1:  (a) Potential energy curves, (b) permanent electric dipole moment
curves, (c) electric quadrupole moment curves for the ground and low-lying
excited electronic states, and (d) transition dipole moment curves from ground
to singlet excited states (A${}^{1}\Pi$ and B${}^{1}\Sigma$) of AlF molecule.

In this work, we identify and attempt to bridge important gaps in the
available data for the spectroscopic properties of these two molecular
systems. We provide a comprehensive study of PECs, permanent dipole moment
(PDM) curves, quadrupole moment (QM) curves, transition dipole moment (TDM)
curves, and spectroscopic constants: $R_{e}$, $D_{e}$, $B_{e}$, $\alpha_{e}$,
$T_{e}$, $\omega_{e}$, $\omega_{e}\,x_{e}$ and $\omega_{e}\,y_{e}$ for not
only ground state but also for low-lying excited states of AlF and AlCl
molecular systems. All calculations are performed at MRCI level of theory
together with the quadruple zeta (QZ) quality basis sets. It is also worth
adding that no previous work has evaluated QMs for all electronic states of
AlCl and excited electronic states of AlF system. Further, this study has been
extended to the calculation of vibrational parameters viz. energy ($E_{v}$),
vibrational-rotational constants ($B_{v}$), TDMs ($\mu_{v^{\prime},v"}$),
spontaneous and black-body radiation (BBR) induced transition rates
($\Gamma_{spon}$ and $\Gamma_{BBR}$), and lifetimes ($\tau$). We have also
studied the rotational parameters in the vibrational levels of each electronic
state. Furthermore, there are some discrepancies in the values of some
parameters and nature of the states, which we tried to address in the current
work. The aim of this work is to compute the diatomic constants, molecular
properties, and spectroscopic parameters for ro-vibrational levels of ground-
and low-lying excited electronic states states of AlF and AlCl molecular
systems at a uniform level of theory, basis sets and configuration space.

The manuscript is structured as follows: Section I provides an introduction
and literature review that then leads us to the motivation of the current
work. The details of our calculations are described in Section II, while the
obtained results are discussed in Section III. Finally, a summary of our work
is given in Section IV.

Table 2:  Spectroscopic constants for the excited electronic states of the AlF
molecule. The results computed in the present work are given in bold fonts.
The abbreviation Exp. is used for the experimental results.

Method |  $R_{e}$ (Å) |  $D_{e}$ (eV) |  $B_{e}$ (cm-1) |  $\alpha_{e}$ (cm-1) |  $T_{e}$ (cm-1) |  $\omega_{e}$ (cm-1) |  $\omega_{e}x_{e}$ (cm-1) |  $\omega_{e}y_{e}$ (cm-1) |
---|---|---|---|---|---|---|---|---|---
$a^{3}\Pi$
MRCI | 1.662 | 3.932 | 0.5450 | 0.0032 | 26593.46 | 826.21 | 5.31 | 0.0150 |
MRCI + Q | 1.663 | 3.743 | 0.5442 | 0.0032 | 26516.33 | 825.80 | 5.43 | 0.0149 |
MRCI [25] | 1.631 | $-$ | $-$ | $-$ | 27607.01 | 845.00 | $-$ | $-$ |
CAM-B3LYP [26] | 1.655 | $-$ | $-$ | $-$ | 25587.86 | 818.72 | $-$ | $-$ |
MRCI [22] | 1.659 | $-$ | $-$ | $-$ | 26035.00 | 855.00 | $-$ | $-$ |
MRCI + Q [22] | 1.664 | $-$ | $-$ | $-$ | 26480.00 | 845.00 | $-$ | $-$ |
CCSD(T) [28] | 1.653 | $-$ | 0.5530 | $-$ | $-$ | 830.00 | $-$ | $-$ |
Fitting∗ [26] | $-$ | $-$ | 0.5574 | 0.0048 | 27239.45 | 830.28 | 4.64 | 0.0123 |
Exp.NIST [30, 31] | 1.6476 | $-$ | 0.5570 | 0.00453 | 27241 | 827.8 | 3.9 | $-$ |
$A^{1}\Pi$
MRCI | 1.661 | 1.715 | 0.5480 | 0.0051 | 44455.44 | 809.03 | 6.89 | 0.0089 |
MRCI + Q | 1.663 | 1.591 | 0.5474 | 0.0049 | 43847.36 | 804.32 | 5.70 | 0.1946 |
MRCI [23] | 1.658 | 1.591 | 0.5501 | $-$ | 43915.18 | 804.25 | 5.59 | $-$ |
MRCI [25] | 1.638 | $-$ | $-$ | $-$ | 44924.57 | 828.00 | $-$ | $-$ |
MRCI [22] | 1.655 | $-$ | $-$ | $-$ | 44873.00 | 845.00 | $-$ | $-$ |
MRCI + Q [22] | 1.664 | $-$ | $-$ | $-$ | 44211.00 | 828.00 | $-$ | $-$ |
Exp. NIST [30, 31] | 1.6485 | $-$ | 0.55640 | 0.00534 | 43949.2 | 803.90 | 5.99 | 0.050 |
$b^{3}\Sigma$
MRCI | 1.656 | 1.654 | 0.5104 | 0.0061 | 44764.65 | 787.20 | 6.42 | 0.2110 |
MRCI + Q | 1.660 | 1.514 | 0.5501 | 0.0053 | 44438.34 | 791.26 | 8.81 | 0.0516 |
MRCI [25] | 1.615 | $-$ | $-$ | $-$ | 44992.77 | 797.00 | $-$ | $-$ |
MRCI [22] | 1.650 | $-$ | $-$ | $-$ | 44106.00 | 813.00 | $-$ | $-$ |
MRCI + Q [22] | 1.659 | $-$ | $-$ | $-$ | 44224.00 | 797.00 | $-$ | $-$ |
Exp.NIST [30, 31] | 1.6391 | $-$ | 0.56280 | 0.00651 | 44813.20 | 786.37 | 7.64 | 0.009 |
$B^{1}\Sigma$
MRCI | 1.667 | $-$ | 0.5441 | $-$ | 66891.64 | $-$ | $-$ | $-$ |
MRCI + Q | 1.613 | $-$ | 0.5812 | $-$ | 49640.25 | $-$ | $-$ | $-$ |
MRCI [25] | 1.593 | $-$ | $-$ | $-$ | 56352.80 | $-$ | $-$ | $-$ |
MRCI [22] | 1.624 | $-$ | $-$ | $-$ | 54211.00 | $-$ | $-$ | $-$ |
MRCI + Q [22] | 1.632 | $-$ | $-$ | $-$ | 54258.00 | $-$ | $-$ | $-$ |
Exp. NIST [30, 31] | 1.6151 | $-$ | 0.57968 | $-$ | 54251.00 | $-$ | $-$ | $-$ |

* Spectroscopic parameters deduced from a fit to all the measured frequencies.

##  II Theory and Methodology

The electronic energy calculations for ground as well as for excited
electronic states of the considered systems are performed using the MOLPRO
[45] quantum chemistry package. The spectroscopic constants and molecular
properties have been calculated at MRCI level of theory in conjunction with
the correlation-consistent polarized valence quadruple zeta (cc-PVQZ) basis
sets. Further, in order to incorporate the unlinked quadruples contributions
which actually improves the energies significantly and hence the potential
energy surfaces, we have also considered the Davidson correction [46].

To carry out MRCI calculations, a complete active space self consistent field
(CASSCF) wave function is considered as the zeroth order wavefunction. The
$C_{2v}$ subgroup of $C_{\infty v}$ point group has been used to perform all
of the structure calculations. The $C_{2v}$ subgroup has four irreducible
representations $A_{1}$, $B_{1}$, $B_{2}$, and $A_{2}$. The $A_{1}$ forms
$\Sigma$, $B_{1}$ and $B_{2}$ together form $\Pi$, and $A_{2}$ forms $\Delta$
electronic states. Six orbitals for AlF and ten orbitals for AlCl form our
frozen core. The complete active spaces (CAS), represented as
($A_{1},B_{1},B_{2},A_{2}$), chosen for AlF and AlCl are (10, 4, 4, 1) and
(12, 5, 5, 1), respectively. As a result, 13 orbitals form the CAS for
electronic energy and property calculations for both the molecules. The
potential energies, PDMs, and QMs are calculated with a step size of 1 Å bond
distance and in order to arrive at the equilibrium geometry, we have chosen a
finer step size of $0.001$ Å around the equilibrium point. The dissociation
limits for AlF and AlCl molecules are $24$ Å and $19.6$ Å, respectively. The
spectroscopic constants are computed using VIBROT program available in
OpenMolcas [47] quantum chemistry software.

The traceless QM tensor, $\Theta_{\alpha\beta}$, can be defined as [48],

$\displaystyle\Theta_{\alpha\beta}=\frac{1}{2}\,\sum_{i}e_{i}(3r_{i_{\alpha}}\,r_{i_{\beta}}\,-\,r_{i}^{2}\delta_{\alpha\beta}),$
(1)

where $\alpha$ and $\beta$ represent the Cartesian components and summation
index $i$ represents the number of charges. By considering $z$ axis as the
internuclear axis of the molecule, the $z$ component of QM, $\Theta_{zz}$, is
related to two other diagonal components by the equation,

$\displaystyle\Theta_{zz}\,=-(\,\Theta_{xx}\,+\,\Theta_{yy}).$ (2)

Further, for linear molecules, $\Theta_{xx}\,=\,\Theta_{yy}$ and therefore,

$\displaystyle\Theta_{zz}\,=\,-2\Theta_{xx}.$ (3)

Using the PECs and PDM curves for ground and excited electronic states at MRCI
level of theory, we have solved the vibrational Schrödinger equation using the
LEVEL [49] program to obtain the vibrational parameters: wavefunctions,
energies, rotational constants, and TDMs between the vibrational levels and
rotational parameters: energies, Einstein coefficients and Franck-Condon
factors (FCFs). We have used cubic spline fitting for interpolation with
suitable step size. Further, the relative energy separation and the TDMs
between the vibrational levels are used to calculate the spontaneous and BBR
induced transition rates at room temperature ($T=300K$) using the following
equations [50, 51]:

$\displaystyle{}\Gamma_{v,J}^{spon}\,=\,\sum\limits_{v^{{}^{\prime\prime}},J^{{}^{\prime\prime}}}\Gamma^{emis}(v,J\,\rightarrow\,v^{{}^{\prime\prime}},J^{{}^{\prime\prime}})$
(4)

and

$\displaystyle{}\Gamma_{v,J}^{BBR}\,$ $\displaystyle=$
$\displaystyle\,\sum\limits_{v^{{}^{\prime\prime}},J^{{}^{\prime\prime}}}\bar{n}(\omega)\,\Gamma^{emis}(v,J\,\rightarrow\,v^{{}^{\prime\prime}},J^{{}^{\prime\prime}})$
(5) $\displaystyle+$
$\displaystyle\sum\limits_{v^{{}^{\prime}},J^{{}^{\prime}}}\bar{n}(\omega)\,\Gamma^{abs}(v,J\,\rightarrow\,v^{{}^{\prime}},J^{{}^{\prime}}),$

where the indices ($v^{{}^{\prime\prime}},J^{{}^{\prime\prime}}$) and
($v^{{}^{\prime}},J^{{}^{\prime}}$) denote the rovibrational levels with
energy smaller and larger than that of ($v,J$) level, respectively. The
average number of photons $\bar{n}(\omega)$ at frequency $\omega$ is given by
the relation,

$\displaystyle{}\bar{n}(\omega)\,=\,\frac{1}{e^{(\hbar\omega/k_{B}T)}-1},$ (6)

where $\hbar\omega\,=\,\arrowvert E_{v,J}-E_{\tilde{v},\tilde{J}}\arrowvert$
is the energy difference between the two rovibrational levels involved, with
($\tilde{v},\tilde{J}$) being ($v^{{}^{\prime\prime}},J^{{}^{\prime\prime}}$)
for emission, and ($v^{{}^{\prime}},J^{{}^{\prime}}$) for absorption. $k_{B}$
is the Boltzmann constant. The emission and absorption rates are calculated
using the equation,

$\displaystyle{}\Gamma^{emis\,or\,abs}(v,J\,\rightarrow\,v^{{}^{\prime\prime}},J^{{}^{\prime\prime}}\,or\,v^{{}^{\prime}},J^{{}^{\prime}})\,=$
$\displaystyle\,\frac{8\pi}{3\epsilon_{0}}\frac{1}{hc^{3}}\,\omega^{3}(TDM_{v,J\rightarrow
v^{{}^{\prime\prime}},J^{{}^{\prime\prime}}\,or\,v^{{}^{\prime}},J^{{}^{\prime}}})^{2}.$
(7)

Finally, the total lifetimes ($\tau$) of the rovibrational states are obtained
as,

$\displaystyle{}\tau\,=\,\frac{1}{\Gamma^{total}},$ (8)

where $\Gamma^{total}(\,=\,\Gamma^{spon}\,+\,\Gamma^{BBR}$) is the of sum of
spontaneous and BBR induced transition rates.

The Franck-Condon factors, $q_{v^{\prime},v"}$, are defined as the square of
the matrix elements of the zeroth-power of the radial variable [49],

$\displaystyle
q_{v^{\prime},v"}\,=\,\mid\langle\Psi_{v^{\prime},J^{\prime}}|\Psi_{v",J"}\rangle|^{2}.$
(9)

##  III Results and Discussion

Tables 1 through 6 and figure 1 to 4 present all of our calculated result, and
their comparison with the existing data, wherever available. In all tables,
the results computed in the current work are shown in bold fonts.

###  III.1 AlF

####  III.1.1 Electronic properties: ground and low-lying excited states

The PECs for the ground and low - lying excited states with $\Sigma$ and $\Pi$
symmetries at MRCI+Q level of theory are shown in Figure 1(a). The
spectroscopic constants for the ground electronic state calculated in the
current work along with the results available in the literature are given in
Table 1. The results for spectroscopic parameters obtained in this work
compares well with the existing values. Except for $\omega_{e}x_{e}$, the
difference between our results for ground spectroscopic constants from that
reported in the recent work by Zhang et al. [23] is less than 1% at the same
level of theory. Although, $\omega_{e}x_{e}$ is extracted from the PECs along
with the other spectroscopic parameters via fitting, it is quite sensitive to
the number of data points used for fitting. The relative difference between
our results obtained using MRCI+Q theory from those given in Ref. [28] at
CCSD(T) level is: $\Delta(R_{e})\,=\,0.44\%$, $\Delta(D_{e})\,=\,0.17\%$,
$\Delta(B_{e})\,=\,1\%$ and $\Delta(\omega_{e})\,=\,0.6\%$ for the ground
electronic state. Comparing our values of $R_{e}$, $D_{e}$, and $B_{e}$ to the
experimental values [17, 29], we observe that the errors are 0.7%, 1.1% and
1.7%, respectively.

Table 3:  Molecular properties (electric dipole and electric quadrupole moments denoted as $\mu_{e}$ and $\Theta_{zz}$, respectively) for the ground and excited states of AlF molecule. State | Method |  $\mu_{e}$ (Debye) |  $\Theta_{zz}$ (au)
---|---|---|---
$X^{1}\Sigma$ | SCF | -1.285 | -5.929
| CASSCF | -1.434 | -5.379
| MRCI | -1.425 | -5.277
| MRCI [23] | -1.079 | $-$
| CCSD(T) [24] | 1.540 | $-$
| SCF [28] | 1.357 | -4.674
| CCSD(T) [28] | 1.486 | -3.766
| ACPF[27] | 1.407 | $-$
| Exp.[29] | 1.53$\pm$ 0.10 | $-$
$a^{3}\Pi$ | MRCI | -1.770 | -2.794
$A^{1}\Pi$ | MRCI | -1.546 | -1.287
| MRCI[23] | 1.497 | $-$
$b^{3}\Sigma$ | MRCI | -1.699 | -9.397
$B^{1}\Sigma$ | MRCI | -1.811 | -5.295
|
---|---
(a) | (b)
|
(c) | (d)
Figure 2:  (a) Relative energy spacings between the vibrational levels (b)
rotational constants (c) lifetimes and (d) transition rates for the
vibrational levels of ground and excited states of AlF molecule at MRCI/QZ
level of theory.

Our value of $D_{e}=7.022$ eV at MRCI+Q level matches well within the error
bars with the measured value of $6.894\,\pm\,0.130$ reported in Ref. [52]. We
could not find any value for $\omega_{e}y_{e}$ in the literature for
comparison. The value of $\alpha_{e}$ reported in this work at MRCI level is
very close to the existing theoretical and experimental results.

Table 2 gives the comparison of results for spectroscopic parameters of low-
lying excited states obtained in this work with the literature. Except for
$A^{1}\Pi$ electronic state, dissociation energies are not available for other
excited states studied in this work. Our value of $D_{e}$ for $A^{1}\Pi$ state
is in excellent agreement with the only result available in Ref. [23]. The
MRCI + Q value of $\omega_{e}x_{e}\,=\,5.70$ cm-1 for $A^{1}\Pi$ state is very
close to the experimental value of $5.99$ cm-1 [31], while for $a^{3}\Pi$
($B^{1}\Sigma$) state, our value is larger by 1.53 cm-1 (1.17 cm-1) from the
experimental data. The $\omega_{e}y_{e}$ value for $a^{3}\Pi$ state is of the
same order as reported in Ref. [26], whereas the $\omega_{e}y_{e}$ values for
$A^{1}\Pi$ and $b^{3}\Sigma$ states are one order of magnitude higher than
those given in NIST database [30, 31]. The computed results for $\alpha_{e}$
in the present work are of the same order as the data given in Refs. [26, 30,
31]. For the other excited state parameters, the relative difference between
the values reported in our work and the experimental data available in
literature lies in the range 0.9 - 1.3% for $R_{e}$, 1.6 - 2.2% for $B_{e}$,
0.2 - 2.7% for $T_{e}$, and 0.05 - 0.6% for $\omega_{e}$.

Table 4:  Spectroscopic constants for the ground electronic state of AlCl molecule. Method |  $R_{e}$ (Å) | $D_{e}(eV)$ |  $B_{e}$ (cm-1) |  $\alpha_{e}$ (cm-1) |  $\omega_{e}$ (cm-1) |  $\omega_{e}x_{e}$ (cm-1) |  $\omega_{e}y_{e}$ (cm-1)
---|---|---|---|---|---|---|---
SCF | 2.149 | 6.590 | 0.2393 | 0.0016 | 476.57 | 2.19 | 0.0051
CASSCF | 2.141 | 7.356 | 0.2314 | 0.0007 | 400.55 | 7.46 | 0.1387
MRCI | 2.143 | 5.539 | 0.2400 | 0.0013 | 478.61 | 1.10 | 0.0115
MRCI + Q | 2.145 | 5.274 | 0.2403 | 0.0015 | 483.47 | 2.20 | 0.0066
Published works |  |  |  |  |  |  |
CASSCF [22] | 2.157 | 4.360 | $-$ | $-$ | 470.00 | $-$ | $-$
MRCI[22] | 2.142 | 5.120 | $-$ | $-$ | 496.00 | $-$ | $-$
MRCI + Q [22] | 2.140 | 5.270 | $-$ | $-$ | 500.00 | $-$ | $-$
ACPF [27] | 2.128 | $-$ | 0.2441 | 0.0016 | 484.81 | 2.07 | $-$
MRCI [11] | 2.145 | 5.220 | 0.2406 | $-$ | 478.36 | 1.95 | $-$
MRCI + Q [43] | 2.137 | 5.262 | 0.2425 | $-$ | 481.83 | 2.01 | $-$
MRCI [12] | 2.137 | 5.214 | 0.2408 | $-$ | 478.13 | $-$ | $-$
MRCI + Q [13] | 2.137 | $-$ | $-$ | $-$ | $-$ | $-$ | $-$
CCSD(T) [24] | 2.159 | $-$ | 0.2360 | $-$ | $-$ | $-$ | $-$
MRCI + Q [44] | 2.140 | 5.250 | 0.2418 | 0.0017 | 484.50 | 6.47 | 0.9800
Exp.[29] | 2.12983 | $-$ | 0.24393 | 0.00160 | $-$ | $-$ | $-$
Exp. [17] | 2.130 | 5.120 | $-$ | $-$ | $-$ | $-$ | $-$
Exp. [42] | $-$ | 5.25 $\pm$ 0.01 | 0.2439 | 0.00161 | $-$ | $-$ | $-$
NIST [30] | 2.130113 | $-$ | 0.24393102 | 0.00161113 | 481.3 | 1.95 | $-$

The $B^{1}\Sigma$ state has a minimum energy at $R_{e}\,=\,1.613$ Å and its
energy attains its maximum value at about $1.8$ Å , after which it starts
decreasing. This trend in the PEC is same as that described in Ref. [25]. It
can be seen from Figure 1(a) that the $B^{1}\Sigma$ state has a shallow
minimum of $1098.8$ cm-1, which renders the computation of other constants via
fitting of PEC meaningless. The values of equilibrium bond length and
rotational constant of this state at MRCI + Q level of theory agrees with the
values in NIST database. However there is a noticeable difference in the
values of transition energies.

The computed first - order molecular properties at the equilibrium bond
lengths are given in Table 3. The variation of PDM as a function of
internuclear distance, for all the electronic states, is shown in Figure 1(b).
The $R$ variation of PDM for the ground state is linear almost upto $9$ Å and
then it drops to zero abruptly. The linear behaviour of the PDM curve
signifies the ionic character of the molecule and the abrupt change indicates
the change from ionic to covalent character of the dipole [25]. Further, it
has been found that using state - averaging over states of symmetry same as
that of the ground state, the PDM for the ground state goes to zero after $4$
Å. A very similar behaviour of PDM curve of this system has been reported in
Ref. [23], though they have not mentioned about the usage of state - averaged
procedure in their work. It has been observed that the values of PDMs are
almost zero at the dissociation limit for all electronic states, which means
that the molecule dissociates into its constituent neutral fragments.

The variation of the $z$ \- component of QM with internuclear distance is
given in Figure 1(c). Similar to PDM, the QM value also drops and attains a
small constant value of about 2.16 au (-3.15 au) after $4$ Å ($9$ Å) with
(without) state - average procedure. The QM values for the excited states
decrease quadratically upto 2Å bond distance for $a^{3}\Pi$, $A^{1}\Pi$, and
$b^{3}\Sigma$ states, while for $B^{1}\Sigma$ state, this behaviour is noticed
upto the bond length of 1.7 Å as shown in the subplot of Figure 1(c). The
quadrupole moment relation for linear diatomic molecules given by eqn. (3) is
satisfied for ground ${}^{1}\Sigma$ states upto $9.5$ Å (4 Å) bond distance
with (without) state average procedure. The TDM curves from ground to singlet
excited states are plotted in Figure 1(d). The TDM goes to zero after $5$ Å
for $X^{1}\Sigma\rightarrow A^{1}\Pi$ transition and for
$X^{1}\Sigma\rightarrow B^{1}\Sigma$, its value is zero $2$ Å onwards due to
the fact that these transitions are forbidden in atomic limits.

####  III.1.2 Vibrational and rotational parameters

We have found $150$ bound vibrational states for $X^{1}\Sigma$, $53$ for
$a^{3}\Pi$, $23$ for $A^{1}\Pi$, and $44$ for $b^{3}\Sigma$ electronic states.
Figure 2(a) and Figure 2(b) presents the behaviour of relative energy spacings
and vibrational - rotational constants against the vibrational quantum number,
respectively. The relative energy separation between the vibrational levels
and the vibrational - rotational constants decreases with the increase in
vibrational quantum number due to anharmonic effects. We have also calculated
the first six centrifugal distortion constants viz.
$D_{v},H_{v},L_{v},M_{v},N_{v}$ and $O_{v}$ for the bound states of AlF. These
results are presented in Tables S1 to S4 of supplementary material. The PDMs
for the vibrational levels (with $J\,=\,0$) of ground and excited electronic
states are tabulated in Supplementary Tables S5 and S6, respectively. We have
obtained the following values of the electric dipole moments $1.421$ D for
$X^{1}\Sigma$, $1.567$ D for $A^{1}\Pi$ and $1.241$ D for $a^{3}\Pi$ in their
lowest ro - vibronic levels. The only measured values of dipole moments for
$X^{1}\Sigma$, $A^{1}\Pi$ and $a^{3}\Pi$ corresponding to $v\,=\,0$
vibrational levels are $1.515\,\pm\,0.004$ D, $1.45\,\pm\,0.02$ D, and
$1.780\,\pm\,0.003$ D, respectively [32]. Our results differ from those
reported in Ref. [32] by $0.094$ D for $X^{1}\Sigma$, $-0.117$ D for
$A^{1}\Pi$ and $0.539$ D for $a^{3}\Pi$.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 3:  (a) Potential energy curves (b) permanent dipole moment curves (c)
quadrupole moment curves for the ground and low-lying excited electronic
states and (d) transition dipole moment curves from ground to singlet excited
electronic states of the AlCl molecule.

Yousefi et al. [27] have obtained the PECs for the ground states of AlF and
AlCl using direct fitting of rotational and ro - vibrational frequencies
together with the dissociation energies taken from the literature [17].
Further, they have studied the vibration - rotation line lists upto $v\,=\,11$
and maximum number of rotational levels, $J_{max}\,=\,200$ for electronic
ground states of these molecules. They have reported $\mu_{v}\,=\,1.439$ D for
the lowest ro - vibrational level of ground electronic state. The difference
between our value and that reported in their work is less than 1.3%.

The variation of total lifetimes against the vibrational quantum number, $v$
is shown in Figure 2(c). The lifetime curve of ground electronic state of AlF
molecule shows several minor oscillations similar to CaLi+ system in Ref.
[50]. This is due to the oscillations in the spontaneous transition rate as
shown in Figure 2 (d) and also observed for CaLi+ ionic system in Ref. [50].
The calculated lifetimes for the lowest vibrational - rotational states
($v\,=\,0$ and $J\,=\,0$) of $X^{1}\Sigma$, $a^{3}\Pi$, $A^{1}\Pi$, and
$b^{3}\Sigma$ are $7.8$ s, $0.59$ s, $10$ s, and $59$ s, respectively.

Further, we have computed the rotational parameters: relative energies,
Einstein coefficients, FCFs, and TDMs. The rotational transitions
corresponding to $\Delta J\,=\,0$, $\Delta J\,=\,+1$ and $\Delta J\,=\,-1$
give the set of lines called $Q$, $R$, and $P$ \- branches, respectively. We
have performed the calculations of rotational parameters for all $P$, $Q$ and
$R$ branches considering $J_{max}$ is equal to 20 in each vibrational level
$v\,=\,0$ to $5$ in ground as well as excited electronic states. All
parameters for the rotational transitions between the vibrational levels of
individual electronic states are given in supplementary Tables S7 to S10. The
Einstein coefficients between the rotational levels of different vibrational
levels in each electronic state are plotted in Figures 1 to 4 of supplementary
file PS1.

Table 5:  Spectroscopic constants for the excited electronic states of the AlCl molecule.  Method |  $R_{e}$ (Å) |  $D_{e}$ (eV) |  $B_{e}$ (cm-1) |  $\alpha_{e}$ (cm-1) |  $T_{e}$ (cm-1) |  $\omega_{e}$ (cm-1) |  $\omega_{e}x_{e}$ (cm-1) |  $\omega_{e}y_{e}$ (cm-1) | Ref.
---|---|---|---|---|---|---|---|---|---
$a^{3}\Pi$
MRCI | 2.112 | 2.591 | 0.2449 | 0.0006 | 24014.24 | 500.68 | 0.25 | 0.0388 |
MRCI + Q | 2.114 | 2.320 | 0.2453 | 0.0008 | 23970.88 | 509.07 | 1.40 | 0.0263 |
NIST [30] | 2.100 | $-$ | 0.2500 | 0.0020 | 24658.00, | 524.35 | 2.18 | $-$ |
|  |  |  |  | 24593.84, 24528 |  |  |  |
MRCI + Q [44] | 2.107 | $-$ | 0.2494 | 0.0015 | 24519.25 | 519.10 | 0.52 | 0.2300 |
MRCI [22] | 2.110 | $-$ | $-$ | $-$ | 23819.00 | 530.00 | $-$ | $-$ |
MRCI + Q [22] | 2.114 | $-$ | $-$ | $-$ | 24194.00 | 519.00 | $-$ | $-$ |
MRCI [11] | 2.112 | 2.240 | 0.2481 | $-$ | 24057.00 | 523.36 | 2.61 | $-$ |
MRCI + Q [43] | 2.104 | 2.261 | 0.2501 | $-$ | 24223.52 | 527.37 | 2.65 | $-$ |
MRCI [12] | 2.105 | 2.246 | 0.2483 | $-$ | 23959.87 | 525.68 | $-$ | $-$ |
Exp. [42] | $-$ | $-$ | 0.2524 | 0.0011 | 24793.10 | $-$ | $-$ | $-$ |
$A^{1}\Pi$
MRCI | 2.146 | 0.730 | 0.2409 | 0.0029 | 38955.52 | 445.76 | 9.15 | 0.0532 |
MRCI + Q | 2.146 | 0.568 | 0.2394 | 0.0016 | 38015.98 | 445.62 | 7.15 | 0.0361 |
MRCI + Q [44] | 2.132 | $-$ | 0.2435 | 0.0027 | 38795.26 | 453.00 | 8.03 | 0.4400 |
NIST [30] | 2.060 | $-$ | 0.2590 | 0.0060 | 38254.00 | 449.96 | 4.37 | 0.2160 |
MRCI [22] | 2.138 | $-$ | $-$ | $-$ | 39630.00 | 464.00 | $-$ | $-$ |
MRCI + Q [22] | 2.152 | $-$ | $-$ | $-$ | 38453.00 | 449.00 | $-$ | $-$ |
MRCI [11] | 2.142 | 0.530 | 0.2412 | $-$ | 38303.00 | 471.83 | 9.61 | $-$ |
MRCI + Q [43] | 2.132 | 0.496 | 0.2435 | $-$ | 38436.37 | 453.43 | 8.48 | $-$ |
MRCI [12] | 2.133 | 0.544 | 0.2397 | $-$ | 38223.98 | 454.24 | $-$ |  |
MRCI + Q [13] | 2.133 | $-$ | 0.24313 | 0.0030 | 38253.72 | 446.26 | 5.04 | $-$ |
Exp. [13] | 2.1220 | $-$ | 0.24535 (2) | 0.002652(7) | 38253.22 (2) | 452.54 (5) | 5.61 (3) | $-$ |
Exp. [42] | $-$ | $-$ | 0.24537 | 0.0025 | 38237.00 | $-$ | $-$ | $-$ |
$B^{1}\Sigma$
MRCI | $-$ | $-$ | $-$ | $-$ | 56763.11 | $-$ | $-$ | $-$ |
MRCI + Q | $-$ | $-$ | $-$ | $-$ | 49649.42 | $-$ | $-$ | $-$ |
MRCI +Q [44] | $-$ | $-$ | $-$ |  | 54119.79 | $-$ | $-$ | $-$ |
MRCI [22] | 2.053 | $-$ | $-$ | $-$ | 53003.00 | 496.00 | $-$ | $-$ |
MRCI + Q [22] | 2.069 | $-$ | $-$ | $-$ | 52800.00 | 468.00 | $-$ | $-$ |

These results for rotational levels, particularly for the excited states, may
useful for the future line list observations in astrophysical environments.

###  III.2 AlCl

####  III.2.1 Electronic properties: ground and low-lying excited states

The potential energies for the electronic states of AlCl molecular system
studied in this work at MRCI + Q level of theory, are plotted against the
internuclear distances in Figure 3(a). The comparison of spectroscopic
constants computed in the current work for ground and excited electronic
states together with the available results in the literature are collected in
Table 4 and Table 5, respectively.

Table 6:  Molecular properties (electric dipole and electric quadrupole moments denoted as $\mu_{e}$ and $\Theta_{zz}$, respectively) for the electronic ground and excited states of AlCl molecule. State | Method | $\mu_{e}(Debye)$ |  $\Theta_{zz}$ (au)
---|---|---|---
$X^{1}\Sigma$ | SCF | -1.604 | -6.387
| CASSCF | -1.549 | -5.379
| MRCI | -1.589 | -5.377
| ACPF [27] | 1.594 | $-$
| MRCI [43] | 0.900 | $-$
| MRCI [12] | -1.308 | $-$
$a^{3}\Pi$ | MRCI | -1.669 | -1.749
| MRCI[43] | 1.770 | $-$
| MRCI [12] | -1.760 | $-$
$A^{1}\Pi$ | MRCI | -1.084 | 0.640
| MRCI[43] | 1.337 | $-$
| MRCI [12] | -0.918 | $-$

The computed results at MRCI + Q level of theory for $R_{e}$, $D_{e}$,
$B_{e}$, and $\alpha_{e}$ of the ground electronic state show very good
agreement with the existing theoretical and experimental data [29, 17, 42].
The values of diatomic constants ($R_{e}$, $D_{e}$, $B_{e}$, and $\omega_{e}$)
reported in this work do not differ by more than 0.92% from those reported
recently in Ref. [43] at the same level of correlation. Further, the value of
$\omega_{e}x_{e}$ which depends on the other diatomic constants shows an
appreciable difference of about 9.4% between the two works. In Ref. [44], the
authors have reported all the spectroscopic parameters and the results for
$R_{e}$, $D_{e}$, $B_{e}$, $\alpha_{e}$ and $\omega_{e}$ show excellent
agreement with those reported in the current work. However, the value of
anharmonic constant, $\omega_{e}x_{e}\,=\,6.47$ cm-1 reported in their work is
large as compared to any other value reported in the literature including ours
and this difference lies in the range 4.27 to 4.52 cm-1. Thus, we refrain from
comparing our value for higher anharmonic constant, $\omega_{e}y_{e}$, with
that reported in Ref. [44].

The computed spectroscopic constants: $R_{e}$, $D_{e}$, $B_{e}$, $T_{e}$ and
$\omega_{e}$ in the present work for excited states compares well with the
published results. The vertical transition energy for $a^{3}\Pi$ ($A^{1}\Pi$)
excited state at MRCI + Q level matches with the experimental result to within
3.3% (0.6%) [42, 13]. The existing values of $\omega_{e}x_{e}$ for $a^{3}\Pi$
and $A^{1}\Pi$ states lie in the range 0.52 to 2.6540 cm-1 and 4.37 to 9.61
cm-1, respectively and our calculated values lies in between them. The
computed $\omega_{e}y_{e}$ values for the excited state are one order of
magnitude smaller than the available results [30, 44].

There is a disagreement between the results of Langhoff et al. [22] and Brites
et al. for the first excited ${}^{1}\Sigma$ state. In the former work, they
have reported equilibrium bond length, harmonic frequency and transition
energy, which clearly gives the idea of attractive the nature of the state
while, the latter work shows the repulsive nature of the PEC. In the present
work, we have found that the $B^{1}\Sigma$ state to be repulsive. Further,
there is a mismatch between the existing results of transition energies for
this state [44, 22]. The value of $T_{e}$ reported in Ref. [22] at MRCI + Q
level of theory is smaller by 1319.8 cm-1 from that reported in [44]. Our
value of $T_{e}$ using MRCI (MRCI + Q) method is larger (smaller) by 3760.11
cm-1 (3150.58 cm-1 ) from that reported in Ref. [22] at a similar level of
theory.

The behaviour of PDMs and $z$ \- component of QMs with the bond distances are
shown in Figure 3(b) and Figure 3(c), respectively. The results for the
molecular properties of AlCl are presented in Table 6. The PDM varies linearly
upto approximately $7$ Å for $X^{1}\Sigma$, $3$ Å for $a^{3}\Pi$ and $2.4$ Å
for $A^{1}\Pi$ state. The PDM curve for excited singlet sigma state shows the
maximum value of $4.236$ D at $2.09$Å bond distance. The absolute values of
PDMs for $X^{1}\Sigma$, $a^{3}\Pi$, $A^{1}\Pi$, and $B^{1}\Sigma$ states at
the equilibrium bond lengths are $1.589$ D, $1.669$ D, $1.084$ D, and $10.314$
D, respectively. Our result for the PDM of electronic ground state is very
close to that reported in Ref. [27] and the difference between the two works
is only $0.32$%. However, our value of PDM is larger by 0.689 D and 0.281 D
from that reported by Xu et al. [43] and Wan et al. [12], respectively at the
same level of correlation. The maximum difference between the current value of
PDM from those reported in literature for $a^{3}\Pi$ ($A^{1}\Pi$) state is
$0.1$ D (0.25 D).

---
(a)
(b)
(c)
Figure 4:  (a) Relative energy spacing (b) rotational constants (c) lifetimes
of the vibrational levels of ground and excited states of AlCl molecule at
MRCI/QZ level of theory.

The results of the QMs for this molecular system is reported for the first
time in this work to the best of our knowledge. The knowledge of multipole
moments is useful for better understanding the internuclear
forces/electrostatic bonding [48, 53]. Figure 3(d) shows the TDM curve from
$X^{1}\Sigma$ to $A^{1}\Pi$ state. The value of TDM reaches to a minimum of
$-3.96$ D at $1.3$ Å and then drops to almost zero at $8$ Å. The TDM value for
$X^{1}\Sigma\rightarrow A^{1}\Pi$ transition at $R_{e}\,(2.143)$Å is found to
be 3.310 D, which is small than $3.392$ D at $R_{e}\,=\,2.137$Å reported in
Ref. [12] by 2.4%.

####  III.2.2 Vibrational and rotational parameters

We have found $158$, $50$ and $19$ bound vibrational states for $X^{1}\Sigma$,
$a^{3}\Pi$ and $A^{1}\Pi$ electronic states, respectively while Xu et al. have
reported 159 vibrational levels for $X^{1}\Sigma$, $53$ for $a^{3}\Pi$, and 10
for $A^{1}\Pi$ state. The difference in number of vibrational states in both
the works is mainly for the $A^{1}\Pi$ state and arises due to the difference
in the depths of potential energy surfaces. The value of potential depth for
$A^{1}\Pi$ state reported in Ref. [43] is smaller than any other calculated
value in the literature including ours. The relatively small value of
dissociation energy accounts for lesser number of vibrational levels in Ref.
[43]. Similar to AlF, the vibrational energy spacings and vibrational -
rotational constants decrease with increase in vibrational quantum number as
shown in Figure 4(a) and Figure 4(b). The centrifugal distortion constants for
the vibrational levels of electronic states are given in supplementary tables
S11 to S13. The difference in zero point energies ($E_{v,J\,=\,0}$) calculated
in our work and that reported in Ref. [43] is 0.06% for $X^{1}\Sigma$, 0.99%
for $a^{3}\Pi$, and $3.25\%$ for $A^{1}\Pi$ state. Supplementary Tables S14
and S15 present the PDMs of ro - vibrational levels of ground and excited
electronic states, respectively. The PDM value of $-1.621$ D for the lowest ro
- vibrational level of $X^{1}\Sigma$ state obtained in the current work
differs by 0.5% from that reported in Ref. [27]. The total lifetimes of
vibrational levels (with rotational quantum number $J\,=\,0$) of the studied
electronic states decrease initially and then begin to increase as shown in
Figure 4(c). We have obtained $2.5$ s, $5.3$ s, and $30$ s lifetimes for
$v\,=\,0,J\,=\,0$ level of $X^{1}\Sigma$, $a^{3}\Pi$, and $A^{1}\Pi$,
respectively. The longer lifetimes for the higher vibrational states have also
been reported in Ref. [54]. The results for $P$, $Q$ and $R$ branches due to
the rotational transitions between the vibrational levels are presented in
Figures 5 to 7 of supplementary file PS1.

##  IV Conclusion

In summary, we have carried out extensive theoretical electronic, vibrational
and rotational structure calculations that include a wide range of properties,
on the AlF and AlCl molecular systems. The ground and low - lying excited
states of both the systems have been studied at MRCI+Q level of theory using
QZ basis sets. Our findings for the diatomic constants (spectroscopic
parameters) agree well with existing results. The calculations performed in
this work report some results for the first time, and also resolve
discrepancies in some existing calculations. We have also studied the
vibrational levels and evaluated their wavefunctions, energies, dipole
moments, spontaneous as well as BBR-induced transition rates, and lifetimes.
Further, we extended this work to the rotational levels corresponding to
vibrational levels of all electronic states. We have reported transitions
between the rotational levels for P, Q, and R branches in the vibrational
levels of all the electronic states. We expect that the $ab$ $initio$ results
that are reported in this work for diatomic constants, molecular properties as
well as vibrational and rotational parameters of ground and excited electronic
states, would be useful for future spectroscopic studies including potential
applications ranging from ultracold molecules to astrophysics.

###### Acknowledgements.

All the structure calculations using MOLPRO program have been performed at the
computation facility available at Indian Association for the Cultivation of
Science (IACS), Kolkata.

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