## Abstract

The present paper aims at *ab initio* and laboratory evaluation of the N_{2} collision-induced absorption band intensity arising from interactions between N_{2} and H_{2}O molecules at wavelengths of around 4 μm. Quantum chemical calculations were performed in the space of five intermolecular coordinates and varying N−N bond length using Møller–Plesset perturbation and CCSD(T) methods with extrapolation of the electronic energy to the complete basis set. This made it possible to construct the intermolecular potential energy surface and to define the surface of the N−N dipole derivative with respect to internal coordinate. The intensity of the nitrogen fundamental was then calculated as a function of temperature using classical integration. Experimental spectra were recorded with a BOMEM DA3-002 FTIR spectrometer and 2 m base-length multipass White cell. Measurements were conducted at temperatures of 326, 339, 352 and 363 K. The retrieved water–nitrogen continuum significantly deviates from the MT_CKD model because the relatively strong nitrogen absorption induced by H_{2}O was not included in this model. Substantial uncertainties in the measurements of the H_{2}O−N_{2} continuum meant that quantification of any temperature dependence was not possible. The comparison of the integrated N_{2} fundamental band intensity with our theoretical estimates shows reasonably good agreement. Theory indicates that the intensity as a function of temperature has a minimum at approximately 500 K.

## 1. Introduction

The study of weak intermolecular interactions between water and nitrogen molecules is strongly motivated by the important role played by both constituents in the Earth's atmosphere. On the experimental side, the microwave spectroscopic probe [1] of the low-temperature H_{2}O−N_{2} complex made it possible to characterize its equilibrium structure in much detail. Planar equilibrium structure of a dimer was found with a nearly collinear arrangement of the N_{2} bond and one of the OH bonds in the H_{2}O molecule. The observation of so-called foreign continuum absorption in ambient water–nitrogen mixtures alludes to a smooth and virtually structureless component of bimolecular absorption, which is proportional to the product of the water vapour and nitrogen partial pressures. This component, which originates from bimolecular absorption of weakly interacting water and nitrogen molecules, may have a notable effect on both the propagation of selective radiation and the global radiative budget of the atmosphere. On the theoretical side, a number of *ab initio* calculations of H_{2}O−N_{2} interaction energy have been carried out [2–6], the most recent of which are aimed at construction of full intermolecular potential energy surface (PES). To the best of our knowledge, the most precise *ab initio* PES for this system is due to Tulegenov *et al.* [6] who used the systematic intermolecular potential extrapolation routine procedure. The quality of the PES in Tulegenov *et al.* [6] was then judged by virtue of comparison of computed second cross virial coefficient (SCVC) with experimental data.

Atmospheric applications require the knowledge of binary absorption coefficients as a function of the wavelength and the temperature. One of the interesting components of absorption by water–nitrogen pairs in the mid-infrared is due to the so-called collision-induced absorption (CIA) in the region of the nitrogen fundamental. In the first approximation, this absorption arises from induction of a transient dipole on a polarizable nitrogen molecule from the permanent dipole field of the neighbouring water molecule. One can expect therefore that this induction may result in much more important absorption band strength than that characterizing conventional CIA in pure nitrogen gas where an induced dipole is generated in a relatively weak quadrupole field of dipoleless nitrogen molecules. Preliminary estimates which were made first by Brown & Tipping [7] indicated that in fact the nitrogen fundamental absorption band, once originated from induction by water molecules, is indeed much stronger than that in pure nitrogen. Temperature variation of the integrated CIA intensity of this band is of particular interest.

It has been demonstrated recently by Baranov *et al.* [8,9] that temperature variations of CIA integrated intensity in pure oxygen or nitrogen are characterized by parabola-like dependence. Although the rate of an intensity increase is significantly different in oxygen in comparison with nitrogen, the position of the minimum roughly corresponds to near room temperature in both cases. This effect was qualitatively interpreted by Vigasin [10] in terms of an increasing role played upon heating of a gas by induced dipole at extremely close intermolecular separations. Later on, the *T*-dependence of the integrated CIA in the nitrogen fundamental was nicely reproduced in Lokshtanov *et al.* [11] on the basis of thorough *ab initio* analysis of the N_{2}−N_{2} PES and complete dipole surface (DS). It was interesting therefore to examine temperature variations of the integrated CIA in the region of the nitrogen fundamental in the H_{2}O−N_{2} system, the potential energy and induced dipole properties of which are significantly at variance with respect to N_{2}−N_{2} system.

The present paper aims at an *ab initio* examination of complete PES and DS in H_{2}O−N_{2}. Armed with the knowledge of these intermolecular characteristics at various levels of theory, we were able to trace how the quality of *ab initio* calculations influences the calculated *T*-dependence of CIA intensity induced in a nitrogen molecule by a water molecule. *Ab initio* computations of weak interactions in multi-electron molecular systems require thorough evaluation of electron correlation energy. Quite reasonable results can be reached in this context with the use of relatively simple second order Møller–Plesset perturbation theory (MP2).^{1} The latter does not permit, however, quantitatively accurate electronic energy for intermolecular pairs to be obtained. An alternative way for the calculation of electron correlation energy is due to coupled clusters method which is presently quite feasible on a level of CCSD(T), i.e. coupled clusters accounting for single, double and (perturbative) triple excitations. Actually, the CCSD(T) method is considered as a ‘gold standard’ of *ab initio* methods for light molecules because it allows accurate and reliable estimates of intermolecular energy to be obtained at a still affordable computation cost. The highest level of calculations in the present work corresponds to CCSD(T) method supplemented by extrapolation of the electronic energy to the limit of a complete basis set (CBS). Our calculated PES was confirmed by computation of the SCVC. Confirmation of our calculated DS, although somewhat indirect, could come from comparison of the calculated integrated CIA intensity with the data retrieved from the measured continuous absorption in water vapour mixed with nitrogen. In terms of experimental data, we have used water vapour continua spectra by Baranov *et al.* [12,13] in pure water vapour and in water–nitrogen mixtures supplemented by CIA spectra of pure nitrogen [9], which were reported recently for several gas temperatures. Laboratory-measured binary absorption spectra in a water–nitrogen mixture are due to weak and diffuse overlapping nitrogen–nitrogen, water–water and water–nitrogen absorptions. The latter one is significantly masked in raw spectra by conventional line absorption by the water monomer and minor admixtures including, for example, the HDO (partially deuterated water) isotopologue. That is why the water–nitrogen absorption can be retrieved only from experimental spectra with appreciable uncertainty. However, we succeeded in deriving the first experimental estimates for the integrated CIA intensity in the N_{2} fundamental for 326, 339, 352 and 363 K temperatures. The comparison of these experimental data with our *ab initio* calculations is given and discussed below.

The organization of this paper is as follows. Sections 2 and 3 outline details of our *ab initio* calculations, which resulted in obtaining multi-dimensional PES and DS for the H_{2}O−N_{2} system. The sensitivity of our *ab initio* computations to the level of quantum chemistry method is briefly commented on. In §4, we outline briefly how experimental spectra were recorded and how it became possible to retrieve N_{2} fundamental integrated intensity owing to H_{2}O−N_{2} binary component from these spectra. Then, the details of our classical calculation of the integrated intensity are presented in §5 along with comparison of theoretical results with available experimental data. Section 6 contains general discussion supplemented by examples of statistical averaging of some representative observables that can be carried out using our calculated PES and DS.

## 2. *Ab initio* method and computational details

Our calculations of H_{2}O−N_{2} PES were carried out assuming rigid water and nitrogen molecules. This means that the lengths of O−H and N−N bonds (as well as HOH angle) were kept at their equilibrium values characteristic to individual monomers. The averaged over ground vibrational state H_{2}O geometry is recommended by Mas & Szalewicz [14]: *r*_{OH}=0.95825 Å and the angle H−O−H=104.69^{°}. The N_{2} bond length is taken from Huber & Herzberg [15]: *r*_{e}=1.10927 Å.

Figure 1 shows the coordinate system adopted in our calculations. The H_{2}O molecule centre of mass lies at the coordinate system origin. The N_{2} molecule centre of mass is situated on the *z*-axis at the distance *R* from the origin. To describe molecular orientations, the set of Euler angles is used: *θ*_{1} and *ψ* angles for H_{2}O, *θ*_{2} and *φ* for N_{2}. Note that *C*_{2} axis for the H_{2}O molecule lies always in the *xz* plane.

The integrated intensity of a vibrational band in the N_{2} unit is known to be determined by the first derivative of induced dipole with respect to interatomic N−N coordinate *r*. To find the derivative, we were obliged to let *r* vary. Thus, the electronic energy of H_{2}O−N_{2} was calculated at three values of *r*: *r*=*r*_{e}, *r*=*r*_{e}+Δ and *r*=*r*_{e}−Δ. The increment Δ was set to 0.08 Å, the value of the order of the zero-point vibrational amplitude.

The H_{2}O−N_{2} PES and DS were calculated for a set of intermolecular coordinates (figure 1) which include 642 symmetry non-equivalent geometries of a complex. The number of quantum chemical calculations for a given basis set was, however, three times in excess of that (i.e. 1926) because the whole set of intermolecular coordinates was supplemented by three values of intramolecular N−N bond length. Given all the symmetry equivalent geometries were counted this would result in 2430 geometries. We consider six different intermolecular separations *R*: 2.5, 3, 3.75, 5, 7 and 10 Å. Angular coordinate varied by a step of 45^{°} within the ranges [0, 45^{°}] for *φ* and [0, 180^{°}] for *θ*_{1}, *θ*_{2} and *ψ*.

As we shall see below, statistical average of observables requires computation of the integrals containing both PES and DS in the integrands. Generally, pointwise representation of the integrands is enough to make numerical integration, which is why no global fit of the calculated PES and DS was attempted in the present work. To obtain correct result while integrating against intermolecular separation, it was indispensable, however, to extrapolate both PES and DS beyond the limiting points on a grid towards larger separations. This has been performed assuming the ‘tails’ of long-range PES and DS dependences to have a form of inverse power series expansions in *R*^{−n}. Here, the power *n* for the leading terms was taken as *n*=4 and *n*=3 for PES and DS, respectively, at any constant set of angular variables. These leading terms guaranteed correct long-range variations of potential energy and asymptotically electrostatic dipole (see §3), whereas other terms in the expansion ensured stepless transition among extrapolated ‘tails’ and *ab initio* values calculated on a grid. The integration against *R* coordinate was performed using adaptive Simpson quadrature method.

The calculations were carried out using PRIRODA quantum chemistry suite of computer codes described by Laikov & Ustynyuk [16]. The use of this package made it possible to significantly decrease the volume of necessary computations because of the option implemented in PRIRODA to calculate the energy derivatives analytically even on a level of CCSD(T) method. Generally speaking, there exist different ways of calculating the dipole properties of molecules using *ab initio* methods. Commonly accepted is that the computation of expectation values in order to find a dipole is insufficiently accurate [17]. An alternative way consists of calculation of the energy derivative with respect to an artificially applied external electric field that requires then finding the limit of the derivative at zero field strength. One of the variants, which is often called ‘finite field method’, proves to be quite accurate, although its numerical realization requires finite-differencing of the electronic energy, i.e. requires an increased number of similar computations. Analytical differentiation is much more preferable in this context because it permits one to reduce the volume of computations, to avoid numerical errors and to skip an arbitrary choice of trial value of an externally applied electric field.

We have used a conventional supermolecular method in which intermolecular energy of a complex is found in terms of the difference between electronic energies of a compound system and its monomeric constituents. The removal of basis set superposition error (BSSE) is indispensable for weakly interacting systems [18]. To account for BSSE at each level of computational methods, we have carried out a series of calculations in which ‘ghost’ basis wave functions were attributed to nuclei centres having zero charges. The resulting BSSE energy correction was found as
where *E*^{BSSE} is BSSE corrected potential energy, *E*^{AB}(*AB*,*R*) is the supermolecular energy at separation *R*, *E*^{A}(*AB*,*R*) and *E*^{B}(*AB*,*R*) are the energies of A and B species calculated in the presence of ‘ghost’ wave functions appropriate, respectively, to B and A centres; labels ‘A’ and ‘B’ refer here to H_{2}O and N_{2}, respectively. Using a similar approach, the equation can be derived for the BBSE corrected dipole moment of a molecular pair (see Czasar *et al.* [18] for further details):
where **μ**^{BSSE} is BSSE corrected dipole moment, **μ**^{AB}(*AB*,*R*) is the supermolecular dipole moment at separation R, **μ**^{A}(*AB*,*R*) and **μ**^{B}(*AB*,*R*) are dipole moments of species A and B calculated with the assumption of ‘ghost’ wave functions centred at B and A components, respectively; **μ**^{A}(A) and **μ**^{B}(B) are dipole moments of isolated species A and B (**μ**^{B}(B) is zero in our case). In our calculations, we have taken care of BSSE for both the energy and the dipole moment, although BSSE correction for a dipole appeared to be negligible.

Another cause of inaccuracy in *ab initio* electronic energy calculations in the frame of supermolecule method is due to the use of a limited basis set. In recent years, it has become conventional to correct the so-called basis set incompleteness error (BSIE) by way of extrapolation of the calculated energy to the CBS with the help of analytical relationships. In the present work, we have tested several procedures of correction for BSIE on the basis of a series of our calculations at Hartree–Fock, MP2, CCSD and CCSD(T) levels using different basis sets. In all cases, it was found that the use of Dunning basis sets [19] results in a better convergence towards the CBS. Our available computer facilities enabled us to carry out extensive calculations with aug-cc-pVXZ (*X*=2(D),3(T)) basis sets only. We used two-point extrapolation formula of Bak *et al.* [20]:
where *E*^{CBS} is the CBS system energy limit, *E*_{X} is the system energy obtained at basis set level *X*. *X* and *X*+1 stand here for 2 and 3, respectively.

Unfortunately, we failed to extrapolate our calculated dipole to CBS limit using a similar approach. Within a limited number of configurations that were examined using the largest basis set, we succeeded to observe no regular convergence of the calculated dipole. The calculations of CIA intensity require the knowledge of the dipole derivative with respect to nitrogen bond length. Although the dipole derivative shows irregularity as a function of the basis set size, we believe that starting from aug-cc-pVTZ basis set its values are already quite close to the CBS limit. This assumption is supported by our test calculations for selected geometries using larger basis sets. We expect the dipole derivative, as calculated using aug-cc-pVTZ base, to deviate from the CBS limit by less than 2 per cent. That is why no BSIE correction for a dipole or its derivatives was made in the present work.

The calculations of observables, be it the CIA intensity, SCVC, etc., have to proceed via successive steps. First, one has to compute potential energy (and dipole, if required) of a system on a chosen coordinate grid. Second, the BSSE correction and extrapolation to CBS limit have to be made. Then, the derived values of potential energy and dipole at preselected points should be subject to inter- and extrapolation procedures to imitate full analytical PES and DS. Finally, the values of interest have to be averaged with appropriate statistical weight (conventionally Boltzmann exponent) to get insight of the ensemble-averaged observables that are functions of temperature. The dominant cause of uncertainty in the course of such a series of computations is the inaccuracy of extrapolation and integration procedures. The use of rare angular grid (45^{°} step relevant to each angular coordinate) results in our estimated uncertainty for the calculated observables of the order of 5 per cent.

## 3. Calculated potential energy surface and dipole surface

Our calculated PES and DS were double-checked using available theoretical and experimental data. Figure 2 demonstrates some potential energy curves of H_{2}O−N_{2} in the vicinity of equilibrium geometry which were obtained by us using CCSD(T) method with aug-cc-pV(D,T)Z basis set supplemented by extrapolation to CBS limit. The results of our work correspond within 5 per cent to those published previously by Tulegenov *et al.* [6].

Temperature variations of the SCVC offer a nice test of the quality of *ab initio* calculated PES *V* (*R*,*Ω*). In a classical approximation, the calculation of SCVC (for diatom-asymmetric top) can be made using the following relationship:
where *B*_{12} is the SCVC, *k* is the Boltzmann constant, *T* is the temperature, 〈〉_{Ω} means averaging over different orientations of both interacting molecules (for the set of angular coordinates *Ω*={*φ*,*θ*_{1},*θ*_{2},*ψ*}, see §2):
3.1On the one hand, it is seen from the above expression that the calculated SCVC should not be too sensitive to details of the PES in the high temperature limit. On the other hand, it is known that the calculation of second virial coefficient at low temperature requires quantum corrections. Usually, these quantum corrections start to be significant at temperatures as low as 100 K or less. In between the low and high temperature limits, the sensitivity of second virial coefficient to details of PES is rather pronounced. Figure 3 demonstrates good agreement of our calculated SCVC using the most precise CCSD(T) PES, extrapolated to CBS, with available experimental data. It is also seen that our calculations are in good agreement with calculations by Tulegenov *et al.* [6] who used a somewhat larger basis set. Note that the error bars for experimental mixed SCVC are by nature much larger than for pure gases.

Figure 4 demonstrates some examples of H_{2}O−N_{2} induced dipole radial variation as calculated at CCSD(T) level with aug-cc-pVTZ basis set. The dipole calculated using the electrostatic approximation (dipole and quadrupole) is shown by thin lines. It is seen that the use of electrostatic approximation is reasonably good at intermolecular separations exceeding approximately 5–7 Å .

Our calculated dipole projection onto *z*-axis can be compared with that measured in equilibrium geometry of a dimer. Planar geometry with *R*=3.8 Å , *θ*_{1}=65^{°} and *θ*_{2}=170^{°} gives rise to a computed value of *μ*_{z}=1.07 D. The experimental value determined in Leung *et al.* [1] equals 0.833 D. *Ab initio* calculations by Kjaergaard *et al.* [5] resulted in the value of 1.04 D. For the total dipole of a complex at equilibrium, Kjaergaard *et al.* [5] obtained the value 2.08 D, which is also close to our calculated 1.96 D.

## 4. Experimental data and retrieval of the H_{2}O−N_{2} collision-induced absorption intensity

The experimental data on the H_{2}O−N_{2} continuum in the 3–5 μm region and the detailed description of experimental and calculation procedures were reported recently in Baranov [13]. Measurements were conducted at the National Institute of Standards and Technology using a 2 m base-length multipass White cell coupled with a BOMEM DA3-002 FTIR spectrometer. Figure 5 represents the water vapour–nitrogen continuum at a temperature of 352 K [13] (solid line). Also shown are the literature data by Burch & Alt [21] and Watkins *et al.* [22] along with the result provided by the MT_CKD continuum model [23]. The strong disagreement between experiment and the model is due to the existence of a broad and structureless feature, which apparently is the N_{2} fundamental band induced by collisions between H_{2}O and N_{2} molecules, as was first pointed out in theoretical calculations by Brown & Tipping [7]. Here, we focus on the method used to retrieve the N_{2} CIA band profile and its integrated intensity.

We assume that the contribution to absorption in the region of the N_{2} fundamental from nearby H_{2}O−N_{2} continuum water vapour bands located at 1620 and 3180 cm^{−1} is nearly exponential, roughly like it is in the CKD continuum model. This contribution, extrapolated towards the window centre, is shown in figure 5 by two dashed lines.

Figure 6 represents residual absorption profile (thin/thick solid curve) obtained after removal of the above-mentioned contributions. It is seen that the profile is not acceptable for direct estimation of the band integrated intensity because of strong contribution from the *ν*_{1} HDO band [12]. The Gaussian centred at about 2400 cm^{−1} (dashed line) was used to reconstruct the H_{2}O−N_{2} CIA band profile in its high frequency wing. It fits experimental data in the region from 2300 to 2540 cm^{−1}. The segment of this Gaussian profile between 2540 and 2800 cm^{−1} (shown by solid line) reconstructs the band wing.

The other Gaussian, centred at 2330 cm^{−1}, was used to reconstruct the H_{2}O−N_{2} CIA band profile in the low frequency wing in order to reduce noise effect. The Gaussian segment (solid line) replaces the band wing in the region from 2000 to 2150 cm^{−1}, where the noise level is substantial. The resulting combined H_{2}O−N_{2} CIA band profile is given in figure 6 as a solid line consisting of the Gaussian wings and noisy central part.

Figure 6 clearly shows the existence of a quite distinct feature (shaded by grey colour) consisting of two maxima at 2335 and 2365 cm^{−1}, respectively, with the gap at 2350 cm^{−1}. We attribute these features to the remnants of the carbon dioxide *ν*_{3} fundamental band. The reasons why these contaminations still persist in our thoroughly processed spectra are out of the scope of the current paper. We assume their nature is similar to the above-mentioned remnant *ν*_{1} HDO rovibrational structure in the vicinity of 2720 cm^{−1} [13]. Table 1 contains our estimated H_{2}O−N_{2} CIA band intensities at four temperatures.

Because of many ill-determined factors that may have an effect on the band shape retrieval as described earlier, it is hard to estimate the true uncertainty of the data reported in table 1. We believe that the inaccuracy of our estimated integrated intensity does not exceed 20–25%.

## 5. Integrated collision-induced absorption intensity

As in our previous work [11], we have calculated the *T*-dependence of the integrated CIA in the N_{2} fundamental in terms of semiclassical approximation. This means that instead of exact quantum summation we have performed classical integration over spatial coordinates of the Boltzmann weighted squared derivative of induced dipole with respect to N−N coordinate. Induced dipole is understood here as the total dipole of a pair at a given set of intermolecular coordinates. Although this dipole contains in our case a virtually permanent component of the water monomer, the water dipole itself has no effect on the N_{2} intensity because the change in the dipole of a pair caused by variations in N−N bond length is intrinsically determined by a dipole induced in the N_{2} molecule. The main impetus for the use of classical statistical integration is conditioned by our belief that at atmospheric temperatures the integrated characteristics of bimolecular absorption are not sensitive to exact positions of the energy levels or the linestrengths. Even true dimer rotational structure which is clearly seen in room temperature CIA spectra (e.g. the spectra of carbon dioxide, obtained by Baranov & Vigasin [24]) is known to be unresolved. The huge density of states and lifetime broadening are expected to smear over fine quantum details in ‘high temperature’ spectra. Note that for the weakly bound system under consideration, atmospheric temperature can be judged as quite high.

In parallel to the above-mentioned SCVC calculation, our adopted statistical average of integrated intensity extends to the totality of phase space of interacting molecules. This note is straightforward to emphasize the difference in the procedure of statistical average over the total phase space or its particular domains. The integration over complete phase space comprises the use of a Boltzmann exponential in terms of statistical weight. An attempt to average the intensity (or any other observable) over a limited domain (e.g. over true-bound dimeric states) in the phase space requires integration in the limits restricting this domain by physical reasons as well as the use of the weighting function which may no longer be dependent on potential energy only. One of the ways of such integration addresses the use of discrimination of pair states in energetic coordinates as was suggested by Vigasin [25,26].

Following the ideas first formulated in Colpa [27] and then used in Lokshtanov *et al.* [11], the integrated CIA intensity in the N_{2} fundamental was calculated using the expression
5.1where *S*_{01} stands for the integrated intensity of the nitrogen fundamental band (normalized to H_{2}O and N_{2} concentrations), *ν* is the transition frequency, *h* is Planck's constant, *c* is the speed of light, 〈〉_{Ω} means averaging over different orientations (see equation (3.1)), *R* is intermolecular distance defined above. Assuming vibration in N_{2} molecule is harmonic the matrix element of the dipole derivative can be written as
where ∂**μ**/∂*r* is vector of dipole moment derivative with respect to nitrogen bond length. Δ_{0} is the zero-point amplitude of vibration in the nitrogen molecule:
Here, *M* is reduced mass of the nitrogen molecule.

Figure 7 shows the calculated intensity in comparison with experimental data. The uncertainty of experimental intensity is too large to allow an explicit retrieval of its temperature variations. However, one can conclude that there is a reasonably good correspondence between the measured data and our calculated ones, at least while the most reliable *ab initio* methods are employed. It is seen that CIA intensity as calculated on MP2 level fails to match experimental values. We believe that the main cause for that is substantial error of reproducing DS within the MP2 method. The dashed line in figure 7 shows the intensity calculated with our best DS and PES from Tulegenov *et al.* [6] which is the best known to-date, whereas the solid line shows the intensity as calculated with both DS and PES from the present work. Because both lines are reasonably close to each other, we suggest the integrated intensity to be less sensitive to details of the PES than to those of DS. In a sense, we can assert our CCSD(T)-CBS data to be the most reliable to-date estimate for H_{2}O−N_{2} binary absorption intensity in the region of N_{2} fundamental.

Although our data are significantly more accurate in comparison with Brown & Tipping [7] data, we have to admit that their estimate^{2} is surprisingly good in spite of its use of simplistic potential and dipole models. The reason for that lies in the principal contribution to the induced dipole coming from the strong water dipole induction on the polarizable nitrogen molecule. Figure 4 demonstrates that in fact the main terms in a multipole expansion nicely correlate with the tails of our calculated dipole at various geometries. In addition, the intensity calculated by us, assuming an *ab initio* PES and electrostatic induced dipole, shows a quite reasonable correspondence to much more sophisticated data at least at the low-temperature end. Of course, such a simplistic model is not able to result in accurate absolute values of intensity.

It is instructive to compare temperature variations of the CIA intensities in H_{2}O−N_{2} with those in N_{2}−N_{2} system borrowed from Lokshtanov *et al.* [11] (figures 7 and 8 and table 2).

## 6. General discussion and statistical average of other observables

Accurate knowledge of the integrated intensity we are discussing in the present work is of crucial importance for evaluation of the so-called foreign continuum absorption by water vapour in the vicinity of 4 μm wavelength. Previous measurements by Watkins *et al.* [21] and Burch & Ault [22] are indicative of relatively strong absorption in this region. It has been clearly established recently by Baranov [13] and then confirmed by Ptashnik *et al.* [30] that the frequently used MT_CKD model of the continuum underestimates the absorption in this particular window by at least two orders of magnitude. Because our calculated *ab initio* data are in quite good agreement with experimental data, we consider them as a concrete support of the importance of the N_{2} CIA contribution within the 4 μm window. Moreover, we can speculate that the more pronounced intensity of the foreign continuum than that suggested previously can also be expected at wavelengths shorter than 4 μm.

Another aim of the present work is to further extend the theoretical understanding using knowledge of PES and DS. One has to keep in mind that any observed CIA band results typically from overlapping contributions from a multitude of molecular pairs that are characterized by a broad distribution over intermolecular coordinates. It is of great interest to average the observables over this distribution. Moreover, as we have mentioned previously, it would be highly desirable to obtain averaged values over preselected distinct domains of phase space with respect to the proportions of true-bound, quasi-bound and free pair states. For the present work, we restrict ourselves to two representative examples of how observables can be averaged over the totality of pair states.

Our first example concerns the position of vibrational frequency in N_{2} molecule subject to perturbation from the neighbouring water molecule. It is known (electronic supplementary material of Kjaergaard *et al.* [5]) that the vibrational shift of N_{2} vibrational band origin in H_{2}O−N_{2} dimer is positive. The question can be raised as to how the vibrational shift in a variety of H_{2}O−N_{2} pairs changes with the temperature on the average. In fact, once the vibrational origin is identified with the band centre of the CIA band, then we have to have two effects. On the one hand, it is clear that the vibrational shift tends to zero as the temperature increases, because of increasing separation between interacting molecules on the average. On the other hand, the induced dipole drops rapidly with increasing separation. That is why the pairs at long intermolecular separation do not give rise to absorption. To account for both effects we can additionally weight the vibrational shift Δ*ν*(*R*,*Ω*)=*ν*_{H2O−N2}−*ν*_{N2} with the calculated induced dipole squared. As a result, we have
Figure 9*a* demonstrates the rate of decrease of the calculated ‘effective’ vibrational shift as a function of temperature that stems from the above equation. The limiting value at zero *T* in this figure corresponds to the nitrogen vibrational shift in a true-bound water–nitrogen complex in its equilibrium geometry. Our relevant value of 1.5 cm^{−1} differs significantly from the value of 5 cm^{−1} reported in electronic supplementary material of Kjaergaard *et al.* [5]. Keeping in mind that the value of calculated vibrational shift is known to strongly vary as a function of both the basis set size and its quality, we can speculate that the use of relatively small basis set might result in the shift being somewhat overestimated in Kjaergaard *et al.* [5].

In reality, one can expect that the position of the CIA band centre can be determined from experimental recordings in terms of the dimer Q-branch rather than by the band centre weighted over the entire absorption band. This is clearly seen for example in the case of CIA absorption bands in carbon dioxide [24]. In this case, the averaging over true-bound domain of pair states is required.

Another example relates to ‘effective’ rotational constant of a pair. Here again, the low temperature extreme has to correspond to a (*B*+*C*)/2 rotational constant in a nearly prolate asymmetric top equilibrium geometry of a dimer. Figure 9*b* shows nice agreement of the calculated rotational constant for the H_{2}O−N_{2} dimer with experiment [1] at *T*→0. As temperature increases, one can expect that ‘effective’ rotational constant in an ensemble of pairs will decrease in agreement to our expectation that intermolecular separation becomes larger and larger on the average. In reality, because the intensity of CIA band has a parabola-like *T*-dependence, we can observe the minimum (figure 9*b*) in the *T*-dependence of the effective rotational constant^{3} as well. This minimum says that on the low temperature end the ‘perceptible’ intermolecular separation is indeed increasing. By contrast, it starts to decrease in a ‘high temperature’ limit because more and more close collisions contribute to the CIA intensity in agreement with rapidly increasing overlap induced dipole.

The above-mentioned examples make allusion to the value of obtaining reliable PES and DS for weakly interacting molecular pairs, which only are able to permit accurate averaging of observables over multitude of pair states which are thermally excited at nearly atmospheric conditions.

## 7. Conclusions

— Accurate PES and complete DS for weakly interacting H

_{2}O−N_{2}system are obtained through the use of*ab initio*CCSD(T) method complemented by CBS procedure.— The ability of reliable

*ab initio*based prediction of the integrated CIA intensity is demonstrated in the case of collision between water and a dipoleless diatomic molecule.— Recent experimental study of the water–nitrogen continuum in the 4 μm spectral region allowed the retrieval of profiles and the estimation of the integrated intensity of the H

_{2}O−N_{2}absorption in the nitrogen fundamental band region. Neither CIA band shapes derived from the experiment nor their integrated intensities make it possible to establish any perceptible*T*-dependence within the substantial error bars characterizing the continuum retrieved from the experimental study.— Theoretical

*ab initio*values of the integrated intensity of absorption in the region of 4 μm nitrogen fundamental are found to be in agreement with recent experimental estimates thus giving new insight on the*T*-dependence of the water vapour foreign continuum absorption.—

*Ab initio*complete PES and DS are indispensable for accurate characterization of observables statistically averaged over a huge number of thermally excited molecular pair states.

## Acknowledgements

Partial support of RFBR grant no. 10-05-93105 in the frame of GDRI ‘SAMIA’ is gratefully acknowledged by Yu.I.B., I.A.B., S.E.L. and A.A.V. The authors appreciate fruitful discussions, invaluable help and suggestions from D. N. Laikov, I. V. Ptashnik and K. P. Shine.

## Footnotes

One contribution of 17 to a Theo Murphy Meeting Issue ‘Water in the gas phase’.

↵1 The details of

*ab initio*nomenclature and abbreviations can be found elsewhere [12].↵2 Note that we are only concerned with the integrated intensity here.

↵3 In fact, we have averaged here the squared intermolecular separation, from which the reciprocal value was taken. This is not exactly the same as to average the squared reciprocal

*R*.

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