Approximately 16 000 electronic structure calculations were previously performed for a dipole moment surface at the MR-CI (multi-reference configuration interaction) level of theory utilizing an aug-cc-pCV6Z basis set and the Douglass–Kroll–Hess Hamiltonian of order two (DKH2) . These calculations span water bond lengths in the range of 1.3–4.0 a0 with angles between 30 and 178∘. Setting the energy at the equilibrium configuration (re=1.8141 a0 and θe=104.52∘) to zero, the maximum energy of these ab initio calculations that we consider is 57 423 cm-1.

These points need to be fitted to a functional form to obtain an ab initio PES; in the fit each data point was weighted as a function of their energy, with weights wi smoothly reducing towards zero as energy increases. The weighting function considered here is similar to the function used by for their 1997 H216O PES. A similar version of this weighting function is also used in an ethylene PES : 1wi(PES)=(tanh[-α(Ei-Vmax)]+1.002002002)2.002002002,α=0.006,Vmax=45000.

While constructing the POKAZATEL potential energy surface, Polyansky et al. found that a single surface could not accurately predict energies from the bottom of the well up to dissociation; hence, they follow the procedure by and define a piecewise potential. The same methodology was recently used to create a PES for the C3 molecule . We are also interested in accurately predicting energies that extend into the near-ultraviolet and so we too use a piecewise defined potential as given by 2V(r1,r2,θ)=Vlow(r1,r2,θ)×χE(r1,r2,θ)+Vup×(1-χE(r1,r2,θ)), where χE is a switching function dependent upon energy (E): 3χE(r1,r2,θ)=121+tanh(Vup(r1,r2,θ)-ζs)1β+ΔE2β3, and r1, r2 and θ are the corresponding values of the bond lengths and inter-bond angle. This function ensures smoothness, and the parameters ζs and β control the range of the switch. Our values are similar to those of the POKAZATEL PES, except our switching point ζs is different. By lowering our ζs from the 35 000 cm-1 value of POKAZATEL to 30 000 cm-1, we allow high-order parameters in Vlow to have a greater influence on the upper levels.

Due to the difficulty of fitting data in different energy regions, it is helpful to begin with a well-defined functional form; hence, the starting point for Vup in our new PES is the Vup function of the POKAZATEL potential. However, for Vlow, we employ a new functional form defined as 4Vlow(r1,r2,θ)=C000G(θ)F(r1,r2)+∑ijkCijkζ1iζ2jζ3kD(θ)F(r1,r2)+D1(1-e-αr1e)2+D1(1-e-αr2e)2+D2e-|r12|, where rie=(ri-re) for i=1,2. r12 is the separation between the two hydrogen atoms, re=1.8141 a0 is the equilibrium bond length and θe=104.52∘ is the angle at equilibrium. α was determined from a series of optimizations, and the optimal value was found to be 1.24. D1 and D2 were also floated during our initial linear least-square fits and are set to 42 778.44 and 683 479.329404 cm-1, respectively. The expansion variables ζ1, ζ2 and ζ3 are defined as 5ζ1=(r1+r2)/2-re,ζ2=(r1-r2)/2,ζ3=cos⁡θ-cos⁡θe. G(θ) and F(r1,r2) are dimensionless damping functions that constrain the potential in the limits of θ→0 and r1,2→∞. These are defined as 6G(θ)=tanh20θθe-3.0020020022.002002002+0.5F(r1,r2)=(0.999821745456)e-0.81r1e2+r1e2.

The number of parameters, Cijk, was optimized to provide the lowest root-mean-square (rms) deviation from the underlying ab initio data such that there are also no holes created from overfitting. A hole is an unphysical feature of a PES that often appears as a continuous (although not always) drop or dip in the surface where it should instead be smooth. We found that using 250 parameters provided the lowest rms deviation of 35 cm-1 from the electronic structure calculations. This value is large due to the large discrepancy between our ab initio data points and Vup from POKAZATEL rather than from our fitting of Vlow. The 250 parameters used here are close to the 241 parameters taken by and , as well as the 245 parameters by . The maximum values of i, j and k that we consider are 10, 8 and 15, respectively. In addition to the fitted ab initio surface, we also include a QED (quantum electrodynamics) correction to our ab initio PES via the one-electron Lamb shift and a second-order relativistic energy correction .

For quanta in ν1 and ν3, i.e. the stretching modes, discovered that his Born–Oppenheimer diagonal corrections (BODC), also known as the adiabatic correction, did not agree with those calculated by . The two calculations did, however, exhibit better agreement for the different quanta of bending in ν2. The adiabatic correction is known to be large for high stretch modes , particularly for those in the visible and near-ultraviolet which we are interested in. However, neither source is well tested nor suited for such energetic states; hence, we chose to omit this correction to our surface and rely on fitting to experiment to incorporate this effect.

The non-adiabatic correction is an important contribution to any high-accuracy potential . For high-temperature spectra, transitions involving high values of the total angular momentum, J, become significantly more prominent and, as the non-adiabatic correction grows approximately as J2 , non-adiabatic effects are more important. For this reason, we follow and embed these corrections within our Hamiltonian as new kinetic energy operators which are functions of operators J^XX, J^YY and J^ZZ. The coefficients before these operators are the values determined from multiplied by a factor of 1.1, which he suggests, times optimized values from In total, this gives (in a.u): (6.48156×10-10)J^XX,(4.86799×10-10)J^YY,7(3.94597×10-10)J^ZZ.

We use the DVR3D suite of programs for solving the nuclear motion problem. For these calculations, we take Radau coordinates with a bisector embedding and use a 55 by 40 discrete variable representation (DVR) grid with Morse oscillator like functions in r and associated Legendre polynomials in θ, respectively. The DVR for these basis sets is constructed using Gaussian quadrature schemes in associated-Laguerre and associated-Legendre polynomials, respectively, in r and θ. For the Morse oscillator-like functions, we take rE=3.0, ω=0.007 and β=0.25 (all in a.u.), which are the values used to compute the POKAZATEL line list. For the vibrational problem, matrices of dimension 3500 are diagonalized and used as a basis for the full rovibrational problem. For this, matrices of dimension 600(J+1-p) are diagonalized, where J is the total angular momentum and p is the parity (p=0 or 1). Nuclear masses have been used throughout.

These parameters have been optimized for the initial J=0 problem such that vibration energies below 27 000 cm-1 are well converged to better than 0.01 cm-1, while for energies at 37 000 cm-1 the convergence error is less than 0.03 cm-1.

PES refinement is a technique where one adjusts the underlying ab initio surface to reproduce measured data to a high degree of accuracy, often to within a fraction of a wavenumber . The method by has been successfully applied to numerous H2O potentials , as well as to TiO , AsH3 , NH3 , CH3Cl and C2H4 . In this procedure, one maintains the overall structure of the underlying ab initio surface while simultaneously optimizing the parameters of the fit. This prevents the development of unwanted holes while refining.

Overall, we are trying to minimize 8X=∑iΔi(obs)2wi(obs)+f∑jΔj(ai)2wj(ai), where Δi(obs) is the typical observed minus calculated DVR3D rovibrational energy and similarly Δj(ai) is the difference between ab initio and calculated potential energies. The factor f is the “weight” of our semi-empirical PES to our initial ab initio surface. Setting f too large can result in overfitting if the sum over j and/or i is too small.

The Hellman–Feynmann theorem allows us to efficiently calculate the derivative of an energy level with respect to a particular parameter in our potential, required for the least-squares fit. With this, we can iterate and optimize the parameters of the PES to reduce the deviation of our semi-empirical energies from the observed levels. The MARVEL (measured active rotational–vibrational energy levels) procedure was originally constructed for a IUPAC study of water spectra . The resulting empirical energy levels for H216O have been subsequently updated in response to both improvements to the MARVEL algorithm and to the availability of new data . We refine our potential to updated MARVEL energy levels with J=0, 2, 5, 10, 15 and 20, representing approximately 4000 states. The more recent potentials for water vapour have been limited to refinement of states with J=0, 2 and 5, which is not sufficient to accurately predict high J levels.

The only near-ultraviolet energy levels available for H216O come from the multiphoton experiments by and span states below J≊7. The reduced number of measurements in the blue–violet and near-ultraviolet makes the Vup particularly difficult to refine accurately. More high-resolution experimental work in these regions would be welcome.

For our initial unrefined ab initio PES, the average deviation from the MARVEL J=0 ab initio vibration band origins (VBOs) below 37 000 cm-1 is approximately 2 cm-1, a figure dominated by overtones in ν2. Refining to the VBOs alone is known to not produce accurate results . However, fits to J=0 levels are significantly faster and provide a good starting point for refining using non-zero J states.

For the first refinement of J=0 VBOs, we set the weight of all levels with energies greater than 26 000 cm-1 to 0.1, while those less than this carry a weight of 1. This ratio of 10 : 1 was chosen such that we can include all states in the refinement without deteriorating the residuals of the lower states. The weight of our semi-empirical PES to the underlying ab initio surface was fixed at 1000, which is large enough to provide accurate results while also small enough to prevent the formation of undesirable holes. For this process, Vup was held constant. Doing this allowed us to reduce our average RMSE from the MARVEL VBOs to only 0.08 cm-1.

For the second step, the ratio of weights for those states below 26 000 cm-1 to those above this limit is now switched compared to the previous refinement of Vlow. Then, 61 of the lowest-order parameters in Vup are optimized to improve the agreement between both our ab initio data points and the MARVEL levels, while Vlow was held fixed. For this refinement of Vup, f carries the same value as the previous step and is 1000.

For the third stage, we return to Vlow and focus on the refinement of energies in higher J states, notably J=2, 5, 10, 15 and 20. The weighting criteria remains the same as in step one, and Vup was not optimized here. The rigorous quantum numbers alone are not enough to uniquely match our calculated states to the correct corresponding states from MARVEL. We therefore need to supplement the rigorous quantum labels with energy differences, which is where it becomes difficult to match and is very often non-trivial, particularly in the near-ultraviolet with the high density of states. To identify the correct match, we add new J states only after the potential was optimized to the previous J states. By doing so, the accuracy of the calculated states in the next J states are always low enough to make a reliable match. For example, we take our previous J=0 optimized surface and calculate all J=2 states using the result of the J=0 optimization and then proceed to match the J=2 states. Next, we refine Vlow to J=0 and 2 energies (as done in step one) and calculate J=5 states using the results of this optimization; we then match these J=5 states to those in MARVEL. The optimization of J=0, 2 and 5 would follow next. This was continued through to J=20. This procedure allowed us to ensure that we optimize the calculated states to the correct empirical values in MARVEL. Outliers were removed from the refinement on a continuous basis and were chosen when their residuals were larger than the band average.

Next, for step four, we apply the weighting criteria of step two; refine Vup to states in J=0, 2, 5, 10, 15 and 20; and hold Vlow fixed. The procedure for adding more J levels to the optimization of Vup was the same as done above in step three. Although there are no known near-ultraviolet states with J=10, 15 or 20, the low-order parameters in Vup potentially interact very weakly with the lower states, and it is important to include these in the optimization such that we do not lose the rotational dependence of these levels. This step is repeated several more times and each time gradually increasing f towards 105. Increasing f above this provided no improvement in the RMSE, and this concluded the refinement of Vup.

For the final optimization of our potential, we refine Vlow to states in J=0, 2, 5, 10, 15 and 20 using the 10 : 1 ratios of step one while also gradually increasing f to 1010. Going beyond this offered no improvement in the final RMSE and only increases the risk of over-refining. This f value is significantly larger than that used in the final refinement of Vup, which is entirely justified by there being significantly fewer states in the near-ultraviolet.

It is common to provide a breakdown of residuals for the VBOs in a large table; however, as already described, these states alone cannot be used to measure how well a potential can calculate energy levels. Hence, we calculate the average deviation of the calculated energy levels using our new potential, the POKAZATEL potential and the PES15K potential to those MARVEL states with J≤20. The calculated states from each potential were matched to the empirical MARVEL values using the same algorithm to facilitate an equal comparison. For states with energies below 26 000 cm-1, a 0.5 cm-1 threshold was used, while for those above 26 000 cm-1, a 1.0 cm-1 limit was used. In Fig. , the average residuals per J are plotted in three sections: (panel a) E≤15000 cm-1, (panel b) E≤26000 cm-1 and (panel c) E≤37000 cm-1. These comparisons include states in MARVEL both refined and not refined. Comparing to the unrefined states is a method of assessing the smoothness of the surface. Firstly, we must acknowledge that PES15K is excellent at reproducing those energy levels below 15 000 cm-1 with J≤9, but above this J threshold, the residuals begin to increase and eventually surpass ours. There is an outlying point at J=9 in Fig. a for PES15K, likely due to the matching algorithm; although, this does not occur for the other data sets. For POKAZATEL, the RMSE increases rapidly with J. This is most likely due to these potentials only being refined to states in J=0, 2 and 5. Our new potential offers lower residuals for those high J states while also providing relatively accurate energies into the near-ultraviolet. However, in Fig. c we see that there is a large amount of noise in both our new surface and POKAZATEL. This is due to an insufficient number of experimental data points to refine. For high values of J, it is also worth noting that, of the three potential surfaces, there are significantly fewer calculated levels from the POKAZATEL PES matched with those in MARVEL despite the same matching criteria being used for all. For the purpose of reproducibility, we provide a VBO comparison in the Supplement as well as a table containing the data used to create Fig. .

Figure plots the same residuals seen in Fig.  but now as a function of energy. The rotational dependence of the POKAZATEL PES is again clear. The Fortran F90 subroutine for our new semi-empirical PES, which we call “HOT_WAT”, is provided in the Supplement.

To generate transition intensities, we require an accurate dipole moment surface. The CKAPTEN surface has previously been shown to provide reliable dipole values ; hence, we will use this DMS to calculate our spectra. We compute a line list for H216O that extends to 41 200 cm-1, i.e. beyond the shortest wavelength that will be accessible by the NASA TEMPO mission, which is 290 nm . The accuracy of this line list is not verified for transitions with frequencies beyond 37 000 cm-1, and this region may be susceptible to basis set convergence issues. In HITRAN units, the minimum intensity considered here is 10-32 cm molecule-1 and Jmax=20, all assuming 296 K. There are no transitions in the near-ultraviolet that include J=20 which have intensities surpassing our 10-32 cm molecule-1 threshold. We then proceed to “MARVELize” this line list, meaning we replace, where possible, our calculated energy levels with empirical ones from MARVEL, which also allows us to add extra quantum labels (Ka, Kc, ν1, ν2, ν3) on top of the rigorous labels J, parity and symmetry. This process is described in more detail in .

In an earlier study , we generated near-ultraviolet spectra with the POKAZATEL potential and CKAPTEN DMS; although, the thresholds used were different to those used here. The maximum transition frequency considered in the previous study was 35 000 cm-1 with Jmax=14, and the minimum intensity considered was 10-30 cm molecule-1. But these criteria should be sufficient for comparison studies in the near-ultraviolet. Comparing these calculations to our new ones will allow us to ascertain how different potential surfaces influence intensities.

In Fig. , we plot transition intensities from our new calculations, the POKAZATEL line list, HITRAN2016 and our old calculations previously described. For transitions in the infrared, shown in Fig. b, the line lists show little deviation from each other; however, as transitions extend further into the blue, differences become significantly more pronounced and, in general, the POKAZATEL intensities appear too weak. At 19 000 cm-1, the first absorption feature not well represented by the POKAZATEL line list appears; see Fig. c. For wavelengths extending from 500 to 400 nm, transition intensities in the HITRAN2016 H216O line list are of comparable magnitude to ours and are, in general, made up from previously published theoretical models, notably BT2 and data. Atmospheric observations by suggest HITEMP2010 (mostly BT2 data) predicts absorption features of water vapour in the visible more accurately than the POKAZATEL line list; hence, it is reasonable to assume POKAZATEL also under-absorbs at 19 000 cm-1. However, at the 400 nm limit of HITRAN2016, we begin to notice larger differences in the intensities; although, our new data agrees much better with POKAZATEL; see Fig. d.

Comparing our new line list to the old calculations indicates that the new potential does not greatly alter the intensities, which was expected as, for stable transitions, the DMS controls the magnitude of the absorption . Hence, the differences which are observed in the near-ultraviolet are due to differences in the underlying dipole surfaces. The POKAZATEL line list was computed with the LTP2011S surface by , where “S” signifies that this surface is a fewer-parameter fit to their ab initio dipoles and is therefore more stable in energetic regions.

evaluated this POKAZATEL line list in the near-ultraviolet and comments that the feature at approximately 363 nm is underestimated by a factor of 2.4±0.7, where the largest contribution to this uncertainty is from the observation. In Fig. a, there is a visible drop in the calculated POKAZATEL cross sections that begin just beyond 25 000 cm-1. To verify that our new line list correctly models this feature, we sum transition intensities in both line lists that are within 27 000–27 800 cm-1. The ratio of our summed intensities to POKAZATEL is 3.08, which is within the uncertainty by . Despite this improvement, further validation is required to verify the entire line list. Future work is planned for this.

In 2013, report measurements of a strong, broadband near-ultraviolet absorption spectrum of water in the 350–290 nm region; these absorptions could not be detected by . The instrumental setup used by enabled them to place an upper limit of absorption in this region of 5×10-26 cm2 molecule-1. also placed several upper limits on the absorption of water vapour in the region of 350–310 nm with different uncertainties. Of these, we consider the weakest upper limit to compare with as it has the lowest uncertainty. This limit is 4.6×10-27 cm2 molecule-1 at a 0.7 nm resolution. More recently, made new measurements in the same region. In order to generate cross sections, we apply approximate air-broadening coefficients (γair) which are computed as functions of J′ and J′′ to our new line list, and we calculate cross sections using the HITRAN API (HAPI) code at resolutions of 0.03 cm-1 and 0.2 nm with the Voigt profile. It is important to note that the cross sections reported by are in 1 nm step sizes and those from are given in 5 nm intervals. Figure  compares our calculations to each of these data sets. The new measurements by give cross sections of comparable magnitude to those by but do not resemble any feature in our line list. The data sets from and are taken directly from their publications and have not been altered by us in any way. Importantly our calculated cross sections do not exceed the upper limit of at any resolution considered, while our 0.2 nm resolution cross sections do not exceed the proposed 0.7 nm resolution upper limit by .

Both and suggest that water vapour absorption in the 290–350 nm window should be on the order of 10-24 cm molecule-1, which is of comparable magnitude to features observed at 20 000–22 750 cm-1 (see Fig. a) (500–450 nm). suggest that this increased water vapour absorption is due to an absorption band between different electronic states; however, the nearest electronic state is an unbound 1B1 state which corresponds to the spectral feature at approximately 170 nm as confirmed by numerous experiments . These experiments show that absorption decreases exponentially with increasing wavelength (i.e. decrease of the wavenumber), as expected considering that the upper state is unbound. In order for these electronic transitions to absorb more in the red, one needs to populate high vibrational levels of the ground state, which is not possible at atmospheric temperatures. At room temperature, this band is unlikely to affect absorption in this 290–350 nm interval to the degree quoted by Conversely, our line list, which predicts greatly reduced cross sections in this region, appears to be in line with atmospheric observations. We are currently collaborating with atmospheric scientists at the Center for Astrophysics | Harvard & Smithsonian to further investigate this near-ultraviolet absorption by water vapour, but this effort would greatly benefit from further experimental research. Initial tests will focus on data obtained from the Ozone Monitoring Instrument (OMI) .

Our calculated line list is available in the Supplement and assumes 100 % H216O isotopic abundance.