Sub-percent accuracy for the intensity of a near infrared water line at 10670 cm$^{-1}$: experiment and analysis

Laser measurements of the intensity of (201) 3$_{22}$ -- (000) 2$_{21}$ near-infrared water absorption line at 10670.1 \cm\ are made using three different Herriott cells. These measurements determine the line intensity with an standard deviation below of 0.3~\% by consideration of the new geometrically derived formula for the opt ical path length without approximations. This determination together with the current accepted value leads to an overall uncertainty of 0.7~\% of the experimentally assessed line intensity which is compared with previous {\it ab initio} predictions. It is found that steady improvements in the both the dipole moment surface (DMS) and potential energy surface (PES) used in the theoretical studies leads to systematic better agreement with the observation, with the most recent prediction agreeing closely with the experiment.


I. INTRODUCTION
All molecules can be arguably divided into three unequal categories. To the first group belong two-electron systems, such as H 2 , HHe + and H + 3 . The second group comprises 10electron systems which includes HF, water, ammonia and methane as well as H 3 O + , NH + 4 and CH + 5 . The third group consists of the great multitude of remaining molecules. Why do we make such an unequal distribution? This division of species is based on the relative simplicity and importance of the species considered.
The fundamental works of Wolniewicz [1,2] on the H 2 molecule represent the beginning of solving the ab initio electronic structure problem for two-electron systems. Modern developments built on work by Wolniewicz [3][4][5] includes the papers by Pachucki, Komasa and co-workers [6][7][8], which demonstrate excellent continuation of Wolniewicz's earlier work.
Ten-electron systems have a particular significance because of the importance of many of them in the atmosphere of the Earth, solar system planets and exoplanets. Clearly the ab initio solution for molecules, belonging to this second group require greater computational efforts and remains much further from a satisfactory or final solution, than the molecules belonging to the two-electron group. The ab initio predictions of the ro-vibrational energy levels of water reached the 1 cm −1 level in [25] and 0.1 cm −1 for stretching states in [26], rising to 0.3 cm −1 when highly excited bending modes are considered. However, there are still many improvements need to reach the level of accuracy achieved by Wolniewicz for H 2 calculations [1,2]. [29]. The LTP dipole moment surface (DMS) was used to compute the water line intensities for comparisons with the first sub-percent accurate measurements by Hodges and Lisak [30]. However, subsequent measurements of line intensities for some other water bands [31] disagreed with the LTP predictions by up to 5 %. Further improvements in the calculations resulted in the sub-percent agreement with these newly measured line intensities [32].
A detailed comparison between theoretical predictions and high accuracy measurements by Birk et al. [33] suggested that while agreement for some bands was satisfactory (about 1%), this was not true for all bands. In particular Birk et al. found that for high overtone transitions in the near infrared or optical theory only agreed with their near IR Fourier Transform Spectrometer (FTS) measurements with an accuracy of about 2 %. Since then further improvements have been achieved in both the theoretical techniques [34] and the corresponding calculations [32,35]. Such improvements need to be tested against ever more accurate experimental results and it is such a test that we present here.
In this paper we present 3 experimental determinations of the intensity of near IR overtone line of H 2 16 O. These measurements are compared to the generally adopted value, from the FTS measurements of Birk et al. [33] leading to a sub per cent uncertainty of the experimentally assessed intensity. The experimental results are also compared to various theoretical predictions.
II. EXPERIMENTAL SET-UP Figure 1 gives a schematic over of the experimental setup use to make high accuracy As illustrated in Fig. 1), the two partial beams running downward were used to characterize the properties of the laser emission. Here, the first partial beam aimed at 'Detector Laser intensity' is used to determine the intensity fluctuations of the laser and the second partial The experimentally determined H 2 O line strengths were measured using three different Herriott cells, to increase sensitivity due to the extended optical path length. Herriott cells consist of two spherical mirrors with concave surfaces facing each other [36,37]. Typically, the coupling of the laser radiation used into and out of cell is done via one or more corresponding holes in the mirrors. In the first cell "HC std " both occurred via the same hole in one of the two mirrors. Transverse coupling, for example via a thin plate with reflective surfaces on both sides, is also possible and was used for the cells two: "HC T AC " and three: describes exactly for which mirror distances closed configurations exist. Here, U describes the number of 2π-twists with respect center line between the two Herriott cell mirrors and s corr is given by s corr = ROC − ROC 2 − r 2 sp , with r sp as the the radius of the spot patterns generated on the mirrors. More details are given by Rubin [38]. The optical path lengths used are specified in Table I.

III. LINE INTENSITY MEASUREMENT RESULTS
The H 2 O absorption line strength measurements were made at 296 K. Accordingly, the line strength was determined with the following formula: The H 2 O pressure range between 0 mbar and 13 mbar was covered.
In this work we compare results of three attempts to make high accuracy predictions of H 2 16 O line intensities. The first of these used the "Bubukina" PES of Bubukina et al. [50] constructed by fitting to ro-vibrational energy levels up to 25 000 cm −1 , which were reproduced with a standard deviation of 0.022 cm −1 . The second PES is the improved PES15K [46] which only fitted to ro-vibrational energy levels below 15 000 cm −1 , which were reproduced with an accuracy of 0.011 cm −1 . The improved PES15K PES has already been shown to result in a significant improvement of the calculated intensities [46]. Finally we consider the recently constructed HOT WAT PES of Conway et al. [35] fitted ro-vibrational energies over the entire range of their availability which is almost up to dissociation [52].
We note that the transition frequency of the line at 10670.1 cm −1 discussed in the present paper is reproduced best by this PES, to within only 0.001 cm −1 .
Experience has shown that it is best to use ab initio DMSs [53]. There has been a steady improvement in both calculation and fitting ab initio DMSs over the years [29,32,[54][55][56]. Here we consider the LTP2011 DMS of LTP [29] which is based on a set of 2000 internally-contracted multi-reference configuration interaction (IC-MRCI) points calculated with an aug-cc-pCV6Z basis set as energy derivatives (ED). Relativistic corrections to the dipoles were obtained in a similar manner by computing the derivatives with respect to the external electric field strength of the mass-velocity, one-electron Darwin (MVD1) relativistic corrections to the IC-MRCI energies. This DMS gives sub-percent accuracy for some bands but has been shown to give predictions a few % off for some bands [33]. Secondly, we consider the CKAPTEN DMS of Conway et al. [32] which was calculated using a similar procedure but with relativistic corrections obtained using a Douglas-Kroll-Hess Hamiltonian to order two (DKH2). The number of points were significantly increased to about 17 500 and an improved fit function used giving a better overall fit. For HOT WAT and CKAPTEN the average deviation of the predicted intensities for the (201) band considered here compared to the measured transition intensities of Birk et al. [33] is only 0.4 %.

V. COMPARISON WITH THEORETICAL CALCULATIONS
The comparison between our measurements and the theoretical calculations described above are given in Table IV  The final three rows of Table IV compare with theoretical predictions. The first two of these both use the LTP2011 DMS [46]; agreement improves with use of the better wavefunctions generated using the more accurate PES15K PES. The final row compares with the most recent result using both an improved PES and the CKAPTEN DMS [32]. The intensity predicted with these calculations lies within the experimental uncertainties and differ by less than 0.1% from the mean measured value. The use of the CKAPTEN DMS gives a significant improvement over LTP2011.
The first step is very important, though clearly we need the expansion in two directions.
First, the measurements of intensities of more lines, belonging to the different vibrational bands. Secondly, the higher overtones journey towards higher frequencies, from near IR towards optical region and even UV with the sub-percent accuracy is necessary.  to provide excellent results for higher stretching overtone although further high accuracy experimental studies would be needed to confirm this situation. The next step is to extend this work to higher overtones and frequencies extending into the optical region and even the near-UV. In this context we note that recent cavity ring down spectroscopy measurements by Vasilchenko et al.
[58] suggest that further work is needed to get equally reliable predicted intensities for the very weak bending overtones in the red region of spectrum.