Rovibrational energies, partition functions and equilibrium fractionation of the CO2 isotopologues

Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

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Rovibrational energies, partition functions and equilibrium fractionation of the CO2 isotopologues J. Cerezo, A. Bastida, A. Requena, J. Zúñiga n Departamento de Química Física, Universidad de Murcia, 30100 Murcia, Spain

a r t i c l e i n f o

abstract

Article history: Received 20 December 2013 Received in revised form 21 May 2014 Accepted 6 June 2014 Available online 14 June 2014

Rovibrational energy levels, partition functions and relative abundances of the stable isotopologues of CO2 in gas phase at equilibrium are calculated using an empirical Morsecosine potential energy surface (PES) refined by fitting to the updated pure (l2 ¼ 0) vibrational frequencies observed for the main 12C16O2 isotopologue. The rovibrational energy levels are calculated variationally using a system of optimized hyperspherical normal coordinates, and from these the vibrational terms Gv and rotational constants Bv of the isotopologues are determined. The refined potential surface is shown to be clearly superior to the original potential surface, with the former reproducing the observed values of the spectroscopic constants Gv and Bv with accuracies of about 0.1 cm  1 and 0.00020 cm  1, respectively, for levels with l2 Z 0 up to 10,000 cm  1 above the ground state. The internal partition functions of the isotopologues are calculated by approximated direct summation over the rovibrational energies and compared with both previous partition sums and values obtained from analytical expressions based on the harmonic oscillator and rigid rotor models. The partition functions calculated by approximated direct summation are then used to determine the abundances of the CO2 isotopologues at thermodynamic equilibrium using the method developed by Wang et al. [74]. Significant variations in the relative abundances of some of the CO2 multiple substituted isotopologues at terrestrial temperatures with respect to those provided by the classical harmonicbased Urey theory are found, which may be of relevance in geochemical processes. & 2014 Elsevier Ltd. All rights reserved.

Keywords: CO2 isotopologues Potential energy surface Vibrational terms and rotational constants Partition functions Relative abundances

1. Introduction The characterization of the infrared spectra of the CO2 molecule continues to receive incessant attention due to the fundamental role that this molecule plays in the chemistry of the Earth's atmosphere [1,2], and that of other planets like Venus [3–5] and Mars [6,7]. Along with the many spectroscopic studies of CO2 and its isotopologues that have been undertaken in the last years [8–35], and that continuously feed the existing spectroscopic databases [36–39] are added the multiple theoretical

n

Corresponding author. Tel.: þ 34 868887439; fax: þ34 868884148. E-mail address: [email protected] (J. Zúñiga).

http://dx.doi.org/10.1016/j.jqsrt.2014.06.003 0022-4073/& 2014 Elsevier Ltd. All rights reserved.

treatments carried out for these molecular species based on the use of both perturbative effective Hamiltonians [12,16,22,40–45] and variational methods for the resolution of the rovibrational Schrödinger equation [39,46–52]. The theoretical approaches pursue the complementary simulation of the infrared spectra of the isotopologues, aimed to give additional information and insights about them; and the success of these simulations rests ultimately on the availability of reliable potential energy and dipole moment surfaces [39,47,49,51–62] able to provide a degree of accuracy compatible with the resolution of the spectroscopic observations. Over the years, our group has developed a theoretical method to enhance the performance of the variational method in the calculation of the rovibrational energy

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levels and wave functions of triatomic molecules. The method is based essentially on the use of optimal coordinate systems designed to speed up the resolution of the rovibrational Schrödinger equation, especially in very demanding computational contexts such as the adjustment of potential energy functions to experimental information [48,53,63–66]. For CO2, in particular, we have derived an accurate Morse-cosine potential surface by fitting to the observed vibrational frequencies of the molecule with angular vibrational quantum number l2 ¼ 0 [53], and later used this empirical potential surface to calculate the vibrational terms Gv and rotational constants Bv of the molecule for levels with l2 4 0 [48]. Motivated by the explosion of ongoing experimental spectroscopic information on the CO2 isotopologues in the last years [8–30], and by the development of increasingly accurate analytical techniques to measure the abundances of the isotopologues in different atmospheric and geological environments [67–73], in this work we undertake the extension of our previous variational calculations of the rovibrational energy levels of CO2 to the stable isotopologues of the molecule. To accomplish this, we have started by refining first our earlier potential surface by fitting again to the pure (l2 ¼ 0) vibrational frequencies observed for the main 12C16O2 isotopologue properly updated. Thus, we obtain a more accurate empirical potential surface that reproduces better the spectroscopic constants Gv and Bv of the main isotopologue for vibrational levels with l2 40 up to 10,000 cm  1 above the ground level, and that does indeed maintain such a degree of accuracy for the rest of the CO2 isotopologues. We have calculated then the internal partition functions of the isotopologues by approximated direct summation over the computed rovibrational energy levels, checking that their values are satisfactorily converged up to temperatures of about 1000 K, and that they are more accurate than those provided by standard analytical expressions based on the harmonic oscillator and rigid rotor approximations. The partition functions computed using the approximated direct summation method have finally been employed to calculate the thermodynamically equilibrated abundances of the isotopologues in gas phase using the methodology developed by Wang et al. [74], and the deviation of the abundances with respect to the stochastic distribution, which is expected to be reached in the limit of infinite temperatures. Our results show significant differences in the relative abundances of some of the multiple substituted CO2 isotopologues at terrestrial temperatures as compared with those provided by the seminal Urey approach based on the use of the harmonic oscillator and rigid rotor models. These differences could be of importance when it comes to interpreting the isotopic fractionation processes in which the CO2 isotopologues are involved and in the proposed use of their temperature-dependent abundances as geothermic markers. 2. Potential energy surface To calculate the rovibrational energy levels of the isotopologues of CO2, we have first refined the empirical potential energy surface previously determined for this

molecule in our group [53] in order to improve further its accuracy. The potential surface is a fourth order Morsecosine (MC) expansion in valence coordinates (R1 ; R2 ; θ) given by V Morse ðy1 ; y2 ; y3 Þ ¼ M 2 y2 þ ∑M ij yi yj þ ∑M ijk yi yj yk þ ∑M ijkl yi yj yk yl

ð1Þ

where y1 ¼ 1  e  a1 ðR1  Re Þ

ð2Þ

y2 ¼ cos θ  cos θe ¼ cos θ þ 1

ð3Þ

y3 ¼ 1  e  a3 ðR2  Re Þ

ð4Þ

The refinement of the surface has been carried out by nonlinear squares fitting of the expansion coefficients M to the observed pure (l2 ¼ 0) vibrational frequencies of the main isotopologue 12C16O2. This is essentially the same strategy which was successfully employed to determine the original potential surface, so the main difference lies basically in the use of properly updated l2 ¼ 0 vibrational terms and a more accurate equilibrium C–O distance (see below). The theoretical frequencies needed in the fit are calculated variationally by solving the vibrational Schrödinger equation using a set of optimal internal coordinates which allows us to employ small reduced basis sets to reach convergence in the desired vibrational levels, thus noticeably accelerating the iterative execution of the fit. The details of the fitting procedure have been already given in our earlier works and will not be repeated here (see, for example, Refs. [53,65,66,75,76]). The optimal internal coordinates used for the CO2 molecule along with the basis set details of the variational calculations are likewise described in Refs. [53,64]. The input data used to refine the potential energy surface are the 49 pure vibrational frequencies observed for the 12C16O2 isotopologue lying below 10,000 cm  1, which were taken from Refs. [10,13,24,25,27,77]. We initiated the refinement starting from our earlier potential surface but changing in it the experimental value of the equilibrium C–O distance previously used, Re ¼ 1:16 Å, which dates from the original papers by Courtoy [78,79], for the more accurate empirical value Re ¼ 1:15995884 Å determined by Graner et al. [80]. The 20 M coefficients of the fourth order Morse-cosine expansion were then allowed to vary in the fit until reaching a minimum in the root-mean-squares (rms) deviation of the vibrational frequencies, while keeping both the equilibrium geometry of the molecule and the Morse parameters ai fixed. We note here that unlike in our previous fit, the MC coefficients M2, M 11 ¼ M 33 , and M13 which account for the polynomial second-order description of the potential were also varied in this case in order to give the refinement some more flexibility. The values obtained for the parameters of the refined CO2 Mose-cosine potential are listed in Table 1, along with those of the previous potential. The most remarkable differences lies as observed in the third-order stretching coefficients M 111 ¼ M 333 and M 113 ¼ M 133 , and the fourthorder coefficients M 1122 ¼ M2233 , M1133 and M2222. The final vibrational frequencies calculated in the fit using the

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

Table 1 Coefficients (in aJ) of the Morse-cosine potentials for CO2. The values of the Morse parameters are a1 ¼ a3 ¼ 2:416 Å  1.

235

Table 2 Comparison between calculated and observed vibrational frequencies (in cm  1) for CO2.

Parameter

Previous

Refined

Pa

V 1 V 2 l2 V 3 rϵ

Obs.

Cal.

Obs.–cal.

M2 M11 ¼M33 M13 M22 M12 ¼ M23 M111 ¼M333 M222 M112 ¼ M233 M122 ¼ M223 M113 ¼ M133 M123 M1111 ¼ M3333 M2222 M1222 ¼M2223 M1122 ¼M2233 M1223 M1112 ¼ M2333 M1123 ¼M1233 M1113 ¼M1333 M1133 ReCO (Å)

0.783081 1.371790 0.214624 0.372488  0.489770 0.060042 0.212785  0.144929  0.363860  0.448250 0.810834 0.119865 0.071605  0.127082  0.024685 1.007238  0.271867  0.071575 0.379820 0.154753 1.16

0.78261538 1.3720526 0.21454954 0.37358775  0.49484340 0.0073175811 0.14882052  0.16136478  0.29025682  0.11295857 0.55126564 0.064550905 0.46938887  0.24406647  0.19654223 0.34665810  0.14255896 0.16795892 0.10434352  0.024495713 1.15995884

2 2 3 4 4 4 5 5 6 6 6 6 6 7 7 7 8 8 8 8 9 9 9 9 9 10 10 10 10 10 10 11 11 11 11 11 11 11 12 12 12 13 13 13 13 13 13 13 13

10002e 10001e 00011e 20003e 20002e 20001e 10012e 10011e 30004e 30003e 30002e 30001e 00021e 20013e 20012e 20011e 40004e 40002e 10022e 10021e 30014e 30013e 30012e 30011e 00031e 50003e 50002e 50001e 20023e 20022e 20021e 40015e 40014e 40013e 40012e 40011e 10032e 10031e 30023e 30022e 00041e 50016e 50015e 50014e 50013e 50012e 20033e 20032e 20031e

1285.408 1388.184 2349.143 2548.367 2671.143 2797.136 3612.841 3714.782 3792.684 3942.543 4064.275 4225.097 4673.325 4853.623 4977.835 5099.661 5197.147 5475.565 5915.212 6016.690 6075.980 6227.916 6347.851 6503.081 6972.577 6725.267 6903.929 7121.742 7133.819 7259.767 7377.705 7283.976 7460.521 7593.690 7734.449 7920.832 8192.550 8293.952 8488.408 8607.133 9246.933 8480.235 8676.707 8831.472 8965.215 9137.790 9388.982 9516.956 9631.339

1285.398 1388.174 2349.189 2548.353 2671.159 2797.137 3612.863 3714.790 3792.674 3942.532 4064.283 4225.127 4673.339 4853.634 4977.851 5099.657 5197.210 5475.561 5915.215 6016.674 6075.983 6227.914 6347.856 6503.079 6972.555 6725.224 6903.940 7121.744 7133.816 7259.761 7377.690 7283.975 7460.499 7593.690 7734.450 7920.817 8192.553 8293.939 8488.400 8607.128 9246.941 8480.244 8676.696 8831.479 8965.249 9137.805 9388.998 9516.974 9631.371

0.010 0.010  0.046 0.014  0.007  0.001  0.022  0.008 0.010 0.011  0.008  0.030  0.014  0.011  0.016 0.004  0.063 0.004  0.003 0.016  0.003 0.002  0.005 0.002 0.022 0.043  0.011  0.002 0.003 0.006 0.015 0.001 0.022 0.000  0.001 0.015  0.003 0.013 0.008 0.005  0.008  0.009 0.011  0.007  0.034  0.015  0.016  0.018  0.032

refined potential are given in Table 2, along with the experimental frequencies and the differences between them. The vibrational frequencies are labeled using the notation V 1 V 2 l2 V 3 r ϵ employed for the CO2 isotopologues in the HITRAN database [36], where 1 r r r V 1 þ 1, V 2 ¼ l2 , ϵ is the Wang symmetry label (ϵ ¼ e; f ), and P is the polyad quantum number P ¼ 2V 1 þV 2 þ 3V 3 extracted from the approximate relationships ω1  2ω2 and ω3  3ω2 existing between the normal modes vibrational frequencies. The rms deviation provided by the refined surface for the 49 vibrational frequencies employed in the fit is 0.018 cm  1, as compared with the value provided by the initial potential surface of 0.211 cm  1. Table 2 also shows that the absolute differences between the observed and the calculated frequencies take values around 0.010 cm  1 in most cases, with the largest absolute residual of 0.063 cm  1 corresponding to the 40004e level. Interestingly, the Gv constant for this level and level 40002e taken from Ref. [77] are really calculated values based on direct numerical diagonalization calculations, and when they are excluded from the analysis, the rms deviation of the refined surface slightly reduces to 0.018 cm  1. We have also checked that the refined Morse-cosine potential surface has no spurious irregularities. 3. Vibrational terms and rotational constants After refining the potential energy surface for the main isotopologue, we have proceeded to calculate the rovibrational energy levels of all the stable CO2 isotopologues in the framework of the Born–Oppenheimer approximation in order to test the validity of the refined potential surface and its extrapolation properties. These calculations have also been carried out variationally, but using a system of optimized hyperspherical coordinates of normal character which facilitates the normal modes assignments of the vibrational energy levels. These type of hyperspherical

a

P ¼ 2V 1 þ V 2 þ 3V 3 .

coordinates are constructed, in general, by making generalized polar transformations of the stretching coordinates of orthogonal coordinate systems [81–84] and for CO2, in particular, the optimal polar transformation is that which applies to Radau coordinates as follows [81]: P 2 ¼ ðs1  ah Þ2 þðs2  bh Þ2

ð5Þ

s2 bh s1  ah

ð6Þ

tan Φ ¼

where s1 and s2 are the stretching Radau coordinates, and ah and bh are the hyperspherical optimization parameters which specify the orientation of the polar coordinates P

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and Φ in the ðs1 ; s2 Þ plane. The second order potential couplings of the polar P and Φ coordinates with the angular Radau coordinate θ cancel due to the linear equilibrium geometry of the molecule, and the second order potential coupling between P and Φ vanishes by giving the displacement parameters the same value, due to symmetry considerations [81]. The hyperspherical coordinates fP; Φ; θg thus form a system of curvilinear normal coordinates for CO2 parametrized by the equal displacements ah ¼ bh . In our recent work on N2O [84], we have discussed in detail the way in which the displacement parameters are calculated to further optimize the generalized hyperspherical coordinates for lineal triatomic molecules. By applying this optimization procedure to the hyperspherical Radau coordinates of the main isotopologue of CO2, we obtain the following values for the displacement parameters: ah ¼ bh ¼ 0:9. These values barely change, moreover, for the rest of the CO2 isotopologues, so we have used them in all the variational calculations. The specific variational calculations of the rovibrational energy levels of the stable CO2 isotopologues have been conducted for rotational quantum numbers J from 0 to 16, using the vibrational product-type three-dimensional basis sets specified in Table 3 (see Ref. [84] for details), which guarantee a convergence within 0.00001 cm  1 in all the levels considered. The calculated rovibrational energy levels Ev;J have then been used to determine the vibrational terms Gv and rotational constants Bv of the CO2 isotopologues using the truncated spectroscopic expression Ev;J ¼ Gv þ Bv JðJ þ 1Þ

ð7Þ

Table 3 Basis set specifications for the variational calculations of rovibrational energies of CO2 and its isotopologues using normal hyperspherical coordinates. J

DVR functionsa NP ; NΦ ; Nθ

Anharmonic functionsb M P ; M Φ ; Mθ

Nc

NðJ þ 1Þd

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

80,40,70 80,40,140 70,40,140 60,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140 50,30,140

39,18,34 38,18,29 32,15,26 28,13,23 25,12,22 25,12,22 23,11,20 21,19,19 21,10,18 29,10,18 19,10,17 19,9,17 18,9,16 18,9,16 17,9,16 17,8,15 16,8,15

5000 4500 3000 2000 1500 1500 1200 1000 1000 900 800 750 690 640 600 560 520

5000 9000 9000 9000 7500 9000 8400 8000 9000 9000 8800 9000 8870 8960 9000 8960 8840

a The number of primitive DVR basis functions used for each vibrational mode. b The maximum number of anharmonic basis functions used for each vibrational mode. c The number of three-dimensional anharmonic vibrational basis functions used for each value of k. d Total basis size of the vibrational–rotational Hamiltonian matrix.

where v is the collective label that accounts for the vibrational quantum numbers vi, the angular vibrational quantum number l2 and the Wang symmetry label ϵ. In this expression, the vibrational terms Gv contain all 2 the contributions depending on l2 , as is being done routinely in the most recent spectroscopic studies of the vibrational bands of the CO2 isotopologues. It is clear that Eq. (7) is the simplest one that can be used to estimate the spectroscopic constants Gv and Bv. A more accurate spectroscopic analysis should include higher order rotational constants such as Dv and Hv. Even in this case, however, the polynomial expansion of the rovibrational energies in JðJ þ 1Þ powers would fail for perturbed levels and effective Hamiltonian should then have to be alternatively used. Comparison of the calculated values for Gv and Bv with those extracted from the analysis of the vibrational bands of the isotopologues experimentally observed is therefore constrained to the proper limitations of Eq. (7) and has to be made cautiously. In any case, inclusion of the centrifugal constants in the energy expansion is not expected to appreciably change the values of the rotational constants within the accuracy with which they are given in this work and their values can be used for identification purposes of isotopologue spectra as illustrated below. To have then a global view of the accuracy with which the refined potential energy surface describes the rovibrational levels structure of the CO2 isotopologues, in Table 4 we give the rms deviations provided by the refined potential surface for the vibrational terms and rotational constants of the 12 isotopologues up to 10,000 cm  1, along with the rms values given by the original surface. We include in this table also the total number of vibrational energy levels with l2 Z 0 employed for each isotopologue to calculate the rms deviations, that is, the number of levels for which the spectroscopic information is available. For the main isotopologue, 12C16O2, the rms deviation for the Gv constants calculated from the refined potential surface goes from the 0.016 cm  1 value reported above for the vibrational levels with l2 ¼ 0 to a value of 0.072 cm  1 for the total number of 192 vibrational levels observed Table 4 Root mean square (rms) deviations for the vibrational frequencies and rotational constants (in cm  1) of the CO2 isotopologues lying below 10,000 cm  1. The rms values of the former CO2 potential energy surface are given in parentheses. Isotopologue

16

O12C16O O13C16O 16 12 18 O C O 16 12 17 O C O 16 13 18 O C O 16 13 17 O C O 18 12 18 O C O 17 12 18 O C O 18 13 18 O C O 17 12 17 O C O 17 13 18 O C O 17 13 17 O C O 16

Gv

Bv

Levels

rms

Levels

rms

192 157 126 85 78 46 100 65 24 52 20 17

0.072 (0.298) 0.112 (0.236) 0.102 (0.223) 0.052 (0.149) 0.044 (0.229) 0.060 (0.233) 0.114 (0.258) 0.068 (0.142) 0.036 (0.132) 0.050 (0.124) 0.035 (0.080) 0.037 (0.068)

193 158 127 86 79 47 101 66 25 53 21 18

0.00021 (0.00073) 0.00022 (0.00099) 0.00017 (0.00067) 0.00016 (0.00071) 0.00016 (0.00096) 0.00015 (0.00069) 0.00013 (0.00057) 0.00013 (0.00026) 0.00012 (0.00068) 0.00014 (0.00031) 0.00011 (0.00066) 0.00008 (0.00023)

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

with l2 Z0. Despite this rms increase, the rms deviation of Gv for all the levels observed, 0.072 cm  1, is, as seen in Table 4, about four times lower than that provided by the original surface of 0.298 cm  1. We see also in Table 4 that the accuracy of the order of 0.1 cm  1 given by the refined surface for the main isotopologue practically holds, and even improves, for the remaining isotopologues. Overall the refined potential energy surface reproduces the vibrational terms Gv observed for the stable CO2 isotopologues with an accuracy of  0:05–0:1 cm  1 . As far as the rotational constants Bv are concerned, Table 4 shows that the rms deviations decrease in this case for all the isotopologues in factors comprised between 2.5 and 6 when going from the original to the refined potential surface. The rms deviations of the rotational constants given by the refined surface for the CO2 isotopologues lie now between 0.00008 and 0.00022 cm  1, with an average value of 0.00015 cm  1. In Table 6 we compare the spectroscopic constants Gv and Bv calculated for the main isotopologue 12C16O2 with their observed values. An inspection of this table shows first that the absolute differences between the calculated and observed vibrational terms Gv steadily increase with the angular vibrational quantum number l2 in no case exceeding 0.4 cm  1. In contrast, the absolute differences for rotational constants Bv remain more uniform on the l2 increase, as shown by the fact that the rms deviation barely changes from 0.00022 cm  1 for levels with l2 ¼ 0 to 0.00021 cm  1 for the complete set of the levels with

237

l2 Z 0. To visualize these variations better, in Fig. 1 we plot the differences between the calculated and observed values of Gv and Bv for the main isotopologue 12C16O2 versus Gv, and to further demonstrate that such differences hold in similar values for the rest of the isotopologues, in Figs. 2–4 we plot them for the isotopologues 16 13 16 O C O, 16O12C18O and 18O12C18O, for which the spectroscopic information available is more abundant. We have also checked the isotopic extrapolation properties of our refined Morse-cosine potential surface versus those of the Ames-1 surface recently published by Huang et al. [39] based on extensive ab initio calculations with subsequent refinement using a large set of 12C16O2 experimental energy levels with J 4 0. In Table 5 we give the rms values provided by both surfaces for the l2 ¼ 0 vibrational terms of the isotopologues up to basically 10,000 cm  1. Our calculations display a higher number of energy levels because they include the most recent experimental values reported. As observed, both potential surfaces give a similar accuracy for the l2 ¼ 0 vibrational terms of the main 12C16O2 isotopologue, although it is clear that the Ames-1 surface performs better for levels higher than 10,000 cm  1. As for the rest of isotopologues, the Ames-1 potential surface clearly surpasses the refined MC surface for the 13C16O2 isotopologue with a rms deviation of 0.035 cm  1 versus our 0.124 cm  1, and provides a practically similar description of the pure vibrational frequencies of the 18O12C18O isotopologue. The remaining nine isotopologues are, however, better

Fig. 1. Differences between observed and calculated vibrational terms (upper panel) and rotational constants (lower panel) for the 16O12C16O isotopologue.

238

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Fig. 2. Differences between observed and calculated vibrational terms (upper panel) and rotational constants (lower panel) for the 16O13C16O isotopologue.

extrapolated by the refined MC potential surface. It is clear that greater improvements of the refined MC potential surface can be achieved by including J 40 rovibrational levels in the fit, as done in the thorough study by Huang et al. [39], and this point is to be considered in further work. For the variational calculations conducted in this work to be of help in the identification and assignment of new vibrational bands of the CO2 isotopologues not observed yet, in Tables S1–S12 of the Supplementary Material we list the values of the spectroscopic constants Gv and Bv calculated for a large number of vibrational levels of the 12 stable isotopologues, along with their experimental values when available.

function is given by Q ðTÞ ¼ g e g n ∑g v g J e  hcEv;J =kT

where Ev;J are the rovibrational energy levels of the molecule relative to the ground level, and ge, gn, gv and gJ are the electronic, nuclear, vibrational and rotational degeneracy factors, respectively. For linear triatomic molecules like CO2, it is convenient to the express the rovibrational levels more explicitly in the form Ev;J  Ev;l2 ;J , where v; l2 together stands for the vibrational quantum numbers fv1 ; vl22 ; v3 ; ϵg, with v2 ¼ l2 ; l2 1; …; 1 or 0, and the rotational quantum number takes values J ¼ l2 ; l2 þ 1; l2 þ 2; …. Moreover, taking into account that g v ¼ 1 and g J ¼ ð2J þ 1Þ, the expression for internal partition for a linear triatomic molecule becomes Q ðTÞ ¼ g e g n ∑ ð2J þ 1Þe  Ev;l2 ;J =kT

4. Partition functions

ð9Þ

v;J

ð10Þ

v;l2 ;J

We consider next the calculation of the partition functions of the CO2 isotopologues. The total partition function of a polyatomic molecule can be written as follows: Q tot: ðTÞ ¼ Q tras: ðTÞQ ðTÞe  hcE0 =kT

ð8Þ

where Q tras: ðTÞ ¼ ð2π mkTÞ =h is the translational partition function, Q(T) is the internal partition function, and E0 is the zero point energy (ZPE). Assuming that only the electronic ground state is populated, the internal partition 3=2

3

The rovibrational energy levels can be expressed as Ev;l2 ;J ¼ Gv;l2 þ F v;l2 ;J

ð11Þ

where Gv;l2 and F v;l2 ;J are the vibrational and rotational terms, respectively. Using this expression in Eq. (10), we obtain Q ðTÞ ¼ g e g n ∑∑e  Gv;l2 =kT ∑ ð2J þ 1Þe  F v;l2 ;J =kT l2 v

ð12Þ

J ¼ l2

where the quantum number l2 takes integer values starting from 0 for the e symmetry levels and from 1 for the f symmetry levels. The nuclear degeneracy factor is, in

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

239

Fig. 3. Differences between observed and calculated vibrational terms (upper panel) and rotational constants (lower panel) for the 16O12C18O isotopologue.

addition, given by 3

g n ¼ ∏ ð2I i þ1Þ

ð13Þ

i¼1

where Ii is nuclear spin of the atoms. On the other hand, the electronic degeneracy factor for the electronic ground state of the CO2 isotopologues is g e ¼ 1. For the symmetric CO2 isotopologues one has to consider the coupling between the rotational motion and the nuclear spins of the end atoms I 1 ¼ I 3 ¼ I. The internal partition sum unfolds then as " Q ðTÞ ¼ g e g n;c ∑∑e  Gv;l2 =kT g n;e l2 v

þg n;o

∑

ð2J þ 1Þe  F v;l2 ;J =kT

∑

J ¼ l2 ;even

#

ð2J þ 1Þe  F v;l2 ;J =kT

ð14Þ

J ¼ l2 ;odd

where g n;c  g 2 ¼ 2I 2 þ 1 is the nuclear degeneracy factor of the central atom, and g n;e and g n;o are the nuclear statistic weights of the even and odd rotational levels, respectively, which are given by expressions g n;e ¼ ð2I þ 1ÞI and g n;o ¼ ð2I þ 1ÞðI þ 1Þ for fermion-type nuclei (half-integer spin), and g n;e ¼ ð2I þ 1ÞðI þ 1Þ and g n;o ¼ ð2I þ 1ÞI for boson-type nuclei (integer spin). The values of the nuclear spin factors for the CO2 isotopologues are listed in Table 7. To compute the internal partition functions of the CO2 isotopologues requires the vibrational, Gv;l2 , and rotational, F v;l2 ;J , terms for all the quantized energy states of the

molecules, whose variational calculation can be quite a daunting task. One common approximation which significantly simplifies these calculations is to ignore the vibration-rotation interactions, in which case the internal partition function can be written as a product of the corresponding vibrational and rotational partition functions, Q ðTÞ ¼ Q vib  Q rot . Within this so-called product approximation, the simplest way to evaluate in turn the partition sums is to model the vibrational motions as harmonic oscillators and the rotational motions as rigid rotators. This is in fact the method used in the HITRAN database to evaluate the partition functions [36,85], employing specifically the harmonic oscillator-based expression derived by Herzberg [86] for the vibrational partition functions, and McDowell's formulas [87] for the rotational partition functions. To go beyond the product approximation in evaluating the internal partition functions of the CO2 isotopologues we have directly employed Eqs. (12) and (14) using the vibrational terms Gv;l2 calculated variationally in this work and approximating the rotational terms F v;l2 ;J by the effective rigid rotor expression, F v;l2 ;J ¼ Bv;l2 JðJ þ 1Þ

ð15Þ

where Bv;l2 are the rotational constants accounting for the interaction between the vibration and the rotation determined for each vibrational level as indicated in Section 3. The expression for the internal partition function of the

240

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

Fig. 4. Differences between observed and calculated vibrational terms (upper panel) and rotational constants (lower panel) for the 18O12C18O isotopologue. Table 5 Comparison of the root mean square (rms) deviations for the l2 ¼ 0 vibrational frequencies (in cm  1) of the CO2 isotopologues lying below 10,000 cm  1 with those provided by the Ames-1 PES [39]. Isotopologue Ames-1 PES

16

O12C16O O13C16O 16 12 18 O C O 16 12 17 O C O 16 13 18 O C O 16 13 17 O C O 18 12 18 O C O 17 12 18 O C O 18 13 18 O C O 17 12 17 O C O 17 13 18 O C O 17 13 17 O C O 16

Refined MC PES

Levels Emax (cm  1)

rms

Levels Emax (cm  1)

rms

48 44 25 24 26 18 20 12 11 10 5 2

0.025 0.035 0.119 0.12 0.12 0.12 0.10 0.10 0.09 0.10 0.09 0.08

51 46 37 29 30 18 27 26 12 16 10 7

0.025 0.124 0.062 0.030 0.048 0.066 0.109 0.059 0.042 0.047 0.040 0.046

10,547 10,731 9180 8255 8009 7695 8147 6894 7930 8215 7872 6723

10,547 10,731 9180 8254 9303 7695 9198 8181 7930 8215 7872 6723

asymmetric isotopologues then becomes Q ðTÞ ¼ g e g n ∑∑e  Gv;l2 =kT ∑ ð2J þ 1Þe  Bv;l2 JðJ þ 1Þ=kT l2 v

ð16Þ

J ¼ l2

and that for the asymmetric isotopologues is " Q ðTÞ ¼ g e g n;c ∑∑e l2 v

 Gv;l2 =kT

g n;e

∑

J ¼ l2 ;even

ð2J þ 1Þe  Bv;l2 JðJ þ 1Þ=kT

# þg n;o

∑

ð2J þ1Þe

 Bv;l2 JðJ þ 1Þ=kT

ð17Þ

J ¼ l2 ;odd

We should note that although these expressions involve summations over rovibrational energies, they are not strictly speaking direct summation formula since the rotational energies are approximated by the spectroscopic equation (15). To avoid any misunderstanding we will refer to Eqs. (16) and (17) as approximated direct summations. The partition functions of the CO2 isotopologues have then been evaluated by approximated summation over all the rovibrational levels with energies up to 10,000 cm  1, so obtaining partition functions that are converged to five significant figures at up to nearly 1000 K temperatures. These calculations required J values of up to about 160 according to the spectroscopic expression (15). In order to check the accuracy of the internal partition functions calculated in this work for the CO2 isotopologues, we have first compared the values obtained for the main isotopologue 12C16O2 with those calculated by Osipov [88]. This author determined the vibrational and internal partition functions of the main isotopologue also by approximated direct summation over the rovibrational energy levels obtained by diagonalization of an effective Hamiltonian whose parameters were fitted to the energy levels observed. In Table 8 we compare our vibrational and internal partition functions for 12C16O2 with those calculated by Osipov for a set of temperatures covering the

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

241

Table 6 Comparison between calculated and observed vibrational frequencies and rotational constants (in cm  1) of CO2. Isotopologue 16-12-16. Pa

V 1 V 2 l2 V 3 rϵ

l2 ¼ 0 states 0 00001e 2 10002e 2 10001e 3 00011e 4 20003e 4 20002e 4 20001e 5 10012e 5 10011e 6 30004e 6 30003e 6 30002e 6 30001e 6 00021e 7 20013e 7 20012e 7 20011e 8 10022e 8 10021e 9 30014e 9 30013e 9 30012e 9 30011e 9 00031e 10 50003e 10 50002e 10 50001e 10 20023e 10 20022e 10 20021e 11 40015e 11 40014e 11 40013e 11 40012e 11 40011e 11 10032e 11 10031e 12 30023e 12 30022e 12 30021e 12 00041e 13 50016e 13 50015e 13 50014e 13 50013e 13 50012e 13 50011e 13 20033e 13 20032e 13 20031e l2 ¼ 1 states 1 01101e 1 01101f 3 11102e 3 11102f 3 11101e 3 11101f 4 01111e 4 01111f 5 21103e 5 21103f 5 21102e 5 21102f 5 21101e 5 21101f 6 11112e 6 11112f

Gv (obs.)

Gv (cal.)

ΔGv

Bv (obs.)

Bv (cal.)

ΔBv

Ref.

0.0 1285.408 1388.184 2349.143 2548.367 2671.143 2797.136 3612.841 3714.782 3792.684 3942.543 4064.275 4225.097 4673.325 4853.623 4977.835 5099.661 5915.212 6016.690 6075.980 6227.916 6347.851 6503.081 6972.577 6725.267 6903.929 7121.742 7133.819 7259.767 7377.705 7283.976 7460.521 7593.690 7734.449 7920.832 8192.551 8293.951 8488.408 8607.133 8756.788 9246.933 8480.235 8676.707 8831.472 8965.215 9137.790 9349.829 9388.982 9516.956 9631.341

0.0 1285.398 1388.174 2349.189 2548.353 2671.159 2797.137 3612.863 3714.790 3792.674 3942.532 4064.283 4225.127 4673.339 4853.634 4977.851 5099.657 5915.215 6016.674 6075.983 6227.914 6347.856 6503.079 6972.555 6725.224 6903.940 7121.744 7133.816 7259.761 7377.690 7283.975 7460.499 7593.690 7734.450 7920.817 8192.553 8293.939 8488.400 8607.128 8756.771 9246.941 8480.244 8676.696 8831.479 8965.249 9137.805 9349.767 9388.998 9516.974 9631.371

0.0 0.010 0.010  0.046 0.014  0.007  0.001  0.022  0.008 0.010 0.011  0.008  0.030  0.014  0.011  0.016 0.004  0.003 0.016  0.003 0.002  0.005 0.002 0.022 0.043  0.011  0.002 0.003 0.006 0.015 0.001 0.022 0.000  0.001 0.015  0.002 0.012 0.008 0.005 0.017  0.008  0.009 0.011  0.007  0.034  0.015 0.062  0.016  0.018  0.030

0.39022 0.39048 0.39019 0.38714 0.39111 0.38956 0.39061 0.38750 0.38706 0.39176 0.38959 0.38959 0.39098 0.38407 0.38820 0.38653 0.38750 0.38453 0.38393 0.38891 0.38671 0.38646 0.38797 0.38099 0.38898 0.39049 0.39139 0.38529 0.38351 0.38438 0.38959 0.38735 0.38558 0.38695 0.38855 0.38156 0.38081 0.38384 0.38324 0.38493 0.37792 0.39038 0.38812 0.38513 0.38590 0.38779 0.38949 0.38238 0.38050 0.38123

0.39020 0.39039 0.39008 0.38721 0.39096 0.38933 0.39045 0.38750 0.38704 0.39156 0.38927 0.38929 0.39079 0.38422 0.38814 0.38639 0.38743 0.38462 0.38400 0.38880 0.38650 0.38624 0.38786 0.38124 0.38850 0.39012 0.39110 0.38533 0.38347 0.38439 0.38943 0.38709 0.38523 0.38670 0.38838 0.38175 0.38096 0.38372 0.38312 0.38490 0.37826 0.39018 0.38781 0.38469 0.38550 0.38749 0.38926 0.38252 0.38055 0.38134

0.00002 0.00009 0.00011  0.00007 0.00015 0.00023 0.00016 0.00000 0.00002 0.00020 0.00032 0.00030 0.00019  0.00015 0.00006 0.00014 0.00007  0.00009  0.00007 0.00011 0.00021 0.00022 0.00011  0.00025 0.00048 0.00037 0.00029  0.00004 0.00004  0.00001 0.00016 0.00026 0.00035 0.00025 0.00017  0.00019  0.00015 0.00012 0.00012 0.00003  0.00034 0.00020 0.00031 0.00044 0.00040 0.00030 0.00023  0.00014  0.00005  0.00009

[77] [13] [13] [13] [77] [77] [77] [13] [13] [77] [77] [77] [77] [13] [13] [13] [13] [10] [10] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [25] [25] [27] [27] [35] [35] [35] [27] [35] [35] [24] [25] [25] [25] [27] [27] [35] [35] [35] [35]

667.380

667.309

0.071

1932.470

1932.427

0.043

2076.856

2076.816

0.040

3004.012

3003.991

0.021

3181.464

3181.440

0.024

3339.356

3339.337

0.019

3500.672

3500.672

0.000

4247.705

4247.694

0.011

0.39064 0.39125 0.39074 0.39169 0.39041 0.39133 0.38759 0.38819 0.39102 0.39235 0.39003 0.39117 0.39039 0.39171 0.38778 0.38870

0.39062 0.39123 0.39065 0.39159 0.39031 0.39123 0.38766 0.38826 0.39086 0.39219 0.38983 0.39096 0.39022 0.39155 0.38778 0.38870

0.00002 0.00002 0.00009 0.00010 0.00010 0.00010  0.00007  0.00007 0.00016 0.00016 0.00020 0.00021 0.00017 0.00016 0.00000 0.00000

[28] [28] [77] [77] [13] [13] [77] [77] [77] [77] [77] [77] [77] [77] [77] [77]

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J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

Table 6 (continued ) Pa

V 1 V 2 l2 V 3 rϵ

Gv (obs.)

Gv (cal.)

6 6 7 7 7 7 7 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 13 13 13 14 14 14 14 14 14 14

11111e 11111f 31104e 31103e 31103f 31102e 31102f 01121e 01121f 21113e 21113f 21112e 21112f 21111e 21111f 41104e 41103e 41102e 41102f 41101e 41101f 11122e 11122f 11121e 11121f 31114e 31114f 31113e 31113f 31112e 31112f 31111e 31111f 01131e 01131f 51104e 51102e 51102f 51101e 21123e 21123f 21122e 21122f 21121e 21121f 41115e 41115f 41114e 41114f 41113e 41113f 41112e 41112f 41111e 41111f 11132e 11132f 11131e 11131f 61103e 01141e 01141f 51115e 51115f 51114e 51114f 51113e 21133e 21133f

4390.629

4390.600

0.029

4416.152 4591.117

4416.139 4591.087

0.013 0.030

4753.450

4753.443

0.007

5315.713

5315.680

0.033

5475.074

5475.065

0.009

5632.765

5632.743

0.022

5790.576

5790.555

0.021

5830.802 6000.532 6179.039

5830.757 6000.491 6179.037

0.045 0.041 0.002

6388.080

6388.099

 0.019

6537.960

6537.935

0.025

6679.706

6679.664

0.042

6688.176

6688.163

0.013

6863.555

6863.524

0.031

7023.674

7023.650

0.024

7203.826

7203.804

0.022

7602.514

7602.479

0.035

7240.003 7616.634

7239.943 7616.640

0.060  0.006

7847.821 7743.696

7847.773 7743.671

0.048 0.025

7901.472

7901.432

0.040

8056.025

8055.988

0.037

7889.101

7889.092

0.009

8081.882

8081.849

0.033

8250.626

8250.604

0.022

8424.999

8424.983

0.016

8628.662

8628.610

0.052

8803.262

8803.254

0.008

8944.132

8944.120

0.012

8842.391 9864.451

8842.442 9864.495

 0.051  0.044

9288.089

9288.097

 0.008

9469.396

9469.412

 0.016

9642.569 9987.363

9642.609 9987.357

 0.040 0.006

ΔGv

Bv (obs.)

Bv (cal.)

ΔBv

Ref.

0.38736 0.38823 0.39136 0.38993 0.39135 0.38971 0.39120 0.38455 0.38513 0.38816 0.38943 0.38701 0.38811 0.38741 0.38862 0.39002 0.38917 0.38959 0.39157 0.39031 0.39270 0.38482 0.38573 0.38431 0.38513 0.38854 0.39026 0.38692 0.38834 0.38675 0.38811 0.38760 0.38915 0.38150 0.38207 0.38884 0.38960 0.39204 0.38988 0.38532 0.38652 0.38413 0.38530 0.38440 0.38551 0.38882 0.39090 0.38713 0.38886 0.38625 0.38782 0.38682 0.38852 0.38824 0.38978 0.38186 0.38275 0.38126 0.38204 0.38909 0.37846 0.37901 0.38743 0.38955 0.38595 0.38784 0.38602 0.38240 0.38361

0.38734 0.38822 0.39114 0.38963 0.39106 0.38943 0.39090 0.38470 0.38528 0.38734 0.38937 0.38689 0.38800 0.38732 0.38854 0.38966 0.38878 0.38926 0.39120 0.39004 0.39235 0.38490 0.38581 0.38438 0.38521 0.38842 0.39014 0.38672 0.38814 0.38655 0.38792 0.38746 0.38902 0.38174 0.38231 0.38835 0.38923 0.39149 0.38957 0.38534 0.38656 0.38421 0.38532 0.38440 0.38552 0.38864 0.39074 0.38684 0.38861 0.38594 0.38752 0.38656 0.38826 0.38805 0.38960 0.38203 0.38293 0.38141 0.38219 0.38860 0.37879 0.37935 0.38711 0.38923 0.38557 0.38746 0.38565 0.38252 0.38375

0.00002 0.00001 0.00032 0.00030 0.00029 0.00028 0.00030  0.00015  0.00015 0.00082 0.00006 0.00012 0.00011 0.00009 0.00008 0.00036 0.00039 0.00033 0.00037 0.00027 0.00035  0.00008  0.00008  0.00007  0.00008 0.00012 0.00012 0.00020 0.00020 0.00020 0.00019 0.00014 0.00013  0.00024  0.00024 0.00049 0.00037 0.00045 0.00031  0.00002  0.00004 0.00002  0.00002 0.00000  0.00001 0.00018 0.00016 0.00029 0.00025 0.00031 0.00030 0.00026 0.00026 0.00019 0.00018  0.00017  0.00018  0.00015  0.00015 0.00049  0.00033  0.00034 0.00032 0.00032 0.00038 0.00038 0.00037  0.00012  0.00014

[77] [77] [15] [18] [18] [18] [18] [13] [13] [13] [13] [13] [13] [13] [13] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [24] [25] [24] [16] [27] [27] [27] [35] [16] [35] [35] [24] [19] [24] [24] [24] [24] [27] [27] [35] [35] [35] [35] [35] [35] [25] [24] [24] [25] [25] [25] [25] [25] [35] [35]

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

243

Table 6 (continued ) Pa

Gv (obs.)

Gv (cal.)

ΔGv

Bv (obs.)

Bv (cal.)

ΔBv

Ref.

l2 ¼ 2 states 2 02201e/f 4 12202e/f 4 12201e/f 5 02211e/f 6 22203e/f 6 22202e/f 6 22201e/f 7 12212e/f 7 12211e/f 8 02221e/f 9 22213e/f 9 22212e/f 9 22211e/f 10 42202e/f 10 42201e/f 10 12222e/f 10 12221e/f 11 32214e/f 11 32213e/f 11 32212e/f 11 32211e/f 11 02231e/f 12 22222e/f 12 22221e/f 13 42215e/f 13 42214e/f 13 42213e/f 13 42212e/f 13 42211e/f 13 12232e/f 13 12231e/f

1335.132 2585.022 2760.725 3659.273 3822.012 4007.915 4197.361 4887.985 5061.778 5958.512 6103.683 6288.493 6474.532 6875.090 7102.137 7166.033 7338.158 7307.647 7505.241 7694.402 7897.550 8232.882 8544.386 8727.307 8501.198 8711.777 8907.403 9108.071 9329.944 9419.183 9589.918

1335.010 2584.951 2760.659 3659.190 3821.973 4007.867 4197.341 4887.933 5061.704 5958.421 6103.636 6288.425 6474.468 6875.067 7102.102 7165.956 7338.068 7307.603 7505.170 7694.334 7897.485 8232.809 8544.295 8727.209 8501.176 8711.729 8907.418 9108.011 9329.815 9419.123 9589.864

0.122 0.071 0.066 0.083 0.039 0.048 0.020 0.052 0.074 0.091 0.047 0.068 0.064 0.023 0.035 0.077 0.090 0.044 0.071 0.068 0.065 0.073 0.091 0.098 0.022 0.048  0.015 0.060 0.129 0.060 0.054

0.39167 0.39194 0.39155 0.38864 0.39235 0.39144 0.39159 0.38894 0.38853 0.38561 0.38942 0.38848 0.38864 0.39123 0.39171 0.38604 0.38550 0.38995 0.38861 0.38830 0.38891 0.38258 0.38551 0.38566 0.39053 0.38890 0.38807 0.38780 0.39029 0.38308 0.38246

0.39164 0.39184 0.39144 0.38870 0.39218 0.39124 0.39142 0.38889 0.38851 0.38576 0.38935 0.38836 0.38855 0.39090 0.39142 0.38613 0.38556 0.38983 0.38841 0.38810 0.38860 0.38282 0.38548 0.38565 0.39034 0.38864 0.38778 0.38758 0.39009 0.38325 0.38262

0.00003 0.00010 0.00011  0.00006 0.00017 0.00020 0.00017 0.00005 0.00002  0.00015 0.00007 0.00012 0.00009 0.00033 0.00029  0.00009  0.00006 0.00012 0.00020 0.00020 0.00031  0.00024 0.00003 0.00001 0.00019 0.00026 0.00029 0.00022 0.00020  0.00017  0.00016

[13] [77] [77] [13] [77] [77] [77] [77] [77] [77] [24] [24] [24] [24] [24] [24] [24] [24] [24] [27] [35] [35] [27] [35] [25] [24] [24] [27] [35] [35] [35]

l2 ¼ 3 states 3 03301e/f 5 13302e/f 5 13301e/f 6 03311e/f 7 23302e/f 7 23301e/f 8 13312e/f 8 13311e/f 9 33301e/f 9 03321e/f 11 13322e/f 11 13321e/f 12 33314e/f 12 33313e/f 12 33312e/f 12 33311e/f 12 03331e/f

2003.246 3240.623 3442.215 4314.914 4676.791 4890.096 5531.303 5730.605 6346.709 6601.713 7797.094 7994.362 7932.225 8149.958 8362.848 8586.647 8863.680

2003.087 3240.524 3442.119 4314.768 4676.714 4890.020 5531.187 5730.468 6346.649 6601.547 7796.940 7994.189 7932.139 8149.837 8362.707 8586.469 8863.529

0.159 0.099 0.096 0.146 0.077 0.076 0.116 0.137 0.060 0.166 0.154 0.173 0.086 0.121 0.141 0.178 0.151

0.39238 0.39266 0.39222 0.38938 0.39219 0.39218 0.38971 0.38925 0.39214 0.38638 0.38677 0.38627 0.39063 0.38946 0.38943 0.38994 0.38337

0.39235 0.39256 0.39211 0.38944 0.39200 0.39207 0.38970 0.38923 0.39190 0.38652 0.38686 0.38634 0.39049 0.38928 0.38932 0.38980 0.38361

0.00003 0.00010 0.00011  0.00006 0.00019 0.00011 0.00001 0.00002 0.00024  0.00014  0.00009  0.00007 0.00014 0.00018 0.00011 0.00014  0.00024

[77] [77] [77] [77] [77] [77] [77] [77] [24] [77] [24] [24] [24] [24] [16] [24] [24]

l2 ¼ 4 states 4 04401e/f 6 14402e/f 6 14401e/f 7 04411e/f 13 04431e/f

2671.715 3898.314 4122.269 4970.928 9494.891

2671.522 3898.187 4122.133 4970.709 9494.626

0.193 0.127 0.136 0.219 0.265

0.39308 0.39336 0.39287 0.39011 0.38417

0.39306 0.39326 0.39277 0.39017 0.38440

0.00002 0.00010 0.00010  0.00006  0.00023

[77] [77] [77] [77] [24]

l2 ¼ 5 states 5 05501e/f 7 15502e/f 7 15501e/f

3340.527 4557.595 4801.365

3340.301 4557.439 4801.179

0.226 0.156 0.186

0.39378 0.39406 0.39352

0.39375 0.39395 0.39342

0.00003 0.00011 0.00010

[77] [77] [77]

l2 ¼ 6 states 6 06601e/f

4009.677

4009.412

0.265

0.39447

0.39443

0.00004

[77]

l2 ¼ 7 states 7 07701e/f

4679.156

4678.841

0.315

0.39513

0.39511

0.00002

[77]

a

V 1 V 2 l2 V 3 rϵ

P ¼ 2V 1 þ V 2 þ 3V 3 .

244

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

Table 7 Nuclear spin factors of the CO2 isotopologues. The nuclear spin values are I(12C)¼ 0, I(13C)¼ 1/2, I(16O)¼ 0, I(17O) ¼5/2 and I(18O)¼ 0. Isotopologue I1

I2

I3

Nuclear system gna gn,eb gn,oc gn;c

16

0 1/2 0 0 1/2 1/2 0 1/2 1/2 0 1/2 1/2

0 0 0 5/2 0 5/2 0 0 0 5/2 0 5/2

Bosons Bosons – – – – Bosons – Bosons Fermions – Fermions

O12C16O O13C16O 16 12 18 O C O 16 12 17 O C O 16 13 18 O C O 16 13 17 O C O 18 12 18 O C O 17 12 18 O C O 18 13 18 O C O 17 12 17 O C O 17 13 18 O C O 17 13 17 O C O 16

a b c

0 0 0 0 0 0 0 5/2 0 5/2 5/2 5/2

1 2 1 6 2 12 1 6 2 36 12 72

1 1 – – – – 1 – 1 15 – 15

0 0 – – – – 0 – 0 21 – 21

1 2 – – – – 1 – 2 1 – 2

(2I1 þ1)(2I2 þ 1)(2I3 þ1). (2Iþ 1)I (fermions); (2Iþ 1)(Iþ 1) (bosons). (2Iþ 1)(Iþ 1) (fermions); (2Iþ 1)I (bosons).

Table 9 Comparison between the internal partition functions for the most abundant CO2 isotopologues calculated using the partial sums (Q ps ) over the spectroscopic constants Gv and Bv up to 9500 cm  1 above the zeropoint energy and from the CDSD-4000 database. T ðKÞ

298 400 600 800 1000 a b

Q

Qv Osipova

Present

ΔQ rel: b

Osipov

Present

ΔQ rel:

1.0867 1.2199 1.6430 2.2874 3.1910

1.0866 1.2199 1.6432 2.2876 3.1911

0.009 0.000  0.012  0.009  0.003

288.27 434.24 877.42 1628.7 2840.3

288.45 434.45 876.99 1626.8 2834.4

 0.06  0.05 0.05 0.12 0.21

Osipov VM. Mol Phys 2004;102:1785–92. ΔQ rel: ¼ 100  ðQ Osipov  Q present Þ=Q Osipov .

interval up to 1000 K. As observed, our vibrational partition functions practically coincide with the Osipov ones for each temperature to the five significant figures with which they are given, as clearly stated by the very low percentage relative differences, ΔQ rel: , existing between them in all cases. Our internal partition functions also agree quite well with those of Osipov, with relative percentage differences that are still quite low, of the order of 0.05% up to 600 K, and somewhat higher above this temperature. A more demanding test of our internal partition functions is to compare them with those calculated by Tashkun and Perevalov [37] for the four most abundant isotopologues (16O12C16O, 16O13C16O, 16O12C17O and 16O12C18O) by approximated direct summations over all energy levels calculated from global fitting of effective Hamiltonians to the observed data collected from the literature. These partition functions, which are included in the CDSD-4000 carbon dioxide databank, are presumably the most accurate ones available for these isotopologues. In Table 9 we show the calculated values for both sets of internal partition functions. As observed, the relative differences between our results and the CDSD ones are quite similar for the four isotopologues, remaining below 0.15% and steadily increasing with temperature. This increase of the relative differences is likely due to the absence in our calculations of the centrifugal distortion contributions to

ΔQ rel: b

290.92 434.45 876.99 1626.8 2834.4

0.04 0.06 0.07 0.10 0.15

O12C18O 618.39 925.90 1879.8 3505.5 6136.8

618.22 925.53 1878.7 3502.5 6128.5

0.02 0.04 0.06 0.09 0.14

Q CDSD O13C16O 586.77 881.67 1798.1 3361.7 5896.1

Q ps

ΔQ rel:

16

16

300 400 600 800 1000

b

T (K)

O12C16O 291.05 434.69 877.67 1628.4 2838.6

Q ps

16

300 400 600 800 1000

a

Table 8 Comparison of vibrational Q v and internal Q partition functions for the 12 16 C O2 isotopologue.

Q CDSD a

O12C17O 3604.1 5389.8 10,914 20,303 35,473

586.52 881.16 1796.7 3358.4 5887.5

0.04 0.06 0.08 0.10 0.15

16

3603.0 5387.7 10,907 20,285 35,422

0.03 0.04 0.06 0.09 0.14

Tashkun SA, Perevalov VI. JQSRT 2011;112:1403–10. ΔQ rel: ¼ 100  (Q CDSD  Q ps )/Q CDSD .

the rotational energy levels (see Eq. (15)), which are expected to have a more pronounced effect on the internal partition function as the temperature rises. We have also compared our internal partition functions with those included in the HITRAN database [36]. In the HITRAN database, specific values of the internal partition functions are tabulated for a set of temperatures from 60 K to 3010 K at 25 K intervals, and a four-point Lagrange interpolation is used to compute the internal partition function for any other temperature [85,89]. The estimated uncertainties of the HITRAN partition functions are 0.1– 0.3% at 296 K, and 0–1% at 1000 K. To avoid interpolation errors, we have selected for comparison the internal partition functions tabulated in the HITRAN database at the temperatures of 210, 410, 610, 810 and 1010 K for all the isotopologues, except for the 13C17O2 one, for which the partition functions have not been reported yet in HITRAN. In Table 10 we include the values calculated in this work for the internal partition functions of the CO2 isotopologues, along with those taken from the HITRAN database. As observed, the corresponding absolute relative differences ΔQ rel: stay now around 0.3% up to 600 K, and increase for higher temperatures. The good results obtained for the internal partition function of the most abundant isotopologues 16O12C16O, 16O13C16O, 16O12C17O and 16O12C18O up to 600 K, as previously compared with those included in the CDSD-4000 databank, combined with the ability of the refined potential surface to reproduce accurately the vibrational terms and rotational constants of the rest of isotopologues, as demonstrated in Section 3, allows us to infer that the internal partition functions calculated for all the stable CO2 isotopologues in such a range of low-to-medium temperatures are quite accurate. For higher temperatures, it is clear that more accurate partition functions would be needed, and the most immediate way to improve them is by including the distortion constant terms Dv, Hv,…in Eq. (15). This method has to be managed however with care, first because, as

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245

Table 10 Comparison between the internal partition functions for the CO2 isotopologues calculated using the partial sums (Q ps ) over the spectroscopic constants Gv and Bv up to 9500 cm  1 above the zero-point energy and from the HITRAN database. T ðKÞ

Q HITRAN

Q ps

O12C16O 191.60 452.80 908.25 1675.9 2907.4

ΔQ rel: a

Q HITRAN

16

210 410 610 810 1010

16

191.12 451.18 906.09 1674.9 2910.1

0.25 0.34 0.24 0.06  0.09

405.43 961.45 1941.5 3606.9 6293.2

0.24 0.28 0.00  0.31  0.58

813.31 1954.1 3981.9 7454.6 13 086

0.24 0.24 0.25 0.11 0.00

215.35 513.03 1042.0 1945.9 3410.0

0.25 0.18  0.26  0.76  1.17

432.04 1042.0 2138.2 4024.5 7096.8

0.23 0.20  0.15  0.53  0.85

16

O12C18O 406.40 964.02 1941.6 3595.6 6256.8

210 410 610 810 1010

O13C18O 815.28 1958.8 3991.8 7462.6 13 086

O13C17O 4755.0 11 402 23 184 43 256 75 729

O13C18O 433.05 1044.1 2135.1 4003.1 7036.4

O12C18O 2516.2 5980.2 12 070 22 393 39 025

0.24 0.31 0.12  0.12  0.33

4743.4 11 359 23 107 43 152 75 600

0.24 0.38 0.33 0.24 0.17

2510.3 5967.4 12 087 22 514 39 372

0.24 0.21  0.14  0.54  0.88

7317.8 17 358 35 063 65 154 113 703

0.24 0.26  0.01  0.33  0.60

17

O12C17O 7335.3 17 403 35 060 64 938 113 020

17

a

O13C18O 5047.7 12 150 24 807 46 449 81 558

2364.8 5596.1 11 271 20 887 36 371

17

18

210 410 610 810 1010

0.25 0.42 0.42 0.37 0.34

16

O12C18O 215.88 513.97 1039.3 1931.2 3370.2

210 410 610 810 1010

383.33 915.63 1857.1 3458.9 6046.2

O12C17O 2370.5 5613.2 11 284 20 861 36 250

18

210 410 610 810 1010

O13C16O 384.29 919.43 1864.9 3471.8 6066.8

ΔQ rel:

16

16

210 410 610 810 1010

Q ps

17

O13C17O

5035.8 12 118 24 795 46 548 81 907

0.24 0.26 0.05  0.21  0.42

14 679 35 243 71 909 134 655 236 445

ΔQ rel: ¼ 100  ðQ HITRAN  Q ps Þ=Q ps .

discussed above, the spectroscopic expansion of the rotational terms Fv in powers of JðJ þ1Þ is not valid for perturbed levels, and second, and more important, because as noticed by Gray and Young [90] some time ago, this kind of polynomial expansions are not free of divergences which can spoil the calculation of the partition functions if sufficiently large values of J are required like for high temperatures. The proper and more computationally demanding way of improving the accuracy of the internal partition functions is by direct summation over the rotational levels determined by variational means, as recently done by Huang et al. [46] for 12C16O2 using the Ames-1 potential energy surface. 5. Equilibrium isotopic fractionation With the internal partition functions in hand, we have determined the isotopic compositions of the CO2 isotopologues in gas phase at equilibrium conditions. The isotopic fractionations of the multiply substituted CO2 isotopologues, in particular, have received much attention in the last ten

years [2,67–74,91–95] due to the ongoing possibility of making increasingly accurate measures of their very low natural abundances, and to the possible use of the temperature dependent abundances as geological thermometers. To calculate the abundances of the CO2 isotopologues, we have used the method developed by Wang et al. [74]. This method proceeds basically by making the concentrations of the isotopologues compatible both with the bulk compositions of their constituent isotopes and with the equilibrium constants of a set of independent isotope exchange reactions in which all the isotopologues participate. The method is described in detail in the work by Wang et al. [74], and only its application to the CO2 isotopologues is, therefore, presented here. Accordingly, the isotope exchange reactions selected for CO2 isotopologues are K1

16

O 12 C 16 O þ 16 O 13 C 17 O⇌16 O 13 C 16 O þ 16 O 12 C 17 O

16

O 12 C 16 O þ 17 O 12 C 17 O⇌16 O 12 C 17 O þ 16 O 12 C 17 O

K2

ð18Þ ð19Þ

246

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Table 11 Theoretical zero point energies (ZPE), harmonic vibrational frequencies and rotational constants (in cm  1) for the CO2 isotopologues. Ma

Isotopologue

ZPE

Δ (ZPE)

ω1

ω2

ω3

Be

B0

44 45 46 45 47 46 48 47 49 46 48 47

16

2535.680 2482.935 2502.429 2518.171 2449.249 2465.199 2468.884 2484.767 2415.278 2500.582 2431.365 2447.386

0.00 52.745 33.251 17.510 86.431 70.482 66.796 50.913 120.402 35.098 104.315 88.295

1353.717 1353.717 1314.631 1333.335 1314.554 1333.314 1276.124 1294.563 1276.124 1313.124 1294.546 1313.124

672.692 653.546 667.564 669.976 648.267 650.751 662.396 664.827 642.944 667.249 645.448 647.943

2396.561 2328.352 2378.768 2387.021 2310.078 2318.543 2359.882 2368.649 2290.581 2377.172 2299.621 2308.390

0.39165 0.39165 0.36951 0.37999 0.36949 0.37998 0.34804 0.35820 0.34804 0.36851 0.35820 0.36851

0.39020 0.39021 0.36817 0.37859 0.36817 0.37864 0.34680 0.35691 0.34682 0.36718 0.35693 0.36719

a

16

16

12 16

O C O O13C16O O12C18O 16 12 17 O C O 16 13 18 O C O 16 13 17 O C O 18 12 18 O C O 17 12 18 O C O 18 13 18 O C O 17 12 17 O C O 17 13 18 O C O 17 13 17 O C O 16 16

Mass number. K3

O 12 C 16 O þ 16 O 13 C 18 O⇌16 O 13 C 16 O þ 16 O 12 C 18 O 12

16

O C Oþ

16

12

16

17

2 O C Oþ 16

12

18

K4 16

12

17

16

12

18

O C O⇌ O C O þ O C O

17

13

17

K5 16

13

16

16

12

17

O C O⇌ O C O þ 2 O C O K6

O 12 C 16 O þ 18 O 12 C 18 O⇌16 O 12 C 18 O þ 16 O 12 C 18 O

ð20Þ ð21Þ ð22Þ ð23Þ

K7

216 O 12 C 16 O þ 17 O 13 C 18 O⇌16 O 13 C 16 O þ 16 O 12 C 17 O þ 16 O 12 C 18 O K8

216 O 12 C 16 O þ 18 O 12 C 18 O⇌16 O 13 C 16 O þ 216 O 12 C 18 O

ð24Þ ð25Þ

Combination then of the equilibrium constant expressions of these eight reactions with the mass-balance equation of all the CO2 isotopologues and with the mass-balance equations of the isotopologues containing the isotopes 13 C, 16O and 18O gives the following set of non-linear equations in the molar fractions xi of the main, 16O12C16O (1), and the three singly substituted, 16O13C16O (2), 16 12 17 O C O (3) and 16O12C18O (4), isotopologues x1 þx2 þ x3 þ x4 þ

x2 x3 x4 x2 x24 þ ¼1 K 7 x21 K 8 x21

ð26Þ

x2 x3 x2 x4 x2 x3 x2 x3 x4 x2 x24 þ þ þ þ ¼ x13 C K 1 x1 K 3 x1 K 5 x21 K 7 x21 K 8 x21

ð27Þ

þ

x2 þ

x2 x2 x2 x3 x2 x4 x3 x4 x2 x3 þ 3 þ þ þ þ 4 K 1 x1 K 2 x1 K 3 x1 K 4 x1 K 5 x21 K 6 x1

2x1 þ x2 þx3 þx4 þ

x4 þ

x2 x3 x2 x4 þ ¼ 2x16 O K 1 x1 K 3 x1

x2 x4 x3 x4 2x24 x2 x3 x4 2x2 x24 þ þ þ þ ¼ x18 O K 3 x1 K 4 x1 K 6 x1 K 7 x21 K 8 x21

ð28Þ

ð29Þ

where x13 C , x16 O and x18 O are the bulk molar fractions of the isotopes 13C, 16O and 18O. These non-linear equations are solved numerically and their solutions are used to compute the molar fractions xi ði ¼ 5 12Þ of the remaining multiply substituted isotopologues 16O13C17O (5), 17 12 17 O C O (6), 16O13C18O (7), 17O12C18O (8), 17O13C17O (9), 18 12 18 O C O (10), 17O13C18O (11) and 18O13C18O (12), using

the equilibrium constant expressions as follows: x23 ; K 2 x1

x5 ¼

x2 x3 ; K 1 x1

x6 ¼

x9 ¼

x2 x23 ; K 5 x21

x10 ¼

x24 ; K 6 x1

x7 ¼

x2 x4 ; K 3 x1

x11 ¼

x8 ¼

x2 x3 x4 ; K 7 x21

x3 x4 K 4 x1

x12 ¼

x2 x24 K 8 x21

ð30Þ

The isotopic fractions needed to solve Eqs. (26)–(29) are chosen as those which satisfy the PDB [96] standard for the isotopes of C (13C/13C¼0.0112372) and the VSMOW [97] standard for the isotopes of O (18O/16O¼0.0020052 and 17 O/16O¼ 0.0003799). As for the equilibrium constants, they are evaluated from statistical thermodynamics, as expressed in terms of the partition functions of reactants and products. For example, for reaction (18) we have, K1 ¼

Q 016 O 13 C 16 O Q 016 O 12 C 17 O Q0

16

O 12 C 16 O

Q0

16

O 13 C 17 O

¼

Q 02 Q 03 Q 01 Q 05

ð31Þ

where primes indicate total partition functions including the zero point energy exponential as given in Eq. (8). The calculation of the molar fractions of the isotopologues ultimately rests therefore on the determination of the total partition functions. One way to proceed then consists of using the seminal method developed by Urey [98] based on the harmonic oscillator and rigid rotor approximations to describe the vibrational and rotational motions of the molecule. In the Urey model, the zero-point energy is written as the sum of the harmonic contributions of the individual vibration normal modes. However, it is possible to improve the calculation of the partition functions within this model by using more accurate zero-point energies which include anharmonic contributions, as proposed and illustrated by Wang et al. in their work [74]. In order to apply in this work the latter so-called ZPE-Urey model to the CO2 isotopologues, in Table 11 we give the zero-point energies computed variationally using the refined potential surface, along with their corresponding normal modes vibrational frequencies and equilibrium rotational constants. A more accurate option for calculating the internal partition functions of the isotopologues consists of summing directly over the vibrational and rotational energy levels, as discussed in Section 4. This possibility was also considered by Wang et al. [74] for CO2, but eventually

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247

Table 12 Comparison of Δ values for the CO2 isotopologues at select temperatures from Wang et al. [74] (I) and calculated in the present work using the Urey-ZPE method (II) and from partition sums (III). 200 K

300 K

1000 K

I

II

III

I

II

III

I

II

III

O13C16O O12C17O Mass 45

 0.0082  0.0117  0.0084

 0.0081  0.0118  0.0083

 0.0081  0.0117  0.0083

 0.0043  0.0060  0.0044

 0.0042  0.0060  0.0043

 0.0043  0.0058  0.0044

 0.0003  0.0003  0.0003

 0.0003  0.0004  0.0003

 0.0002  0.0001  0.0002

16

 0.0225 0.9347 0.2370  0.0205

 0.0226 0.9421 0.2365  0.0205

 0.0224 0.9457 0.2003  0.0204

 0.0114 0.4888 0.0848  0.0104

 0.0115 0.4939 0.0841  0.0104

 0.0110 0.5014  0.0229  0.0099

 0.0007 0.0306  0.0007  0.0006

 0.0007 0.0318  0.0014  0.0006

 0.0002 0.0263  0.0841  0.0001

O13C18O O12C18O 17 13 17 O C O Mass 47

1.7977 0.4674 2.1273 1.7552

1.8082 0.4670 2.1383 1.7644

1.8130 0.4243 2.1072 1.7676

0.9384 0.1681 1.0738 0.9138

0.9449 0.1673 1.0804 0.9195

0.9532 0.0409 0.9823 0.9234

0.0582  0.0010 0.0616 0.0563

0.0592  0.0017 0.0620 0.0572

0.0511  0.0995  0.0298 0.0462

18

O12C18O O13C18O Mass 48

0.9217 3.2220 0.9311

0.9211 3.2355 0.9309

0.8373 3.1986 0.8473

0.3329 1.6078 0.3381

0.3314 1.6150 0.3369

0.0876 1.4966 0.0936

 0.0013 0.0892  0.0010

 0.0027 0.0890  0.0023

 0.1869  0.0193  0.1862

18

4.5322

4.5505

4.4654

2.2192

2.2290

1.9819

0.1169

0.1168

 0.0768

16 16

O12C18O O13C17O 17 12 17 O C O Mass 46 16

16 17

17

O13C18Oa a

Mass 49.

Fig. 5. Variation of Δ values of multiply substituted isotopologues of CO2 with 1000/T calculated using Urey method (left panel) and approximated direct summation method (right panel).

ruled out, due essentially to the lack, at the time, of enough spectroscopic information for some of the multiply substituted isotopologues. In our case, however, the refined potential surface gives sufficiently accurate rovibrational energy levels for all the isotopologues of CO2, in principle, so as to have reliable and complete sets of the partition functions available for all of them, and to eventually undertake the full analysis of the equilibrium fractionations of the multiply substituted CO2 isotopologues confidently beyond the ZPE-Urey model. One of the main issues in determining the equilibrium compositions of the CO2 isotopologues is to find out how they vary with temperature and therefore how they deviate from the stochastic concentrations, which are those corresponding to the random distribution of the

isotopes in the isotopologues reached in the limit of infinity temperature. To quantify the deviations of the isotopologue concentrations from the stochastic distribution, Wang et al. [74] introduced the variable Δi defined as   R Δi ¼ i  e 1  1000 ð32Þ Ri  r where Ri  e is the abundance of the isotopologue i of interest divided by the abundance of the main non substituted isotopologue, when both are evaluated at thermodynamic equilibrium, and Ri  r is the same ratio for the stochastic abundances. Positive values of Δi are indicative of enrichments of the isotopologue i relative to the stochastic distribution, and negative values of Δi are indicative of depletions. Due to the instrumental difficulties in

248

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

Fig. 6. Variation of Δ values of isotopologues of CO2 versus 1000/T calculated using the approximated direct summation method.

distinguishing the isotopologues with the same mass number, it is also necessary to calculate the variable Δmass given by   ∑i Ri  e Δmass ¼ 1  1000 ð33Þ ∑i Ri  r which takes in the contributions from all isotopologues with the same mass number. By using molar fractions for the abundances, we write Ri  e ¼ xi =x1 and Ri  r ¼ xi  r =x1  r , where xi  r are the molar fractions of the stochastic distribution. The latter molar fractions are calculated by multiplying the bulk concentration of the isotopes that the isotopologue contains, if the isotopologue is symmetric, and by adding a factor of 2 if the isotopologue is asymmetric [74]. The values of the equilibrium constants of the isotopic exchange reactions (18)–(25) corresponding to the stochastic distribution are then 1, 4, 1, 2, 4, 4, 2 and 4. In Table 12 we give the Δ values calculated by Wang et al. [74] for the CO2 isotopologues at temperatures 200, 300 and 1000 K using the ZPE-Urey model with zero-point energies and normal modes vibrational frequencies

extracted from our previous potential energy surface (I), and the results obtained in this work based on the data provided by the refined potential surface using both the ZPE-Urey model (II) and the approximated direct summation method to calculate the partition functions (III). We see first in this Table that the Δ values from the ZPE-Urey model calculated using both the previous and the refined potential surfaces differ very little. The largest differences appear at 200 K, but do not reach 0.02‰, and they rapidly decrease as the temperature raises. Since Δ values can be currently measured to precisions reaching 0:01–0:02‰ [67,93,99,100], we conclude that the two potential surfaces give essentially the same description of the equilibrium fractionation of the isotopologues within the framework of the ZPE-Urey model. Table 12 also shows that the differences between the Δ values calculated using the ZPE-Urey model and the approximated direct summation method with data taken in both cases from refined potential surface are, however, larger, more significant, and susceptible in principle to experimental verification. It is therefore clear that the way

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249

in which more accurate partition functions are calculated is dominant against the refinement of the potential energy function, in the calculation of the relative abundances of the CO2 isotopologues. The Δ differences between the two methods are more marked, in particular, at the terrestrial temperature of 300 K, for which they exceed 0:1‰ for the isotopologues 17O12C17O, 17O12C18O and 17O13C18O, and 0:2‰ for the isotopologues 18O12C18O and 18O13C18O. To appreciate these variations better, in Fig. 5 we plot the Δ values versus 1000/T for the multiply substituted isotopologues calculated using both methods. As observed, the Δ values for the 18O12C18O isotopologue, with mass number 48, are those which behaves most differently when passing from one method to the other. Specifically, the Δ values for this isotopologue calculated using the approximated direct summation method drop faster than the ZPE-Urey values as temperature rises, entering negative depletion values at some temperature higher than 300 K and going through a minimum between 500 and 600 K in which Δ   0:2‰. The Δ values of the isotopologues 17O12C17O and 17O12C18O, with mass numbers 46 and 47 respectively, calculated by the approximated direct summation method also exhibits similar, although smoother, minima. In Fig. 6 we show specifically the variations of the Δ values calculated using the approximated direct summation method, with 1000/T gathering the isotopologues with the same mass number separately. We also plot in each group the variation of Δmass which, as expected, follows most closely the Δ curve of the most abundant isotopologue in the group. In general, the Δ values of the multiply substituted isotopologues of CO2 decrease faster with temperature when the partition functions are calculated more accurately by summing directly over the rovibrational energy levels of the isotopologues, exhibiting in some cases negative depletion minima at intermediate temperatures of around 500 K. We should finally emphasize that the significant variations obtained in the relative abundances of the CO2 isotopologues result from the use of much more accurate partition functions than those provided by the ZPE-Urey model rather than from the refinement of the potential energy function

comparison. Calculated values of Gv and Bv are then given for a large number of vibrational energy levels of every isotopologue, with the aim of helping in the identification and assignment of new unobserved vibrational bands of them. The Gv and Bv values have subsequently been used to calculate the internal partition functions of the isotopologues by approximated direct summation over the computed rovibrational energy levels, obtaining converged values up to a temperature of about 1000 K. These partition functions have in turn been employed to calculate the abundances of the CO2 isotopologues in gas phase relative to the stochastic distribution, using the method developed by Wang et al. [74]. Significant differences in the relative abundances of the multiple substituted isotopologues with respect to those calculated using the seminal harmonic-rigid based Urey method have thus been found, especially for the isotopologues 18O12C18O, 17O12C17O and 17O12C18O. These differences exceed the experimental uncertainty with which the natural abundances of the isotopologues are measured and could therefore be relevant for verification, interpretation and practical use of the isotopic fractionations of the CO2 isotopologues in different geochemical environments.

6. Conclusions

References

In this work, we have calculated the rovibrational energy levels of the stable isotopologues of CO2 using a refined Morse-cosine potential energy surface obtained by fitting to the observed pure (l2 ¼ 0Þ vibrational frequencies of the main isotopologue 12C16O2 properly updated. The calculations have been carried out variationally using a system of optimal normal hyperspherical coordinates which facilitates the assignment of the normal modes quantum numbers to the vibrational energy levels. The refined potential energy surface reproduces the observed experimental vibrational terms Gv of the main isotopologue up 10,000 cm  1 with an rms deviation of 0.093 cm  1, and the corresponding rotational constants Bv with an rms deviation of 0.00021 cm  1. This degree of accuracy holds, and even improves, for the rest of the isotopologues, as far as the smaller amount of spectroscopic data available for them allows us to make a proper

Acknowledgments This work was partially supported by the Spanish Ministerio de Ciencia e Innovación under Projects CTQ2011-25972 and CONSOLIDER CSD2009-00038, and by the Fundación Séneca del Centro de Coordinación de la Investigación de la Región de Murcia under Project 08735/PI/08. J.C. acknowledges a FPU fellowship provided by the Ministerio de Educación of Spain. Appendix A. Supplementary material Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j. jqsrt.2014.06.003.

[1] Miller CE, Brown LR, Toth RA, Benner DC, Devi VM. Spectroscopic challenges for high accuracy retrievals of atmospheric CO2 and the orbiting carbon observatory (OCO) experiment. C R Phys 2005;6: 876–87. [2] Eiler JM. Clumped-isotope geochemistry—the study of naturallyoccurring multiply-substituted isotopologues. Earth Planet Sci Lett 2007;262:309–27. [3] Robert S, Borkov YG, Vander-Auwera J, Drummond R, Mahieux A, Wilquet V, et al. Assignment and rotational analysis of new absorption bands of carbon dioxide isotopologues in venus spectra. J Quant Spectrosc Radiat Transfer 2013;114:29–41. [4] Wilquet V, Mahieux A, Vandaele AC, Perevalov VI, Tashkun SA, Fedorova A, et al. Line parameters for the 01111–00001 band of 12 16 18 C O O from soir measurements of the Venus atmosphere. J Quant Spectrosc Radiat Transfer 2008;109:895–905. [5] Bertaux JL, Vandaele AC, Wilquet V, Montmessin F, Dahoo R, Villard E, et al. First observation of 628 CO2 isotopologue band at 3:3 μm in the atmosphere of Venus by solar occultation from Venus express. Icarus 2008;195:28–33. [6] Villanueva GL, Mumma MJ, Novak RE, Hewagama T. Identification of a new band system of isotopic CO2 near 3:3 μm: implications for remote sensing of biomaker gases on Mars. Icarus 2008;195:34–44.

250

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

[7] Villanueva GL, Mumma MJ, Novak RE, Hewagama T. Discovery of multiple bands of isotopic CO2 in the prime spectral regions used when searching for CH4 and HDO on Mars. J Quant Spectrosc Radiat Transfer 2008;109:883–94. [8] Teffo JL, Claveau C, Valentin A. Infrared fundamental bands of O13C17O isotopic variants of carbon dioxide. J Quant Spectrosc Radiat Transfer 1998;59:151–64. [9] Claveau C, Teffo JL, Hurtmans D, Valentin A. Infrared fundamental and first hot bands of O12C17O isotopic variants of carbon dioxide. J Mol Spectrosc 1998;189:153–95. [10] Claveau C, Teffo JL, Hurtmans D, Valentin A, Gamache RR. Line positions and absolute intensities in the laser bands of carbon-12 oxygen 17 isotopic species of carbon dioxide. J Mol Spectrosc 1999;193:15–32. [11] Ding Y, Perevalov VI, Tashkun SA, Teffo JL, Liu AW, Hu SM. 16 13 18 O C O: high-resolution absorption spectrum between 4000 and 9500 cm  1 and global fitting of vibration-rotational line positions. J Mol Spectrosc 2003;220:276–83. [12] Ding Y, Macko P, Romanini D, Perevalov VI, Tashkun SA, Teffo JL, et al. High sensitive CW-cavity ringdown and Fourier transform absorption spectroscopies of 13CO2. J Mol Spectrosc 2004;226:146–60. [13] Miller CE, Brown LR. Near infrared spectroscopy of carbon dioxide I, 16 12 16 O C O line positions. J Mol Spectrosc 2004;228:329–54. [14] Miller CE, Montgomery MA, Onorato RM, Johnstone C, McNicholas TP, Kovaric B, et al. Near infrared spectroscopy of carbon dioxide II. 16 13 16 O C O and 16O13C18O line positions. J Mol Spectrosc 2004;228: 355–74. [15] Garnache A, Liu A, Cerruti L, Campargue A. Intracavity laser absorption spectroscopy with a vertical external cavity surface emitting laser at 2:3 μm: application to water and carbon dioxide. Chem Phys Lett 2005;416:22–7. [16] Majcherova Z, Macko P, Romanini D, Perevalov VI, Tashkun SA, Teffo JL, et al. High-sensitivity CW-cavity ringdown spectroscopy of 12 16 C O2 near 1:5 μm. J Mol Spectrosc 2005;230:1–21. [17] Ding Y, Campargue A, Bertseva E, Tashkun SA, Perevalov VI. Highly sensitive absorption spectroscopy of carbon dioxide by ICLASVECSEL between 8800 and 9530 cm  1. J Mol Spectrosc 2005;231: 117–23. [18] Toth RA, Brown LR, Miller CE, Devy VM, Benner DC. Line strengths of 12C16O2: 4550–7000 cm  1. J Mol Spectrosc 2006;239:221–42. [19] Perevalov BV, Kassi S, Romanini D, Perevalov VI, Tashkun SA, Campargue A. CW-cavity ringdown spectroscopy of carbon dioxide isotopologues near 1:5 μm. J Mol Spectrosc 2006;238:241–5. [20] Toth RA, Miller CE, Brown LR, Devy VM, Benner DC. Line positions and strengths of 16O12C18O, 18O12C18O and 17O12C18O between 2200 and 7000 cm  1. J Mol Spectrosc 2007;243:43–61. [21] Toth RA, Miller CE, Brown LR, Devy VM, Benner DC. Line strengths of 16O13C16O, 16O13C18O, 16O13C17O and 18O13C18O between 2200 and 6800 cm  1. J Mol Spectrosc 2008;251:64–89. [22] Wang L, Perevalov VI, Tashkun SA, Song KF, Hu SM. Fourier transform spectroscopy of 12C18O2 and 16O12C18O in the 3800– 8500 cm  1 region and the global modeling of the absorption spectrum of 12C18O2. J Mol Spectrosc 2008;247:64–75. [23] Perevalov BV, Perevalov VI, Campargue A. A (nearly) complete experimental linelist for 13C16O2, 16O13C18O, 16O13C17O, 13C18O2 and 17 13 18 O C O by high-sensitivity CW-CRDS spectroscopy between 5851 and 7045 cm  1. J Quant Spectrosc Radiat Transfer 2008;109: 2371–462. [24] Perevalov BV, Kassi S, Perevalov VI, Tashkun SA, Campargue A. High sensitivity CW-CRDS spectroscopy of 12C16O2, 16O12C17O and 16O12C18O between 5851 and 7045 cm  1: line positions analysis and critical review of the current data bases. J Mol Spectrosc 2008;252:143–59. [25] Kassi S, Song KF, Campargue A. High sensitivity CW-cavity ringdown spectroscopy of 12CO2 near 1:35 μm (II): line positions. J Quant Spectrosc Radiat Transfer 2009;110:1801–14. [26] Campargue A, Song KF, Mouton N, Perevalov VI, Kassi S. High sensitivity CW-cavity ring down spectroscopy of five 13CO2 isotopologues of carbon dioxide in the 1:26–1:44 μm region (I): line positions. J Quant Spectrosc Radiat Transfer 2010;111:659–74. [27] Song KF, Kassi S, Tashkun SA, Perevalov VI, Campargue A. High sensitivity CW-cavity ring-down spectroscopy of 12CO2 near 1:35 μm (II): new observations and line intensities modeling. J Quant Spectrosc Radiat Transfer 2010;111:332–44. [28] Song KF, Lu Y, Tan Y, Gao B, Liu AW, Hu SM. High sensitivity cavitycavity ring-down spectroscopy of CO2 overtone bands near 790 μm. J Quant Spectrosc Radiat Transfer 2011;112:761–2. [29] Vaernewijck AGE, Kassi S, Herman M. 17O12C17O and 18O12C17O overtone spectroscopy in the 1:64 μm region. Chem Phys Lett 2011;514:29–31.

[30] Lyulin OM, Karlovets EV, Jacquemart D, Lu Y, Liu AW, Perevalov VI. Infrared spectroscopy of 17O- and 18O-enriched carbon dioxide in the 1700–8300 cm  1 wavenumber region. J Quant Spectrosc Radiat Transfer 2012;113:2167–81. [31] Karlovets EV, Campargue A, Mondelain D, Beguier S, Kassi S, Tashkun SA, et al. High sensitivity cavity ring down spectroscopy of 18O enriched carbon dioxide between 5850 and 7000 cm  1: I. Analysis and theoretical modeling of the 16O12C18O spectrum. J Quant Spectrosc Radiat Transfer 2013;130:116–33. [32] Borkov YG, Jacquemart D, Lyulin OM, Tashkun SA, Perevalov VI. Infrared spectroscopy of 17O- and 18O-enriched carbon dioxide: line positions and intensities in the 3200–4700 cm  1 region. Global modeling of the line positions of 16O12C17O and 17O12C17O. J Quant Spectrosc Radiat Transfer 2014;137:57–76. [33] Karlovets EV, Campargue A, Mondelain D, Kassi S, Tashkun SA, Perevalov VI. High sensitivity cavity ring down spectroscopy of 18O enriched carbon dioxide between 5850 and 7000 cm  1: II. Analysis and theoretical modeling of the 12C18O2, 13C18O2 and 16O13C18O spectra. J Quant Spectrosc Radiat Transfer 2014;136:71–88. [34] Karlovets EV, Campargue A, Mondelain D, Kassi S, Tashkun SA, Perevalov VI. High sensitivity cavity ring down spectroscopy of 18O enriched carbon dioxide between 5850 and 7000 cm  1: III. Analysis and theoretical modeling of the 12C17O2, 16O12C17O, 17O12C18O, 16 13 17 O C O and 17O13C18O spectra. J Quant Spectrosc Radiat Transfer 2014;136:89–107. [35] Karlovets EV, Kassi S, Tashkun SA, Perevalov VI, Campargue A. High sensitivity cavity ring down spectroscopy of carbon dioxide in the 1:19–1:26 μm region. J Quant Spectrosc Radiat Transfer 2014;144: 137–53. [36] Rothman LS, Gordon IE, Babikov Y, Barbe A, Benner DC, Bernath PF, et al. The HITRAN 2012 molecular spectroscopic database. J Quant Spectrosc Radiat Transfer 2013;130:4–50. [37] Tashkun SA, Perevalov VI. CDSD-4000: high-resolution, hightemperature carbon dioxide spectroscopic databank. J Quant Spectrosc Radiat Transfer 2011;112:1403–10. [38] Jacquinet-Husson N, Crepeau L, Armante R, Boutammine C, Chedin A, Scott NA, et al. The 2009 edition of the GEISA spectroscopic database. J Quant Spectrosc Radiat Transfer 2011;112:2395–445. [39] Huang X, Schwenke DW, Tashkun SA, Lee TJ. An isotopic-independent highly accurate potential energy surface for CO2 isotopologues and an initial 12C16O2 infrared line list. J Chem Phys 2012;136:124311. [40] Perevalov VI, Tashkun SA, Song KF, Campargue A. Global modeling of 16O12C17O and 16O12C18O absolute line intensities. J Mol Spectrosc 2010;263:183–5. [41] Perevalov BV, Kassi S, Romanini D, Perevalov VI, Tashkun SA, Campargue A. Global effective Hamiltonians of 16O13C17O and 16 13 18 O C O improved from CW-CRDS observations in the 5900– 7000 cm  1 region. J Mol Spectrosc 2007;241:90–100. [42] Perevalov BV, Campargue A, Gao B, Kassi S, Tashkun SA, Perevalov VI. New CW-CRDS measurements and global modeling of 12C16O2 absolute line intensities in the 1:6 μm region. J Mol Spectrosc 2008;252:190–7. [43] Perevalov BV, Deleporte T, Liu AW, Kassi S, Campargue A, VanderAuwera J, et al. Global modeling of 13C16O2 absolute line intensities from CW-CRDS and FTS measurements in the 1:6 μm and 2:0 μm regions. J Quant Spectosc Radiat Transfer 2008;109:2009–26. [44] Sánchez-Castllanos M, Lemus R, Carvajal M, Pérez-Bernal F. The potential energy surface of CO2 from an algebraic approach. Int J Quant Chem 2012;112:3498–507. [45] Sánchez-Castllanos M, Lemus R, Carvajal M, Pérez-Bernal F, Fernández JM. A study of the Raman spectrum of CO2 using an algebraic approach. Chem Phys Lett 2012;554:208–13. [46] Huang X, Freedman RS, Tashkun SA, Schwenke DW, Lee TJ. Semiempirical 12C16O2 IR line lists for simulations up to 1500 K and 20000 cm  1. J Quant Spectrosc Radiat Transfer 2013;130:134–46. [47] Aguir MBE, Perrin MY, Taine J. Variational calculation of energies of highly excited rovibrational states of 12C16O2. J Mol Spectrosc 2002;215:234–43. [48] Zúñiga J, Bastida A, Alacid M, Requena A. Variational calculations of rovibrational energies of CO2. J Mol Spectrosc 2001;205:62–72. [49] Lu Y, Xie D, Yan G. A potential energy surface for the electronic ground state of CO2. Int J Quant Chem 2000;78:269–80. [50] Wattson RB, Rothman LS. Direct numerical diagonalization: wave of the future. J Quant Spectrosc Radiat Transfer 1992;48:763–80. [51] Wattson RB, Rothman LS. Determination of vibrational energy levels and parallel band intensities of 12C16O2 by direct numerical diagonalization. J Mol Spectrosc 1986;119:83–100. [52] Lacy M. The anharmonic force field of carbon dioxide. Mol Phys 1982;45:253–8.

J. Cerezo et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 147 (2014) 233–251

[53] Zúñiga J, Alacid M, Bastida A, Carvajal FJ, Requena A. Determination of a potential energy surface for CO2 using generalized internal vibrational coordinates. J Mol Spectrosc 1999;195:137–46. [54] Zúñiga J, Bastida A, Alacid M, Requena A. Global potential energy surfaces for the CO2 and CS2 molecules. Chem Phys Lett 1999;313: 670–8. [55] Mardis KL, Sibert EL. The effectiveness of newton's method for improving ab initio force fields with application of CO2 and H2CO. J Mol Spectrosc 1998;187:167–78. [56] Martin JML, Taylor PR, Lee TJ. Accurate ab initio quartic force fields for the N2O and CO2 molecules. Chem Phys Lett 1993;205:535–42. [57] Rakauskas RJ, Sulskus JK, Pivovarn SMZVA. The ab initio surfaces of þ the potential energy and dipole moment of 1 Σ g CO2. Stretching vibrational states. Chem Phys Lett 1989;183:381–6. [58] Romanowski H, Gerber RB, Ratner MA. The anharmonic stretchingbending potential of CO2 from inversion of spectroscopic data. J Chem Phys 1988;88:6757–87. [59] Murrell JN, Guo H. Potential-energy functions for the ground states of CO2, CS2, and OCS, and dynamical calculations on the reaction þ þ Oð1 DÞ þ CSð1 Σ Þ-Sð1 DÞ þ COð1 Σ Þ. J Chem Soc Faraday Trans 1987;83:683–92. [60] Chedin A, Teffo JL. The carbon dioxide molecule: a new derivation of the potential, spectroscopic and molecular constants. J Mol Spectrosc 1984;107:333–42. [61] Suzuki I. Dipole moment functions of carbon dioxide and nitrous oxide. J Mol Spectrosc 1980;80:12–22. [62] Chedin A. The carbon dioxide molecule. Potential, spectroscopic, and molecular constants from its infrared spectrum. J Mol Spectrosc 1979;76:430–91. [63] Zúñiga J, Bastida A, Requena A. Optimization of vibrational coordinates. J Chem Soc Faraday Trans 1997;93:1681–90. [64] Zúñiga J, Bastida A, Alacid M, Requena A. Optimal generalized internal vibrational coordinates for symmetrical lineal triatomic molecules. Chem Phys Lett 1998;298:36–42. [65] Zúñiga J, Alacid M, Bastida A, Carvajal FJ, Requena A. Determination of highly excited rovibrational states for N2O using generalized internal coordinates. J Chem Phys 1999;110:6339–52. [66] Zúñiga J, Bastida A, Requena A. Optimal generalized internal vibrational coordinates and potential energy surface for the ground electronic state of SO2. J Chem Phys 2001;115:139–48. [67] Yoshida N, Vasilev M, Ghosh P, Abe O, Yamada K, Morimoto M. Precision and long-term stability of clumped-isotope analysis of CO2 using a small-sector isotope ratio mass spectrometer. Rapid Commun Mass Spectrom 2013;27:207–15. [68] He B, Olack GA, Colman AS. Pressure baseline correction and highprecision CO2 clumped-isotope (Δ47) measurements in bellows and micro-volume modes. Rapid Commun Mass Spectrom 2012;26: 2837–53. [69] Eagle RA, Schuble EA, Tripati AK, Tütken T, Hulbert RC, Eiler JM. Body temperatures of modern and extinct vertebrates from 13C–18O bond abundances in bioapatite. Proc Natl Acad Sci 2010;107: 10377–82. [70] Yeung LY, Affek HP, Hoag KJ, Guo W, Wiegel AA, Atlas EL, et al. Large and unexpected enrichment in stratospheric 16O13C18O and its meridional variation. Proc Natl Acad Sci 2009;106:11496–501. [71] Affek HP, Xu X, Eiler JM. Seasonal and diurnal variations of 13 18 16 C O O in air: initial observations from Pasadena, ca. Geochim Cosmochim Acta 2007;71:5033–43. [72] Affek HP, Eiler JM. Abundance of mass 47 CO2 in urban air, car exhaust, and human breath. Geochim Cosmochim Acta 2006;70: 1–12. [73] Eiler JM, Schauble EA. 18O13C16O in earth's atmosphere. Geochim Cosmochim Acta 2004;68:4767–77. [74] Wang Z, Schauble EA, Eiler JM. Equilibrium thermodynamics of multiply substituted isotopologues of molecular gases. Geochim Cosmochim Acta 2004;68:4779–97. [75] Zúñiga J, Bastida A, Alacid M, Requena A. Excited vibrational states and potential energy function for OCS determined using generalized internal coordinates. J Chem Phys 2000;113:5695–704. [76] Zúñiga J, Picón JAG, Bastida A, Requena A. A spectroscopic potential energy surface for FCN. J Quant Spectrosc Radiat Transfer 2012;113: 1155–69.

251

[77] Rothman LS, Hawkins RL, Wattson RB, Gamache RR. Energy levels, intensities, and linewidths of atmospheric carbon dioxide bands. J Quant Spectrosc Radiat Transfer 1992;48:537–66. [78] Courtoy CP. Spectres de vibration-rotation de molecules simples diatomiques ou polyatomiques avec long parcours d'absorption. 1 XII. Le spectre de C12O16 et les constantes 2 entre 3500 et 8000 cm moleculaires de cette molecule. Can J Phys 1957;35:608–48. [79] Courtoy CP. Spectre infrarouge a grande dispersion et constantes moleculaires du CO2. Ann Soc Sci Brux Ser I: Math Astron Phys 1959;73:5. [80] Graner G, Rossetti C, Bailly D. The carbon dioxide molecule. A test case for the re and rm structures. Mol Phys 1986;58:627–36. [81] Requena A, Bastida A, Zúñiga J. Curvilinear Jacobi and Radau normal coordinates for linear triatomic molecules. Application to CO2. Chem Phys 1993;175:255–64. [82] Bastida A, Requena A, Zúñiga J. Generalized hyperspherical coordinates for molecular vibrations. J Phys Chem 1993;97:5831–5. [83] Zúñiga J, Bastida A, Alacid M, Requena A. Hyperspherical kinematic vibrational coordinates for linear triatomic molecules. J Phys Chem 1995;99:11051–60. [84] Zúñiga J, Bastida A, Requena A. Theoretical vibrational terms and rotational constants for the 15N substituted isotopologues of N2O calculated using normal hyperspherical coordinates. J Quant Spectrosc Radiat Transfer 2012;113:26–46. [85] Laraia A, Gamache RR, Lamouroux J, Gordon IE, Rothman LS. Total internal partition sums to support planetary remote sensing. Icarus 2011;215:391–400. [86] Herzberg G. Molecular spectra and molecular structure: II. Infrared and raman spectra of polyatomic molecules. New York: Van Nostrand-Reinhold; 1966. [87] McDowell RS. Rotational partition functions for linear molecules. J Chem Phys 1988;88:356–61. [88] Osipov VM. Partition sums and dissociation energy for 12C16O2 at high temperatures. Mol Phys 2004;102:16–7. [89] Fischer J, Gamache RR, Goldman A, Rothman LS, Perrin A. Total internal partition sums for molecular species in the 2000 edition of the HITRAN database. J Quant Spectrosc Radiat Transfer 2003;82: 401–12. [90] Gray LD, Young AT. Relative intensity calculations for carbon dioxide—IV. Calculations of the partition function for isotopes of CO2. J Quant Spectrosc Radiat Transfer 1969;9:569–89. [91] Came RE, Eiler JM, Veizer J, Azmy K, Brand U, Weidman CR. Coupling of surface temperatures and atmospheric CO2 concentrations during the paleozoic era. Nature 2007;449:198–201. [92] Huntington KW, Eiler JM, Affek HP, Guo W, Bonifacie M, Yeung LY, et al. Methods and limitations of clumped CO2 isotope (Δ47) analysis by gas. Source isotope ratio mass spectrometry. J Mass Spectrom 2008;44:1318–29. [93] Dennis KJ, Affek HP, Passey BH, Schrag DP, Eiler JM. Defining and absolute reference frame for clumped isotope studies of CO2. Geochim Cosmochim Acta 2011;75:7117–31. [94] Cao X, Liu Y. Theoretical estimation of the equilibrium distribution of clumped isotopes in nature. Geochim Cosmochim Acta 2012;77: 292–303. [95] Webb MA, Miller TF. Position-specific and clumped stable isotope studies: comparison of the Urey and path-integral approaches for carbon dioxide, nitrous oxide, methane, and propane. J Phys Chem A 2014;118:467–74. [96] Craig H. Isotopic standards for carbon and oxygen and corrections factors for mass-spectrometric analysis of carbon dioxide. Geochim Cosmochim Acta 1957;12:133–49. [97] IUPAC. Isotopic compositions of the elements 2009 (IUPAC technical report). Pure Appl Chem 2011;83:397–410. [98] Urey HC. The thermodynamic properties of isotopic substances. J Chem Soc (London) 1947:562–81. [99] Eiler JM, Clog M, Magyar P, Piasecki A, Sessions A, Stolper D, et al. A high-resolution gas–source isotope ratio mass spectrometer. Int J Mass Spectrom 2013;335:45–56. [100] Grauel AL, Schmid TW, Hu B, Bergami C, Capotondi L, Zhou L, et al. Calibration and application of the clumped isotope thermometer to foraminifera for high-resolution climate reconstructions. Geochim Cosmochim Acta 2013;108:125–40.