Hidden physics in molecular rovibrational spectrum

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 132 (2014) 32–37

Contents lists available at ScienceDirect

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa

Hidden physics in molecular rovibrational spectrum Weiguo Sun a,c,⇑, Yi Zhang a,b, Qunchao Fan a, Hao Feng a, Jia Fu a,b, Huidong Li a,c, Jie Ma d, Liantuan Xiao d, Suotang Jia d a

School of Physics and Chemistry, Research Center for Advanced Computation, Xihua University, Chengdu, Sichuan 610039, PR China School of Physics, Sichuan University, Chengdu, Sichuan 610065, PR China c Institute of Atomic and Molecular Physics, Sichuan University, Chengdu, Sichuan 610065, PR China d State Key Laboratory of Quantum Optics and Quantum Optics Devices, Laser Spectroscopy Laboratory, Shanxi University, Taiyuan 030006, PR China b

 An algebraic method for rotational

energies (AMr) is proposed.  The accurate rotational energies {eJ}

of 7 Li2 and NaF are obtained at first time.  The accurate rovibrational interaction energies eint tJ have been studied at first time.  The error of the well known rigid rotor rotational energies has been studied.

g r a p h i c a l a b s t r a c t Pþ 7 3 The rovibrational interaction energies eint tJ for the state a u of Li2 molecule, and indicates that the absolute values of rovibrational interactions are getting greater as either vibrational state t or rotational state J increases. The fact that all interaction energies eint tJ are negative confirms that the rovibrational interaction energies are attractive, and that it is the rovibrational coupling makes the rovibrational energy EtJ smaller than the sum EtþJ of its corresponding vibrational energy et and the rotational energy eJ and makes the rovibrational system stable

Rovibrational interaction energies cm-1

h i g h l i g h t s

0 -5 -10 -15 -20 -25 -30 -35 0

1

2

3

4

5

6

7

8

9

J

a r t i c l e

i n f o

Article history: Received 20 January 2014 Received in revised form 2 April 2014 Accepted 13 April 2014 Available online 30 April 2014 Keywords: Algebraic method Diatomic molecules Rotational energies Rovibrational spectrum Interaction energies

10

9

8

7

6

5

4

3

2

1

0

V

a b s t r a c t An algebraic method for rotational energies (AMr) is proposed to unearth the rotational spectrum {eJ} and the rovibrational interaction energies eint tJ that are hidden in the rovibrational energies EtJ. The applica1 + 7 tions to the excited electronic state a3 Rþ u of Li2 and the ground state X R of NaF molecules show that: (1) the rotational energies eJ of the lighter 7 Li2 molecule have better accuracies than the widely used rigid rotor rotational energies err J particularly for the lowest two rotational states, while the rigid rotor model produces satisfied rotational energies for the heavier NaF molecule and (2) the attractive rovibrational interaction energies eint tJ stabilize a molecular rovibrational system. Ó 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author at: School of Physics and Chemistry, Research Center for Advanced Computation, Xihua University, Chengdu, Sichuan 610039, PR China. Tel.: +86 28 87728001; fax: +86 28 87728002. E-mail addresses: [email protected], [email protected] (W. Sun). http://dx.doi.org/10.1016/j.saa.2014.04.137 1386-1425/Ó 2014 Elsevier B.V. All rights reserved.

11

Introduction The knowledge of molecular rotational, vibrational levels and rotation–vibration interactions is very important in many studies.

W. Sun et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 132 (2014) 32–37

For example, the cooling mechanism of cold molecules depends strongly on the molecular rovibrational states [1–4], the accurate knowledge of molecular rotational and vibrational levels is necessary in the studies of the product energy distribution and the channel couplings of the exit dynamics of atom–molecule collisions [1–5], the high-lying vibrational levels can provide accurate information on the long-range molecular potentials [1,6], the small rotation energy gap ensures a small molecular electric field that makes it possible to control molecular polarization with high precision and speed [7], the accurate rotational and vibrational energies are indispensable in the assignment of molecular bands, in the collisional resonance studies [8], and in the numerical computations of rotational and vibrational partition functions of statistical mechanics. There have been many experimental studies [9–15] and theoretical investigations [16–21] on the vibrational energies and rovibrational energies for many diatomic states. However, it is hardly known the molecular rotational spectrum {eJ} (as expressed in Eq. (4) below) and rovibrational interaction energies (as expressed in Eq. (5) below) experimentally and/or theoretically to our knowledge. People have used a rigid rotor approximate rotational energy err J as in Eq. (7) below in many important studies for nearly a century. The questions are: (1) can we obtain correct rotational energies {eJ} based on known experimental rovibrational information? (2) How much error does this rigid rotor approximate rotational energy have? (3) How do the rovibrational interaction energies behave both qualitatively and quantitatively? This study proposes an algebraic method for rotational energies (AMr) to obtain correct diatomic rotational spectrum based on accurate experimental rovibrational energies. The accurate rovibrational interaction energies are also obtained. Section ‘The algebraic method for rotational energy (AMr)’ describes the method, Section ‘Applications and discussions’ presents the applications and discussions, and Section ‘Summary’ summarizes this study. The algebraic method for rotational energy (AMr) The physical law that diatomic rovibrational energies satisfy can be accurately expressed as Herzberg equation or Dunham formula [22,23]

 n X  n X 1 1 EtJ ¼ Y nm ½JðJ þ 1Þm t þ ¼ BJn t þ 2 2 n;m n

ð1Þ

which can be alternatively written as

EtJ ¼ et þ eJ þ eint tJ

ð2Þ

where the vibrational energies for m = 0

et ¼

n X  1 Y n0 t þ 2 n¼0

ð3Þ

the rotational energies for n = 0

eJ ¼

X Y 0m ½JðJ þ 1Þm

ð4Þ

m>0

and the rovibrationl interaction energies

eint tJ ¼

X

Y nm ½JðJ þ 1Þm

n>0;m>0



tþ

n 1 2



  2  3  4 1 1 1 1 þ DJ t þ þ HJ t þ þ LJ t þ 2 2 2 2  5  6  7 1 1 1 þ PJ t þ þ QJ t þ þ SJ t þ þ  2 2 2

¼ BJ

tþ

ð5Þ

33

The diatomic rotational energies in Eq. (4) can be rewritten as h eJ ¼ m Y 0m ½JðJ þ 1Þm ¼ errJ þ ecd J þ eJ m>0

ð6Þ

which is equivalent to the series expansion of the non-rigid rotor rotational energy in the chapter 3 of Ref. [23]. In Eq. (6) the well known rigid rotor energy is

errJ ¼ Y 01 ½JðJ þ 1Þ ¼ Boe ½JðJ þ 1Þ

ð7Þ

the centrifugal distortion energy is 2 2 ~o ecd J ¼ Y 02 ½JðJ þ 1Þ ¼ De ½JðJ þ 1Þ

ð8Þ

and the energy contributions from other high-order physical effects is

ehJ ¼

X cd Y 0m ½JðJ þ 1Þm ¼ eJ  err J  eJ

ð9Þ

m¼3

Therefore once the rotational constants (Y0m) of a rotational system are obtained, the rotational energies eJ, the frequently used approxcd imate rotational energies err J and eJ , and the corresponding highorder rotational energies ehJ can be easily obtained.The Eq. (1) may be rewritten as an algebraic equation for a given rotational state J,

AX J ¼ EJ

ð10Þ

where the coefficient matrix element

Aik ¼



ti þ

1 2

k

k ¼ 0; 1; 2; 3; . . . ; ðn  1Þ

ð11Þ

the solution vector XJ and the energy vector EJ

1 BJ0 B BJ1 C C B C XJ ¼ B B BJ2 C A @ .. . 0

1 Et1 J B Et J C B 2 C C EJ ¼ B B Et3 J C A @ .. . 0

; nX1

ð12Þ nX1

The relationships between the energy expansion coefficients {BJn} in Eq. (1) and the rovibrational interaction coefficients ðBJ ; DJ ; HJ ; . . . ; Q J ; SJ Þ in Eq. (5) are

BJn ¼ F J þ Y n0 ;

F ¼ e; B; D; H; L; P; Q ; S; . . .

for n ¼ 0; 1; 2; . . .

ð13Þ

where Yn0 are the vibrational spectroscopic constants (including Y00) and can be obtained using the algebraic method for vibrational spectrum (AMv) [24,25]. Rotational energies eJ in Eq. (4) satisfy Eq. (13) too,

BJ0 ¼ eJ þ Y 00

ð14Þ

where the energy expansion coefficient BJ0 can be obtained by solving Eq. (10). Since each of the m experimental rovibrational energies {EtJ} may contain particular physical information and inevitable error message, if one uses all m data to solve Eq. (10), different defects embedded in these experimental data are mixed with their merits and might poison the physics included in these data during computational processes. Therefore, this only solution obtained using all m data may not be the one containing the best physics. On the other hand, one may choose n of the m(> n) data and solve the algebraic Eq. (10) of the known subset ½Eti J  using standard algebraic method to obtain a set of rotational dependent energy constants (RDECs) X J ¼ ðBJ0 ; BJ1 ; BJ2 ; . . .Þ for a certain rotational state J. There are N ¼ C nm sets of known subset ½Eti J  and will have N sets of solutions XJ’s for the given J. There must be one that best satisfy the following physical requirements [24,25]:

34

W. Sun et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 132 (2014) 32–37

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 Xm1  exp cal  DEðe; cÞ ¼ E  E   !0 t J t J t¼0 m

ð15Þ

DEtmax ;tmax 1;J ¼ Etmax J  Etmax 1;J ! small enough

ð16Þ

cal

d ðtmax ; JÞ ffi Etmax J þ cal

DE2tmax ;tmax 1;J DEtmax ;tmax 2;J  DEtmax ;tmax 1;J

ð17Þ

energy defined in Eq. (4). The rovibrationl interaction energy eint tJ defined in Eq. (5) can be easily obtained from Eq. (2), eint tJ ¼ EtJ  et  eJ . It can be seen that the rotational energies eJ in Eq. (4) may also be written as an algebraic equation

HY r ¼ er

ð20Þ

where the coefficient matrix element

cal

d ðtmax ; J þ 1Þ P d ðtmax ; JÞ 6 Dexp e ðJÞ

ð18Þ

HJm ¼ ½JðJ þ 1Þm ;

m ¼ 1; 2; 3; . . . ; K

ð21Þ

the solution vector Yr and the rotational energy vector er cal

d ðtmax ; JÞ  Etmax J 0< 61 DEtmax ;tmax 1;J

ð19Þ

where ‘‘exp’’ denotes the experimental data, and ‘‘cal’’ denotes the calculated data. For a given rotational state J that corresponds to a stable system potential, Dexp e ðJÞ is the experimental dissociation energy of this J, dcal(tmax, J) is the maximum energy that is slightly greater than the highest rovibrational energy Etmax J , and for J = 0, cal d ðtmax ; J ¼ 0Þ  Dcal is the calculated molecular dissociation e energy [24,25] of this rotational state. The physical requirements in Eqs. (15)–(19) work as a physical standard to judge the quality of the measured data and to select a best combination with the best accuracy from the data. Such determined solution X J ¼ ðBJ0 ; BJ1 ; BJ2 ; . . .Þ will be the best representation of the profound physics embedded in all known m experimental rovibrational energies {EtJ}, and is the one having minimum error combination in rotational state J of the system. The eJ(= BJ0  Y00) is the rotational

1 Y 01 C B B Y 02 C C B B Y 03 C C B C B Y 04 C Yr ¼ B C B B .. C B . C C B C B @ Y 0;k1 A Y 0k K1; 0

0

er

eJ

1

Be C B Jþi C C B B eJþk C C B Be C Jþl C ¼B C B B .. C B . C C B C B @ eJþp A

eJþq

ð22Þ

K1

Therefore, one may also solve the algebraic Eq. (20) using the algebraic method with a set of convergence requirements, and find the best representation of rotational constants Y r ¼ ðY 01 ; Y 02 ; Y 03 ; Y04 ; . . . ; Y 0k Þ of this system. The method proposed here may be called as the algebraic method for rotational energy (AMr). The algebraic method for vibrational energies (AMv) [24,25] have been successfully used to

Table 1 1 7 Experimental based rovibrational energies* EtJ for each rotational state J of the a3 Rþ ). u state of Li2 molecule (all data are in cm

t

J=0

1

2

3

4

5

6

7

8

9

10

11

32.435

33.501

35.099

37.229

39.886

43.071

46.778

51.006

55.750

61.006

66.768

1 2 3 4 5 6

31.901 90.500 142.567 188.283 227.727 260.884 287.708

90.997 143.026 188.707 228.113 261.227

91.988 143.942 189.545 228.872 261.904

288.035

288.623

93.473 145.315 190.802 230.010 262.918 289.475

95.450 147.143 192.475 231.524 264.268 290.648

97.919 149.423 194.562 233.413 265.949 292.106

100.876 152.155 197.061 235.673 267.960 293.847

104.318 155.333 199.968 238.301 270.295 295.865

108.242 158.955 203.280 241.292 272.950 298.153

112.644 163.016 206.992 244.643 275.919 300.703

117.519 167.512 211.099 248.346 279.195 303.507

122.863 172.436 215.595 252.396 282.769 306.549

7

308.145

308.418

308.908

309.640

310.590

311.800

313.236

314.891

316.755

318.839

321.095

323.540

8

322.203

322.407

322.786

323.351

324.090

325.006

326.091

327.340

328.727

330.233

331.842

333.526

9

330.214

330.342

330.596

330.970

331.463

332.058

332.734

333.476

334.223

334.965

10

333.316

0

*

These experimental based energies are obtained by shifting the term energies in Table 5 of Ref. [34] with Et=0,J=0 = 31.901. Those underlined are not given in Table 5 but generated using the spectroscopic constants in Table 4 of Ref. [34]. Those in italic bold are the ones used in the converged AMr studies.

Table 2 1 7 Rotational energies eJ, rovibrational interaction coefficients for each rotational state J, and the vibrational constants Yn0 of the a3 Rþ ). u state of Li2 molecule (all data are in cm J=

eJ

BJ

DJ

HJ

LJ

PJ

QJ

SJ

MJ

0 1 2 3 4 5 6 7 8 9 10 11

0 0.760759 1.934380 3.621046 5.902389 8.664885 12.001072 15.850432 20.321654 25.251699 30.697196 36.727525

0 0.724440 1.044524 1.211737 1.697247 1.933334 2.297516 2.610535 3.240336 3.630647 4.009270 4.663259

0 0.736932 1.016420 1.051164 1.500975 1.555599 1.718917 1.799911 2.163248 2.217509 2.226932 2.544561

0 0.373742 0.526345 0.543455 0.815636 0.844271 0.936888 0.995641 1.179476 1.212721 1.229668 1.437466

0 0.102879 0.148642 0.155813 0.243056 0.251259 0.280180 0.302310 0.350852 0.363423 0.374224 0.449380

0 1.6209E2 2.4154E2 2.6037E2 4.1890E2 4.3293E2 4.8564E2 5.3250E2 6.0224E2 6.3122E2 6.6533E2 8.2308E2

0 1.4566E3 2.2512E3 2.5173E3 4.1546E3 4.2983E3 4.8551E3 5.4128E3 5.9311E3 6.3112E3 6.8744E3 8.7714E3

0 6.9242E5 1.1173E4 1.3042E4 2.2001E4 2.2836E4 2.6009E4 2.9501E4 3.1124E4 3.3727E4 3.8437E4 5.0553E4

0 1.3456E6 2.2857E6 2.7986E6 4.8143E6 5.0265E6 5.7815E6 6.6744E6 6.7289E6 7.4432E6 9.0079E6 1.2180E5

Y00

Y10

Y20

Y30

Y40

Y50

Y60

Y70

Y80

0.330269

66.762743

5.047188

0.905665

0.252155

4.1845E2

4.0308E3

2.0513E4

4.1942E6

35

W. Sun et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 132 (2014) 32–37 Table 3 cd Rigid rotor rotational energies err J , centrifugal distortion energies eJ , high order cd rr contributions ehJ ð¼ eJ  err J  eJ Þ;the corresponding percent errors eJ % for each 7 rotational state J, and the rotational constants Y0m of the a3 Rþ state of Li2 molecule u 1 (all data but err ). J % are in cm J

errJ

ecd J

ehJ

errJ %

Y 0m

0 1 2 3 4 5 6 7 8 9 10 11

0.000000 0.551912 1.655737 3.311474 5.519123 8.278685 11.590159 15.453545 19.868844 24.836055 30.355178 36.426214

0.000000 0.000075 0.000679 0.002716 0.007543 0.016973 0.033267 0.059141 0.097764 0.152756 0.228190 0.328594

0.000000 0.208922 0.279322 0.312288 0.390809 0.403173 0.444180 0.456028 0.550574 0.568400 0.570208 0.629906

0.000000 72.547591 85.595228 91.450756 93.506600 95.542929 96.576030 97.496050 97.771784 98.353995 98.885833 99.179603

Y 01 Y 02 Y 03 Y 04 Y 05 Y 06 Y 07 Y 08 Y 09 Y 0;10 Y 0;11

|εcd | J

0.6

Energies in cm-1

4.42882E01 3.96869E02 4.82756E03 3.31157E04 1.38717E05 3.71071E07 6.45063E09 7.23246E11 5.02876E13 1.96613E15 3.29451E18

εhJ 0.4

0.2

0.0 0

1

2

3

4

5

6

7

8

9

10

11

12

J     Fig. 1. Absolute values of centrifugal distortion energies ecd J  and high-order 7 rotational energies ehJ of the a3 Rþ state of Li molecule. 2 u

7 [et] of the a3 Rþ u of Li2 molecule had been given in Ref. [34]. The similar data of the X1R+ of NaF molecule for t = 0, . . ., 9 and J = 0, . . ., 30 had been given in Ref. [35,36], and the dissociation energy De is 40,567.0 cm1 [36]. Table 1 gives the experimental based rovibrational energies EtJ for every rotational state J of the 7 a3 Rþ u state of Li2 molecule [34], and those in italic bold are the ones used in the converged AMr studies. The first column of Table 1 is the vibrational energies {et}. Table S1 in the supplemental information (SI) gives those for the state X1R+ of NaF molecule. All energies and constants in this study are in cm1. Table 2 lists the rotational energies eJ and rovibrational interaction coefficients obtained using the AMr, and Eqs. (10) and (13) for each rotational state J, and the vibrational constants Yn0 of the a3 Rþ u state of 7 Li2 molecule. The Yn0 are obtained using the AMv [24,25]. Table S2 in the SI gives the similar data for the stateX1R+of NaF molecule. Tables 2 and S2 show respectively that the rotational energies eJ become greater as the rotational quantum number J increases, and so does the absolute value of each rotational constant. It can be seen from these tables that the rovibrational interaction coefficients (BJ, DJ, HJ, LJ, . . ., ) of the rovibtational interaction energies 7 feint tJ g of the lighter molecule Li2 converge slowly while those of the heavier molecule NaF converge quickly for each rotational state. Table 3 gives the rigid rotor rotational energies err J as expressed in Eq. (7), and the centrifugal distortion energies ecd J as in Eq. (8),   cd as in Eq. (9) and the high-order contributions ehJ ¼ eJ  err J  eJ 7 for the excited state a3 Rþ u of Li2 molecule. The rigid rotor rotarr tional energies eJ have been widely used in a lot of accurate studies involving rotational motions for nearly a century. The .  Boe f¼ 1=ð2Ie Þ ¼ 1 2lR2e g of err J is the rigid rotor constant and n . o o 3 2 6 cd ~ of eJ is the centrifugal distortion contrithe De ¼ 1 2l xe Re

bution while the AMr rotational energies eJ contain other high order contributions ehJ involving much complicated physics. The

3. Applications and discussions

Boe ~o (= 0.275956 cm1) and the centrifugal distortion constant D e (= 1.88587  105 cm1) are (reduced mass) l = 3.508 amu, (equilibrium internuclear distance) Re = 4.173  108 cm, and (harmonic vibrational constant) xe = Y10 = 66.762743 cm1 respec  rr tively. The err J % ¼ 100  eJ =eJ in Table 3 is the percent ratio of the

The AMr is applied to study the pure rotational energies {eJ} and the rovibrational interaction energies feint tJ g of the excited elec1 + 7 tronic state a3 Rþ u of Li2 molecule [34] and the ground state X R of NaF molecule [35,36]. The experimental molecular dissociation 1 energy Dexp e ð¼ 333:69 cm Þ and a set of molecular rovibrational energies [EtJ; t = 0, . . ., 10; J = 0, . . ., 11] and vibrational energies

of the rotational constants Y0m. It is shown from Table 3 that the lowest two rigid rotor rotational energies err J are 72.5% and 85.6% of the correct rotational energies eJ of this system for J = 1 and 2 respectively. Since these two rotational states are the most populated in many situations, people may get more than 20% and 10% errors for J = 1 and 2

obtain a lot of correct vibrational spectra {Et} and molecular dissociation energies Dcal for many diatomic systems in our previous e studies [26–33].

Table 4 Rovibrational interaction energies

constants

used

to

evaluate

the

rigid

rotor

constant

errJ relative to the eJ. The last two columns of Table 3 give the values

1 3 þ 7 eint tJ of the a Ru state of Li2 molecule (in cm ).

eint tJ

J¼0

1

2

3

4

5

6

7

8

9

10

11

t¼0

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.219 0.277 0.294 0.345 0.383 0.405 0.438 0.495 0.556 0.615 0.756

0.325 0.463 0.550 0.681 0.797 0.897 1.022 1.180 1.350 1.536 1.823

0.402 0.669 0.868 1.107 1.341 1.578 1.846 2.147 2.468 2.844 3.339

0.561 0.965 1.315 1.721 2.111 2.507 2.962 3.470 4.008 4.640 5.443

0.669 1.261 1.798 2.397 2.984 3.587 4.265 5.025 5.855 6.810 7.889

0.820 1.640 2.402 3.234 4.058 4.910 5.860 6.925 8.106 9.468 10.945

0.962 2.043 3.073 4.176 5.278 6.424 7.695 9.115 10.710 12.577 14.576

1.207 2.593 3.918 5.341 6.763 8.240 9.880 11.722 13.794 16.303

1.392 3.123 4.788 6.556 8.339 10.202 12.259 14.569 17.216 20.486

1.580 3.692 5.740 7.895 10.082 12.372 14.898 17.759 21.054 24.789

1.850 4.380 6.848 9.431 12.060 14.832 17.887 21.348 25.401 30.130

1 2 3 4 5 6 7 8 9 10

36

W. Sun et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 132 (2014) 32–37

Rovibrational interaction energies cm-1

0 -5 -10 -15 -20 -25 -30 -35 0

1

2

3

4

5

6

7

8

9

11

10

9

7

8

J

5

6

4

3

1

2

0

V

Fig. 2. Rovibrational interaction energies

Table 5 The rovibrational energies EtJ , the sum EtþJ ð¼ et þ eJ Þ and the rovibrational interation 1 energies eint tJ for the vibrational state t ¼ 2½et¼2 ¼ 142:554 cm  of the electronic state 1 7 a 3 Rþ ). u of Li2 molecule (in cm J

EtJ

EtþJ

eint tJ

0 1 2 3 4 5 6 7 8 9 10 11

142.554 143.021 143.938 145.307 147.141 149.421 152.153 155.331 158.958 163.018 167.511 172.434

142.554 143.315 144.488 146.175 148.456 151.219 154.555 158.404 162.876 167.806 173.251 179.282

0 0.294 0.550 0.868 1.315 1.798 2.402 3.073 3.918 4.788 5.740 6.848

Note: the vibrational energy ½et¼2 ¼ 142:554 cm1  is not the one in Table 1, but from present AM calculation.

þ u

3 7 eint tJ of the a R state of Li2 molecule.

respectively if one uses err J of this system in their studies. The energy contributions ehJ from fine physics other than rigid rotor and centrifugal distortion are not trivial for all rotational quantum states of this lighter molecule, particularly for J = 1 and 2. It is also shown in Fig. 1 that although the absolute values of the centrifugal distortion rotational energies jecd J j are much smaller than the highorder effects ehJ of this molecule at lower rotational states, they rise sharply as J increases. Table S3 in the SI shows the rigid rotor energies err J , the centrifh ugal distortion energies ecd J , the high order contributions eJ and the 1 + percent ratio err % for the ground state X R of NaF molecule. It is J seen that the rigid rotor energies err J well represent the rotational energies eJ of this heavier molecule for all rotational states, and that the centrifugal distortion energy ecd J is only 0.003% of eJ at J = 3 and 0.25% at J = 30. These results agree with the well known facts that all non-rigid rotor effects of the rotational motion of a heavier system can be neglected while those in a lighter molecule like 7 Li2 play notable role particularly at lower rotational states.

Rovibrational interaction energies cm-1

0 -50 -100 -150 -200 -250 -300 -350 -400 -450 0

25

50

75 100

V

125 150

175

Fig. 3. Rovibrational interaction energies

30 int tJ

e

15

20

25

10

J 1

+

of the X R state of NaF molecule.

5

0

W. Sun et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 132 (2014) 32–37

Table 4 and Fig. 2 show the rovibrational interaction energies 3 þ 7 eint tJ obtained using Eq. (2) for the state a Ru of Li2 molecule, and

indicate that the absolute values of rovibrational interactions are getting greater as either vibrational state t or rotational state J increases. The fact that all interaction energies eint tJ are negative confirms that the rovibrational interaction energies are attractive, and that it is the rovibrational coupling makes the rovibrational energy EtJ smaller than the sum Et+J of its corresponding vibrational energy et and the rotational energy eJ as shown in Table 5, and makes the rovibrational system stable. Table S4 in the SI shows the similar information for the state X1R+ of NaF molecule and the similar properties as those in the lighter molecule 7 Li2 . Fig. 3 and Table S5 demonstrate that the rovibrational interaction energies eint tJ of this heavier NaF are also attractive. To further analyze the qualities of the AMr studies, the AMr 1 + rovibrational energies fEAMr tJ g of the X R state of NaF molecule 7 are listed in Table S6, and those of the a3 Rþ u state of Li2 molecule are given in Table S7. It can be seen from Tables S1 and S6 that the agreement between experimental rovibrational energies and AMr ones is until the 5th or 6th significant digit for all energies and the largest difference is 0.042 cm1 with 8.8  104% error in the v ¼ 9; J ¼ 1 state of NaF  X1R+ and, and from Tables 1 and S7 that the agreement is until the 4th or 5th significant digit for all EtJ and the largest difference is 0.024 cm1 with 7.2  103% error in the v ¼ 10; J ¼ 0 state of 7 Li2  a3 Rþu respectively. Summary This study analyzes the microstructure of the energy expansion that best represents the physical nature of the rovibrational energies EtJ of a diatomic system, and proposes an algebraic method for rotational energies (AMr) to excavate the rotational spectrum {eJ} and the rovibrational interaction energies eint tJ that are hidden in the accurate experimental rovibrational energies EtJ. The rotational 7 energies eJ of the electronic state a3 Rþ u of the lighter Li2 molecule have better accuracies than the widely used rigid rotor rotational energies err J in Eq. (7) particularly for the lowest two rotational states, while the rigid rotor model produces satisfied rotational energies for the X1R+ state of the heavier NaF molecule. The numerical results of this study agree with the well known facts that all non-rigid rotor effects may play important role in the rotational motion of a lighter diatomics but may be neglected in a heavier molecule. Both examples show that the rovibrational interaction energies eint tJ are physically attractive, and stabilize these diatomic rovibrational systems. Acknowledgements This study is supported by the Chinese National Natural Science Foundation (Grant Numbers 11074204, 11174236, 11204244, 61008012 and 20101401120004) and the Science Foundation of

37

the Chinese Educational Ministry (Grant Number 201001811 10085). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.saa.2014.04.137. References [1] R. Wynar, R.S. Freeland, D.J. Han, C. Ryu, D.J. Heinzen, Science 287 (2000) 1016–1019. [2] J.G. Danzl, E. Haller, M. Gustavsson, M.J. Mark, R. Hart, N. Bouloufa, O. Dulieu, H. Ritsch, H.C. Nägerl, Science 321 (2008) 1062–1066. [3] M. Viteau, A. Chotia, M. Allegrini, N. Bouloufa, O. Dulieu, D. Comparat, P. Pillet, Science 321 (2008) 232–234. [4] S. Ospelkaus, K.K. Ni, D. Wang, M.H.G. de Miranda, B. Neyenhuis, G. Quéméner, P.S. Julienne, J.L. Bohn, D.S. Jin, J. Ye, Science 327 (2010) 853–857. [5] K. Bolton, H.B. Schlegel, W.L. Hase, K. Song, Phys. Chem. Chem. Phys. 1 (1999) 999–1011. [6] B.J. Xiao, S. Kado, S. Kajita, D. Yamasaki, Plasma Phys. Contr. Fusion 46 (2004) 653–668. [7] D. DeMille, E.R. Hudson, Nat. Phys. 4 (2008) 911–912. [8] C. Puzzarini, M.P. de Lara-Castells, R. Tarroni, P. Palmieri, J. Domaison, Phys. Chem. Chem. Phys. 1 (1999) 3955–3960. [9] A.J. Ross, P. Crozet, C. Linton, F. Martin, I. Russier, A. Yiannopoulou, J. Mol. Spec. 191 (1998) 28–31. [10] Y.M. Liu, H.M. Chen, J. Li, D.Y. Chen, L. Li, R.W. Field, J. Mol. Spectrosc. 192 (1998) 32–40. [11] J.Y. Seto, Z. Morbi, F. Charron, S.K. Lee, P.F. Bernath, R.J. Le Roy, J. Chem. Phys. 110 (1999) 11756–11767. [12] V.S. Ivanov, V.B. Sovkov, L. Li, A.M. Lyyra, G. Lazarov, J. Huennekens, J. Mol. Spectrosc. 194 (1999) 147–155. [13] S. Motohiro, S. Nakajima, T. Ishiwata, J. Mol. Spectrosc. 212 (2002) 194–202. [14] D. Li, F. Xie, L. Li, A. Lazoudis, A.M. Lyyra, J. Mol. Spectrosc. 246 (2007) 180–186. [15] A. Grochola, J. Szczepkowski, W. Jastrzebski, P. Kowalczyk, Chem. Phys. Lett. 535 (2012) 17–20. [16] S. Kotochigova, E. Tiesinga, P.S. Julienne, New J. Phys. 11 (2009) 055043– 055057. [17] Y.M. Liu, J. Li, D.Y. Chen, L. Li, J. Chem. Phys. 111 (1999) 3494–3497. [18] J.A. Coxon, P.G. Hajigeorgiou, J. Chem. Phys. 132 (2010) 094105. [19] Y. Huang, R.J. Le Roy, J. Chem. Phys. 119 (2003) 7398–7416. [20] R.H. Skelton, H. Li, C.D. Boone, R.J. Le Roy, P.F. Bernath, C. Focsa, B. Pinchemel, J. Mol. Spectrosc. 219 (2003) 1–12. [21] A. Shayesteh, R.J. Le Roy, T.D. Varberg, P.F. Bernath, J. Mol. Spectrosc. 237 (2006) 87–96. [22] J.L. Dunham, Phys. Rev. 41 (1932) 721–731. [23] G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules, Van Nostrand, New York, 1953. [24] W.G. Sun, S.L. Hou, H. Feng, W.Y. Ren, J. Mol. Spectrosc. 215 (2002) 93–105. [25] Q.C. Fan, W.G. Sun, Spectrochim. Acta Part A 72 (2009) 298–305. [26] W.G. Sun, W.Y. Ren, S.L. Hou, Mol. Phys. 103 (2005) 2335–2345. [27] W.G. Sun, X.Y. Liu, Y.J. Wang, Y. Zhan, Q.C. Fan, Prog. Phys. 27 (2007) 151–200. [28] W.Y. Ren, W.G. Sun, Phys. Sin. 54 (2005) 594–605. [29] W.G. Sun, X.Y. Liu, Y.J. Wang, Y. Zhan, Q.C. Fan, Front. Phys. China 3 (2008) 382–413. [30] Q.C. Fan, W.G. Sun, H. Feng, Spectrochim. Acta Part A 74 (2009) 911–916. [31] Q.C. Fan, H. Feng, W.G. Sun, Acta Phys. Sin. 59 (2010) 203–209. [32] Q.C. Fan, W.G. Sun, H.D. Li, H. Feng, Phys. Scripta 83 (2011) 025304. [33] Y. Zhang, W.G. Sun, J. Fu, Q.C. Fan, J. Ma, L.T. Xiao, S.T. Jia, H. Feng, H.D. Li, Spectrochim. Acta Part A 117 (2014) 442–448. [34] C. Linton, F. Martin, A.J. Ross, I. Russier, P. Crozet, A. Yannopoulou, L. Li, A.M. Lyyra, J. Mol. Spectrosc. 196 (1999) 20–28. [35] A. Muntianu, B. Guo, P.F. Bernath, J. Mol. Spectrosc. 176 (1996) 274–279. [36] J.A. Coxon, P.G. Hajigeorgiou, Chem. Phys. 167 (1992) 327–340.