417
Theoretical and experimental study of infrared spectra of He2-CO21 Jian Tang, A.R.W. McKellar, Xiao-Gang Wang, and Tucker Carrington Jr.
Abstract: Additional high-resolution lines in the n3 CO2 fundamental band of the rovibrational spectrum of He2-CO2 have been observed using a tunable infrared laser to probe a pulsed supersonic jet expansion. The ro-vibrational spectrum was calculated using Euler angles and five vibrational coordinates that specify the positions of the He atoms with respect to the CO2 molecule, a product basis, and a Lanczos eigensolver. Rotational states with J = 0, 1, 2, and 3 associated with the vibrational ground state and different states (nt) of the torsional motion of the two He atoms about the CO2 axis are identified and assigned. The assignment is consistent with having different principle axis orientiations in nt = 0 and nt = 1 states. Dnt = 1 infrared transistions are forbidden due to the symmetry of CO2, but Dnt = 1 microwave transistions are possible. Theoretical line positions and intensities are predicted. Good agreement between experiment and theory was obtained. The calculated energy levels and intensities were crucial in assigning some of the weaker observed transitions. PACS No: 78.30.–j Re´sume´ : Utilisant un laser infrarouge accordable et un fasiceau supersonique pulse´, nous avons observe´ de lignes additionnelles de haute re´solution dans la bande v3 fondamentale du CO2 du spectre rovibrationnel du He2-CO2. Le spectre rovibrationnel a e´te´ calcule´ en utilisant les angles d’Euler et les cinq coordonne´es vibrationnelles qui de´crivent la position des atomes de He par rapport a` la mole´cule de CO2, ainsi qu’une base produit et l’algorithme de Lanczos. Nous identifions et attribuons les e´tats rotationnels J = 0, 1, 2 et 3 associe´s au fondamental vibrationnel ainsi que les e´tats (vt) dus au mouvement de torsion des deux atomes de He par rapport a` l’axe du CO2. L’attribution est cohe´rente avec un choix d’orientation des axes principaux diffe´rent pour les e´tats vt = 0 et vt = 1. Les transitions infrarouges Dvt = 1 sont interdites par la syme´trie du CO2, mais les transitions Dvt = 1 micro-ondes sont possibles. Nos positions et intensite´s des lignes calcule´es sont en bon accord avec les donne´es expe´rimentales. Nous pre´sentons aussi des pre´dictions. Les e´nergies et intensite´s calcule´es sont cruciales dans l’assignation de certaines des transitions les plus faibles.
1. Introduction The study of helium clusters provides a means of linking the microscopic world of individual molecules with the macroscopic world of bulk matter (in this case, liquid helium). Helium nanodroplet isolation (HENDI) spectroscopy [1] has been used to study vibrational and rotational dynamics of individual molecules (e.g., OCS or CO2) [2, 3] embedded in large but still finite scale clusters (N >> 103–104 He atoms), while direct infrared or microwave spectroscopy can now also probe small clusters containing 1 to 10 or more He atoms [4–8]. In liquid 4He, Bose–Einstein statistics are reReceived 24 October 2008. Accepted 6 November 2008. Published on the NRC Research Press Web site at cjp.nrc.ca on 8 July 2009. J. Tang2 and A.R.W. McKellar. Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, ON K1A 0R6, Canada. X. Wang and T. Carrington, Jr..3 Chemistry Department, Queen’s University, Kingston, ON K7L 3N6, Canada. 1This
article is part of a Special Issue on Spectroscopy at the University of New Brunswick in honour of Colan Linton and Ron Lees. 2Present address: Department of Chemistry, Faculty of Science, Okayama University, 3-1-1 Tsushima-naka, Okayama 7008530, Japan. 3Corresponding author (e-mail:
[email protected]). Can. J. Phys. 87: 417–423 (2009)
sponsible for superfluidity, while in small clusters some levels are missing because of the fact that 4He nuclei are bosons. By studying the evolution of the spectra as a function of cluster size, we hope to obtain a more complete understanding of these symmetry effects. In a recent paper, we presented theoretical and experimental results on the rovibrational energy levels of the He2-N2O complex and its resulting spectrum [9]. The present paper extends this study to a closely related system, He2-CO2, for which we have observed the infrared spectrum in the region of the n3 band of CO2 (2350 cm–1), calculated in detail the rotation–torsion energy levels and spectrum, and compared theory with experiment to understand the observations. Although He2-CO2 and He2-N2O are similar in many respects, there is a crucial difference: the higher symmetry of CO2 means that half of the rotational levels of He2-CO2 are missing, relative to He2-N2O. But half the He2-N2O levels are already missing because of 4He interchange symmetry, so in fact only one quarter of the normally expected rotational levels are allowed for He2-CO2. This sparse energy level structure leads to a sparse spectrum, which is therefore easier to assign but ultimately carries less information. The infrared spectrum of He-CO2 in the 2350 cm–1 region was studied in 1994 by Weida et al. [10]. More recently, Xu and Jaeger [11] observed a forbidden microwave transition, and McKellar [12] studied the infrared spectrum with isotopically substituted CO2. This complex was found to have a T-shaped structure ˚ . When a secwith an intermolecular distance of about 3.6 A
doi:10.1139/P08-119
Published by NRC Research Press
418
Can. J. Phys. Vol. 87, 2009
Fig. 1. Part of the observed infrared spectrum of HeN-CO2 clusters, with transitions labelled by the number of helium atoms, N, given in circles. Approximate experimental conditions were: slit jet nozzle; nozzle temperature, –30 8C; backing pressure, 26 atm (1 atm = 101.325 kPa). The lines labelled ‘‘2’’ (for He2-CO2) are those studied in this paper.
ond He atom is added to form He2-CO2, it assumes an equivalent position to the side of the linear CO2 entity. The equilibrium structure is dihedral, due to weak attraction between the He atoms, but the barrier to planarity is small. The resulting low frequency vibrational dynamics are dominated by the torsional motion of the two He atoms on a ring around the equator of the CO2, giving rise to a series of torsional states, nt = 0, 1, 2, etc. In the somewhat analogous complexes He2-N2O [9] and He2-Cl2 [13, 14], the splitting between nt = 0 and 1 is only about 0.5 cm–1, and the same turns out to be true in the case of He2-CO2. The torsional states have different symmetry properties, and so the allowed rotational levels depend on whether nt is even or odd. The infrared spectra of HeN-CO2 clusters with N = 2 to 17 were studied recently by Tang and McKellar [7, 8]. They reported 7 transitions for He2-CO2, of which 5 were assigned to nt = 0 and 2 to nt = 1. The experimental results in the present paper are a more detailed extension of that work [7]. With the help of the theoretical results reported here, we have now assigned a total of 12 transitions for He2-CO2, divided equally between nt = 0 and 1. The rotational assignments given previously [7] are correct for nt = 0 but not (quite) for nt = 1, due to subtle symmetry and axis-switching effects. The intermolecular potential energy surface for the HeCO2 interaction has been the subject of a number of studies [15–19], in three of which the ab initio surfaces were used to calculate He-CO2 spectra for comparison with experiment [16, 18, 19]. The most recent surface of Li and Le Roy [19] is the only one that considers CO2 antisymmetric vibration. In this study, we choose to use the surface of Korona et al. [18]. We plan to use the surface of Li and Le Roy in the future to explicitly study the effect of the dopant vibration. All of these potentials have a global minimum with a depth of 45–50 cm–1 at a T-shaped configuration with an intermo-
˚ and a secondary minimum lecular distance of R * 3.1 A –1 with a depth of about 29 cm at a linear configuration with ˚ . Recently a path integral Monte Carlo method R * 4.25 A was used to study the shift of the CO2 antisymmetric vibration band origin and the effective rotational constant in HeNCO2 (N £ 17) as a function of the cluster size at low temperatures [20].
2. Experiment Experimental details are the same as in our paper on HeNCO2 clusters [7]. A tunable diode laser used in a rapid scan mode probes a pulsed supersonic jet expansion. The laser beam is passed many times (>100) through the jet, and spectra are calibrated by simultaneously recording signals from a temperature-stabilized etalon and a room temperature CO2 gas cell. Very dilute (0.1% to 0.002%) mixtures of CO2 in helium are expanded through a slit-shaped jet nozzle, which can be cooled by circulating gas from a liquid nitrogen dewar. To minimize the strong absorption by atmospheric CO2 in this region (2350 cm–1), the infrared optical path is purged with N2. The spectral lines due to He2-CO2 were distinguished from those due to smaller (CO2, He-CO2) and larger (He3-CO2, He4-CO2, etc.) clusters mainly by observing their dependence on the jet backing pressure and nozzle temperature. Other clusters containing more than one CO2 molecule (including the CO2 dimer) [21] were effectively eliminated by using very dilute gas mixtures. Figure 1 shows a portion of the observed spectrum, including 9 out of our 12 He2-CO2 transitions. The numbers enclosed in circles in Fig. 1 denote N, the number of He atoms in the cluster HeNCO2, and the experimental conditions were chosen to give strong He2-CO2 lines without too many larger clusters. By far the strongest line in this spectrum is the R(0) transition of the CO2 molecule itself, and the next strongest are those Published by NRC Research Press
Tang et al.
419
Table 1. Calculated energy levels of n3 = 0 state of He2-CO2. Aþ 1 J=0 0.0000(0) 1.6554(2) 6.3768(4) 9.0775 11.6422 J=1
J=2 1.0418(0;202) 1.4828(0;220) 2.7931(2;202) 3.1918(2;220) J=3 2.5843(0;322) 4.2630(2;322)
Aþ 2
Bþ 1
Bþ 2
A 1
A 2
B 1
B 2
8.2358 10.0055
9.4733 12.3681
8.1546 10.9776 12.6218
9.7196
12.8012
10.4809
0.5007(1) 3.7560(3) 9.6403(5) 12.5323
0.8902(1;101) 4.1075(3;101)
0.9072(1;110) 4.1590(3;110)
0.4897(0;110) 2.1337(2;110)
0.8954(1;111) 4.1566(3;111)
0.4648(0;111) 2.1183(2;111)
0.3535(0;101) 2.0365(2;101)
1.7249(1;221) 5.0514(3;221)
1.6812(1;212) 4.8982(3;212)
1.1396(0;212) 2.8552(2;212)
1.6976(1;211) 4.8785(3;211)
1.2230(0;211) 2.8750(2;211)
1.4850(0;221) 3.1931(2;221)
1.6687(1;202) 1.7293(1;220) 4.8339(3;202) 5.0642(3;220)
2.8207(1;303) 2.9059(1;321) 5.9347(3;303) 6.1391(3;321)
2.8472(1;312) 2.9543(1;330) 5.9533(3;312) 6.4846(3;330)
2.3100(0;312) 2.8622(0;330) 4.0000(2;312) 4.9930(2;330)
2.8394(1;313) 2.9530(1;331) 5.9916(3;313) 6.4760(3;331)
2.1090(0;313) 2.8589(0;331) 3.9938(2;313) 4.9889(2;331)
2.0333(0;303) 2.5923(0;321) 3.9349(2;303) 4.2705(2;321)
2.9033(1;322) 6.1208(3;322)
Note: The assignments (nt; JKaKc) give the torsional (nt) and rotational (JKaKc) quantum numbers. Only nt assignments are given for the J = 0 levels. Only levels labelled here as having Aþ 1 and A1 symmetry are allowed
due to He-CO2 [10]. Next in intensity come the R(0) lines of He2- and He3-CO2, followed by a number of additional He2CO2 lines. Note how the R(0) lines form a smooth compact progression for N = 0 to 4; the further evolution of this series is described in refs. 7 and 8. Compared to the case of He2N2O [9], the present spectrum is cleaner and less crowded. The symmetry of CO2 (and zero spin of the 16O nucleus) mean that He2-CO2 and all sizes of cluster have fewer possible transitions, making it easier to detect weak transitions. The observed He2-CO2 transitions are listed and compared with theory in Sect. 5 below.
3. Theory The method used to compute the rovibrational spectrum of He2-CO2 is essentially the same as the one used for He2N2O and is described in detail in ref. 9. In this section, we present a brief summary and discuss the changes in the method required to account for the fact that the CO2 dopant has two identical atoms for which permutation is feasible. The shape of He2-CO2 is represented by three vectors: (r0, r1, r2). r0 is along the (assumed linear) CO2 axis. r1 and r2 are Radau vectors from a canonical point, whose position is determined by the centre of mass of CO2 and the positions of the two He nuclei, to the two He nuclei. These vectors are sometimes called orthogonalized satellite vectors. See, for example, ref. 22 and Fig. 1b of ref. 23. The polyspherical angles of these three vectors (q1, q2, f) are used as bend coordinates [22]. The kinetic energy operator (KEO) in terms of these coordinates is simple because the vectors are ‘‘orthogonal’’ [22, 24]. Since we are only interested in the slower intermolecular motions, the fast internal vibration of CO2 is frozen. The effect of the antisymmetric stretch mode of CO2 (n3) is, however, included by using different rotational constants for the n3 = 0 and 1 states. The basis functions we use are products of a parity
adapted bend-rotation function and potential optimized discrete variable representation [25, 26] (PODVR) functions for the stretches. Each basis function is of the form, fa1 ðr1 Þfa2 ðr2 ÞuJMP l1 l2 m2 K ðq 1 ; q 2 ; f2 ; a; b; gÞ
ð1Þ
The parity-adapted bend-rotation function uJMP l1 l2 m2 K is a combination of primitive functions that are products of an associated Legendre function for q1, a spherical harmonic for (q2, f2), and a Wigner function for the Euler angles (a, b, g). The molecular symmetry group of He2-CO2 includes the inversion operation E*; permutation of the two He nuclei, denoted as (12); and permutation of two O nuclei, denoted as (34). Therefore, He2-CO2 belongs to permutation-inversion group G8 ¼ fE; Eg fE; ð12Þg fE; ð34Þg. The irrep labels are defined as follows: A and B label symmetry with respect to permutation of the He nuclei; subscripts 1 and 2 label symmetry with respect to permutation of the O nuclei; and supersripts + and – label parity labels. Because 4He and 16O are both bosons with 0 nuclear spin, the rovibrational wave functions are required to be symmetric under permutation of 4He nuclei and permutation of 16O nuclei. Therefore, only Aþ 1 and A1 symmetry states are physically allowed. This removes roughly 75% percent of computed levels, simplifying the spectra. The effect of the symmetry operations on the uJMP l1 l2 m2 K functions is discussed in ref. 23. The results are summarized in Table 4 of ref. 23 and reproduced here, P JMP E uJMP l1 l2 m2 ;K ¼ ð1Þ ul1 l2 m2 K JMP ð12ÞuJMP l1 l2 m2 ;K ¼ ul2 l1 m1 ;K
ðif K > 0Þ
JþP JMP ð12ÞuJMP ul2 l1 m 1 ;K l1 l2 m2 ;K ¼ ð1Þ
ðif K ¼ 0Þ
l1 þl2 þP JMP ul1 l2 m1 ;K ð34ÞuJMP l1 l2 m2 ;K ¼ ð1Þ
ð2Þ
The product basis is very large, and it is necessary to use Published by NRC Research Press
420
Can. J. Phys. Vol. 87, 2009
Table 2. The relationship between (n3, nt, JKaKc) labels and permutation inversion group symmetries for various vibrational states. State |n3 = 0, |n3 = 0, |n3 = 1, |n3 = 1,
nt nt nt nt
= = = =
0,2> 1,3> 0,2> 1,3>
eea A1 þ B 2 Aþ 2 B 1
oe Bþ 2 Bþ 1 Bþ 1 Bþ 2
oo A 2 A1 A1 A 2
eo B 1 Aþ 2 B 2 A1 þ
Note: The results are derived from J = 0 and 1 levels and expected to hold for higher J levels. The symmetries of |n3 = 1> and |nt = 1> are Aþ 2 and B 2 , respectively. The physically allowed symmetries are in bold face. a
Evenness and oddness of KaKc.
Fig. 2. Diagram of lowest few energy levels of n3 = 0 state (lower panel) and n3 = 1 state (upper panel). Levels marked by black (dashed) bars are for nt = 0 and 1 states, respectively. Infrared transitions are shown. Note also that levels of six physically forbidden symmetries are not shown.
over l2 would depend on l1. In addtion, because summations are done sequentially, intermediate vectors labelled by grid indices would be as large as the full grid, and there would be no memory savings. It is much simpler to instead use the SAL to obtain levels for all of the even (odd) irreps using one set of matrix–vector products. To do so, one needs to þ þ construct projection operators for the irreps Aþ 1 , A2 , B1 , þ B2 , in the even parity case. This is easily done using (2). It is a good approximation to assume that the dipole moment is oriented along the dopant axis, but constant, and therefore independent of the intermolecular coordinates (r1, r2, q1, q2, f2) associated with the motions of the two He atoms. Transitions between states whose wave functions have different intermolecular vibrational (in particular the He torsion) excitations occur because of rovibrational coupling. However, due to the symmetry of CO2, Dnt = 1 infrared transitions are forbidden and Dnt = 2 transitions are possible. See Sect. 5 for details. The line strengths of the rovibrational transitions are computed using the computed rovibrational wave functions. Details have been presented in ref. 9.
4. Calculation details The potential we use for He2-CO2 is a sum of two He-CO2 potentials and a He-He potential. The He-CO2 potential of ref. 18 and the He-He potential of ref. 30 were used. The masses are 4.0026 u for 4He and 15.99491 u for 16O. The CO2 rotational constants are 0.390219 cm–1 and 0.389135 cm–1 for the n3 = 0 and v3 = 1 states, respectively, following ref. 18. The potential is zero when all three of the constituents of the complex are far apart. The global minimum geometry is r1e = r2e = 5.45325 bohr, q1e = q2e = 908, f2e = 61.788. At the minimum, the potential is –108.36 cm–1. The PODVR basis functions in (1) were obtained from eigenfunctions of a 1D cut potential in the range [2.0 bohr to 20.0 bohr]. We use 35 functions for both r1 and r2. The maximum value of l1, l2, and m is 25. The number of q quadrature points is 30 and the number of f quadrature points is 64. The basis size and the number of quadrature points have been chosen to ensure that the levels we report are converged within 0.001 cm–1. A potential ceiling of 1000 cm–1 is imposed to reduce the spectral range (without generating significant errors) and hence accelerate the convergence of the iterative eigensolver [31]. The size of the product basis defined above is 7.6 million for the case of J = 0 even parity. The size grows by a factor of 2J + 1 if J > 0. To compare our computed energy levels directly with experimental results, we calculate rovibrational transition wavenumbers from, n~ ¼ n~ 0 þ Eðv ¼ 1Þ Eðv ¼ 0Þ an iterative method to obtain the energy levels and wave functions. We choose a symmetry adapted variant (SAL algorithm) [27, 28] of the the Cullum and Willoughby [29] Lanczos method. To exploit permutational symmetry without the SAL, it would be necessary to divide basis functions into groups and to do matrix–vector products for different groups separately. For example, for (34), basis functions with l1 + l2 even and with l1 + l2 odd would be put into different groups. If this grouping were used, limits of the sum
ð3Þ
where n~ 0 ¼ 2349:37 cm1 [7] is the n = 1 band centre and E(n = 0) and E(n = 1) are calculated wavenumbers measured from the n3 = 0 and n3 = 1 ground states. To reduce the differences between the calculated and observed transitions, we could have adjusted n~ 0 , but we have not done so. In Table 1, we report n3 = 0 levels for J = 0, 1, 2, 3 and give torsional and rotational assignments |nt; JKaJKc>. The corresponding n3 = 1 levels are similar and not presented. The lowest vibrational frequency of the complex is that of the He2 torsion. The torsional levels for the n3 = 0 calculaPublished by NRC Research Press
Tang et al.
421
Table 3. Observed and calculated transitions in the infrared spectrum of He2-CO2 (in cm–1). vt0 0 1 0 1 1 0 0 1 0 1 1 0
JK0 a0 Kc0
JK00a00 Kc00
111 101 111 101 221 211 111 221 313 303 321 331
220 211 202 111 211 202 000 111 202 211 211 220
sobsb 2348.3354a 2348.5632 2348.8060 2349.3609 2349.3950 2349.5601 2349.8395 2350.1921 2350.4255 2350.4762 2350.5642 2350.7485
scalc 2348.3493 2348.5603 2348.7903 2349.3625 2349.3917 2349.5487 2349.8322 2350.1939 2350.4313 2350.4882 2350.5698 2350.7345
sobs – scal –0.0139 0.0029 0.0157 –0.0016 0.0033 0.0114 0.0073 –0.0018 –0.0058 –0.0120 –0.0056 0.0140
Iobsd 0.07a 0.04 0.08 0.09 0.01 0.10 1.00 0.19 0.11 0.01 0.02 0.07
Icale 0.04 0.02 0.06 0.08 0.01 0.13 1.00 0.23 0.31 0.02 0.04 0.10
a
This line was observed with a poor signal-to-noise ratio. Observed line position in cm–1. c Calculated line position in cm–1. d Approximate observed line intensity, relative to the 111-000 transition. e Calculated line intensity, relative to the 111-000 transition, for an assumed temperature of 0.5 K. b
Fig. 3. Calculated (T = 0.5 K) and observed infrared spectra of He2-CO2. The intensity of R(0) transition is 1 for both the calculation and the experiment. There are two types of transitions: transitions from nt = 0 to nt = 0 and transitions (marked by a star) from nt = 1 to nt = 1.
tion are assigned by examining the wave function along the f2 coordinates. Specifically, J = 0 levels at 0.501, 1.655, 3.756, 6.377, 9.640 cm–1 are assigned as the nt = 1, 2, 3, 4, 5 torsional states. For each J, the lowest set of 2J + 1 levels is assigned to the nt = 0 state. The next set is assigned to the nt = 1 levels, and so on. Within each nt set, (Ka, Kc) = (J,0), (J,1), . . .(0,J) are assigned to levels in order of decreasing energy. In this fash-
ion, irrep labels are linked to n3, nt, JKaKc labels. This works well at least up to nt = 3, see Table 1. The relationship between (n3, nt, JKaKc) labels and permutation inversion group symmetries for various vibrational states is given in Table 2. According to these assignments, levels with the same nt and n3 and the same KaKc evenness and oddness have the identical symmetries irrespective of J. Knowing the symmetries of |n3, nt, JKaKc states, and the fact that the symmetry of the nt = Published by NRC Research Press
422
Can. J. Phys. Vol. 87, 2009 Table 4. Fitted asymmetric rotor parameters for the calculated rotational levels (0 £ J £ 3) of the nt = 0, 1, and 2 states of He2-CO2 (in cm–1).
A B C DK DJK DJ dK dJ
nt = 0 0.2999 0.1940 0.1615 0.0087 –0.0038 0.0006 0.0028 –0.0003
nt = 1 0.2088 0.2037 0.1893 0.0013 –0.0011 0.0003 0.0006 0.0002
nt = 2 0.2801 0.1988 0.1768 –0.0070 0.0033 –0.0002 0.0023 0.0003
1 torsional factor is B 2 and assuming that each state is a product of a torsional factor and a rotational factor, one can determine symmetries of the rotational factors for each vibrational state. We find that for nt = 1 state, the (ee, oe, oo, eo) þ are associated with symmetries ðAþ 1 ; A2 ; B2 ; B1 Þ, while for nt = 0 state the (ee, oe, oo, eo) are associated with symmetries þ ðAþ 1 ; B2 ; A2 ; B1 Þ. From the symmetry of the rotational factors and asymmetric top symmetry rule [32], we can deduce principal axes. These axes are those that, if used with the asymmetric top symmetry rule, would give the relation between JKaKc and irrep labels that we report in Table 2. Using this procedure, one finds that for nt = 0 the b axis is along the CO2 axis, but that for nt = 1 it is the c axis that is along the CO2 axis. Therefore, exciting from nt = 0 to nt = 1 switches the principal axes. In comparison, in our study of He2-N2O [9], the assignment of all nt = 0 and nt = 1 states is consistent with the b axis being along the linear dopant. However for the He2-N2O nt = 1 state, we did find that 110 level is slightly lower than 111 level. Switching the axes would make 111 lower than 110. It is rovibrational coupling that causes the need to switch the axes to maintain the (Ka, Kc) order. If 111 had been significantly below 110 in He2-N2O, we would have switched the axes. Note that principal axes for each vibrational state are only meaningful to the extent that rovibrational coupling is weak. The differences between the n3 = 0 levels and the n3 = 1 levels are solely due to the difference in the rotational constants for CO2. In principle, it would be better to use potential surfaces for the n3 = 0 and n3 = 1 calculations, obtained by averaging the full potential with n3 = 0 and n3 = 1 wave functions. These potentials are expected to be very similar. The symmetries of the n3 = 0 levels are the same as the symmetries of the (intermolecular vibration + rotation 8d) wave functions we compute. The symmetries of the n3 = 1 states are obtained by by multiplying the symmetry of the n3 = 1 computed 8d eigenfunction with Aþ 2 (symmetry of n3 = 1 state). The physically allowed n3 = 1 and n3 = 0 states therefore have quite different patterns. For n3 = 0, it is the Aþ 1 and A1 8d states that are allowed. For n3 = 1, it is the þ A2 and A 2 8d states that are allowed. This would not be the case if permutation of the two ideal nuclei of the dopant was not a feasible operation (and is not, for example, the case for He2-N2O). The ZPEs for n3 = 0 and n3 = 1 calculations are –36.3520 and –36.3773 cm–1, respectively. Figure 2 shows the lowest n3 = 0 and n3 = 1 energy levels and infrared transitions between these two states.
5. Comparison of experiment and theory Seven of the twelve observed lines here were listed in [7], with the same rotational assignments for nt = 0 and slightly different ones for nt = 1. Most of the five new lines in Table 3 belong to nt = 1, and they are all relatively weak (less than 0.1 of the intensity of the strongest line). The agreement between observed and theoretical line positions is excellent, showing an rms deviation of just under 0.01 cm–1. The band origin was not re-adjusted. The experimental intensities in Table 3 are very approximate, being limited by the fact that our spectra were made up of numerous short scans obtained with varying experimental conditions and that the laser power varied within each scan (e.g., Fig. 1 is composed of 3 separate scans). With this in mind, the agreement of the observed and calculated (for T = 0.5 K) relative intensities in Table 3 is quite satisfactory. This is also demonstrated in Fig. 3, which shows the observed and calculated spectra in the form of a stick diagram. A line originally suspected to belong to He2-CO2 was observed at 2350.1746 cm– 1, but there is no calculated transition of sufficient intensity close to this. We now believe that this transition may actually be due to He3-CO2. Weak Dnt = 1 and 2 transitions were observed in He2N2O [9]. The situation is different for He2-CO2 because of the symmetry of CO2. Transitions can occur only between physically allowed states with different parities. This means that only transitions between Aþ 1 and A1 levels are permitted. All possible symmetry labels for several (n3, nt) states are given in Table 2. According to the table, transitions between states with n3 = 0 and nt = even(odd) and states with n3 = 1 and nt = even(odd) are possible, but transitions between states with n3 = 0 and nt = even(odd) and states with n3 = 1 and nt = odd(even) do not occur. Transitions within a n3 manifold (i.e., microwave transitions) and between nt = even(odd) and nt = odd(even) are permitted by symmetry. The fact that infrared transitions with Dnt = 1 are forbidden is due to the permutation symmetry of the O atoms in CO2. Without this symmetry, there would be no 1 and 2 subscripts in Table 2, and therefore only four symmetry species (A+, A–, B+, B–), and two allowed species, A+ and A–, between which transitions would occur. Infrared transitions with Dnt = 1 would be permitted. This is why such transitions are permitted in He2-N2O, but not in He2-CO2. If the dipole moment were along the dopant axis but depended not only on Q3 but also on the intermolecular coordinates then the integral would not be zero for all values of the intermolecular coordinates, and even if the wave function is a product, |n3>|F>, the microwave transitions would occur. The He atoms may induce a dependence of the dipole on the intermolecular coordinates. The best possibility for directly measuring the torsional splitting of He2-CO2 is therefore in the very high sensitivity of Fourier transform microwave spectroscopy, as demonstrated by the detection [11] of the 101-000 transition of He-CO2, whose transition intensity depends entirely on an induced dipole moment. An induced dipole of roughly the same magnitude in He2-CO2 should make it possible to observe forbidden rovibrational transitions with Dnt = 1. For example, we would predict the low-lying transitions (nt; JKaKc) = (1;111)-(0;000) and (0;202)-(1;111) to lie at about 26 840 and 4 390 MHz, rePublished by NRC Research Press
Tang et al.
spectively. As in the case of He2-N2O [9], we tried fitting the calculated energy levels with a conventional asymmetric rotor Hamiltonian to see how well this works and how much the parameters vary as a function of nt. The fits involved all the calculated levels in Table 1 (including forbidden ones), and the resulting parameters are listed in Table 4. The overall results are very similar to those for He2-N2O [9]. For example, the quality of the fits was worse for nt = 0 (largest residual
