ExoMol line lists XV: H{}_{2}O{}_{2}

ExoMol line lists XV: A new hot line list for hydrogen peroxide

Ahmed F. Al-Refaie, Oleg L. Polyansky, Roman I. Ovsyannikov, Jonathan Tennyson and Sergei N. Yurchenko,
Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, UK
Institute of Applied Physics, Russian Academy of Sciences, Ulyanov Street 46, Nizhny Novgorod, Russia 603950
July 13, 2019Accepted XXXX. Received XXXX; in original form XXXX
July 13, 2019Accepted XXXX. Received XXXX; in original form XXXX
Abstract

A computed line list for hydrogen peroxide, HO, applicable to temperatures up to  K is presented. A semi-empirical high accuracy potential energy surface is constructed and used with an ab initio dipole moment surface as input TROVE to compute 7.5 million rotational-vibrational states and around 20 billion transitions with associated Einstein- coefficients for rotational excitations up to . The resulting APTY line list is complete for wavenumbers below 6 000 cm ( m) and temperatures up to 1250 K. Room-temperature spectra are compared with laboratory measurements and data currently available in the HITRAN database and literature. Our rms with line positions from the literature is 0.152 cm and our absolute intensities agree better than 10%. The full line list is available from the CDS database as well as at www.exomol.com.

keywords:
molecular data; opacity; astronomical data bases: miscellaneous; planets and satellites: atmospheres
pagerange: ExoMol line lists XV: A new hot line list for hydrogen peroxideReferencespubyear: 2016

1 Introduction

Terrestrial hydrogen peroxide exists as a trace molecule in the Earth’s atmosphere and contributes to the atmospheres oxidising budget as well as ozone production and water chemistry (Davis, 1974; Chance et al., 1991; Allen et al., 2013; Zins & Krim, 2014) and its concentration is now being routinely observed (Allen et al., 2013). Astrophysically there have been multiple detections of HO in the atmosphere of Mars (Clancy et al., 2004; Encrenaz et al., 2004, 2012; Aoki et al., 2015) with seasonal variation, possibly formed by triboelectricity in dust devils and dust storms (Encrenaz et al., 2012) and may well act as an agent in the oxidization of the Martian surface. Hydrogen peroxide has also been detected in the atmosphere of Europa (Hand & Brown, 2013) in the 3.5 region. The first detection of interstellar HO was made by Bergman et al. (2011) and is believed to play an important role in astrophysical water chemistry similar to that on Earth. Du et al. (2012) suggest that HO is produced on dust-grains via the hydrogenation of grain HO and released into the gas-phase through surface reactions. On the dust-grain, HO acts as an intermediate in the formation of water and aids in the production of other species such as HCO, CHOH, and O.

Hydrogen peroxide belongs to the peroxide group of molecules with an HO-OH bond dissociation enthalpy of 17050 cm(Bach et al., 1996) at 0 K. HO is an asymmetric prolate rotor molecule and is the simplest molecule that exhibits internal rotation. This torsional motion gives rise to a double minimum potential curve with respect to its internal rotation co-ordinates as well as two alignments of the O-H bonds: cis and trans. The consequence of this motion means that there are four sub-levels for each torsional excitation which are characterized by their symmetry. This necessitates the use of an additional quantum number, , to unambiguously describe its motion. The molecular states can be classified using the (M) symmetry group which best describes the torsional splitting caused by the cis and trans tunneling (Hougen, 1984). HO has six vibrational modes: and represent the symmetric and asymmetric O-H stretching respectively, and represent the O-H bending modes, represents the O-O stretch and the mode represents the torsional excitation with the more common notation of .

Experimental studies of ro-vibrational HO spectra have mostly probed the torsional motion in the ground (Olson et al., 1988), the (Camy-Peyret et al., 1992) and (Perrin et al., 1990, 1995) vibrational modes. Conversely, the higher-lying O-H stretching modes, and , are poorly studied using high resolution techniques. The difference between the two stretching bands is about 8 – 10 cm-1 and torsional splitting from the double minimum of the potential gives rise to doubling (Giguere & Srinivasan, 1974) in the form of ’quasi’-degenerate states (Rauhut et al., 2009) that are difficult to resolve with a degree of accuracy. Olson et al. (1988) give an estimate of 3610 - 3618 cm-1 for and 3601 - 3617 cm-1 for whilst a Raman study gives a lower value of 3607 cm-1 for the    band-centre (Giguere & Srinivasan, 1974) but determining the accuracy to better than 0.1 cm-1 is difficult.

HO has been a benchmark system for developing methods aiming to treat large amplitude motion (Luckhaus, 2000; Mladenovic, 2002; Yu & Muckerman, 2002; Carter et al., 2009). Recent calculations on the ro-vibrational states for HO include the ab initio computation using CCSD(T)-F12 electronic structure calculations of band frequencies accurate to about 4.0 cm-1 by Rauhut et al. (2009), models of the peroxide stretches by Bacelo & Binning (2005), a discrete variable representation (DVR) calculation for levels up to 6000 cm-1 by Chen et al. (2001); Lin & Guo (2003) and finally, a potential energy surface (PES) calculations by Koput et al. (1998) and Kuhn et al. (1999). Calculation which also consider transition intensities are rather rarer but a recent example is provided by Carter et al. (2011). The peroxide system was used to benchmark the large amplitude calculations of MULTIMODE (Bowman et al., 2003) up to and showed good agreement against HITRAN line intensities but the PES used had an rms of 20 cm-1 against experimental band centers. However, this PES has been superceded by the higher accuracy ab initio potential energy surface (PES) of Małyszek & Koput (2013) which was further modified by Polyansky et al. (2013). This modified PES was used for our room-temperature line list (Al-Refaie et al., 2015a) and provides the starting point for the refinements performed here.

Experimental transitions frequencies and intensities for HO are available in the HITRAN 2012 database (Rothman et al., 2013) but only for room temperature modelling up to 1800 cm. This region covers the torsional, O-H bending modes and O-O stretch but misses the O-H stretches in the 3750 cm region. Only a few studies deal with absolute intensities of HO in the far-infrared (Zumwalt & Giguere, 1941; Perrin et al., 1996, 1995) with only PNNL-IR (Sharpe et al., 2004) data providing integrated intensities in the mid-infrared region (Johnson et al., 2009). The thermal decomposition of hydrogen peroxide at 423 K makes it difficult and dangerous to study at higher temperatures.

Theoretical line lists can be used to fill in gaps in the experimental data both in terms of wavelength and temperature coverage. The ExoMol project (Tennyson & Yurchenko, 2012) aims to produce comprehensive theoretical molecular line lists to aid in studies of the atmospheres of exoplanets, cool stars and other (hot) bodies. A room temperature line list for HO was previously computed by us (Al-Refaie et al., 2015a) using the PES of Polyansky et al. (2013) and a new ab initio dipole moment surface (DMS). It provides about 1 billion transitions at up to 8,000 cm-1. However it is limited as the rotational excitation of makes it inadequate for high temperature modelling and the lower energy cut-off means that coverage above 4,000 cm-1 rapidly becomes incomplete. This work aims to build upon this line list by refining the PES towards spectroscopic accuracy and extending the temperature and frequency range for which the resulting line list is applicable.

2 Method

2.1 Potential energy Surface Refinement

Our previous room-temperature HO line list (Al-Refaie et al., 2015a) was computed using the ab initio PES of Małyszek & Koput (2013) with the small adjustment of the ab initio equilibrium geometry and height of the torsional barrier proposed by Polyansky et al. (2013). This PES reproduces the known empirical energy levels with a root mean square (rms) of about 1–2 cm-1. However, by utilising empirical band-centre shifting (Yurchenko et al., 2009) during the computation of the Hamiltonian, this was significantly reduced it to 0.001 – 0.1 cm-1 in the room temperature line list. The empirical shifting can be thought of as an addition to the ab initio PES producing an ’ad-hoc’ PES which we will refer to as ’H2O2-2015’ in comparisons below. Whilst band-centre shifting can reproduce experimental energies simply, its accuracy and predictive ability is limited to vibrational bands whose band centre positions are already experimentally characterised. A more robust method of correcting the PES is through fitting or refining to experimental energies. The TROVE nuclear motion program used by us here, see below, provides PES refinement capabilities and has been successful in producing accurate PES for molecules such as HCO (Yachmenev et al., 2011), NH (Yurchenko et al., 2011a), SO Underwood et al. (2014), PH (Sousa-Silva et al., 2015) and CH(Yurchenko & Tennyson, 2014).

The procedure implemented in TROVE describes a correction surface to the initial (ab initio) surface . The new refined surface can therefore be written as and the new Hamiltonian as , where is the Hamiltonian if the starting point for the refinement. The wavefunctions from , are used as basis-functions for . is expanded in Taylor series and the expansion coefficients, , are obtained in a variational least-squares fit to spectroscopic data via the objective function :

(1)

where is the number of observed energies , are the calculated energies and are the weights.

Here the original 282 expansion coefficients of the ab initio PES are reduced to 163 by removing the symmetry-related O-H stretching () and bending () terms from the input PES and simply linking them in the computation of potential energy terms in the Hamiltonian. This ensures that the symmetry of these terms is preserved during the fitting process. The quality of the fit is determined by the quality and vibrational diversity of the input dataset. Two sources of experimental data come from line-positions provided in the literature and transitions from HITRAN. The HITRAN dataset sources come from observations by Perrin et al. (1995), Perrin et al. (1996), Perrin et al. (1990) and Klee et al. (1999) with literature line-positions from Flaud et al. (1989), Olson et al. (1988) Giguere (1950), Zumwalt & Giguere (1941) and Camy-Peyret et al. (1992). This empirical dataset provides the , , and vibrational terms. Unfortunately there is little to no reliable data on the and energy levels, their reported band-centre values vary significantly in literature making them unsuitable for the refinement. This hampers the vibrational diversity that would aid in construction of an extensive fitted PES. However, these terms can be indirectly improved by including higher values from other vibrational states.

Our input dataset includes all energies for up to 4000 cm-1. The weights used have an arbitrary range of values that are normalized in the fit. The energies given in literature are the simplest to include in the refinement process and are given the highest weighting. Here the pure torsional band at from Camy-Peyret et al. (1992) and Olson et al. (1988) are given the highest weighting of . The , , , , energies and HO band centers (except for and ) from Camy-Peyret et al. (1992), Giguere (1950), Perrin et al. (1990), Flaud et al. (1989) and Zumwalt & Giguere (1941) are given weights .

Transitions from HITRAN require additional work. In order to determine the upper state of a transition requires the assignment of the lower state energy. Fortunately HITRAN provides the lower state energy for all transitions in the database. However, lower state energies require corroboration from literature data and/or the ab initio energies for the upper state energies to be included in the fit with based on confidence of the datum. Each input datum must be correlated with a theoretically computed energy level which, in this present work, was straightforward due to the good agreement given by the initial ab initio PES.

Special measures must be taken in order to ensure that the refinement process does not lead to unphysical shapes for the new PES due to a limited sampling of the experimental data not covering all the complexity of the potential energy surface of HOOH. For example the high stretching or bending overtones are poorly represented in the experimental and therefore it is important to retain the ab initio quality of the original PES by Polyansky et al. (2013). To this end we constrain the PES around ab initio energies at each geometry (Yurchenko et al., 2003, 2011b; Yachmenev et al., 2011; Sousa-Silva et al., 2015).

The new potential energy surface is called H2O2-2016. Table 1 describes the rms for states of a particular weight. The high quality of the H2O2-2016 energies computed without any empirical band shifts shows that this new semi-empirical PES performs better overall than the ab initio band-shifted PES especially for the lower weighted states. Weights 10 relate to vibrational states that were involved in the band-shifting which gives H2O2-2015 its low rms values. Comparing weights lower than 10 suggests that the predictive ability of the H2O2-2016 PES is greatly enhanced. The overall comparison as a function of the rotational quantum number with a weighted rms is given in Table 2.

H2O2-2016 H202-2015
Weight N rms (cm-1) rms (cm-1) Comment
100 43 0.001 0.000 pure torsional states
10-20 144 0.004 0.007 Band centers and , states
1-9 186 0.539 1.369 Upper state extracted from HITRAN with corroborated lower states
Table 1: Comparison of weighted experimental data-points in the fit and non-weighted root mean squared deviation of both H2O2-2016 (this work) and H2O2-2015 (Al-Refaie et al., 2015a) for each dataset.
H2O2-2016 H202-2015
J N wrms (cm-1) wrms (cm-1)
0 34 0.238 0.254
1 47 0.079 0.320
2 81 0.096 0.345
3 116 0.183 0.404
4 132 0.154 0.287
Total 0.150 0.321
Table 2: Comparision of experimental data-points in the fit and weighted root mean squared deviation of both H2O2-2016 (this work) and H2O2-2015 (Al-Refaie et al., 2015a).
Obs Calc O-C Obs Calc O-C
30 0 0 0 1 0 789.58 789.64 -0.06 31 11 0 0 2 0 1950.60 1950.55 0.05
30 1 0 0 2 0 793.05 793.12 -0.07 31 11 0 0 3 0 1963.12 1963.07 0.05
30 2 0 0 1 0 829.29 829.34 -0.05 31 5 0 1 2 1 2577.85 2577.91 -0.06
30 2 0 0 4 0 841.34 841.32 0.02 31 1 0 2 2 1 2694.38 2694.48 -0.10
30 3 0 0 2 0 876.03 876.07 -0.04 31 2 0 2 1 1 2729.02 2729.07 -0.05
30 3 0 0 2 0 876.03 876.07 -0.04 31 3 0 2 2 1 2775.74 2775.66 0.08
30 4 0 0 1 0 940.03 940.07 -0.04 31 9 0 0 3 1 2877.92 2877.84 0.08
30 6 0 0 4 0 1134.61 1134.58 0.03 31 7 0 1 3 1 2940.33 2940.25 0.08
30 2 0 1 4 0 1198.50 1198.58 -0.08 31 9 0 1 3 1 3232.19 3232.21 -0.02
30 7 0 0 2 0 1241.30 1241.33 -0.03 32 1 0 0 2 0 898.70 898.78 -0.08
30 7 0 0 2 0 1241.30 1241.33 -0.03 32 2 0 0 1 0 936.15 936.21 -0.06
30 7 0 0 3 0 1253.29 1253.25 0.04 32 2 0 0 4 0 948.32 948.29 0.02
30 4 0 1 4 0 1308.34 1308.43 -0.09 32 3 0 0 2 0 983.10 983.16 -0.05
30 1 0 2 2 0 1362.75 1362.83 -0.08 32 4 0 0 1 0 1047.03 1047.07 -0.04
30 8 0 0 4 0 1390.06 1390.01 0.05 32 2 0 1 4 0 1305.21 1305.30 -0.09
30 5 0 1 3 0 1390.16 1390.27 -0.11 32 7 0 0 2 0 1347.93 1347.96 -0.03
30 2 0 2 1 0 1396.42 1396.52 -0.10 32 1 0 2 2 0 1468.52 1468.61 -0.09
30 9 0 0 3 0 1544.80 1544.75 0.05 32 8 0 0 4 0 1496.82 1496.77 0.05
30 0 1 0 1 0 1645.15 1645.20 -0.05 32 9 0 0 3 0 1651.43 1651.37 0.06
30 2 1 0 2 0 1688.11 1688.15 -0.03 32 0 1 0 1 0 1749.83 1749.89 -0.06
30 10 0 0 1 0 1707.36 1707.33 0.03 32 1 1 0 2 0 1752.86 1752.92 -0.07
30 10 0 0 4 0 1717.16 1717.10 0.06 32 2 1 0 1 0 1794.43 1794.46 -0.03
30 3 1 0 1 0 1731.55 1731.59 -0.04 32 10 0 0 4 0 1823.55 1823.48 0.07
30 11 0 0 2 0 1898.17 1898.11 0.06 32 3 1 0 2 0 1837.22 1837.26 -0.04
30 11 0 0 3 0 1910.47 1910.41 0.06 32 5 0 1 2 1 2631.92 2632.00 -0.08
30 6 1 0 2 0 1978.18 1978.19 0.00 32 9 0 0 3 1 2931.74 2931.70 0.04
30 5 0 1 2 1 2525.44 2525.50 -0.06 32 9 0 1 3 1 3285.71 3285.72 -0.01
30 0 0 2 1 1 2638.21 2638.21 0.00 33 4 0 0 1 0 1103.04 1103.09 -0.05
30 2 0 2 1 1 2678.98 2678.96 0.02 33 7 0 0 2 0 1403.73 1403.76 -0.03
30 6 0 1 4 1 2768.27 2768.35 -0.08 33 9 0 0 3 0 1707.24 1707.18 0.06
30 9 0 0 3 1 2825.75 2825.74 0.01 33 10 0 0 4 0 1879.23 1879.16 0.07
30 9 0 1 3 1 3180.32 3180.35 -0.03 33 5 0 1 2 1 2687.68 2687.77 -0.09
30 11 0 0 3 1 3196.72 3196.74 -0.02 33 7 0 2 2 1 3252.13 3252.10 0.03
30 10 0 1 4 1 3355.50 3355.61 -0.11 33 9 0 1 3 1 3340.86 3340.86 0.00
31 3 0 0 2 0 928.92 928.97 -0.04 34 2 0 0 4 0 1061.96 1061.93 0.03
31 4 0 0 1 0 992.69 992.73 -0.04 34 3 0 0 2 0 1096.87 1096.93 -0.06
31 6 0 0 4 0 1187.24 1187.21 0.03 34 4 0 0 1 0 1160.72 1160.77 -0.05
31 1 0 1 3 0 1230.77 1230.86 -0.09 34 2 0 1 4 0 1418.58 1418.69 -0.11
31 2 0 1 4 0 1253.76 1253.86 -0.10 34 7 0 0 2 0 1461.18 1461.23 -0.05
31 7 0 0 2 0 1293.78 1293.81 -0.03 34 10 0 0 4 0 1936.55 1936.48 0.07
31 7 0 0 2 0 1293.78 1293.81 -0.03 34 7 0 2 2 0 2026.09 2026.08 0.01
31 0 0 2 1 0 1410.64 1410.73 -0.09 34 0 0 0 4 1 2287.94 2288.05 -0.11
31 0 0 2 1 0 1410.64 1410.73 -0.09 34 5 0 1 2 1 2745.11 2745.21 -0.10
31 8 0 0 4 0 1442.61 1442.56 0.05 34 9 0 1 3 1 3397.65 3397.63 0.02
31 9 0 0 3 0 1597.29 1597.23 0.06 35 1 0 0 2 0 1091.72 1091.77 -0.05
31 1 1 0 2 0 1717.38 1717.60 -0.23 35 4 0 0 1 0 1220.07 1220.13 -0.06
31 2 1 0 1 0 1736.75 1736.79 -0.05 35 7 0 0 2 0 1520.29 1520.34 -0.05
31 10 0 0 4 0 1769.53 1769.46 0.07 35 1 0 0 2 1 2324.23 2324.16 0.07
31 3 1 0 2 0 1783.76 1783.79 -0.03 35 1 0 0 2 1 2324.23 2324.16 0.07
31 4 1 0 1 0 1847.46 1847.49 -0.03 36 2 0 0 4 0 1182.24 1182.21 0.03
Table 3: Residuals in cm-1 for energies computed from the H2O2-2016 PES. Observed data is from Camy-Peyret et al. (1992) and Perrin et al. (1990). The overall rms is 0.0642 cm-1

Overall H2O2-2016 improves the rms deviations of H2O2-2015 by more than a factor of 2. Table 3 highlights residuals for for the and line positions from Camy-Peyret et al. (1992) and Perrin et al. (1990) and shows excellent agreement with an overall rms of 0.064 cm. The rms deviation for all 2734 states in HITRAN up to and energy up to 3461.02 cm-1 is 0.834 cm-1. Vibrational terms that correspond to the highest weighted states have an rms of 0.192 cm-1. Around 12 states related to higher excited torsional modes have an rms of 5.2 cm-1 and may well be due to misassignments. This PES is therefore of there of improved accuracy and is the one used below. The coefficients defining this PES are given in the Supplementary Information.

2.2 Variational computation

Theoretical ROVibrational Energies (TROVE) (Yurchenko et al., 2007) was employed to compute the ro-vibrational energies of HO. TROVE is a variational nuclear motion solver and was used to successfully produce the room temperature HO line list as well as hot line lists for NH (Yurchenko et al., 2011a), CH (Yurchenko & Tennyson, 2014), PH (Sousa-Silva et al., 2015), HCO (Al-Refaie et al., 2015b) and SO (Underwood et al., 2016).

TROVE can operate with any co-ordinate system of our choosing by utilizing an approximate kinetic energy operator (KEO). For HO, this approximate KEO was shown by Polyansky et al. (2013) to produce results that are in very good agreement with the exact KEO code WAVR4 (Kozin et al., 2004). However, the computational cost is greatly reduced for rotationally excited states by using TROVE. Convergence of the KEO usually requires an expansion order of 6 or 8 (Yurchenko et al., 2007); 6 being chosen for this work and 8 for the potential energy expansion as suggested from a previous HO calculation by Polyansky et al. (2013).

Basis sets and wave functions are symmetrized to the (M) molecular symmetry group (Bunker & Jensen, 1998) which is isomorphic to the (M) symmetry group. This has the benefit of factorizing the Hamiltonian into smaller independent blocks for diagonalization. The irreducible representations of this group are A, A, B, B, B, B, B and B. However, the states corresponding to B, B, B and B have zero nuclear statistical weight and therefore these matrix blocks are not needed for . The corresponding energies from these representations are also unphysical, but computed and kept as vibrational band centres for reference purposes.

A symmetry-adapted basis-set is constructed by a multi-step contraction scheme by solving the 1D Schrödinger equation via the Numerov-Cooley method (Noumerov, 1924; Cooley, 1961) for the basis-functions , where represents the O-O stretching co-ordinate, and represent the O-H stretching co-ordinates, and represents the O-H bending modes and represents the torsional mode. is the local mode quantum number assigned by TROVE. The basis functions are then used to form a product-type basis set, which is truncated by the polyad number :

(2)

We use as it was found to give good convergence (Polyansky et al., 2013). The second step requires computing a contracted basis set by reducing the six dimensional co-ordinates into four dimensions. This accomplished by reducing the 6-dimensional co-ordinate system into four subspaces: , , and based on their permutation properties. The reduced Hamiltonian is solved using the primitives as basis functions to obtain the symmetrised vibrational eigenfunctions , , and . The vibrational contracted basis set is then formed from symmetrized products of these eigenfunctions which is also truncated via Eq. (2). A final contraction step is performed by solving the problem and replacing the bulky primitive and contracted vibrational basis set with the more compact wavefunctions truncated at 12 000 cm-1. The benefit of this is that the computation of the Hamiltonian matrix elements for is more efficient and reduces the size of the matrix. Whilst this form means we can replace the diagonal vibrational terms with experimental band centers, the low rms error of H2O2-2016 PES band centers makes this procedure unnecessary. Our final ro-vibrational wavefunction has the form:

(3)

where are rigid rotor functions with parity defined by Yurchenko et al. (2005a) and are the eigenvector coefficients obtained by diagonalization of the Hamiltonian matrix. The linear algebra libraries LAPACK (Anderson et al., 1999) and SCALAPACK (Blackford et al., 1997) were employed to solve for the eigenvalues and eigenvectors.

TROVE assigns six local mode quantum numbers to each state by finding the largest contributing . These can be reassigned to match the more standard normal modes generally used in literature. However, TROVE only provides six quantum numbers for reassignment, when seven are required by the inclusion of . The quantum number can be preserved in the reassignment in TROVE by utilizing the following form:

(4)

where is the excitation and is the symmetry where is A, B, B and A respectively. To retrieve and simply requires:

(5)

2.3 Dipole moment surface and intensities

An electric dipole moment surface (DMS) is required in order to compute transition intensities. The ab initio DMS of Al-Refaie et al. (2015a) was utilized; this was computed at the CCSD(T)-f12b (Bartlett & Musiał, 2007) level of theory in the frozen-core approximation and is applicable to energies up to   cm-1.

The eigenvectors, obtained by diagonalization, are used in conjunction with the DMS to compute the required linestrengths and Einstein- coefficients of transitions that satisfy the rotational selection rules

(6)

and the symmetry selection rules:

(7)

applied to para and ortho states, respectively. States with and symmetry are forbidden due their zero nuclear statistical weights.

The Einstein- coefficient for a particular transition from the initial state to the final state is given by:

(8)

where is the rotational quantum number for the initial state, is Planck’s constant, is the transition frequency (), and represent the eigenfunctions of the final and initial states respectively, is the electronically averaged component of the dipole moment along the space-fixed axis (see also Yurchenko et al. (2005b)). From this the absolute absorption intensity is determined by:

where is the second radiation constant, is the energy term value, the absolute temperature and is the nuclear spin statistical weight factor. , the partition function, is given by:

(9)

where is the degeneracy of a particular state with the energy term value . For HO, is with for and symmetries and for and symmetries. The transitions were computed using the energy limits 6 000 and 12 000 cm for the lower and upper states, respectively giving a complete coverage of the region 0 – 6 000 cm ( m).

The intensities were computed using an enhanced version of TROVE that utilizes the nVidia graphics processing units (GPU) allowing for the computation of 5,000–30,000 transitions per second on a single GPU. The GPUs utilized were the nVidia M2090, K20 and the K40 models. A paper discussing this will be published elsewhere (Al-Refaie et al., 2016).

3 Results

The final hot line list named APTY contains 7 560 352 states and almost 20 billion transitions that completely cover the 0 – 6 000 cm-1 region. An extended line list is provided which contains an additional 8 billion transitions in the 6 000– 8 000 cm-1 region with reduced completeness for higher temperature. Radiative lifetimes are also computed via the methodology presented by Tennyson et al. (2016a). Figure 1 presents the lifetimes computed for states up to 6 000 cm-1. Here the complicated rotational structure of HO gives rise to long lived lower state energies with lifetimes that can reach up to 30 million years. Extract from the ExoMol-form (Tennyson et al., 2013, 2016b) states and transition files are given in Tables 4 and 5. Spectra at arbitrary temperatures can be computed using the Einstein- coefficients from the transition files.

,cm
1 0.0 1 0 inf 1 0 0 0 0 0 0 1 1 0 0 1 1 0.95 0 0 0 0 0 0
2 254.54376 1 0 6.9464D-01 1 0 0 0 1 0 0 1 1 0 0 1 2 0.94 0 0 0 0 0 4
3 569.726553 1 0 3.1256D-01 1 0 0 0 2 0 0 1 1 0 0 1 3 0.94 0 0 0 0 0 8
4 865.942367 1 0 3.1317D+01 1 0 0 1 0 0 0 1 1 0 0 1 4 0.87 1 0 0 0 0 0
5 1000.872029 1 0 1.7264D-01 1 0 0 0 3 0 0 1 1 0 0 1 5 0.93 0 0 0 0 0 12
6 1119.712431 1 0 6.8795D-01 1 0 0 1 1 0 0 1 1 0 0 1 6 0.85 1 0 0 0 0 4
7 1391.868926 1 0 6.8101D+00 1 0 1 0 0 0 0 1 1 0 0 1 7 0.44 0 0 0 1 0 0
8 1432.296785 1 0 2.9134D-01 1 0 0 1 2 0 0 1 1 0 0 1 8 0.81 1 0 0 0 0 8
9 1477.148868 1 0 1.2710D-01 1 0 0 0 4 0 0 1 1 0 0 1 9 0.84 0 0 0 0 0 16
10 1679.479144 1 0 5.6793D-01 1 0 1 0 1 0 0 1 1 0 0 1 10 0.42 0 0 0 1 0 4
11 1714.839900 1 0 1.5294D+01 1 0 0 2 0 0 0 1 1 0 0 1 11 0.80 2 0 0 0 0 0
12 1862.945393 1 0 1.7244D-01 1 0 0 1 3 0 0 1 1 0 0 1 12 0.82 1 0 0 0 0 12
13 1945.069104 1 0 1.1596D-01 1 0 0 0 5 0 0 1 1 0 0 1 13 0.74 0 0 0 0 0 20
14 1967.624310 1 0 6.7322D-01 1 0 0 2 1 0 0 1 1 0 0 1 14 0.77 2 0 0 0 0 4
15 1973.269531 1 0 2.8904D-01 1 0 1 0 2 0 0 1 1 0 0 1 15 0.39 0 0 0 1 0 8
16 2240.375911 1 0 4.8422D+00 1 0 1 1 0 0 0 1 1 0 0 1 16 0.35 1 0 0 1 0 0
17 2280.037817 1 0 2.8806D-01 1 0 0 2 2 0 0 1 1 0 0 1 17 0.72 2 0 0 0 0 8

(M) represention: 1 is , 2 is , 3 is , 4 is , 5 is , 6 is , 7 is and 8 is
: State counting number;
: State term energy in cm;
: State degeneracy;
: State rotational quantum number;
: Lifetime (s)
: Total symmetry in (M) (, , and only)
,,,,: Normal mode vibrational quantum numbers;
: torsional mode excitation;
: torsional symmetry;
: Symmetry of vibrational contribution in (M);
: State projection of the rotational quantum number;
: Rotational parity;
: Symmetry of rotational contribution in (M);
: State number in block;
: Largest coefficient used in the assignment;
: TROVE vibrational quantum numbers.

Table 4: Extract from the HO state file. The full table is available at http://cdsarc.u-strasbg.fr/cgi-bin/VizieR?-source=J/MNRAS/.
4893324 4865894 2.5558e-33
6064947 6004042 5.0185e-24
978337 813317 2.3963e-29
5920140 5983274 9.1885e-25
4838267 4920666 1.5740e-25
2328674 2173645 7.2614e-33

: Upper state counting number;

: Lower state counting number;

: Einstein- coefficient in s.

Table 5: Extracts from the HO transitions file. The full table is available at http://cdsarc.u-strasbg.fr/cgi-bin/VizieR?-source=J/MNRAS/
Figure 1: Radiative lifetimes computed (Tennyson et al., 2016a) for HO states up to 6,000 cm-1

An estimate of the temperature applicability of the line list can be performed by checking the partition function convergence of Eq. (9), which is computed via explicit summation (Vidler & Tennyson, 2000). The convergence can be measured by computing where is the partition function for all energy levels up to rotational excitation . For 296 K, the partition function converges to 0.001% at which matches the room temperature line list’s limit of . At higher temperatures, it is well-converged up to at least 1500 K where the estimated error is only 0.2 % at . This can be attributed to the good coverage of states computed that contribute to the overall population. A second partition can be evaluated by only including states that fall below the 6 000 cm-1 lower state energy limit of the line list and compared against by computing the ratio to assess the completeness of the full line list. Figure 2 shows that at up to 800 K, the partition functions are essentially the same. At 1250 K about 90% of the population of states is represented by but this falls to 80% at 1500 K giving the upper temperature for which APTY is reasonably complete as 1250 K. Usage of the line list at higher temperatures runs the risk of losing opacity due to missing contributions. The ratio can be used to estimate this. However, the decomposition of HO means that it is unlikely to be an important species above 1000 K. The partition function tabulated in steps of 1 K, alongside an associated cooling function, is given as Supplementary Information to this article.

Figure 2: / against temperature where is the partition function computed using only energy levels below our lower state threshold of 6000 cm and is our estimate of the full partition function.

Table 6 compares our partition function against HITRAN; we see that at temperatures less than 1000 K we agree better than 1. For temperatures 1000 – 1500 K, the APTY partition function is greater by 2–4% suggesting that the explicit summation method gives higher, and probably better values, than the more approximate method used by HITRAN (Fischer et al., 2003). However, at 3000 K APTY’s is 30% lower than the HITRAN value which can be attributed to the eigenvalue cutoff of 12 000 cm and . Studies on ammonia and phosphine have shown that considerably extended lists of energy levels are required to get converged partition sums at these elevated temperatures (Sousa-Silva et al., 2014).

/ K APTY HITRAN
75 895.506 894.866
150 2 815.866 2 811.187
255 7 360.598 7 336.856
300 10 126.961 10 087.090
500 31 246.17 30 990.11
1000 232 439.8 226 152.5
1500 1 031 673.6 993 983.8
3000 21 847 680 15 151 254
Table 6: Comparisons of HO partition functions as function of temperature for this work those used in HITRAN (Fischer et al., 2003).

Figure 3 is a simulated spectrum of the APTY line list computed at K. This highlights the coverage and sheer number and density of transitions available compared to the current edition of the HITRAN database (Rothman et al., 2013). Figure 4 compares our results with specific regions in the HITRAN database, the torsional and bands. Comparisons of the two show excellent agreement in replicating both line position and intensities.

Figure 3: Overview of our synthetic spectrum at K against HITRAN data (Rothman et al., 2013).
Figure 4: The fundamental bands compared to the HITRAN database (Rothman et al., 2013) at K: (a) Torsional and (b) bands.

Our line list in the and band regions can be validated by simulating absorption cross sections in the the 2.7  region and comparing against PNNL-IR data (Sharpe et al., 2004). The structure and positions are in good agreement with the overall integrated intensity for APTY in this region being 3% stronger than PNNL. The improvement given by the H2O2-2016 PES can be demonstrated by comparing with our previous room temperature line list and with the PNNL-IR data; see Figure 6. The wavelengths of the largest two peaks in this band at 2.738  and 2.736  are correctly reproduced by APTY but are shifted by about 0.001  for H2O2-2015, showing the improvement in this band due to use of the refined PES. Overall the integrated cross-sections for this band differs only by 3% from PNNL; indeed the entire spectrum up to 6,000 cm-1 only differs by 3%.

Figure 5: The and band region with APTY against PNNL-IR data at 323.15 K (Sharpe et al., 2004) with HWHM = 0.300 cm
Figure 6: Cross-section comparison of peaks in the and band region with APTY (this work) and H2O2-2015 (Al-Refaie et al., 2015a) against PNNL-IR data at 323.15 K (Sharpe et al., 2004) with HWHM = 0.300 cm

Band intensities can be computed by explicit summation of all transitions within a band and compared against available data. Table 7 shows that for the limit available empirical band intensities we agree with most regions to 7.3% which is below the estimated experimental uncertainty of 10%. Two discrepancies are with the band from Perrin et al. (1995) and the band from Johnson et al. (2009). The former conflicts with other measurements due to Johnson et al. (2009) and Klee et al. (1999) where integrated absorption intensities were measured directly, whilst the band intensities of Perrin et al. (1995) were obtained by summing a synthetic spectrum of only 27 276 transitions. Our band intensity is 33.78% weaker than the experimentally derived value. Johnson et al. (2009) suggests that the assignment of this band is compared to the assignment by Giguere (1950). This is based on a Q-branch peak observed at 2,658.62 cm-1. The assignments from APTY suggest that the peak observed is actually a convolution of Q-branches of the , , , , and transitions with an average separation between them at 0.02  cm-1. Computing the band intensities of all of these bands in this region give an answer that agrees with value given by Johnson et al. (2009) to 3.97%.

Figure 7: The 2.77 m band of HO at room temperature. Upper display: APTY cross-sections (296 K) generated using a Doppler profile; Lower display: an uncalibrated Kitt Peak spectrum of HO (Archive Name is 800628R0.002, Date is 28/06/1980; Range is 1599.010271–6421.422198 cm-1, Observer is Hunt; 1.77 m cell).

The Kitt Peak Archive provides FTIR spectra of HO covering the wavenumber region up to 6422 cm-1, which is only partly assigned. Figure 7 (lower display) shows an uncalibrated spectrum of HO in the 1.78 m region (800628R0.002) recorded by R. H. Hunt in 1980, which covers the and fundamental bands of the hydrogen peroxide. To the best our knowledge these two bands have not been spectroscopically analysed. The upper display of this figures presents our absorption spectrum at 296 K simulated using the Doppler line profile. Our synthetic spectrum resembles all the main features of the experimental data. We would like to encourage a spectroscopic analysis of the Kitt Peak HO spectra in the IR and near-IR regions currently not present in HITRAN. We believe that our theoretical line list with a capability of providing absolute intensities and quantum numbers can assist in the assignment of these spectra.

Band Frequency range (cm-1) Ref. Obs Calc (O-C)/O (%)
Torsional 0–1,427 Perrin et al. (1996) 4.0400 3.7450 7.3
750–1,100 Johnson et al. (2009) 0.0157 0.0165 -5.43
1,135–1,393 Johnson et al. (2009) 1.7458 1.7651 -1.10
1,170–1,380 Klee et al. (1999) 1.8500 1.7633 4.68
1,170–1,380 Perrin et al. (1995) 1.0030 1.7633 -75.80
2,300–2,900 Johnson et al. (2009) 0.0830 0.055 33.78
Multiple bands 2,300–2,900 Johnson et al. (2009) 0.0830 0.0797 3.97
, region 3,300–3,800 Johnson et al. (2009) 0.8356 0.8724 -4.40
Table 7: Band intensities, in 10 cm/(molecule cm).
Figure 8: Cross-sections using a Doppler profile for the APTY line list as a function of temperature. The depth of the minima (windows) decreases monotonically with temperature.

Figure 8 presents integrated absorption cross-sections computed using a Doppler profile (Hill et al., 2013) for a range of temperatures. The figure shows how the opacity changes with increasing temperature. We note the particularly dramatic effect raising the temperature has on the absorption by HO in the 13.7 m region. This smoothing in the overall spectra can only be modelled if there is adequate coverage and population of rotationally and vibrationally excited states. We also note the strength of the OH stretch feature at about 2.75 m; these features are absent from line databases such as HITRAN because of the absence of assigned spectra in this region. Hopefully APTY can be used to help analyse spectra in this region, as the BYTe NH line list is being used to analyse ammonia spectra (Barton et al., 2015, 2016).

4 Conclusion

The frequency and Einstein- coefficients of almost 20 billion transitions of hydrogen peroxide are computed. These transitions cover wavelengths longer than 1.6 m and include all rotational excitations up to , making the line list applicable for temperatures up to 1250 K. The line list gives a room-temperature spectrum in excellent agreement with available experimental data and has good predictive ability for bands and line-positions not available experimentally. The new line list may be accessed via www.exomol.com or http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/MNRAS/. The cross-sections of HO can be also generated at www.exomol.com as described by Tennyson et al. (2016b).

5 Acknowledgements

This work was supported by the ERC under the Advanced Investigator Project 267219 and made use of the DiRAC@Darwin, DiRAC@COSMOS HPC cluster and Emerald CfI cluster. DiRAC is the UK HPC facility for particle physics, astrophysics and cosmology and is supported by STFC and BIS. The authors would like to acknowledge the work presented here made use of the EMERALD High Performance Computing facility provided via the Centre for Innovation (CfI). The CfI is formed from the universities of Bristol, Oxford, Southampton and UCL in partnership with STFC Rutherford Appleton Laboratory. We would also like to thank nVidia for supplying our group with a Tesla K40 GPU for usage in intensity and cross-section computation. AFA would also like to thank Dr. Faris N. Al-Refaie, Lamya Ali, Sarfraz Ahmed Aziz, and Rory and Annie Gleeson for their support.

References

  • Al-Refaie et al. (2015a) Al-Refaie A. F., Ovsyannikov R. I., Polyansky O. L., Yurchenko S. N., Tennyson J., 2015a, J. Mol. Spectrosc., 318, 84
  • Al-Refaie et al. (2016) Al-Refaie A. F., Tennyson J., Yurchenko S. N., 2016, Comput. Phys. Commun., (to be submitted)
  • Al-Refaie et al. (2015b) Al-Refaie A. F., Yurchenko S. N., Yachmenev A., Tennyson J., 2015b, MNRAS, 448, 1704
  • Allen et al. (2013) Allen N. D. C., Abad G. G., Bernath P. F., Boone C. D., 2013, J. Quant. Spectrosc. Radiat. Transf., 115, 66
  • Anderson et al. (1999) Anderson E. et al., 1999, LAPACK Users’ Guide, 3rd edn. Society for Industrial and Applied Mathematics, Philadelphia, PA
  • Aoki et al. (2015) Aoki S., Giuranna M., Kasaba Y., Nakagawa H., Sindoni G., Geminale A., Formisano V., 2015, Icarus, 245, 177
  • Bacelo & Binning (2005) Bacelo D. E., Binning R. C., 2005, Intern. J. Quantum Chem., 105, 740
  • Bach et al. (1996) Bach R. D., Ayala P. Y., Schlegel H. B., 1996, J. Am. Chem. Soc., 118, 12758
  • Bartlett & Musiał (2007) Bartlett R. J., Musiał M., 2007, Rev. Mod. Phys., 79, 291
  • Barton et al. (2016) Barton E. J., Yurchenko S. N., Tennyson J., Béguier S., Campargue A., 2016, J. Mol. Spectrosc., (submitted)
  • Barton et al. (2015) Barton E. J., Yurchenko S. N., Tennyson J., Clausen S., Fateev A., 2015, J. Quant. Spectrosc. Radiat. Transf., 167, 126
  • Bergman et al. (2011) Bergman P., Parise B., Liseau R., Larsson B., Olofsson H., Menten K. M., Güsten R., 2011, A&A, 531, L8
  • Blackford et al. (1997) Blackford L. S. et al., 1997, ScaLAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA
  • Bowman et al. (2003) Bowman J. M., Carter S., Huang X. C., 2003, Intern. J. Quantum Chem., 22, 533
  • Bunker & Jensen (1998) Bunker P. R., Jensen P., 1998, Molecular Symmetry and Spectroscopy, 2nd edn. NRC Research Press, Ottawa
  • Camy-Peyret et al. (1992) Camy-Peyret C., Flaud J.-M., Johns J. W. C., Noel M., 1992, J. Mol. Spectrosc., 155, 84
  • Carter et al. (2009) Carter S., Handy N. C., Bowman J. M., 2009, Mol. Phys., 107, 727
  • Carter et al. (2011) Carter S., Sharma A. R., Bowman J. M., 2011, J. Chem. Phys., 135, 014308
  • Chance et al. (1991) Chance K. V., Johnson D. G., Traub W. A., Jucks K. W., 1991, Geophys. Res. Lett., 18, 1003
  • Chen et al. (2001) Chen R. Q., Ma G. B., Guo H., 2001, J. Chem. Phys., 114, 4763
  • Clancy et al. (2004) Clancy R. T., Sandor B. J., Moriarty-Schieven G. H., 2004, Icarus, 168, 116
  • Cooley (1961) Cooley J. W., 1961, Math. Comp., 15, 363
  • Davis (1974) Davis D. D., 1974, Can. J. Chem., 52, 1405
  • Du et al. (2012) Du F., Parise B., Bergman P., 2012, A&A, 538, A91
  • Encrenaz et al. (2004) Encrenaz T. et al., 2004, Icarus, 170, 424
  • Encrenaz et al. (2012) Encrenaz T., Greathouse T. K., Lefevre F., Atreya S. K., 2012, Planet Space Sci., 68, 3
  • Fischer et al. (2003) Fischer J., Gamache R. R., Goldman A., Rothman L. S., Perrin A., 2003, J. Quant. Spectrosc. Radiat. Transf., 82, 401
  • Flaud et al. (1989) Flaud J.-M., Camy-Peyret C., Johns J. W. C., Carli B., 1989, J. Chem. Phys., 91, 1504
  • Giguere (1950) Giguere P. A., 1950, J. Chem. Phys., 18, 88
  • Giguere & Srinivasan (1974) Giguere P. A., Srinivasan T. K. K., 1974, J. Raman Spectrosc., 2, 125
  • Hand & Brown (2013) Hand K. P., Brown M. E., 2013, ApJL, 766, L21
  • Hill et al. (2013) Hill C., Yurchenko S. N., Tennyson J., 2013, Icarus, 226, 1673
  • Hougen (1984) Hougen J. T., 1984, Can. J. Phys., 62, 1392
  • Johnson et al. (2009) Johnson T. J., Sams R. L., Burton S. D., Blake T. A., 2009, Anal. Bioanal. Chem., 395, 377
  • Klee et al. (1999) Klee S., Winnewisser M., Perrin A., Flaud J.-M., 1999, J. Mol. Spectrosc., 195, 154
  • Koput et al. (1998) Koput J., Carter S., Handy N. C., 1998, J. Phys. Chem. A, 102, 6325
  • Kozin et al. (2004) Kozin I. N., Law M. M., Tennyson J., Hutson J. M., 2004, Comput. Phys. Commun., 163, 117
  • Kuhn et al. (1999) Kuhn B., Rizzo T. R., Luckhaus D., Quack M., Suhm M. A., 1999, J. Chem. Phys., 111, 2565
  • Lin & Guo (2003) Lin S. Y., Guo H., 2003, J. Chem. Phys., 119, 5867
  • Luckhaus (2000) Luckhaus D., 2000, J. Chem. Phys., 113, 1329
  • Małyszek & Koput (2013) Małyszek P., Koput J., 2013, J. Comput. Chem., 34, 337
  • Mladenovic (2002) Mladenovic M., 2002, Spectra Chimica Acta A, 58, 809
  • Noumerov (1924) Noumerov B. V., 1924, MNRAS, 84, 592
  • Olson et al. (1988) Olson W. B., Hunt R. H., Young B. W., Maki A. G., Brault J. W., 1988, J. Mol. Spectrosc., 127, 12
  • Perrin et al. (1990) Perrin A., Flaud J.-M., Camy-Peyret C., Goldman A., Murcray F. J., Blatherwick R. D., 1990, J. Mol. Spectrosc., 142, 129
  • Perrin et al. (1996) Perrin A. et al., 1996, J. Mol. Spectrosc., 176, 287
  • Perrin et al. (1995) Perrin A., Valentin A., Flaud J. M., Camy-Peyret C., Schriver L., Schriver A., Arcas P., 1995, J. Mol. Spectrosc., 171, 358
  • Polyansky et al. (2013) Polyansky O. L., Kozin I. N., Maĺyszek P., Koput J., Tennyson J., Yurchenko S. N., 2013, J. Phys. Chem. A, 117, 7367
  • Rauhut et al. (2009) Rauhut G., Knizia G., Werner H.-J., 2009, J. Chem. Phys., 130, 054105
  • Rothman et al. (2013) Rothman L. S. et al., 2013, J. Quant. Spectrosc. Radiat. Transf., 130, 4
  • Sharpe et al. (2004) Sharpe S. W., Johnson T. J., Sams R. L., Chu P. M., Rhoderick G. C., Johnson P. A., 2004, Appl. Spectrosc., 58, 1452
  • Sousa-Silva et al. (2015) Sousa-Silva C., Al-Refaie A. F., Tennyson J., Yurchenko S. N., 2015, MNRAS, 446, 2337
  • Sousa-Silva et al. (2014) Sousa-Silva C., Hesketh N., Yurchenko S. N., Hill C., Tennyson J., 2014, J. Quant. Spectrosc. Radiat. Transf., 142, 66
  • Tennyson et al. (2013) Tennyson J., Hill C., Yurchenko S. N., 2013, in AIP Conference Proceedings, Vol. 1545, 6 international conference on atomic and molecular data and their applications ICAMDATA-2012, AIP, New York, pp. 186–195
  • Tennyson et al. (2016a) Tennyson J., Hulme K., Naim O. K., Yurchenko S. N., 2016a, J. Phys. B: At. Mol. Opt. Phys., 49, 044002
  • Tennyson & Yurchenko (2012) Tennyson J., Yurchenko S. N., 2012, MNRAS, 425, 21
  • Tennyson et al. (2016b) Tennyson J. et al., 2016b, J. Mol. Spectrosc.
  • Underwood et al. (2016) Underwood D. S., Tennyson J., Yurchenko S. N., Clausen S., Fateev A., 2016, MNRAS, (submitted)
  • Underwood et al. (2014) Underwood D. S., Yurchenko S. N., Tennyson J., Jensen P., 2014, J. Chem. Phys., 140, 244316
  • Vidler & Tennyson (2000) Vidler M., Tennyson J., 2000, J. Chem. Phys., 113, 9766
  • Yachmenev et al. (2011) Yachmenev A., Yurchenko S. N., Jensen P., Thiel W., 2011, J. Chem. Phys., 134, 11
  • Yu & Muckerman (2002) Yu H. G., Muckerman J. T., 2002, J. Mol. Spectrosc., 214, 11
  • Yurchenko et al. (2011a) Yurchenko S. N., Barber R. J., Tennyson J., 2011a, MNRAS, 413, 1828
  • Yurchenko et al. (2011b) Yurchenko S. N., Barber R. J., Tennyson J., Thiel W., Jensen P., 2011b, J. Mol. Spectrosc., 268, 123
  • Yurchenko et al. (2009) Yurchenko S. N., Barber R. J., Yachmenev A., Thiel W., Jensen P., Tennyson J., 2009, J. Phys. Chem. A, 113, 11845
  • Yurchenko et al. (2003) Yurchenko S. N., Carvajal M., Jensen P., Herregodts F., Huet T. R., 2003, Chem. Phys., 290, 59
  • Yurchenko et al. (2005a) Yurchenko S. N., Carvajal M., Jensen P., Lin H., Zheng J. J., Thiel W., 2005a, Mol. Phys., 103, 359
  • Yurchenko & Tennyson (2014) Yurchenko S. N., Tennyson J., 2014, MNRAS, 440, 1649
  • Yurchenko et al. (2005b) Yurchenko S. N., Thiel W., Carvajal M., Lin H., Jensen P., 2005b, Adv. Quant. Chem., 48, 209
  • Yurchenko et al. (2007) Yurchenko S. N., Thiel W., Jensen P., 2007, J. Mol. Spectrosc., 245, 126
  • Zins & Krim (2014) Zins E.-L., Krim L., 2014, RSC Adv., 4, 22172
  • Zumwalt & Giguere (1941) Zumwalt L. A., Giguere P. A., 1941, J. Chem. Phys., 9, 458
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
169771
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description