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6

The twofold diabatization of the KRb (1 ∼ 2)1Π complex in the framework of

ab initio and deperturbation approachesS. 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 coupled-channel deperturbation analysis of experimental energies. The po-tential energy curves (PECs) of the diabatic states and their scalar coupling were constructed from the ab

initio adiabatic PECs by analytically integrating the radial 〈ψad1 |∂/∂R|ψad

2 〉 matrix element obtained by afinite-difference 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 spin-orbit 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 elec-tronic states plays a key role in understanding the de-tailed mechanism of the photo and collisionally inducedchemical reactions. The singlet-triplet 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 sponta-neous transitions to the low-lying states a coherent stimu-lated Raman adiabatic passage3 (STIRAP) is often used.

The photoassociative production and trapping of ul-tracold KRb molecules has been performed4. The res-onance coupling of the B(1)1Π and 21Π states of KRb(see, Fig. 1) is found to be a promising pathway for di-rect photoassociative formation of the X(0, 0) ultracoldmolecules5. The rigorous multi-channel 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), spin-orbit 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 opti-mal 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 stud-ies of the KRb electronic states can be found in the e-book12. The mutually perturbed 11Π and 21Π statesconverging to the second K(42S)+Rb(52P) and thirdK(42P)+Rb(52S) dissociation thresholds were investi-gated13,14 using Doppler-free optical-optical double reso-nance 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

23

33 +

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 vibra-tional quanta. The experimental rovibronic term values

2

of the 11Π ∼ 21Π complex were reduced to the ”effec-tive” band Ev′ , Bv′ and conventional Dunham Yij molec-ular constants14,15. The Rydberg-Klein-Rees (RKR) po-tential 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 spec-troscopic analysis of the MB experiment17. The groundsinglet X1Σ+ and triplet a3Σ+ states were comprehen-sively studied by means of high resolution LIF spectra18

coming from the spin-orbit 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, (

cm-1)

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 avail-able 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 E

exp

v′J′

termvalues data sets15,16 included in the present CC depertur-bation 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 non-relativistic PECs, perma-nent 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 ion-ization potential, electronic affinity and the polarizabilityof K and Rb atoms in their ground states as well as the al-most degenerate energies of the first excited K(42P) andRb(52P) states27–29. The high density of the low-lyingcovalence and ion-pair states apparently leads to the pro-nounced 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 depen-dence of the relevant electronic matrix elements on theinternuclear distance R. The sharp R-dependence of thespin-orbit, angular and radial coupling matrix elementsembarrasses a deperturbation analysis while the abruptR-variation 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 so-calledadiabaticity parameter30 γ ≡ V12/

√ω1ω2 is close to 3.

Here, ωi are harmonic frequencies of the interacting adi-abatic states. To our best knowledge, a global depertur-bation 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 mu-tually complementary methods, namely: ab initio elec-tronic structure calculations and direct coupled-channel(CC) deperturbation treatment of experimental termval-ues12–17.

II. AB INITIO DIABATIZATION OF THE (1 ∼ 2)1ΠTWIN STATES

The simplest two-state transformation (diabatization)of the adiabatic electronic wavefunctions ψ1,2(R) to theirdiabatic counterparts ϕ1,2(R) can be realized by the uni-tary transformation30

(

ϕ1

ϕ2

)

=

(

cos θ sin θ− sin θ cos θ

)(

ψ1

ψ2

)

(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|∂/∂R|ψ2〉 =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 point-wise B12(R)function in the vicinity of a dominant maximum (which

3

is located near the avoided crossing point Rc of the cor-responding adiabatic potentials) by the simplest Lorentzform30:

B12(R) ≈w

4(R−Rc)2 + w2(3)

with the two R-independent 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 ac-complish 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 + sin2 θU2

V2 = sin2 θU1 + cos2 θU2

V12 = sin 2θ|U1 − U2|/2 (5)

The adiabatic PECs Uabi (R) were evaluated for the

low-lying excited (1 − 3)1,3Σ+ and (1, 2)1,3Π states con-verging to the lowest three dissociation limits (see Fig. 1)in the basis of the zeroth-order (spin-averaged) electronicwavefunctions corresponding to pure (a) Hund’s couplingcase. To diminish the systematic R-depended error (firstof all, basis set superposition error) the originally calcu-lated adiabatic potentials Uab

i for the excited states werecorrected due to the semi-empirical relation2:

Ui(R) = [Uabi (R)− Uab

X (R)] + UempX (R) (6)

where the highly accurate empirical PEC UempX of the

ground X1Σ+ state was borrowed from Ref. 18.All electronic structure calculations were performed

in a wide range of internuclear distances on the den-sity 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 finite-difference method.The details of the computational procedure used can

be found elsewhere24. Briefly, the inner core shellof both potassium and rubidium atoms was replacedby energy-consistent non-empirical effective core poten-tials32 (ECP), leaving 9 outer shell (8 sub-valence plus1 valence) electrons for explicit treatment. The relevantspin-averaged and spin-orbit 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 state-averaged complete active space self-consistent field problem for all 18 electrons on the lowest(1-10)1,3Σ+ and (1-5)1,3Π electronic states taken withequal weights33. The dynamical correlation was intro-duced via the internally contracted multi-reference con-figuration interaction (MRCI) method34 which was ap-plied for only two valence electrons keeping the remain-ing 16 sub-valence electrons frozen. The l-independent

core-polarization potentials (CPPs) of both atoms (seeTable I) were employed to implicitly account for the resid-ual core-valence correlation effects35. The correspondingCPP cut-off radii of both atoms were adjusted to repro-duce the experimental fine-structure splitting of the low-est excited K(42P) and Rb(52P) states29.

TABLE I. The static dipole polarizability28 of the cation andits cut-off radii implemented in the CPP potentials of the Kand Rb atoms. All parameters in a.u..

αcore rcut−off

K 5.354 0.247Rb 9.096 0.379

The resulting MRCI wave functions were used to eval-uate the permanent dipole functions d1,2(R) of adia-batic 11Π and 21Π states as well as the corresponding11Π − 21Π transition dipole moment d12(R). The adia-batic matrix elements were transformed to the relevantdiabatic moments µ1,2(R), µ12(R) as

µ1 = cos2 θd1 + sin2 θd2 − sin 2θd12

µ2 = sin2 θd1 + cos2 θd2 + sin 2θd12

µ12 = cos 2θd12 + sin 2θ|d1 − d2|/2 (7)

Finally, we have calculated spin-orbit 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 mo-ments 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 COUPLED-CHANNEL DEPERTURBATION

ANALYSIS OF THE (1 ∼ 2)1Π COMPLEX

In the framework of the rigorous coupled-channel(CC) deperturbation model24,36,37, the non-adiabaticrovibronic energy ECC of the (1, 2)1Π complex is deter-mined 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 cou-pling matrix element V12(R) are the mass-invariant func-tions of internuclear distance R. The two-component

4

vibrational eigenfunction Φ in Eq.(9) is normalized forthe bound states as

∑

i Pi = 1, where Pi = 〈φi|φi〉(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

〈φi|1/R2|φi〉 (12)

The rovibronic energies ECC and vibrational wavefunctions φi(R) were obtained through solving the CCequations (9) by the finite-difference 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/Long-Range(MLR)40–42 function

V2(R) ≡ UMLR = [T dis2 −De] + (13)

De

[

1− uLR(R)

uLR(Re)e−β(R)yeq

p (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 en-ergy T dis

2 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 center-of-gravity)DXe = 4217.822 cm−1

and the corresponding non-relativistic energy of the D-lines 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,q

ref

Rp,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 con-strained to be ln {2De/uLR(Re)}, where De is the thewell depth and uLR is the long-range potential

uLR(R) =∑

n=6,8

DnCn

Rn(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 double-exponential/long-range (DELR) po-tential46:

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 dis

1 = DXe +EK(42P)−EK(42S) =

17241.481 cm−1 was fixed during the fit. The non-relativistic energy of the D-lines 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 pre-exponential coefficients A and B weredetermined from the conditions UDELR(Re) = 0 anddUDELR/dR|R=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 po-tentials as well as electronic coupling matrix elementwere determined simultaneously in the framework of theweighted nonlinear least-squared fitting (NLSF) proce-dure:

χ2 =

Nexp

∑

j=1

(

Eexpj − ECC

j )/σexpj

)2(22)

+

Nab

∑

j=1

(

[V ab1 (Rj)− UDELR(Rj)]/σ

abj

)2

+

Nab

∑

j=1

(

[V ab2 (Rj)− UMLR(Rj)]/σ

abj

)2

+

Nab

∑

j=1

(

[V ab12 (Rj)− V emp

12 (Rj)]/σabj

)2

5

where Eexpj denote the experimental term values of the

(1 ∼ 2)1Π complex and the σexpj -values mean their un-

certainties. The V ab1,2 and V ab

12 are the diabatic functions

evaluated by Eq.(5) at the point Rj , and σabj are their

uncertainties obtained by averaging the present and pre-ceding11 ab initio curves. The theoretical curves wereincorporated in the NLSF procedure in order to propa-gate 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, (

cm-1)

Internuclear distance, R (A)

21

FIG. 3. The ”difference-based” 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 ”difference-based” PECs obtained byEq.(6) for adiabatic B(1)1Π and 21Π states from thepresent and preceding11 ab initio calculations demon-strates overall good agreement as can be seen in Fig. 3.The most significant deviations are observed in the vicin-ity of the avoided crossing point Rc ≈ 5.36 (A). The cor-responding 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, respec-tively. The inset demonstrates that the simplest two-parameters 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 transi-tion dipole moment are given on Fig. 5a. The adiabaticfunctions d1,2(R), d12(R) were obtained during the elec-tronic structure calculations while the diabatic momentsµ1,2(R), µ12(R) were evaluated according to Eq.(7). Asfollows from the charge density of the electronic wave-functions of the twin (1, 2)1Π states19, the d1,2(R) func-tions have a ”mirror” R-dependance d1 ≈ −d2 with asharp global extremum about ±2 a.u. located near thepoint Rc. It should be noticed, that the adiabatic transi-tion moment d12 becomes zero at the same point. In con-trast to sharp adiabatic functions, their diabatic counter-parts demonstrate rather smooth R-behavior. Further-more, the absolute magnitudes of the diabatic |µ1,2(R)|functions significantly decrease at intermediate internu-clear distances.The adiabatic (1, 2)1Π − (X,A)1Σ+ transition dipole

moments are depicted on Fig. 5b along with the dia-batic 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 R-distances.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≈ 8π2

3~ǫ0〈φJ′

1 |[∆V1X ]3[µ1X ]2|φJ′

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 experimen-tal counterpart13 of 11.6 ns. It should be noted, thatthe contribution of the 11Π − A1Σ+ transition into theτ11Π-estimate 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 R-behavior 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 intermediateR-ranges 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+ non-adiabatic 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 q-factors

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. Unfortu-nately, there are no experimental q-values for comparisonso far.The resulting ab initio potential energy curves, per-

manent and transition dipole moments as well as elec-tronic, spin-orbit and angular coupling matrix elementsare given in pointwise form in the Supplementary mate-rial48.

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 11Π 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 A).(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 20-2

-1

0

1

2

3

4

2 3 4 5 6 7 8 9 10 11 12-2,0

-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 ini-tio TDM functions between the (1, 2)1Π and X,A1Σ+ states.The dashed line denotes the adiabatic dB1Π−X1Σ+ (R) func-tion 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 Eexp

v′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 mea-

surements13,14 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] lev-els.

All energies above correspond to the most abundant39K85Rb isotopologue while the 29 rovibronic termval-ues16 of 39K87Rb held in reserve for confirmation of themass-invariant 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 Eexp

v′ since the difference of theempirical data13,14 and Ref. 17 reach few reciprocal cen-

7

2 4 6 8 10 12 14-80

-70

-60

-50

-40

-30

-20

-10

0

10

20

302 4 6 8 10 12 14

-40-30-20-10

01020304050

607080

Rc

11 -b3

21 -b3

11 -23

21 -23

b)Spin

-orb

it co

uplin

g m

atrix

ele

men

ts (c

m-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 cir-cles) ab initio spin-orbit coupling matrix elements ξij(R) be-tween 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 mass-invariant parameters of the DELR(18) and MLR (13) potentials obtained for the dia-batic 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 Ta-ble IV. The resulting parameters are duplicated in non-truncated 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 Eexp

v′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 lev-els of v′1 = 4, 5 vibrational states from the CC estimates

2 4 6 8 10 12 14-0,5

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 ob-served 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 BCC

v′ (12) and empir-ical Bexp

v′ 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 termvaluesECC

v′J′ for a pair of the closely lying 11Π(v′1 = 54) ∼21Π(v′2 = 15) levels of the (1 ∼ 2)1Π complex (seeFig. 10) experimentally studied in Ref. 14. The vibra-tional numbering used above corresponds to the adiabaticrepresentation of the 11Π ∼ 21Π complex applied for theassignment of the OODRPS X1Σ+ → 11Π ∼ 21Π spec-tra14. The calculated ECC

v′J′ positions are found to be in agood agreement with their experimental counterparts. Inparticular, the minimal distance ∆CC = 4.3 cm−1 pre-dicted 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 admix-ture 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 poten-tials. 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)+52P1/2(Rb) asymptotic undergo a predissoci-

ation effect14.

Thus, raw experimental termvalues corresponding tohigh v′, J ′-levels of the (1 ∼ 2)1Π complex would be cer-tainly useful for the comprehensive deperturbation anal-ysis since the ”effective” band constants indispensably”absorb” the spin0orbit perturbation effect in the highenergy region.

TABLE II. The resulting mass-invariant parameters of theUDELR(R) potential (18) obtained for the diabatic 11Π state.

fitted

De, cm−1 2214.757

Re , A 4.3715A , cm−1 -10280.36B , cm−1 4577.51β0, A

−1 0.47730β1, A

−1 0.22714β2, A

−1 0.11574β3, A

−1 -0.04563β4, A

−1 -0.20030β5, A

−1 0.19131β6, A

−1 0.73518β7, A

−1 -0.15280β8, A

−1 -0.72517fixed

q 3Rref , A 5.6781T dis, cm−1 17241.481C6, cm

−1·A6 −2339× 105

C8, cm−1·A8 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 ”abnor-mal” 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 16800-6-4-2024615000 15200 15400 15600 15800 16000 16200 16400-8-6-4-2024

c)

Termvalues, Eexp (cm-1)

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 (

cm-1)

Eex

pvJ

- E

CC

vJ (

cm-1)

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. (b-c) The circlesdenote the OODRPS termvalues13,14 while the triangles de-note the MB data17. The bars denotes the differences betweenthe empirical terms and their estimates evaluated by the cor-responding RKR potentials.

The absorbtion intensities from the ground X1Σ+

state to a pair of adjoining levels of the (1 ∼ 2)1Π com-plex were evaluated according to the relation:

ICCk1Π−X1Σ+ ∼ |MkX|2 (25)

= |〈φJ′

k1|µ1X |χJ′′

X 〉|2 + |〈φJ′

k2|µ2X |χJ′′

X 〉|2

+ 2〈φJ′

k1|µ1X |χJ′′

X 〉〈φJ′

k2|µ2X |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 X-state (see Fig. 11) calculated with thehighly accurate empirical potential18.The rovibronic 〈φJ′

k1|µ1X |χJ′′

X 〉 and 〈φJ′

k2|µ2X |χJ′′

X 〉 ma-trix elements calculated for the X1Σ+(v′′X = 12) →11Π(v′1 = 54) ∼ 21Π(v′2 = 15) transitions are given on

Fig. 12a. It is seen that the 〈φJ′

k2|µ2X |χJ′′

X 〉 terms givea negligible contribution to the total |MkX|2 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 vx100

(cm

-1)

v2

21

Rot

atio

nal c

onst

ants

Bvx1

00 (

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 squares

on the inset denoteBexp

v′ -values calculated for the v′2 ≤ 6 levels

by the Dunham constants15.

TABLE III. The resulting mass-invariant parameters of theUMLR(R) (13) potential obtained for the diabatic 21Π state.

fitted

De, cm−1 1111.961

Re, A 5.2872β0 -0.92350β1 0.09744β2 0.09023β3 -0.71368β4 -1.08251β5 0.12502β6 0.35828

fixedq 4p 4Rref , A 6.8511T dis, cm−1 16955.169C6, cm

−1·A6 3025× 105

C8, cm−1·A8 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

m-1)

CC

FIG. 10. (a) The rovibronic termvalues ECCv′J′ predicted for

the 11Π(v′ = 54) ∼ 21Π(v′ = 15) levels under the present CCdeperturbation model. The straight lines denotes the adia-batic energies calculated as E

exp

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 ter-mvalues above.

TABLE IV. The resulting mass-invariant parameters of theelectronic 11Π ∼ 21Π coupling V

emp12 (R) function (21).

fittedβ0 114.87β1 -2.4087β2 -98.392

fixedq 3Rref , A 5.36

probability since |µ2X | ≪ |µ1X | (see Fig. 5b). Further-

more, 〈φJ′

21|µ1X |χJ′′

X 〉 matrix elements demonstrate ab-normally 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'11|

b)

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 coupled-channel treatment ofexperimental term values of the (1 ∼ 2)1Π complex. Thepresent CC deperturbation model, based on a diabaticrepresentation, provides the almost spectroscopic (exper-imental) accuracy of the approximation. The empiricalPECs and electronic coupling function, along with ab

initio spin-orbit and angular coupling matrix elements,could be utilized in further deperturbation analysis car-ried 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. 16-03-

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 34-0,2

-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| 1X|

J"X >; < J'

12| 2X|J"X >

FIG. 12. (a) The rovibronic transition matrix elements cal-culated 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 inten-sity ratios. The experimental points were digitized from Fig.8of Ref. 14.

00529a.

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