ab ad i arXiv:1608.00225v1 [physics.chemph] 31 Jul 2016 · dipole moment (TDM) functions. The...
Embed Size (px)
Transcript of ab ad i arXiv:1608.00225v1 [physics.chemph] 31 Jul 2016 · dipole moment (TDM) functions. The...

arX
iv:1
608.
0022
5v1
[ph
ysic
s.ch
emp
h] 3
1 Ju
l 201
6
The twofold diabatization of the KRb (1 ∼ 2)1Π complex in the framework ofab initio and deperturbation approaches
S. V. Kozlov, E. A. Pazyuk, A. V. StolyarovDepartment of Chemistry, Lomonosov Moscow State University, 119991 Moscow, Leninskie gory 1/3,Russia
(Dated: 15 October 2018)
We performed a diabatization of the mutually perturbed 11Π and 21Π states of KRb based on both electronicstructure calculation and direct coupledchannel deperturbation analysis of experimental energies. The potential energy curves (PECs) of the diabatic states and their scalar coupling were constructed from the abinitio adiabatic PECs by analytically integrating the radial 〈ψad1 ∂/∂Rψad2 〉 matrix element obtained by afinitedifference method. The diabatic potentials and electronic coupling function were refined by the leastsquares fitting of the rovibronic termvalues of the 11Π ∼ 21Π complex. The empirical PECs combined withthe coupling function as well as the diabatized spinorbit coupling and transition dipole matrix elements areuseful for further deperturbation treatment of both singlet and triplet states manifold.
I. INTRODUCTION
The accurate representation of the interacting electronic states plays a key role in understanding the detailed mechanism of the photo and collisionally inducedchemical reactions. The singlettriplet levels of alkalimetal dimers serve as an intermediate state in the twostep optical transformation of the weakly bound atomicpairs into the absolute ground X1Σ+(v = 0, J = 0)molecular state1,2. To suppress the undesired spontaneous transitions to the lowlying states a coherent stimulated Raman adiabatic passage3 (STIRAP) is often used.
The photoassociative production and trapping of ultracold KRb molecules has been performed4. The resonance coupling of the B(1)1Π and 21Π states of KRb(see, Fig. 1) is found to be a promising pathway for direct photoassociative formation of the X(0, 0) ultracoldmolecules5. The rigorous multichannel modeling of thelaser formation of vibrationally cold KRb molecules hasbeen accomplished in Ref. 6. The a3Σ+ → A1Σ+ ∼b3Π → X1Σ+ and a3Σ+ → B1Π ∼ c3Σ+ → X1Σ+optical schemes to create ultracold KRb molecules havebeen studied7 by using the ab initio potential energycurves (PECs), spinorbit coupling (SOC) and transitiondipole moment (TDM) functions. The combination of amolecular beam (MB) and an ultracold molecule (UM)excitation spectroscopy8 was used to identify the optimal a3Σ+(v′′a = 21) → B1Π(v′1 = 8) → X1Σ+(v′′X = 0)STIRAP pathway for the 39K85Rb molecule assembling.The magnetoassociated fermion 40K87Rb molecules havebeen STIRAP transferred9,10 to the lowest X(0, 0) levelthrough the B1Π ∼ c3Σ+ levels located near the seconddissociation threshold.
A comprehensive review of modern spectroscopic studies of the KRb electronic states can be found in the ebook12. The mutually perturbed 11Π and 21Π statesconverging to the second K(42S)+Rb(52P) and thirdK(42P)+Rb(52S) dissociation thresholds were investigated13,14 using Dopplerfree opticaloptical double resonance polarization spectroscopy (OODRPS). In the sub
3 4 5 6 7 8 9 10 11 120
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
2333 +
31 +
b3
A1 +
c(2)3 +
21
B(1)1
a3 +
X1 +
K(42S) + Rb(52S)
K(42P) + Rb(52S)
Ene
rgy
(cm
1)
Internuclear distance, R (Å)
K(42S) + Rb(52P)
FIG. 1. Scheme of the ab initio adiabatic potential energycurves11 of the KRb electronic states correlated to the lowestthree dissociation limits. The arrows denote a possible twostep stimulated Raman adiabatic passage9.
sequent laser induced 31Π → 21Π fluorescence (LIF)studies15,16 of both 39K85Rb and 39K87Rb isotopologuesby Fourier transform spectroscopy (FTS) the vibrationalnumbering of the 21Π(v′2) state was corrected by 6 vibrational quanta. The experimental rovibronic term values
http://arxiv.org/abs/1608.00225v1

2
of the 11Π ∼ 21Π complex were reduced to the ”effective” band Ev′ , Bv′ and conventional Dunham Yij molecular constants14,15. The RydbergKleinRees (RKR) potential was constructed for the adiabatic B(1)1Π state(see, Fig. 2). The vibrational termvalues of the lowestB1Π(v′1 ∈ [0, 21]) levels were obtained during the spectroscopic analysis of the MB experiment17. The groundsinglet X1Σ+ and triplet a3Σ+ states were comprehensively studied by means of high resolution LIF spectra18
coming from the spinorbit coupled B1Π ∼ c3Σ+ levels.
4 5 6 7 8 9 10 1115,00k
15,25k
15,50k
15,75k
16,00k
16,25k
16,50k
16,75k
4,5 5,0 5,5 6,0 6,5 7,015,6k
15,7k
15,8k
15,9k
16,0k
16,1k
16,2k
V1
11
21
V2
Rc
11
Ener
gy, (
cm1)
Internuclear distance, R (A)
21
RKR adiabatic diabatic
FIG. 2. The empirical adiabatic (open symbols  present work,closed symbols  RKR) and diabatic (solid lines) PECs available for the (1, 2)1Π twin states of KRb. The RKR potentialfor the 21Π state was built using the Dunham coefficients15,while the RKR points of theB(1)1Π state were borrowed fromRef. 14. The shadowed regions indicate the rovibronic Eexp
v′J′
termvalues data sets15,16 included in the present CC deperturbation analysis. The inset enlarges the region in the vicinityof the crossing point of V1(R) and V2(R) diabatic PECs.
The adiabatic potentials, permanent and transitiondipole moments for the radially coupled 11Π and 21Πstates were first ab initio calculated in Refs 19 and 20.The comprehensive set of nonrelativistic PECs, permanent and transition dipole moments for the ground andexcited states of KRb are available11,21–25 as well. TheSOC effect has been included into ab initio calculationsin Refs. 6, 11, 24, and 26.
Among other alkali diatomics, the KRb moleculestands out because of the high density of the electronicstates belonging to both singlet and triplet manifolds.This is attributed to the accidental close values of the ionization potential, electronic affinity and the polarizabilityof K and Rb atoms in their ground states as well as the almost degenerate energies of the first excited K(42P) andRb(52P) states27–29. The high density of the lowlyingcovalence and ionpair states apparently leads to the pronounced radial coupling effect between the states of thesame spatial and spin symmetry. This appears (see, forexample, Figs. 1 and 2) as an avoided crossing of thecorresponding adiabatic PECs as well as a sharp dependence of the relevant electronic matrix elements on theinternuclear distance R. The sharp Rdependence of thespinorbit, angular and radial coupling matrix elementsembarrasses a deperturbation analysis while the abruptRvariation of the adiabatic TDM functions prevents toa straightforward simulation of radiative properties.The electronic coupling matrix element V12 ≈ U1(Rc)−
U2(Rc) estimates near the avoided crossing point Rc ofthe adiabatic PECs warn that a conventional adiabaticapproximation is not ideally suitable for representationof the twin (1, 2)1Π states of KRb, since the socalledadiabaticity parameter30 γ ≡ V12/
√ω1ω2 is close to 3.
Here, ωi are harmonic frequencies of the interacting adiabatic states. To our best knowledge, a global deperturbation analysis of the 11Π ∼ 21Π complex has not beenperformed yet in the framework of either adiabatic ordiabatic approximation.In the present work, we performed a twofold diabati
zation of the KRb 11Π ∼ 21Π complex by means of mutually complementary methods, namely: ab initio electronic structure calculations and direct coupledchannel(CC) deperturbation treatment of experimental termvalues12–17.
II. AB INITIO DIABATIZATION OF THE (1 ∼ 2)1ΠTWIN STATES
The simplest twostate transformation (diabatization)of the adiabatic electronic wavefunctions ψ1,2(R) to theirdiabatic counterparts ϕ1,2(R) can be realized by the unitary transformation30
(
ϕ1ϕ2
)
=
(
cos θ sin θ− sin θ cos θ
)(
ψ1ψ2
)
(1)
where the rotation angle θ(R) is evaluated as a functionof the internuclear distance R by integration of the radialcoupling matrix element:
B12 ≡ 〈ψ1∂/∂Rψ2〉 =dθ
dR. (2)
The integration of the ab initio calculated radial couplingmatrix element (2) is performed implicitly by means ofa smooth interpolation of the original pointwise B12(R)function in the vicinity of a dominant maximum (which

3
is located near the avoided crossing point Rc of the corresponding adiabatic potentials) by the simplest Lorentzform30:
B12(R) ≈w
4(R−Rc)2 + w2(3)
with the two Rindependent parametersRc and w. Then,the required rotation angle function
θ(R) =1
2arctan
[
2(R−Rc)w
]
+π
4(4)
is prolonged to the R ∈ [0,+∞)range in order to accomplish a diabatization of the corresponding electronicwave functions (1).The diabatic potentials V1,2(R) and electronic coupling
matrix element V12(R) are calculated from the adiabaticPECs U1,2(R) via the relations:
V1 = cos2 θU1 + sin
2 θU2
V2 = sin2 θU1 + cos
2 θU2
V12 = sin 2θU1 − U2/2 (5)The adiabatic PECs Uabi (R) were evaluated for the
lowlying excited (1 − 3)1,3Σ+ and (1, 2)1,3Π states converging to the lowest three dissociation limits (see Fig. 1)in the basis of the zerothorder (spinaveraged) electronicwavefunctions corresponding to pure (a) Hund’s couplingcase. To diminish the systematic Rdepended error (firstof all, basis set superposition error) the originally calculated adiabatic potentials Uabi for the excited states werecorrected due to the semiempirical relation2:
Ui(R) = [Uabi (R)− UabX (R)] + UempX (R) (6)
where the highly accurate empirical PEC UempX of theground X1Σ+ state was borrowed from Ref. 18.All electronic structure calculations were performed
in a wide range of internuclear distances on the density grid by means of the MOLPRO program package31.The radial coupling matrix element B12(R) between the11Π and 21Π states was evaluated by three points finitedifference method.The details of the computational procedure used can
be found elsewhere24. Briefly, the inner core shellof both potassium and rubidium atoms was replacedby energyconsistent nonempirical effective core potentials32 (ECP), leaving 9 outer shell (8 subvalence plus1 valence) electrons for explicit treatment. The relevantspinaveraged and spinorbit Gaussian basis sets used foreach atom (ECP10MDF for K and ECP28MDF for Rb,respectively) were taken from the above reference. Theoptimized molecular orbitals were constructed from thesolutions of the stateaveraged complete active space selfconsistent field problem for all 18 electrons on the lowest(110)1,3Σ+ and (15)1,3Π electronic states taken withequal weights33. The dynamical correlation was introduced via the internally contracted multireference configuration interaction (MRCI) method34 which was applied for only two valence electrons keeping the remaining 16 subvalence electrons frozen. The lindependent
corepolarization potentials (CPPs) of both atoms (seeTable I) were employed to implicitly account for the residual corevalence correlation effects35. The correspondingCPP cutoff radii of both atoms were adjusted to reproduce the experimental finestructure splitting of the lowest excited K(42P) and Rb(52P) states29.
TABLE I. The static dipole polarizability28 of the cation andits cutoff radii implemented in the CPP potentials of the Kand Rb atoms. All parameters in a.u..
αcore rcut−offK 5.354 0.247Rb 9.096 0.379
The resulting MRCI wave functions were used to evaluate the permanent dipole functions d1,2(R) of adiabatic 11Π and 21Π states as well as the corresponding11Π − 21Π transition dipole moment d12(R). The adiabatic matrix elements were transformed to the relevantdiabatic moments µ1,2(R), µ12(R) as
µ1 = cos2 θd1 + sin
2 θd2 − sin 2θd12µ2 = sin
2 θd1 + cos2 θd2 + sin 2θd12
µ12 = cos 2θd12 + sin 2θd1 − d2/2 (7)
Finally, we have calculated spinorbit ξij(R) (j ∈(1, 2)3Π; (2, 3)3Σ+) and angular L±ij(R) (j ∈ (1− 3)1Σ+)coupling matrix elements as well as transition dipole moments dij(R) (j ∈ (1, 2)1Σ+) for adiabatic i ∈ 11Π, 21Πstates. The resulting adiabatic matrix elements Wij ∈ξij , L
±
ij, dij were unitary transformed to their diabaticcounterparts Wij as
W1j = cos θW1j + sin θW2j
W2j = − sin θW1j + cos θW2j (8)
III. THE COUPLEDCHANNEL DEPERTURBATION
ANALYSIS OF THE (1 ∼ 2)1Π COMPLEX
In the framework of the rigorous coupledchannel(CC) deperturbation model24,36,37, the nonadiabaticrovibronic energy ECC of the (1, 2)1Π complex is determined by the solution of the two coupled radial equations
(
−I ~2d2
2µdR2+V(R)− IECC
)
Φ(R) = 0 (9)
Φ(0) = Φ(∞) = 0,
where µ is the reduced molecular mass, I is the identitymatrix andV(R) is the symmetric matrix of the potentialenergy given by
V11Π = V1; V21Π = V2; V11Π−21Π = V12 (10)
where the diabatic potentials V1,2(R) and electronic coupling matrix element V12(R) are the massinvariant functions of internuclear distance R. The twocomponent

4
vibrational eigenfunction Φ in Eq.(9) is normalized forthe bound states as
∑
i Pi = 1, where Pi = 〈φiφi〉(i ∈ 11Π, 21Π) is the fractional partition of the level withthe energy ECC .The diabatic matrix elements are related to the adia
batic PECs as
U1,2 = (V1 + V2)/2±√
(V1 − V2)2/4 + V 212 (11)
The corresponding ”effective” rotational constant BCC
is defined as the expectation value
BCC =~2
2µ
∑
i=1,2
〈φi1/R2φi〉 (12)
The rovibronic energies ECC and vibrational wavefunctions φi(R) were obtained through solving the CCequations (9) by the finitedifference boundary valuemethod38. The adaptive analytical mapping procedure39
was exploited to decrease the required number of gridpoints.To perform a direct fit of the experimental data
we represented the diabatic interatomic potentials andthe relevant electronic coupling matrix element in theirfully analytical forms. In particular, the Morse/LongRange(MLR)40–42 function
V2(R) ≡ UMLR = [T dis2 −De] + (13)
De
[
1− uLR(R)uLR(Re)
e−β(R)yeqp (R)
]2
is used to approximate the diabatic PEC of the 21Πstate converging to the K(42S)+Rb(52P) dissociationthreshold. The fixed parameter of the dissociation energy T dis2 involved in Eq.(13) was determined as T
dis2 =
DXe + ERb(52P) − ERb(52S) = 16955.169 cm−1, where the
experimental dissociation energy of the ground state43
(taken at the hfs centerofgravity)DXe = 4217.822 cm−1
and the corresponding nonrelativistic energy of the Dlines of the Rb atom29.The coefficient β(R) in the MLR function
βMLR(R) = yrefp β∞ +
[
1− yrefp]
N∑
i=0
βi[
yrefq]i, (14)
is the polynomial function of the reduced coordinatesyrefp,q :
yrefp,q (R) =Rp,q −Rp,qrefRp,q +Rp,qref
, (15)
where Rref is the reference distance and the parametersq and p are integers. The reduced variable yeqp in Eq.(13)is defined by Eq.(15) where the Rref is substituted forthe equilibrium distance Re. The parameter β∞ is constrained to be ln {2De/uLR(Re)}, where De is the thewell depth and uLR is the longrange potential
uLR(R) =∑
n=6,8
DnCnRn
(16)
with fixed sets of the dispersion coefficients Cn and thedamping functions44,45 Dn :
Dn(R) =
[
1− exp(
−3.3(ρ ·R)n
− 0.423(ρ · R)2
√n
)]n−1
,
(17)where the scaling parameter ρ = 0.46145.To approximate the diabatic PEC of the 11Π state con
verging to the K(42P)+Rb(52S) dissociation thresholdwe used the doubleexponential/longrange (DELR) potential46:
V1(R) ≡ UDELR = T dis1 − uLR(R) + (18)Ae−2β(R)(R−Re) −Be−β(R)(R−Re)
which allowed us to represent a rotationless barrier abovethe third asymptote at distances R > Re (see, Fig. 1).The dissociation energy T dis1 = D
Xe +EK(42P)−EK(42S) =
17241.481 cm−1 was fixed during the fit. The nonrelativistic energy of the Dlines of the K atom was takenfrom Ref. 29.The exponent coefficient β(R) of the DELR potential
was defined as
βDELR(R) =
N∑
i=0
βi[
yrefq]i
(19)
while the preexponential coefficients A and B weredetermined from the conditions UDELR(Re) = 0 anddUDELR/dRR=Re = 0 which lead to
A = De − uLR(Re)− u′LR(Re)/βDELR(Re) (20)B = De − uLR(Re) +A
where u′LR ≡ duLR/dR.Finally, the electronic coupling matrix element
V emp12 (R) between the diabatic 11Π and 21Π states was
represented by the polynomial:
V emp12 (R) = (1− yrefq )N∑
i=0
βi[
yrefq]i
(21)
The optimal parameters of the MLR and DELR potentials as well as electronic coupling matrix elementwere determined simultaneously in the framework of theweighted nonlinear leastsquared fitting (NLSF) procedure:
χ2 =
Nexp∑
j=1
(
Eexpj − ECCj )/σexpj
)2(22)
+
Nab∑
j=1
(
[V ab1 (Rj)− UDELR(Rj)]/σabj)2
+
Nab∑
j=1
(
[V ab2 (Rj)− UMLR(Rj)]/σabj)2
+
Nab∑
j=1
(
[V ab12 (Rj)− V emp12 (Rj)]/σabj)2

5
where Eexpj denote the experimental term values of the
(1 ∼ 2)1Π complex and the σexpj values mean their uncertainties. The V ab1,2 and V
ab12 are the diabatic functions
evaluated by Eq.(5) at the point Rj , and σabj are their
uncertainties obtained by averaging the present and preceding11 ab initio curves. The theoretical curves wereincorporated in the NLSF procedure in order to propagate the empirical functions outside of the experimentaldata region.
IV. RESULTS AND DISCUSSION
IV.1. Ab initio data
3 4 5 6 7 8 9 10 11 12 1314750
15000
15250
15500
15750
16000
16250
16500
16750
17000
17250
17500
K(42P) + Rb(52S)
K(42S) + Rb(52P)
c(2)3 +
b3
B(1)1
Ener
gy, (
cm1)
Internuclear distance, R (A)
21
FIG. 3. The ”differencebased” adiabatic PECs obtained fromthe present (open symbols) and preceding11 (solid symbols) abinitio calculations. The solid lines denote the correspondingdiabatic ab initio PECs of the 11Π and 21Π states.
The resulting ”differencebased” PECs obtained byEq.(6) for adiabatic B(1)1Π and 21Π states from thepresent and preceding11 ab initio calculations demonstrates overall good agreement as can be seen in Fig. 3.The most significant deviations are observed in the vicinity of the avoided crossing point Rc ≈ 5.36 (Å). The corresponding diabatic V1(R) and V2(R) PECs obtained by
the unitary transformation (5) are depicted as well.The ab initio radial B12(R) and electronic V12(R) cou
pling matrix elements are given on Fig. 4a and b, respectively. The inset demonstrates that the simplest twoparameters Lorentz curve (3) perfectly fits a peak of theab initio B12(R) function.The permanent dipole moments of the 11Π and 21Π
states as well as the corresponding 11Π − 21Π transition dipole moment are given on Fig. 5a. The adiabaticfunctions d1,2(R), d12(R) were obtained during the electronic structure calculations while the diabatic momentsµ1,2(R), µ12(R) were evaluated according to Eq.(7). Asfollows from the charge density of the electronic wavefunctions of the twin (1, 2)1Π states19, the d1,2(R) functions have a ”mirror” Rdependance d1 ≈ −d2 with asharp global extremum about ±2 a.u. located near thepoint Rc. It should be noticed, that the adiabatic transition moment d12 becomes zero at the same point. In contrast to sharp adiabatic functions, their diabatic counterparts demonstrate rather smooth Rbehavior. Furthermore, the absolute magnitudes of the diabatic µ1,2(R)functions significantly decrease at intermediate internuclear distances.The adiabatic (1, 2)1Π − (X,A)1Σ+ transition dipole
moments are depicted on Fig. 5b along with the diabatic moments evaluated by Eq.(8). A good agreementof the present B(1)1Π − X1Σ+ transition moment andthe preceding estimate23 is observed. The Fig. 5b alsoshows that the diabatic 11Π − A1Σ+ and 21Π → X1Σ+moments are very small at short and intermediate Rdistances.The diabatic 11Π → X1Σ+ transition moment µ1X(R)
was used to evaluate a radiative lifetime for the lowestvibrational levels of the 11Π state by the approximatesum rule47:
1
τ11Π≈ 8π
2
3~ǫ0〈φJ′1 [∆V1X ]3[µ1X ]2φJ
′
1 〉 (23)
where ∆V1X(R) = V1(R)−UX(R) is the difference of thediabatic PEC of the 11Π state and ground state PEC.The resulting τ = 11.3 ns predicted for the B1Π(v′1 =2, J ′ = 41) level is remarkably close to its experimental counterpart13 of 11.6 ns. It should be noted, thatthe contribution of the 11Π − A1Σ+ transition into theτ11Πestimate could be neglected since µ1A ≪ µ1X  and∆V1A ≪ ∆V1X .The resulting SOC matrix elements obtained during
the present ab initio calculations are depicted on Fig. 6.As expected, the diabatization procedure provides asmooth Rbehavior of most SOC functions. However, thediabatic (1, 2)1Π − 33Σ+ functions are still not smoothenough since the radial coupling of the 33Σ+ state withthe higher (n ≥ 4)3Σ+ states takes place. It should bealso noted that the present B1Π− c3Σ+ and B1Π− b3Πfunctions deviate significantly at short and intermediateRranges from the preceding result6, which has been usedin modeling the optimal a3Σ+ → B1Π ∼ c3Σ+ → X1Σ+STIRAP cycle7.

6
The angular coupling matrix elements L±ij(R) obtained
for the (1, 2)1Π − (1 − 3)1Σ+ nonadiabatic transitionsare presented on Fig. 7. It is seen that Van Vleck’s pureprecession hypothesis30 L±ij ≈
√
l(l + 1) = const works
perfectly for the 11Π−A1Σ+ pair with l = 1. With l = 2,it works fairly well for the 21Π− 31Σ+ pair at short andintermediate distances.Under unique perturber approximation30 the qfactors
of the doubly degenerate 1Π states are estimated as
q1Π ≈(
~2
2µR2e
)2∑
1Σ+
2l(l+ 1)
T 1Πe − T 1Σ+e(24)
yielding, for the 39K85Rb diabatic states, q11Π ≈ 3.2 ×10−5 and q21Π ≈ 1.5×10−4 cm−1, respectively. Unfortunately, there are no experimental qvalues for comparisonso far.The resulting ab initio potential energy curves, per
manent and transition dipole moments as well as electronic, spinorbit and angular coupling matrix elementsare given in pointwise form in the Supplementary material48.
3 4 5 6 7 8 9 10 11 120
40
80
120
160 3 4 5 6 7 8 9 10 11 12
0,0
0,5
1,0
1,5
2,0
2,5
b)Rc
ab initio CC fit
V12
(cm
1)
R (Å)
4,5 5,0 5,5 6,0 6,50,0
0,5
1,0
1,5
2,0
2,5
B 12 (
a.u.
/Å)
ab initio Lorentzian
R (Å)
Rc
B12
(a.u
./Å)
a)
FIG. 4. (a) The radial coupling electronic matrix elementB12(R) calculated between adiabatic 1
1Π and 21Π states. Onthe inset the red line depicts the fitting Lorentz curve (3)with the parameters Rc = 5.36 and w = 0.4036 (both in Å).(b) The ab initio and empirical electronic coupling V12(R)function between diabatic 11Π and 21Π states.
2 4 6 8 10 12 14 16 18 202
1
0
1
2
3
4
2 3 4 5 6 7 8 9 10 11 122,0
1,5
1,0
0,5
0,0
0,5
1,0
1,5
2,0
2,5
b)
Rc
d1A, d2A 1A, 2A
R(Å)
d1X, d2X 1X, 2X
a)
Rc
1 2 12
Perm
anen
t and
tran
sitio
n di
pole
mom
ents
(a.u
.)
d1 d2 d12
FIG. 5. (a) The adiabatic and diabatic ab initio permanentdipole moments of the 11Π and 21Π states as well as thecorresponding 11Π− 21Π transition moment. (b) The ab initio TDM functions between the (1, 2)1Π and X,A1Σ+ states.The dashed line denotes the adiabatic dB1Π−X1Σ+ (R) function from Ref. 23.
IV.2. CC deperturbation data
Experimental input data of the 11Π ∼ 21Π complexused in the NLSF fitting procedure (22) consists of (i)the 110 original rovibronic termvalues16 Eexpv′J′ obtainedduring the (3)1Π → (1, 2)1Π FTS LIF experiment15 forthe rotational levels J ′ ∈ [24, 145] in the short range ofvibrational quantum numbers v′1, v
′2 ∈ [0, 6]; (ii) the re
duced termvalues Eexpv′ obtained by the OODRPS measurements13,14 for the v′1 ∈ [0, 69] and v′2 ∈ [6, 19] levels,respectively; (iii) the rotationless (J ′ = 0) termvaluesextracted from the MB experiment17 for v′1 ∈ [0, 20] levels.
All energies above correspond to the most abundant39K85Rb isotopologue while the 29 rovibronic termvalues16 of 39K87Rb held in reserve for confirmation of themassinvariant properties of the fitting functions. Theuncertainty σexp of the raw rovibronic termvalues16 wastaken as 0.05 cm−1 while the σexp = 2 cm−1 was adoptedfor the vibronic terms Eexpv′ since the difference of theempirical data13,14 and Ref. 17 reach few reciprocal cen

7
2 4 6 8 10 12 1480
70
60
50
40
30
20
10
0
10
20
302 4 6 8 10 12 14
40302010
01020304050
607080
Rc
11 b3
21 b3
11 23
21 23
b)Spi
nor
bit c
oupl
ing
mat
rix e
lem
ents
(cm
1)
B1 b3
Rc
11 33 + 21 33 +
B1 c3 +
a)
R(Å)
11 c3 + 21 c3 +
FIG. 6. The adiabatic (solid lines) and diabatic (open circles) ab initio spinorbit coupling matrix elements ξij(R) between the (1, 2)1Π and (2, 3)3Σ+ (a), (1, 2)3Π (b) states.The dashed lines denote the adiabatic ξB1Π−c3Σ+(R) andξB1Π−b3Π(R) SOC functions borrowed from Ref. 6.
timeters (see, Fig. 8b).
The adjusted massinvariant parameters of the DELR(18) and MLR (13) potentials obtained for the diabatic 11Π and 21Π states are presented on Table II andIII, respectively. The fitting parameters of the empiric11Π ∼ 21Π coupling function (21) are given on the Table IV. The resulting parameters are duplicated in nontruncated ASCII form in the Supplementary material48,where both experimental and CC term values along withtheir residuals (see Fig.8) and fractional partitions arecollected as well. The adiabatic PECs obtained by thetransformation (11) from the empirical diabatic functionsagree very well with the corresponding RKR potentialsin the low energy region (see Fig. 2).
Fig. 8a demonstrates that the present deperturbationmodel allows one to reproduce the most experimentalrovibronic termvalues16 of the 39K85Rb isotopologue andto predict the Eexpv′J′ values of the
39K87Rb isotopologuewith an uncertainty close to 0.05 cm−1. However, thereare several pronounced deviations of the experimentaltermvalues corresponding to the particular rotational levels of v′1 = 4, 5 vibrational states from the CC estimates
2 4 6 8 10 12 140,5
0,0
0,5
1,0
1,5
2,0
2,52 4 6 8 10 12 14
0,0
0,5
1,0
1,52 4 6 8 10 12 14
0
1
2
3
4
Rc
c)
a)
11 31 + 21 31 +
R(Å)
Angu
lar c
oupl
ing
mat
rix e
lem
ent (
a.u.
)
Rc
b)
11 A1 + 21 A1 +
Rc
11 X1 + 21 X1 +
FIG. 7. The adiabatic (solid lines) and diabatic (open circles)ab initio angular coupling matrix elements L±ij(R) calculated
between the (1, 2)1Π and X1Σ+ (a), A1Σ+ (b), 31Σ+ (c)states. The horizontal dashed lines correspond to
√2 (b) and√
6 (c) values, respectively.
(see the Supplementary material48 for details). The observed outliers are attributed to the local SOC effect withthe lower lying c(2)3Σ+ state (see, Fig. 3).The CC diabatic model also improves the represen
tation of the vibronic Eexpv′ termvalues13,14,17 up to the
excitation energies ∼ 16400 cm−1 (see Fig. 8b). Overallgood agreement of the calculated BCCv′ (12) and empirical Bexpv′ rotational constants is observed on Fig. 9 forthe vibrational v′1 ≤ 40 and v′2 ≤ 6 levels.To test extrapolation possibilities of the deperturba
tion model we have calculated rovibronic termvaluesECCv′J′ for a pair of the closely lying 1
1Π(v′1 = 54) ∼21Π(v′2 = 15) levels of the (1 ∼ 2)1Π complex (seeFig. 10) experimentally studied in Ref. 14. The vibrational numbering used above corresponds to the adiabaticrepresentation of the 11Π ∼ 21Π complex applied for theassignment of the OODRPS X1Σ+ → 11Π ∼ 21Π spectra14. The calculated ECCv′J′ positions are found to be in agood agreement with their experimental counterparts. Inparticular, the minimal distance ∆CC = 4.3 cm−1 predicted at J ′ = 27 is remarkably close to the empiricalestimate ∆exp = 2 × 2.2 cm−1 obtained in Ref. 14. The

8
fraction partition Pki ≡ 〈φJ′
ki φJ′
ki〉 of the CC vibrationaleigenfunctions highlights a strong dependance of admixture of states on the rotational quantum number in theinterval J ′ ∈ [11, 35].The divergence of the present CC estimates and empir
ical band constants generally increases as the vibrationalexcitation increases (see Fig. 8c). The same effect takesplaces in the empirical adiabatic PECs and RKR potentials. It can be attributed to the monotonically growingSO coupling with the b3Π state correlated with the samedissociation limit (see, Fig. 3). Furthermore, the highvibrational 11Π(v′1 ≥ 63) levels lying just above the fine42S1/2(K)+5
2P1/2(Rb) asymptotic undergo a predissoci
ation effect14.
Thus, raw experimental termvalues corresponding tohigh v′, J ′levels of the (1 ∼ 2)1Π complex would be certainly useful for the comprehensive deperturbation analysis since the ”effective” band constants indispensably”absorb” the spin0orbit perturbation effect in the highenergy region.
TABLE II. The resulting massinvariant parameters of theUDELR(R) potential (18) obtained for the diabatic 1
1Π state.
fitted
De, cm−1 2214.757
Re , Å 4.3715A , cm−1 10280.36B , cm−1 4577.51β0, Å
−1 0.47730β1, Å
−1 0.22714β2, Å
−1 0.11574β3, Å
−1 0.04563β4, Å
−1 0.20030β5, Å
−1 0.19131β6, Å
−1 0.73518β7, Å
−1 0.15280β8, Å
−1 0.72517fixed
q 3Rref , Å 5.6781T dis, cm−1 17241.481C6, cm
−1·Å6 −2339× 105
C8, cm−1·Å8 9536 × 105
IV.3. Intensity anomalies of the
X1Σ+(v′′ = 12) → 11Π(v′ = 54) ∼ 21Π(v′ = 15) transition
The present deperturbation model (9) combined withdiabatic (1, 2)1Π − X1Σ+ transition moments µ1X(R),µ2X(R) (see Fig. 5b) was used to elucidate the ”abnormal” intensity distribution observed for the X1Σ+(v′′X =12, J ′′ = J ′ + 1) → 11Π(v′1 = 54, J ′) ∼ 21Π(v′2 = 15, J ′)rovibronic transition14.
16400 16500 16600 16700 16800642024615000 15200 15400 15600 15800 16000 16200 164008642024
c)
Termvalues, Eexp (cm1)
b)
a)15000 15200 15400 15600 15800 16000 16200 16400
0,2
0,1
0,0
0,1
0,2
Eex
pv
 E
CC
v (
cm1)
Eex
pvJ
 E
CC
vJ (
cm1)
FIG. 8. The residual of the experimental termvalues of the11Π ∼ 21Π complex and CC estimates (9). Black and redcircles mark the 11Π and 21Π states, respectively. (a) Solidsymbols correspond to the rovibronic termvalues16 of 39K85Rbisotopologue while open circles  39K87Rb. (bc) The circlesdenote the OODRPS termvalues13,14 while the triangles denote the MB data17. The bars denotes the differences betweenthe empirical terms and their estimates evaluated by the corresponding RKR potentials.
The absorbtion intensities from the ground X1Σ+
state to a pair of adjoining levels of the (1 ∼ 2)1Π complex were evaluated according to the relation:
ICCk1Π−X1Σ+ ∼ MkX2 (25)= 〈φJ′k1µ1X χJ
′′
X 〉2 + 〈φJ′
k2µ2X χJ′′
X 〉2
+ 2〈φJ′k1µ1X χJ′′
X 〉〈φJ′
k2µ2X χJ′′
X 〉
where k = 1, 2 is the index of the states represented onFig. 10, φJ
′
k1(R), φJ′
k2(R) are the CC rovibrational wave
functions and χJ′′
X (R) are the vibrational wavefunctionsof the ground Xstate (see Fig. 11) calculated with thehighly accurate empirical potential18.The rovibronic 〈φJ′k1µ1X χJ
′′
X 〉 and 〈φJ′
k2µ2X χJ′′
X 〉 matrix elements calculated for the X1Σ+(v′′X = 12) →11Π(v′1 = 54) ∼ 21Π(v′2 = 15) transitions are given onFig. 12a. It is seen that the 〈φJ′k2µ2X χJ
′′
X 〉 terms givea negligible contribution to the total MkX2 transition

9
0 10 20 30 40 50 60 701,0
1,2
1,4
1,6
1,8
2,0
2,2
2,4
2,6
2,8
3,0
3,2
0 4 8 12 16 201,4
1,6
1,8
2,0
2,2
2,4
2,6
B vx1
00 (
cm1)
v2
21
Rot
atio
nal c
onst
ants
Bvx
100
(cm
1)
v1
11
FIG. 9. Comparison of the empirical (solid circles) rotationalconstants Bexp
v′of the 11Π ∼ 21Π complex13,14 and theoretical
BCCv′ estimates (open circles) derived by Eq.(12). Red squareson the inset denoteBexp
v′values calculated for the v′2 ≤ 6 levels
by the Dunham constants15.
TABLE III. The resulting massinvariant parameters of theUMLR(R) (13) potential obtained for the diabatic 2
1Π state.
fitted
De, cm−1 1111.961
Re, Å 5.2872β0 0.92350β1 0.09744β2 0.09023β3 0.71368β4 1.08251β5 0.12502β6 0.35828
fixedq 4p 4Rref , Å 6.8511T dis, cm−1 16955.169C6, cm
−1·Å6 3025× 105
C8, cm−1·Å8 7875× 105
0 200 400 600 800 1000 1200 1400 16000
10
20
30
40
50
60
70
80
90
100
0 200 400 600 800 1000 1200 1400 160016660
16665
16670
16675
16680
16685
16690
16695
16700
P21; P22
b)
J'(J'+1)
P11; P12
Frac
tiona
l par
titio
n, P
ki (%
)
ECCv'J' (k=1)
ECCv'J' (k=2)
a)
21 (v'=15)
11 (v'=54)
Term
val
ue, E
v'J' (c
m1)
CC
FIG. 10. (a) The rovibronic termvalues ECCv′J′ predicted forthe 11Π(v′ = 54) ∼ 21Π(v′ = 15) levels under the present CCdeperturbation model. The straight lines denotes the adiabatic energies calculated as Eexp
v′+ Bexp
v′× J ′(J ′ + 1) by the
”effective” band constants14. The symbol ∆CC means theminimal distance predicted at J ′ = 27. (b) The fractionpartition Pki of the CC eigenfunctions corresponding to termvalues above.
TABLE IV. The resulting massinvariant parameters of theelectronic 11Π ∼ 21Π coupling V emp12 (R) function (21).
fittedβ0 114.87β1 2.4087β2 98.392
fixedq 3Rref , Å 5.36
probability since µ2X  ≪ µ1X  (see Fig. 5b). Furthermore, 〈φJ′21µ1X χJ
′′
X 〉 matrix elements demonstrate abnormally strong J ′dependance, and they accidentallybecome very small in the vicinity of J ′ ≈ 25 due to theinterference effect taking place in the overlap integral ofthe upper and ground rovibrotional wavefunctions.

10
3,5 4,0 4,5 5,0 5,5 6,00
2
4
6
83,5 4,0 4,5 5,0 5,5 6,0
0
2
4
6
83,5 4,0 4,5 5,0 5,5 6,0
0
2
4
6
8
R(Å)
 J"X 
c)
 J'11b)
a) J'21
Rov
ibra
tiona
l wav
efun
ctio
n (a
rb.u
.) J'=11 J'=25 J'=35
FIG. 11. The nodal structure of vibrational wavefunctionscalculated for the particular rotational J ′ = J ′′ − 1 levels ofthe upper v′1 = 54 (a), v
′2 = 15 (b) and ground v
′′X = 12 (c)
vibrational states.
V. CONCLUDING REMARKS
We performed the diabatization of the twin (1, 2)1Πstates of KRb based on an ab initio electronic structurecalculation and the direct coupledchannel treatment ofexperimental term values of the (1 ∼ 2)1Π complex. Thepresent CC deperturbation model, based on a diabaticrepresentation, provides the almost spectroscopic (experimental) accuracy of the approximation. The empiricalPECs and electronic coupling function, along with abinitio spinorbit and angular coupling matrix elements,could be utilized in further deperturbation analysis carried out in the framework of both adiabatic and diabaticapproximation. The diabatic transition dipole momentsare appropriated for radiative property estimates.
ACKNOWLEDGMENTS
Authors are indebted to Claude Amiot for providingthe raw rovibronic termvalues of the (1 ∼ 2)1Π complexand Ekaterina Bormotova for fruitful discussion. Thework was partly supported by RFBR grant No. 1603
12 14 16 18 20 22 24 26 28 30 32 340,0
0,2
0,4
0,6
0,8
1,0 c)I[11 (v'=54)]I[21 (v'=15)]
Inte
nsity
ratio
I[21 (v'=15)]I[11 (v'=54)]
Rotational quantum number, J'
CC Exp
12 14 16 18 20 22 24 26 28 30 32 340,00
0,02
0,04
0,06
0,08
0,10
0,12
b)
Tran
sitio
n pr
obab
ility
(a.u
.)2
M21X M22X
12 14 16 18 20 22 24 26 28 30 32 340,2
0,1
0,0
0,1
0,2
0,3 < J'21 1X
J"X >; <
J'22 2X
J"X >
a)
Mat
rix e
lem
ent (
a.u.
)
< J'11 1XJ"X >; <
J'12 2X
J"X >
FIG. 12. (a) The rovibronic transition matrix elements calculated for the v′′X = 12 → v′1 = 54 ∼ v′2 = 15 transition; (b)The total X1Σ+ → 11Π ∼ 21Π transition probabilities; (c)Comparison of the theoretical (25) and experimental14 intensity ratios. The experimental points were digitized from Fig.8of Ref. 14.
00529a.
REFERENCES
1R. Krems, B. Friedrich, and W. C. Stwalley, Cold molecules:theory, experiment, applications (CRC press, 2009).
2E. A. Pazyuk, A. V. Zaitsevskii, A. V. Stolyarov, M. Tamanis,and R. Ferber, Russ. Chem. Rev. 84, 1001 (2015).
3K. Bergmann, N. V. Vitanov, and B. W. Shore, J. Chem. Phys.142, 170901 (2015).
4D. Wang, J. Qi, M. Stone, O. Nikolayeva, H. Wang, B. Hattaway,S. Gensemer, P. Gould, E. Eyler, and W. Stwalley, Phys. Rev.Lett. 93, 243005 (2004).
5W. Stwalley, J. Banerjee, M. Bellos, R. Carollo, M. Recore, andM. Mastroianni, J. Phys. Chem. A 114, 81 (2010).
6S. Kotochigova, E. Tiesinga, and P. S. Julienne, New J Phys 11,055043 (2009).
7D. Borsalino, B. LondoñoFlorèz, R. Vexiau, O. Dulieu,N. BouloufaMaafa, and E. LucKoenig, Phys. Rev. A 90, 033413(2014).
8J.T. Kim, Y. Lee, B. Kim, D. Wang, W. C. Stwalley, P. L.Gould, and E. E. Eyler, Phys. Rev. A 84, 062511 (2011).

11
9K.K. Ni, S. Ospelkaus, D. Wang, G. Quéméner, B. Neyenhuis,M. De Miranda, J. Bohn, J. Ye, and D. Jin, Nature 464, 1324(2010).
10K. Aikawa, D. Akamatsu, M. Hayashi, K. Oasa, J. Kobayashi,P. Naidon, T. Kishimoto, M. Ueda, and S. Inouye, Phys. Rev.Lett. 105, 203001 (2010).
11S. Rousseau, A. Allouche, and M. AubertFrécon, J. Mol. Spectrosc. 203, 235 (2000).
12J.T. Kim, B. Kim, and W. C. Stwalley, Analysis of the AlkaliMetal Diatomic Spectra (Morgan & Claypool Publishers, 2014).
13N. Okada, S. Kasahara, T. Ebi, M. Baba, and H. Katô, J. Chem.Phys. 105, 3458 (1996).
14S. Kasahara, C. Fujiwara, N. Okada, H. Katô, M. Baba, et al.,J. Chem. Phys. 111, 8857 (1999).
15C. Amiot, J. Verges, C. Effantin, and J. d’Incan, Chem. Phys.Lett. 321, 21 (2000).
16C. Amiot, private communication (2016).17J.T. Kim, Y. Lee, B. Kim, D. Wang, W. C. Stwalley, P. L.Gould, and E. E. Eyler, Phys. Chem. Chem. Phys. 13, 18755(2011).
18A. Pashov, O. Docenko, M. Tamanis, R. Ferber, H. Knöckel, andE. Tiemann, Phys. Rev. A 76, 022511 (2007).
19T. Leininger and G.H. Jeung, Phys. Rev. A 51, 1929 (1995).20T. Leininger, H. Stoll, and G.H. Jeung, J. Chem. Phys. 106,2541 (1997).
21A. Yiannopoulou, T. Leininger, A. M. Lyyra, and G.H. Jeung,International Journal of Quantum Chemistry 57, 575 (1996).
22S. J. Park, Y. J. Choi, Y. S. Lee, and G.H. Jeung, Chem. Phys.257, 135 (2000).
23R. Beuc, M. Movre, T. Ban, G. Pichler, M. Aymar, O. Dulieu,and W. E. Ernst, J. Phys. B: Atom. Mol. Phys. 39, S1191 (2006).
24K. Alps, A. Kruzins, M. Tamanis, R. Ferber, E. A. Pazyuk, andA. V. Stolyarov, J. Chem. Phys. 144, 144310 (2016).
25M. Shundalau, G. Pitsevich, A. Malevich, A. Hlinisty, A. Minko,R. Ferber, and M. Tamanis, Computational and TheoreticalChemistry 1089, 35 (2016).
26S. Kotochigova, P. Julienne, and E. Tiesinga, Phys. Rev. A 68,022501 (2003).
27A. A. Radzig and B. M. Smirnov, Reference data on atoms,molecules, and ions, Vol. 31 (Springer Science & Business Media,2012).
28J. Mitroy, M. S. Safronova, and C. W. Clark, J. Phys. B: Atom.Mol. Phys. 43, 202001 (2010).
29A. Kramida, Yu. Ralchenko, J. Reader, and NISTASD Team, NIST Atomic Spectra Database (ver. 5.3):http://physics.nist.gov/asd. National Institute of Standardsand Technology, Gaithersburg, MD.
30H. LefebvreBrion and R. W. Field, The Spectra and Dynamicsof Diatomic Molecules (Academic Press, 2004).
31H.J. Werner, P. J. Knowles, G. Knizia, F. R. Manby, M. Schütz,et al., “Molpro, version 2010.1, a package of ab initio programs,”(2010), see http://www.molpro.net/.
32I. S. Lim, P. Schwerdtfeger, B. Metz, and H. Stoll, J. Chem.Phys. 122, 104103 (2005).
33H.J. Werner and P. J. Knowles, J. Chem. Phys. 82, 5053 (1985).34P. J. Knowles and H.J. Werner, Theor. Chim. Acta 84, 95(1992).
35I. S. Lim, W. C. Lee, Y. S. Lee, and G.H. Jeung, J. Chem.Phys. 124, 234307 (2006).
36T. Bergeman, C. Fellows, R. Gutterres, and C. Amiot, Phys.Rev. A 67, 050501 (2003).
37V. V. Meshkov, E. A. Pazyuk, A. Zaitsevskii, A. V. Stolyarov,R. Brühl, and D. Zimmermann, J. Chem. Phys. 123, 204307(2005).
38S. N. Yurchenko, L. Lodi, J. Tennyson, and A. V. Stolyarov,Comp. Phys. Comm. 202, 262 (2016).
39V. V. Meshkov, A. V. Stolyarov, and R. J. Le Roy, Phys. Rev.A 78, 052510 (2008).
40R. J. Le Roy, Y. Huang, and C. Jary, J. Chem. Phys. 125,164310 (2006).
41R. J. Le Roy and R. D. E. Henderson, Mol. Phys. 105, 663 (2007).42H. Salami, A. J. Ross, P. Crozet, W. Jastrzebski, P. Kowalczyk,and R. J. Le Roy, J. Chem. Phys. 126, 194313 (2007).
43D. Wang, J.T. Kim, C. Ashbaugh, E. Eyler, P. Gould, andW. Stwalley, Phys. Rev. A 75, 032511 (2007).
44R. J. Le Roy, N. S. Dattani, J. A. Coxon, A. J. Ross, P. Crozet,and C. Linton, J. Chem. Phys. 131, 204309 (2009).
45R. J. Le Roy, C. C. Haugen, J. Tao, and H. Li, Mol. Phys. 109,435 (2011).
46Y. Huang and R. J. Le Roy, J. Chem. Phys. 119, 7398 (2003).47E. A. Pazyuk, A. V. Stolyarov, and V. I. Pupyshev, Chem. Phys.Lett. 228, 219 (1994).
48See EPAPS Document No. ... for data files associated with thispaper. This document can be reached via a direct link in theonline article’s HTML reference section or via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html).
http://dx.doi.org/10.1016/j.cpc.2015.12.021