Astron. Astrophys. 335, 1070-1076 (1998)

2. The water dimer

2.1. The structure of (H₂O)₂

The water dimer is a simple type of a hydrogen bonded complex. The binding energy of this kind of intermolecular bond is between the pure van der Waals bonds, which are typically in the range of 10-20 meV and chemical bonds (e.g. CO: 11.2 eV). The binding energy for (H₂O)₂ is 0.22 eV (2500 K), and it corresponds to half the binding energy of the weakest chemical bond (i.e. that of the Cs₂ molecule). The structure of the water dimer has been deduced from measurements by Dyke et al. (1977) and Odutola et al. (1988). The structural information was obtained from microwave and radiofrequency measurements using isotopic substitution effects, Stark effect measurements and measurements of the hyperfine structure. The resultant structure, which is shown in Fig. 1 together with an energy level diagram, has an oxygen-oxygen distance, r, of 2.98 Å. The proton accepting and donating water axes have an angle of and degrees with respect to the intermolecular axis connecting the two oxygen molecules and the proton donating axis, respectively. The structure deduced from the measurements is in excellent agreement with that predicted by ab initio calculations (Matsuoka et al. 1976, Odutola et al. 1988, Amos 1986).

Fig. 1. The structure of the water dimer, and its energy level diagram. The () asymmetric top levels are shown for each of the six tunneling states (A, B, E). For the and levels, the asymmetry doubling is neglected.

This structure is consistent with a linear hydrogen bond and the proton acceptor tetrahedrally oriented to the hydrogen bond. Stark effect measurements on different rotational line yielded the dipole moment of (H₂O)₂ (Dyke 1977). A least-squares fit of the second order stark coefficient led to D and D (Coudert et al. 1987), which is in good agreement with that determined by Dyke et al. (1977).

2.2. Tunneling modes and level pattern of (H₂O)₂

Although one might think of the water dimer as a molecule which has a definite structure, e.g. binding length and angles, the spectrum is greatly complicated by the effect of tunneling motions of the constituent H₂O molecules. The observed spectrum could only be interpreted in terms of a theoretical model which includes all feasible proton-exchange tunneling motions in the dimer (Dyke et al. 1977). It is assumed that all tunneling splittings are small compared to the vibrational frequencies of the dimer. As a consequence each rotational level of the symmetry of the water dimer is split into six sublevels (Fig. 1). Each of these six vibrational tunneling levels has its own statistical weight and rotational structure, and only a small number of them are following a near rigid-rotor type pattern.

It was possible to fit all measured data using a model as developed by Hougen (1985), Coudert et al. (1987) and Coudert & Hougen (1988, 1990), which shows the existence of four large amplitude motions. The tunneling with the lowest barrier corresponds to a rotation of the hydrogen accepting H₂O monomer about its two-fold axis of symmetry during which the hydrogen bond in the axis is not changed. This motion leads to a splitting into two states, separated by approximately 200 GHz. The next most likely motion is an interconversion tunneling motion, in which the hydrogen donor/acceptor roles of the two water monomers are interchanged. Formally, this motion corresponds to a geared rotation of each monomer about its two-fold axis of symmetry, accompanied by simultaneous readjustment of the wagging angles. This motion splits each of the levels into three states, one of them being doubly degenerate. The magnitude of the splitting between the upper and lower level in the pattern amounts to 19 GHz. The other two tunneling motions are the following: A geared rotation of both monomers about axes perpendicular to the plane of symmetry of the dimer by which the hydrogen in the hydrogen bond is changed and a anti-geared rotation of the two monomer units about their two-fold axis of symmetry accompanied by wagging readjustments. These further tunneling motions had to be proposed as a consequence of measurements (Hougen 1985).

The high resolution data set (see Coudert et al. 1987 for an overview) include microwave measurements as obtained by Dyke (1977), Coudert et al. (1987), Martinache et al. (1988), and Fraser et al. (1989a,b) as well as FIR data of Busarow et al. (1989). We searched for water dimer lines in the interstellar medium at transitions around 24 and 74-92 GHz as listed in Coudert & Hougen (1990).

2.3. The partition function

Since only a small number of transitions have been measured so far, most of the levels had to be predicted to obtain the partition function. This is given by ( statistical weight, see Fig. 1, E: energy above the ground, : excitation temperature). We followed the model developed in Dyke (1977) and Coudert & Hougen (1988, 1990) neglecting the K asymmetry doubling. This corresponds to an extrapolation of the measured energy pattern toward higher energies. In addition, we considered all low lying van der Waals modes of (H₂O)₂. These are expected to be at 125 cm^-1, 142 cm^-1, 149 cm^-1, 174.5 cm^-1, 365.5 cm^-1 and 597 cm^-1 following the calculation of Amos (1986). In Table 1, we list , the fraction of all water dimer molecules which are in the lower state of the observed transition for typical values of . At high excitation, many levels are populated. This tends to decrease f for a given state. The 3 mm transitions chosen for our search are typically 20 or 30 K above the ground state. Therefore they are quite well matched to be observed toward hot cores. We chose not to search for lines with a higher J because their frequencies are less well determined and because excitation conditions are less clear.

Table 1. Observed (H₂O)₂ lines

2.4. Dipole matrix elements

The dipole matrix elements are required to estimate the column density from the observed line intensities (see Eq. 1). is the rotational quantum number of the lower level involved and µ is the dipole moment. We assumed D. S is the line strength, which is tabulated in the Appendix V of Townes & Schawlow (1975).

Since (H₂O)₂ is far from being a rigid asymmetric top molecule, the rotational constants A, B, and C lose their meaning as structural parameters. The E-type symmetry is the symmetry which follows most closely a rigid rotor type pattern. We have therefore fitted A, B and C values for E-type transitions and have calculated for all symmetries using these constants. However, these values are only rough estimates since for transitions of the A and B symmetries, a tunneling motion is involved. For a rigorous calculation of the line strength, the tunneling path would have also to be taken into account. Since (H₂O)₂ is close to being a prolate molecule, we assumed that the value (see Townes & Schawlow 1975) is about -1. Therefore, the quantum number K in our nomenclature equals approximately in Townes & Schawlow (1975). The derived estimates for based on this structure are given in Table 1.

© European Southern Observatory (ESO) 1998

Online publication: June 26, 1998
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