Chemical Physics Letters 565 (2013) 5–11

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Chemical Physics Letters

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New dipole moment surfaces of methane

Andrei V. Nikitin a,b,⇑, Michael Rey b, Vladimir G. Tyuterev b

a Laboratory of Theoretical Spectroscopy, V.E. Zuev Institute of Atmospheric Optics, SB RAS, 1 Academician Zuev Square, 634021 Tomsk, Russiab Groupe de Spectrométrie Moléculaire et Atmosphérique, UMR CNRS 6089, Université de Reims, U.F.R. Sciences, B.P. 1039, 51687 Reims Cedex 2, France

a r t i c l e i n f o

Article history:Received 12 December 2012In final form 12 February 2013Available online 26 February 2013

0009-2614/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.cplett.2013.02.022

⇑ Corresponding author at: Laboratory of TheoretInstitute of Atmospheric Optics, SB RAS, 1 AcademicianRussia.

E-mail address: avn@lts.iao.ru (A.V. Nikitin).

a b s t r a c t

New dipole moment surfaces (DMS) of methane are constructed using extended ab initio CCSD(T) calcu-lations at 19882 nuclear configurations. The DMS analytical representation is determined through anexpansion in symmetry adapted products of internal nonlinear coordinates involving 967 parametersup to the 6th order. Integrated intensities of seven lower polyads up to J = 30 for 12CH4 and 13CH4 arein a good agreement with the HITRAN 2008 database, and with other available experimental data.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Methane is a high symmetry hydrocarbon, which is importantin numerous fields of science. Acting as a greenhouse gas of theearth atmosphere, CH4 is also a significant constituent of variousplanetary atmospheres, like those of the Giant Planets [1,2] (Jupi-ter, Saturn, Uranus and Neptune) and of Titan [3,4] (Saturn’s mainsatellite). The knowledge of the CH4 opacity is also very importantfor the modeling of brown dwarfs and for other astrophysicalapplications. As the remote sensing via infrared spectroscopy isgenerally the best diagnostic tool to study CH4 in these environ-ments, it appears essential to be able to model its absorption veryprecisely. The analysis of highly excited vibration–rotation energylevels and transitions of the methane molecule is a difficult prob-lem due to complex structures of vibrational polyads, numerousresonance couplings and high dimensionality of the calculationmodels [5–8].

Knowledge of the molecular dipole moment surface (DMS)would help resolving many of related issues. High temperaturemeasurements [9,10] of methane spectra were not yet fully ana-lyzed as this requires much more accurate theoretical predictionsof ro-vibrational spectra. A certain progress in the dipole momentcalculation from ab initio theory has been achieved [11–14]. Signo-rell et al. [15] have determined six parameters in an approximationof the methane DMS containing four valence depending and twoangle depending terms. This DMS form has been applied by Marqu-ardt and Quack [13] for the calculation of some integrated cross sec-tions of deuterated methane isotopologues. Warmbier et al. [16]calculated an ab initio methane DMS and methane spectra atT = 1000 K using MULTIMODE program [17]. Recently Cassam-Chenaiand Lievin [18] computed a third order dipole moment normal

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ical Spectroscopy, V.E. ZuevZuev Square, 634021 Tomsk,

mode expansion and calculated line intensities for rotational tran-sitions of methane within the vibrational ground state. However aquantitatively accurate modeling of intensities of excited ro-vibrational states of methane still remains a difficult problem tobe solved. Several issues specific to the methane molecule have tobe addressed in the context of current spectroscopic applications:

(a) The first challenge is to extend accurate ab initio calculationsto sufficiently dense grids of nuclear configurations and toobtain a precise fit of ab initio points with an appropriatereliable analytical DMS representation accounting formolecular symmetry properties;

(b) Another one is to assure the convergence of calculations ofro-vibrational levels and eigenfunctions from molecularpotential energy surface (PES) as well as of transitionmoments using quantum mechanical, variationally-basedapproaches up to higher energy ranges.

This Letter is a part of a long term effort to extend spectroscopicdata analyses and calculations for isotopologues of the methanemolecule in the infrared range [19–21]. The aim of this Letter isan accurate calculation of the methane DMS in internal coordinatesfor a large extent of nuclear configurations. In order to check thevalidity of our DMS the integrated intensities of seven lower poly-ads were calculated, and compared with experimental databases.

2. Electronic structure calculations and determination of abinitio DMS

In order to obtain accurate calculations of intensities of vibra-tion–rotation transitions from theoretical DMS surfaces whichwould be useful for spectroscopic analyses, it is necessary to com-bine high level ab initio methods with sufficiently large basis sets inelectronic structure calculations. The coupled cluster approachincluding single and double excitations and the perturbative

6 A.V. Nikitin et al. / Chemical Physics Letters 565 (2013) 5–11

treatment of triple excitations CCSD(T) [22] was employed in thisLetter. The same method was used in our previous study [20].The validity of the CCSD(T) method for the methane moleculehas been confirmed by Schwenke and Partridge [23,24] via thecomparison with the internally contracted averaged pair functional(icACPF) method. We used well established correlation consistentpolarized core valence basis sets cc-pCVQZ of Dunning and co-workers [25,26]. The relativistic effects are taken into account viaone-electron Douglas–Kroll correction [27]. All the ab initio calcu-lations were carried out using the MOLPRO program package [28]version 2009.1. All dipole moments were computed as the deriva-tive of the energy with respect to the weak external uniform elec-tric field using the finite difference scheme with the field variationof 0.0001 a.u. around at the zero field strength. The dependence offinal dipole moment values on the external field was not significantin the range 0.001–0.00005 a.u. of the field variations. As a firststep of the DMS construction, the dipole moment values in theelectronic ground state were calculated with four basis sets onthe grid of 19882 nuclear configurations described in [20]. The dis-tribution of the density of the DMS geometrical configurationswhich correspond to the same grid choice as for the PES has beengiven in Fig. 1 of [20].

The best known equilibrium geometry parameter valuere = 1.08601 Å calculated in [20] was applied in this Letter. As thisparameter is very important for accurate descriptions of the rota-tional spectra, the re was held fixed to the same value for all calcu-lations involving different basis sets. So-called Molecular-Bond(MB) representation [29,13,30,31] for the dipole moment in inter-nal coordinates was used:

~l ¼X

i¼1���4lið~r1;~r2;~r3;~r4Þ~ei; ð1Þ

where four functions lið~r1;~r2;~r3;~r4Þ depend on internal coordinatesl1(r1,r2,r3,r4,cos(q12),cos(q13),cos(q14),cos(q23),cos(q24),cos(q34)),and~ei are the unit vectors along the C–Hi bonds. Due to the symme-try considerations three functions lið~r1;~r2;~r3;~r4Þ for i = 2,3,4 can beobtained from l1 by cyclic permutation (1234), for instancel2ð~r2;~r3;~r4;~r1Þ ¼ l1ð~r1;~r2;~r3;~r4Þ. Such representation of ~l corre-sponds to the body fixed frame which depends on instantaneouspositions of the nuclei. In order to understand the symmetry prop-erties of the function l1 let us choose the z axis of the orthogonalmolecular fixed frame (C,xyz) in the direction of the C–H1 bond.The y axis is chosen so that the plane yz contains the carbon atomand the protons H1, H2. The same nuclear orientations were usedduring our calculations with MOLPRO program [28]. A permutationof the atoms 2,3,4 does not change z-component of the dipole mo-ment. This means that the l1 function has to be invariant undersuch permutations. On the other hand the l1 function is not invari-ant with respect to the permutations involving the H1 proton be-cause the position of H1 is distinguished by the choice of thedirection ~e1 in Eq. (1). With this choice of the body fixed framethe function l1 in Eq. (1) has the same symmetry properties as aPES for AB3C type molecules (A1 representation of the C3v pointgroup) though the full dipole ~l transforms according to the F2 rep-resentation of the full point group Td. Using the algorithm of ourprevious work [32–34] a full set of 967 (or 156) totally symmetricirreducible tensors up to the 6th (or 4th) order expansion was con-structed with MIRS computational suite of codes [35]. Every tensorwas represented as the sum of symmetrized products

Rpi ¼ Sp1

RH;A1Sp2

RH;E

� �Sp3

RH1 ;A1

� �Sp4

QH;A1Sp5

QH;E

� �Sp6

TH;E

� �� �A1ð2Þ

of symmetry adapted C3v coordinates Si. Here SRH1 stands for the axiscorresponding to the selected hydrogen atom i = 1, and SRH for otherhydrogen atoms i = 2,3,4 whereas the notations SQH, STH are used forangular coordinates (see [32] for more detail). The upper index

p = p1 + p2 + p3 + p4 + p5 + p6 is the total power of the term. The fol-lowing one-dimensional elementary functions were used

uðrÞ ¼ 1� exp½�1:9ðri � reÞ�; uðqÞ ¼ cosðqÞ � cosðqeÞ

in order to construct the symmetry adapted coordinates SRH, SRH1 ,SQH, STH. The use of Morse elementary functions in the DMS expan-sion of sixth order (p = 6) allows fitting analytical surfaces to ab ini-tio points a little better than the use of linear functions /0(r) = ri � re.The standard definition of the direct product of irreducible tensors[36,37] with the sub-indices corresponding to irreducible represen-tations of the C3v point group was applied. The first step corre-sponds to the construction of the symmetrized powers of Si, andthe second step to the coupling of the symmetrized powers of dif-ferent symmetrized coordinates in irreducible balanced treesaccording to the algorithm of [38]. A set of all possible trees ofthe totally symmetric A1 representation gives a final set of the 9Dexpansion terms for the l1 function. This DMS component was fi-nally developed in power series of irreducible tensors (2) dependingon internal coordinates

l1ðr1; r2; r3; r4; q12; q13; q14; q23; q24; q34Þ

¼X

i

KiRpi ðr1; r2; r3; r4; q12; q13; q14; q23; q24; q34Þ ð3Þ

Note that in our recent work [32] on CH3F calculations exactly thesame form has been used for the PES but not for the DMS l1 com-ponent as here. In the similar case of a pyramidal AB3 type molecule[31] (symmetry group C3v) the dipole moment function l1 has beenconstructed as a function invariant under a certain part of groupoperations only. This suggests a general rule that the l1 componentof the DMS representation of Eq. (1) should have a behavior corre-sponding to a lower symmetry group than the full dipole momentobtained as a sum of all components. However the advantage ofthe form (1) is that other DMS components could be easily obtainedfrom the l1 component by hydrogen permutations.

These symmetry properties of the DMS were tested in the VTZbasis set involving independent computations of all components.Six calculations were made for each geometry with the VTZ basis,the positive and negative external field variations being applied inthe x, y and z directions of molecular fixed axes. Three body fixedcomponents of the VTZ DMS surface were calculated using MOLPRO

program and then fitted independently. The standard deviationsof the DMS fits were very closed in all cases. The resulting DMSparameters for x, y components gave nearly the same values asthose re-computed from the z component. For this reason forhigher basis sets we were able to use two external field calcula-tions for every geometry but not six calculations as in Ref. [31].The possibility to find x and y Cartesian DMS components fromz components becomes obvious in the instantaneous configura-tion where any three of four vectors~ei are orthogonal. For the ba-sis sets CVQZ, VQZ_F12, VQZ_F12_Rel (first order relativisticcorrection was taken in account) only positive and negative exter-nal field in z direction were applied for each geometry and onlyparameters of z component for CVQZ, VQZ_F12, VQZ_F12_RelDMS were fitted. Two other x, y components of these DMS weresystematically recalculated from the z component because in gen-eral case all three Cartesian DMS components in the molecularfixed frame are necessary for rotating the DMS to the Eckartembedding [39]. It is necessary to stress that the number of ab ini-tio geometries was considerably larger than the number of fittedDMS parameters. The grid of 19882 points in the nuclear config-urations space has been already built in our previous work onmethane PES calculations [20]. As a first step we fitted our ab ini-tio dipole moment z component corresponding to the ~e1 vector inEq. (1) by using the analytical symmetry adapted representation(3) and the weight function (depending on energy E in cm�1) em-ployed by Schwenke and Partridge in Ref. [23]

Table 1Statistics for the fit of ab initio points with analytical DMS representations fordifferent basis set.

Ab initio basis set DMS Order St.Dev.Internal coord.a.u./C3v

St.Dev.Normal coord.a.u./Td

VTZ 4 0.00013 0.0001726 0.000033 0.000026

CVQZ 4 0.000068 0.0001256 0.0000129 0.0000160

VQZ_F12 6 0.0000131 0.0000155VQZ_F12 + DK1 6 0.0000128 0.0000157

A.V. Nikitin et al. / Chemical Physics Letters 565 (2013) 5–11 7

wðEÞ ¼ tanhð�0:0005ðE� 12000Þ þ 1:002002002Þ2:002002002

: ð4Þ

In total 816 parameters (of 967 belonging to the order 6 expan-sion) were statistically well determined in this fit on the entire gridof all 19882 ab initio points with the weighted standard deviationSt.Dev. = 0.0000129 a.u. (dipole moment atomic units [28]). Theroot-mean-squares (RMS) deviation of the ab initio DMS fit was con-siderably lower than that of Ref. [16]. Figure 1 shows the CVQZ DMSerror distribution of the final fit. The errors (defined as ab initio di-pole value minus fitted dipole value) are quite small up to energies�12000–13000 cm�1. A larger scatter of points above this rangeoccurs because the weighting function (4) rapidly de-emphasizesenergies above this threshold [23]. The statistics of the DMS fit forvarious basis sets and for 4th and 6th expansion orders in internalcoordinates is given in the third column of Table 1. Other compo-nents of the dipole moment as a function of nuclear displacementswere computed from the z-component using the permutation sym-metry properties of our representation (1). As explained above thisgave the same results as an independent fit of all three Eckart framedipole moment components to the ab initio points.

As already mentioned above, a simplified analytical form of DMSwith only six parameters has been applied in [13]. In the work [30]the radial part of DMS of AB4 type molecule has been fitted in twodifferent models. The first model (Cartesian model) which is relateddirectly to the transition selection rules expands the Cartesian com-ponents employing symmetrized basis sets. This formula reads

~Mðr1; r2; r3; r4Þ ¼X

n¼x;y;z

Mnðr1; r2; r3; r4Þ~en; ð5Þ

where Mnðr1; r2; r3; r4Þ; n ¼ x; y; z have been constructed as a sumof some symmetric functions [40,41]. The second model imple-mented in [30] is the bond dipole model in form:

~Mðr1; r2; r3; r4Þ ¼ M1ðr1; r2; r3; r4Þ~e1 þM2ðr2; r1; r3; r4Þ~e2

þM3ðr3; r1; r2; r4Þ~e3 þM4ðr4; r1; r2; r3Þ~e4; ð6Þ

Figure 1. The errors of the methane DMS fit to 19882 ab i

where Mi is the bond dipole function for the ith bond whichtakes the same functional forms for all four bonds due tosymmetry. In Ref. [30] it has been noted that the expansions inthe bond dipole and Cartesian models are related to each otherby: ~e1 ¼ ð~ex þ~ey þ~ezÞ=

ffiffiffi3p

; ~e2 ¼ ð�~ex �~ey þ~ezÞ=ffiffiffi3p

; ~e3 ¼ ð�~ex þ~ey

�~ezÞ=ffiffiffi3p

; ~e4 ¼ ð~ex �~ey �~ezÞ=ffiffiffi3p

, and two forms are equivalent forthe radial dipole moment. On the other hand in general case(if angles change) three vectors ~ex ¼ ð~e1 �~e2 þ~e3 �~e4Þ=2;~ey ¼ ð~e1 �~e2 �~e3 þ~e4Þ=2; ~ez ¼ ð~e1 þ~e2 �~e3 �~e4Þ=2 are not orthog-onal, and the form (6) is preferable because the symmetry proper-ties of form (5) are more difficult to evidence. Our form (1), (3) isapplicable for a more general case involving angle variations.

3. Intensity calculations

In order to check our DMS, the ro-vibrational intensities for the12CH4 and 13CH4 methane isotopologues were calculated in thespectral range 0–10000 cm�1. In all calculations of ro-vibrationalenergies and wave functions we have used our recent methanePES [20] which is hereafter referred to as NRT PES. The ro-vibrational model is based on the complete normal-mode nuclearHamiltonian in the Eckart frame [39]. The vibration–rotation Ham-iltonian is transformed systematically to a full symmetrized form

nitio values computed with the CVQZ/CCSD(T) ansatz.

8 A.V. Nikitin et al. / Chemical Physics Letters 565 (2013) 5–11

using irreducible tensor operators (ITO) as described in [21,42].This technique allows casting each term of the Hamiltonian expan-sion in the normal mode ITO form whatever the order of the devel-opment. The six order Hamiltonian using the reduction–truncationscheme H10/6 (in notations of [21]) has been calculated in our pre-vious work [21]. In order to make ro-vibrational calculations up toJ = 30 feasible, only medium-size basis set F6 (in notations of [21])of the nuclear motion was applied in the present Letter. With thissize the ro-vibrational line positions are not yet fully converged atthe high edge of our very large wave number range. But a furtherstudy (in progress [43]) of the nuclear basis convergence allowsconcluding that this error in individual line positions should notconsiderably affect integrated intensities of methane polyads.

Because all our ro-vibrational basis functions were developed interms of normal modes, the ab initio dipole moment values werealso refitted in the same normal mode representation. The DMSin normal mode ITO representation (symmetry type F2 of the fullmethane point group Td [37]) was developed using the same tech-nique as for the Hamiltonian [21,35]. This normal mode DMSexpansion of the order 4 (or 6) contains 102 (or 680) parameters.The fit statistics are given in the last column of Table 1. Then nor-mal coordinates/momenta were converted to symmetrized combi-nations of creation, annihilation operators [21,42] that allowsusing standard formulas for effective dipole transition momentmatrix elements [37]. Because the DMS defined in the molecular-fixed frame as a function of internal coordinates does not involveangular momentum operators, only a computation of 9C symbols[44] was sufficient for dipole matrix element calculations (contraryto the case of effective transition moments which require 12C sym-bols in a general case). All calculations of energies, wave functions,matrix elements and line intensities were carried out with theMIRS computational code, the recent version of which [35] hasbeen extended for global vibration–rotation predictions using abinitio surfaces. The details of individual line intensity calculationsand extended comparisons with experimental spectra will be re-ported elsewhere [43]. Estimations for the related calculation er-rors are briefly discussed in Section 5.

Figure 2. Comparison of spectra calculated from ab initio DMS using four basis sets withand vibration–rotation basis functions.

4. Integrated intensities for seven lower methane polyadscalculated from ab initio DMS

Four normal mode frequencies of CH4 exhibit an approximaterelation of stretching and bending frequencies with x1 �x3 -� 2x2 � 2x4 resulting in vibrational levels being grouped intopolyads with levels of similar energy [5]. Though we do not usehere any polyad approximations specific to effective models it isconvenient to use the spectroscopic terminology in terms of polyadspectral ranges for the comparison with experimental data. Anexample of the comparison of predicted spectra using PES of [20]and our various ab initio DMS with empirically based HITRAN lin-elist of 12CH4 in the Octad range is given in Figure 2. This polyadsystem contains eight bands with 24 vibrational sub-bunds. Glob-ally all four calculations are in a very good agreement with exper-imental intensities. In order to compare integrated intensities weextended line calculations up to rotational quantum numberJ = 30. In Tables 2 and 3 integrated intensities in the ranges of low-er polyads are collected at 296 K. The integrated intensity in arange R is the sum S ¼

PSiðminÞi2R Si of line intensities Si for all calcu-

lated or observed transitions it this range. The cutoff Si(-min) = 1.E�28 was applied for weak bands in the first rangebelow 1000 cm�1 and the cutoff Si(min) = 3.E�26 was applied forall other ranges. In all cases we use the HITRAN intensity unitscm�1/(molecule cm�2).The HITRAN integrated polyad intensitiesof 12CH4 (13CH4) were normalized by dividing on the natural abun-dance factor 0.988274 (0.0111031). For four lower polyads (seeranges in Table 2): corresponding to the ground state (G.S.), Dyad,Pentad, Octad and for Triacontad (the polyad in the range 7900–9300 cm�1) only HITRAN [7] database was used in our comparison.In the range 0–1000 cm�1 both G.S. ? G.S. and Dyad ? Dyad tran-sitions give comparable contributions at room temperature. For theTetradecad (4900–6250 cm�1) of 12CH4 the number of HITRANlines in the range 5550–6238 cm�1 is not sufficiently complete.Four time more lines in this particular range is available in theempirical GOSAT line list [45]. For a reliable test the HITRAN12CH4 integrated intensity in the range 4900–5550 cm�1 was

HITRAN 2008 database. In all calculations the PES of [20] was used for line positions

Table 2Integrated band intensity predictions with various ab initio DMS surfaces up to J = 30 in the ranges of seven lower polyads of 12CH4.

Regioncm�1

Major polyad contributions at 296 K Common Sv factora VTZ VQZ_F12 VQZ_F12 + relativistic DK1 CVQZ Observed

0–1000 G.S. ? G.S. and Dyad ? Dyad E�23 5.25 5.08 5.08 5.25 5.36b

1000–2000 G.S. ? Dyad E�18 5.08 5.06 5.03 5.44 5.27b

2000–3400 G.S. ? Pentad E�17 1.26 1.18 1.19 1.16 1.14b

3400–4900 G.S. ? Octad E�19 9.40 9.33 9.35 9.23 9.20b

4900–6300 G.S. ? Tetradecad E�19 1.33 1.44 1.43 1.47 1.40c

6300–7900 G.S. ? Icosad E�20 4.30 4.18 4.17 4.17 4.09d

7900–9300 G.S. ? Triacontad E�20 2.08 2.32 2.32 2.46 2.55b

NRT PES [20] was used for energies and wave functions calculations.a Empirical databases were used for the intensity comparison in units cm�1/(molecule cm�2) normalized at 100% abundance.b HITRAN2008 [7].c HITRAN2008 [7] + GOSAT [45].d WKMC [46].

Table 3Integrated band intensity predictions with various ab initio DMS surfaces up to J = 30 in the ranges of six lower polyads of 13CH4.

Region cm�1 Major polyad contributions at 296 K Common Sv factora VTZ CVQZ HITRAN

0–1000 G.S. ? G.S. and Dyad ? Dyad E�23 5.56 5.52 5.041000–2000 G.S. ? Dyad E�18 4.98 5.34 5.052000–3400 G.S. ? Pentad E�17 1.27 1.17 1.113400–4900 G.S. ? Octad E�19 9.43 9.27 –4900–6300 G.S. ? Tetradecad E�19 1.24 1.38 –6300–7900 G.S. ? Icosad E�20 4.36 4.23 –

The HITRAN2008 [7] database is used for the intensity comparison normalized at 100% abundance.a Same notations and units as in Table 1 are used.

A.V. Nikitin et al. / Chemical Physics Letters 565 (2013) 5–11 9

added to the 12CH4 GOSAT line list integrated intensity in the5550–6238 cm�1 region. In the range of the Icosad (6300–7900 cm�1) the HITRAN 12CH4 integrated intensity value is veryclose to that of recent very accurate and complete WKMC 296 Kline list experimentally obtained from two-temperature laser mea-surements [46,47] and containing more weak lines. The integratedintensity of the WKMC line list was used for the final comparisonin the Icosad range. The relativistic effects are taken into accountvia one-electron Douglas–Kroll correction [27] that influencedquite weakly (less than 0.5%) the integrated polyad intensities.The relative errors of calculated integrated polyad intensities ver-sus observed integrated polyad intensities are shown in Figure 3.The polyad intensities calculated from the surfaces VQZ_F12 andCVQZ are very close to each other except for the case of pure

GS

Dyad

Pentad

Octad

Tetradecad

Icosad

Triacontad

-20 -16 -12 -8 -4 0 4 8 12

Errors in % (Icalc- Iobs)*100 /Iobs

CVQZ VQZ-F12 VTZ

Figure 3. Diagram of relative deviations between calculated and observedintegrated polyad intensities using three basis sets. Relative error of integralpolyad intensity (IC � IO)/IO in %.

rotational ‘forbidden’ transitions G.S. ? G.S. The spectra calculatedfrom the DMS expansions of orders 4 and 6 are very similar forlower polyads of CH4 except for recently assigned five-quantabands. For the 5v4 band [48] (at the lower edge of the Icosad) the4th order DMS significantly overestimates line intensities. Over-view of spectra in the region of 5v4 band is shown on the Figure 4.

5. Discussion and summary

The overall comparison of first-principles intensities computedfrom our new ab initio DMS with experimental data looks surpris-ingly good. The HITRAN database integrated intensities in theTetradecad range are certainly too low: we have much betteragreement by including more extensive experimental data recentlyobtained via GOSAT data [45]. In the Icosad range which is consid-ered as an extremely complicated for theoretical line-by-line anal-ysis (no full ro-vibrational assignment is available so far except forthe 5v4 band at the lower edge [48]) the agreement in terms ofintegrated intensities is excellent both with HITRAN and with re-cent very thoroughly measured experimental WKMC line list [46].

The fourth order DMS expansion was found not sufficient for 5-vibration-quanta bands in particular in the 1.56 lm ‘transparencywindow’ [6] which is crucially important for quantifying the meth-ane absorption in planetologic applications [3,4]. The medium-sizeF6 vibration basis set (notations of [21]) is not fully converged athigher polyads, this might affect line positions and the couplingwithin a polyad leading to some re-distribution within the polyads.As the interaction between polyads are quite weak, this should notaffect integrated intensities in Tables 2 and 3 and in Figure 3. Forthe integrated intensities the basis convergence error is estimatedas 1–3% except for the highest triacontad above 8000 cm�1wherethe errors could be somewhat more significant. Note that for thetriacontad range there are no any assignments available and thereis no discrimination between12CH4 and 13CH4 lines in the HI-TRAN2008 database. The normalization by dividing on the naturalabundance factor 0.988 should result then in �1% error the

Figure 4. Comparison of spectra calculated from two DMS of orders 4 and 6 with observed line lists WKMC_296K and HITRAN 2008.

10 A.V. Nikitin et al. / Chemical Physics Letters 565 (2013) 5–11

experimental integrated intensity for12CH4 in this range (last lineof Table 2) which is not significant for the comparison.

This Letter was focused essentially on accurate DMS calcula-tions and on the analytical surface model parameterization. Thenuclear motion calculation as well as the convergence issues forline positions and intensities will be presented in the forthcomingstudy [43]. According to more detailed study for vibration–rotationcalculations [43] the band centers could be converged in average to0.01–0.1 cm�1 up to 5000 cm�1 and to �0.5–1 cm�1 up to7000 cm�1 with the F(11) basis set. With the F(8) basis set a typicalbasis convergence error for the line positions of strong and med-ium lines (at J �8) with respect to the band center is estimatedas 0.15 cm�1 (or less) and as 1–3% for the line intensities, at leastup to the tetradecad range. These more extended calculations donot affect significantly the integrated polyad intensities given inTables 2 and 3.

We provide two surfaces VQZ_F12, CVQZ as the electronic Sup-plementary materials of this Letter. The DMS involving relativisticcorrections are also available, but we did not find an impact ofthese corrections very significant for intensities. More detailedcomparisons with recent experimental data [49,50] and analysesare planned in future together with the study of nuclear basis con-vergence effects [43]. From general considerations supported byFigure 3 the VTZ DMS would be less accurate. We observe thatthe CVQZ DMS gives intensity predictions somewhat closer toexperimental data, but more detailed spectroscopic work is neces-sary for final conclusions, in particularly concerning data assign-ments and analyses at high wave number ranges.

Acknowledgments

The supports of ANR (France) through the Grant ‘CH4@Titan’(Ref.: BLAN08-2_321467), of the contract LEFE-CHAT CNRS, andof Groupement de Recherche International SAMIA between CNRS(France), RFBR (Russia) and CAS (China) are acknowledged. Weacknowledge the support from IDRIS computer centre of CNRS,and CINES computer centre of France as well as the Clovis

computer centre Reims-Champagne-Ardenne. A.N. thanks the Uni-versity of Reims for the invitation to work on this project.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.cplett.2013.02.022.

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