# Fourier-transform spectroscopy and global deperturbation treatment of the
$A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$ states of K2 in the entire bound
energy range

I. Klincare  A. Lapins  M. Tamanis  R. Ferber<EMAIL_ADDRESS>Laser Center,
Faculty of Physics, Mathematics and Optometry, University of Latvia, 19 Rainis
blvd, Riga LV-1586, Latvia  A. Zaitsevskii  Department of Chemistry, Lomonosov
Moscow State University, 119991, Moscow, Leninskie gory 1/3, Russia  E.A.
Pazyuk  A.V. Stolyarov<EMAIL_ADDRESS>Department of Chemistry,
Lomonosov Moscow State University, 119991, Moscow, Leninskie gory 1/3, Russia

###### Abstract

Rotationally-resolved Fourier-transform spectra of laser-induced fluorescence
(LIF) $A^{1}\Sigma^{+}_{u}\sim b^{3}\Pi_{u}\rightarrow X^{1}\Sigma^{+}_{g}$ of
K2 molecules were recorded and analyzed yielding 4053 term values of spin-
orbit (SO) coupled $A\sim b$ complex of 39K2 isotopologue with ca 0.01 cm-1
accuracy. Their compilation with 1739 term values from previously published
sources allowed to cover the energy range [9955, 17436] cm-1 from the bottom
of the lower-lying $b^{3}\Pi_{u}$ state up to the vicinity of the atomic
asymptote $4s\;^{2}{\rm S}_{\frac{1}{2}}$ \+ $4p\;^{2}{\rm P}_{\frac{1}{2}}$,
with rotational quantum number $J\in[0,149]$. The experimental data were
processed by a direct 6$\times$6 coupled-channel (CC) deperturbation
treatment, which accounted explicitly for both SO and electronic-rotational
interactions between all six $e$-symmetry states:
$A^{1}\Sigma^{+}_{u}$($0^{+}_{u}$),
$b^{3}\Pi_{u}$($0^{+}_{u}$,$1_{u}$,$2_{u}$), $c^{3}\Sigma_{u}$($1_{u}$), and
$B^{1}\Pi_{u}$($1_{u}$). The initial parameters of the global deperturbation
model have been estimated in the framework of ab initio electronic structure
calculations applying multi-referenced configuration-interaction and coupled-
clusters methods. The interatomic potentials analytically defined for $A$ and
$b$ states, as well as SO-splitting of the triplet $b$ state and $A\sim b$ SO-
coupling functions have been particularly refined to fit 5792 term values of
the 39K2 isotopologue, whereas the rest parameters were fixed on their ab
initio values. The resulting mass-invariant parameters of the 6$\times$6 CC
model reproduced the overall rovibronic term energies of the $A\sim b$ complex
of 39K2 with the accuracy, which is well within the experimental errors. The
quality of the deperturbation analysis was independently confirmed by
comparison with the present obtained 705 and 14 term values of respective
39K41K and 41K2 isotopologues, as well as by agreement between measured and
predicted relative intensity distributions in long $A\sim b\rightarrow
X(v_{X})$ band progressions. This deperturbation analysis provided the refined
dissociation energy $T_{dis}$=17474.569(5) cm-1 and the long-range coefficient
$C_{3}^{\Sigma}$ = 5.501(4) $\times 10^{5}$cm-1Å3 relevant to the non-
relativistic atomic limit $4s$+$4p$. The derived $T_{dis}$ yielded the
accurate well depth $D_{e}$ = 4450.910(5) cm-1 for the ground
$X^{1}\Sigma^{+}_{g}$ state, whereas the new $C_{3}^{\Sigma}$ value yielded
the improved estimates for atomic K($4p\;^{2}{\rm
P}_{\frac{1}{2};\frac{3}{2}}$) radiative lifetimes,
$\tau_{\frac{1}{2}}$=26.67(3) and $\tau_{\frac{3}{2}}$=26.32(3) ns.

##  I Introduction

The low-lying excited $A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$ states ($A$ and
$b$ for short) of a homonuclear diatomic alkali molecule are attracting a lot
of attention for several reasons. First, they open a window to the state
manifold of g-parity, which is not directly accessible from the ground state
of the same parityBernath2005book . In addition, the triplet states become
accessible due to an admixture of the $b^{3}\Pi_{u}$ stateField2004book . More
recent interest is connected with producing the stable ultracold molecular
ensemblesJulienne2012  since these states can successfully serve as the
intermediates to transfer translationally cold molecules produced from
ultracold atoms in the weakly bound ground state levels by magneto- or photo-
association to the deeply bound ground $X^{1}\Sigma^{+}_{g}$ state Tiemann2020
, in particular, to the lowest rovibrational level with $\textit{v}_{X}$ =
$\textit{J}_{X}$ = 0.

The challenging issue in studying the singlet $A^{1}\Sigma^{+}_{u}$ and
triplet $b^{3}\Pi_{u}$ states relates to the spin-orbit (SO) interaction,
which rapidly increases as the mass of a dimer increasesPazyuk15_RCR . In
respect to that, among alkali dimers one can obviously distinguish light (Li2,
where the fine structure Li($2p\;^{2}{\rm P}$) splitting NIST  $\Delta
E_{fs}=E_{2p^{2}{\rm P}_{\frac{3}{2}}}-E_{2p^{2}{\rm P}_{\frac{1}{2}}}$ is
small, being 0.336 cm-1), intermediate (Na2 and K2, the respective $\Delta
E_{fs}$ are 17.1963 cm-1 and 57.7103 cm-1) and heavy (Rb2 and Cs2, where the
respective $\Delta E_{fs}$ values reach 237.595 cm-1 and even 554.0393 cm-1).
For Li2 it is often not necessary at all to consider the mutual SO interaction
between $A$ and $b$ states in an explicit form Linton1996 ; Russier1997  . The
situation becomes completely different for the heaviest Rb2 and Cs2 dimers, in
which the $A$ and $b$ states should be considered as a fully SO-mixed $A\sim
b$ complex. For these molecules, the 2$\times$2
$A^{1}\Sigma^{+}_{u}(0^{+}_{u})\sim b^{3}\Pi_{u}(0^{+}_{u})$ or even
4$\times$4 $A^{1}\Sigma^{+}_{u}(0^{+}_{u})\sim
b^{3}\Pi_{u}(0^{+}_{u},1_{u},2_{u})$ coupled-channel (CC) deperturbation
treatment has unavoidably to be applied yielding the interatomic potential
energy curves (PECs) and SO coupling matrix elements allowing to reproduce the
measured rovibronic term values of the $A\sim b$ complex with the experimental
accuracy, close to 0.01 cm-1, see PRA2013  for Rb2 and PRA2011 ; PRA2019  for
Cs2. This allows, in particular, to reliably predict transition probabilities
and laser frequencies necessary to excite a selected rovibronic level.

The intermediate case of sodium and potassium dimers still leaves some hope
for a possibility to separately focus on the singlet $A$ state alone; this
simplified approach was used, for instance, by Lyyra et al. Lyyra1990  and
Falke et al. Lisdat2006  who presented an effective interatomic PEC for the
SO-deperturbed (more precisely, the SO-decoupled from the $b$ state) $A$ state
of K2. The first deperturbative analysis of the $A$ and $b$ states in K2 was
accomplished in the framework of the conventional effective Hamiltonian
approach (EHA) Brown79  by Ross et al. Ross87 , then by Jong et al.
JongJMS1992  and Lisdat et al. Lisdat2001 . The similar band-by-band
deperturbation treatment of the locally perturbed $A^{1}\Sigma^{+}_{u}$ and
$b^{3}\Pi_{u}$ states in Na2 was performed using the magnetic Landé factors
measured by Hanle effect Stolyarov92_Na2 ; Stolyarov93_Na2 . For Na2 the most
accurate 4$\times$4 CC deperturbation analysis of $A$ and $b$ states was
performed in a more recent study Qi2007 , which accumulates the experimental
term values data from 20 sources.

The so far most comprehensive deperturbative study of the coupled $A\sim b$
states in K2 was performed by Manaa et al. Manaa2002 . The authors had
incorporated the experimental term values available at that time, see, Ross87
; Lyyra1990 ; JongJMS1992 ; Lisdat2001 ; AmiotJMS1991 ; Amiot1995 ; Kim95 , as
well as a considerable amount of the new data Manaa2002 . The deperturbative
treatment Manaa2002  involved 1802 term values and combination differences of
39K2 isotopologue covering the $A\sim b$ complex region up to about 16 500
cm-1, which is, however, still too far from the second relativistic
42S${}_{\frac{1}{2}}$+42P${}_{\frac{1}{2}}$ dissociation threshold being about
17 436 cm-1. These data were fitted based on the reduced 4$\times$4 CC model,
which takes into account explicitly the SO coupling between the singlet $A$
and triplet $b$ states, as well as the equidistant SO splitting of the triplet
state. The unweighted fit yielded an r.m.s. residual of 0.018 cm-1 comparable
with the accuracy of spectroscopic measurements limited by the Doppler effect.
The authors finally presented: (i) the fitted set of empirical Dunham
coefficients used for Rydberg-Klein-Rees (RKR) potentials construction of
$A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$ states up to $v_{A}=88$ and
$v_{b}=93$ vibrational level, respectively, as well as (ii) the properly
scaled (morphed) diagonal and off-diagonal SO matrix elements obtained due to
all-electron multi-reference configuration interaction (MR-CI) electronic
structure calculations. The most reliable sets of both non-relativistic
Magnier2004  (${}^{1,3}\Lambda_{u,g}$) and relativistic Magnier2009
($\Omega_{u,g}^{+/-}$) ab initio PECs for singlet and triplet states manifold
of K2 have been obtained in the framework of full CI calculations performed
for two valence electrons in combination with $l$-dependant core-polarization
potentials (CPP-$l$) construction.

The following considerations motivate the present study of the $A$ and $b$
states in K2. As already mentionedManaa2002 , over significant energy region
the experimental term values are still rather sparse. Also, the data analyzed
by Manaa et al. Manaa2002  involve only the most abundant 39K2 isotopologue in
the limited energy range of the singlet $A$ state ($E_{A}$<16037 cm-1) far
from dissociation threshold. We therefore to include into the global
deperturbation analysis the extremely accurate term values of the $A\sim b$
complex available up to the 42S${}_{\frac{1}{2}}$+42P${}_{\frac{1}{2}}$
dissociation threshold Lisdat2006 ; Falke2007 . These data mainly belonging to
the highly excited rovibronic levels of the $A$ state were obtained by Falke
et al.Falke2007PhD  by applying the sub-Doppler laser absorption spectroscopy
of K2 produced in a molecular beam. So, there is hope that addressing these
issues may allow providing the global uniform description of the
$A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$ states of K2, which will reproduce
the overall term values set with the experimental (both Doppler and sub-
Doppler) spectroscopic accuracy up to the second non-relativistic dissociation
limit $4s\;^{2}{\rm S}+4p\;^{2}{\rm P}$. To reach this goal, we aimed to
considerably enlarge the amount and abundance of experimental term values for
39K2, as well as to obtain the $A\sim b$ data for the 39K41K and 41K2
isotopologues.

We performed in the University of Latvia Laser Center in Riga the rotationally
resolved Fourier-transform (FT) measurements of $A^{1}\Sigma^{+}_{u}\sim
b^{3}\Pi_{u}\rightarrow X^{1}\Sigma^{+}_{g}$ fluorescence. The significantly
enlarged data field was compiled with the previously published ones, including
low-lyingManaa2002 ; Kim95  and highly-excitedLisdat2006 ; Falke2007  term
values. The overall data set has been then treated in the framework of the
comprehensive 6$\times$6 CC deperturbation model, which consists of the six
$e$-symmetry $\Omega_{u}=(2)0^{+}_{u},(3)1_{u},(1)2_{u}$ states and takes into
account explicitly both SO and electronic-rotational non-adiabatic
interactions between four $A^{1}\Sigma^{+}_{u}$, $b^{3}\Pi_{u}$,
$c^{3}\Sigma^{+}_{u}$, and $B^{1}\Pi_{u}$ states, which correspond to pure
Hund’s "a" case coupling and converge to the common non-relativistic
$4s^{2}$S+$4p^{2}$P limit (see Fig.1). The initial structure parameters
required as the functions of the internuclear distance $R$, have been
estimated in the framework of alternative scalar-relativistic MR-CI method
with subsequent account for the SO interactions of scalar states and fully
relativistic coupled cluster approach. To clime the mass-invariant property of
the derived structure parameters the rovibronic term values predicted for the
39K41K and 41K2 isotopologues were directly compared with their experimental
values. A high reliability of the calculated multi-channel vibrational
wavefunctions of the $A\sim b\sim c\sim B$ complex and ab initio $A-X$
transition dipole moment (TDM) borrowed from Ref. Krumins2020  was
independently confirmed by a good agreement of the estimated Einstein emission
coefficients with the relative intensity distributions measured for the long
$A\sim b\rightarrow X(v_{X})$ band progressions of 39K2 and 39K41K
isotopologues.

Figure 1: The schema of ab initio scalar-relativistic potential energy curves
of the K2 molecule involved into the present consideration.

##  II Experiment

###  II.1 Experimental setup

The rovibrational term values of the $A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$
states in K2 were obtained from the FT spectra of the $A\sim b\rightarrow
X^{1}\Sigma^{+}_{g}$ laser induced fluorescence (LIF) recorded with the Bruker
IFS-125HR spectrometer. The 39K2, 39K41K, and 41K2 molecules were produced in
a linear heat-pipe oven according to the natural abundance of the stable K
isotopes Tiecke2019 : 93.26% of 39K and 6.73 % of 41K. For detection of near
infra-red LIF spectra the InGaAs diode was used. About one third of data was
picked up from previously recorded FT-LIF spectra during our studies of the
$A\sim b$ complex of KCs and KRb molecules Kruzins2010 ; Alps2016 . As is
clear, since molecular vapour in the respective heat-pipe operated at about
300oC contained also potassium dimer, the accidentally excited $A\sim
b\rightarrow X$ LIF of K2 was also observed and recorded. In the KCs
experiments, the tunable home-made external cavity diode lasers with 930, 950,
980, and 1020 nm laser diodes were used for excitation. In the KRb experiments
the $A\sim b$ complex of K2 molecules was excited by a Ti:Sapphire laser
(Coherent MBR 110). The laser excitation frequencies in these experiments were
set within a range 9800–13 000 cm-1, which predominantly excited the K2 levels
with energy below 13 500 cm-1.

In order to get more systematic term values data on both $A$ and $b$ states of
K2 in the higher energy range we performed a new measurement session in the
heat-pipe filled only with potassium metal. For this purpose, 10 g of
potassium (from _Alfa Aesar_) were heated to about 350oC. The pressure of the
buffer gas Ar in the heat-pipe was increased to about 6 mbar to ensure
efficient collisional population transfer between rotational levels in the
upper excited state. A single mode frequency stabilized Ti:Sapphire laser
(Equinox/SolsTis(MSquared)) was exploited, its power at the entrance of the
heat-pipe being about 600 mW. Laser frequency was measured with a wavemeter
WS7. Backward LIF was recorded with a typical instrumental resolution of 0.03
cm-1. The spectral calibration of FT spectrometer has been checked using the
Ar and Ne lines from the low current hollow cathode lamps by comparing to the
respective data NIST ; Whaling . In order to reach a higher energy range of
the $A\sim b$ complex than in our previous measurements, the excitation laser
frequency was set within 13 000–13 800 cm-1 range, which was the upper limited
by the laser performance. Overall about 200 new $A\sim b\rightarrow X$ FT-LIF
spectra of K2 were recorded.

###  II.2 LIF spectra analysis

The recorded $A\sim b(J^{\prime})\rightarrow
X(v^{\prime\prime},J^{\prime\prime}=J^{\prime}\pm 1)$ emission spectra were
processed applying a homemade program, which searches and assigns the
rotational lines belonging to a particular LIF progression basing on
coincidences between the observed and calculated rovibrational differences.
The program operates with the comprehensive ground state term values databases
$E^{Calc}_{X}(v^{\prime\prime},J^{\prime\prime})$ prepared in advance for all
molecules under consideration. The output file reports the following
information about the assigned progressions: the molecule (including
particular isotopologue), rotational quantum number $J^{\prime}$ of the upper
level and its term value $E^{\prime}$, the calculated absorption (excitation)
transition, the positions of the calculated and experimental lines assigned to
a particular progression, the differences of their positions, the line
intensities. In case of K2 the ground $X$ state term values for 39K2, 39K41K,
and 41K2 isotopologues have been generated using the properly mass-scaled
Dunham’s constants borrowed from Ref.Amiot1995 . The assignment of particular
progression was accepted after checking for possible errors, e.g. bad signal-
to-noise ratio (SNR), non-realistic absorption transition, etc. Along with the
brand-new spectra collection, we have reanalyzed the FT-LIF spectra previously
obtained in KCs and KRb heat-pipes focusing on the K2 progressions.

In a number of FT-LIF spectra, the satellite lines appeared around strong
("mother’s") lines due to depopulation transfer from a directly excited
rovibronic level to neighboring rotational levels in collisions with buffer Ar
gas atoms. The data processing of satellite lines had provided a great number
of additional rovibronic term values of the upper state. Several spectra
contained the satellite lines originating from collision-ally populated levels
with dominant $b^{3}\Pi_{u}(0^{+}_{u})$ character. An example of such
spectrum, which was recorded at the KRb study is shown in Fig. 2. The full
spectrum (Fig. 2a) contains several progressions: 3 for K2, 10 for KRb, and 6
for Rb2. The progression marked by blue points originates from the 39K2 upper
level $E_{A\sim b}(J^{\prime}=57)$ = 12031.243 cm-1. A zoomed fragment of
spectrum for this progression in the range $v_{X}$ = 28 is presented in Fig.
2b. The additional satellite lines marked as $P^{\prime}$(58) and
$R^{\prime}$(56) demonstrate transfer of population to the neighboring
$b^{3}\Pi_{u}(0^{+}_{u})$ level with $J^{\prime}$ = 57, which lies below the
$A^{1}\Sigma^{+}_{u}(0^{+}_{u})$ level with $J^{\prime}$ = 57. Overall more
than 6000 term values were obtained. After averaging over energies obtained
for the same levels from different spectra the final amount was 4053 term
values for 39K2, see Fig. 3a, 705 term values for 39K41K, and 14 term values
for 41K2, see Fig. 3b. All present term values are referred to the minimum of
the ground state potential by adding the Dunham’s correction $Y^{X}_{00}$ =
-0.022 cm-1 from Ref. Amiot1995 .

Figure 2: The $A^{1}\Sigma^{+}_{u}\sim b^{3}\Pi_{u}(J^{\prime})\rightarrow
X^{1}\Sigma^{+}_{g}(v_{X},J^{\prime\prime}=J^{\prime}\pm 1)$ FT LIF spectrum
recorded at excitation laser frequency $\nu_{laser}$ = 11625.540 cm-1: (a)
full spectrum. Red squares mark 39K2 doublet $P,R$ progression from the upper
state level $E_{A\sim b}(J^{\prime}=57)$ = 12031.243 cm-1 ( $A\sim b$ mixing:
$P_{A}$ = 67% and $P_{b0}$ = 31%). Green points mark the progression from
collisionally populated level ($A\sim b$ mixing: $P_{A}$ = 33% and $P_{b0}$ =
65%); (b) zoomed fragment with assigned satellite lines, see dotted and solid
red lines below the spectrum for $P$ and $R$ branches, respectively. The
mutual local perturbation of levels with $J^{\prime}$ = 57 is clearly seen in
the relaxation lines pattern. The inset presents term values of the respective
rovibronic levels as dependent on $J^{\prime}(J^{\prime}+1)$. Figure 3:
Experimental term values of the K2 $A\sim b$ complex as dependent on
rotational quantum number $J^{\prime}$: (a) for 39K2 included in the fit,
black points - present work (PW), blue Manaa2002 , magenta Lisdat2001 ;
Lisdat2001PhD , red Lisdat2006 ; Falke2007PhD , purple Kim95 ; (b) for other
isotopologues not included in the fit - 39K41K: black points PW, magenta
Lisdat2001 ; Lisdat2001PhD , red Falke2007 ; for 41K2: red empty circles PW .
Green stars mark the levels, for which relative intensity distributions in LIF
progressions are presented, see Sec. IV.3.

##  III Direct Deperturbation Fit

Direct deperturbation fit (DDF) machinery used for the present treatment of
the rotationally resolved $e$-symmetry levels corresponding to the mutually
perturbed $A^{1}\Sigma^{+}_{u}$, $b^{3}\Pi_{u}$, $c^{3}\Sigma^{+}_{u}$ and
$B^{1}\Pi_{u}$ states of potassium dimer, which are belonging to the pure
Hund’s "a" coupling case Field2004book  and converging to the non-relativistic
dissociation threshold (see Fig. 1), consists of the several items described
below.

The developed deperturbation model neglects completely the hyperfine structure
(HFS) of the K2 terms since the HFS is not resolved in the Doppler effect
limited spectra which were assigned to both $\Omega=0^{+}_{u}$-components of
the $A\sim b$ complex, singlet $B^{1}\Pi_{u}$ state and high rotational levels
of both triplet $b^{3}\Pi_{u}$ and $c^{3}\Sigma^{+}_{u}$ states. In the
Doppler free spectraLisdat2001 ; Lisdat2001PhD ; Falke2007 ; Falke2007PhD  the
observed HFS was previously extracted from the measured lines using the HFS
version of the effective Hamiltonian approachBrown2003book . The
deperturbation model can not be formally applied for the mixed 39K41K
isotopologue due to a breakdown of the $u\leftrightarrow u$ and
$g\leftrightarrow g$ selection rules for intramolecular perturbations. The
model tacitly ignores the implicit dependence of the empirically refined
adiabatic interatomic PECs on the reduced molecular mass $\mu$.

###  III.1 The molecular Hamiltonian model

Basically, the vast majority of the observed levels belonging to $A$ and $b$
states can be properly described in the framework of the simplest 2$\times$2
model accounting only for the SO interaction between the $A^{1}\Sigma^{+}$
state and the $0^{+}_{u}$-component of the triplet $b^{3}\Pi_{u}$ state since
the higher-lying $B^{1}\Pi_{u}$ and $c^{3}\Sigma^{+}_{u}$ electronic terms do
not cross the $A$ and $b$ states at any internuclear distance $R$.
Nevertheless, the $B$ and $c$ states were still involved explicitly into the
present consideration in order to account precisely for the SO-coupling effect
on the quasi-degenerate levels located in a vicinity of the common non-
relativistic dissociation limit. For intermediate and high rotational levels
the diagonal SO-splitting of the triplet $b$ state and spin-rotational
interaction between its components $b^{3}\Pi(0^{+}_{u},1_{u},2_{u})$ should be
taken into account leading to the 4$\times$4 deperturbation model
corresponding to the intermediate between Hund’s "a" and "b" coupling
casesField2004book .

The non-adiabatic eigenvalues $E^{CC}_{j}(J)$ and eigenfunctions
$\Phi^{CC}_{j}(R)$ corresponding to the $j$-level of the $A\sim b\sim c\sim B$
complex of the K2 molecule were described within the framework of the
6$\times$6 CC deperturbation model based on a numerical solution of the six
coupled radial Schrödinger equations

$\displaystyle\left(-{\bf I}\frac{\hbar^{2}d^{2}}{2\mu dR^{2}}+{\bf
V}(R;\mu,J)-{\bf I}E^{CC}_{j}(J)\right)\Phi^{CC}_{j}(R)=0$ (1)

with the conventional boundary $\Phi^{CC}_{j}(0)=\Phi^{CC}_{j}(+\infty)=0$ and
normalization $\sum_{i}P_{i}=1$ conditions, where
$i\in[A^{1}\Sigma^{+}_{u};b^{3}\Pi_{0^{+}_{u},1_{u},2_{u}};c^{3}\Sigma^{+}_{1_{u}};B^{1}\Pi_{u}]$
is the index of electronic $\Omega$-state under the consideration, namely:
$\Omega\in[(2)0^{+}_{u},(3)1^{+}_{u},(1)2^{+}_{u}]$. ${\bf I}$ is the identity
matrix and $P_{i}=\langle\phi_{i}|\phi_{i}\rangle$ is the fractional partition
of the multi-channel wavefunction $\Phi^{CC}_{j}$.

Hereafter, the ${\bf V}(R;\mu,J)$ is the symmetric 6$\times$6 matrix of
potential energy having the following diagonal

$\displaystyle\langle A^{1}\Sigma^{+}|H|A^{1}\Sigma^{+}\rangle$
$\displaystyle=$ $\displaystyle U_{A}+B(X+2)$ (2) $\displaystyle\langle
b^{3}\Pi_{0^{+}}|H|b^{3}\Pi_{0^{+}}\rangle$ $\displaystyle=$ $\displaystyle
U_{b0}+B(X+2)$ $\displaystyle\langle b^{3}\Pi_{1}|H|b^{3}\Pi_{1}\rangle$
$\displaystyle=$ $\displaystyle U_{b0}+A_{so}+B(X+2)$ $\displaystyle\langle
b^{3}\Pi_{2}|H|b^{3}\Pi_{2}\rangle$ $\displaystyle=$ $\displaystyle
U_{b0}+2A_{so}+B(X-2)$ $\displaystyle\langle B^{1}\Pi|H|B^{1}\Pi\rangle$
$\displaystyle=$ $\displaystyle U_{B}+BX$ $\displaystyle\langle
c^{3}\Sigma^{+}_{1}|H|c^{3}\Sigma^{+}_{1}\rangle$ $\displaystyle=$
$\displaystyle U_{c}+B(X+2)$

and the non-vanishing off-diagonal matrix elements

$\displaystyle\langle A^{1}\Sigma^{+}|H|b^{3}\Pi_{0^{+}}\rangle$
$\displaystyle=$ $\displaystyle-\sqrt{2}\xi_{Ab0}$ (3) $\displaystyle\langle
A^{1}\Sigma^{+}|H|b^{3}\Pi_{1}\rangle$ $\displaystyle=$
$\displaystyle-\eta_{Ab1}B\sqrt{2X}$ $\displaystyle\langle
b^{3}\Pi_{0}|H|b^{3}\Pi_{1}\rangle$ $\displaystyle=$
$\displaystyle-B(1-\gamma_{bb})\sqrt{2X}$ $\displaystyle\langle
b^{3}\Pi_{1}|H|b^{3}\Pi_{2}\rangle$ $\displaystyle=$
$\displaystyle-B(1-\gamma_{bb})\sqrt{2(X-2)}$ $\displaystyle\langle
B^{1}\Pi|H|b^{3}\Pi_{1}\rangle$ $\displaystyle=$ $\displaystyle-\xi_{Bb}$ (4)
$\displaystyle\langle B^{1}\Pi|H|c^{3}\Sigma^{+}_{1}\rangle$ $\displaystyle=$
$\displaystyle\xi_{Bc}$ $\displaystyle\langle
b^{3}\Pi_{1}|H|c^{3}\Sigma^{+}_{1}\rangle$ $\displaystyle=$
$\displaystyle\xi_{bc}$ $\displaystyle\langle
A^{1}\Sigma^{+}|H|B^{1}\Pi\rangle$ $\displaystyle=$ $\displaystyle-
BL_{AB}\sqrt{2X}$ (5) $\displaystyle\langle
b^{3}\Pi_{0^{+}}|H|c^{3}\Sigma^{+}_{1}\rangle$ $\displaystyle=$
$\displaystyle-BL_{bc}\sqrt{X}$ $\displaystyle\langle
b^{3}\Pi_{2}|H|c^{3}\Sigma^{+}_{1}\rangle$ $\displaystyle=$ $\displaystyle-
BL_{bc}\sqrt{(X-2)},$

where

$\displaystyle X\equiv J(J+1);\quad B\equiv\frac{\hbar^{2}}{2\mu R^{2}}.$ (6)

All electronic structure parameters involved in the model are assumed to be
the mass-invariant functions of the internuclear distance $R$. In particular,
$U_{i}(R)$ in Eqs.(2) are the effective (SO-deperturbed) interatomic
potentials for the $A^{1}\Sigma^{+}_{u}$, $b^{3}\Pi_{0^{+}_{u}}$,
$c^{3}\Sigma^{+}_{1_{u}}$ and $B^{1}\Pi_{u}$ states, $A_{so}(R)$ is the on-
diagonal SO splitting function of the triplet $b$ state. The $\xi_{ij}(R)$ in
Eq.(4) and $L_{ij}(R)$ in Eq.(5) functions are the off-diagonal SO coupling
and $L$-uncoupling electronic matrix elements, respectively. The dimensionless
parameters $\eta_{Ab1}$ and $\gamma_{bb}$ from Eq.(3) are considered here as a
small constants arisen from the 2-nd order SO$\times$$L$-uncoupling
interaction with the remote singlet and triplet $u$-states manifold (excluding
a contributions of $B^{1}\Pi_{u}$ and $c^{3}\Sigma^{+}_{u}$ states,
respectively):

$\displaystyle\eta_{Ab1}\approx\sum_{j\in^{1}\Pi_{u}}\frac{L_{Aj}\xi_{jb}}{(U_{A}+U_{b})/2-U_{j}};\quad\eta_{bb}\approx\sum_{j\in^{3}\Sigma^{+}_{u}}\frac{L_{bj}\xi_{jb}}{U_{b}-U_{j}}.$
(7)

###  III.2 Ab initio electronic structure calculations

Interatomic potentials and non-adiabatic electronic matrix elements involved
into the applied deperturbation 6$\times$6 CC model are obviously needed to be
estimated in a wide range of $R$ since we intend to deperturb the $A\sim b\sim
c\sim B$ complex in the entire boundary range starting from the bottom of the
$b$ state and ending on its common non-relativistic $4s\;^{2}{\rm S}$ \+
$4p\;^{2}{\rm P}$ dissociation limit. It has been done below by means of
alternative scalar-relativistic and fully-relativistic electronic calculations
accomplished by multi-referenced configuration-interaction and coupled-
clusters methods, respectively.

####  III.2.1 Scalar-relativistic approximation $-$ Hund’s "a" coupling case

The scalar-relativistic (sr) adiabatic PECs $U^{sr}_{i}(R)$ and respective
electronic eigenfunctions, $\Psi^{sr}_{i}(\textbf{r},R)$, corresponding to a
pure Hund’s "a" coupling case were obtained within the framework of the two-
valence electrons configuration interaction (CI) calculations combined with
the model treatment of core-valence interaction by means of core polarization
potentials (CPPs) Magnier2004 ; Magnier2009 ; CPP . The computational details
of the CI-CPP procedure can be found in Refs Pazyuk2015 ; Pazyuk2016  on the
heavier Rb2 and Cs2 dimers. Briefly, the atomic core of potassium was replaced
by the averaged relativistic ECP10MDF effective core potentials Lim2005 ;
Sadlej1991 , leaving 1 valence and 8 sub-valence electrons of each atom for
explicit treatment. The molecular orbitals (MO) were generated using the
state-averaged complete active space self-consistent field (SA-CASSCF) method
Werner85 , taking the (1-5)${}^{1,3}\Sigma^{+}_{u/g}$,
(1-3)${}^{1,3}\Pi_{u/g}$, and (1)${}^{1,3}\Delta_{u/g}$ states with equal
weights. The dynamic correlation was treated explicitly only for the two
valence electrons within the multi-reference configuration interaction (MR-
CISD) calculations Knowles92 . At both CASSCF and CI steps all (eight) sub-
valence MOs were kept doubly occupied. The rest core-polarization and core-
valence correlations were taken into account using the $l$-independent CPPs
CPP  which the cutoff parameter was adjusted to represent exactly the
experimental energy $\nu^{exp}_{4S-4P}$ of the $4s\;^{2}{\rm S}\to
4p\;^{2}{\rm P}$ transition in K atom. All scalar-relativistic calculations
were conducted using the MOLPRO package MOLPRO .

####  III.2.2 Fully relativistic model $-$ Hund’s "c" coupling case

Alternatively, adiabatic (avoided crossing) and SO-deperturbed (quasidiabatic)
interatomic potentials for both $A^{1}\Sigma^{+}_{u}(A0^{+}_{u})$ and
$b^{3}\Pi_{u}(b0^{+}_{u})$ states of K2 were obtained (together with the off-
diagonal $\xi^{rel}_{Ab0}(R)$ SO coupling function and SO-splitting between
$b0^{+}_{u}$, $b1^{+}_{u}$ and $b2^{+}_{u}$ states) within the framework of
fully relativistic electronic structure calculations corresponding to a pure
Hund’s "c" coupling case. For K atom the relativistic (two-component)
semilocal shape-consistent effective core potentials (RECP) properly
accounting for spin-dependent relativistic effects were used Mosyagin:10a ;
GRECP . Eight sub-valence (outer core) electrons, $3s^{2}\,3p^{6}$, and one
valence electron of each K atom were treated explicitly. To solve the
relativistic 18-electron problem the Fock-space relativistic coupled-cluster
(FS-RCC) method Visscher:01 ; Eliav:22  has been employed. The Gaussian basis
$[7s\,7p\,6d\,4f\,2g]$ adapted to the use with the mentioned ECP was
constructed via modification and contraction of the diffuse primitive
functions from Ref. Widmark:04 . The Fermi vacuum was defined by the ground-
state SCF determinant for the doubly charged molecular ion, so that the
neutral states corresponded to the two-particle Fock space sector ($0h2p$).
The cluster operator expansion comprised single and double excitations (FS-
RCCSD approximation). The model space was spanned by all possible
distributions of two valence electrons among 52 Kramers pairs of lowest-energy
K${}_{2}^{++}$ spinors. To suppress instabilities caused by intruder states,
we employed the dynamic denominator shift technique combined with the
extrapolation to the zero-shift limit using matrix Padé approximants
Zaitsevskii:18a . The calculations were performed with the appropriately
modified DIRAC17 program package DIRAC:17 ; DIRACpaper .

To transform the resulting adiabatic potentials for both $A0^{+}_{u}$ and
$b0^{+}_{u}$ relativistic states to their quasidiabatic SO-deperturbed
counterparts corresponding to “a” Hund’s coupling case and extract the
relevant SO coupling function, we employed the technique based on projecting
the solutions of scalar-relativistic (SO-free) problem on the subspace of
strongly SO-mixed eigenstates of the full relativistic Hamiltonian
Zaitsevskii:17 . At this stage the many-electron wavefunctions were replaced
by their FS-RCCSD model space parts, i. e. by the eigenvectors of the
effective Hamiltonians. It is worth noting that the resulting SO-deperturbed
states, in a strict analogy with their empirical counterparts, incorporate
implicitly the effects of SO coupling with the scalar state outside of the
chosen set.

###  III.3 Analytical representation of the interatomic potentials and spin-
orbit functions

The overwhelming number of experimental terms investigated in the present work
can be unambiguously assigned to $A^{1}\Sigma^{+}_{u}(0^{+}_{u})$ or
$b^{3}\Pi_{u}(0^{+}_{u})$ state of the $A\sim b\sim c\sim B$ complex, so it is
not surprising that the global CC deperturbation model depends most strongly
on interatomic $U_{A}(R)$ and $U_{b0}(R)$ potentials of the interacting $A$
and $b$ states, as well as on the relevant SO-splitting $A_{so}(R)$ and SO-
coupling $\xi_{Ab0}(R)$ functions.

To improve the optimization and extrapolation facilities of the global DDF
both $A$ and $b$ state PECs were represented in fully analytical “Double-
Exponential/Long-Range” (DELR) form LeRoyDELR :

$\displaystyle U^{DELR}(R)=[T_{dis}+(Az-B)z]-U^{LR},$ (8)

where coefficients $A$ and $B$ are determined as

$\displaystyle A$ $\displaystyle=$ $\displaystyle
D_{e}-U^{LR}(R_{e})-\frac{dU^{LR}/dR|_{R_{e}}}{\beta(R_{e})};$ (9)
$\displaystyle B$ $\displaystyle=$ $\displaystyle D_{e}-U^{LR}(R_{e})+A,$

$D_{e}=T_{dis}-U^{DELR}(R_{e})$ is the well depth of the potential and
$T_{dis}$ is the dissociation energy with respect to the minimum of the ground
$X$-state. The symbol $R_{e}$ involved into the $z=e^{-\beta(R-R_{e})}$
variable of Eq.(8) and into Eq.(9) means the equilibrium distance (minimum) of
DELR potential while the coefficient $\beta$

$\beta(R)=\sum^{N}_{i=0}\beta_{i}y^{i};\quad
y\equiv\frac{(R/R_{ref})^{p}-1}{(R/R_{ref})^{p}+1}$ (10)

is the polynomial function of the reduced radial coordinate $y\in[-1,1]$,
where $R_{ref}$ is the reference distance and $p$ is the integer parameter.

The $T_{dis}$ value in Eq.(8) relates on the dissociation limit $T_{dis}^{b}$
of the $b0^{+}_{u}$ state and the well depth $D_{e}^{X}$ of the ground state

$\displaystyle T_{dis}\equiv
T_{dis}^{A}=T_{dis}^{b}+\xi^{K}_{so}=D_{e}^{X}+\nu^{K}_{4p-4s},$ (11)

through the well known energy Tiecke2019  of the K($4p$)-K($4s$) atomic
transition $\nu^{K}_{4p-4s}=13023.6587$ cm-1 and the SO-splitting of the
K($4p$) state $\xi^{K}_{so}=[E^{K}_{4p\;^{2}{\rm
P}_{\frac{3}{2}}}-E^{K}_{4p\;^{2}{\rm P}_{\frac{1}{2}}}]/3=19.2368$ cm-1.

The long-range part of DELR potential was approximated by the leading term

$\displaystyle
U^{LR}(R)=\frac{C^{\Sigma/\Pi}_{3}}{R^{3}}f_{ret}^{\Sigma/\Pi},$ (12)

which accounts for the retardation effect

$\displaystyle f_{ret}^{\Sigma}(R)$ $\displaystyle=$
$\displaystyle\cos\left(\frac{R}{\lambdabar}\right)+\left(\frac{R}{\lambdabar}\right)\sin\left(\frac{R}{\lambdabar}\right);$
(13) $\displaystyle f_{ret}^{\Pi}(R)$ $\displaystyle=$ $\displaystyle
f_{ret}^{\Sigma}(R)-\left(\frac{R}{\lambdabar}\right)^{2}\cos\left(\frac{R}{\lambdabar}\right);\quad
2\pi\lambdabar=\frac{c}{\nu^{K}_{4p-4s}}$

where the leading dispersion coefficients $C^{\Sigma/\Pi}$ relate on the
dipole matrix element of non-relativistic K($4p$)-K($4s$) transition and the
radiative lifetime of the K($4p$) state:

$\displaystyle|d^{K}_{4p-4s}|^{2}=C^{\Pi}_{3}=\frac{C^{\Sigma}_{3}}{2};\qquad\tau^{K}_{4p}=\frac{3\hbar\lambdabar}{2C^{\Sigma}_{3}}.$
(14)

Both diagonal $A_{so}$ and off-diagonal $\xi_{Ab0}$ SO functions involved into
Eq.(2) and Eq.(3), respectively, were represented as the sum of the fixed
$\xi^{ab}_{so}$ and variable $\Delta\xi_{so}$ parts:

$\displaystyle\xi_{so}(R)=\xi^{ab}_{so}(R)+\Delta\xi_{so}(R).$ (15)

The fixed part $\xi^{ab}_{so}$ consists of the relevant _ab initio_ point-wise
data, which were smoothly interpolated by cubic splines with the appropriate
boundary conditions NR , namely:
$d^{2}\xi_{so}/dR^{2}|_{R_{min}}=d\xi_{so}/dR|_{R_{max}}=0$.

The variable part $\Delta\xi_{so}$ was defined as the Chebyshev polynomial
expansion (CPE) with the correct long-range behaviour at the dissociation
limitBormotova2019 :

$\displaystyle\Delta\xi_{so}(R)=\frac{\sum_{i=1}^{N}\alpha_{i}T_{i-1}(y)}{1+(R/R_{ref})^{p}},$
(16)

where $T_{i}$ are the 1-st kind Chebyshev polynomials with respect to the
reduced variable

$\displaystyle
y\equiv\frac{(R/R_{ref})^{p}-1}{(R/R_{ref})^{p}+1-2(R_{min}/R_{ref})^{p}},$
(17)

which transforms the conventional semi-interval $R\in[R_{min},+\infty)$ into
the finite domain $y\in[-1,+1]$.

###  III.4 The least-squares processing

The trial parameters of the $U^{DELR}_{A}(R)$ and $U^{DELR}_{b0}(R)$
analytical potentials (see Eq.(8)) for the $A0^{+}_{u}$ and $b0^{+}_{u}$
states, as well as the CPE coefficients $\alpha_{i}$ of both
$\Delta\xi_{Ab0}(R)$ and $\Delta A_{so}(R)$ correction functions (see Eqs.(15)
and (16)) were refined in the framework of the weighted nonlinear least-square
fitting (NLSF) procedure:

$\displaystyle\chi^{2}=\chi^{2}_{expt}+w\chi^{2}_{ab},$ (18)

which simultaneously accounts for the contribution of the experimental term
values and the ab initio data in the object $\chi^{2}$-function. The _ab
initio_ data were incorporated into the NLSF procedure in order to maintain a
physically correct behaviour of the empirical functions outside the
experimentally available region of the internuclear distance. During the
iterative minimization of the functional (18) the weighting factor $w$ is
smoothly varied to provide a proper balance between the contributions of
experimental and ab initio data into to the total $\chi^{2}$-value.

The first term in the expression (18) has been evaluated as the sum

$\displaystyle\chi^{2}_{expt}=\sum_{i=1}^{N_{set}}\left[\sum_{j=1}^{N_{expt}}\left(\frac{E^{expt}_{j}(J^{\prime})-E^{CC}_{j}(J^{\prime})+\Delta^{shift}_{i}}{\sigma^{expt}_{i}}\right)^{2}\right],$
(19)

where $N_{expt}$ is the total number of rovibronic levels of the particular
experimental data set (see Table 1) included in the NLSF procedure. Only
assigned to the most abundant 39K2 isotopologue term values were used in the
fit. $E^{CC}_{j}(J^{\prime})$ are the non-adiabatic eigenvalues calculated
according to the CC equations (1), while $E^{expt}_{j}(J^{\prime})$ are the
experimental term values with their uncertainties $\sigma^{expt}_{i}$. Here,
$\Delta^{shift}_{i}$ is the systematic energy shift presumably attributed to
the tacit uncertainty of the experimental term values corresponding to the
$i$-th data set.

The last term in Eq.(18) was estimated as the double sum:

$\displaystyle\chi^{2}_{ab}=\sum_{i\in
A0^{+}_{u},b0^{+}_{u}}\left[\sum_{j=1}^{N_{ab}}\left(\frac{U^{DELR}_{i}(R_{j})-U^{ab}_{i}(R_{j})}{\sigma^{ab}_{j}}\right)^{2}\right],$
(20)

where $N_{ab}$ is the total number of the internuclear distance points $R_{j}$
where the relevant ab initio point-wise potentials $U^{ab}_{i}(R)$ were
evaluated within the framework of scalar-relativistic calculations (see Sec.
III.2). The uncertainties $\sigma^{ab}_{j}$ were estimated by a comparison of
the results of scalar-relativistic and fully relativistic calculations.

###  III.5 Numerical recipes

The main computational challenge of the DDF procedure is the multiple
numerical solution of the six CC radial equations in the entire bound spectral
range, starting from the practically isolated triplet levels in a vicinity of
the bottom of the $b0^{+}_{u}$-state located near 9885 cm-1 (with respect to
the minimum of the ground $X$-state) and ending with fully mixed
$A0^{+}_{u}\sim b0^{+}_{u}$ levels closely-lying to the
$4s^{2}$S${}_{\frac{1}{2}}$+$4p^{2}$P${}_{\frac{1}{2}}$ dissociation threshold
with energy 17436.075 cm-1. It should be also noted that due to the smooth
long-range asymptotic behaviour of interatomic potentials
$U^{LR}(R\to+\infty)\approx T_{dis}-C^{\Sigma/\Pi}_{3}/R^{3}$ the highly-
oscillating multi-component wavefunctions for the highest vibrational levels
are distributed in a very wide range of $R$. For instance, the classical left
$R_{-}$ and right $R_{+}$ turning points for the last observed $v_{A}$=245
level Lisdat2006 ; Falke2007PhD  of the $A\sim b$ complex with the bound
energy only about -0.0164 cm-1 seems to be located at $R_{-}\approx$ 2.6 Å and
$R_{+}\geq$ 300 Å, respectively.

To improve the efficiency of the iterative solution of the CC equations we
implemented the analytical mapping procedure Meshkov2008  based on replacement
of the conventional radial coordinate $R$ by the reduced counterpart

$\displaystyle y\equiv\frac{(R/R_{ref})^{p}-1}{(R/R_{ref})^{p}+1};\qquad
y\in[-1,+1],$ (21)

where the optimal parameters of mapping $R_{ref}$=5.0 Å and $p$=1.5 were
adopted to minimize a truncation error in the merged eigenvalues evaluated in
the entire energy range of the complex treated. The analytically (exactly)
transformed CC equations were then solved on the finite interval
$R\in[2.3,10~{}000]$ Å by the central five-points finite-difference (5FD)
boundary value method, which enforced the fixed number $M$ of the uniformly
distributed $y$-points. The integral form of the error correction to
eigenvalues appeared due to the 5FD method Meshkov2008  was applied for
extrapolating the calculated energies to infinite number of grid points
$E^{CC}_{j}(M\to+\infty)$. Thus, the mapping procedure had reduced the overall
absolute error in all calculated energies below 0.0005 cm-1 by using only
$M$=10 000 uniformed grid $y$-points.

The particular eigenvalue and eigenfunction problem of the resulting band
matrix was solved by the implicitly restarted Lanczos method in the shift-
inverted spectral transformation mode implemented in ARPACK package ARPACK .
The minimum of the functional (18) was searched by the modified Levenberg-
Marquardt algorithm realized by MINPACK software MINPACK . A quantum phase
formalism ABARENOV  was applied for solving the single-channel problem
corresponding to the isolated ground $X$-state.

###  III.6 Simulation of intensity distributions in the $A\sim b\to X$ LIF
progressions

Since DDF was accomplished according to Sec. III by exploiting exclusively the
energy data $E^{CC}_{j}(J^{\prime})$ of the single 39K2 isotopologue, the
reliability of the corresponding multi-component vibrational eigenfunctions
$\Phi^{CC}_{j}(R)$ simultaneously obtained from the solution of CC equations
(1) can be independently validated by comparing the relative intensity
distributions in the $A\sim b\to X$ LIF spectra with their experimental
counterparts.

The line intensity $I^{calc}_{jX}(v_{X})$ in the $A\sim b\to X(v_{X})$ LIF
progressions of 39K2 and 39K41K isotopologues from the rovibronic $j$-level of
the $A\sim b\sim B\sim c$ complex to rovibronic levels of the isolated
$X^{1}\Sigma^{+}_{g}$ state can be evaluated as

$\displaystyle I^{calc}_{jX}\sim\nu^{4}_{jX}\left|M_{jX}\right|^{2}$
$\displaystyle\approx$
$\displaystyle\nu^{4}_{jX}|\langle\phi_{A}^{J^{\prime}}|d^{ab}_{AX}|\phi_{X}^{J^{\prime}\pm
1}\rangle|^{2};$ (22)
$\displaystyle\nu_{jX}=E_{j}^{CC}(J^{\prime})-E_{X}^{J^{\prime}\pm 1}$ ;
$\displaystyle\quad
M_{jX}=\sum_{i=1}^{6}\langle\phi_{i}^{J^{\prime}}|d_{iX}|\phi_{X}^{J^{\prime}\pm
1}\rangle,$

where the contribution of both spin-forbidden $b-X$, $c-X$ and spin-allowed
$B-X$ transitions was completely neglected. The latter assumption is justified
by the extremely small admixture of the $B$ state for all levels of the
complex selected for the intensity testing, see Fig. 3. Thus,
$\phi_{A}^{J^{\prime}}(R)$ in the right side of Eq.(22) is the singlet $A$
state fraction of the multi-component wavefunction corresponding to the
eigenvalue $E_{j}^{CC}(J^{\prime})$, whereas $d^{ab}_{AX}(R)$ is the spin-
allowed $A-X$ transition dipole moment evaluated in Ref.Krumins2020  using the
robust ab initio FS-RCC finite-field (FS-RCCSD-FF) method ZaitsevskiiTDM . The
adiabatic energies $E_{X}^{J^{\prime}\pm 1}$ and eigenfunctions
$\phi_{X}^{J^{\prime}\pm 1}(R)$ for the ground state were obtained by solving
the radial equation with empirical $X$-state PECTiemann2020 .

##  IV Results and discussion

Figure 4: Residuals $E^{expt}_{j}-E^{CC}_{j}$ between the experimental and
fitted term values as dependent on the $A\sim b$ complex energy for 39K2:
black points - present (PW), magenta Lisdat2001 ; Lisdat2001PhD , red
Lisdat2006 ; Falke2007PhD , blue Manaa2002 . The legend shows relation to the
data sets in Table 1. Figure 5:  (a) Residuals $E^{expt}_{j}-E^{CC}_{j}$
between the highly-excited term values of the 39K2 $A\sim b$ complex precisely
measured by the 2s-LES-MB setup Falke2007PhD  and their theoretical
counterparts calculated in Lisdat2006  under the simple adiabatic
approximation (the isolated SO-deperturbed $A$ state) and presently by using
the advanced non-adiabatic 6$\times$6 CC approach; vertical dashed line
denotes the dissociation threshold with energy 17436.075 cm-1; (b) the
fraction partition $P_{A}$, $P_{b0}$ of the rovibronic levels of the $A\sim b$
complex with the vibrational quantum numbers $v_{A}\in[88,245]$ and
$v_{b0}\in[88,236]$. These $v_{A}$ and $v_{b0}$ values have been determined as
a number of nodes of the corresponding non-adiabatic vibrational wavefunctions
$\phi_{A}(R)$ and $\phi_{b0}(R)$, respectively.

###  IV.1 Experimental and calculated rovibronic term values of the $A\sim b$
complex

The K2 rovibronic term values data considered in present study are summarized
in Table 1; all data sets from different sources can be find in Supplementary
Material SM . Only the term values data for the most abundant 39K2
isotopologue from the sets I to V in Table 1 were included in the fit, see
Fig. 3a; they contain more than 5700 term values, from which 4053 values (I)
are obtained in the present work (PW). The compiled data set covers the whole
energy range from 9955 cm-1 to 17436 cm-1, that is, from the bottom of the
$b^{3}\Pi_{u}(b0^{+}_{u})$ state (data III from Ref. Kim95 ) up to the
dissociation limit (data V from Ref. Falke2007PhD ). The term values of highly
excited levels were derived by us for both main and mixed istopologues by
adding the original sub-Doppler line positions measured in Ref. Falke2007PhD
to the ground $X$-state levels calculated using the highly accurate adiabatic
empirical PEC from Ref. Tiemann2020 . These values are found to be close
within 0.003-0.005 cm-1 to their previous Dunham-based counterparts Lisdat2006
; Falke2007PhD  corrected for the $Y^{X}_{00}$-value Amiot1995 .

It should be noted that, as mentioned in Sec. III.4, the fitting routine,
which exploited data sets from different sources, contained the systematic
energy shift $\Delta_{i}^{shift}$, see Eq.(19). The term values of sets I, IV,
and V were fixed at the DDF procedure ($\Delta_{i}^{shift}\equiv 0$). For two
other sets II and III the systematic $\Delta_{i}^{shift}$-value was applied as
described in Sec. III.4. The obtained correction $\Delta_{i}^{shift}$ = +0.031
cm-1 has been adjusted to fit the PF-OODR data from Ref. Manaa2002 . Though
the reason for this shift is not clear, it is worth to mention that the
calculated $E^{expt}_{j}$ value for the level $J^{\prime}$ = 66 excited in the
first step of the PF-OODR experiment is perfectly reproduced, see Sec. IV.3.3
below.

The most significant correction of -2.1 cm-1 applied for set III can be at
least partially attributed to the large experimental uncertainty. The reason
of including the data from Kim95  is to cover the energy range close to the
very bottom of the "dark" $b^{3}\Pi_{u}$ state with $v_{b0}$ = 0, as well as
to provide the experimental data for all three fine $b0^{+}_{u}$, $b1^{+}_{u}$
and $b2^{+}_{u}$ components of the triplet state.

Table 1: Summary of the experimental term values of the $A\sim b$ complex of
K2 considered in the present work (PW). Symbol † marks the sets of the
$E_{j}^{expt}$ and $\Delta_{i}^{shift}$ values used in Eq.(19) of the DDF
procedure. The columns $Mean$ and $SD$ (standard deviation) provide an
appropriate statistic of the residual data:
$E^{expt}_{j}-E^{CC}_{j}+\Delta^{shift}_{i}$. All energies are in cm-1.

Nset | isotope | Nexpt |  $J^{\prime}$-values |  $E_{j}^{expt}$-region | Method | Ref. | $\sigma_{i}^{expt}$ | $\Delta_{i}^{shift}$ | $Mean$ | $SD$
---|---|---|---|---|---|---|---|---|---|---
I |  39K2 | 4054† | [2,149] | [11145,16651] | FT-LIF | PW | 0.015 | - | 0.000 | 0.007
|  39K41K | 705 | [1,162] | [11155,15863] |  |  |  | - | 0.003 | 0.009
|  41K2 | 14 | [12,86] | [11637,12385] |  |  |  | - | -0.007 | 0.015
II |  39K2 | 188† | [60,74] | [10180,16497] | PF-OODR | Manaa2002  | 0.02 | +0.031† | 0.000 | 0.003
III |  | 17† | [20,22] | [9955,10433] | PF-OODR | Kim95  | 0.50 | -2.1† | 0.002 | 0.323
IV |  39K2 | 511† | [0,37] | [12019,12616] | 2s-LES-MB | Lisdat2001PhD ; Lisdat2001 ; Manaa2002  | 0.005 | - | 0.000 | 0.002
|  39K41K | 62 | [0,10] | [12408,12849] |  | Lisdat2001PhD ; Falke2007PhD  | 0.005 | - | 0.000 | 0.001
V |  39K2 | 1023† | [2,16] | [15870,17436] | 2s-LES-MB | Falke2007PhD  | 0.001-0.005 | - | 0.001 | 0.003
|  39K41K | 29 | [8,10] | [17435.37,17436.06] |  | Falke2007  | 0.001 | - | 0.000 | 0.001
VI |  39K2 | 327 | [3,177] | [11857,16037] | FT-LIF | AmiotJMS1991 ; Amiot1995  | 0.010 | - | 0.008 | 0.008
VII |  | 308 | [4,64] | [11146,12064] | AOTR | JongJMS1992  | 0.015 | - | 0.005 | 0.008
VIII |  | 315 | [4,96] | [11975,12655] | FT-LIF | Ross87  | 0.013 |  | -0.001 | 0.009
IX |  | 64 | [12,34] | [12557,13765] | AOTR | Lyyra1990 ; JongJMS1992  | 0.015 | - | 0.006 | 0.012
X |  | 65 | [6,52] | [11307,12066] | AOTR | Manaa2002  | 0.017 | - | 0.004 | 0.016
XI |  | 6 | [18,20] | [14889,14990] | AOTR | Manaa2002  | 0.017 | - | -0.01 | 0.03

FT-LIF - Fourier-transform of the laser-induced fluorescence spectra; PF-OODR
- perturbation facilitated optical-optical double resonance spectroscopy; AOTR
- all-optical triple-resonance spectroscopy; 2s-LES-MB - two steps laser
excitation spectroscopy of a molecule beam

Table 2: The resulting parameters of the analytical DELR potentials obtained for the deperturbed singlet $A^{1}\Sigma^{+}_{u}(0_{u}^{+})$ and triplet $b^{3}\Pi_{u}(0_{u}^{+})$ states of K2. The dispersion coefficient $C^{\Sigma}_{3}$ is defined in cm-1/Å3, internuclear distances in Å, and energies in cm-1. The polynomial coefficients $\beta_{i}$ are dimensionless and their total number for both states is $N$=17. The parameters with more digits are presented in Supplementary Material SM . Parameter |  $b^{3}\Pi_{u}(0_{u}^{+})$-state |  $A^{1}\Sigma^{+}_{u}(0_{u}^{+})$-state
---|---|---
$T_{dis}$ | 17474.5688
$C^{\Sigma}_{3}=2C^{\Pi}_{3}$ | 550100
$R_{\textrm{ref}}$ | 5.0
$p$ | 3
${\mathfrak D}_{e}$ | 7570.4715 | 6367.5309
$R_{e}$ | 3.888657 | 4.550638
$\beta_{0}$ | 0.68284220 | 0.54006196
$\beta_{1}$ | 0.20408495 | 0.06155514
$\beta_{2}$ | 0.19374658 | 0.09689141
$\beta_{3}$ | 0.18735892 | 0.08927925
$\beta_{4}$ | 0.09070268 | 0.10889245
$\beta_{5}$ | -0.02173617 | 0.36613441
$\beta_{6}$ | 0.00039517 | -0.00000448
$\beta_{7}$ | -0.73601292 | -2.03437500
$\beta_{8}$ | -2.86990662 | 0.10673692
$\beta_{9}$ | 0.91874942 | 9.23970194
$\beta_{10}$ | 9.57662145 | 0.00595872
$\beta_{11}$ | -1.32818589 | -23.0068544
$\beta_{12}$ | -17.1303243 | 1.65936885
$\beta_{13}$ | 0.84419525 | 30.1200697
$\beta_{14}$ | 16.3070732 | -7.48445214
$\beta_{15}$ | -0.20286255 | -18.2492960
$\beta_{16}$ | -6.27975991 | 7.38151726
$\beta_{17}$ | 0.00237189 | 1.52291107
$T_{e}$ | 9884.8605 | 11107.0378
Table 3: The resulting (both fitted and constrained) parameters obtained for the variable parts of diagonal $\Delta A_{so}$ and off-diagonal $\Delta\xi_{Ab0}$ SO functions of K2. The fitting coefficients $\alpha_{i}$ are in cm-1, the fixed parameters $R_{\rm{min}}$ and $R_{\rm{ref}}$ are in Å. The fitting parameters $\eta_{Ab1}$ and $\eta_{bb}$ involved in Eq.(3) are dimensionless. The parameters with more digits are presented in Supplementary Material SM . | $\Delta\xi_{Ab0}$ | $\Delta A_{so}$
---|---|---
$p$ | 3
$R_{\rm{min}}$ | 1.99
$R_{\rm{ref}}$ | 4.0 | 5.0
$\alpha_{1}$ | 0.19985938 | 4.13434831
$\alpha_{2}$ | 2.11715093 | -4.10868122
$\alpha_{3}$ | 0.00396364 | 2.90669862
$\alpha_{4}$ | 2.34082411 | -2.55234792
$\alpha_{5}$ | 0.36215827 | 0.75114018
| $\eta_{Ab1}$ | $\eta_{bb}$
| -0.005175 | 0.006823
Table 4: Empirical and ab initio adiabatic electronic energies $T_{e}$ and equilibrium distances $R_{e}$ of the $A^{1}\Sigma^{+}_{u}(A0_{u}^{+})$ and $b^{3}\Pi_{u}(b0_{u}^{+},b1_{u},b2_{u})$ states of K2. Here SO-dep means SO-deperturbed method. Symbol † corresponds to the SO-deperturbed FS-RCC results. State |  $T_{e}$(cm-1) |  $R_{e}$(Å) | Method | Ref.
---|---|---|---|---
$A^{1}\Sigma^{+}_{u}$ | 11107.038 | 4.55064 | 6$\times$6 CC | PW
| 11107.019 | 4.551 | SO-dep | Lisdat2006
| 11107.112 | 4.54658 | 4$\times$4 CC | Manaa2002
| 11107.141 | 4.55078 | 4$\times$4 CC | Lisdat2001
| 11107.92 | 4.5498 | EHA | JongJMS1992
| 11105.9 | 4.557 | EHA | Ross87
| 11041 | 4.53 | CI-CPP-$l$ | Magnier2004
| 11256 | 4.57 | CI-CPP | PW
| 11053 | 4.555 | FS-RCC† | PW
$A0_{u}^{+}$ | 11050 | 4.561 | FS-RCC | PW
| 11045 | 4.55 | CI-CPP-$l$ | Magnier2009
$b^{3}\Pi_{u}(0_{u}^{+})$ | 9884.861 | 3.8887 | 6$\times$6 CC | PW
| 9903 | 3.891 | FS-RCC† | PW
$b0_{u}^{+}$ | 9903 | 3.891 | FS-RCC | PW
| 9805 | 3.86 | CI-CPP-$l$ | Magnier2009
$b^{3}\Pi_{u}(1_{u})$ | 9907.20 | 3.8905 | 6$\times$6 CC | PW
| 9907.50 | 3.8437 | 4$\times$4 CC | Manaa2002
| 9912.95 | 3.8945 | 4$\times$4 CC | Lisdat2001
| 9911.8 | 3.875 | EHA | JongJMS1992
| 9910.0 | 3.883 | EHA | Ross87
| 9827 | 3.88 | CI-CPP-$l$ | Magnier2004
| 9993 | 3.91 | CI-CPP | PW
$b1_{u}$ | 9925 | 3.892 | FS-RCC | PW
| 9832 | 3.86 | CI-CPP-$l$ | Magnier2009
$b^{3}\Pi_{u}(2_{u})$ | 9929.531 | 3.8931 | 6$\times$6 CC | PW
$b2_{u}$ | 9948 | 3.893 | FS-RCC | PW
| 9859 | 3.86 | CI-CPP-$l$ | Magnier2009

The data for presently measured isotopologues 39K41K and 41K2 have been used
for verifying the quality of the present fit, see Sec. IV.3.1. The data for
39K2 isotopologue not included in the fit, see sets from VI to XI in Table 1,
belong to the energy range, which is well covered by the present experiment.
Therefore these data have been used for verifying the prediction capability of
the present fit, see Sec. IV.3.3. All term values of data sets summarized in
Table 1 can be found in Supplementary Material SM .

The residuals $E_{j}^{expt}-E_{j}^{CC}$ between the experimental and fitted
term values are presented in Fig. 4. As is seen, most of residuals (ca. 95 %)
lay within $\pm$0.01 cm-1 interval, which corresponds to SD about 0.007 cm-1
(see the SD values in Table 1). It should be noted that the data (set III)
from Kim95  are not depicted in the figure since the residuals would be
markedly out of scale. It is important that no systematic shift (see the
$Mean$ value) of residuals exploiting the fixed values of data sets from
different sources (PW, IV and V, see Table 1) has been observed.

To characterize the particular improvement of description of the highly-
excited rovibronic K2 $A\sim b$ levels by the current 6$\times$6 CC DDF
procedure, we present in Fig. 5a the residuals between the term values
produced by the present fit and the experimental values given in Falke2007PhD
. For a comparison the residuals between the same experimental values
Falke2007PhD  and the adiabatic model predictions obtained in the framework of
the isolated SO-deperturbed $A$ state approximation Falke2007 , is also
depicted.

As can be seen, the present deperturbation model describes all data over 16
000 cm-1, including the levels under strong local perturbations, with $SD$
about 0.003 and $Mean$ value of only 0.001 cm-1, see Fig. 4 and Table 1. The
positions of local perturbations are clearly exposed in the respective
dependencies of fraction partitions $P_{A}$ and $P_{b0}$, see Fig. 5b. As
expected, in a vicinity of the dissociation threshold the mixing between $A$
and $b0^{+}$ states steeply increases as the energy of the $A\sim b$ complex
increases, and the fraction partitions $P_{A}$ and $P_{b0}$ monotonically
approach their asymptotic values of $\frac{100}{3}\approx 33.3$% and
$\frac{200}{3}\approx 66.6$%, respectively.

Table 5: Comparison of experimental dissociation energies $D_{e}^{X}$ (cm-1) available for the ground $X$-state of 39K2. PW | Falke2006  | Asen2008  | Tiemann2020  | Falke2008
---|---|---|---|---
4450.910(5) | 4450.904(4) | 4450.906(50) | 4450.906(3) | 4450.899(4)
Table 6: Comparison of long-range coefficients $C_{3}^{\Sigma}$ ($\times 10^{5}$ cm-1Å3) related to the K($4s$)+K($4p$) asymptotes and radiative lifetimes $\tau^{K}$ (ns) of the $4p\;^{2}{\rm P}_{\frac{1}{2}}$ and $4p\;^{2}{\rm P}_{\frac{3}{2}}$ fine components of the K atom. $C_{3}^{\Sigma}$ | $\tau^{K}_{4p\;^{2}{\rm P}_{\frac{1}{2}}}$ | $\tau^{K}_{4p\;^{2}{\rm P}_{\frac{3}{2}}}$ | Ref.
---|---|---|---
Empirical
5.501(4) | 26.65(3) | 26.30(3) | PW
5.483(6) | 26.74(3) | 26.39(3) | Falke2006
5.493(9) | 26.69(5) | 26.34(5) | WangPRA97
5.487(9) | 26.72(5) | 26.37(5) | Wang97
Ab initio
5.536 | 26.52 | 26.17 | ZaitsevskiiTDM ; Krumins2020
Table 7: Comparison of some experimental rovibronic term values $E_{j}^{expt}$ obtained for the 39K2 $A\sim b$ complex with their theoretical counterparts $E_{j}^{CC}$ predicted by using the present 6$\times$6 CC deperturbation model. All energies $E^{expt}_{j}$ presented here include the $Y^{X}_{00}$ = ${-}$0.022 cm-1 correction. The energies are given in cm-1 and the fraction partition in %. Ref. | $J^{\prime}$ | $E_{j}^{expt}$ | $E_{j}^{CC}$ | $E_{j}^{expt}-E_{j}^{CC}$ | $P_{A}$ | $P_{b0}$ | $P_{b1}$ | $P_{b2}$
---|---|---|---|---|---|---|---|---
Chu2005  | 24 | 11718.182 | 11718.197 | -0.015 | 35.0 | 64.6 | 0.4 | 0.0
| 26 | 11726.741 | 11726.750 | -0.009 | 48.0 | 51.5 | 0.5 | 0.0
| 28 | 11732.098 | 11732.088 | 0.010 | 30.2 | 69.0 | 0.8 | 0.0
| 30 | 11738.064 | 11738.054 | 0.010 | 17.7 | 81.3 | 1.0 | 0.0
| 32 | 11744.578 | 11744.582 | -0.004 | 10.6 | 88.2 | 1.2 | 0.0
| 34 | 11751.607 | 11751.617 | -0.010 | 6.8 | 91.8 | 1.4 | 0.0
| 36 | 11612.841 | 11612.808 | 0.033 | 8.0 | 0.7 | 89.8 | 1.5
| 36 | 11759.110 | 11759.123 | -0.013 | 4.7 | 93.8 | 1.5 | 0.0
| 38 | 11621.363 | 11621.348 | 0.015 | 24.3 | 3.7 | 70.7 | 1.3
| 40 | 11629.743 | 11629.714 | 0.029 | 3.3 | 2.7 | 92.1 | 1.9
| 22 | 12044.480 | 12044.476 | 0.004 | 18.3 | 81.3 | 0.4 | 0.0
| 22 | 11985.501 | 11985.482 | 0.019 | 19.1 | 0.1 | 80.3 | 0.5
Manaa2002  | 66 | 12415.216 | 12415.220 | -0.004 | 5.6 | 89.6 | 4.7 | 0.1
Sherstov2007  | 25 | 12297.748 | 12297.743 | 0.005 | 1.9 | 97.4 | 0.7 | 0.0
| 26 | 12300.464 | 12300.460 | 0.004 | 2.0 | 97.2 | 0.8 | 0.0
Asen2008  | 24 | 11718.210 | 11718.197 | 0.013 | 35.0 | 64.5 | 0.5 | 0.0
| 26 | 11726.761 | 11726.750 | 0.011 | 48.0 | 51.5 | 0.5 | 0.0
| 28 | 11732.101 | 11732.088 | 0.013 | 30.2 | 69.0 | 0.7 | 0.0
| 30 | 11738.059 | 11738.054 | 0.005 | 17.7 | 81.3 | 1.0 | 0.0
| 32 | 11744.591 | 11744.582 | 0.009 | 10.6 | 88.2 | 1.2 | 0.0
| 26 | 11726.760 | 11726.750 | 0.010 | 48.0 | 51.5 | 0.5 | 0.0
| 24 | 11718.211 | 11718.197 | 0.014 | 35.0 | 64.5 | 0.4 | 0.0
| 26 | 11726.761 | 11726.750 | 0.011 | 48.0 | 51.5 | 0.5 | 0.0
| 32 | 11744.593 | 11744.582 | 0.011 | 10.6 | 88.2 | 1.2 | 0.0
| 24 | 11718.211 | 11718.197 | 0.014 | 35.0 | 64.6 | 0.4 | 0.0
| 28 | 11732.101 | 11732.088 | 0.013 | 30.2 | 69.0 | 0.8 | 0.0
Falke2008  | 6 | 17261.487 | 17261.486 | 0.001 | 95.5 | 4.5 | 0.0 | 0.0
| 8 | 17261.787 | 17261.786 | 0.001 | 95.5 | 4.5 | 0.0 | 0.0
| 10 | 17262.167 | 17262.166 | 0.001 | 95.5 | 4.5 | 0.0 | 0.0
| 2 | 17292.516 | 17292.516 | 0.000 | 93.7 | 6.3 | 0.0 | 0.0
| 6 | 17292.839 | 17292.839 | 0.000 | 93.7 | 6.3 | 0.0 | 0.0
| 10 | 17293.449 | 17293.449 | 0.000 | 93.8 | 6.2 | 0.0 | 0.0
| 10 | 17361.247 | 17361.242 | 0.005 | 84.5 | 15.5 | 0.0 | 0.0
| 6 | 17360.821 | 17360.816 | 0.005 | 84.6 | 15.4 | 0.0 | 0.0

###  IV.2 Ab initio and empirical interatomic potentials and non-adiabatic
coupling functions of the $A\sim b\sim c\sim B$ complex

Figure 6: The ab initio $L$-uncoupling (a) and SO coupling (b) matrix elements
between the electronic states of the $A\sim b\sim c\sim B$ complex, which were
derived within the framework of the CI-CPP approach and were kept in the fixed
form during the present DDF. At large internuclear distance $L$-uncoupling
functions asymptotically converge to the $\sqrt{l(l+1)}$-value (where $l=1$
for $p$ electron), while the SO matrix elements converge towards the
asymptotic $\xi^{K}_{4p}$-value corresponding to the atomic SO constant of the
K($4p^{2}$P) state. The $L$-uncoupling matrix element between the singlet $A$
and $B$ states obey the Van Vleck’s hypothesisField2004book  of "pure
precession": $L_{AB}(R)\approx\sqrt{L(L+1)-\Lambda(\Lambda\pm 1)}=\sqrt{2}$.
Figure 7: Comparison of diagonal $A_{so}$ and off-diagonal $\xi_{Ab0}$ SO
functions of K2 obtained within the framework of ab initio CI-CPP and FS-RCC
approaches, respectively, and described in Secs. III.2.1 and III.2.2 with
their semi-empirical Manaa2002  and empirical counterparts.
$R_{e}^{b0^{+}_{u}}\approx 3.89$ Å is the equilibrium distance of the
$b^{3}\Pi_{u}(0_{u}^{+})$ state, $R_{c}^{Ab0}\approx 4.76$ Å is the crossing
point of the $A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$ Hund’s "a" coupling case
PECs, and $\xi^{K}_{4p}$=19.24 cm-1 is the asymptotic value corresponding to
the atomic SO constant of K($4p^{2}$P) state. $\Delta U_{01}\equiv
U_{b1_{u}}-U_{b0^{+}_{u}}$ and $\Delta U_{12}\equiv U_{b2_{u}}-U_{b1_{u}}$ are
the differences of the Hund’s "c" coupling case PECs obtained for
$b0_{u}^{+}$, $b1_{u}$, and $b2_{u}$ components of the triplet $b$ state
during the present fully-relativistic FS-RCCSD calculations.

The interatomic potentials for all singlet and triplet $u/g$ states
corresponding to the pure Hund’s "a" coupling case and converging to the first
four non-relativistic dissociation limits of K2 dimer (see, for instance,
Fig.1) have been obtained on the density grid in the range of $R\in[2.3,25]$ Å
within the framework of the scalar-relativistic CI-CPP electronic structure
calculations, which was described in Sec. III.2.1. The relevant electronic
wavefunctions $\Psi^{sr}_{i}(\textbf{r};R)$ were then applied for the
evaluating of all non-vanishing SO
$\xi^{sr}_{ij}(R)=\langle\Psi^{sr}_{i}|\hat{h^{so}}|\Psi^{sr}_{j}\rangle_{\textbf{r}}$
and $L$-uncoupling $L^{sr}_{ij}(R)=\langle\Psi^{sr}_{i}|[\hat{L_{x}}\pm
i\hat{L_{y}}]/\sqrt{2}|\Psi^{sr}_{j}\rangle_{\textbf{r}}$ matrix elements
between the states treated.

Since the eigenvalues of the Hamiltonian matrix depend on a relative sign of
the non-diagonal matrix elements we took particularly care on the correct
$R$-behaviour of the coupling functions in the entire interatomic range. At
the dissociation limit the signs and normalized coefficients of the present SO
matrix elements were defined to coincide with those of Ref. Kato1993 . The
correct relative sign and smooth behaviour of the CI-CPP matrix elements are
guaranteed by getting them from the same ab initio calculation in order to
keep a phase of the required electronic wavefunctions unchangeable.

All relevant CI-CPP data are picked up in the Supplementary Material SM .
Furthermore, $A_{so}$, $\xi_{Ab0}$, $\xi_{Bb}$, $\xi_{Bc}$, and $\xi_{bc}$ SO
functions, as well as the $L$-uncoupling $L_{AB}$ and $L_{bc}$ matrix
elements, which were explicitly involved in the 6$\times$6 CC molecular
Hamiltonian model in Sec. III.1, are depicted in Figs. 6 and 7. It should be
reminded that only $b^{3}\Pi_{u}(0_{u}^{+})$ and $A^{1}\Sigma^{+}_{u}$ state
PECs, along with the on-diagonal $A_{so}(R)$ and off-diagonal $\xi_{Ab0}(R)$
SO-functions (more precisely, only their corrections) have been adjusted
during the global DDF procedure, while all other interatomic potentials and
non-adiabatic matrix elements were constrained on their ab initio values, see
Fig. 7.

The adiabatic interatomic potentials for (1, 2)$0_{u}^{+}$, (2)$1_{u}$, and
(1)$2_{u}$ states of K2 corresponding to the pure Hund’s "c" coupling case
have been evaluated on the sparse grid in the range of $R\in[2.5,10]$ Å within
the framework of the fully relativistic FS-RCCSD calculations presented in
Sec. III.2.2. The PECs for the SO-deperturbed $A0^{+}_{u}$ or $b0^{+}_{u}$
states together with the relevant off-diagonal $\xi_{Ab0}(R)$ SO matrix
element were constructed during the FS-RCC calculations as well. The resulting
FS-RCC point-wise functions are given in the Supplementary Material SM , along
with the original fully relativistic PECs.

The interatomic PECs for the excited K2 states were finally constructed
through the so-called "difference-based" and ground-state potentials
Zaitsevskii:05

$U^{*}_{i}=[U^{ab}_{i}-U^{ab}_{X}]+U^{emp}_{X}$ (23)

implicitly allowing to diminish the $R$-depended systematic errors in the
original ab initio CI-CPP or/and FS-RCCSD PECs $U^{ab}_{i}(R)$. Hereafter,
$U^{emp}_{X}(R)$ is the empirical ground-state PEC determined in the
analytical form in Ref. Tiemann2020 .

The FS-RCC PECs of the SO-deperturbed $b^{3}\Pi_{u}(0_{u}^{+})$ and
$A^{1}\Sigma^{+}_{u}(0_{u}^{+})$ states were used to get the initial sets for
the non-linear parameters of the analytical DELR potentials, which were then
optimized in the global DDF procedure.

The resulting parameters of the DELR potentials defined in Sec. III.3 and
empirically adjusted during the DDF procedure for the $A^{1}\Sigma^{+}_{u}$
and $b^{3}\Pi_{u}(0_{u}^{+})$ states of K2 are given in Table 2 and in
Supplementary Material SM . The dispersion coefficient $C^{\Sigma}_{3}$ of the
singlet $A$ state, the dissociation energy $T_{dis}\equiv T_{dis}^{A}$ of the
$A$ state, as well as the well depths $D_{e}$, the equilibrium distances
$R_{e}$ and the $\beta_{i}$ coefficients for both $A$ and $b$ states are
considered as the adjustable fitting parameters during the least-squares
processing (see Sec. III.4), while $R_{ref}$ and $p$ parameters were kept
fixed during the fit, see Table 2.

The fitted parameters of on-diagonal $\Delta A_{so}(R)$ and off-diagonal
$\Delta\xi_{Ab0}(R)$ SO-correction functions are presented in Table 3 and
Supplementary Material SM . The total resulting empirical, semi-empirical, and
ab initio SO $A_{so}(R)$ and $\xi_{Ab0}(R)$ functions are compared in Fig. 7.
It is clearly seen that the present empirical SO-functions agree very well
(within 0.3 - 0.5 cm-1) with their both CI-CCP and FS-RCC ab initio
counterparts, as well as with the semi-empirical SO-functions. The latter were
derived in Ref. Manaa2002  by the smooth scaling of their original ab initio
MR-CI data on the appropriate experimental molecular and atomic data.
Furthermore, the present empirical $A_{so}(R_{e}^{b0^{+}_{u}})=22.34$ and
$\xi_{Ab0}(R_{c}^{Ab0})=13.28$ (in cm-1) values agree very well with their
previous counterparts 21.5/21.75 and 13.0/13.18 cm-1, respectively, derived in
Ref. Ross87  and Ref.JongJMS1992  using the conventional EHA method. At the
same time, both present empirical and theoretical $A_{so}$($R_{e}$)-values
diverse from their ab initio CI-CPP-$l$ estimates Magnier2009  by more than 5
cm-1 (>20%), see Table 4. It should be also noted that the reference FS-RCCSD
calculation of the K atom provides the K($4p$) atomic SO constant
$\xi^{K}_{so}$ = 18.85 cm-1 which is only about 0.4 cm-1 less than its
experimental counterpart Tiecke2019 .

The resulting empirical energies $T_{e}$ of $A$ and $b$ states of K2 obtained
by the elaborated 6$\times$6 CC DDF procedure are presented in Table 4. The
present $A^{1}\Sigma^{+}_{u}$ state $T_{e}$-value is very close to $T_{e}$
from Lisdat2006 , being slightly smaller than the ones given in Refs.
Manaa2002  and Lisdat2001 , by ca 0.07 and 0.1 cm-1, respectively. The present
empirical $T_{e}$-value for the $b(1_{u})$ state are also smaller, the
respective differences being 0.3 and 0.75 cm-1. The present empirical
$R_{e}$-values are very close to both 4$\times$4 CC and EHA data presented in
Refs.Lisdat2006 ; Ross87 ; Lisdat2001 ; JongJMS1992 . A comparison of the ab
initio CI-CPP, CI-CPP-$l$, and FS-RCC electronic energies and equilibrium
distances of the $A^{1}\Sigma^{+}_{u}(0_{u}^{+})$ and
$b^{3}\Pi_{u}(0_{u}^{+},1_{u},2_{u})$ states of K2 obtained within both Hund’s
"a" and "c" coupling cases representation of the mutually perturbed states is
also given in Table 4. The present ab initio FS-RCC calculations seam to be
closer to their empirical counterparts than their respective CI-CPP and CI-
CPP-$l$ calculations. As can be seen from Table 4, the FS-RCC $T_{e}$ energies
obtained for the $A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}(0_{u}^{+})$ states
are only by 54(<0.5%) and 18(<0.2%) cm-1, respectively, lower than their
empirical values. Meanwhile the relative accuracy of the CI-CPP-$l$ energies
Magnier2004 ; Magnier2009  is slightly higher than those of the very similar
current CI-CPP estimates.

The empirical ground state dissociation energy $D^{X}_{e}$ given in Table 5
was estimated from the present $T_{dis}$-value in Table 2 and the well known
non-relativistic $4p^{2}$P-$4s^{2}$S transition $\nu^{K}_{4p-4s}$ energy of K
atom Tiecke2019  by using the rearranged relation (11):
$D_{e}^{X}=T_{dis}-\nu^{K}_{4p-4s}$. The present $D^{X}_{e}$-value agrees
within the error intervals with the empirical values reported in Refs.
Tiemann2020 ; Falke2006 ; Asen2008 . The $D_{e}^{X}$-value in Ref.Falke2006
was obtained with a help of the long-range deperturbation analysis of the
highly excited rovibronic levels of the $A\sim b$ complex Falke2007PhD . This
SO-deperturbation of the $A\sim b$ complex has been accomplished under the
conventional adiabatic approximation based on the analytical PEC construction
for the artificially isolated $A^{1}\Sigma^{+}_{u}$ state. The alternative
$D_{e}^{X}$-values were obtained in Refs.Asen2008 ; Tiemann2020 ; Falke2008
from high resolution molecular spectroscopy of the ground $X$-state and
scattering experiments on the ultracold potassium atoms.

Table 6 contains the long-range coefficients $C^{\Sigma}_{3}$ corresponding to
the K($4s$)+K($4p$) asymptote and the radiative lifetimes $\tau^{K}_{4p}$ for
the ${}^{2}{\rm P}_{\frac{1}{2}}$ and ${}^{2}{\rm P}_{\frac{3}{2}}$ fine
components of the K($4p$) atom. The elaborated $C^{\Sigma}_{3}$-value in Table
2 was used to evaluate the present $\tau^{K}_{4p}$-values in accordance with
Eq.(14). All data in Table 6 are compared with the empirical values from other
sources; the present $C^{\Sigma}_{3}$ and $\tau$ values fit the best with the
data given in Ref. WangPRA97 . The ab initio $C_{3}^{\Sigma}$ and
$\tau^{K}_{4p}$ values in Table 6 were obtained using the theoretical
transition dipole moment $|d^{K}_{4p-4s}|$ = 2.918 $a.u.$ and the relativistic
$\nu^{K}_{4p\;^{2}{\rm P}_{\frac{1}{2}}-4s\;^{2}{\rm S}_{\frac{1}{2}}}$ =
12979.7 and $\nu^{K}_{4p\;^{2}{\rm P}_{\frac{3}{2}}-4s\;^{2}{\rm
S}_{\frac{1}{2}}}$ = 13036.3 (in cm-1) transition energies derived within the
framework of the reference FS-RCCSD-FF atomic calculations Krumins2020 . It
should be stressed that the ab initio FS-RCCSD-FF method ZaitsevskiiTDM
provided $C_{3}^{\Sigma}$ and $\tau^{K}_{4p}$ values, which coincide within
their most accurate empirical counterparts with a systematic error only about
0.5%.

###  IV.3 Verification of the applied DDF procedure

####  IV.3.1 Isotopic substitution effect

The molecular Hamiltonian construction presented in Sec. III.1 depends on the
reduced molecular mass $\mu$ in an explicit form, and, hence, one can expect
that the empirically adjusted electronic structure parameters (excluding,
perhaps, adiabatic DELR potentials of both $A$ and $b$ states) of the present
6$\times$6 CC model should be mass-invariant. To prove this assumption, we
used the empirical parameters derived during the least-squares processing (see
Sec.III.4) of the experimental term values belonging exclusively to the most
abundant isotopologue 39K2 to evaluate the rovibronic term-values and the
respective multi-component (non-adiabatic) vibrational wavefunctions for other
isotopologues just by substituting the appropriate molecular mass in the CC
equation (1).

A comparison of the term values predicted for the minor 39K41K and 41K2
isotopologues with their experimental counterparts is given in Fig. 8. As can
be seen the residuals between the experimental term values obtained in present
work and their predicted values agree within ca. $\pm$ 0.01 cm-1, see Fig.8a,
which corresponds to the experimental error $\sigma^{expt}_{i}$ = 0.015 cm-1
in Table 1. However, a week systematic shift 0.003 cm-1 (see column $Mean$) is
obvious. The residuals with respect to the experimental values from
Falke2007PhD ; Lisdat2001PhD ; Pazyuk2015 , see Fig. 8b, are even smaller
lying within ca. $\pm$ 0.002 cm-1.

####  IV.3.2 Relative intensity distributions

The experimental relative intensity distribution for $A\sim b\to X$ LIF
progressions originating from a particular rovibronic $J^{\prime}$-level of
the $A\sim b$ complex are presented in Figs. 9 and 10, see bars. The
intensities are normalized to the most intensive line of the LIF progression
and compared with the respective theoretical calculations in accordance with
Eq.(22). We have selected the long enough LIF progressions, which belong to
different isotopologues and demonstrate the cases with different partial
contributions of the $0^{+}_{u}$ component of the triplet $b$ state. As is
seen, the figures demonstrate quite satisfactory agreement between measured
and calculated relative intensity values for different isotopologues, see Fig.
9 for 39K2 and Fig. 10 for 39K41K. It is important to stress that this relates
also to the cases with markedly strong fraction partition of the $b0^{+}_{u}$,
$b1_{u}$ and $b2_{u}$ states, being above 40% for $b0^{+}_{u}$ in Fig. 9 and
Fig. 10b,c. For the $b1_{u}$ it is 8% in Fig. 10c and 13% for $b2_{u}$ in Fig.
10a. It should be noted that the detector sensitivity drops rapidly for
transition frequencies higher than 12 000 cm-1, which may result in larger
experimental uncertainties for low $v_{X}$, see, for instance, the transitions
to $v_{X}$<4 in Fig. 10a.

####  IV.3.3 Comparison with 39K2 term values from other sources not included
in the DDF procedure

Let us analyze how the resulting 6$\times$6 CC model describes the data sets
in Table 1 from VI to XI not included in the present DDF procedure, which are
exposed in Fig. 11a. The dispersion for majority of respective residuals, see
Fig. 11b, demonstrate the agreement within ca. $\pm$ 0.015 cm-1 or even
better, which agrees with the experimental uncertainties $\sigma^{expt}_{i}$
in Table 1. Note that the Dunham correction $Y^{X}_{00}$ has been applied
(except the set VIII) to account for differing values of the ground $X$-state
rovibronic energies Heinze1987  applied to obtain VII, IX, X, and XI data
sets. A small systematic shift can be noticed, see the respective values in
$Mean$ column.

Another way of verification is presented in Table 7, where the $E^{expt}_{j}$
values were obtained by us using the information given in the respective
references. For Ref. Chu2005  the $E^{expt}_{j}$ values were obtained from
two-step laser excitation experiment 2${}^{3}\Pi_{g}\leftarrow A\sim
b\leftarrow X$ (Table 1 in Ref. Chu2005 ) by subtraction the probe laser
frequencies $\nu_{probe}$ from the energy of the 2${}^{3}\Pi_{g}$ level:
$E$(2${}^{3}\Pi_{g}$) – $\nu_{probe}$. The term value from the one-colour PF-
OODR experiment Manaa2002  was obtained by adding laser frequency
$\nu_{probe}$ = 12024.581 cm-1 to the ground state energy level:
$E_{X}$($v_{X}$ = 1, $J^{\prime\prime}$ = 67) + $\nu_{probe}$. In the same way
other presented $E^{expt}_{j}$-values have been obtained from the experimental
data in Table 1 of Ref. Sherstov2007 , Table 1 of Ref. Asen2008 , and Table 2,
of Ref. Falke2008 . All energies $E^{expt}_{j}$ presented in Table 7 include
the correction $Y^{X}_{00}$ = ${-}$0.022 cm-1.

Figure 8: Residuals $E^{expt}_{j}-E^{CC}_{j}$ between the experimental and
calculated term values not included in the DDF procedure as dependent on the
$A\sim b$ complex energy: (a) present work (PW) - black points for 39K41K, red
circles for 41K2; (b) from the sub-Doppler 2s-LES-MB experiments: magenta
Lisdat2001PhD  (set IV in Table 1) and red circles Falke2007PhD  (set V in
Table 1). Figure 9: Experimental (bars) and calculated (points) relative
intensity distributions normalized to the most intensive transition in the
$A\sim b\rightarrow X$ LIF $P,R$-progressions from the $E_{A\sim
b}$($J^{\prime}$ = 52) = 15063.108 cm-1 level of 39K2 (green - $P$-branch,
orange - $R$-branch). The fraction partition $P_{i}$ of $A$/$b0$/$b1$/$b2$
components is evaluated as 58/41/1/0 (in %). The inset presents the obtained
multi-component wavefunctions $\phi_{i}(R)$ (see text). Figure 10:
Experimental (bars) and calculated (points) relative intensity distributions
normalized to most intensive transition in the $A\sim b\rightarrow X$ LIF
$P,R$-progressions (green - $P$-branch, orange - $R$-branch): (a) from the
$E_{A\sim b}$($J^{\prime}$ =59) = 12853.512 cm-1 level of 39K41K, the fraction
partition $P_{i}$ of $A$/$b0$/$b1$/$b2$ components is evaluated as
85/0.7/0.4/13 (in %); (b) from the $E_{A\sim b}$($J^{\prime}$=76) = 12694.294
cm-1 level of 39K41K, $P_{i}$ is 49/47/4/0; (c) from the $E_{A\sim
b}$($J^{\prime}$ = 110) = 12618.06 cm-1 level of 39K41K, $P_{i}$ is
41/50/8/0.3. The inset presents the obtained multi-component wavefunctions
$\phi_{i}(R)$ (see text). Figure 11: The 39K2 $A\sim b$ complex data from the
sources not explicitly included in the DDF procedure: (a) - experimental term
values; (b) $E^{expt}_{j}-E^{CC}_{j}$ \- residuals between experimental values
and the values calculated in present work.

##  V Summary and Conclusions

A considerable amount of high resolution rotationally resolved Fourier
Transform LIF spectra of $A\sim b\rightarrow X$ transitions of K2 molecules in
natural mixture of potassium vapor has been recorded and analyzed. The
obtained 39K2 term values were compiled with the previously published ones and
applied for the direct coupled-channel (CC) deperturbation analysis of the
discrete levels of the entire $A\sim b$ complex. This made it possible to
cover the $A\sim b$ energy range from the very bottom of the $b^{3}\Pi_{u}$
state up to the dissociation energy of K($4s\;^{2}{\rm
S}_{\frac{1}{2}}$)+K($4p\;^{2}{\rm P}_{\frac{1}{2}}$) of separated
relativistic atoms. This task proposed to apply the global 6$\times$6 CC
deperturbation model, which took into account explicitly both SO and
electronic-rotational (Coriolis) interactions between all four
$A^{1}\Sigma^{+}_{u}$, $b^{3}\Pi_{u}$, $c^{3}\Sigma^{+}_{u}$, and
$B^{1}\Pi_{u}$ scalar-relativistic states converging to the common non-
relativistic K($4s$)+K($4p$) dissociation limit. The explicit consideration of
the higher lying $B^{1}\Pi_{u}$ and $c^{3}\Sigma^{+}_{u}$ states in the
applied DDF procedure looks somewhat redundant, however, it guarantees the
precise simultaneous treatment all quasi-degenerate rotational $J^{\prime}>0$
levels located in vicinity of the dissociation threshold, where electronic-
rotational interaction indeed is very weak but not negligible. Furthermore,
the current 6$\times$6 CC deperturbation model is ready in principle to fit
$0^{-}_{u}$, $1_{u}$, and $2_{u}$ terms of the $A\sim b\sim c\sim B$ complex,
as well as to include the quasi-bound rovibronic levels located above the
first relativistic threshold.

The initial parameters of the 6$\times$6 CC deperturbation model as a
functions of the internuclear distance have been estimated in the framework of
alternative scalar-relativistic and fully-relativistic ab initio electronic
calculations accomplished by multi-referenced configuration-interaction and
coupled-clusters methods. Among the considered ab initio methods the most
robust interatomic potentials, SO coupling functions and transition dipole
moments seem to be obtained in the framework of the FS-RCCSD-FF method.

The analytical DELR interatomic potentials of the
$A^{1}\Sigma^{+}_{u}(0^{+}_{u})$ and $b^{3}\Pi_{u}(0^{+}_{u})$ states along
with the on-diagonal $A_{so}(R)$ and off-diagonal $\xi_{Ab0}(R)$ spin-orbit
functions have been globally refined to fit the experimental 39K2 data massive
non-uniformly distributed between the bottom of the $b^{3}\Pi_{u}$ state and
the K($4s\;^{2}{\rm S}_{\frac{1}{2}}$)+K($4p\;^{2}{\rm P}_{\frac{1}{2}}$)
dissociation limit within the uncertainty of their spectroscopic measurements.
The resulting parameters of the deperturbation model allowed us to predict the
energies of both locally and regularly perturbed levels of the
$A^{1}\Sigma^{+}_{u}$ and $b^{3}\Pi_{u}$ states with an accuracy of about 0.01
cm-1, or better, which agrees with a usually reached experimental accuracy.
Moreover, the present CC model reproduces the sub-Doppler term values derived
from the 2s-LES-MB measurements with an accuracy of about 0.003-0.005 cm-1,
which is mainly limited by the uncertainty in the ground $X$-state potential.

The applied deperturbation analysis provided the refined values of empirical
dissociation energy $T_{dis}$ and the long-range coefficient $C_{3}^{\Sigma}$
= 2$C_{3}^{\Pi}$, which are both related to the non-relativistic atomic
threshold K($4s$)+K($4p$). The derived $T_{dis}$=17474.569(5) cm-1 value
yielded an accurate well depth $D_{e}$ = 4450.910(5) cm-1 for the ground
$X^{1}\Sigma^{+}_{g}$ state of 39K2, whereas the $C_{3}^{\Sigma}$ =
5.501(4)$\times 10^{5}$cm-1Å3 value yielded the improved atomic
K($4p^{2}$P1/2;3/2) radiative lifetimes $\tau_{\frac{1}{2}}$=26.67(3) and
$\tau_{\frac{1}{2}}$=26.32(3) $ns$.

To clime the mass-invariant property of electronic structure parameters of the
applied deperturbation model the rovibronic term values predicted for 39K41K
and 41K2 isotopologues were directly compared with their experimental
counterparts. A high reliability of nodal structure and fraction partition of
the non-adiabatic wavefunctions of the $A\sim b$ complex was independently
confirmed by a good agreement of the estimated Einstein emission coefficients
with the relative intensity distributions measured for the long $A\sim
b\rightarrow X^{1}\Sigma^{+}_{g}(v_{X})$ band progressions of both 39K2 and
39K41K isotopologues.

The present analysis of the experimental data from different sources revealed
a necessity in some cases to apply a systematic uniform correction most
probably due to some differences of the exploited Dunham molecular constants
and empirical potentials for the ground $X$-state. A further improvement of
description of the entire state manifold converging to both first
$4s\;^{2}{\rm S}_{\frac{1}{2}}+4p\;^{2}{\rm P}_{\frac{1}{2}}$ and second
$4s\;^{2}{\rm S}_{\frac{1}{2}}+4p\;^{2}{\rm P}_{\frac{3}{2}}$ relativistic
dissociation limits would need additional empirically-based information on all
$0^{\pm}_{u}$, $1^{\pm}_{u}$ and $2^{\pm}_{u}$ components belonging to $A$,
$b$, $B$ and $c$ states of K2.

###### Acknowledgements.

Riga team acknowledges the support from the University of Latvia Faculty of
Physics, Mathematics an Optometry, project: "A detailed description of the
strongly mixed first excited states of the K2 molecule: challenges and
solutions", the Laserlab-Europe EU-H2020 project No 871124 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; Project Nos.
121031300173-2 and 121031300176-3).

## Data Availability Statement

The data that support the findings of this study are available within the
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