Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 99 (2012) 57–61

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Spectrochimica Acta Part A: Molecular andBiomolecular Spectroscopy

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Analytical potential energy functions and spectroscopic properties for theground and low-lying excited states of KRb

Kun Chen, Chuan-Lu Yang ⇑, Mei-Shan Wang, Xiao-Guang Ma, Wen-Wang LiuSchool of Physics and Optoelectronics Engineering, Ludong University, Yantai 264025, People’s Republic of China

h i g h l i g h t s

" The PECs and APEFs of the X1R+ and13R+ and 13G of KRb molecule areobtained.

" The spectroscopic parameters De, Re,Be, xe, xexe, xeye, etc. aredetermined.

" The obtained vibrational energylevels correspond with theexperimental data.

1386-1425/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.saa.2012.08.088

⇑ Corresponding author. Tel./fax: +86 535 6672870E-mail address: yangchuanlu@126.com (C.-L. Yang

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Article history:Received 27 June 2012Received in revised form 20 August 2012Accepted 31 August 2012Available online 8 September 2012

Keywords:Multi-reference interaction configurationPotential energy curveSpectroscopic parametersVibrational energy level

a b s t r a c t

The potential energy curves (PECs) of the ground state X1R+ and two low-lying excited states 13R+ and13G of KRb molecule have been calculated using the multireference configuration interaction methodand the effective core potential basis set. The PECs are fitted into analytical potential energy functions(APEFs) using the Morse long-range potential. The spectroscopic parameters for the states are determinedusing the analytical derivatives of APEFs. The vibrational energy levels have been calculated by solvingthe radial Schrödinger equation of nuclear motion based on the APEFs, and compared with the theoreticaland experimental works available at present.

� 2012 Elsevier B.V. All rights reserved.

Introduction

The NaK, NaRb, KRb molecule is the most extensively studiedamong the mixed alkali-metal dimers. A number of experimentaland theoretical investigations have been performed on the mixedalkali-metal dimers during the past several decades. Many spectro-scopic experiments were reported for the ground and low-lying ex-cited states of NaK [1,2], NaRb [3–5] and RbCs [6–11]. The quantumchemical calculations have been reported for the NaK [1,12]. ForKRb, Ross et al. [13] analyzed the A1R+–X1R+ system of KRb usinga titanium-doped sapphire laser. The rotational and vibrational

ll rights reserved.

.).

constants covering the range 0 6 v 6 44 and J < 141 are reportedfor the ground state X1R+ in 1990. Negative ion photoelectronspectra were reported for the heteronuclear alkali dimer and tri-mer anions (NaK�, KRb�, KCs�, RbCs�, Na2K�, and K2Cs�) at488 nm by Eaton et al. [14] in 1992. For 39K85Rb molecule, the highresolution spectra of the 11G and 21G transitions have been mea-sured with the technique of Doppler-free optical–optical doubleresonance polarization spectroscopy by Kasahara et al. [15] in1999. The dissociation energies of the X1R+ state have been deter-mined to be 4217.4 ± 0.8 cm�1. Pashov et al. [16] presented a com-prehensive study of the electronic states at the 4s + 5s asymptoteand reported a more accurate one as 4217.815 ± 010 cm�1 in2007. High resolution spectra of the A1R+ ? X1R+ system of theKRb molecule, obtained after excitation using a titanium-dopedsapphire laser, were recorded on a Connes-type Fourier transform

58 K. Chen et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 99 (2012) 57–61

interferometer by Amiot et al. [17] in 1999. Then the molecularconstants of the X1R+ state are determined from the first 88 vibra-tional levels. Abundant spectroscopic data on the a 3R+ state werecollected by Fourier-transform spectroscopy.

Theoretical investigations are also performed for KRb. Yiannop-oulou et al. [18] calculated the molecular spectroscopic constantsxe and Re for 13P of KRb in 1996. Leininger et al. [19] investigatedthe ground and the two first 1P excited states of the KRb dimer usingthe multireference configuration interaction (MRCI) [20,21] in 1997,obtained De, Re and xe, respectively. Park et al. [22] also performedMRCI calculations for the molecule in 2000. However, the potentialenergy curves (PECs) at large internuclear distance have not been re-ported up to date. Although the analytical potential energy functions(APEFs) of the states based on spectroscopic experiments has beenobtained, it is a combination of piecewise function. This type of APE-Fs often results discontinuous derivative, which is unfavorable tofurther study, such as constructing potential energy surface formany-body system and performing dynamics calculations.

In this paper, we increase the significant figures of the total en-ergy by modifying and re-compiling the MOLPRO codes [23]. Then,we calculated the PECs for the ground (X1R+) and two low-lyingexcited states (13R+, 13G) of KRb in a range of large internucleardistance, i.e., 2.5–52.45 Å. We fit the PECs into APEFs with a non-piecewise function suggested by Le Roy et al. [24,25]. The spectro-scopic parameters are determined using the derivatives of APEF,and the vibrational levels are calculated by solving Schrödingerequation based on the APEF.

6000

8000

10000

12000

14000

16000

18000

20000

al E

nerg

y(

)

cm-1

X1Σ+

13Σ+

13Π

Theoretical details

The PECs for the ground and two low-lying excited states of KRbare calculated by using high level ab initio MRCI method. Theeffective-core-potential (ECP) basis set for K atom is ECP10MDFwhich describes the inner 10 core electrons with pseudopotentialand the 3s23p64s1 valence electrons with basis set(11s11p5d3f) ? [8s8p5d3f] [26]. For Rb atom, we use the ECP basisset ECP28MDF which describes the inner 28 core electrons withpseudopotential and the other electrons with basis set(13s10p5d3f) ? [8s7p5d3f]. MOLPRO package [23] only usesAbelian point group symmetry, here C2v point group is usedthrough the CASSCF/MRCI calculations. The active space for theCASSCF calculations is 2 electrons in 8 orbitals (4a1, 2b1, 2b2,0a2). However, all the doubly occupied orbitals in all configura-tions are optimized in CASSCF and MRCI calculations. At the sametime, we have performed contrastive calculations without optimizingthe doubly occupied orbitals. The results show that the optimiza-tion for the doubly occupied orbitals can significantly improvethe calculational spectroscopic properties. The energies of PECsare calculated in the range of internuclear distances from 2.50 to52.45 Å with a step of 0.05 Å. APEFs are deduced by using theMorse long-range (MLR) potential function suggested by Le Roy[24,25] and the non-linear least-square fitting method. The spec-troscopic parameters are determined from the derivatives of APEF.The vibrational energy levels calculated by solving the Schrödingerequation of nuclear motion with LEVEL program package [27].

0 10 20 30 40 50 60

-6000

-4000

-2000

0

2000

4000

Tot

R(Angstom)

Fig. 1. PECs of the ground and two low-lying excited states of KRb.

Expression for analytical potential energy function

Many suggested potential functions in the literature includingno long-range effect cannot accurately reproduce the PECs of thepresent states because they involve large internuclear distance.Fortunately, MLR was designed to reproduce long-range behaviorof PEC, has been proposed by Le Roy [24,25]. Therefore, MLR is usedto fit the present PECs to analytical potential function. MLR poten-tial has the general form

VMLRðRÞ ¼ De 1� uLRðRÞuLRðReÞ

e�UMLRðRÞYP ðRÞ� �2

ð1Þ

De is the dissociation energy and Re is the equilibrium internucleardistance. The desired long-range behavior is defined by

uLRðRÞ ¼Cn

Rn þCm

Rm þCk

Rk; ð2Þ

In the fitting process, Cn, Cm and Ck are fixed as those used by Pashovet al. [16] to keep their physical significance. uLR (R) is the value ofthis function at Re, and

ypðRÞ � ðRp � RpeÞ=ðR

p þ RpeÞ ð3Þ

is a dimensionless radial expansion variable. This function implic-itly incorporates the two leading inverse power terms in the long-range potential, since at long range it takes on the form

VðRÞ � De � uLRðRÞ ¼ De �Cn

Rn� Cm

Rm� Ck

Rk: ð4Þ

The power p in the definition of yp (R) is typically a small positiveinteger chosen to minimize the possibility of irregular potentialfunction behavior at long-range [28–31]. However, if the potentialis to achieve the long-range behavior, necessarily p > (m�n). Theexponent coefficient is a polynomial in yp(R) which is constrainedto asymptotically approach a specified limiting value of

U1 ¼ ln 2De=uLRðReÞf g ¼ ln 2De= Cn=ðReÞn þ Cm=ðReÞm� �� �

ð5Þ

and is expressed in the form

UMLRðRÞ ¼ ½1� ypðRÞ�XN

i¼0

UiypðRÞi þ ypðRÞU1 ð6Þ

While there is only a single set of Ui coefficients, the parameters Ui

is determined by fitting.The root of mean square (RMS) error can be used to assess

quantitatively the quality of the fitting process. RMS can be calcu-lated with

RMS ¼ 1N

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXN

i¼1

ðVAPEF � Vab initioÞ2vuut ð7Þ

Where VAPEF and Vab initio are energies given by the fitted and ab ini-tio calculations, respectively. N is the number of points (N is 1000 inthe present paper).

Table 1The parameters of MLR potential energy functions for the ground and two low-lying states of KRb.

X1R+ 13R+ 13P

U1 �0.2943593345369 � 101 �0.5906676444932 � 101 �0.5433427331726 � 101

U2 �0.8264315075157 0.2677866157253 � 101 �0.3068533567163 � 101

U3 �0.2057353497547 � 101 �0.7588447299946 0.1185656577180 � 101

U4 0.3347272218471 � 101 0.1956139712787 � 101 0.1790974584659 � 102

U5 0.1422243396219 � 102 0.3655753032782 � 101 �0.4990127570037 � 102

U6 �0.2759762316936 � 102 �0.7067866883244 � 102 �0.2797256557265 � 103

U7 �0.1177647635541 � 103 �0.3807306800468 � 103 0.2997086314759 � 103

U8 0.1170274370812 � 103 �0.3405113161358 � 103 0.2144063759054 � 104

U9 0.5250048237286 � 103 0.1352980033889 � 104 �0.1212932395801 � 104

U10 �0.2905483434834 � 103 0.2453095812471 � 104 �0.9636245592753 � 104

U11 �0.1381144207976 � 103 �0.1682239295692 � 104 0.4082754539379 � 104

U12 0.4211671125153 � 103 �0.6738419534153 � 104 0.2570704375620 � 105

U13 0.2131452714752 � 104 �0.5387032576314 � 103 �0.1159985916843 � 105

U14 �0.3271326857177 � 103 0.9553203925686 � 104 �0.3799196684758 � 105

U15 �0.1782264044283 � 104 0.1723896721541 � 104 0.2121922344501 � 105

U16 0.1089582514821 � 103 �0.6067744286498 � 104 0.2403963792106 � 105

U17 0.6269387817737 � 103 0.2466056437639 � 103 �0.1685814810647 � 105

p 0.4015629089859 � 101 0.1313376772864 � 101 0.2527310853656 � 101

De (cm�1) 0.4128752723820 � 104 0.2377023971274 � 103 0.6681863147626 � 104

Re (Å) 0.4093267300226 � 101 0.5920266013559 � 101 0.4081094105119 � 101

C6 (cm�1 Å6) 0.2072097000000 � 108 0.2072097000000 � 108 0.2072097000000 � 108

C8 (cm�1 Å8) 0.6509487000000 � 109 0.6509487000000 � 109 0.6509487000000 � 109

C10 (cm�1 Å10) 0.2575245000000 � 1011 0.2575245000000 � 1011 0.2575245000000 � 1011

RMS (cm�1) 0.40 0.09 0.95

K. Chen et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 99 (2012) 57–61 59

After obtaining the APEFs of the states, the quadratic, cubic, andquartic force constants can be easily calculated. Then, the spectro-scopic parameters are determined with the formula given by Dun-ham et al. [32].

Table 2Spectroscopic parameter for the ground and two low-lying excited states of KRb.

KRb X1R+ 13R+ 13P

Results and discussion

PECs and APEFs

The obtained PECs for the considered states of KRb are plottedin Fig. 1. The fitted parameters of MLR as well as RMSs for thestates are shown in Table 1. In the process of fitting APEF, we keepthe C6, C8 and C10 as those used by Pashov et al. [16]. From the verysmall RMS shown in Table 1 we can conclude that the present APEFperfectly reproduce the PEC.

De (cm�1) 4128.752 237.702 6681.8633790.806a,* – –4217.400b – –4113.427c,* 241.966c,* 7017.023c,*

4008.575d,* – –4217.815g – –

Re (Å) 4.0933 5.9203 4.08114.07e – –4.09c 5.94c 4.06c

4.09d – 4.26f

xe (cm�1) 75.3947 17.9308 65.973076e – 6976c 18c 78c

76d – –Be (cm�1) 3.7505 � 10�2 1.7931 � 10�2 3.7732 � 10�2

ae (cm�1) �1.9253 � 10�4 �3.6562 � 10�4 �1.0952 � 10�4

xexe (cm�1) �0.4319 �0.3302 0.4737xeye (cm�1) 3.8827 � 10�2 2.8989 � 10�4 �2.8670 � 10�2

xeze (cm�1) �3.7556 � 10�2 �1.1430 � 10�6 8.3031 � 10�4

Fe (cm�1) 1.0922 � 10�14 �4.3218 � 10�13 7.4818 � 10�14

He (cm�1) �1.2876 � 10�19 �7.8503 � 10�18 7.1110 � 10�19

Drot (cm�1) �3.7121 � 10�8 �7.9319 � 10�8 �4.9434 � 10�8

a Ref. [14].b Ref. [15].c Ref. [22].d Ref. [19].e Ref. [13].f Ref. [18].g Ref. [16].

* The original value in eV.

Spectroscopy parameters

To understand the three states completely, we calculate thespectroscopic parameters and compare them with experimentaldata as well as other theoretical results available in the literature.The spectroscopic parameters of the ground state X1R+ and twolow-lying excited states 13R+,13G of KRb based on APEFs, togetherwith previous experimental [13,16] and theoretical [18,22] data,are collected in Table 2.

For the ground state X1R+ of KRb, the present spectroscopicparameters are in excellent agreement with the previous experi-mental results. Previous theoretical calculations [19,22] only re-ported De, Re and xe. The present De value of 4128.752 cm�1 is alittle better than the theoretical 4113.427 cm�1 of Park et al. [22]compared with the experimental 4217.400 cm�1 by Kasaharaet al. [15].The present Re and xe are close to previous theoreticaland experimental values [13–15,19,22]. We also noticed that theRe and xe of Leininger et al. [19] and Park et al. [22] are a little bet-ter than ours because their calculations focus on the equilibriumposition and the two physical quantities are primarily determinedby the properties of equilibrium position. On the other hand, ourAPEF can describe the large internuclear distance or higher orderforce constants. This will inevitably induce somewhat change ofthe physical quantities based on the information of equilibrium

position. No theoretical or experimental results in the literaturecan be used to compare with the other spectroscopic parametersreported in the present paper. Therefore, they can be used as aguide for future investigation.

We also determined the spectroscopic parameters for the low-lying excited states 13R+ and 13P of KRb. Table 2 shows our weDe, Re and xe for 13R+ are close to the reported data of Park et al.in the literature. For 13P state, our Re is close to that of Parket al. [22] but xe close to that of Yiannopoulou et al. [18]. Unfortu-nately, no more theoretical or experimental results for the twostate are found to be compared with. Therefore, more investiga-

Table 3Theoretical and experimental values vibrational levels (J = 0) of the for KRb(X1R+).

m Present Expt [17] m Present Expt [17]

0 37.5366 37.850 44 2862.9442 2865.5411 112.3429 113.236 45 2913.5263 2916.2602 186.5462 188.158 46 2963.3388 2966.2483 260.2490 262.614 47 3012.3684 3015.4934 333.5123 336.601 48 3060.6008 3063.9835 406.3698 410.117 49 3108.0217 3111.7046 478.8369 483.159 50 3154.6161 3158.6447 550.9175 555.725 51 3200.3684 3204.7908 622.6080 627.812 52 3245.2625 3250.1269 693.9006 699.416 53 3289.2816 3294.64010 764.7846 770.534 54 3332.4084 3338.31511 835.2483 841.163 55 3374.6247 3381.13712 905.2796 911.298 56 3415.9118 3423.09013 974.8662 980.938 57 3456.2502 3464.15814 1043.9969 1050.077 58 3495.6199 3504.32415 1112.6607 1118.712 59 3534.0002 3543.57016 1180.8477 1186.839 60 3571.3697 3581.88117 1248.5487 1254.454 61 3607.7068 3619.23618 1315.7550 1321.551 62 3642.9891 3655.61919 1382.4587 1388.128 63 3677.1940 3691.01020 1448.6522 1454.178 64 3710.2987 3725.39121 1514.3283 1519.697 65 3742.2801 3758.74022 1579.4799 1584.680 66 3773.2827 3773.115023 1644.0998 1649.121 67 3802.9494 3802.780124 1708.1807 1713.016 68 3831.4205 3831.252425 1771.7155 1776.357 69 3858.6723 3858.508926 1834.6962 1839.141 70 3884.6812 3884.526627 1897.1151 1901.359 71 3909.4240 3909.282828 1958.9638 1963.006 72 3932.8782 3932.755229 2020.2336 2024.076 73 3955.0228 3954.921630 2080.9154 2084.561 74 3975.8383 3975.760031 2140.9999 2144.455 75 3995.3071 3995.249132 2200.4774 2203.749 76 4013.4150 4013.368333 2259.3377 2262.437 77 4030.1479 4030.098134 2317.5706 2320.511 78 4045.4812 4045.421235 2375.1654 2377.962 79 4059.3766 4059.323736 2432.1113 2434.783 80 4071.8467 4071.796837 2488.3972 2490.964 81 4082.9418 4082.839938 2544.0117 2546.497 82 4092.6553 4092.464239 2598.9435 2601.372 83 4101.0170 4100.697440 2653.1808 2655.580 84 4108.0426 4107.589241 2706.7117 2709.111 85 4113.4995 4113.216242 2759.5242 2761.956 86 4117.4260 4117.684643 2811.6059 2814.103 87 4121.0419 4121.1241

Table 5Theoretical al values vibrational levels (J = 0) of the low-lying excited state 13P forKRb (in cm�1).

v E v E v E v E

0 32.7105 36 2477.5613 72 4572.8068 108 6066.87161 99.6917 37 2541.5048 73 4623.9140 109 6096.10032 167.3890 38 2605.1546 74 4674.5636 110 6124.53463 235.6694 39 2668.5094 75 4724.7488 111 6152.16674 304.4185 40 2731.5677 76 4774.4625 112 6178.98955 373.5381 41 2794.3275 77 4823.6979 113 6204.99766 442.9431 42 2856.7871 78 4872.4478 114 6230.18637 512.5597 43 2918.9443 79 4920.7048 115 6254.55318 582.3236 44 2980.7966 80 4968.4617 116 6278.09679 652.1783 45 3042.3416 81 5015.7108 117 6300.817710 722.0744 46 3103.5763 82 5062.4445 118 6322.718911 791.9685 47 3164.4978 83 5108.6550 119 6343.804612 861.8220 48 3225.1027 84 5154.3342 120 6364.081613 931.6012 49 3285.3876 85 5199.4740 121 6383.558414 1001.2762 50 3345.3486 86 5244.0659 122 6402.245215 1070.8207 51 3404.9819 87 5288.1014 123 6420.154416 1140.2116 52 3464.2832 88 5331.5717 124 6437.299417 1209.4285 53 3523.2483 89 5374.4677 125 6453.695118 1278.4535 54 3581.8727 90 5416.7803 126 6469.357519 1347.2711 55 3640.1515 91 5458.4999 127 6484.302820 1415.8679 56 3698.0800 92 5499.6169 128 6498.547921 1484.2321 57 3755.6531 93 5540.1214 129 6512.109722 1552.3537 58 3812.8656 94 5580.0032 130 6525.004923 1620.2241 59 3869.7123 95 5619.2519 131 6537.249624 1687.8362 60 3926.1876 96 5657.8570 132 6548.859625 1755.1837 61 3982.2861 97 5695.8076 133 6559.850226 1822.2616 62 4038.0021 98 5733.0928 134 6570.235927 1889.0656 63 4093.3298 99 5769.7014 135 6580.030828 1955.5923 64 4148.2632 100 5805.6222 136 6589.248729 2021.8388 65 4202.7966 101 5840.8438 137 6597.903230 2087.8028 66 4256.9237 102 5875.3548 138 6606.007931 2153.4822 67 4310.6385 103 5909.1439 139 6613.576432 2218.8755 68 4363.9348 104 5942.1997 140 6620.610033 2283.9814 69 4416.8062 105 5974.5112 141 6627.013734 2348.7984 70 4469.2463 106 6006.067235 2413.3254 71 4521.2487 107 6036.8574

60 K. Chen et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 99 (2012) 57–61

tions are expected to understand the spectroscopic propertiescompletely.

Vibrational and rotational energy levels

To test the APEFs more widely, we solve the one dimensionalSchrödinger equation of nuclear motion by using the obtained APE-Fs. The calculations are carried out with the LEVEL7.7 programpackage [27]. The found vibrational energy levels are 88 for X1R+

Table 4Theoretical al values vibrational levels(J = 0)of the low-lying excited state 13R+ forKRb (in cm�1).

v E v E

0 8.4104 12 162.23151 24.8368 13 170.77872 40.6046 14 178.66333 55.7154 15 185.88454 70.1704 16 192.44315 83.9707 17 198.34176 97.1167 18 203.58587 109.6087 19 208.18478 121.4463 20 212.15309 132.6287 21 215.511310 143.1548 22 218.275711 153.0229

state, 22 for 13R+ state and 141 for 13G state. Table 3–5 presentthe vibrational levels for X1R+, 13R+ and 13G state, respectively.The vibrational levels of the X1R+ state experimentally detectedby Amiot et al. [16] are the same as ours, and they are also pre-sented in Table 3. In order to exactly appreciate the levels, we havecalculated the percent error between the experimental values andours for every level. The maximum error is only 0.91% while theaverage percent error is 0.15%. This shows that our APEF for theground state X1R+ of KRb molecule can provide accurate vibra-tional energy levels. It is also implied that the APEF reproducesthe interaction between K and Rb correctly and can be safety touse to more investigations. For the vibrational energy levels ofthe low-lying excited states 13R+ and 13G, there is no experimentalor theoretical report available to compare with. Considering the re-sults for the ground is in good agreement with the experimentalvalues, we hope that the present predicted result for states 13R+

and 13G could be a helpful reference to further study.

Conclusion

The PECs of the ground state X1R+ and two low-lying excitedstates 13R+, 13P of KRb molecule have been obtained and fittedto APEFs using MLR potential function. The present spectroscopicparameters for the three state based on the new APEFs are in goodagreement with the experimental data available at present. More-over, the vibrational energy levels of the ground state determinedfrom the present APEF are also in good agreement with the exper-imental values in all the considered levels. It implies that the pres-ent APEF can accurately reproduce the interaction energy betweenK and Rb.

K. Chen et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 99 (2012) 57–61 61

Acknowledgment

This work was supported by the National Science Foundation ofChina under Grant Nos. NSFC-11174117 and NSFC-10974078.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.saa.2012.08.088.

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