# Observation and modelling of bound-free transitions to the $X^{1}\Sigma^{+}$
and $a^{3}\Sigma^{+}$ states of KCs

V. Krumins  A. Kruzins  M. Tamanis  R. Ferber<EMAIL_ADDRESS>Laser
Center, Faculty of Physics, Mathematics and Optometry, University of Latvia,
19 Rainis blvd, Riga LV-1586, Latvia  V. V. Meshkov  E. A. Pazyuk  A. V.
Stolyarov  Department of Chemistry, Lomonosov Moscow State University, 119991
Moscow, Leninskie gory 1/3, Russia  A. Pashov  Faculty of Physics, Sofia
University, 5 James Brouchier blvd., 1164 Sofia, Bulgaria

###### Abstract

The oscillation continuum in laser-induced fluorescence spectra of bound-free
$c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ and (4)${}^{1}\Sigma^{+}\to
X^{1}\Sigma^{+}$ transitions of the KCs molecule were recorded by Fourier-
transform spectrometer and modelled under the adiabatic approximation. The
required interatomic potentials for ground $a^{3}\Sigma^{+}$ and
$X^{1}\Sigma^{+}$ states were reconstructed in an analytical Chebishev-
polynomial-expansion form in the framework of the regularization direct-
potential-fit procedure based on the simultaneous consideration of
experimental line positions from [R. Ferber et al, Phys. Rev. A, 80, 062501
(2009)] and the present _ab initio_ calculation of short-range repulsive
potential data. It was proved that the repulsive part over dissociation limit
of the derived $a^{3}\Sigma^{+}$ potential reproduces the experiment better
than the potentials reported in literature. It is also shown that all
empirical and semi-empirical potentials available for the $X^{1}\Sigma^{+}$
state reproduce the bound-free (4)${}^{1}\Sigma^{+}\to X^{1}\Sigma^{+}$
spectrum with equal quality in the range of observations.

††preprint: AIP/123-QED

##  I Introduction

The growing interest in increasing the accuracy of potential energy curves of
polar alkali diatomics relates to their successful production in ultracold
conditions. Fast developing research of cold and ultracold molecular gases
revealed their prospective application in various fields of physics and
chemistry, see [1] for a review. Among them, the ground-state ultracold KCs
molecules have become an object of recent studies [2] due to its large
electric dipole moment and their favorable collisional properties [3]. A usual
approach is, first, to form the weakly-bound molecules from atom pairs by
magneto-association using Feshbach resonances and, second, to transfer them to
a deeply-bound rovibrational state by stimulated Raman adiabatic passage
(STIRAP) [4]. This procedure requires precise knowledge of the potential
energy curves (PECs) of the ground electronic $X^{1}\Sigma^{+}$ and
$a^{3}\Sigma^{+}$ states. The experiment-based singlet and triplet potentials
of these states have been determined in [5, 6] by Fourier-transform
spectroscopy in the bound part of interacting $X^{1}\Sigma^{+}$ and
$a^{3}\Sigma^{+}$ states applying a coupled-channel deperturbation procedure.
Based on these data the authors of [7] performed coupled-channel calculations
to predict positions and widths of the interspecies Feshbach resonances in the
ultracold K-Cs mixture. These resonances were recently detected in ultracold
K-Cs mixture [2]. It has been revealed that the observed positions of the
interspecies Feshbach resonances were by about 20 G higher than the respective
predictions in [6, 7]. The authors [2] refined the repulsive part of the
ground $a^{3}\Sigma^{+}$-state PEC located closely to dissociation threshold,
using the same form $A+B/R^{n}$, as in [5, 6], to match the observed positions
of Feshbach resonances. They concluded that it was possible to reproduce the
low-energy scattering properties by adjusting only the short-range part of the
potential while retaining the intermediate and long-range parts to change the
spectroscopic $a^{3}\Sigma^{+}$ PEC as little as possible. The resulting
analytical PEC allowed to predict with required accuracy the positions of the
observed Feshbach resonances [2]. Recently [8] the $a^{3}\Sigma^{+}$ state
PECs of KCs, NaCs, and RbCs were constructed employing the Tang-Toennies type
model, which is continuous in the entire range of internuclear distances $R$.
This model has been used also for KRb triplet $a^{3}\Sigma^{+}$ state [9].
Within this model the repulsive part, including short range, is described in
form $(A+BR^{n})exp(-bR-cR^{2})$ by five fitting parameters. Though the
reported [2, 5, 6, 8] PECs describe the intermediate bound part of the
$a$-state of KCs with high accuracy, a comparision of the short-range parts
from different sources reveals substantial differences between them when $R$
is diminishing.

These studies motivated the necessity to obtain a more direct experimental
information on the short-range repulsive PEC of the $a^{3}\Sigma^{+}$ state,
including the part above the dissociation limit. As is known [10], such data
may be obtained by recording and analysing the bound-free radiative
transitions in laser-induced fluorescence (LIF) from a bound upper-state level
that terminate to the lower-state with energies above the dissociation limit.
Such transitions produce the modulated continuous (’oscillating’) part of a
LIF spectrum, which, provided the upper electronic state’s PEC is known well
enough, is sensitive to the short-range repulsive part of the lower electronic
state. The bound-free transitions were observed in
(2)$d^{3}\Sigma^{+}-a^{3}\Sigma^{+}$ system of the NaK molecule [11] and later
developed [12] for the $(1)^{3}\Pi\rightarrow a^{3}\Sigma^{+}$ system to
determine the $a^{3}\Sigma^{+}$ state repulsive wall above the dissociation
limit, which was found to be in reasonable agreement with a prescription
suggested in [13]. In [14] the repulsive part of $a^{3}\Sigma^{+}$ state of
NaK was revisited using bound-free $c^{3}\Sigma^{+}\rightarrow
a^{3}\Sigma^{+}$ transitions. Later the authors of [15] applied the bound-free
part of $(4)^{3}\Sigma\to a^{3}\Sigma^{+}$ emission in NaK to obtain the
improved transition dipole moment functions. In a homonuclear alkali dimer Cs2
LIF to the short-range repulsive part of the $a^{3}\Sigma^{+}_{u}$ state was
observed [16]. More recently, Ashman and co-authors [17] used the recorded
bound-free $(5)^{3}\Pi_{0}\rightarrow(1)a^{3}\Sigma^{+}$ transitions in NaCs
to determine the repulsive wall of the $a^{3}\Sigma^{+}$ state and a
$(5)^{3}\Pi_{0}\rightarrow a^{3}\Sigma^{+}$ relative transition dipole moment
function. They demonstrated that applying a simple short-range repulsive
wall’s shape extrapolated in [18] as $V(R)=A+B/R^{3}$ revealed lack of
agreement with the recorded bound-free spectra, both in positions and
amplitudes of oscillations. By adjusting the parameters of exponential
analytical representation of the repulsive wall potential and the transition
dipole moment function the authors of [17] achieved excellent agreement
between observed and simulated oscillation structure in wide range of
rovibrational levels of the $a^{3}\Sigma^{+}$ state in NaCs.

In present study we report on spectroscopic observation and analysis of
continuous oscillation structure of bound-free transitions to the repulsive
part of both ground singlet $X^{1}\Sigma^{+}$ and triplet $a^{3}\Sigma^{+}$
states of KCs above the dissociation limit. For $a^{3}\Sigma^{+}$ state we
applied the $c^{3}\Sigma^{+}\rightarrow a^{3}\Sigma^{+}$ bound-free LIF to the
repulsive part of the $a^{3}\Sigma^{+}$ state potential. The bound-free
transitions to the $X$-state were recorded in (4)${}^{1}\Sigma^{+}\rightarrow
X^{1}\Sigma^{+}$ LIF. To model the recorded bound-free spectra we have
refitted spectroscopic data from [5] on $a/X$ states in bound region and
constructed the respective potentials in a fully analytical Chebychev
polynomial expansion (CPE) form, smoothly combined with the short-range part
of _ab initio_ potentials above dissociation limit, which were calculated in
present work. The obtained present short-range part of $a^{3}\Sigma^{+}$ state
potential as well as the potentials predicted in [5, 6, 2, 8] were used to
calculate the oscillation structure of bound-free transitions, which was
compared with the experiment. Modelling of the LIF signal became possible due
to recent extended experiment-based data on the $c^{3}\Sigma^{+}$-state in
[19, 20]. The (4)${}^{1}\Sigma^{+}$ state of KCs has been studied in detail in
[22, 21], hence, its properties are well known. The bound-free part of the
(4)${}^{1}\Sigma^{+}\rightarrow X^{1}\Sigma^{+}$ system was modelled with
present $X$-state PEC as well as with PECs from [5, 6, 23]. The required for
simulations spin-allowed (4)${}^{1}\Sigma^{+}-X^{1}\Sigma^{+}$,
$c^{3}\Sigma^{+}-a^{3}\Sigma^{+}$ and spin-forbidden
(4)${}^{1}\Sigma^{+}-a^{3}\Sigma^{+}$ transition dipole moment functions are
borrowed from Refs.[24, 21, 25, 20], in which they were obtained in the
framework of scalar-relativistic and fully relativistic _ab initio_ electronic
structure calculations.

##  II Experiment

###  II.1 Details

The continuous bound-free transitions in $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$
LIF, see Fig. 1, were recorded with the same experimental set-up as described
in our recent study of the $c^{3}\Sigma^{+}$ state in KCs [20]. The TiSph
laser Solstis/Msqured was used as excitation source. The LIF spectra were
recorded by Fourier-transform (FT) spectrometer using an InGaAs diode as
detector. KCs molecules were produced in a heat-pipe loaded with K and Cs
metals. Buffer gas Ar was added into heat-pipe to prevent condensation of
metal vapour on the windows. The main difficulty in the present experiment was
to find such excitation transitions, which ensure the most selective
excitation of the $c$-state rovibronic levels. Hence, we recorded, at first, a
great number of $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ LIF spectra, see [20], at
high resolution of 0.03 cm-1 and analyzed them from the point of view of
excitation selectivity of a rovibronic level. Unfortunately, it was not
possible to excite only one rovibronic level. At best we have chosen the
spectra, in which one very strong progression was dominating while other
disturbing progressions were very weak. To observe the bound-free transitions,
LIF spectra from selected suitable rovibronic levels were recorded at
diminished spectral resolution of 0.2 cm-1 since at high resolution the
continuous part of LIF spectra recorded by FT spectrometer is suppressed.
These measurements were performed at lowest possible buffer-gas-pressure of
about 0.6 mbar at diminished working temperature about 260 oC to minimize the
effect of buffer gas induced collisional population transfer between
rotational levels (so called rotational relaxation) in the excited state. The
estimated contribution of the latter into obtained bound-free spectra is about
4%.

Figure 1: Excitation and LIF observation schemes used in present study of
bound-free transitions in KCs: solid lines for the $a^{3}\Sigma^{+}$ state,
dotted lines for the $X^{1}\Sigma^{+}$ state. Term scheme is borrowed from
[26].

In order to record bound-free spectra to the singlet ground $X^{1}\Sigma^{+}$
state we used LIF transitions from (4)${}^{1}\Sigma^{+}$ state, see Fig. 1,
which was studied in [22]. To excite this state a dye laser Coherent 699-21
with Rhodamin 6G dye was exploited. A photomultiplier tube was used for LIF
detection. According to theoretical estimates [21] the bound-free transitions
(4)${}^{1}\Sigma^{+}\to X^{3}1\Sigma^{+}$ can be observed starting from upper
state vibrational levels $v^{\prime}$ = 44 and higher. Hence, we searched for
a selective excitation of high vibrational levels with possibly smaller
rotational quantum numbers $J^{\prime}$. We recorded and analysed LIF spectra
at high resolution and chose the ones, in which only one LIF progression was
dominating. For detection of bound-free transitions we recorded preliminary
spectra with different spectral resolution, 0.2, 0.5, and 1 cm-1. We found
that spectral noise diminishes with lower resolution, while the oscillation
contrast remains almost unchanged, therefore the bound-free spectra in this
case were recorded with resolution 0.5 cm-1.

###  II.2 The spectra

The LIF spectra for $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ transitions were
recorded in spectral range from 8000 cm-1 to 12000 cm-1. Some examples of the
recorded bound-free spectra with oscillation structure are presented in Figs.
2 and 3, see black traces. For better representation of a bound-free spectrum
the discrete part of the spectrum is not shown in these figures. Fig. 2.(a)
presents the bound-free part of LIF spectrum from the $c$-state $e$-parity
level $v^{\prime}$ = 24, $J^{\prime}$ = 47. It represents a sum of two
oscillating $P$, $R$ branches with $N_{a}$ = $J^{\prime}$ $\pm$ 1, $N_{a}$ is
a rotational quantum number of the $a^{3}\Sigma^{+}$. Besides, a weak
accidentally excited progression from $c$-state level $v^{\prime}$ = 28,
$J^{\prime}$ = 130 was also recorded in the discrete spectrum. Fig. 2 (b)
shows another example for LIF from the $f$-parity level with $v^{\prime}$ =
24, $J^{\prime}$ = 50. The oscillating structure represents a sum of three
branches $P^{Q}$, $Q$, and $R^{Q}$ with $N_{a}$ = $J^{\prime}$\+ 2, $N_{a}$ =
$J^{\prime}$, and $N_{a}$ = $J^{\prime}$\- 2, with normalized intensity ratios
determined from the discrete spectrum as 0.6:1:0.9, respectively. A very weak
fragmentary $P$, $R$ progression from the B${}^{1}\Pi$ state level
$v^{\prime}$ = 7, $J^{\prime}$ = 130, $E^{\prime}$ = 14 791.666 cm-1 was also
recorded in high-frequency range. Fig. 2 (c) demonstrates the bound-free
spectrum from $e$-level $v^{\prime}$ = 25, $J^{\prime}$ = 36. Two weak
progressions from levels $v^{\prime}$ = 33, $J^{\prime}$ = 111 and
$v^{\prime}$ = 40, $J^{\prime}$ = 19 were also recorded in the high frequency
part. In all three examples the bound-free spectrum of dominating progressions
is very weakly contaminated by weak branches from neighbour rotational levels
$J^{\prime}$ due to rotational relaxation, being of about 4% from main signal.

Figure 2: Recorded bound-free LIF spectra (black trace) of $c^{3}\Sigma^{+}\to
a^{3}\Sigma^{+}$ transition from the levels: (a) $v^{\prime}$ = 24,
$J^{\prime}$ = 47 $E^{\prime}$ = 13 762.000 cm-1, (b) $v^{\prime}$ = 24,
$J^{\prime}$ = 50 ($f$-level), $E^{\prime}$ = 13 767.304 cm-1; (c)
$v^{\prime}$ = 25, $J^{\prime}$ = 36, $E^{\prime}$ = 13 838.207 cm-1. Green
vertical line at high frequency side marks the end of the discrete spectrum.
The bound-free spectra simulated using the present $a^{3}\Sigma^{+}$ PEC and
the $c$-state PEC from [19] are shown by red solid lines. The intensity of
calculated oscillation spectrum is matched with experiment at the largest peak
at 8650 cm-1. Blue line in (a) depicts the calculated weak accidentally
excited bound-free spectrum from the $v^{\prime}$= 28, $J^{\prime}$ = 130,
$E^{\prime}$ = 14194.319 cm-1 level. The corresponding integrals over Franck-
Condon density, $\int_{\varepsilon_{min}}|\langle
v^{\prime}_{c}|E_{a}^{\prime\prime}\rangle_{R}|^{2}d\varepsilon$, are equal to
0.7807, 0.7814, and 0.7845 for progressions (a), (b), and (c), respectively.
Figure 3: Recorded bound-free LIF spectra (black trace) of spin-allowed
(4)${}^{1}\Sigma^{+}\to X^{1}\Sigma^{+}$ and spin-forbidden
(4)${}^{1}\Sigma^{+}\to a^{3}\Sigma^{+}$ transitions. Vertical line at high
frequency side marks the end of discrete spectrum of the dominating
progression. Solid red lines are the simulated spectra using the present
$X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ PECs and the (4)${}^{1}\Sigma^{+}$ PEC
from [22]. The curves are obtained accounting for rotational relaxation and
contribution of accidentally excited ($accid$) progressions (blue lines) and
normalized to the experiment at the highest peak. Calculated bound-free
spectra to the $a^{3}\Sigma^{+}$ state normalized to the respective calculated
(4)${}^{1}\Sigma^{+}\to X^{1}\Sigma^{+}$ spectrum are shown by green lines:
(a) $v^{\prime}$ = 48, $J^{\prime}$ = 16, $E^{\prime}$ = 17 921.9 cm-1; (b)
$v^{\prime}$ = 50, $J^{\prime}$ = 19, $E^{\prime}$ = 17 957.1 cm-1, $accid$ –
$v^{\prime}$ = 58, $J^{\prime}$ = 110, $E^{\prime}$ = 18 237.889 cm-1; (c)
$v^{\prime}$ = 52, $J^{\prime}$ = 20, $E^{\prime}$ = 17 991.349 cm-1, $accid$
– $v^{\prime}$ = 56, $J^{\prime}$ = 16, $E^{\prime}$ = 18 056.672 cm-1. The
corresponding integrals over Franck-Condon density,
$\int_{\varepsilon_{min}}|\langle
v^{\prime}_{(4)^{1}\Sigma^{+}}|E_{X}^{\prime\prime}\rangle_{R}|^{2}d\varepsilon$,
are equal to 0.3662, 0.3724, and 0.4530 for cases (a), (b), and (c),
respectively.

Fig. 3 shows examples of the recorded bound-free LIF spectra from the
(4)${}^{1}\Sigma^{+}$ state. To focus on the bound-free structure the spectra
in Fig. 3 are cut off from both sides. The very end of the discrete part is
marked by vertical lines at the upper frequency edge. Along with spin-allowed
(4)${}^{1}\Sigma^{+}\to X^{1}\Sigma^{+}$ transitions the spin-forbidden
(4)${}^{1}\Sigma^{+}\to a^{3}\Sigma^{+}$ transitions are observed as well due
to triplet state admixture of the (4)${}^{1}\Sigma^{+}$ state, see [5, 21] for
details. As can be seen, the continuous bound-free (4)${}^{1}\Sigma^{+}\to
X^{1}\Sigma^{+}$ part markedly differs from the oscillation structure in
transitions to the $a$-state in Fig. 2. Namely, the spectra are substantially
shorter and contain several peaks with sharply dropping amplitude. The
oscillating structure typical to transitions to the $a^{3}\Sigma^{+}$ state is
observed at lower frequencies where transition to $X$-state expires. The
spectra in Fig. 3 (a, b, c) include transitions from (4)${}^{1}\Sigma^{+}$
state vibrational levels 48, 50, and 52 with rather small $J^{\prime}$ equal
to 16, 19, and 20, respectively.

##  III Modelling

###  III.1 Intensity distribution in the observed LIF spectra

The transition probabilities in the observed (4)${}^{1}\Sigma^{+}\to
X^{1}\Sigma^{+}$ and $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ emission spectra
(see Fig. 1) have been modelled in the framework of the conventional adiabatic
approximation. In particular, this means that a very weak hyperfine
interaction between the ground $X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ states,
as well as the spin-orbit coupling (SOC) of the excited $c^{3}\Sigma^{+}$
state with the nearby $b^{3}\Pi$ and $B^{1}\Pi$ states were completely
neglected. Though the impact of the SOC effect on the energies of the
$c$-state monotonically grows as the vibrational quantum number
$v_{c}^{\prime}$ increases, it does not affect very much the nodal structure
of the upper state wavefunctions [27] outside the crossing point of $c$\- and
$b$-states. Regarding the ground states it should be noted that the hyperfine
(HFS) $a/X$ mixing is expected to be important only for the highly-excited
bound and quasi-bound rovibronic levels located in a vicinity of the
dissociation limit.

The relative intensity distribution in the bound-bound (bb) and bound-
continuum (bc) parts of the observed (4)${}^{1}\Sigma^{+}\to X^{1}\Sigma^{+}$
and $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ LIF spectra were evaluated according
to the relation

$\displaystyle I_{ij}^{\textrm{bb}}$ $\displaystyle\sim$
$\displaystyle\nu_{ij}^{4}|\langle
v^{\prime}_{i}|d^{\textrm{ab}}_{ij}|v_{j}^{\prime\prime}\rangle_{R}|^{2}\frac{S_{ij}}{2J^{\prime}+1};$
(1) $\displaystyle I_{ij}^{\textrm{bc}}$ $\displaystyle\sim$
$\displaystyle\sum_{J_{j}^{\prime\prime}}\nu_{ij}^{4}|\langle
v^{\prime}_{i}|d^{\textrm{ab}}_{ij}|E_{j}^{\prime\prime}\rangle_{R}|^{2}\frac{S_{ij}}{2J^{\prime}+1},$
(2)

where $\nu_{ij}=E_{i}^{\prime}-E_{j}^{\prime\prime}$ are the transition wave
numbers, $S_{ij}(J^{\prime},J^{\prime\prime})$ are the Hönl-London factors
known in the analytical form [28], $i\in[(4)^{1}\Sigma^{+},c^{3}\Sigma^{+}]$,
and $j\in[X^{1}\Sigma^{+},a^{3}\Sigma^{+}]$. The structural (oscillation)
continuum $I_{ij}^{\textrm{bc}}$ was calculated on an uniform grid of energy
points $E_{j}$ (with the step $\Delta\varepsilon$ = 0.5 cm-1) covering the
experimental transition wave numbers, $\nu_{ij}^{expt}$ and satisfied to the
complete vibrational basis condition

$\displaystyle\sum_{v^{\prime\prime}=0}^{v^{\prime\prime}_{max}}|\langle
v^{\prime}_{i}|v_{j}^{\prime\prime}\rangle_{R}|^{2}+\int_{\varepsilon_{min}}|\langle
v^{\prime}_{i}|E_{j}^{\prime\prime}\rangle_{R}|^{2}d\varepsilon=1,$ (3)

where the minimal $\varepsilon_{min}$-value corresponds to the energy of the
last found (highest) quasi-bound level, $v^{\prime\prime}_{max}$.

Hereafter, $d^{\textrm{ab}}_{ij}(R)$ are the electronic transition dipole
moments which were borrowed for spin-allowed (4)${}^{1}\Sigma^{+}\to
X^{1}\Sigma^{+}$, $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ and spin-forbidden
(4)${}^{1}\Sigma^{+}\to a^{3}\Sigma^{+}$ transitions from electronic structure
calculations performed in the framework of both scalar-relativistic [24, 21]
and fully relativistic [25, 20] approximations. The achieved results have
demonstrated that the absolute accuracy of the state-of-art _ab initio_
$d^{\textrm{ab}}_{ij}(R)$ functions derived, at least, for strong spin-allowed
transitions becomes comparable with the uncertainty of intensity measurements.

The eigenvalues and eigenfunctions required for the particular (bound, quasi-
bound and continuum) rovibronic levels of upper and lower electronic states
were obtained from the numerical solution of the 1D radial equation [29]:

$\displaystyle\left(\frac{\hbar^{2}d^{2}}{2\mu
dR^{2}}+U^{ad}(R)+\frac{\hbar^{2}J(J+1)}{2\mu
R^{2}}-E_{i}^{calc}\right)|v_{i}\rangle=0$ (4)

with the corresponding adiabatic PEC, $U^{ad}(R)$. The ro-vibrational
wavefunctions $|v_{i}\rangle$ belonging to bound and quasi-bound levels were
conventionally normalized as $|\langle v_{i}|v_{i}\rangle_{R}|^{2}=1$ while
the energy-normalized continuum wavefunctions $|E_{j}^{\prime\prime}\rangle$
were obtained by scaling a numerically propagate wavefunctions on the relevant
amplitudes of their analytical second-order semi-classical counterparts [30]
taken in the vicinity of equilibrium distance $R_{e}$ of the corresponding
$U^{ad}(R)$ potential.

The adiabatic potential of the upper (4)${}^{1}\Sigma^{+}$ state determined in
the fully analytical Chebychev-polynomial-expansion (CPE) form was taken from
[22], while the point-wise inverted-perturbation-approach (IPA) potential [19]
was implemented for the excited $c^{3}\Sigma^{+}$ state.

###  III.2 Semi-empirical potentials for $X^{1}\Sigma^{+}$ and
$a^{3}\Sigma^{+}$ states

Interatomic $U^{ad}(R)$ PECs for both singlet and triplet ground states were
defined on the semi-interval $R\in[R_{min},+\infty)$ in the CPE form [22]:

$\displaystyle
U_{CPE}(R)=T_{dis}-\frac{\sum_{k=0}^{m}c_{k}T_{k}(y_{p})}{1+\left(R/R_{ref}\right)^{n}},$
(5)

where $T_{k}(y)$ are the Chebyshev polynomials of the first kind depending on
the reduced radial variable $y_{p}(R)\in[-1,1]$:

$\displaystyle
y_{p}(R)=\frac{R^{p}-R_{ref}^{p}}{R^{p}+R_{ref}^{p}-2R_{min}^{p}},$ (6)

in which $p$ is a small positive integer and $R_{ref}>R_{min}\geq 0$ is a
reference distance.

The CPE coefficients $c_{k}$ were obtained in the framework of the
regularization direct-potential-fit (DPF) procedure based on the iterative
non-linear minimization of the sum of squares:

$\displaystyle\sum_{l=1}^{N_{expt}}\left(\frac{\nu^{expt}_{l}-\nu^{calc}_{l}}{\sigma^{expt}_{l}}\right)^{2}+w\sum_{l=1}^{N_{ab}}\left(\frac{U^{CBS}(R_{l})-U^{CPE}(R_{l})}{\sigma^{CBS}(R_{l})}\right)^{2},$
(7)

where the weight factor $w$ was selected to provide the proper balance between
experimental and _ab initio_ data contributions into the total sum. $N_{expt}$
and $N_{ab}$ are the numbers of experimental lines and _ab initio_ PEC points
involved in the sum, respectively. The input _ab initio_ data mainly determine
the repulsive part of the $U^{CPE}(R)$ potential (above dissociation limit)
while and the experimental data provide an accurate reconstruction of the
well-bound (below dissociation limit) part of the resulting semi-empirical
potential.

The CPE analytical form (5) possesses the correct long range asymptotic
behavior

$\displaystyle U_{CPE}(R\to+\infty)=T_{dis}-\sum_{n}\frac{C_{n}}{R^{n}},$ (8)

where the leading dispersion coefficients $C_{n}$ are explicit functions of
CPE parameters [22]:

$\displaystyle C_{n}=R_{ref}^{n}\sum_{k=0}^{m}c_{k},$ (9) $\displaystyle
C_{n+p}=2(R_{min}^{p}-R_{ref}^{p})\sum_{k=0}^{m}k^{2}c_{k}.$ (10)

Hence, the fitting coefficients $c_{k}$ were constrained owing to the
relations (9) - (10) in order to fit the $C_{6}$, $C_{8}$, and $C_{10}$ values
available from the relevant _ab initio_ atomic calculations [31, 32].

In expression (7), $\nu^{expt}_{l}$ are the experimental wave numbers
corresponding to the bound-bound part of the LIF spectra below the
dissociation limit. These values, together with their uncertainties
$\sigma^{expt}_{l}$ were borrowed from the Supplemented Material of Ref. [5].
All mutually HFS-perturbed levels of both singlet and triplet states closely-
lying to their common dissociation threshold were excluded from the present
fit. The calculated transition energy $\nu^{calc}_{l}\equiv\nu^{calc}_{ij}$ is
$E^{\prime}_{i}-E_{j}^{\prime\prime}$, where $E^{\prime}_{i}$ and
$E_{j}^{\prime\prime}$ are rovibronic energies of the upper and lower
electronic states, respectively. The $E_{i}^{\prime}$-values are considered to
be the adjusted fitting parameters while $E_{j}^{\prime\prime}\equiv
E^{calc}_{i}$ are the eigenvalues of the radial equation (4) solved with the
present CPE potential (5).

The $U^{CBS}(R)$ PEC involved in Eq.(7) was obtained by an extrapolation to
the complete basis set (CBS) electronic energy $U^{CCSD(T)}$, which was
calculated by means of the spin-restricted close-shell coupled-cluster
approach with perturbative treatment of triple excitation (CCSD(T)) and the
pseudo-potentials of the _aug-cc-pwCVnZ-PP_ (with $n$ = 4 and 5) quality for
both K and Cs atoms [33, 34]. The electronic structure calculation was
conducted by means of the MOLPRO 2010.1 package [35]. All 18 (2 valence + 16
sub-valence) electrons of the molecule were explicitly correlated. For the CBS
extrapolation the two-point formula was adopted:

$\displaystyle U^{CCSD(T)}(n)=U^{CBS}+\frac{A}{n^{\alpha}},$ (11)

where the degree parameter $\alpha$ was adjusted, by the substitution of
$U^{CBS}$ for $U^{emp}$ in Eq.(11). This allowed to achieve that the resulting
extrapolated energy perfectly fits the repulsive part of the corresponding
empirical PECs $U^{emp}(R)$, which have been already well-established in the
well-bound region (below the dissociation limit) in the ground states studies
[5, 6]. Then, the linear extrapolation of the derived $\alpha(R)$ functions to
smaller $R$-values (for $R\leq R_{min}$, where $U^{CPE}(R_{min})=T_{dis}$) was
performed and the extrapolated $\alpha$-values were applied for a direct
estimate, by Eq.(11), of the repulsive limb of the $U^{CBS}(R\leq R_{min})$
PEC above the dissociation limit. The uncertainties $\sigma^{CBS}$ in the
extrapolated PEC $U^{CBS}$ were roughly estimated as a difference between the
$U^{CBS}$ values obtained with the fixed $\alpha$-values taken as $\alpha$ = 3
and 4. These point-wise $U^{CBS}(R_{l})$ and $\sigma^{CBS}(R_{l})$ functions
were finally involved in the sum (7).

##  IV Results

###  IV.1 Potential energy curves

The resulting parameters of obtained semi-empirical CPE interatomic potentials
of the $a^{3}\Sigma^{+}$ and $X^{1}\Sigma^{+}$ states are presented in Table
1. These fully analytical PECs were used to describe the recorded bound-free
LIF spectra to $a^{3}\Sigma^{+}$ and $X^{1}\Sigma^{+}$ states. The short-range
part of $a^{3}\Sigma^{+}$ and $X^{1}\Sigma^{+}$ potentials over dissociation
limit in the energy range covered by present experiment is presented in Fig. 4
along with the previously constructed empirical potentials borrowed from
previous works. As can be seen in Fig. 4 (a), the present $a$-state potential
(pw) is very close to the analytical semi-empirical potential recently
obtained in Ref. [8]; they start to differ slightly only at energies about 600
cm-1. According to Fig.4 (b) the present $X$-state potential (pw) defined in
the CPE form practically coincides with its Morse/Long-Range (MLR) counterpart
[23]. Robert LeRoy has used to build the MLR potential (given in the
Supplementary Material [36]) exactly the same LIF line positions [5] as we did
but he did not involve in the DPF procedure any _ab initio_ data. The slope of
both CPE and MLR potentials is slightly smaller than the slope of potentials
constructed in the hybrid, so-called Hannover’s, form in Refs [5, 6].

###  IV.2 The $a^{3}\Sigma^{+}$ state

Let us analyze how the repulsive part of the present semi-empirical $a$-state
CPE potential (5) describes the experimental bound-free $c-a$ spectra. The
calculated bound-free spectra are depicted in Fig. 2 by solid red curves,
which amplitudes are matched with experiment at the largest peak located at
about 8650 cm-1; note that the calculated oscillation spectra are averaged
over rotational branches. The curves include weak rotational relaxation, which
contributes by about 4% to the total oscillation spectrum. According to our
analysis the rotational relaxation does not cause any shifting of peak
positions leading only to a slight broadening of oscillation structure. The
simulated signal in Fig.2 (a) contains also a contribution from an
accidentally excited level $v^{\prime}$ = 28, $J^{\prime}$ = 130 of the
$c$-state with energy $E^{\prime}$= 14 194.284 cm-1, see the small-amplitude
blue curve. The simulated curve in Fig.2 (b) is averaged over three rotational
branches $N_{a}$ = $J^{\prime}$ \+ 2, $N_{a}$ = $J^{\prime}$, and $N_{a}$ =
$J^{\prime}$\- 2, with normalized intensity ratios 0.6:1:0.9, respectively. We
have not included here the 5% contribution from the $B^{1}\Pi$ state level
$v^{\prime}$ = 7, $J^{\prime}$= 130, $E^{\prime}_{B}$ = 14 791.666 cm-1 since
a correct accounting would need the knowledge of non-adiabatic multi-channel
structure of upper state wavefunction. For the same reason we did not include
in Fig.2 (c) the 7% contribution from the $c$-state level $v^{\prime}$ = 33,
$J^{\prime}$ = 111, $E^{\prime}$ = 14 314.418 cm-1. As can be seen in these
examples the calculated intensities describe satisfactory the oscillation
peaks’ positions. However, it is clearly seen that oscillations’ amplitude in
the higher frequency range markedly exceeds the experimental one; the related
discussion is postponed to Sec. V.

It is of interest to compare the calculated oscillation patterns obtained
applying the $a^{3}\Sigma^{+}$ state short-range potentials above dissociation
limit from different sources. Fig. 5 presents an example of simulated bound-
free part of the $c-a$ spectrum from $v^{\prime}$ = 24, $J^{\prime}$ = 47
level; the pattern obtained with present CPE potential is depicted with by red
line. For visibility the simulated oscillations are divided in two parts. Fig.
5 (a) includes the frequency range just above the $a$-state dissociation limit
where the first oscillation starts. It is obvious that meaningful differences
between the signals obtained by the present potential and by PECs borrowed
from the literature appear even at second oscillation for Ref. [5], and for
the third oscillation for Ref. [2] and the differences are increasing with
oscillation number. The differences become overwhelming at low frequency part
below 9000 cm-1, see Fig. 5 (b).

Figure 4: Comparison of short-range semi-empirical potentials of the
$a^{3}\Sigma^{+}$ (a) and $X^{1}\Sigma^{+}$ (b) states of KCs above
dissociation limit taken from different sources; pw - present work. Figure 5:
Comparison of calculated bound-free $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$
spectrum from the upper level $v^{\prime}$ = 24, $J^{\prime}$ = 47 using
different $a^{3}\Sigma^{+}$ semi-empirical PECs: red – present, blue– [2],
green– [8], orange– [5], and black– [6]. (a) first four oscillations of the
continuous spectrum just over dissociation limit; (b) the lower frequency part
of continuous spectrum.

To compare quantitatively the potentials agreement with the experiment, we
have determined the differences between experimental and calculated positions
of oscillations peaks. These differences were transformed from frequency scale
to energy differences $\Delta E=E^{expt}-E^{calc}$ with respect to energy
$\varepsilon$, Eq.(3), above dissociation limit for a measured progression,
see green vertical lines at high frequency side in Fig. 2. The differences are
represented in Fig. 6 as dependent on $\varepsilon$. The differences $\Delta
E(\varepsilon)$ obtained with the present CPE potential are depicted by full
circles. The respective values are distributed within $\pm$ 1.5 cm-1 corridor
in the range up to 400 cm-1, then they start to decline slightly from the
experiment. For instance, for progression with $v^{\prime}$ = 25, $J^{\prime}$
= 36 the difference reaches the value of 10 cm-1 at energy of 1050 cm-1 over
dissociation limit. The differences $\Delta E(\varepsilon)$ obtained with the
alternative forms of potential taken from Refs [2, 5, 6, 8] are depicted by
empty squares for progression with $v^{\prime}$ = 24, $J^{\prime}$ = 47. As it
can be seen, the PECs [2, 5, 6] yield the respective values, which start to
decline from the experiment very fast. Note that the last point at 1000 cm-1
for PEC taken from Ref. [6] is out of scale by 158 cm-1. The $\Delta
E(\varepsilon)$ values obtained with the semi-empirical PEC recently
constructed in Ref. [8] are much closer to the one obtained by the present
semi-empirical potential, however, they still start to decline at energies
over 500 cm-1.

###  IV.3 The $X^{1}\Sigma^{+}$ state

The simulations of observed bound-free spectra to the ground singlet $X$-state
are presented in Fig. 3 using the present $X$-state CPE potential and the
(4)${}^{1}\Sigma^{+}$ CPE PEC borrowed from Ref. [22]. The simulated signal
accounts for the weaker bound-free spectrum to the triplet $a$-state as well.
The intensities calculated for the spin-forbidden
(4)${}^{1}\Sigma^{+}-a^{3}\Sigma^{+}$ transition (depicted by a green line)
were normalized to the theoretical intensities of the spin-allowed
(4)${}^{1}\Sigma^{+}\to X^{1}\Sigma^{+}$ transition (red line). The
simulations account for rotational relaxation, which contributes overall of
about 10% to the bound-free spectrum. The simulations have demonstrated that
this effect causes some broadening while it does not shift the oscillation
peak positions. The simulated signals in Fig. 3 (b), (c) account for a
contribution from accidentally excited levels depicted by small-amplitude blue
curves. As can be seen, the simulated spectra describe satisfactorily the
recorded spectra. The tiny non-systematic deviations of peaks maxima are most
probably caused by additional background ignored in the modeling. The
simulation proves the appearance of additional peaks, which are becoming more
pronounced with increasing upper state vibrational quantum number,
$v^{\prime}$, see the signals in Fig. 3, (b) and (c).

It should be mentioned that for $v^{\prime}$ = 52 we also have performed
testing calculations of the expected (4)${}^{1}\Sigma^{+}-X^{1}\Sigma^{+}$
bound-free spectra using different empirical potentials which are available
for the $X^{1}\Sigma^{+}$ state from previous works, namely [5, 6, 23]. The
oscillation pattern was practically identical for all four exploited PECs; the
corresponding figure is presented in Supplemented Material.

Table 1: The resulting mass-invariant parameters of the CPE potential (5) defined for the $X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ states of KCs on the semi-interval $[R_{min},+\infty)$. The dissociation energy $T_{dis}=4069.076$ (in cm-1) referring to the minimum of the ground $X^{1}\Sigma^{+}$ state. The Chebyshev expansion coefficients $c_{k}$ are given in cm-1 while the fixed $R_{min}$ and $R_{ref}$ parameters in Å. The common $n$ = 6 and $p$ = 2 parameters were also fixed for both states. The FORTRAN subroutine generating the present CPE potentials for both $X^{1}\Sigma^{+}$ and $a^{3}\Sigma^{+}$ states of KCs is provided in the Supplementary Material [36]. | $X^{1}\Sigma^{+}$ | $a^{3}\Sigma^{+}$
---|---|---
$c_{0}$ | -373.624 | 1557.978
$c_{1}$ | -5391.769 | -3532.255
$c_{2}$ | 6442.491 | 2448.655
$c_{3}$ | -2248.684 | -1132.530
$c_{4}$ | 307.482 | 376.473
$c_{5}$ | -679.865 | -82.734
$c_{6}$ | 295.480 | -7.879
$c_{7}$ | 30.805 | 16.981
$c_{8}$ | 87.482 | -15.873
$c_{9}$ | -42.874 | 20.270
$c_{10}$ | -46.075 | -16.752
$c_{11}$ | -0.775 | 8.659
$c_{12}$ | 17.062 | -4.080
$c_{13}$ | 6.304 | 1.238
$c_{14}$ | 10.069 | 0.049
$c_{15}$ | -4.478 |
$c_{16}$ | 5.343 |
$c_{17}$ | -3.195 |
$c_{18}$ | 1.150 |
$c_{19}$ | -2.588 |
$c_{20}$ | -0.434 |
$c_{21}$ | -0.685 |
$R_{min}$ | 2.5 | 3.0
$R_{ref}$ | 5.0 | 6.4

##  V Concluding remarks

Both Fig. 2 and Fig. 6 demonstrate that the conventional adiabatic
approximation used in present work describes rather well the observed
oscillatory structure of the bound-free $c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$
LIF spectrum. However, some discrepancies should be mentioned. First, the
calculated oscillations amplitude in the beginning part of the bound-free
spectrum exceeds the experimental one, see Fig. 2. Second, the small
systematic difference between present calculated and experimental energies of
the short-range part of repulsive potential is seen in Fig. 6. A possible
reason, which can influence the current results is the fact that the
$c^{3}\Sigma^{+}$ state is perturbed by the close-lying (see Fig. 1) singlet
$B^{1}\Pi$ state and also by the triplet $b^{3}\Pi$ state, forming the so-
called spin-orbit $B\sim b\sim c$ complex, see [20] and references therein.
Accordingly, the $c$-state wavefunction should be affected by these
perturbations, hence, the usage of upper state adiabatic vibrational
wavefunction strictly speaking is not correct. For example, Fig. 7 illustrates
a relative intensity distribution measured for the bound-bound part of the
$c\to a$ LIF progression from the levels $v^{\prime}$ = 25, $J^{\prime}$ = 36
and $v^{\prime}$ = 24, $J^{\prime}$ = 47. As can be seen, though the
intensities calculated under adiabatic approximation nicely reproduce the
experiment for transitions to higher $v_{a}$ above the matching vibrational
level, they completely fail for transitions to lower levels. The apparent
reason for the observed discrepancy is the spin-orbit interaction of $c$\- and
$b$-states in the vicinity of their crossing point (see Fig. 1). Therefore, a
comprehensive deperturbation analysis of the spin-orbit coupled $B\sim b\sim
c$ complex is indispensably required to refine a nodal structure of
vibrational multi-channel wavefunction of the upper $c$-state. This extremely
challenging and complicated task is still to be realized. The respective work
is currently in progress.

As is shown by Fig. 6, though the Feshbach resonances are a very sensitive
probe of the quality of the $a$-state PEC in the vicinity of dissociation
limit (see, for instance, Ref. [2]) they do not guarantee the correct PEC
behaviour for higher energies above the dissociation limit. Here, the
informations based on bound-free transitions experiments is useful.

Figure 6: Energy differences $\Delta E(\varepsilon)=E^{expt}-E^{calc}$ between
experimental energies $E^{expt}$ for measured LIF progressions and the
respective energies $E^{calc}$ calculated using the $a^{3}\Sigma^{+}$ PECs
taken from different works as dependent on energy over dissociation limit,
$\varepsilon$. Dashed horizontal lines depict experimental uncertainty as
$\pm$ 1.5 cm-1 limits. Figure 7: Relative intensity distribution in
$c^{3}\Sigma^{+}\to a^{3}\Sigma^{+}$ bound - bound LIF progression coming from
upper levels (a) $v^{\prime}$ = 24, $J^{\prime}$ = 47 and (b) $v^{\prime}$ =
25, $J^{\prime}$= 36; columns - experiment, full circles - calculation. The
calculated intensities are matched with experiment at $v_{a}$ = 8 and $v_{a}$
= 7 bands for the progressions of (a) and (b), respectively.

## Acknowledgements

We are indebted to Michael Schwarzer and Jan Peter Toennies for providing
their $a^{3}\Sigma^{+}$ state potential in the appropriated format and to Ilze
Klincare for numerous helpful pieces of advice. Riga team acknowledges the
support from the Latvian Council of Science, project No. lzp-2020/2-0215:
"Interatomic potentials of alkali atom pairs in a wide range of internuclear
distances" and from the University of Latvia Base Funding No A5-AZ27. Moscow
team is grateful for the support by the Russian government budget (section
0110), projects No.121031300173-2 and 121031300176-3.

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