Intermolecular potential and rovibrational states of the H2O–D2 complex

Chemical Physics 399 (2012) 28–38

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Intermolecular potential and rovibrational states of the H2O–D2 complex Ad van der Avoird a,⇑, Yohann Scribano b, Alexandre Faure c, Miles J. Weida d, Joanna R. Fair e, David J. Nesbitt f a

Theoretical Chemistry, Institute for Molecules and Materials, Radboud University Nijmegen, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands Laboratoire Interdisciplinaire Carnot de Bourgogne-UMR 5209, CNRS-Université de Bourgogne, 9 Av. Alain Savary, B.P. 47870, F-21078 Dijon Cedex, France c UJF-Grenoble 1/CNRS, Institut de Planétologie et d’Astrophysique de Grenoble (IPAG) UMR 5274, Grenoble F-38041, France d Daylight Solutions, 15378 Avenue of Science, San Diego, CA 92128, USA e Department of Radiology, MSC10 5530, 1 University of New Mexico, Albuquerque, NM 87131-0001, USA f JILA, University of Colorado and National Institute of Standards and Technology, and Department of Chemistry and Biochemistry, University of Colorado, Boulder, CO 80309-0440, USA b

a r t i c l e

i n f o

Article history: Available online 25 July 2011 Keywords: Intermolecular potential Hydrogen bonding H2O Water H2 Rovibrational states Infrared spectrum

a b s t r a c t A five-dimensional intermolecular potential for H2O–D2 was obtained from the full nine-dimensional ab initio potential surface of Valiron et al. [P. Valiron, M. Wernli, A. Faure, L. Wiesenfeld, C. Rist, S. Kedzˇuch, J. Noga, J. Chem. Phys. 129 (2008) 134306] by averaging over the ground state vibrational wave functions of H2O and D2. On this five-dimensional potential with a well depth De of 232.12 cm1 we calculated the bound rovibrational levels of H2O–D2 for total angular momentum J = 0–3. The method used to compute the rovibrational levels is similar to a scattering approach—it involves a basis of coupled free rotor wave functions for the hindered internal rotations and the overall rotation of the dimer—while it uses a discrete variable representation of the intermolecular distance coordinate R. The basis was adapted to the permutation symmetry associated with the para/ortho (p/o) nature of both H2O and D2, as well as to inversion symmetry. As expected, the H2O–D2 dimer is more strongly bound than its H2O–H2 isotopologue [cf. A. van der Avoird, D.J. Nesbitt, J. Chem. Phys. 134 (2011) 044314], with dissociation energies D0 of 46.10, 50.59, 67.43, and 73.53 cm1 for pH2O–oD2, oH2O–oD2, pH2O–pD2, and oH2O–pD2. A rotationally resolved infrared spectrum of H2O–D2 was measured in the frequency region of the H2O bend mode. The ab initio calculated values of the rotational and distortion constants agree well with the values extracted from this spectrum. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Collisions between H2O and H2 have been investigated rather extensively, both by experiment [1–6] and theory [7–15,5,6]. The main motivation for these studies is that the phenomena to which such collisions may give rise are of great astrophysical interest. H2 is the most abundant molecule in the universe and also H2O occurs in interstellar clouds as well as in star regions. Theoretical studies of molecular collisions require the knowledge of a global intermolecular potential surface and possibly—depending on the phenomena to be investigated—its dependence on the intramolecular coordinates. A nine-dimensional (9D) potential surface for H2O– H2 that depends on both the inter- and intramolecular coordinates has been obtained from high quality ab initio calculations by Valiron et al. [16]. A 5D H2O–H2 potential for rigid monomers in their vibrationally averaged geometries was calculated by Hodges ⇑ Corresponding author. E-mail address: [email protected] (A. van der Avoird). 0301-0104/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2011.06.008

et al. [17]. Also Valiron et al. obtained a 5D intermolecular potential from their 9D potential, either by averaging over the ground state vibrational wave functions of the monomers or by freezing the monomers at their ground state vibrationally averaged geometries. Recently, these 5D potentials have been used by Wang and Carrington [18] and by van der Avoird and Nesbitt [19] to calculate the bound rovibrational levels of H2O–H2. The results of these calculations were compared with rotationally resolved diode laser spectra of this weakly bound complex in the HOH bending region [20]. In earlier work on water dimers [21–29], in particular, it has been established that the comparison of rovibrational and tunneling levels calculated on a given intermolecular potential surface with high-resolution spectroscopic data provides a very critical test of the potential. The levels obtained [18,19] from the 5D potentials for H2O–H2 of Valiron et al. and Hodges et al. agree well with the experimental data of Weida and Nesbitt [20], so it was concluded that both these potentials are fairly accurate. Although D2 is much less abundant in interstellar clouds than H2, the study

A.van der Avoird et al. / Chemical Physics 399 (2012) 28–38

of the H2O–D2 isotopologue will provide a further check of the quality of the H2O–H2 potential surface. Here, we present experimental data from infrared spectroscopy for this isotopologue. Furthermore, we generated a 5D intermolecular potential specifically for H2O–D2, either by averaging the 9D potential of Valiron et al. [16] over the vibrational ground state wave functions of H2O and D2, or by freezing H2O and D2 at their vibrationally averaged geometries. Both of these 5D potentials were used in accurate calculations of the rovibrational levels of H2O–D2, which were then compared with the spectroscopic data. The organization of this paper is as follows. Section 2 describes the 5D potential surface for H2O–D2 and the way it was obtained from the 9D H2O–H2 potential of Valiron et al. Section 3 outlines the method applied to calculate the bound states of H2O–D2 on this potential. Section 4 describes the measurements. In Section 5 we present and discuss the calculated results and compare them with the spectroscopic data. Our conclusions are given in Section 6.

2. Potential surface A potential surface (PES) for H2O–H2 that includes all nine internal degrees of freedom was calculated ab initio by Valiron et al. [9,16] with the use of the CCSD (T)-R12 method (coupled-cluster with singles, doubles, and perturbative triples, explicitly correlated). This PES is independent of the nuclear masses and can be employed for any pair of water–hydrogen isotopologues in various vibrational states. A number of rigid-rotor, five dimensional, PES’s have been obtained so far from the full dimensional PES in the two following ways: either by averaging the 9D potential over vibrational wave functions of H2O and H2 [9,16] or by fixing the internal geometry of the monomers at their vibrationally averaged ground state values, as done for HDO–H2 [15], D2O–H2 [14], and D2O–D2 [6]. In the case where both H2O and H2 are in their ground vibrational states, Valiron et al. have shown that the rigid-rotor PES at the average vibrational ground state (VGS) geometries is in very good agreement with the explicitly vibrationally averaged potential (VAP). The corresponding effects on scattering cross sections were examined by Scribano et al. [14]; the VGS and VAP potentials were shown to provide very similar cross sections, even at collision energies below 1 cm1. We note that the high accuracy of the PES of Valiron et al. has been confirmed recently by a number of comparisons between theory and experiment including inelastic differential cross sections [5], pressure broadening cross sections [4,13], elastic integral cross sections [6], and the spectrum of the complex [18,19]. In the present work, the rigid-rotor 5D PES of H2O–D2 was obtained both by explicitly averaging the 9D potential over the ground state vibrational wave functions of H2O and D2 (VAP potential) and by fixing H2O and D2 at their internal vibrationally averaged ground state geometries (VGS potential). It should be noted that the non-rigid-rotor part of the 9D H2O–H2 potential of Valiron et al. was constructed as a correction, d9D, to a rigid-rotor PES, see Eq. (1) of Valiron et al. [16]. This vibrational correction depends on the inter- and intramolecular coordinates and was expanded as the product of angular functions and first- and second-order Taylor polynomials, as expressed in Eqs. (12) and (13) of Ref. [16]. Here, d9D was both explicitly averaged and evaluated at the averaged geometries of the monomers. For the VAP potential, the H2O vibrational ground state wave function was taken from Polyansky et al. [30] while the D2 wave function was computed using the Fourier Grid Hamiltonian (FGH) method [31,32] based on the H2 potential of Schwartz and Le Roy [33]. For the VGS potential, the averaged geometries were determined from the employed wave functions, resulting in rOH = 1.843a0, HOH = 104.41° and rDD = 1.435a0. The VGS potential is probably less accurate than the VAP potential,

29

but we used both potentials in the computation of the rovibrational levels, for comparison. Finally, both 5D potentials were expressed as a 149 term angular expansion, as explained by Valiron et al. [16]. The expansion functions are coupled spherical harmonics in the polar angles of the vector R that points from the center of mass of the H2O monomer to that of D2 and the polar angles of the D2 axis. These angles are defined with respect to a frame fixed to the H2O monomer with the z axis parallel to the C2 symmetry axis and the xz plane parallel to the plane of the molecule. These polar angles are not the same as the body-fixed (BF) angular coordinates used in the rovibrational level calculations; the z axis of the BF frame is parallel to the intermolecular vector R. It was explicitly derived in Ref. [19] that one can analytically transform the angular expansion functions of the potential to the corresponding angular functions in our BF coordinates and directly use the R-dependent coefficients in the expansion of the potential to compute the matrix elements of the Hamiltonian in our BF basis. A view of the potential for planar geometries is shown in Fig. 1. The angles bH2 O and bD2 are the polar angles of the H2O symmetry axis and D2 bond axis in the BF frame. The global minimum corresponds to a planar geometry with C2v symmetry, see Fig. 1(a) of Ref. [19], at bH2 O ¼ 0 and bD2 ¼ 0 or 180°. There is also a metastable non-planar structure, a local minimum, with bH2 O ¼ 119 and bD2 ¼ 90 and dihedral angle a = 90°. The potential valley that gradually rises from the global minimum to the planar structure with a = 0° near to the local minimum can be clearly seen in Fig. 1. The potential is similar to the potential of H2O–H2 illustrated in Fig. 2 of Ref. [19], both minima occur for practically the same geometries as the global and local minimum in the H2O–H2 potential, but they are slightly shallower. The global minimum in the VAP potential for H2O–D2 corresponds to a binding energy De = 232.12 cm1, while De = 235.14 cm1 for the VAP potential of H2O–H2; the center-of-mass distance Re is 5.82a0 in both potentials. The local minimum has De = 197.30 cm1 instead of De = 199.40 cm1 for the H2O–H2 potential; its geometry is shown in Fig. 1(b) of Ref. [19]. For the VGS potential of H2O–D2 the values of De for the global and local minimum are 232.68 and 199.64 cm1. Both of these geometries may be considered as

Fig. 1. View of the 5D potential surface (in cm1) for planar geometries with R optimized to find the energy minimum for all angles. A potential value of zero implies that it is repulsive for all values of R. The angles bH2 O and bD2 are the polar angles of the H2O symmetry axis and D2 bond axis in the BF frame.

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hydrogen bonded: in the global minimum structure the D2 monomer is the donor and H2O the acceptor, in the local minimum structure H2O is the donor and D2 the acceptor. The fact that the binding energy is slightly smaller for H2O–D2 than for H2O–H2, in line with the Ubbelohde effect [34] for hydrogen bonded systems, is related to the amplitude of the monomer stretch vibration which, of course, is smaller for D2 than for H2. One finds in general for Van der Waals and hydrogen-bonded complexes that an increase of the amplitude of a monomer vibration increases the binding energy, through an increase of the monomer’s multipole moments and polarizability. So it is observed, for example, that the binding energy for an excited vibrational state of one of the monomers is larger than the ground state binding energy, which leads to a red shift of the monomer vibration frequency in the complex [35,36]. In the calculations on H2O–H2 [19] we also used another potential, calculated by Hodges et al. [17] with the use of scaled perturbation theory. The results on this potential were not very different from the results on the 5D potential of Valiron et al. and, since the potential of Hodges et al. was only calculated for H2O–H2 at fixed monomer geometries, we did not use it in the present work.

3. Bound state calculations The method used to calculate the rovibrational levels of H2O–D2 was described and applied to H2O–H2 in Ref. [19]. It is based on a general computational method [23] developed for weakly bound molecular dimers with large amplitude internal motions, and is similar to a coupled-channel scattering approach. It was previously applied, for example, to the ammonia [37–39] and water [23–27] dimers. For all details on the form of the Hamiltonian and the basis in body-fixed (BF) coordinates, etc., we refer to the paper on H2O– H2 [19]. We used the ground state experimental values for the monomer rotational constants, i.e., A0 = 27.8806 cm1, B0 = 14.5216 cm1, and C0 = 9.2778 cm1 for H2O [40] and B0 = 29.9068 cm1 for D2 [41]. The atomic masses are 1.007825 u for H, 2.014102 u for D, and 15.994915 u for O. The grid in the discrete variable representation (DVR) for the intermolecular distance R ranges from R = 4 to 26a0 and contains 96 equidistant points. In the calculations we used a radial basis of 20 functions contracted in the same way as in Ref. [19]. Also the angular basis containing symmetric rotor functions—Wigner D functions [42]—and spherical harmonics for the internal rotations of H2O and D2, respectively, and Wigner D functions for the end-over-end rotation of the dimer was truncated at the same level (internal rotor quantum numbers jAmax ¼ 10 for H2O and jBmax ¼ 8 for D2) as in Ref. [19]. In principle, convergence of the hindered internal rotations of D2 requires a larger jBmax than for H2, but the value of 8 is largely sufficient also for D2. The permutation–inversion (PI) or molecular symmetry group G8  D2h(M) [43] of H2O–D2 is, of course, the same as for H2O– H2. It is generated by the permutation P12 that interchanges the H nuclei in H2O, the permutation P34 that interchanges the D nuclei in D2, and inversion E⁄. The nuclear spin statistical weights are different, however, since deuterons have nuclear spin I = 1 while protons have I = 1/2. Table 1 lists the nuclear spin weights and shows the relation between the irreducible representations of G8 and the quantum numbers kA, which determines the para/ortho (p/o) H2O nature of the states, and jB, which determines whether the states belong to ortho or para D2. The quantum number kA is the projection of the H2O angular momentum jA on the C2 symmetry axis of H2O. Other (approximate) quantum numbers that are important to understand the nature of the rovibrational states are mA and mB, the projections of the monomer angular momenta jA and jB on the dimer axis R, and the projection K = mA + mB of the total angular

Table 1 Irreducible representations of G8, quantum numbers kA and jB relevant for symmetry, para/ortho (p/o) nature of the monomers in H2O–D2, and nuclear spin statistical weights. Irrep

kA

H2O

jB

D2

Aþ 1 A 1

even

p

even

o

6

even odd

p o

even even

o o

6 18

odd even even odd odd

o p p o o

even odd odd odd odd

o p p p p

18 3 3 9 9

Aþ 2 A 2 Bþ 1 B 1 Bþ 2 B 2

Weight

momentum J on this axis. Finally, we observe that the total angular momentum J and the parity p = ±1 under E⁄ are exact quantum numbers. In our analysis of the rovibrational states we use the spectroscopic parity , which is related to the inversion parity by p = (1)J. We follow the convention to label states of even/odd spectroscopic parity by e/f. We use the absolute value of K as a label and distinguish the states with K > 0 by their parity e/f. For further details we refer to Ref. [19]. When comparing the rovibrational states of H2O–D2 with those of H2O–H2, one must remember that the relation between the ortho/para labels of D2 and the even/odd quantum numbers jB— which determine the PI symmetry properties—is opposite to that for H2: both oD2 and pH2 have a j = 0 ground state, while pD2 and oH2 both have a j = 1 ground state. The reason for this o/p reversal is that the ortho label is always given to states with the highest nuclear spin statistical weight and the para label to states with a lower weight, irrespective of their spatial or permutation symmetry. The comparison of Table 1 with Table 1 of Ref. [19] illustrates the above point. 4. Experimental Jet cooled H2O–D2 complexes are formed by pulsed supersonic expansion through a 4 cm  120 lm slit valve and detected via direct absorption of high resolution IR light in the 6.2 lm region corresponding to m2 bending of the H2O chromophore. The source of the high resolution IR laser is a homostructure Pb–salt diode with a lasing junction formed via diffusion, which yields reasonably good spectral coverage (80%) with 4 cm1 of continuous tuning followed by a 1 cm1 gap. This performance differs from the molecular beam epitaxially (MBE) grown heterostructure diode structures used in our laboratories for the 4.3 lm spectral region, for which continuous coverage is nearly 100%. Although these tuning gaps do obscure some spectral regions of interest at higher frequencies, they do not seriously influence our ability to record the HOH bending fundamental region of the H2O–D2 complex. The stagnation gas is formed by passing a mixture of H2 (99.999%) and ‘‘first run’’ Ne (70/30 Ne/He) at 260 kPa through a distilled H2O bubbler held at 273 K. Previous studies determined that optimum formation of H2O–H2 occurs for a 1:2 ratio of H2 to the Ne/He diluent mixture. For the present work, this ratio is intentionally reduced to 1:4 due to cost considerations of D2 vs H2. The resulting mixture contains 0.25% H2O and is throttled with a needle valve to yield the appropriate stagnation pressure P0. We find no evidence for Ne–H2O or He–H2O complexes when using this mixture. This is not surprising, as the binding energy of D2 on H2O (D0 = 46–76 cm1, depending on nuclear spin modification) though still quite weak, is significantly stronger than anticipated for Ne–H2O or He–H2O. Indeed, it is worth noting that even small differences in this binding energy prove to be, by detailed balance arguments, exponentially important at the cold (T = 5  10 K)

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temperatures of the slit jet expansion. For example, though we start with a 6:3 ratio of ortho (even j) vs para (odd j) D2 in our expansion, the spectra are completely dominated by complexes with only the nuclear spin para D2 isomer. Specifically, we see no evidence for spectra of complexes with ortho D2, which are bound by DD0 = 22 cm1 less than the corresponding para D2 isotopomers (vide infra). Note that this does not imply slow rates for collisional formation of ortho D2 complexes with H2O, but rather the importance of subsequent ‘‘competition’’ by collisions with para D2 species

Table 2 Transitions measured; J 00 ; K 00a ; K 00c and J 0 ; K 0a ; K 0c are the rotational quantum numbers of the lower and upper levels, Ka is the same as K. J0

pH2O–pD2 (K = 0 3 0 2 0 1 0 0 0 1 0 2 0 3 0 4 0 oH2O–pD2 (K = 1 3 1 3 1 2 1 2 1 1 1 1 1 3 1 2 1 1 1 1 1 2 1 3 1 2 1 2 1 3 1 3 1 4 1 4 1

H2 O  oD2 ðj ¼ 0Þ þ pD2 ðj ¼ 1Þ H2 O  pD2 ðj ¼ 1Þ þ oD2 ðj ¼ 0Þ; ð1Þ 1

which are weakly exothermic (DE = 22 cm ) and therefore, by detailed balance under 10 K conditions, strongly shift the equilibrium in the forward direction (Keq = 23.7). Indeed, this so called ‘‘chaperone mechanism’’ for pre-formation of a more weakly bound complex which then stabilizes by bimolecular displacement is likely to be a dominant formation pathway. Typical optimal expansion conditions are found empirically to be P0 = 200 kPa, with the gas mixture precooled (280 K) in the stagnation region to further enhance cluster formation. Since direct absorption methods are used, the signals can be converted to integrated absorbances, from which absolute concentrations of H2O–D2 can be determined. If we assume that the infrared band strength of the m2 bend of the H2O monomer is unchanged upon complexation with H2, then the measured absorbances correspond to approximately 0.5–1% of the total H2O monomer concentration being incorporated into H2O–D2, which represents a rather high efficiency for clustering into a single target complex species. Relative frequencies for the H2O–D2 complex spectra are measured via linear interpolation from an actively stabilized confocal Fabry–Perot cavity (250 MHz free spectral range), which in turn is locked onto a polarization stabilized HeNe laser. The quality of the servo loop is such that the frequency of each cavity transmission fringe is stable to a less than 8 MHz root mean square (rms) deviation over multiple days. The cavity fringes can thus be used to link together individual spectral scans even in regions where H2O and HOD reference transitions are sparse. The rms precision and accuracy of measured frequencies are found to be comparable to the cavity stability (i.e. 8 MHz), as directly confirmed by repeated measurements over several days. Absolute frequencies 18 are determined from H17 2 O and H2 O m2 bend transitions measured in the slit jet expansion, which at our sensitivity levels are readily observable in natural isotopic abundance. The results of the measurements are summarized in Table 2 and Fig. 2. 5. Results

K 0a

K 0c 0) 3 2 1 0 1 2 3 4 1) 2 3 1 2 0 1 3 2 1 0 1 2 2 1 3 2 4 3

J00

K 00a

K 00c

Frequency (cm1)

4 3 2 1 0 1 2 3

0 0 0 0 0 0 0 0

4 3 2 1 0 1 2 3

1592.2612 1593.1237 1593.9911 1594.8610 1596.5970 1597.4592 1598.3124 1599.1552

4 4 3 3 2 2 3 2 1 1 2 3 1 1 2 2 3 3

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3 4 2 3 1 2 2 1 0 1 2 3 1 0 2 1 3 2

1592.2685 1592.5074 1593.1181 1593.3172 1593.9751 1594.1204 1595.2088 1595.4497 1595.6162 1595.7794 1595.9325 1596.1546 1597.2542 1597.4287 1598.0239 1598.2873 1598.7883 1599.1385

Fig. 2. Overview of the transitions measured, see also Table 2.

Table 3 Rovibrational levels of pH2O–oD2 (in cm1), dissociation limit 0, D0 = 46.10 cm1. In parentheses the R or P character; if not indicated it is higher than 99%. The parity e/f is the spectroscopic parity.

5.1. Calculated energy levels and comparison with experiment

R(K = 0) parity

J=0

J=1

J=2

J=3

Tables 3–6 show the energy levels calculated for total angular momentum J = 0–3 on the VAP potential surface. They contain all bound levels for these values of J; blank entries in the tables imply that the corresponding states are not bound. The levels on the VGS potential are not shown because they are very similar. The global minimum in this potential is about 0.6 cm1 deeper than in the VAP potential and the lower rovibrational levels are 0.2–0.3 cm1 lower in energy. Although K is not an exact quantum number one can see in these tables that many states can be labeled with specific (approximate) K values. States with K = 0, K = 1, and K = 2 are called R, P, and D, respectively. The percentage of R, P, or D character of a state was obtained by summation of the squared eigenvector components with a given K over all other basis set labels. In addition we extracted effective mA and mB values for each

e e e

46.1027 13.9449 7.5745

45.2749 13.3260 (98%) 6.6057 (83%)

43.6227 12.0664 (94%) 4.6873 (64%)

41.1529 10.1433 (91%) 1.8837 (53%)

parity

J=1

J=2

J=3

e f

8.4717 (80%) 8.2522

7.2806 (58%) 6.6004

5.5247 (43%)a 4.1311

P(K = 1)

a

For J = 3 this state gets 57% R character.

K = mA + mB from the eigenvectors by transforming the coefficients of the coupled BF angular basis functions used in our calculations to an uncoupled product basis labeled by mA and mB.

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Table 4 Rovibrational levels of oH2O–oD2 (in cm1), dissociation limit 23.7994 cm1, D0 = 50.59 cm1. In parentheses the R or P character; if not indicated it is higher than 99%.

R(K = 0) parity J=0

J=1

J=2

e e e f

26.1312 (97%) 11.6965 (93%) 23.4352 (70%) 4.6948

24.7933 12.5431 23.7066 6.4859

parity

J=1

J=2

J=3

e e e f f f

19.1026 (97%) 6.6466 14.9933 (94%) 19.2817 6.7466 14.7832

17.1203 (93%) 5.0070 16.4682 (85%) 17.6323 5.3019 (98%) 15.9055

14.2055 (89%) 2.5548 18.5470 (79%) 15.1676 3.1300 (95%) 17.5616

26.7890 11.3019 23.4033 3.7947

Table 6 Rovibrational levels of oH2O–pD2 (in cm1), dissociation limit 83.6130 cm1, D0 = 73.53 cm1. In parentheses the R, P, or D character; if not indicated it is higher than 99%.

J=3 (93%) (85%) (60%) (97%)

22.7449 (89%) 13.8916 (78%) 9.1502 95%)

P(K = 1)

J=0

J=1

J=2

J=3

e e e e f

7.6174 37.1776 44.3875 57.7512 47.3823

6.7564 37.8643 45.2141 58.0076 (90%) 48.1764 (97%)

5.0370 39.2280 46.8655 (97%) 58.6078 (86%) 49.7369 (91%)

2.4645 41.2483 (99%) 49.3342 (94%) 59.4930 (82%) 51.9795 (75%)

J=1

J=2

J=3

P(K = 1) parity e e e e f f f f

10.2394 40.2930 50.5107 58.9432 (88%) 10.2626 40.2814 50.5351 (99%) 59.0286

11.8519 41.7948 (98%) 51.7202

14.2645 44.0357 (95%) 53.5174 (98%)

11.9211 41.7637 (98%) 51.8198 (94%)

14.4013 43.9824 (96%) 53.8241 (82%)

a b c

D(K = 2) parity

J=2

J=3

e f

48.3975(99%) 48.3939(99%)

50.6738(96%) 50.6549(96%)

Before discussing the nature of the bound states in H2O–D2 and comparing them with H2O–H2 let us recall that dimer species with oD2 have the same symmetry (even jB) as those with pH2, while the symmetry (odd jB) of pD2 dimers is the same as that of dimers with oH2. This is reflected, for instance, by the dissociation energies D0 of the four different species of H2O–D2, which are 46.10, 50.59, 67.43, and 73.53 cm1 for pH2O–oD2, oH2O–oD2, pH2O–pD2, and oH2O–pD2, respectively. These values are in the same order as the D0 values of the H2O–H2 species of the same symmetry, but considerably larger: by about 40% for the oD2 species and by about 25% for the pD2 species, cf. Ref. [19]. By contrast, the well depth De = 232.12 cm1 of the H2O–D2 is smaller by about 3 cm1 than that of the H2O–H2 potential. As we will illustrate when discussing the wave functions in Section 5.2, the greater stability of H2O–D2 is related to the larger mass and smaller rotational constant of D2 which allow the dimer states to be better localized in the region of the potential well than those of H2O–H2. On the other hand, it is noteworthy that the nature of the ground states in H2O–D2 is the same as in H2O–H2: the oH2O–pD2 ground state has P character, just as the oH2O–oH2 ground state, while all other dimer species have R ground states.

J=0

J=1

J=2

e e e e e f f f f f

21.4509 46.0244 65.6070 74.5807 80.4833 27.9739 52.5716 60.6378 71.8431 82.7968

22.3789 46.7433 66.3361 75.2596 80.9401 28.2730 53.3580 61.3951 72.4980 83.1126

24.2288 48.1827 67.7774 76.6005 81.8388 29.4874 54.9235 62.9011 73.7794

(99%) (99%) (94%) (53%) (98%) (97%) (98%) (78%)

J=3 (98%) (98%) (97%) (97%) (90%) (51%) (92%) (96%) (93%)

26.9887 50.3445 69.9001 78.5557 83.1417 31.5167 57.2525 65.1363 75.6359

(97%) (96%) (94%) (93%) (88%) (50%)a (83%) (89%) (86%)

P(K = 1)

Table 5 Rovibrational levels of pH2O–pD2 (in cm1), dissociation limit 59.8136 cm1, D0 = 67.43 cm1. In parentheses the R, P or D character; if not indicated it is higher than 99%.

R(K = 0) parity

R(K = 0) parity

parity

J=1

J=2

e e e e e e e f f f f f f f

10.0795 28.8512 54.7053 61.9904 69.2047 79.8645 82.5989 10.1545 29.3650 54.7096 62.0490 69.1658 79.9853 82.4950

11.6449 30.4377 55.9573 63.3258 70.3407 80.4804 83.3468 11.8668 31.3575 55.9786 63.4839 70.2314 80.7936 83.0071

(99%) (96%) (95%) (53%) (98%) (99%) (98%) (86%)

J=3 (98%) (52%) (96%) (98%) (91%) (86%) (51%) (46%)c (96%) (93%) (98%) (63%)

13.9937 32.8142 58.1375 65.3776 72.0460 81.4315

(97%) (99%) (60%) (93%) (96%) (85%)

14.4290 34.1225 58.2243 65.6444 71.8468 81.9608

(50%)b (46%) (90%) (88%) (96%)

D(K = 2) parity

J=2

J=3

e e e f f f

41.2206 56.5714 75.5139 41.2227 56.5831 75.5311

43.6205 (99%) 59.1159 (63%) 77.2365(95%) 43.6308 (99%) 59.1563 (58%) 77.3160 (91%)

(53%) (98%) (51%) (97%)

For J = 3 this state gets 50% P character. For J = 3 this state gets 50% R character. For J = 2 this state gets 49% D character.

In Table 3 one observes that pH2O–oD2 has bound states of both

R and P character and that the P states have both parities e and f. The corresponding H2 isotopologue, pH2O–pH2, has only R states of spectroscopic parity e [19]. A larger number of bound states is found for oH2O–oD2, see Table 4, both R and P and of both parities e and f. Here, the contrast with the H2 isotopologue is less pronounced since such states were also found for oH2O–pH2. When calculating the dissociation energy D0 it should be remembered that oH2O and pH2O have different nuclear spin states and that the monomer nuclear spins are conserved at the time scale of the experiment, so that the ground state oH2O–oD2 dimer dissociates into oH2O with jA = 1, kA = 1 and energy 23.7994 cm1 and ground state oD2. For pH2O–pD2 the number of bound levels is still larger than for oH2O–oD2, cf. Tables 5 and 4. Here, again, we find a contrast with the H2 isotopologue since pH2O–oH2 has about the same number of bound states as oH2O–pH2, see Tables 3 and 4 in Ref. [19]. Also D states are bound in pH2O–pD2, whereas pH2O–oH2 only has bound R and P states. The ground state pH2O–pD2 dimer dissociates into ground state (jA = 0) pH2O and pD2 with jB = 1, energy 59.8136 cm1. Table 6 shows that oH2O–pD2 has by far the largest number of bound states of all species, just as its oH2O–oH2 isotopologue. This is, of course, related with the fact that oH2O with jA = 1 and oH2 and pD2 with jB = 1 can be easily aligned and adopt the most favorable structure. Ground state oH2O–pD2 dissociates into jA = 1, kA = 1 oH2O and jB = 1 pD2 with energy 23.7994 + 59.8136 = 83.6130 cm1.

33

A.van der Avoird et al. / Chemical Physics 399 (2012) 28–38

In Ref. [19] we investigated the nature of the bound states by looking at the approximate quantum numbers mA, mB, the projections of the monomer angular momenta jA, jB on the intermolecular axis R, and kA, the projection of jA on the H2O monomer symmetry axis. We explained the relative stabilities of the R and P states, the observation that the ground state is a P state for oH2O–oH2 and has R character for the other species, and the nature of the stable D states in oH2O–oH2 by making the link between these approximate quantum numbers and the orientations of the H2O and H2 monomers in the complex. Stabilization occurs when these orientations are such that they provide the most attractive configuration of the H2O dipole and the (positive) H2 quadrupole and yield the strongest hydrogen bond with H2 as the donor and H2O as the acceptor. The same arguments explain the relative stabilities of most states in the H2O–D2 dimer species. The differences should be more pronounced than for H2O–H2 since it is easier to align D2 than H2, due to smaller zero-point energy of vibration and internal rotation. This will be illustrated in Section 5.2 when we show the wave functions. The P and D levels occur in pairs, of e and f parity. In our calculations the e  f parity splittings originate from off-diagonal Coriolis terms in the Hamiltonian that couple basis functions with DK = ±1. The P states split because the e and f components mix with different R states of the same parity. The mixing is strong when the energy gap between the P state and the R state of the correct parity is small. Stronger mixing gives rise to larger parity splittings, which explains why the parity splittings of the P states in different species show considerable differences. This also explains why the parity splittings of the D states are much smaller than those of the P states, because the D states can only mix indirectly with the R states, through the P states. For some of the excited states we find very strong K mixing, due to a near-degeneracy of the R and P or P and D states. If one considers the dimer as a (prolate) near-symmetric rigid rotor the parity splitting becomes manifest as an asymmetry doubling. Table 7 lists the parameters for H2O–D2 that were extracted from the levels calculated for J = 0–3 by fitting them with the same model as applied to the spectrum of H2O–H2 in Ref. [19]. We refer to this reference for details. The off-diagonal Coriolis coupling between the lowest R level and the lowest P level of the same parity as the R level was only included for the two oH2O complexes, where the energy gap between the R and P states is smallest. We listed the data obtained on the VAP potential; those on the VGS potential are very similar. It was found for H2O–H2 [19] that the values of EP  ER and b agreed very well with the values extracted from the measured spectrum. For H2O–D2, however, no R P transitions were observed, so the values of EP  ER and b obtained from the fit of the calculated levels cannot be compared with experimental data. It was derived in Ref. [19] that the size of the Coriolis coupling parameter b yields information about the internal rotor behavior of the complex: b = 2B if the monomers are free internal rotors,

where B is the end-over-end rotational constant of the dimer. The results in Table 7 show that this is nearly so (b = 1.88B) for oH2O–oD2 and that the internal rotations are more strongly hindered (b = 1.58B) in oH2O–pD2. This situation is qualitatively similar to that in the corresponding H2O–H2 complexes [19], but the higher values found there: b = 1.98B for oH2O–pH2 and b = 1.65B for oH2O–oH2 show that, as expected, the internal rotations are more strongly hindered in the D2 isotopologue. In the experimental spectrum of H2O–D2 only K = 0 0 and K=1 1 transitions were observed. This implies that there is essentially no information on the Coriolis coupling between the R(K = 0) and P(K = 1) states. Since we still wanted to compare the results of our calculations with the experimental data, we considered the dimer as a (nearly) rigid asymmetric rotor and represented the e/f parity splitting of the P levels as an asymmetry doubling reflected by the difference in the rotational constants B and C. Least squares fits of the K = 1 1 transitions in oH2O–pD2 were performed using the Watson asymmetric top Hamiltonian in the A reduction, Ir representation [44]. We adjusted the ground and excited state rotational constants B00 , C00 and B0 , C0 and the lowest order quartic centrifugal distortion parameters D00J ¼ D0J and 00 0 dJ ¼ dJ , but kept the latter constrained to be the same in ground and vibrationally excited state. A similar fit was made to the K = 1 levels calculated for J = 0  3. Since there is also practically no information on the long axis rotational constant A of the dimer in the experimental spectrum, the parameter A00 = A0 in the fit of the experimental data was fixed at the value of 17.34 cm1 obtained from the fit of the calculated levels. Finally, one overall band origin m0 is floated, which can be subtly different from the m0 value of the K=0 0 band, where the shift in band origin represents the change in A between the upper and lower state. The K = 0 0 band in pH2O–pD2 contains less information, and the transitions in this band are thus fit to a simpler expression BJðJ þ 1Þ  DJ ½JðJ þ 1Þ2 with B ¼ ðB þ CÞ=2. The fit parameters are the rotational constants B00 ; B0 and distortion constants D00J ; D0J of the upper and lower states, with one band origin m0. In order to be able to compare with experiment, exactly the same fits were made to the calculated K = 0 and K = 1 levels, using the same Hamiltonian and floating the same set of parameters. The results are listed in Table 8. The oH2O–pD2 species has a P(K = 1) ground state and, indeed, no K = 0 0 transitions were observed experimentally. The lowest R(K = 0) level of this species is more than 11 cm1 above the ground state. The same kind of detailed balance conditions that

Table 8 Calculated and experimental rotational and distortion constants for the ground and H2O bend excited state (in cm1) from fits to an asymmetric rotor model, see text. Experiment pH2O–pD2(K = 0) Ground state Excited state

oH2O–pD2(K = 1) Ground state

Table 7 Rotational and distortion constants B and D, P  R energy gap, and Coriolis coupling constant b (in cm1) from fits to the energy levels for J = 0–3 with the model of Ref. [19]. For the dimers with oH2O the R and P levels p were fitted simultaneously with ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the inclusion of the off-diagonal Coriolis coupling b JðJ þ 1Þ.

pH2O–oD2 oH2O–oD2 pH2O–pD2 oH2O–pD2

R R P R P R

B

D

0.4142 0.4173 0.4112 0.4307 0.4251 0.4275

1.42  104

EP  ER

7.098

b

Excited stateb

a b

11.725

0.675

B0 D0

0.43500(1)

0.43072(1)

1.065(29)  104 0.43367(1)

1.095(3)  104

m0

1.097(30)  104 1595.7304(3)

A00 B00 C00 00 dJ

(17.34)a 0.45205(8) 0.37358(8) 6.56(14)  105

17.34 0.44774(2) 0.37233(2) 5.96(7)  105

D00J

3.07(26)  105

4.18(13)  105

A0 B0 C0

(17.34)a 0.45398(8) 0.36849(8) 1595.6993(2)

m0

0.780

1.09  104

B00 D00

Calculated

Fixed at the value obtained from the calculated levels. 0 The parameters dJ ; D0J in the fit were constrained to be the same as in the ground state.

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A.van der Avoird et al. / Chemical Physics 399 (2012) 28–38

overpredict the DJ values, which would suggest that our H2O–D2 potential is slightly too shallow.

explain the preferential formation of the pD2 species over the oD2 species, see Eq. (1), make it plausible that transitions starting from this R level are too weak to be observed. The value of A = 17.34 cm1 from the fit of the (ground state) calculated levels is close to, but greater than, the C2 axis rotational constant B0 = 14.5216 cm1 of H2O. It is greater due to the large amplitude internal motion in the dimer that samples geometries with a smaller H2O moment of inertia around the intermolecular axis. The same value of A was used for the upper state in the fit of the K = 1 1 band in oH2O–pD2; the small difference of 0.03 cm1 between the pH2O–pD2 (K = 0 0) and oH2O–pD2 (K = 1 1) band origins m0 confirms that this assumption is probably correct. The agreement between the experimental and calculated data, shown in Table 8, is good. The B and C values from the calculations are all within 1% of the experimental data, and even the weak DJ centrifugal terms are well predicted (within 20–30%). The calculations seem to systematically underpredict the B and C values and

5.2. Wave functions The character of the various bound states of the different nuclear spin species can be determined by looking at their wave functions. Wave function plots also give insight in the nature of the excited intermolecular vibrations. Contour plots of the wave functions of the H2O–H2 isotopologue at R = 6.3a0 are shown in Ref. [19]. Fig. 3 shows similar J = 0 wave functions for the four para/ ortho H2O and ortho/para D2 nuclear spin species of H2O–D2. The nodal character which is dictated by the symmetry related to the para/ortho nature of the monomers is the same as in the H2O–H2 species, with oD2 corresponding to pH2 and pD2 to oH2, as explained above. The wave functions of H2O–D2 are also qualitatively similar to those of H2O–H2, cf. Fig. 3 of Ref. [19], but they are considerably more localized in the region of the potential well,

180

1

0. 09

(degrees)

90

0.01

0

0.06 0.0 0.08 7 0.0 9

0.06

3

30

0.0

4

0 0

60

0.05

0.0

07

0.1 0.1 1 0.1 2

0.

30

0.02

β

β

D

D

2

2

4

0.15

0

120

5

6 0.0 0.07

05

8

0.

0.0

0.1

6

11

(degrees)

0.

0.

60

30

60

β

H O

90 120 (degrees)

150

0 0

180

8 0.00.09 0.1

30

60

β

H O

2

90 120 (degrees)

150

180

2

180 0.0

0.1

180

5

−0.25

150

5

−0.2

0

0.05

−0.15

120

0.2

150

0.2

120

−0.1

0.15

(degrees)

−0.05

90

90

β

D

2

2

βD (degrees)

0.02

3

0.1 0.1 3 4

0.12

150 06

0.01

90

03

6

0.0

04

0.11 2 0.1

0.0

0.0

0.0

0.

0.09

0.07

120

07

0.11

0.

0.0

0.

150

0.

04 0. 05

08

0.

0.12

0.

0.03 0.04 0.05 0.07 0.08 0.09 0.1

180

0 0.05

60

60

0.1

30

−0

.05

.1

0.2

0 0

5

0.2

−0

30

0

0.1

5

30

60

90 120 βH O (degrees) 2

150

180

0 0

.15

−0

−0.2

30

60

βH

O

90 120 (degrees)

150

180

2

Fig. 3. Wave functions for J = 0, planar geometries, R = 6.30a0, bH2 O and bD2 are the angles of the H2O symmetry axis and the D2 bond axis with the intermolecular axis R. At the global minimum they are both zero, at the local minumum bH2 O ¼ 119 and bD2 ¼ 90 .

A.van der Avoird et al. / Chemical Physics 399 (2012) 28–38

35

Fig. 4. Wave functions for J = 0, planar geometries, R = 6.30a0, bH2 O and bD2 are the angles of the H2O symmetry axis and the D2 bond axis with the intermolecular axis R. At the global minimum they are both zero, at the local minumum bH2 O ¼ 119 and bD2 ¼ 90 .

see Fig. 1. A contour plot of the potential is shown in Fig. 2 of Ref. [19]. In contrast with H2O–H2 we found in H2O–D2 at least two excited R states in each of the four species. Fig. 4 shows the wave functions of the first excited R states. For pH2O–pH2 it was found [19] that the first (and only) excited R state is excited in the intermolecular stretch mode, but it is clear from the nodal character of the wave functions shown in Fig. 4 that in the corresponding pH2O–oD2 species the first excited R state is excited in the H2O internal rotation. Also in the second R state of pH2O–pD2 the H2O internal rotation is excited. A very interesting feature is seen in oH2O–pD2; here the second R state is not an excited state in the region of the global minimum, but it is localized near the local minimum in the potential that corresponds to the hydrogenbonded structure with H2O as the donor. In Fig. 5 we plotted the wave functions of the lowest P states with J = 1. As mentioned above, the ground state is of P type in oH2O–pD2. There is some arbitrariness in the plots of J = 1 wave

functions since they depend on the overall rotation angles (U, H, aA). The angles were defined in Ref. [19]; H and U are the polar angles of the vector R with respect to a space-fixed frame. The angles aA and aB, with A and B denoting the H2O and D2 monomers, are the azimuthal angles of the H2O symmetry axis and D2 bond axis in the two-angle embedded BF frame with its z axis along R. The angle aA is considered as an overall rotation angle of the dimer and the dihedral angle a = aB  aA is an internal angle. We chose M = 0 and H = 90°, aA = 0°, since the real part of the overall rotation wave function with (J, M, K) = (1, 0, 1) has its maximum amplitude at these angles and does not depend on U. When we compare these plots with Fig. 4 of Ref. [19] we see that the P wave functions look similar to those of the corresponding H2O–H2 species, except that they are more localized in the valley around the global minimum of the potential. Note that the upper left panel in Fig. 4 of Ref. [19] contains a R state since there are no bound P states in pH2O–pH2, so it cannot be compared to Fig. 5. As it was already noticed in Ref. [19], the amount of localization of the

36

A.van der Avoird et al. / Chemical Physics 399 (2012) 28–38

Fig. 5. Wave functions for J = 1, planar geometries, R = 6.30a0, bH2 O and bD2 are the angles of the H2O symmetry axis and the D2 bond axis with the intermolecular axis R. At the global minimum they are both zero, at the local minumum bH2 O ¼ 119 and bD2 ¼ 90 .

wave functions and the geometries where they have their maximum amplitudes are determined to some extent by the nodal plane structure imposed by the symmetry. This illustrates that the change of the overall rotation quantum number K from 0 to 1 significantly affects the (hindered) internal rotations of the monomers. Fig. 6 displays the wave functions in the region of the nonplanar metastable (local minimum) structure illustrated in Fig. 1(b) of Ref. [19]. These are the same wave functions as shown in Figs. 3 and 4. Here, they are plotted for geometries that start from a planar structure with the same coordinates bA = 119°, cA = 0°, bB = 90° as the metastable geometry. We recall that bA and bB are the polar angles of the H2O symmetry axis and D2 bond axis in the BF frame and cA describes the rotation of H2O about its own symmetry axis. The system is planar for dihedral angle a = 0° and arrives at the local minimum in the potential for a = 90°. One observes that the wave functions of oD2 complexes have considerable amplitude at the local minimum, slightly higher than at the corresponding planar structure. The wave

functions of the pD2 complexes have a nodal plane that, due to symmetry, must pass through the local minimum at a = 90°. The amplitudes of the lowest R states are slightly larger than those of the corresponding H2O–H2 isotopologues, also in this region, cf. Fig. 5 of Ref. [19]. The amplitudes of the first excited R states are smaller for the oD2 complexes and larger for the pD2 complexes. Especially for oH2O–pD2 the amplitude is very large; we saw already in Fig. 4 that this state is localized in the region of the local minimum. Wave functions of spectroscopic parity f have a nodal plane for planar geometries, including the equilibrium geometry. For this reason the f states with K = 0 may be considered as excited outof-plane intermolecular vibrations, but it should be realized that they have rather low energies and look more like hindered rotations. As already mentioned, the e/f parity doublets for K > 0 are split by (off-diagonal) Coriolis coupling between the internal and overall rotations of the dimer, but may also be considered as asymmetry doublets if we regard the dimer as a prolate near-symmetric rotor.

A.van der Avoird et al. / Chemical Physics 399 (2012) 28–38

Fig. 6. Wave functions (J = 0) for nonplanar geometries in the region of the local minimum. Closed lines represent the lowest R states, dashed lines the first excited R states; R = 6.30 a0; a is the dihedral angle.

6. Conclusions This paper describes the calculation of a 5D intermolecular potential, VAP, for H2O–D2 by averaging the full 9D H2O–H2 potential surface of Valiron et al. [16] over the ground state vibrational wave functions of H2O and D2. Another 5D potential, VGS, is obtained from the 9D potential by freezing the monomer geometries at their ground state vibrationally averaged values. The two 5D potentials are very similar. On both potentials we calculated the rovibrational states of the H2O–D2 dimer for J = 0–3. The variational method employed involves a discrete variable representation of the intermolecular distance R and a basis of coupled free rotor wave functions for the hindered internal rotations and the overall rotation of the dimer. The basis is adapted to the permutation symmetry associated with the para/ortho (p/o) nature of both H2O and D2, as well as to inversion symmetry. All bound rovibrational levels of the four different nuclear spin species: pH2O–oD2, oH2O–oD2, pH2O–pD2, and oH2O–pD2, are computed for both parities. The results are compared with the levels calculated [19] for the corresponding species of H2O–H2. We found that a large fraction of the binding energy De— which is 232.12 cm1 in the VAP potential—goes into the intermolecular vibrational zero-point energy, although not as much as for H2O–H2. The dissociation energies D0 on this potential are 46.10, 50.59, 67.43, and 73.53 cm1 for pH2O–oD2, oH2O– oD2, pH2O–pD2, and oH2O–pD2, respectively. These values are 25 to 40% larger than the D0 values of the corresponding H2O– H2 species. Wave functions, which are also calculated, show that the complex is floppy, and that the intermolecular vibrations look more like hindered internal rotations. Most of the states of H2O–D2 are to some extent localized in the valley of the potential around the global minimum, slightly more so than the corresponding states of the H2O–H2 dimer. Still, they also have substantial amplitude in the region of the local minimum. For oH2O–pD2 we also found a state that is localized in the region of the local minimum. Let us recall that the global minimum corresponds to a hydrogen bonded structure with D2 as the donor and H2O as the acceptor, while the local minimum structure is hydrogen-bonded with H2O

37

as the donor and D2 as the acceptor. The observation that oH2O and pD2 give stronger binding than pH2O and oD2 is not surprising, because pH2O and oD2 can only orient themselves under the influence of the anisotropic interaction potential when their j = 0 ground states mix with excited rotational states. On the contrary, both oH2O and pD2 have degenerate j = 1 ground states in which the monomers can be aligned without any mixing. The ortho/para difference is most pronounced for D2, the monomer with the largest rotational constant. In contrast with H2O–H2, all four nuclear spin species of H2O–D2 have bound states of R(K = 0) and P(K = 1) character, and of both e and f parities. Both species containing pD2 also have bound D(K = 2) states. The ground state is R, except for oH2O–pD2 where it is of P type. Keeping in mind that o/pD2 has the same spatial symmetry as p/oH2, this situation is the same as in the H2O–H2 isotopologue. A difference is, however, that only the oH2O–oH2 species has a bound D state. We measured a high-resolution spectrum of the H2O–D2 dimer, in the region of the H2O bend mode. The bands observed correspond to K = 0 0 transitions in pH2O–pD2 and K = 1 1 transitions in oH2O–pD2. This is in line with the calculations, which show that pD2 complexes are more strongly bound than oD2 complexes and that pH2O–pD2 has a R (K = 0) ground state and oH2O– pD2 has a P(K = 1) ground state. With the aid of some information derived from the calculated energy levels, we could extract rotational and distortion constants of the ground and excited R and P states from the experimental data. These were compared with the ground state parameters extracted in the same manner from the levels calculated for J = 0–3. The rotational constants B and C agree to within about 1% with the experimental data, the small distortion constants DJ to 20–30%, which shows that the ab initio potential surface used to calculate the rovibrational levels is accurate. The calculations seem to systematically underpredict the B and C values and overpredict the DJ values. Similar (small) deviations were found for H2O–H2 [19], which would suggest that our 5D H2O–D2 potential and the 5D potential for H2O–H2 and, probably, also the 9D potential of Valiron et al. [16] from which they were derived are slightly too shallow. Also the experimental results of Belpassi et al. [6] seem to point into this direction. The H2O–H2 potential of Hodges et al. [17] that was also used in Ref. [19] overpredicted the B values, by about the same amount as they were underpredicted by the potential of Valiron et al., so the potential of Hodges et al. is probably too attractive. Let us remind the reader that the global minima in the H2O–H2 5D potentials of Valiron et al. and of Hodges et al. correspond to De values of 235.14 and 241.18 cm1, and Re values of 5.82 and 5.77a0, respectively. This difference in Re will probably make a difference in the rotational constant B of about 2%. Hence, we might speculate that the real PES lies in between the potentials of Valiron et al. and of Hodges et al. This is supported by the best available estimate for De, 236.2 cm1, from a single point CCSD (T) calculation by Hodges et al. in which the interaction energy was extrapolated to the complete basis set limit. The above discussion demonstrates that current ab initio electronic structure methods can provide intermolecular potentials accurate to about 1% in the well depth and 0.5% in geometrical parameters for molecules of the size of H2O.

Acknowledgments This work has been supported in part (DJN) by funds from the National Science Foundation and the Department of Energy. AvdA and DJN also acknowledge additional assistance through the Senior Alexander von Humboldt Award program for providing the opportunity to work together.

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