Computational and Theoretical Chemistry 963 (2011) 130–134

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Computational and Theoretical Chemistry

journal homepage: www.elsevier .com/locate /comptc

A theoretical study of the accurate analytic potential energy curveand spectroscopic properties for AlF (X1R+)

Jun Zhao a,⇑, Hui Zeng a, Zhenghe Zhu b

a School of Physical Science and Technology, Yangtze University, Jingzhou 434023, Chinab Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China

a r t i c l e i n f o

Article history:Received 9 August 2010Received in revised form 10 October 2010Accepted 10 October 2010Available online 16 October 2010

Keywords:Potential energy functionDissociation energySpectroscopic constantsCCSD(T)AlF

2210-271X/$ - see front matter � 2010 Elsevier B.V.doi:10.1016/j.comptc.2010.10.013

⇑ Corresponding author.E-mail address: zhaojun@yangtzeu.edu.cn (J. Zhao

a b s t r a c t

In this paper, the energy, equilibrium geometry and harmonic frequency of the ground electronic stateX1R+ of AlF have been calculated utilizing two quantum chemical methods (CCSD(T) and QCISD(T)) withfour different basis sets (cc-pVQZ, cc-pV5Z, 6-311+G(3df) and 6-311G(3df)). Comparing various compu-tational results mentioned above with the experimental values, it can be concluded that reliable equilib-rium geometry calculations can be obtained at CCSD(T)/cc-pVQZ computational level of AlF (X1R+)molecule. The whole potential curves for the ground electronic state are further scanned using theCCSD(T)/cc-pVQZ method. The potential energy functions and relevant spectroscopic constants of thisstate are then obtained by least square fitting to the Murrell–Sorbie function and the modified Mur-rell–Sorbie+c6 function, respectively. It is shown that the Murrell–Sorbie function and the modified Mur-rell–Sorbie function are both very suitable for reproducing the accurate PEC of AlF (X1R+). Besides, ourcalculations are also more accurate than other theoretical results.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Since quantitative molecular spectroscopic knowledge can bederived from the molecular analytic potential energy function(APEF), one of the most important problems in atomic and molec-ular physics is to obtain reliable physical models of bonding poten-tials so as to thoroughly understand the molecular spectroscopicproperties [1]. Thus, it is not a surprise that many attempts havebeen focused on the ‘‘excellent” APEF, especially for diatomic mol-ecules, whose spectroscopic parameters are often of fundamentalimportance in astrophysics and other fields. Moreover, becausethe APEF only depends on the internuclear separation R, the Schro-dinger equation is separable. If insignificant effects such as mag-netic interactions are neglected, the diatomic APEF can provideall of the molecular properties, including the spectroscopic param-eters and etc. [2]. Since the relatively slow nuclear motion is verysensitive to the details of the APEF [3,4], accurate diatomic APEFis of the essence in understanding collision processes betweentwo atoms in molecular reaction dynamics.

In recent years, several new fluoride of group 13 elements havereceived extensive attention because of their structural richnessand their potential to act as metal sources in the physical andchemical reaction processes [5–8]. Aluminum monofluoride, AlF,is a high-temperature molecule that exists in the gas-phase only

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).

at high temperatures. As an interstellar molecule, its radiationcharacteristics and spectroscopic properties in astrophysics andother fields have attracted special interests of researchers, aboutwhich there have been a large number of theoretical and experi-mental investigations [9–29]. Maki and Lovas [9] studied the firsthigh-resolution vibrational–rotational spectroscopy of AlF for them = 1–0 to 3–2 bands with an infrared diode laser spectrometer.Hedderich and Bernath [10] extended the observation up to the m= 5–4 band with a Fourier transform spectrometer. Measurementsof the rotational lines of AlF was reported by Lide [11,12], Wyseet al. [13] and Hoeft et al. [14]. Wyse and coworkers [13] observedthe rotational transitions for the vibrational states up to m = 4 andthe rotational levels up to J = 14–13 (for m = 0). Song and coworkers[15] discussed the effects of electric fields on the system energy,bond distance, dipole moment, HOMO–LUMO gaps, charge distri-bution and the infrared spectrum for the ground states of AlF mol-ecule. Horiai and cooperators [16] reported the first observation ofthe Dm = 2 transitions of AlF with an infrared diode laser spectrom-eter. Kumar and coworkers [17] evaluated Franck–Condon factorsand r-centroids by the more reliable numerical integration proce-dure for the bands of B1R+–X1R+, C1R+–X1R+, c3R+–b3R+, f3P–b3R+, F1P–B1R+ and G1R+–B1R+ systems of the AlF molecule. Kellöet al. [18] determined the nuclear quadrupole moment of 27Al bycombining molecular measurements on AlF and AlCl with electricfield gradients calculated at CCSD(T) level as well as by combiningatomic measurements on Al(3p 2P3/2) with the electric field gradi-ent obtained at numerical MCHF level. Zhang and cooperators [19]

J. Zhao et al. / Computational and Theoretical Chemistry 963 (2011) 130–134 131

recorded the infrared emission spectroscopy of AlF. They extendedtheir research to higher m and J in order to obtain improved Dun-ham coefficients and an improved internuclear potential.

Up to now, however, to the best of our knowledge, there are fewreport [30] on accurate analytical potential energy function (APEF)of AlF (X1R+) which could be used in molecular reaction dynamics.Either, no theoretical researches have systematically studied thespectroscopic properties and the potential energy curve (PEC) ofAlF molecule. In literature [30], using SAC/D95(d) method andbased on the Murrell–Sorbie potential function, Luo and collabora-tors obtained the whole potential curve and the correspondingspectroscopic constants (Re, De, xe, xeve, Be and ae, with the frac-tional errors being 1.69%, 30.82%, 7.86%, 22.80%, and 34.85%,respectively), which seemed not very accurate compared withthe available experimental values. On the other hand, in our previ-ous work [31], by least square fitting to the Murrell–Sorbie func-tion (M–S function) and the modified Murrell–Sorbie+c6 function(M–S+c6 function), we have successfully studied the potentialenergy function for the ground state of KH(D) molecule. Whetherthe spectroscopic constants can be more accurate by utilizing otherquantum chemical methods and whether we can also get reliableresults for the ground state of AlF molecule based on the twopotential energy functions motivate us to carry out the presentcalculations.

In this paper, in order to obtain spectroscopic constants as closeto the experimental data as possible, the dissociation energy De,the equilibrium internuclear distance Re, the harmonic frequencyxe and the adiabatic analytic potential energy curve (APEF) ofthe AlF (X1R+) molecule are calculated using two quantum chem-ical methods (CCSD(T) and QCISD(T)) with four different basis sets(cc-pVQZ, cc-pV5Z, 6-311+G(3df) and 6-311G(3df)). Furthermore,for the sake of testing whether the computational results can alsobe very reliable based on the M–S+c6 function for the X1R+ of AlFmolecule, we also develop our analytic potential energy curve cal-culations by least square fitting to the M–S+c6 function. All the cal-culated spectroscopic properties are compared with those obtainedby other theory and the experimental values to examine whetherthe present calculations represent an improvement and give thedirection for the experimental research.

2. Computational methods

All calculations were performed using GAUSSIAN03 programpackage [32]. Quantum chemical methods CCSD(T) and QCISD(T)in conjunction with cc-pVQZ, cc-pV5Z, 6-311+G(3df) and6-311G(3df) basis sets are employed to compute the harmonic fre-quency, equilibrium internuclear separation of the AlF (X1R+) mole-cule. Comparing the computational results with the experimentalvalues available, we can find that CCSD(T)/cc-pVQZ method can beused to produce reliable results for the energy calculation of AlF mol-ecule. The whole analytic potential energy functions (APEFs) for theground electronic state is further scanned using the above method,the potential energy functions and relevant spectroscopic constantsof this state are then obtained by least square fitting to M–S functionand the M–S+c6 function, respectively.

In order to describe the analytical potential energy function ofAlF molecule, the four-parameter Murrell–Sorbie function andthe five-parameter Murrell–Sorbie+c6 function [33,34] which arewidely-used potential energy functions and can accurately repro-duce interaction potential energies of neutral and cationic diatomicmolecule well, are used in the present work. We must indicate theMurrell–Sorbie potential function used here for the ground state ofAlF molecule is hardly the only good one. There also exists otherexcellent choices, such as the London inverse-sixth-power energybuilt by Cahill [35] for the ground state of a pair of neutral atoms

from their internuclear separation, the depth and curvature of theirpotential at its minimum, and from their van der Waals coefficientC6 and the Morse/long-range (MLR) potential function form uti-lized by LeRoy and his coworkers [36] for 7,7Li2, 6,6Li2, and 6,7Li2 sys-tem and an accurate hybrid potential which is a combination of theRydberg potential. In the present work, the general expressions forMurrell–Sorbie function and Murrell–Sorbie+c6 function are asfollows:

V ¼ �Deð1þ a1qþ a2q2 þ a3q3Þ expð�a1qÞ ð1ÞV ¼ �Deð1þ a1qþ a2q2 þ a3q3Þ expð�a1qÞ � c6=R6 ð2Þ

where q = R � Re, R is the diatomic internuclear distance. Re is itsequilibrium internuclear distance and is regarded as a fixed param-eter in the fitting process in this paper. The dissociation energy De

and the parameter ai and c6 are determined by fitting. For the Mur-rell–Sorbie function (M–S function), the quadratic, cubic and quarticforce constants can be derived from the equations as follows:

f2 ¼ Deða21 � 2a2Þ ð3Þ

f3 ¼ �6Deða3 � a1a2 þ13

a21Þ ð4Þ

f4 ¼ Deð3a41 � 12a2

1a2 þ 24a1a3Þ ð5Þ

For the modified Murrell–Sorbie+c6 function (M–S+c6 function),

f2 ¼ Deða21 � 2a2Þ � 42c6=Re8 ð6Þ

f3 ¼ �6Deða3 � a1a2 þ13

a21Þ þ 336c6=Re9 ð7Þ

f4 ¼ Deð3a41 � 12a2

1a2 þ 24a1a3Þ � 3024c6=Re10 ð8Þ

And then, the spectroscopic parameters are

Be ¼h

8pclR2e

ð9Þ

xe ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2

4p2lc2

sð10Þ

ae ¼ �6B2

e

xe1þ f3Re

3f 2

� �ð11Þ

xeve ¼Be

8� f4R2

e

f2þ 15 1þxeae

6B2e

!224

35 ð12Þ

wherel is the reduced mass of diatomic AlF, and c is the speed of lightin vacuity. In the equations above, Be, ae, xe and xeve represent rigid-rotational factor, nonrigid-rotational factor, harmonic vibrationalfrequency and inharmonic vibrational factor, respectively.

3. Results and discussion

In the present work, the dissociation energy De, the equilibriuminternuclear distance Re, the harmonic frequency xe and the adia-batic analytic potential energy curve (APEF) of the AlF (X1R+) mol-ecule are calculated using two quantum chemical methodsincluding CCSD(T) and QCISD(T) together with four different basissets, cc-pVQZ, cc-pV5Z, 6-311+G(3df) and 6-311G(3df). All thecomputational results are tabulated in Table 1. All the availableexperimental data [37] and the published values [30] are alsolisted in Table 1 for comparison. Utilizing the chosen computa-tional level CCSD(T)/cc-pVQZ, the PEC of the AlF (X1R+) moleculeis calculated at 0.05 Å intervals over the internuclear separationranging from 0.5 Å to 5.0 Å, totally 90 points are calculated forthe ground state X1R+ of AlF based on the M–S function and theM–S function+c6 function. It is well known that the spectroscopicproperties are mainly dominated by the potential energies near theequilibrium position. For convenient employment in further

Table 1The comparison of molecular parameters at equilibrium for the ground state of AlF(X1R+).

Method Basis set Re/nm De/eV xe/cm�1

CCSD(T) 6-311+G(3df) 0.1677 7.0067 773.26-311+G(3df)a 0.1670 7.0953 775.16-311G(3df) 0.1668 7.0756 799.4cc-pVQZ 0.1667 7.1248 802.5cc-pV5Z 0.1663 7.0312 800.2

QCISD(T) 6-311+G(3df) 0.1678 7.0114 770.96-311+G(3df)a 0.1672 7.0999 774.86-311G(3df) 0.1669 7.0686 796.7cc-pVQZ 0.1667 7.0216 801.7cc-pV5Z 0.1664 7.0396 798.0

Theory [30](SAC/D95(d)) 0.1682 8.9568 865.4Exptb. 0.1654 7.8966 802.3

a The values are those in which BSSE corrections are included.b Experimental values were taken from Ref. [35].

132 J. Zhao et al. / Computational and Theoretical Chemistry 963 (2011) 130–134

investigations, the complete APEF parameters are compiled in Ta-ble 2. In order to give a clear comparison, other theory [30] resultsare also displayed in Table 2. Based on the APEF obtained, the forceconstants, quadratic f2, cubic f3 and quartic f4 are estimated andshown in Table 2. From Eqs. (3)-(8), the relevant spectroscopicparameters (Be, ae, xe and xeve) are then calculated and tabulatedin Table 3. The calculated PECs based on the M–S function and theM–S+c6 function are plotted in Fig. 1.

3.1. Analytical potential energy functions for the ground state of AlF

Table 1 shows the present calculated results and the correspond-ing theoretical and experimental values of equilibrium positions Re,dissociation energies De and harmonic vibrational frequency xe. Ascan be seen in Table 1, CCSD(T) and QCISD(T) calculations are verysimilar when the same basis set is used. This is to be expected. Thecloseness of our QCISD(T) and CCSD(T) results for each fixed basisset is expected from theoretical analysis given in Refs. [38–40].Moreover, the computational results with 6-311G(3df) basis setare in better agreement with the experimental values than thosewith 6-311+G(3df) basis set, which indicates that 6-311+G(3df) ba-sis calculations may suffer from basis set superposition errors (BSSE)[41]. So in order to test this assumption for added diffuse functions,6-311G+(3df), we also performed equilibrium geometry, harmonicvibrational frequency and dissociation energy in which BSSE isunder consideration. The results are also listed in Table 1. It is notedthat the calculations intend to be more accurate when BSSE correc-tions are taken into account. Such as CCSD(T) results, the dissocia-tion energy De changes from 7.0067 to 7.0953 eV. The latter is inbetter accordance with the experimental values (7.8966 eV). Inaddition, this BSSE corrected De is also more reliable than that with6-311G(3df) basis set result (7.0756 eV). Compared with the exper-imental data, it is clear that the computational quantities obtained atCCSD(T)/cc-pVQZ level are better than the other calculations in thisarticle. Take xe for example. The computed value of 802.5 cm�1 hasan extraordinary agreement with the experimental values [37](802.3 cm�1), with the fractional error of only 0.025%. As for De,

Table 2Parameters of Murrell–Sorbie and Murrell–Sorbie+c6 potential functions and force consta

Potential function Re/nm De/eV a1/nm�1 a2/nm�2 a3

M–S 0.1667 7.1248 31.242 198.63 25M–S+c6 0.1667 7.1248 30.109 195.72 25Theory [30] 0.1682 8.9568 28.469 212.16 35Expta. 0.1654 7.8966 32.000 192.93 29

a Experimental values were taken from Ref. [35].

the calculated value (7.1248 eV) also agrees well with the experi-mental result (7.8966 eV). In the case of Re, it can be noted that thecomputational values is 0.1667 nm, with the corresponding experi-mental value of 0.1654 nm, in which the discrepancy is only0.0013 nm and the fractional error is of 0.79%. By the way, it is note-worthy that the calculated results of Re, De,xe in theory [30] employ-ing SAC/D95(d) method are 0.1682 nm, 8.9568 eV and 865.4 cm�1,respectively, with the fractional errors of 1.69%, 13.42% and 7.86%,respectively. Obviously, our calculations utilizing CCSD(T)/cc-pVQZlevel are more reliable and accurate than the published values [30],which means that our present work represents some improvementsfor the geometric calculations. Whether the computational lever wechoose can still be used to reproduce more accurate spectroscopicconstants is to be tested in the following segment.

By exact fits of computational energies to the Murrell–Sorbiefunction and the modified Murrell–Sorbie function, we obtain theanalytical potential energy functions (APEFs) for the X1R+ state ofthe AlF molecule. The complete values of parameters are compiledin Table 2. And in order to test the accuracy of analytical potentialenergy functions for the X1R+ state of the AlF molecule, we alsocalculated the quadratic, cubic and quartic force constants basedon Eqs. (2)–(8) listed in Table 2. For the sake of comparison, theexperimental values and other theoretical treatment [30] are alsotabulated in Table 2. To appreciate the fitting quality of APEF atCCSD(T), we calculate the root means square (RMS) error in the fit-ting process, which can be used to estimate quantitatively thequality of fitting process [42,43]. The RMS for the Murrell–Sorbiefunction and the modified Murrell–Sorbie function is 0.0262 eVand 0.0257 eV, respectively, which are much smaller than thechemical accuracy (1.0 kcal/mol). It consequently proves that ourfitting process is of high quality, and the M–S function and themodified M–S function are both very suitable for reproducing theaccurate PEC of AlF (X1R+).

3.2. Spectroscopic constants (Be, ae, xe and xeve)

Table 3 shows the spectroscopic parameters calculated using Eqs.(9)–(12) of AlF, while results of other theory as well as the experi-mental data are also displayed in Table 3. From Table 3, we can easilysee that the spectroscopic parameters obtained in this paper are inexcellent accordance with the experimental data, and our resultsare better than other works compared with the available experimen-tal values. For example, in the present paper, the dissociation ener-gies De based on the the Murrell–Sorbie function and the modifiedMurrell–Sorbie function is 8.638 � 10�7 cm�1 and 9.717 �10�7 cm�1, respectively, just slightly smaller than the experimentalvalue 10.464 � 10�7 cm�1. However, Luo and coworkers’ computa-tional result is 7.239 � 10�7 cm�1 at SAC/D95 (d) level, whose frac-tional error is as large as 30.82%. This situation also exists in xeve,with the calculated value in this paper 6.29 cm�1 and 6.22 cm�1

respectively, which are in agreement with the experimental result4.77 cm�1. As shown in Table 3, the published value 9.09 cm�1 forxeve seems quite inaccurate, nearly twice as much as the experi-mental value. Furthermore, according to the M–S function, the devi-ations of the spectroscopic parameterae andxe in this work from the

nts for AlF (X1R+).

/nm�3 c6/eV nm6 f2/aJ nm�2 f3/aJ nm�3 f4/aJ J nm�4

10.7 – 663.50 �44499.5 276740576.4 �2.869 � 10�6 590.74 �39784.8 252080736.0 – 554.18 �44665.2 333404326.0 – – – –

Table 3Spectroscopic constants of the ground states of AlF (X1R+).

Potential function Re/nm xe/cm�1 xeve/cm�1 Be/cm�1 ae/cm�1 De/cm�1

M–S 0.1667 802.5 6.29 0.544 0.00482 8.638 � 10�7

M–S+c6 0.1667 802.5 6.22 0.547 0.00510 9.717 � 10�7

Theory [30] 0.1682 918.5 9.09 0.535 0.00657 7.239 � 10�7

Expta. 0.1654 802.3 4.77 0.552 0.00498 10.464 � 10�7

a Experimental values were taken from Ref. [35].

0.1 0.2 0.3 0.4 0.5

-10

-5

0

5

10

15

20

25

30

35

V(r

)/(e

V)

r(nm)

CCSD(T) Murrell-Sorbie Murrell-Soribe+c6

Fig. 1. The Murrell–Sorbie and the Murrell–Sorbie+c6 potential energy curves ofthe ground states of AlF (X1R+).

J. Zhao et al. / Computational and Theoretical Chemistry 963 (2011) 130–134 133

experiments [37] are of 3.20% and 2.41%, respectively. And for theM–S+c6 potential function, the deviations discussed above decreaseto 1.45% and 0.91%, respectively. Thus, we can clearly see that twopotential energy functions are both very suitable for reproducingthe reliable PEC of the ground state X1R+ for AlF molecule. Compar-ing with those obtained by other theory [30] and the experimentalvalues [35], all the calculated spectroscopic parameters in our pres-ent work prove that the present calculations represent an improve-ment in theoretical computations and may give the direction for theexperimental research.

Based on the M–S function and the M–S+c6 function, the ab ini-tio calculation points and the fitting results over the internucleardistance ranging from 0.5 to 5.0 Å are depicted in Fig. 1. As canbe seen in Fig. 1, two PECs of the ground state X1R+ for AlF mole-cule are both very smooth. Besides, the dissociation energies ob-tained by ab initio calculations are in good accordance with thefitting results. The M–S function and the M–S+c6 function are bothsuitable for the description of the potential energy curves for AlF(X1R+) here, which demonstrates that both the M–S function andthe M–S+c6 function can be used to give credible spectroscopicparameters.

4. Conclusion

In this paper, the dissociation energy De, the equilibriuminternuclear distance Re, the harmonic frequency xe and the adi-abatic analytic potential energy curve (APEF) of the AlF (X1R+)molecule have been calculated using quantum chemical methodsCCSD(T) and QCISD(T) together with four different basis sets cc-pVQZ, cc-pV5Z, 6-311+G(3df) and 6-311G(3df). Compared withthe available experimental values, the CCSD(T) method in combi-nation with cc-pVQZ basis set can be utilized to produce reliableequilibrium geometry calculations of AlF (X1R+) molecule. At the

same level, analytical potential energy functions (APEFs) of theX1R+ state for AlF molecule have been obtained over the internu-clear separation ranging from 0.5 to 5.0 Å and have been fitted tothe M–S function and the M–S+c6 function respectively to com-pute the spectroscopic parameters. The force constants, includingquadratic f2, cubic f3 and quartic f4 and spectroscopic constantsthen have been precisely estimated. Based on the fitting curves,it is indicated that both potential energy functions are suitablefor giving the credible spectroscopic parameters. Moreover, allthe calculated spectroscopic parameters in our present work indi-cate that the present calculations represent an improvement intheoretical computations and give the direction for the experi-mental research. The computational results here can also be usedin further investigation about the dynamic properties of diatomicmolecules.

Acknowledgement

This work is supported by the National Natural Science Founda-tion of China under No. 101776017, by the Research Foundation ofEducation Bureau of Hubei Province of China under Grant No.B20101303, and by the Research Fund of School of Physical Scienceand Technology in Yangtze University under No. 201001A.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.comptc.2010.10.013.

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