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Rovibrational bound states of the Ne–OCS complex
24 April 1998
Chemical Physics Letters 287 Ž1998. 162–168
Rovibrational bound states of the Ne–OCS complex Guosen Yan, Minghui Yang, Daiqian Xie Department of Chemistry, Sichuan UniÕersity, Chengdu 610064, China Received 24 June 1997; in final form 21 November 1997
Abstract Ab initio potential energy surface of the Ne–OCS complex is used to calculate the rovibrational bound states for the five isotopomers ŽNe–OCS, 22 Ne–OCS, Ne–OC 34 S, NeO13 CS, Ne– 18 OCS. with a vibrational SCF-CI procedure. The ab initio potential supports 20 bound vibrational states for all of the five isotopomers. The rovibrational ground state of Ne–OCS is bounded by 59.49 cmy1, corresponding to a zero-point energy of 23.45 cmy1. Most of the calculated frequencies for pure rotational transitions of the five isotopomers are within 0.01 cmy1 of the observed values. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Van der Waals ŽvdW. dimers containing triatomic molecules Žsuch as CO 2 , OCS, N2 O. have been studied extensively. These triatomic molecules have a cylindrically shape in contrast with HX ŽX s F, Cl, Br. which is virtually spherical. It is interesting to inspect what has been changed with HX replaced by these linear triatomic molecules in vdW dimers w1x. The rare gas ŽRg. –OCS interaction is of astrophysical interest w2–5x. The early studies were mostly the scattering of the He–OCS system w6–8x and the recent studies were focused on infrared and microwave spectroscopy of He, Ne, Ar–OCS w9–13x. Danielson et al. obtained an empirical anisotropic potential energy surface ŽPES. of He–OCS by fitting differential cross-sections w6x. This PES ignores the asymmetry of OCS and has a single minimum in the perpendicular direction. Some ab initio calculations of He–OCS w14x and Ar–OCS w15x were limited to Hartree–Fock ŽHF. level in providing short-range repulsive interactions. Recently we have obtained an ab initio PES of Ne–OCS w16x. The ab initio poten-
tial is characterized by an asymmetry T-shaped global minimum and two linear local minima with considerable well-depth. Because of the asymmetry of OCS, the shape of the potential is different from the potential of the vdW complexes containing CO 2 , which is isovalent with OCS. Several studies have indicated that the potentials of He–CO 2 and Ar–CO 2 have no linear secondary minimum or a secondary minimum with very shallow well-depth w17–19x. There are abundant spectroscopy data for Ne– OCS. Hayman et al. w9,10x studied the infrared absorption spectrum of Ne–OCS. Lovas and Suenram w11x, Xu and Gerry w12x measured the pure rotation spectra of isotopomers of Ne–OCS. The ground state average structure of Ne–OCS has been derived from the rotational and centrifugal distortion constants. The Ne–OCS has an asymmetric T-shaped structure with neon atom close to the oxygen of the OCS molecule w9–12x. From a harmonic force field suggested by Xu and Gerry w12x, the vdW stretching and bending mode frequency are predicted to be 32 and 16 cmy1 , respectively. The rovibrational bound states of a vdW complex
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 0 6 8 - 2
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
play an important role in explaining the details of the intermolecular forces responsible for the bonding and stability of the complex. In order to test the PES further and provide theoretical information about the rovibrational bound states of the Ne–OCS complex, we have studied the rovibrational energy levels and rotational transition frequencies using the vibrational SCF-CI procedure described in Section 2, with the numerical results and discussions given in Section 3.
2. Computational details The Jacobi coordinate system is used to describe the geometry of Ne–OCS. The coordinate R is the distance between the neon atom and center of mass of OCS and u is the angle between R and the OCS axis with u s 08 corresponding to the linear Ne–OCS. The geometry of OCS is fixed at the values of: ˚ and r ŽC–S. s 1.5643 A. ˚ In our r ŽC–O. s 1.1629 A previous work w16x, the ab initio intermolecular potential energy surface for Ne–OCS has been calculated using the second-order Møller–Plesset perturbation theory with a basis set containing the nucleus-centered basis set 6-31 G q Ž2df. and the midpoint bond function set 3s3p2d suggested by Tao and Pan w20x. The calculations were performed for nine bending angles u ranging from 0 to 1808 and 15 ˚ A total of distances R ranging from 2.5 to 8.0 A. 135 discrete potential energy points were calculated using Boys and Bernardi’s full counterpoise method w21x to correct the basis set superposition errors. MP4 calculations were also carried out for the three local minimum to ascertain the MP2 calculations. In this Letter, the values of the potential V Ž R, u . are calculated as follows. Nine new grid points V Ž R, u k . are first extracted from the cubic spline interpolation in the R direction for each input angle ˚ R ( 8.0 A. ˚ For other values of R, the u k for 2.5 A potential V Ž R, u k . is calculated using the following forms:
˚ V Ž R , u k . s A Ž u k . eya Ž u k . R , R ( 2.5 A
Ž 1.
V Ž R , u k . s C6 Ž u k . Ry6 q C 7 Ž u k . Ry7
˚ qC8 Ž u k . Ry8 , R 0 8.0 A.
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the ab initio values of V Ž2.50, u k . and V Ž3.00, u k ., C6, C7 and C8 are determined from the values of V Ž6.00, u k ., V Ž7.00, u k . and V Ž8.00, u k .. Then, the cubic spline interpolation in the u direction from the nine grid points V Ž R, u k . is performed to produce the potential energy V Ž R, u .. The cubic spline interpolation for Vm Ž u . indicates that the surface has three local energy minima and two barriers: the first local minimum is the global minimum, corresponding to the geometry of R s ˚ and u s 71.888. The second and third min3.554 A ˚ u s 1808, and R s ima are located at R s 4.427 A, ˚ u s 08. Between the three energy minima, 4.928 A, there are two energy barriers. The first barrier is located between the first and second energy mini˚ and u and mum with the geometry of R s 4.318 A u s 1248 and the barrier height is 43.25 cmy1 . And ˚ and u s 278 the second one is located at R s 4.670 A with the height of 38.74 cmy1 . Numerous procedures were suggested to calculate the bound states of a vdW complex. Peet and Yang w22x suggested the collocation method. Choi and Light w23x implemented the discrete variable representation ŽDVR. method. Korambath et al. w24x modified the discrete variable representation method. Clary and Knowles w25x presented a self-consistent field–configuration-interaction ŽSCF-CI. approach. In this Letter, we used an alternative vibrational SCF-CI procedure developed from our previous work w26,27x on stable triatomic molecules. In this procedure, the SCF method is used to optimize the basis set for bending and stretching motions in the vdW complex and then the CI method is employed to produce the converged results. For a given Ž JK ., where J is the total angular momentum and K is its projection on the body-fixed z-axis, the vibrational eigenfunction FnJK can be expanded as:
FnJK Ž R , u . s Ý c mJKn wm 1Ž R . wmK2Ž u .
Ž 3.
m
where, m is a collective vibrational index denoting the values of m1 and m 2 , the expansion coefficient c mJKn can be obtained by diagonalizing the following matrix
Ž 2.
Here, the parameters A and a are determined from
² wm Ž R . wmK Ž u . FMJ pK < H < wm Ž R . wmK Ž u . FMJ pK : 1
2
X 1
X 2
Ž 4.
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
164
in which H is the complete rovibrational Hamiltonian of the complex and FMJ pK is the parity-adapted rotation functions. wm 1 Ž R . and wmK2 Ž u . are the optimal independent-mode wavefunctions which are determined by using the following one-dimensional independent-mode Hamiltonians, 0 Js0, ps0 < < Ks 0 h1 Ž R . s² wmKs Ž u . FMs0, Ks0 H wm 2 s0 Ž u . 2 s0
The final rovibrational energy levels and wavefunctions can be obtained by diagonalizing the following Hamiltonian matrix X
²FnJK Ž R ,u . FMJ pK < H
X
3. Results and discussions
Js 0, ps0 =FMs 0, Ks0 :
Ž 5.
h 2JK Ž u . s² wm 1s0 Ž R . FMJ pK < H < wm 1s0 Ž R . FMJ pK :
Ž 6.
In the case of J s 0, the above two secular problems are coupled to each other, so the SCF procedure should be used. In the case of J ) 0, only the h 2JK Ž u . and the corresponding eigenfunctions need to be determined. In this Letter, we use a set of associated Legendre functions to solve variationally the bending one-dimensional eigenvalue problem, and use the renormalized Numerov–Johnson w28x method to solve numerically the stretching one-dimensional eigenvalue problem in the region w R min , R max x. Although the radial basis function thus determined is not an analytical function, it is convenient to evaluate the related integrals over the radial basis functions by use of the HEG quadrature w29x.
In the SCF-CI calculations, the integration limits ˚ and R max s 14.0 A˚ 100 are taken as R min s 2.3 A numerical radial basis functions are obtained by solving Eq. Ž5. with 5000 grid points, and 100 angular basis functions are determined by solving Eq. Ž6. variationally with associated Legendre functions. The vibrational CI calculations are performed with 1800 lowest configurations. The vibrational energy levels for Ne–OCS and its four isotopomers Ž 22 Ne–OCS, Ne–OC 34 S, Ne–O 13 CS, Ne– 18 OCS. are listed in Table 1 together with assignments of the quantum numbers Ž n,b . for the vdW stretching and bending modes. It should be noted that the assignments are only loosely defined, since there are considerable mixing between the different modes in the excited vibrational states. One can see from Table 1 that the
Table 1 Vibrational energy levels of isotopomers of Ne–OCS Žin cmy1 . n
b
Ne–OCSa
Ne–OCSb
22
0 0 1 0 1 2 1 0 2 2 3 3 3 4 3 4 4 4
0 1 0 2 1 0 2 3 1 2 0 1 2 0 3 1 2 3
y59.491 y42.518 y38.972 y30.554 y29.000 y24.508 y22.282 y17.032 y16.390 y14.924 y12.960 y10.213 y8.920 y6.109 y5.328 y4.295 y2.921 y2.470
y59.491 y42.521 y38.978 y30.573 y29.066 y24.765 y22.324 y17.244 y16.545 y15.306 y13.000 y10.405 y9.239 y6.157 y5.422 y4.910 y3.154 y2.621
y60.068 y43.137 y39.875 y31.224 y29.934 y25.358 y23.309 y18.097 y17.71 y16.211 y14.285 y11.237 y10.467 y6.913 y6.477 y5.776 y4.144 y3.107
a b
Ž 7.
Calculations with 1700 lowest configurations. Calculations with 1800 lowest configurations.
Ne–OCS
Ne–OC 34 S
Ne–O13 CS
Ne– 18 OCS
y59.614 y42.740 y39.120 y30.778 y29.176 y25.341 y22.692 y18.295 y16.646 y15.719 y13.266 y11.149 y10.213 y6.635 y6.164 y5.852 y3.553 y3.304
y59.547 y42.582 y39.056 y30.646 y29.173 y24.730 y22.363 y17.169 y16.715 y15.319 y13.106 y10.356 y9.242 y6.230 y5.400 y4.783 y3.224 y2.578
y59.852 y43.033 y39.388 y31.180 y29.765 y24.754 y22.549 y17.716 y17.237 y15.310 y13.682 y10.289 y9.376 y6.820 y5.448 y4.350 y3.570 y2.426
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
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Table 2 Coefficients of the CI wavefunctions for the four lowest vibratonal states No.
E Žcmy1 .
n
b
Coefficient
1 2
y59.491 y42.521
3
y38.978
4
y30.573
0 0 1 1 0 1 0 1 1 0 1 0 1 1
0 1 0 1 1 0 2 2 0 2 2 3 3 4
0.992 y0.849 y0.208 y0.311 0.295 y0.781 y0.334 y0.260 0.382 y0.478 y0.363 0.214 0.236 0.278
Fig. 1. The probability density for finding Ne relative to the OCS molecule for the vibrational ground state.
Only the values larger than 0.20 are given.
ab initio potential supports 20 bound vibrational states for all of the five complexes. The energy levels of Ne–OCS using 1700 configurations are also listed in Table 1 to show the convergence of the state energies. Since the strong anisotropy of the potential, the convergence for higher energy states is slower. The three lowest levels are converged to within 0.01 cmy1 , other higher levels are converged to within 0.5 cmy1 or better. Table 2 gives the coefficients of the CI wavefunctions for the four lowest vibrational states of Ne–OCS. Figs. 1 and 2, and 3 plot the probability densities
Fig. 2. The probability density for finding Ne relative to the OCS molecule for the first vibrational excited state.
Fig. 3. The probability density for finding Ne relative to the OCS molecule for the second vibrational excited state.
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
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vibrational ground state, the value of ² P2 Žcos u .: is y0.342, corresponding to a ² u : of 71.058. This is close to the experimental value of 70.448 w12x and deviated by 38 from the value of 748 for the asymmetric T-shaped minimum of the ab initio potential w16x. This shows the rigid behavior of vdW bending motion in the angular direction. Fig. 1 also shows that the ground state vibrational wavefunction is well localized in the T-shaped configuration. It is shown in Table 2 that the first vibrational excited state is the vdW bending state with the frequency of 17.0 cmy1 . The value of ² P2 Žcos u .: is y0.319, corresponding to ² u : s 69.678. Fig. 2 shows that two maxima of the CI wavefunction are
located at u s 62 and 848, respectively, and the node plane is located at u s 728. The second excited vibrational state is the vdW stretching state with the frequency of 20.5 cmy1 . The value of ² P2 Žcos u .: is y0.318, corresponding to ² u : s 69.618. Fig. 3 shows that the wavefunction has three maxima at 62, 72 and 908, respectively. The two node planes are at u s 65 and 808. The first and second vibrational excited states are localized in the region of the global minimum. In addition, we calculated the rovibrational energy levels Žavailable upon request. for five isotopomers of Ne–OCS ŽNe–OCS, 22 Ne–OCS, Ne–OC 34 S, Ne–O 13 CS, Ne– 18 OCS.. In these calculations, for a
Table 3 Calculated transition frequencies of isotopomers of Ne–OCS Žin cmy1 . a JKX X a K X c 111 2 02 2 11 2 12 2 12 2 20 2 21 3 03 3 03 312 312 313 313 3 21 3 22 3 21 3 22 4 04 4 04 4 14 4 13 4 13 4 23 4 22 4 22 505 514 514 523 6 15 6 24 a
JKY Y a K Y c 0 00 1 01 110 111 1 01 2 11 2 12 2 02 2 12 2 11 303 2 12 2 02 2 20 2 21 312 313 303 313 313 4 04 312 322 4 13 321 4 14 505 4 23 514 6 06 6 15
22
Ne–OCS
Ne–OCS
Ne–OC 34 S
Ne–O13 CS
Ne– 18 OCS
yobs
ycal
yobs
ycal
yobs
ycal
yobs
ycal
yobs
ycal
0.308 0.293 0.318 0.272 0.432 0.483 0.549 0.433 0.293 0.475 0.250 0.406 0.546 0.451 0.442 0.459 0.585 0.568 0.454 0.539 0.312
y0.004 y0.007 y0.009 y0.006 y0.006 y0.002 y0.006 y0.010 y0.010 y0.013 y0.007 y0.008 y0.008 y0.012 y0.011 y0.001 y0.009 y0.012 y0.014 y0.011 y0.013 0.613 y0.014 y0.001 0.592 y0.016 y0.020 y0.024 y0.002 y0.029 y0.006
0.303 0.277 0.299 0.258 0.422 0.493 0.553 0.411 0.266 0.448 0.244 0.386 0.531 0.425 0.418 0.470 0.585 0.539 0.419 0.512 0.298
y0.004 y0.007 y0.008 y0.006 y0.006 y0.001 y0.005 y0.010 y0.010 y0.013 y0.006 y0.008 y0.008 y0.012 y0.011 0.000 y0.008 y0.012 y0.014 y0.011 y0.011 0.578 y0.014 0.000 0.557 y0.017 y0.017 0.292 y0.001 y0.025 0.431
0.303 0.287 0.312 0.267 0.425 0.476 0.541 0.426 0.288 0.467 0.246 0.400 0.538
y0.004 y0.007 y0.009 y0.005 y0.006 y0.002 y0.006 y0.009 y0.010 y0.013 y0.007 y0.008 y0.007 0.432 y0.011 y0.001 y0.009 y0.011 y0.013 y0.010 y0.012 0.602 0.564 y0.001 0.582 y0.015 0.371 0.345 0.419 0.467 0.421
0.306 0.292 0.317 0.271 0.431 0.479 0.546 0.432 0.293 0.474 0.249 0.405 0.544
y0.004 y0.007 y0.009 y0.006 y0.006 y0.001 y0.006 y0.010 y0.011 y0.013 y0.007 y0.008 y0.008 0.438
0.290 0.290 0.317 0.268 0.412 0.433 0.502 0.428 0.305 0.473 0.240 0.401 0.523
y0.004 y0.007 y0.009 y0.006 y0.006 y0.001 y0.005 y0.009 y0.010 y0.013 y0.007 y0.008 y0.007 0.437
0.587 0.438 0.611 0.397 0.376 0.427 0.505 0.433
0.555 0.449 0.570 0.373 0.435 0.469
0.435 0.453 0.576 0.558 0.446 0.530 0.307
0.432 0.600
0.430 0.456 0.581 0.566 0.454 0.538 0.311
0.000 y0.008 y0.012 y0.014 y0.011 y0.013 0.611 0.572 0.434 0.591 0.594 0.377 0.354 0.422 0.476 0.424
0.428 0.410
0.463 0.532
0.532 0.548 y0.013 y0.010 0.294 0.610 0.568 0.391 0.591 0.600 0.379 0.399 0.384 0.485 0.394
The observed values are taken from Ref. w12x. For observed transitions, the calculated values are given as calculated-observed, otherwise the actual calculated values are given.
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
given value of Ž JK ., the vibrational SCF-CI method with 1000 lowest configurations is used to calculate to vibrational energy levels and wavefunctions. The 100 lowest vibrational states thus obtained are then used to calculated the Hamiltonian matrix elements that are given in Eq. Ž7.. The rovibrational energy levels and wavefunctions are obtained by diagonalizing this Hamiltonian matrix. In order to test the convergence of the state energies, we performed calculations with 1500 lowest configurations. The results show that the rovibrational energy levels for the ground vibrational states are converged within 0.01 cmy1 or better. Microwave frequencies associated with the observed rotational transitions can be obtained directly from the energy difference between appropriate energy levels. The calculated transition frequencies for the five isotopomers of Ne–OCS are listed in Table 3 along with the observed values w12x. It is seen from Table 3 that most of the calculated frequencies are within 0.01 cmy1 of the observed values. The results show that the ab initio PES agrees with the measured rotational frequencies.
4. Conclusions This study presents theoretical calculations of the rovibrational bound states of the Ne–OCS complex using our ab initio potential energy surface of the complex. The ab initio potential supports 20 bound vibrational states. The rovibrational ground state of Ne–OCS is bounded by 59.49 cmy1 , corresponding to a zero-point energy of 23.45 cmy1 . The rigid behavior of vdW bending motion in the angular direction was found by analyzing the average value of u over the bound state. From the given probability densities of Ne relative to OCS, it was also found that the wavefunctions for the lowest two excited vibrational states are localized in the region of the global minimum. Most of the calculated frequencies for pure rotational transitions of the five isotopomers are in good agreement with the observed values. It is expected that the global description of the rovibrational bound states might provide useful information for further observation of the spectroscopy of the Ne–OCS complex.
167
Acknowledgements Project 29673029 was supported by National Natural Science Foundation of China. We wish to thank Professor Auchin Tang for his stimulation of this work. We also appreciate most gratefully the reviewer for his very helpful suggestions on the manuscript.
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Chemical Physics Letters 287 Ž1998. 162–168
Rovibrational bound states of the Ne–OCS complex Guosen Yan, Minghui Yang, Daiqian Xie Department of Chemistry, Sichuan UniÕersity, Chengdu 610064, China Received 24 June 1997; in final form 21 November 1997
Abstract Ab initio potential energy surface of the Ne–OCS complex is used to calculate the rovibrational bound states for the five isotopomers ŽNe–OCS, 22 Ne–OCS, Ne–OC 34 S, NeO13 CS, Ne– 18 OCS. with a vibrational SCF-CI procedure. The ab initio potential supports 20 bound vibrational states for all of the five isotopomers. The rovibrational ground state of Ne–OCS is bounded by 59.49 cmy1, corresponding to a zero-point energy of 23.45 cmy1. Most of the calculated frequencies for pure rotational transitions of the five isotopomers are within 0.01 cmy1 of the observed values. q 1998 Elsevier Science B.V. All rights reserved.
1. Introduction Van der Waals ŽvdW. dimers containing triatomic molecules Žsuch as CO 2 , OCS, N2 O. have been studied extensively. These triatomic molecules have a cylindrically shape in contrast with HX ŽX s F, Cl, Br. which is virtually spherical. It is interesting to inspect what has been changed with HX replaced by these linear triatomic molecules in vdW dimers w1x. The rare gas ŽRg. –OCS interaction is of astrophysical interest w2–5x. The early studies were mostly the scattering of the He–OCS system w6–8x and the recent studies were focused on infrared and microwave spectroscopy of He, Ne, Ar–OCS w9–13x. Danielson et al. obtained an empirical anisotropic potential energy surface ŽPES. of He–OCS by fitting differential cross-sections w6x. This PES ignores the asymmetry of OCS and has a single minimum in the perpendicular direction. Some ab initio calculations of He–OCS w14x and Ar–OCS w15x were limited to Hartree–Fock ŽHF. level in providing short-range repulsive interactions. Recently we have obtained an ab initio PES of Ne–OCS w16x. The ab initio poten-
tial is characterized by an asymmetry T-shaped global minimum and two linear local minima with considerable well-depth. Because of the asymmetry of OCS, the shape of the potential is different from the potential of the vdW complexes containing CO 2 , which is isovalent with OCS. Several studies have indicated that the potentials of He–CO 2 and Ar–CO 2 have no linear secondary minimum or a secondary minimum with very shallow well-depth w17–19x. There are abundant spectroscopy data for Ne– OCS. Hayman et al. w9,10x studied the infrared absorption spectrum of Ne–OCS. Lovas and Suenram w11x, Xu and Gerry w12x measured the pure rotation spectra of isotopomers of Ne–OCS. The ground state average structure of Ne–OCS has been derived from the rotational and centrifugal distortion constants. The Ne–OCS has an asymmetric T-shaped structure with neon atom close to the oxygen of the OCS molecule w9–12x. From a harmonic force field suggested by Xu and Gerry w12x, the vdW stretching and bending mode frequency are predicted to be 32 and 16 cmy1 , respectively. The rovibrational bound states of a vdW complex
0009-2614r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 8 . 0 0 0 6 8 - 2
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
play an important role in explaining the details of the intermolecular forces responsible for the bonding and stability of the complex. In order to test the PES further and provide theoretical information about the rovibrational bound states of the Ne–OCS complex, we have studied the rovibrational energy levels and rotational transition frequencies using the vibrational SCF-CI procedure described in Section 2, with the numerical results and discussions given in Section 3.
2. Computational details The Jacobi coordinate system is used to describe the geometry of Ne–OCS. The coordinate R is the distance between the neon atom and center of mass of OCS and u is the angle between R and the OCS axis with u s 08 corresponding to the linear Ne–OCS. The geometry of OCS is fixed at the values of: ˚ and r ŽC–S. s 1.5643 A. ˚ In our r ŽC–O. s 1.1629 A previous work w16x, the ab initio intermolecular potential energy surface for Ne–OCS has been calculated using the second-order Møller–Plesset perturbation theory with a basis set containing the nucleus-centered basis set 6-31 G q Ž2df. and the midpoint bond function set 3s3p2d suggested by Tao and Pan w20x. The calculations were performed for nine bending angles u ranging from 0 to 1808 and 15 ˚ A total of distances R ranging from 2.5 to 8.0 A. 135 discrete potential energy points were calculated using Boys and Bernardi’s full counterpoise method w21x to correct the basis set superposition errors. MP4 calculations were also carried out for the three local minimum to ascertain the MP2 calculations. In this Letter, the values of the potential V Ž R, u . are calculated as follows. Nine new grid points V Ž R, u k . are first extracted from the cubic spline interpolation in the R direction for each input angle ˚ R ( 8.0 A. ˚ For other values of R, the u k for 2.5 A potential V Ž R, u k . is calculated using the following forms:
˚ V Ž R , u k . s A Ž u k . eya Ž u k . R , R ( 2.5 A
Ž 1.
V Ž R , u k . s C6 Ž u k . Ry6 q C 7 Ž u k . Ry7
˚ qC8 Ž u k . Ry8 , R 0 8.0 A.
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the ab initio values of V Ž2.50, u k . and V Ž3.00, u k ., C6, C7 and C8 are determined from the values of V Ž6.00, u k ., V Ž7.00, u k . and V Ž8.00, u k .. Then, the cubic spline interpolation in the u direction from the nine grid points V Ž R, u k . is performed to produce the potential energy V Ž R, u .. The cubic spline interpolation for Vm Ž u . indicates that the surface has three local energy minima and two barriers: the first local minimum is the global minimum, corresponding to the geometry of R s ˚ and u s 71.888. The second and third min3.554 A ˚ u s 1808, and R s ima are located at R s 4.427 A, ˚ u s 08. Between the three energy minima, 4.928 A, there are two energy barriers. The first barrier is located between the first and second energy mini˚ and u and mum with the geometry of R s 4.318 A u s 1248 and the barrier height is 43.25 cmy1 . And ˚ and u s 278 the second one is located at R s 4.670 A with the height of 38.74 cmy1 . Numerous procedures were suggested to calculate the bound states of a vdW complex. Peet and Yang w22x suggested the collocation method. Choi and Light w23x implemented the discrete variable representation ŽDVR. method. Korambath et al. w24x modified the discrete variable representation method. Clary and Knowles w25x presented a self-consistent field–configuration-interaction ŽSCF-CI. approach. In this Letter, we used an alternative vibrational SCF-CI procedure developed from our previous work w26,27x on stable triatomic molecules. In this procedure, the SCF method is used to optimize the basis set for bending and stretching motions in the vdW complex and then the CI method is employed to produce the converged results. For a given Ž JK ., where J is the total angular momentum and K is its projection on the body-fixed z-axis, the vibrational eigenfunction FnJK can be expanded as:
FnJK Ž R , u . s Ý c mJKn wm 1Ž R . wmK2Ž u .
Ž 3.
m
where, m is a collective vibrational index denoting the values of m1 and m 2 , the expansion coefficient c mJKn can be obtained by diagonalizing the following matrix
Ž 2.
Here, the parameters A and a are determined from
² wm Ž R . wmK Ž u . FMJ pK < H < wm Ž R . wmK Ž u . FMJ pK : 1
2
X 1
X 2
Ž 4.
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
164
in which H is the complete rovibrational Hamiltonian of the complex and FMJ pK is the parity-adapted rotation functions. wm 1 Ž R . and wmK2 Ž u . are the optimal independent-mode wavefunctions which are determined by using the following one-dimensional independent-mode Hamiltonians, 0 Js0, ps0 < < Ks 0 h1 Ž R . s² wmKs Ž u . FMs0, Ks0 H wm 2 s0 Ž u . 2 s0
The final rovibrational energy levels and wavefunctions can be obtained by diagonalizing the following Hamiltonian matrix X
²FnJK Ž R ,u . FMJ pK < H
X
3. Results and discussions
Js 0, ps0 =FMs 0, Ks0 :
Ž 5.
h 2JK Ž u . s² wm 1s0 Ž R . FMJ pK < H < wm 1s0 Ž R . FMJ pK :
Ž 6.
In the case of J s 0, the above two secular problems are coupled to each other, so the SCF procedure should be used. In the case of J ) 0, only the h 2JK Ž u . and the corresponding eigenfunctions need to be determined. In this Letter, we use a set of associated Legendre functions to solve variationally the bending one-dimensional eigenvalue problem, and use the renormalized Numerov–Johnson w28x method to solve numerically the stretching one-dimensional eigenvalue problem in the region w R min , R max x. Although the radial basis function thus determined is not an analytical function, it is convenient to evaluate the related integrals over the radial basis functions by use of the HEG quadrature w29x.
In the SCF-CI calculations, the integration limits ˚ and R max s 14.0 A˚ 100 are taken as R min s 2.3 A numerical radial basis functions are obtained by solving Eq. Ž5. with 5000 grid points, and 100 angular basis functions are determined by solving Eq. Ž6. variationally with associated Legendre functions. The vibrational CI calculations are performed with 1800 lowest configurations. The vibrational energy levels for Ne–OCS and its four isotopomers Ž 22 Ne–OCS, Ne–OC 34 S, Ne–O 13 CS, Ne– 18 OCS. are listed in Table 1 together with assignments of the quantum numbers Ž n,b . for the vdW stretching and bending modes. It should be noted that the assignments are only loosely defined, since there are considerable mixing between the different modes in the excited vibrational states. One can see from Table 1 that the
Table 1 Vibrational energy levels of isotopomers of Ne–OCS Žin cmy1 . n
b
Ne–OCSa
Ne–OCSb
22
0 0 1 0 1 2 1 0 2 2 3 3 3 4 3 4 4 4
0 1 0 2 1 0 2 3 1 2 0 1 2 0 3 1 2 3
y59.491 y42.518 y38.972 y30.554 y29.000 y24.508 y22.282 y17.032 y16.390 y14.924 y12.960 y10.213 y8.920 y6.109 y5.328 y4.295 y2.921 y2.470
y59.491 y42.521 y38.978 y30.573 y29.066 y24.765 y22.324 y17.244 y16.545 y15.306 y13.000 y10.405 y9.239 y6.157 y5.422 y4.910 y3.154 y2.621
y60.068 y43.137 y39.875 y31.224 y29.934 y25.358 y23.309 y18.097 y17.71 y16.211 y14.285 y11.237 y10.467 y6.913 y6.477 y5.776 y4.144 y3.107
a b
Ž 7.
Calculations with 1700 lowest configurations. Calculations with 1800 lowest configurations.
Ne–OCS
Ne–OC 34 S
Ne–O13 CS
Ne– 18 OCS
y59.614 y42.740 y39.120 y30.778 y29.176 y25.341 y22.692 y18.295 y16.646 y15.719 y13.266 y11.149 y10.213 y6.635 y6.164 y5.852 y3.553 y3.304
y59.547 y42.582 y39.056 y30.646 y29.173 y24.730 y22.363 y17.169 y16.715 y15.319 y13.106 y10.356 y9.242 y6.230 y5.400 y4.783 y3.224 y2.578
y59.852 y43.033 y39.388 y31.180 y29.765 y24.754 y22.549 y17.716 y17.237 y15.310 y13.682 y10.289 y9.376 y6.820 y5.448 y4.350 y3.570 y2.426
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
165
Table 2 Coefficients of the CI wavefunctions for the four lowest vibratonal states No.
E Žcmy1 .
n
b
Coefficient
1 2
y59.491 y42.521
3
y38.978
4
y30.573
0 0 1 1 0 1 0 1 1 0 1 0 1 1
0 1 0 1 1 0 2 2 0 2 2 3 3 4
0.992 y0.849 y0.208 y0.311 0.295 y0.781 y0.334 y0.260 0.382 y0.478 y0.363 0.214 0.236 0.278
Fig. 1. The probability density for finding Ne relative to the OCS molecule for the vibrational ground state.
Only the values larger than 0.20 are given.
ab initio potential supports 20 bound vibrational states for all of the five complexes. The energy levels of Ne–OCS using 1700 configurations are also listed in Table 1 to show the convergence of the state energies. Since the strong anisotropy of the potential, the convergence for higher energy states is slower. The three lowest levels are converged to within 0.01 cmy1 , other higher levels are converged to within 0.5 cmy1 or better. Table 2 gives the coefficients of the CI wavefunctions for the four lowest vibrational states of Ne–OCS. Figs. 1 and 2, and 3 plot the probability densities
Fig. 2. The probability density for finding Ne relative to the OCS molecule for the first vibrational excited state.
Fig. 3. The probability density for finding Ne relative to the OCS molecule for the second vibrational excited state.
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
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vibrational ground state, the value of ² P2 Žcos u .: is y0.342, corresponding to a ² u : of 71.058. This is close to the experimental value of 70.448 w12x and deviated by 38 from the value of 748 for the asymmetric T-shaped minimum of the ab initio potential w16x. This shows the rigid behavior of vdW bending motion in the angular direction. Fig. 1 also shows that the ground state vibrational wavefunction is well localized in the T-shaped configuration. It is shown in Table 2 that the first vibrational excited state is the vdW bending state with the frequency of 17.0 cmy1 . The value of ² P2 Žcos u .: is y0.319, corresponding to ² u : s 69.678. Fig. 2 shows that two maxima of the CI wavefunction are
located at u s 62 and 848, respectively, and the node plane is located at u s 728. The second excited vibrational state is the vdW stretching state with the frequency of 20.5 cmy1 . The value of ² P2 Žcos u .: is y0.318, corresponding to ² u : s 69.618. Fig. 3 shows that the wavefunction has three maxima at 62, 72 and 908, respectively. The two node planes are at u s 65 and 808. The first and second vibrational excited states are localized in the region of the global minimum. In addition, we calculated the rovibrational energy levels Žavailable upon request. for five isotopomers of Ne–OCS ŽNe–OCS, 22 Ne–OCS, Ne–OC 34 S, Ne–O 13 CS, Ne– 18 OCS.. In these calculations, for a
Table 3 Calculated transition frequencies of isotopomers of Ne–OCS Žin cmy1 . a JKX X a K X c 111 2 02 2 11 2 12 2 12 2 20 2 21 3 03 3 03 312 312 313 313 3 21 3 22 3 21 3 22 4 04 4 04 4 14 4 13 4 13 4 23 4 22 4 22 505 514 514 523 6 15 6 24 a
JKY Y a K Y c 0 00 1 01 110 111 1 01 2 11 2 12 2 02 2 12 2 11 303 2 12 2 02 2 20 2 21 312 313 303 313 313 4 04 312 322 4 13 321 4 14 505 4 23 514 6 06 6 15
22
Ne–OCS
Ne–OCS
Ne–OC 34 S
Ne–O13 CS
Ne– 18 OCS
yobs
ycal
yobs
ycal
yobs
ycal
yobs
ycal
yobs
ycal
0.308 0.293 0.318 0.272 0.432 0.483 0.549 0.433 0.293 0.475 0.250 0.406 0.546 0.451 0.442 0.459 0.585 0.568 0.454 0.539 0.312
y0.004 y0.007 y0.009 y0.006 y0.006 y0.002 y0.006 y0.010 y0.010 y0.013 y0.007 y0.008 y0.008 y0.012 y0.011 y0.001 y0.009 y0.012 y0.014 y0.011 y0.013 0.613 y0.014 y0.001 0.592 y0.016 y0.020 y0.024 y0.002 y0.029 y0.006
0.303 0.277 0.299 0.258 0.422 0.493 0.553 0.411 0.266 0.448 0.244 0.386 0.531 0.425 0.418 0.470 0.585 0.539 0.419 0.512 0.298
y0.004 y0.007 y0.008 y0.006 y0.006 y0.001 y0.005 y0.010 y0.010 y0.013 y0.006 y0.008 y0.008 y0.012 y0.011 0.000 y0.008 y0.012 y0.014 y0.011 y0.011 0.578 y0.014 0.000 0.557 y0.017 y0.017 0.292 y0.001 y0.025 0.431
0.303 0.287 0.312 0.267 0.425 0.476 0.541 0.426 0.288 0.467 0.246 0.400 0.538
y0.004 y0.007 y0.009 y0.005 y0.006 y0.002 y0.006 y0.009 y0.010 y0.013 y0.007 y0.008 y0.007 0.432 y0.011 y0.001 y0.009 y0.011 y0.013 y0.010 y0.012 0.602 0.564 y0.001 0.582 y0.015 0.371 0.345 0.419 0.467 0.421
0.306 0.292 0.317 0.271 0.431 0.479 0.546 0.432 0.293 0.474 0.249 0.405 0.544
y0.004 y0.007 y0.009 y0.006 y0.006 y0.001 y0.006 y0.010 y0.011 y0.013 y0.007 y0.008 y0.008 0.438
0.290 0.290 0.317 0.268 0.412 0.433 0.502 0.428 0.305 0.473 0.240 0.401 0.523
y0.004 y0.007 y0.009 y0.006 y0.006 y0.001 y0.005 y0.009 y0.010 y0.013 y0.007 y0.008 y0.007 0.437
0.587 0.438 0.611 0.397 0.376 0.427 0.505 0.433
0.555 0.449 0.570 0.373 0.435 0.469
0.435 0.453 0.576 0.558 0.446 0.530 0.307
0.432 0.600
0.430 0.456 0.581 0.566 0.454 0.538 0.311
0.000 y0.008 y0.012 y0.014 y0.011 y0.013 0.611 0.572 0.434 0.591 0.594 0.377 0.354 0.422 0.476 0.424
0.428 0.410
0.463 0.532
0.532 0.548 y0.013 y0.010 0.294 0.610 0.568 0.391 0.591 0.600 0.379 0.399 0.384 0.485 0.394
The observed values are taken from Ref. w12x. For observed transitions, the calculated values are given as calculated-observed, otherwise the actual calculated values are given.
G. Yan et al.r Chemical Physics Letters 287 (1998) 162–168
given value of Ž JK ., the vibrational SCF-CI method with 1000 lowest configurations is used to calculate to vibrational energy levels and wavefunctions. The 100 lowest vibrational states thus obtained are then used to calculated the Hamiltonian matrix elements that are given in Eq. Ž7.. The rovibrational energy levels and wavefunctions are obtained by diagonalizing this Hamiltonian matrix. In order to test the convergence of the state energies, we performed calculations with 1500 lowest configurations. The results show that the rovibrational energy levels for the ground vibrational states are converged within 0.01 cmy1 or better. Microwave frequencies associated with the observed rotational transitions can be obtained directly from the energy difference between appropriate energy levels. The calculated transition frequencies for the five isotopomers of Ne–OCS are listed in Table 3 along with the observed values w12x. It is seen from Table 3 that most of the calculated frequencies are within 0.01 cmy1 of the observed values. The results show that the ab initio PES agrees with the measured rotational frequencies.
4. Conclusions This study presents theoretical calculations of the rovibrational bound states of the Ne–OCS complex using our ab initio potential energy surface of the complex. The ab initio potential supports 20 bound vibrational states. The rovibrational ground state of Ne–OCS is bounded by 59.49 cmy1 , corresponding to a zero-point energy of 23.45 cmy1 . The rigid behavior of vdW bending motion in the angular direction was found by analyzing the average value of u over the bound state. From the given probability densities of Ne relative to OCS, it was also found that the wavefunctions for the lowest two excited vibrational states are localized in the region of the global minimum. Most of the calculated frequencies for pure rotational transitions of the five isotopomers are in good agreement with the observed values. It is expected that the global description of the rovibrational bound states might provide useful information for further observation of the spectroscopy of the Ne–OCS complex.
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Acknowledgements Project 29673029 was supported by National Natural Science Foundation of China. We wish to thank Professor Auchin Tang for his stimulation of this work. We also appreciate most gratefully the reviewer for his very helpful suggestions on the manuscript.
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