An ab initio potential energy surface and infrared spectra for Kr–N2O in the v3 stretching region of N2O

Chemical Physics Letters 626 (2015) 43–48

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An ab initio potential energy surface and infrared spectra for Kr–N2 O in the v3 stretching region of N2 O Zhongquan Wang a,∗ , Eryin Feng b , Chunzao Zhang a , Chunyan Sun a a b

Department of Physics, Huainan Normal University, Huainan 232001, People’s Republic of China Department of Physics, Anhui Normal University, Wuhu 241000, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 29 November 2014 In final form 11 March 2015 Available online 20 March 2015

a b s t r a c t A three-dimensional ab initio potential energy surface of the Kr–N2 O complex was calculated using the CCSD(T) method and avqz-pp + 33221 basis set. The potential includes explicit dependence on the v3 stretching coordinate of the N2 O molecule. The potential energy surface is characterized by a near Tshaped minimum. The global minimum locates at R = 6.70a0 and  = 96.00◦ with a depth of −270.29 cm−1 . Using the fitted potential energy surface, we have calculated bound energy levels of the Kr–N2 O complexes. The resulting the potential provides a good representation of the experimental data. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Much attention has been devoted to the interaction of rare gases with the N2 O molecule in experimental [1–14] and theoretical [15–23] studies. In experiments, the pure rotational spectra of six isotopomers of Ne–N2 O were measured in the frequency range between 5 and 18 GHz by Ngari and Jäger using a pulsed beam cavity Fourier transform microwave spectrometer [1]. Based on a simple ‘stick and ball’ model [2], the structures of the He–N2 O complexes were shown to be roughly T-shaped. Recently, Zhu and co-workers studied the rovibrational spectrum of the He–N2 O van der Waals complex in the N2 O-monomer v1 region using an infrared tunable diode laser spectrometer [3]. The Ar–N2 O complex was at first studied by Joyner et al. [4] using the molecular beam electric resonance technique. The results indicated that the complex has a T-shaped structure with the argon atom titled slightly toward the oxygen atom. Further experiments have been carried out on this complex in the microwave region with Fourier-transform microwave spectrometers [5–7] as well as in the infrared region with tunable diode laser spectrometers [8–10]. The structure of the Ar–N2 O complex is almost unchanged upon vibrational excitation of the N2 O-monomer v3 (∼2224 cm−1 ) [8,9] and v1 (∼1285 cm−1 ) [10] normal vibrational mode. Due to the extreme weakness of the He van der Waals bonds, the infrared spectrum of He–N2 O was not reported in the N2 O v3 region until 2002 [11]. Then, the infrared spectra of Ne–N2 O [12] and Kr–N2 O [13] in the same region were also recorded by them. In 1998, Herrebout et al. [14] investigated

∗ Corresponding author. E-mail address: [email protected] (Z. Wang). http://dx.doi.org/10.1016/j.cplett.2015.03.013 0009-2614/© 2015 Elsevier B.V. All rights reserved.

the infrared spectra of Ne–N2 O, Kr–N2 O and Xe–N2 O in the v3 region of the N2 O monomer with a diode laser absorption spectrometer. The results showed that the structures of these three complexes are very similar to that of Ar–N2 O. On the theoretical side, for He–N2 O complex, several ab initio PESs were reported owing to its close relevance to our understanding of the superfluidity in the helium nano-droplet [15–18]. It is worth noting that the first attempt to explicitly include the dependence on the antisymmetric stretching normal mode Q3 of the N2 O molecule in the He–N2 O interaction potential was made by Zhou and co-workers [18] in 2006. The calculated band shifts for 4 He–N O and 3 He–N O agreed well with the observed values. In 2 2 2002, Zhu et al. [19] calculated the intermolecular potential energy surface (PES) for Ne–N2 O using the fourth-order Moller–Plesset (MP4) perturbation theory. The calculated rovibrational transition frequencies for both 20 Ne–N2 O and 22 Ne–N2 O isotopomers were in good agreement with the experiment. The two-dimensional ab initio intermolecular PESs for Ar–N2 O [20], Kr–N2 O [21] and Xe–N2 O [22] were also reported. In 2012, Wang et al. [23] calculated the bound states of the Kr–N2 O complex dependence on the antisymmetric stretching normal mode Q1 of the N2 O molecule. Up to now, there has been no report on a three-dimension surface of the Kr–N2 O complex with the stretching normal mode Q3 . 2. Computational details 2.1. Ab initio calculations The interaction potential energy surface of the Kr–N2 O complex is calculated by the single and double excitation coupledcluster theory with noniterative perturbational treatment of

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Z. Wang et al. / Chemical Physics Letters 626 (2015) 43–48

potential energies V (Q3i , R, ) was fitted to an analytical form which has been used by Jankowski et al. [29], V (R, ) = Vsh (R, ) + Vas (R, ).

(2)

The short-range part of the potential is expressed as Vsh (R, ) = G(R, )eB()+D()R ,

(3)

where D(), B() and G(R,) are all expansions in Legendre polynomials Pl0 (cos ): Figure 1. Jacobi coordinates for the Kr–N2 O complex employed in the ab initio calculations.

triple excitations [CCSD(T)] [24–26]. We employ the augmented correlation-consistent polarized valence quadruple zeta (aug-ccpVQZ) basis set for the N and O atom, and the quasirelativistic 10-core-electron pseudopotential and augmented correlationconsistent polarized valence quadruple-␨ (aug-cc-pVQZ-PP) basis set for the Kr atom [27,28]. Midbond functions (3s3p2d2f1g) (for 3s and 3p, ˛ = 0.9, 0.3 and 0.1; for 2d and 2f, ˛ = 0.6 and 0.2; for g, ˛ = 0.3) were used to eliminate the need for higher angular momentum functions in the atom-centered basis set. The CCSD(T) calculations were performed using MOLPRO 2012 package. The geometry of the Kr–N2 O complex is defined by the Jacobi coordinates: (R, , Q3 ). R refers to the distance between the Kr atom and the center of mass of N2 O molecule, and  is the angle between R and rN2 O with  = 0◦ referring to the linear configuration Kr–N–N–O. The coordinate Q3 represents the normal mode for the v3 stretching vibration of N2 O monomer, while the other two normal modes of N2 O are fixed at their equilibrium values. The geometry optimization provided the expression for the Q3 normal mode coordinate, Q3 = −0.552717N1z + 0.803839N2z − 0.219877Oz ,

(1)

where N1z , N2z , Oz refer to unit vector displacements from equilibrium of N1 , N2 , and O, respectively(see Figure 1). In the calculation of the full Kr–N2 O PES V (Q3 , R, ), we considered seven different values of Q3 : −0.2, −0.15, −0.1, 0, 0.1, 0.15, 0.2 relevant to extensions of the molecule, with Q3 = 0 corresponding to the N2 O equilibrium geometry. From Eq. (1), one can arrive at actual values of r1 , r2 and r (r = r1 + r2 ) corresponding to different values of Q3 as listed in Table 1. For each of the seven Q3 values, the ab initio PES grids were chosen as follows: 17 values of the intermolecular separation R: {6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5, 10, 10.5, 11, 11.5, 12, 12.5, 13, 15, 17} a0 , and over 13 angles : {0, 20, 40, 60, 80, 90, 100, 110, 120, 130, 150, 170, 180}◦ , resulting in a potential energy surface consisting of a total of 1547 configurations.

B() =

5 

bl Pl0 (cos ),

(4)

l=0

D() =

5 

dl Pl0 (cos ),

(5)

l=0

G(R, ) =

5 

(g0l + g1l R + g2l R2 + g3l R3 )Pl0 (cos ).

(6)

l=0

Here, bl , dl and gkl , are adjustable parameters. The term Vas (R, ) represents the long-range part of the potential and is expressed as Vas (R, ) =

9 

n−4 

n=6

fn (D()R) ×

Cnl 0 P (cos ). Rn l

(7)

l = 0, 2, . . . or l = 1, 3, . . .

The sum over l starts from 0 when n is even; otherwise, it begins with 1. The fn (x) is the Tang–Toennies damping function [30]. To obtain the most reliable fit, in particular for the long-range part of the potential, we follow a two-step nonlinear least squares procedure [31]. In the first step, the n = 6 and 7 terms of the longrange function Vas (R, ) were determined using only the ab initio values of R ≥ 11a0 with the damping function set to one. In the second step, all other parameters including the n = 7–9 long range coefficients were obtained with fixed the long-range coefficients of n = 6. Those grid points with energy more than 500 cm−1 were excluded from the fit. Second, we used the seven two-dimensional PESs to construct the three-dimensional PES by interpolating along Q3 using a sixorder polynomial, V (Q3i , Rj , k ) =

6 

Ai,n an (Rj , k ),

(8)

n=0 n

2.2. Construction of the PES In this work, the full three-dimensional PES V (Q3 , R, ) was constructed in two steps. First, the calculated two-dimensional ab initio Table 1 The symmetric stretching coordinate Q3 correspond sets of r1 , r2 , r values of the internal coordinates (all values in a0 ). Q3

r1

r2

r

−0.2 −0.15 −0.1 0 0.1 0.15 0.2

1.85899 1.92681 1.99465 2.13030 2.26595 2.33379 2.40161

2.44425 2.39306 2.34187 2.23950 2.13713 2.08594 2.03475

4.30324 4.31987 4.33652 4.36980 4.40308 4.41973 4.43636

where Ai,n = (Q3i ) . For a given values of (Rj ,  k ), the an (n = 0,. . .,6) is thus determined from seven two-dimensional PESs V (Q3i , R, ) by solving a 7-variables linear equations, an (Rj , k ) =

6 

[Ai,n ]−1 V (Q3i , Rj , k ),

(9)

i=0

where Q3i = (−0.2, −0.15, −0.1, 0, 0.1, 0.15, 0.2). The an ’s are then used to calculate the V (Q3i , Rj , k ) according to Eq. (8). We consider here the N2 O molecule in its ground (v3 = 0) and first (v3 = 1) excited vibrational states along the Q3 normal coordinate. The corresponding vibrationally adiabatic potential Vv3 v3 (R, ) was calculated as Vv3 v3 (R, ) = v3 (Q3 )|V (Q3 , R, )|v3 (Q3 ),

(10)

Z. Wang et al. / Chemical Physics Letters 626 (2015) 43–48

45

locates at R = 6.70a0 and  = 96.00◦ with a depth of −270.29 cm−1 . The PES shows a fairly large angular anisotropy, which means that Kr atom is strongly hindered from free motion around the N2 O molecule. Contour plots of the V11 surfaces shows the same overall features, although it has slightly shallower minimum. In order to qualitatively justify the adiabatic potential for bound state calculations, we have plotted the elements of the vibrational matrix Vv v3 (R,  = 96◦ ) in Figure 3. It should be noted that the dia3

batic couplings are important only in regions that are inaccessible to the bound states. In addition, since the vibrational wave functions are only determined up to a phase, the signs of the off diagonal coupling elements are arbitrary. 3.2. Rovibrational energy levels

Figure 2. Contour plot of the potential energy surface for V00 of the Kr–N2 O complex (in cm−1 ).

where xv3 (Q3 ) are the corresponding vibrational wave functions. The wave functions were obtained by solving the one-dimensional N2 O Schrödinger equation:



−



1 d2 + VN2 O (Q3 ) xv (Q3 ) = Ev xv (Q3 ), 2M dQ 2 3

(11)

with M being the N2 O reduced mass along the Q3 normal coordinate and VN2 O (Q3 ) being the corresponding potential energy curve. VN2 O (Q3 ) was calculated at the CCSD(T)-F12 level employing the aug-cc-pVQZ basis set at 51 points of Q3 ranging from +0.5 to −0.5. 2.3. Calculation of the rovibrational energy levels In order to obtain the explicit dependence of the Kr–N2 O interaction on a specific vibrational state of the N2 O molecule, we make use of the well-known feature that in weakly bound van der Waals complexes the intramolecular vibrations can, to a good approximation, be decoupled from the intermolecular modes because of the much higher frequencies of the former. Given this assumption, the Hamiltonian for the Kr–N2 O system can be written as 2

ˆ =− H

1 ∂2 (ˆJ − ˆj2 ) + + bv3 ˆj2 + Vv3 v3 (R, ), 2 ∂R2 2R2

(12)

where  is the reduced mass of the complex, ˆJ and ˆj are the rotational angular momentum operators corresponding to the total and the N2 O monomer, respectively. Vv3 v3 (R, ) is the adiabatic potential in a particular vibrational state v3 (v3 = 0, 1) of the isolated N2 O and bv3 is the corresponding rotational constant. The method used to solve Eq. (12) has been described in detail in our previous letter [31]. 3. Results and discussion 3.1. Features of the PES The global rms error is 0.068 cm−1 and the maximum absolute error is 0.12 cm−1 . In the vicinity of the potential well, 8.5a0 ≤ R ≤ 10.5a0 and V < 0, the absolute error is less than 0.1 cm−1 and the relative error is less than 0.08%. In the long range the relative error is within 1.6%. The contour plot of V00 (R, ) is displayed in Figure 2. It is clear that the CCSD(T) potential is characterized by a near T-shaped minimum. In the fittings, the global minimum

A self-written Fortran code [32] is used to calculate the rovibrational energy levels and wave functions of the Kr–N2 O complex. The values of the N2 O rotational constants for v3 = 0 and v3 = 1 employed in the calculations were fixed at the experimental values [3] (b0 = 0.419011 cm−1 and b1 = 0.417255 cm−1 ). The wave functions of the first four J = 0 vibrational levels are presented in Figure 4. In the case of V00 the ground vibrational energy is E = −230.5248 cm−1 , which corresponds to a zero-point energy of 39.7652 cm−1 , while for the V11 surface the corresponding value is E = −230.5644 cm−1 . It can be seen that the contours of this state are mainly localized in the near T-shaped geometry. The first excited vibrational level, at −196.0328 cm−1 for the V00 and −196.1143 cm−1 in case of V11 , corresponds basically to a bending excitation observed from the node structure. We thus deduce the bending ground frequency is 34.4920 cm−1 for vN2 O = 0 state and 34.4501 cm−1 for vN2 O = 1 state. The second excited level, at −194.3041 cm−1 for V00 and −194.3607 cm−1 for V11 is a stretching excitations. The next level, at −166.2782 cm−1 for V00 and −166.2523 cm−1 for V11 , can be assigned to two quanta of bending excitation. From the bound state energies computed with N2 O in both v3 = 0 and v3 = 1 vibrational states, we then calculated the rovibrational transition energies of Kr–N2 O complexes according to v3 =0 , v = v03 + EJv 3,K=1 ,K − EJ,K a ,Kc a

c

(13)

where v03 is the symmetric vibrational frequency of the free N2 O, v3 and EJ,K is the rotational levels computed with corresponding a ,Kc vibrationally adiabatic potentials. Ka and Kc denote the projections of total angular momentum J onto the a and c axes in the principal axes of inertia. The rovibrational energies of Kr–N2 O were assigned by the asymmetric rotor quantum numbers JKa ,Kc . 3.3. Comparison with experiments In 1998, Herrebout et al. [14] using a diode laser absorption spectrometer measured the infrared spectra of the Kr–N2 O complex in the region of the v3 N2 O monomer vibrational band. In Table 2, we present a comparison of ab initio values with experimental results. It can be seen that the theoretical results are in good agreement with the experimental data. The v3 band origin shift of Kr–N2 O dimer can be easily calculated by the readers: it is simply the difference between the reported ground vibrational energies calculated by the V00 and V11 surfaces. The calculated value −0.0396 cm−1 is a relatively large quantitative discrepancy with the experimental [14] observations −0.10131 cm−1 . One possible reason of the large discrepancy is that only a 1D intramolecular coordinates used in the construction of PES. For He van der Waals dimer with N2 O, it has been proved that the two-dimensional intramolecular coordinates would improve the accuracy of PES [33].

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Z. Wang et al. / Chemical Physics Letters 626 (2015) 43–48

Figure 3. Vibrational matrix elements of the three-dimension PES as a function of R for  = 96◦ .

Table 2 Observed and calculated infrared spectra of Kr–N2 O complex, and compare with the experimental results. The (O–C) is (Expt. [14] − Calc.) (in cm−1 ). Transition

Calc. (O–C)

Transition

Calc. (O–C)

Transition

Calc. (O–C)

321 –432 322 –431 220 –331 221 –330 321 –330 322 –331 422 –431 423 –432 523 –532 615 –726 515 –624 514 –625 414 –523 413 –524 313 –422 312 –423 212 –321 211 –322 111 –220 110 –221 818 –827 717 –726 616 –625 10010 –11111 515 –524 414 –423 313 –322 212 –221 211 –220 312 –321 413 –422 514 –523 615 –624 716 –725 817 –826 808 –919 707 –818 606 –717 505 –616 404 –313

2221.333 (0.049) 2221.332 (0.050) 2221.418 (0.049) 2221.418 (0.049) 2221.676 (0.049) 2221.676 (0.050) 2221.677 (0.049) 2221.676 (0.050) 2221.678 (0.048) 2221.901 (0.043) 2221.902 (0.043) 2221.973 (0.044) 2222.001 (0.045) 2222.048 (0.046) 2222.097 (0.047) 2222.124 (0.045) 2222.190 (0.047) 2222.203 (0.048) 2222.280 (0.047) 2222.285 (0.048) 2222.373 (0.041) 2222.391 (0.043) 2222.407 (0.045) 2222.419 (0.034) 2222.421 (0.036) 2222.432 (0.046) 2222.441 (0.046) 2222.448 (0.048) 2222.461 (0.047) 2222.468 (0.047) 2222.477 (0.046) 2222.487 (0.047) 2222.499 (0.046) 2222.512 (0.045) 2222.527 (0.044) 2222.552 (0.036) 2222.619 (0.038) 2222.689 (0.039) 2222.760 (0.040) 2223.586 (0.043)

313 –404 505 –414 212 –303 606 –515 111 –202 707 –616 808 –717 211 –202 312 –303 413 –404 514 –505 615 –606 716 –707 111 –000 625 –716 918 –909 212 –101 10010 –919 524 –615 313 –202 11011 –10110 423 –514 414 –303 12012 –11111 322 –413 515 –404 616 –505 717 –606 14014 –13113 818 –707 1129 –11110 1028 –1019 927 –918 826 –817 725 –716 624 –615 523 –514 422 –413 321 –312 220 –211

2223.643 (0.032) 2223.680 (0.042) 2223.736 (0.033) 2223.776 (0.043) 2223.826 (0.033) 2223.874 (0.042) 2223.973 (0.042) 2224.007 (0.033) 2224.014 (0.034) 2224.023 (0.033) 2224.035 (0.033) 2224.050 (0.032) 2224.067 (0.032) 2224.084 (0.035) 2224.103 (0.016) 2224.111 (0.031) 2224.166 (0.034) 2224.173 (0.041) 2224.205 (0.018) 2224.245 (0.034) 2224.274 (0.040) 2224.304 (0.019) 2224.322 (0.034) 2224.375 (0.039) 2224.401 (0.020) 2224.397 (0.033) 2224.469 (0.033) 2224.540 (0.032) 2224.578 (0.037) 2224.609 (0.032) 2224.648 (0.014) 2224.664 (0.014) 2224.679 (0.017) 2224.695 (0.018) 2224.709 (0.019) 2224.723 (0.020) 2224.735 (0.021) 2224.746 (0.021) 2224.754 (0.021) 2224.761 (0.022)

221 –212 322 –313 423 –414 524 –515 625 –616 726 –717 827 –818 928 –919 1029 –10110 634 –725 11210 –11111 221 –110 322 –211 321 –212 423 –312 422 –313 524 –413 523 –414 625 –514 624 –515 726 –615 928 –817 1138 –1129 432 –423 11210 –1019 927 –818 12211 –11110 330 –221 331 –220 431 –322 432 –321 532 –423 533 –422 634 –523 734 –625 735 –624 937 –826 936 –827 1037 –928

2224.775 (0.021) 2224.781 (0.022) 2224.790 (0.022) 2224.801 (0.022) 2224.815 (0.021) 2224.831 (0.020) 2224.849 (0.020) 2224.869 (0.019) 2224.891 (0.019) 2224.925 (0.002) 2224.916 (0.019) 2224.938 (0.023) 2225.019 (0.022) 2225.033 (0.022) 2225.098 (0.022) 2225.126 (0.022) 2225.175 (0.021) 2225.221 (0.022) 2225.249 (0.022) 2225.320 (0.022) 2225.321 (0.021) 2225.459 (0.019) 2225.509 (−0.000) 2225.532 (0.006) 2225.587 (0.017) 2225.634 (0.022) 2225.648 (0.015) 2225.790 (0.007) 2225.790 (0.007) 2225.875 (0.007) 2225.875 (0.007) 2225.961 (0.007) 2225.961 (0.007) 2226.046 (0.007) 2226.133 (0.005) 2226.130 (0.008) 2226.298 (0.006) 2226.306 (0.006) 2226.393 (0.006)

Z. Wang et al. / Chemical Physics Letters 626 (2015) 43–48

Figure 4. Contour plots of the wave functions for the first four vibrational levels (J = 0) of Kr–N2 O complex.

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Z. Wang et al. / Chemical Physics Letters 626 (2015) 43–48

Table 3 The calculated molecular constants (in cm−1 ) for the Kr–N2 O complex.

other observable quantities like scattering cross sections and second virial coefficients.

Experiment [14]

This work

12 666.7 1360.3 1222.6

12 874.9 1357.4 1221.6

12 770.8 1362.3 1224.8

12 873.9 1356.4 1221.3

Acknowledgments

Ground A B C

v3 A B C

We fitted the rotational energy levels of Kr–N2 O with the Watson asymmetric rotor Hamiltonian employing the A-type reduction in the Ir representation, ˆ = H





1 1 (B + C)ˆJ 2 + A − (B + C) ˆJZ2 − DJ ˆJ 4 − DJK ˆJ 2 ˆJZ2 − DK ˆJZ4 2 2 +

1 2 2 (B − C)(ˆJ+ + ˆJ− ). 4

(14)

The fitted molecular constants are reported in Table 3. Our calculated spectroscopic constants are in good agreement with the observed values. This overall agreement indicates that the present potential energy surface accurately describes the interaction between the N2 O molecule and the Kr atom. The rotational constants A > B ≈ C indicate that the Kr–N2 O complex is a prolate near-symmetric rotor. It also found the B and C rotational constants were very accurate, but the A constant appears to be a bit large. Thus, the radial potentials are very good, but the angular potential (or angular dynamics) have displaced the complex too far from T-shaped. 4. Summary A three-dimensional PES for the weakly bound Kr–N2 O complex was constructed with the v3 stretching vibrational Q3 normal coordinate of N2 O. The calculations are performed by CCSD(T) method with a large basis set plus (3s3p2d2f1g) midbond functions. It is clear that the CCSD(T) potential is characterized by a near T-shaped minimum. The global minimum locates at R = 6.70a0 and  = 96.00◦ with a depth of −270.29 cm−1 . The two vibrational adiabatic surfaces are further used to calculate the bound rovibrational states and the infrared spectrum for the Kr–N2 O complex. The theoretically predicted results are in good agreement with the observed values. These comparisons demonstrate that the present potential is rather accurate, and may be very useful in the computation of

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11404126 and 11374014), the Natural Science Foundation of Anhui Province (the Grant No. 1208085MA08), the Natural Science Foundation of Anhui Educational Committee (Grant Nos. KJ2012Z372 and KJ2013B262) and the Natural Science Foundation of Huainan Normal University (Grant No. 2012LK01). References [1] M.S. Ngari, W. Jäger, J. Mol. Spectrosc. 192 (1998) 320. [2] G.D. Hayman, J. Hodge, B.J. Howard, J.S. Muenter, T.R. Dyke, J. Mol. Spectrosc. 133 (1989) 423. [3] D.S. Zhu, R.B. Wang, R. Zheng, G.M. Huang, C.X. Duan, J. Mol. Spectrosc. 253 (2009) 88. [4] C.H. Joyner, T.A. Dixon, F.A. Baiocchi, W. Klemperer, J. Chem. Phys. 75 (1981) 5285. [5] H.O. Leung, Chem. Commun. 1996 (1996) 2525. [6] H.O. Leung, D. Gangwani, J.-U. Grabow, J. Mol. Spectrosc. 184 (1997) 106. [7] M.S. Ngari, W. Jäger, J. Mol. Spectrosc. 192 (1998) 452. [8] H. Hodge, G.D. Hayman, T.R. Dyke, B.J. Howard, J. Chem. Soc. Faraday Trans. 82 (1986) 1137. [9] T.A. Hu, E.L. Chappell, S.W. Sharpe, J. Chem. Phys. 98 (1993) 6162. [10] G. Gimmler, M. Havenith, J. Mol. Struct. 599 (2001) 117. [11] J. Tang, A.R.W. McKellar, J. Chem. Phys. 117 (2002) 2586. [12] R. Zheng, D.S. Zhu, Y. Zhu, C.X. Duan, J. Mol. Spectrosc. 263 (2010) 174. [13] R. Zheng, Y. Zhu, S. Li, C.X. Duan, Mol. Phys. 109 (2011) 823. [14] W.A. Herrebout, H.-B. Qian, H. Yamaguchi, B.J. Howard, J. Mol. Spectrosc. 189 (1998) 235. [15] X.G. Song, Y.J. Xu, P.N. Roy, W. Jäger, J. Chem. Phys. 121 (2004) 12308. [16] B.T. Chang, O. Akin-Ojo, R. Bukowski, K. Szalewicz, J. Chem. Phys. 119 (2003) 11654. [17] Y.Z. Zhou, D.Q. Xie, J. Chem. Phys. 120 (2004) 8575. [18] Y.Z. Zhou, D.Q. Xie, D.H. Zhang, J. Chem. Phys. 124 (2006) 144317. [19] H. Zhu, D.Q. Xie, G.S. Yan, Chem. Phys. Lett. 351 (2002) 149. [20] H. Zhu, D.Q. Xie, G.S. Yan, J. Comput. Chem. 24 (2003) 1839. [21] R. Chen, H. Zhu, J. Theor. Comput. Chem. 7 (2008) 1093. [22] J.X. Chen, H. Zhu, D.Q. Xie, G.S. Yan, Acta Chim. Sin. 62 (2004) 5. [23] J. Wang, Y. Han, Z. Li, E. Feng, W. Huang, Mol. Phys. 111 (2012) 771. [24] K. Raghavachari, G.W. Trucks, J.A. Pople, M. Head-Gordon, Chem. Phys. Lett. 157 (1989) 479. [25] C. Hampel, K. Peterson, H.-J. Werner, Chem. Phys. Lett. 190 (1992) 1. [26] M.J.O. Deegan, P.J. Knowles, Chem. Phys. Lett. 227 (1994) 321. [27] K.A. Peterson, J. Chem. Phys. 119 (2003) 11099. [28] K.A. Peterson, D. Figgen, E. Goll, H. Stoll, M. Dolg, J. Chem. Phys. 119 (2003) 11113. [29] P. Jankowski, K. Szalewicz, J. Chem. Phys. 108 (1998) 3554. [30] K.T. Tang, J.P. Toennies, J. Chem. Phys. 80 (1984) 3726. [31] Z. Wang, M. Gong, Y. Zhang, E. Feng, Z. Cui, J. Chem. Phys. 128 (2008) 044309. [32] E.Y. Feng, W.Y. Huang, Z.F. Cui, W.J. Zhang, J. Mol. Struct.: THEOCHEM 724 (2005) 195. [33] L. Wang, D.Q. Xie, R.J. Le Roy, P.N. Roy, J. Chem. Phys. 137 (2012) 104311.