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Journal of Molecular Spectroscopy 291 (2013) 33–47

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Journal of Molecular Spectroscopy

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Analysis of the rovibrational spectrum of 13CH4 in the Octad range

0022-2852/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jms.2013.06.003

⇑ Corresponding authors. Fax: +613 533 6669 (T. Carrington), Address: ETHZürich, Laboratorium für Physikalische Chemie, Wolfgang-Pauli-Strasse 10, CH-8093 Zürich, Switzerland. Fax: +41 44 632 10 21 (M. Quack).

E-mail addresses: [email protected] (T. Carrington Jr.), [email protected] (M. Quack).

Hans-Martin Niederer a, Xiao-Gang Wang b, Tucker Carrington Jr. b,⇑, Sieghard Albert aSigurd Bauerecker a,c, Vincent Boudon d, Martin Quack a,*a Laboratorium für Physikalische Chemie, ETH Zürich, CH-8093 Zürich, Switzerlandb Chemistry Department, Queen’s University, Kingston, Ontario K7L 3N6, Canadac Institut für Physikalische und Theoretische Chemie, Technische Universität Braunschweig, D-38106 Braunschweig, Germanyd Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Université de Bourgogne, 9 Av. A. Savary, BP 47870, F-21078 Dijon Cedex, France

a r t i c l e i n f o

Article history:Available online 26 June 2013

Keywords:High resolution spectroscopyResonanceFTIR spectroscopyInfraredIsotopesAtmospheric trace gasesMethane13CH4

a b s t r a c t

We have measured the infrared spectrum of methane 13CH4 from 1100 cm�1 (33 THz), below the funda-

mental range, to about 12000 cm�1 (360 THz) in the high overtone region at temperatures ranging from80 K to 300 K by high resolution Fourier transform infrared (FTIR) spectroscopy. With instrumental band-widths between 0.0027 cm�1 (80 MHz) and 0.01 cm�1 (300 MHz) this provides close to Doppler-limitedspectra, using the Zürich prototype spectrometer (ZP2001, Bruker 125HR) combined with a multipathcollisional cooling cell. Using perturbation theory and an accurate empirically adjusted potential we havecomputed ro-vibrational energy levels of 13CH4 and

12CH4 in the same energy range. Exploiting the syn-ergy between theory and experiment, we analyze here specifically the experimental spectra in the Octadrange (�3700–4700 cm�1, or 110 to 140 THz), using the theoretical results to guide the fitting of param-eters of a Dijon effective Hamiltonian theory. With the aid of the theoretical results it is possible to ana-lyse the Octad of 13CH4 with much less effort than without such information. In the end 1144 purelyexperimental line positions were fitted with root mean square deviations drms 6 2.6 � 10�3 cm�1 (5548data including theoretical results, with similar drms).

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Methane with its substituted derivatives is one of the mostimportant prototypical molecules in numerous fields of science:in structural chemistry (including the history of three dimensionalstructure [1–3], chirality, and its relation to parity violation [4–6]),in theoretical chemistry (e.g. the construction of accurate potentialhypersurfaces for polyatomic molecules [7–13] and quantumdynamical time–dependent wave packet propagation [14,15] aswell as nuclear spin symmetry conservation [16–18]) and in thephysics of many planetary or substellar objects (giant planets ofthe Solar System, Titan, exoplanets, brown dwarfs), which maycontain large amounts of methane in their atmospheres [19–34].Methane is also important as a major natural energy resource forcombustion and as a greenhouse gas in the Earth’s atmosphere[35]. Many of these fields require a highly accurate and detailedknowledge of rovibrational levels from high resolution spectros-copy over a wide energy range. Although the spectroscopy of themajor isotopomer 12CH4 has been well studied (see Refs. [36–43]

and literature cited therein), much less is known about the secondmost abundant natural isotopomer 13CH4. Recent investigations of13CH4 have begun to close this gap [44–47].

We have studied 13CH4 at high resolution providing an analysisup to the Pentad [44–46] and a number of more highly excitedvibrational levels at high accuracy [42]. An important next stepin the analysis of the spectra of 13CH4 is thus the analysis of the Oc-tad with a total of 24 vibrational sublevels (see the survey in Ta-ble 1, which provides also a definition of the Polyads Pn). It isthus the goal of the present work to provide a first analysis up tothe Octad.

It is difficult to analyze and fit an effective Hamiltonian to thetransitions of the 13CH4 Octad, because hundreds of parametersare required. To facilitate the analysis we use starting values forthe parameters that are determined from rovibrational energy lev-els of 12CH4 and 13CH4 computed with fourth order perturbationtheory. Using the methods outlined in Section 2 (see Refs. [48–51]) we compute rovibrational levels of both molecules as well as12CD4 [52], the latter with results from an effective hamiltoniannot being reported in detail here. For 12CH4 and 13CH4 the Dijon fit-ting procedure [39] is used to fit the level sets. These fits are fairlystraightforward because the assignments of the computed levelsare known and no levels are missing. From these fits we obtainenergy level shifts. Using these shifts and accurate 12CH4 levels,

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Table 1The vibrational polyads of methane.

Polyada Range (cm�1) Levels nb n(Cv)c

A1 A2 E F1 F2

P0 Monad 0 1 1 1 0 0 0 0P1 Dyad 1300–1500 2 2 0 0 1 0 1P2 Pentad 2600–3100 5 9 3 0 2 1 3P3 Octad 3900–4600 8 24 4 2 5 5 8P4 Tetradecad 5200–6200 14 60 11 2 14 13 20P5 Icosad 6500–7700 20 134 18 11 28 34 43P6 Triacontad 7800–9300 30 280 41 20 58 71 90P7 Tetracontad 9100–10800 40 538 64 45 112 148 169P8 Pentacontakaipentad 10400–12300 55 996 126 81 204 272 313

a The vibrational polyads Pn of methane are defined by the polyad quantum number n = 2v1 + v2 + 2v3 + v4 with the vi being the vibrational quantum number related to thefundamentals mi as summarized in Table 4.

b Total number of vibrational (sub) levels.c Number of vibrational (sub) levels with symmetry Cv.

34 H.M. Niederer et al. / Journal of Molecular Spectroscopy 291 (2013) 33–47

obtained from a previous fit to experimental data for 12CH4, wedetermine a list of corrected 13CH4 levels which we refer to asapproximate 13CH4 levels. Using the Dijon procedure [39,44], aneffective Hamiltonian is fitted to approximate transitions of 13CH4.In order to obtain fitting parameters for the experimental spectrumwe replace, in groups, approximate 13CH4 levels with their experi-mental counterparts and refit. By this approach we thus finally ob-tain a balanced set of effective Hamiltonian parameters based onexperiment but including some theoretical information. The pres-ent analysis, indeed, provides an excellent example of a fruitfulinterplay of theory with experiment. Our results should provide afairly good starting point for simulating the spectra of 13CH4 in thisspectral range and provides a good basis for further studies.

2. Theory

2.1. Calculating rovibrational 12CH4–13CH4 level shifts using CVPT

It is possible, but costly, to determine vibrational and ro-vibra-tional energy levels of small molecules by computing eigenvaluesof a basis representation of a Hamiltonian matrix. The most straight-forward approach uses a product basis set [53–61]. In many cases,using contracted basis functions is more efficient [62–70]. Thewell-known 2:2 stretch-stretch (Darling-Dennison) and 1:2stretch-bend (Fermi) resonances make it difficult to compute anumerically exact vibrational spectrum of methane. Computingrovibrational levels is harder, for methane, and for any molecule,because the required basis is much larger. Wang and Carringtonand Wu et al. have nonetheless computed J = 1 rovibrational levelsusing variational methods [71,72]. Wang and Carrington use poly-spherical coordinates associated with orthogonal vectors and Wuet al. use normal coordinates. Using the Lanczos algorithm and anEckart-frame normal-coordinate kinetic energy operator (KEO) itis possible to converge energy levels for higher values of J [73].Recently we have also shown that it is possible to compute high-ly-ing ro-vibrational levels with an Eckart-frame internal coordinateKEO [74]. Another problem is the accuracy of the potential energysurface (PES). It is still difficult to determine an accurate PES formethane. For these reasons we employ perturbation theory and aPES that has been fitted, using perturbation theory, to experimentalband centers. For methane, we expect the perturbation theory re-sults to be accurate for low-lying levels. This is because methaneis a fairly rigid molecule and the PES we use has been fitted, withthe same perturbation theory, to experimental results. Differencesbetween energy levels of different isotopomers, in this case 12CH4to 13CH4, are expected to be even more accurate. We thereforeexpect perturbation theory on the fitted surface to be useful forthe purpose of analysing the spectrum of 13CH4.

The version of perturbation theory we use in this paper is oftencalled Canonical Van Vleck perturbation theory (CVPT) [48,49]. TheHamiltonian is written as a sum of sequentially smaller terms andthen transformed with a succession of canonical transformationsas defined in Eq. (12) below. The transformations remove couplingbetween zeroth-order basis functions that are not coupled by res-onance terms to generate effective Hamiltonians. Energy levels areobtained by diagonalising small matrices representing the effectiveHamiltonians [75]. In this paper we use the fitted potential energysurface of Ref. [50], curvilinear normal coordinates, and truncatethe Hamiltonian at fourth order. Details of the CVPT calculationof rovibrational levels have been presented in [51]. To aid theassignment and fitting of the experimental levels we have com-puted levels up to the Octad and up to J = 8 for both 12CH4 and13CH4. A succinct summary of the method is given in this section.Special attention is given to the way Eckart axes are used withinternal coordinates.

The starting point for any calculation of a spectrum is the Ham-iltonian operator. The rovibrational KEO in the curvilinear normalcoordinates we use (or in any set of internal coordinates) can bewritten [50,51,76–79]

T ¼ 12

X3N�6k;l¼1

PkGk;lPl þ12

Xx;y;za;b

Jaðla;b þKa;bÞJb

� 12

X3N�6k¼1

Xx;y;zaðPkAa;kJa þ JaAa;kPkÞ þ V 0ðQ Þ ð1Þ

where Qk is a coordinate, and Pk is a conjugate vibrational momen-tum. Ja is a molecule-fixed component of the angular momentum.At this stage the definition of the frame is not specified. The vol-ume element for this kinetic energy operator is unity, which en-ables the use of harmonic oscillator basis functions. V0 is apotential-like term that appears because the Jacobian has beenmodified.

The KEO depends on four matrices G (the vibrational Wilson G-matrix [78]), l (the inverse of the moment of inertia matrix), Kand A. The last two matrices can be written in terms of G, l, andC and X,

C ¼ lX ð2ÞA ¼ CG ð3ÞK ¼ CGCt ð4Þ

where the matrix elements of X are,

Xa;k ¼XNi¼1

mi�abcxbi@xci@Q k

ð5Þ

Table 2Rotational Td symmetry adapted functions (SAF) up to J = 3.

J sym SAFa

J = 1 F1x �ffiffi2p

2 j11�iF1y �

ffiffi2p

i2 j11þi

F1z j10i

J = 2 Ea j20iEb

ffiffi2p

2 j22þiF2x

ffiffi2p

2 j21þiF2y

ffiffi2p

2 j21�iF2z �

ffiffi2p

i2 j22�i

J = 3 A2ffiffi2p

i2 j32�i

F1x � 14ffiffiffi5pj33�i �

ffiffiffi3pj31�i

h iF1y þ i4

ffiffiffi5pj33þi þ

ffiffiffi3pj31þi

h iF1z j30iF2x 1

4

ffiffiffi3pj33�i þ

ffiffiffi5pj31�i

h iF2y i

4

ffiffiffi3pj33þi �

ffiffiffi5pj31þi

h iF2z

ffiffi2p

2 j32þi

a j J;Kþi ¼j J;Kiþ j J;�Ki; j J;K�i ¼j J;Ki� j J;�Ki.

H.M. Niederer et al. / Journal of Molecular Spectroscopy 291 (2013) 33–47 35

with mi the mass of atom i, xai the Cartesian coordinates of atom i,(a,b,c) the three Cartesian components, and �abc the Levi–Civitaantisymmetric symbol. Following Wilson, Decius and Cross [78],the G matrix can be written as the product

G ¼ BM�1Bt; ð6Þ

where M�1 is the diagonal matrix with diagonal elementsm�11 ;m

�11 ;m

�11 ;m

�12 ;m

�12 ;m

�12 ; . . . and the B matrix is the Cartesian

derivative of the internal coordinate Qk:

Bk;ai ¼@Q k@xai

: ð7Þ

In our calculations it is convenient to compute elements of G, l, andA numerically. Various schemes for doing so have been presented inthe literature [79–84]. A numerical KEO in an Eckart frame was firstused in Ref. [79]. We follow the notation and ideas of Refs.[51,77,79]. G and X can be easily determined if the full 15 � 15 Bmatrix is known. The first nine rows of B, are obtained from centralfinite difference derivatives. To compute these derivatives at a givenset of curvilinear normal coordinates (i.e. molecular shape), onemust attach a frame to the molecule and thereby obtain xai. Thenext step is to compute finite difference derivatives by extendingCartesian coordinates (xai) and calculating the change in the inter-nal coordinates (Qk). We use a six-point finite difference equationbut the details of the finite difference implementation appear tobe unimportant.

In this fashion, one obtains the first 9 rows of the full B matrix(which are used to compute G matrix using Eq. (6)). In this paperwe use the Eckart frame associated with the equilibrium geometry[85]. It reduces the Coriolis coupling and improves the accuracy ofthe levels obtained with perturbation theory. It is difficult to derivean Eckart KEO in internal coordinates [86], but the numerical pro-cedure completely obviates the need for an analytic KEO. Havingopted for the Eckart frame, the remaining 6 rows of B are easilycomputed from translational and rotational coordinates:

Ta ¼XNi¼1

mixai ð8Þ

Ra ¼XNi¼1

mi�abcabixci ð9Þ

where (axi,ayi,azi) are the Cartesian coordinates at the referenceshape in a reference frame and a, b, c = x, y, or z. The reference shapeand frame we use is the equilibrium geometry and the principal axisframe. From the inverse of the full B matrix, one can readily com-pute X.

When all operators are written in terms of creation and annihi-lation operators it is possible to systematize the transformationsrequired in CVPT [48]. For this reason it is advantageous to usethe Schwinger transformation [87] to write angular momentumoperators, Jx, Jy, Jz, in terms of creation-annihilation operators fordegenerate harmonic oscillators [51,79]. The effective Hamiltonianis diagonalized in a product ro-vibrational basisjn1ijn2aijn2bij~n3xn3y~n3zij~n4xn4y~n4zijnr1nr2i, where jni is a harmonicoscillator basis function and jnr1nr2i is for rotation. j~ni ¼ injni is amodified harmonic oscillator basis function used with imaginarycreation-annihilation operators ~ay ¼ iay and ~a ¼ �ia. The imaginarycreation-annihilation operators are used for the F2x and F2z modesto ensure that the Hamiltonian matrix elements are real.

In [51] a C3v symmetry-adapted rotational basis was used. To dothe calculations reported in this paper we have used a Td symme-try-adapted rotational basis. Symmetry adapted basis functions(SAF) for J values up to 3 are given in Table 2. The SAF for J + 1are obtained from those for J by applying promotion operators

and Gram-Schmidt orthogonalization [88,89]. This procedure is ap-plied repeatedly starting from J = 1.

The final basis is made from products of Td symmetry-adaptedvibrational functions for each degenerate mode and Td symme-try-adapted rotational functions. Products are decomposed intofunctions that transform like irreducible representations of Td, withthe help of vector coupling coefficients [90]. Using the Td basis al-lows us to assign Td labels to levels directly.

The potential surface of [50] was obtained by refining 12parameters so that vibrational levels of methane and its isotopo-mers with Td, C3v and C2v symmetry, obtained from a fourth orderCVPT calculation, nearly reproduce their experimental counter-parts. The potential is a force field in symmetry coordinates. Therefined force constants are given in Table III of [50] and force con-stants that are not refined are in Tables I and II of [91]. A total of130 experimental levels from 9 isotopomers were fitted with a rootmean square deviation drms of 0.70 cm�1. Some of the experimen-tal levels used in [50] are now known to be inaccurate. Two ofthem (for CH4) have large errors: the level at 4105.15 cm�1 (privatecommunication from Champion cited in Ref. [92]) is now correctlydetermined to be 4101.39 cm�1 [37] and the level at 4446.41 cm�1

(private communication from Champion cited in Ref. [92]) is nowknown to be at 4435.12 cm�1 [37]. The fourth-order CVPT calcu-lated levels are 4102.19 cm�1 and 4435.81 cm�1. The new experi-mental values of [37] are in much better agreement with thecalculated levels, indicating that the fitted levels were not affectedvery much by the previously used incorrect experimental data. Wetherefore decided to use the potential of [50] and fourth-orderCVPT for the calculations reported here. Although some of thehigher bending states computed with fourth-order CVPT on thesurface we use are known to be in error (e.g. an error 7.6 cm�1

for 3m4) in the Octad [50], 13CH4–12CH4 shifts should be ratheraccurate. The original potential of Ref. [91] was written in symme-try coordinates. To make the potential of Wang and Sibert, whichwe use in this paper, the symmetry coordinates are written aslinear combinations of stretch and bend coordinates and thestretch coordinates are then replaced with expansions in termsof the Morse variable y = (1 � e�a x). Curvilinear normal coordi-nates are linear combinations of symmetry coordinates. Thestretch symmetry coordinates are linear combinations of y. Theredundancy among the six bending coordinates is dealt with asexplained in Ref. [50].

Table 3Rovibrational Td SAF (symmetry adapted functions) for J = 3 and polyad P1 (see also footnote to table 2).

sym SAF

1A1 14ffiffi3p þ

ffiffiffi3pj~v4xij33�i þ

ffiffiffi5pj~v4xij31�i �

ffiffiffi3pjv4yij33þi þ

ffiffiffi5pjv4yij31þi þ 2

ffiffiffi2pj~v4zij32þi

h i1A2 1

4ffiffi3p þ




ffiffiffi3pjv4yij31þi � 4j~v4zij30i

h i1Eb 1

4ffiffi2p �



ffiffiffi3pjv4yij33þi �

ffiffiffi5pjv4yij31þi

h i2Eb 1

4ffiffi6p þ




ffiffiffi3pjv4yij31þi þ 8j~v4zij30i

h i3Eb

ffiffi2p

2 jv2aij32�i1F1z 1

4ffiffi2p �

ffiffiffi3pj~v4xij33þi þ


ffiffiffi3pjv4yij33�i þ

ffiffiffi5pjv4yij31�i

h i2F1z 1

4ffiffi2p þ


ffiffiffi3pj~v4xij31þi �



h i3F1z

ffiffi2p

2 j~v4zij32�i4F1z

ffiffi2p

2 jv2bij32þi5F1z jv2aij30i1F2z 1

4ffiffi2p þ

ffiffiffi3pj~v4xij33þi �




h i2F2z 1

4ffiffi2p þ



ffiffiffi5pjv4yij33�i �


h i3F2z

ffiffi2p

2 jv2aij32þi4F2z jv2bij30i


There are two sources of error in the CVPT calculation. The firstis the Hamiltonian expansion error which is larger for the bendingcoordinates (it is smaller for the stretching coordinates because theMorse variable is used for these). The second is the convergence er-ror of the perturbation theory. The second error has a larger effecton high bending modes. For the Octad, it is known, for instance,that differences between fourth and sixth order CVPT vibrationallevels and eighth order CVPT levels are 7.6 and 1.1 cm�1, respec-tively [50]. However, these errors can be largely avoided, if thefourth order CVPT is consistently used in both adjusting the poten-tial energy surface and calculating energy levels. The 12CH4–13CH4shifts agree very well with experiment (see Section 4).

2.2. The Dijon effective Hamiltonian

The Dijon effective Hamilonian theory is a useful tool for calculat-ing the spectrum of a spherical top molecule [39]. It is often used byadjusting parameters of blocks, one for each polyad, of a matrix rep-resentation of the effective Hamilonian, so that differences betweeneigenvalues of the blocks agree with experimental results. Once a fithas been achieved, one can generate the complete set of rovibra-tional energy levels from the parameters. The tetrahedral tensorialeffective Hamiltonian formalism is described in Refs. [93–95].

Polyads are groups of levels that arise because of the simplerelationship among the fundamental frequencies of the four nor-mal modes of methane [3,96]

m1 � 2m2 � m3 � 2m4: ð10Þ

Table 4 provides a survey of the current best values of the funda-mentals for the normal modes of 13CH4, including results from thispaper and a comparison with previous results for 12CH4. A combina-tion band (v1,v2,v3,v4) is assigned to polyad Pn if

ðm1;m2;m3;m4Þð2;1;2;1Þt ¼ n: ð11Þ

The superscript t denotes transposition. The vector (2,1,2,1)t is re-ferred to as the polyad structure of methane. Alternative definitionsof polyads exist [8]. The set of quantum numbers (v1,v2,v3,v4) de-fines a ‘‘level’’ and the numbers of levels for each polyad are givenin Table 1 (8 for the Octad, the Greek prefix indicating this number).Many levels are highly degenerate and this degeneracy is lifted inpart, given that the Td group allows for A, E, F symmetries withdegeneracies 1, 2, 3 only. The number of sublevels arising from thissplitting is also indicated in Table 1 (24 for the Octad). Table 1 lists

the polyads of 13CH4, and shows that there is considerable overlapnear and above 12000 cm�1.

To derive an effective Hamiltonian for methane one applies aunitary transformation [97],

eH ¼ expðiSÞH expð�iSÞ: ð12Þwhere S is traceless and Hermitian, and chooses S so that matrix ele-ments of eH coupling states in different polyads are small. The trans-formed Hamiltonian can be written,

eH ¼ eHfP0g þ eHfP1g þ � � � þ eHfPig þ � � � : ð13ÞMatrix elements of eHfPig between basis functions in a polyad Pj withj < i are zero. eHfPig is a linear combination of symmetrized tensoroperators, [98]

eHfPig ¼X~ts1s2rXrðJ;nCrÞCþv C�v TXrðJ;nCrÞCþv C�v ðA1Þs1s2r : ð14ÞAlthough the coefficients ~ts1s2rXrðJ;nCrÞCþv C�v can in principle be calculated,they are usually treated as fitting parameters. The sum in Eq. (14)extends over all indices [95] and the coupling vectors s1; s2 2 N4of the Pi polyad. The 4-tuple s1 = (s11,s12,s13,s14) indicates the poly-nomial degree of the operator in elementary vibrational creationoperators (+ as a superscript) for the vibrational band ( v1,v2,v3,v4) -2 Pi. s2 = (s21,s22,s23,s24) is the 4-tuple for the polynomial degree ofthe vibrational operator in elementary annihilation operators (� asa superscript). Xr is the rotational degree of the effective rotation–vibration operator, which is related to the total order X of the oper-ator via its sum with the vibrational degree Xv:

X ¼ 2X4i¼1

s1i þXr � 2; ð15Þ

sinceP4

i¼1s1i ¼P4

i¼1s2i. We use a notation different from ref. [41],here. J is the rank of the rotation operator and Cr the symmetry spe-cies of this component according to the decomposition of the 2J + 1-dimensional spherical tensor operator within the group chain O(3) � Td. n = 0, 1, . . . enumerates Cr up to its multiplicity in thedecomposition of the Wigner rotation matrix DJ in Td [99]. C

þv C

�v

� �denotes the symmetry species of the vibrational creation (annihila-tion) operator. A1 is the overall symmetry species of the operatorcomponent r, which must be totally symmetric in a Hermitian the-ory, i.e. for tetrahedral molecules the A1 species must be containedin the decomposition of the product Cr � Cv, hence, Cr = Cv. Cv is

Table 4The fundamental band centers ~miðJ ¼ 0Þ ¼ eEi ¼ Ei=hc of the normal modes of methane.

Assignment Fundamentals ~miðJ ¼ 0Þ=cm�1 Description13CH4 12CH4

Level C(Td) eEa (Fit) eEb (Exp.) eEc (Fit)m1 A1 2915.442036 2915.442641 2916.482 sym. stretchingm2 E 1533.492779 1533.333 bending (E)m3 F2 3009.545581 3009.545514 3019.493 deg. stretchingm4 F2 1302.780778 1302.780788 1310.761 bending (F2)

a J = 0 term values calculated from the fitted effective Hamiltonian with respectto the ground state energy. The effective Hamiltonian parameters are from the fit of[44–46].

b Experimental term values, derived from allowed P(1) transitions from theJ00 ¼ 1;C00 ¼ F�1� �

level of the ground state at 10.482132 cm�1 [45,46,120] tovibrational levels J0 ¼ 0;C0 ¼ Fþ1

� �(see also footnote to table 8).

c The values for 12CH4 are bandcentres from the fit of Ref. [37]. One recognizesthe large isotope shifts for m3 and m4.

Table 5Summary of symmetry species of 12 CH4 in the groups T; Td ; S

4.

C(T) C(Td) CðTdÞ " S4a[partition]parity I bP (12CH4) bP (13CH4)

A A1 Aþ1 + A�2 [4]

+ + [14]� 2 A�2 Aþ2

A A2 Aþ2 + A�1 [1

4]+ + [4]� 2 Aþ2 A�2

E E E+ + E� [22]+ + [22]� 0 E+ + E� E+ + E�

F F1 Fþ1 + F�2 [2,1

2]+ + [3,1]� 1 Fþ1 F�1

F F2 Fþ2 + F�1 [3,1]

+ + [2,12]� 1 F�1 Fþ1

a [�, �] Gives the partition in the permutation group with parity + or � as upperindex [44,104].

b Pauli allowed rovibrational species for 12CH4 and 13CH4 [44,104].


the overall symmetry species of the vibration operator. For polyad Pjthe ~ts1s2rXrðJ;nCrÞCþv C�v constants are determined by diagonalisingeHhPji ¼ Pj eHPj ¼ eHhPjifP0g þ � � � þ eHhPjifPig þ � � � þ eH hPjifPjg: ð16Þwhere Pj is the projection operator onto states in polyad Pj.

In principle the fitting process is straightforward but in practiceit is difficult. One problem is the sheer number of parameters. Forstates in P3 one needs to fit 896 parameters if the expansion in-cludes order X = 6, 6, 6 and 5 for P0, P1, P2 and P3, respectively(see Table 7 below in Section 4). Another problem is the difficultyof assigning experimental transitions. A third problem arises fromcorrelations between the parameters. If one uses only experimen-tal data, fitting is, in practice, extremely laborious for the higherpolyads.

2.3. Symmetry and Notation

The symmetry labels in Tables 1, 4, 5 and 8 are explained in thissubsection. The vibrational sublevels in each polyad have the sym-metry species A1, A2, E, F1 and F2 in Td with degeneracy 1 for A-lev-els, 2 for E and 3 for F as given in Table 1. Because of the electricdipole selection rules [3] strongly allowed vibrational transitionsconnect the vibrational ground state with excited states of symme-try F2 but because of rovibrational couplings the actual absorptionspectrum is much more complex at high resolution. Also the rovi-brational levels can be classified by the point group species of Td[100,101]. However, when one wishes to consider the parity ofthe wavefunction and inversion tunneling it is appropriate to con-sider, following Longuet–Higgins [102], the permutation inversiongroup S4 ¼ S4 � S

, which is the direct product of the symmetricgroup S4 of permutations for the four protons and the inversiongroup S⁄. In this paper we use the character table and notationsof Ref. [103,104], which is not the same as the notation in Ref.

[105,106] (see also the critical discussion of notation in [104]and the recommendation in [107]).

Following Longuet–Higgins [102] one can associate the pointgroup Td with an isomorphous molecular symmetry group MS24of order 24, which is a subgroup of permutation and permuta-tion–inversion operations of S4 (of order 48). By inversion tunnel-ing and due to further effects each sublevel of symmetry speciesC(Td) in Td (or MS24) splits into two sublevels with species in S

4

being given by the induced representation CðTdÞ " S4 as shown inTable 5, which also defines the notation used here. The upper indexof the symmetry species denotes parity (+ or �) [103,104].

Because of the four equivalent protons with nuclear spin IH = 1/2,there are three nuclear spin isomers with total nuclear spin from thefour protons being I = 0, 1 or 2. The representation of the correspond-ing 24 = 16 nuclear spin functions from the protons is reducible to5Aþ1 ðI ¼ 2Þ

1E+ (I = 0) and 3Fþ2 ðI ¼ 1Þ, where the upper left indexindicates both the frequency of the species in the reduction andthe multiplicity due to the proton nuclear spins. Because ofapproximate conservation of nuclear spin symmetry one identifies3 nuclear spin isomers in methane as ortho 3Fþ2 ; I ¼ 1

� �, meta

5Aþ1 ; I ¼ 2� �

and para (1E+,I = 0). According to the generalized Pauliprinciple only the nurovibrational (i.e. rovibration and nuclear spin)states that transform as A2 are allowed [104] and as a consequencethe rovibrational states of the ortho isomer transform as F1 , those ofthe meta isomer as A2 and of the para isomer as E

±, as summarized inTable 5.

The 12C nucleus has angular momentum (‘‘spin’’) I(12C) = 0 andpositive parity, which gives the rovibrational symmetries and par-ities including the 12C nucleus in the second last column of Table 5.The 13C nucleus has angular momentum I(13C) = 1/2 and negativeparity [104] resulting in the Pauli allowed rovibrational species,including the 13C wavefunction given in the last column of Table 5,of reversed parity compared to 12CH4. Because the proton wave-function has positive parity, these last two columns give also thetotal parity for the two isotopomers of CH4. The spin multiplicityis (2I + 1) for 12CH4 and 2(2I + 1) for 13CH4. This is not resolved inour spectra. The general electric dipole selection rule in S4 is for al-lowed transitions conservation of nuclear spin symmetry andchange of parity, thus

Aþ2 $ A�2 ; E

þ $ E�; Fþ1 $ F�1 ð17Þ

while A1 and F

2 rovibrational levels do not exist for CH4. These con-

siderations with Table 5 provide the complete definitions of sub-level structures and symmetries.

3. Experimental

All spectra were taken on our Zürich prototype FTIR interfero-metric spectrometer (ZP2001) Bruker IFS 125HR (described in[38,108,109]). The sample of 13CH4 was purchased from CambridgeIsotope Laboratories. The isotopic purity was specified to be 98%and the identity of the sample was obvious from the spectra.

Here we summarize the spectroscopic data which are relevantfor the present work on the Octad region of 13CH4. Fig. 1 showsthree spectra in this spectral region recorded at differentconditions.

The experimental details are summarized in Table 6. The effec-tive instrumental resolution used is d~meff ¼ 0:9=L ¼ 0:0047 cm�1[38,44], resulting in almost Doppler limited spectra. L denotesthe interferometer optical path length. For 13CH4 typical Dopplerwidths in the Octad range are between 6 � 10�3 cm�1 at3800 cm�1 and 80 K and 14 � 10�3 cm�1 at 4700 cm�1 and 300 K,for example. The spectra are self-apodized with an aperture diam-eter of 1 mm. About 100–200 spectra were added in each spectralregion. All spectra were taken at 80 K with a total pressure of

Fig. 1. Spectra of 13CH4 at 80 K measured with the Bruker IFS 125 HR ZP 2001spectrometer in the Octad region. Decadic absorbance as lg (I0/I) is shown. Thespectrum in the lower part of the Octad (top frame) shows strong signatures fromH2O inside the spectrometer. The spectrum numbers indicated in the Figure aredefined in Table 6. For the recording of spectrum 2 the cell with a pressure of4.9 hPa was partially pumped out (⁄) such that strong absorption lines could bemeasured without saturation (thus the actual pressure in the measurement waslower).

Fig. 2. Details of the Octad spectrum (see Table 6 and Spectrum 2 in Fig. 1)expanded in a small section around the P(1) transition from the lowerJ00 ¼ 1;C00 ¼ F�1 ;n00 ¼ 1� �

rovibrational level at 10.482132 cm�1 [45,46,120] to theupper Fþ1 component of the m1 + m4 combination band at 4213.824035 cm

�1

J0 ¼ 0;C0 ¼ Fþ1 ;n0 ¼ 4� �

. The index n labels the sublevels of given symmetry withrespect to energy in ascending order. The superscripts of the lables refer to thelower (00) and upper state (0). Figs. 5, 7 and 8 show larger sections of this spectrum.Decadic absorbance as lg (I0/I) is shown for both the experimental and simulatedspectrum of 13CH4 recorded at 80 K. The symbol (;) is shown above lines which areassigned in this spectrum while (j) is shown above all assigned line positions in thisspectral region. The top frame shows the uncalibrated experiment. The data for thefit are calibrated (for calibration factors see Table 6) so that the simulation in thisfigure is slightly shifted compared to the experiment.


approximately 5 hPa, representing the total pressure p of a mixtureof the sample 13CH4 and He in the cell with p(CH4)/p � 0.1. For therecording of spectrum 2 the cell with a pressure of 4.9 hPa was par-tially pumped out such that strong absorption lines could be mea-sured without saturation (thus the actual pressure in themeasurement was lower). Weak absorption lines could be detectedin partially saturated spectra. Pressure broadening can be ne-glected under the conditions used here.

For the recording of cold spectra we used a collisional cooling(enclosive flow) cell based on White cell [110] optics embeddedin a glass Dewar which recently has been applied to the spectros-copy of other isotopomers of methane [111,112]. Its design hasbeen described in detail in [113,114]. The set up of Ref. [115]was used here in the static mode (without permanent flow) withoptical path lengths of approximately 10 m. The effect of coolingthe sample gas is very clear: the Doppler width of a single spectral

Table 6Experimental details for the sample spectra of 13CH4 at 80 K.

pa/hPa T/K db/mm d~meff c/cm�1 ld/m Scans

4.3 80 1.0 0.0040 10.0 1504.9⁄ 80 1.0 0.0047 10.0 1004.9 80 1.0 0.0047 10.0 200

a Pressure p of a mixture of methane and helium where p(CH4)/p � 0.1.b Diameter d of the circular aperture.c Effective instrumental resolution d~meff ¼ 0:9=L.d Optical path length l in the multireflexion cell.e See Fig. 1.f Calibrated ð~meÞ and uncalibrated ð~mueÞ experimental wavenumber ð~me ¼ m � ~mueÞ.

⁄ Denotes experiments, in which the sample cell was partially pumped out, such that t

line of the molecule narrows with the square root of the tempera-ture. Compared to spectra at room temperature, the Doppler fullwidth is reduced by a factor of about 1.9 at 80 K (to 0.0065 cm�1

around 4200 cm�1). The quality of the line shapes obtained is illus-trated in Fig. 2.

In addition spectral lines for high rotational quantum numbersare attenuated and complex polyad patterns are simplified result-ing in an easier analysis [111]. The overall widths of the rotation–vibration bands decrease about proportionally with temperature.For calibration we used H2O line positions from the HITRAN molec-ular spectroscopic data-base [116], which is based largely on theoriginal calibration work of Toth [117,118]. The calibration factorsfrom the least square fits are summarized in Table 6. The calibratedexperimental wavenumbers ~me reported in this work are given by~me ¼ m~mue.

At 80 K the relative wavenumber accuracy of nonblended,unsaturated and reasonably strong lines is estimated to be betterthan 10�5 cm�1 in the spectral range of the Octad. The absolute

Range/cm�1 Spectrume Calibration factorsf m ¼ ~me=~mue

3300–4400 Spectrum 1 1–2.68 � 10�74000–6000 Spectrum 2 1–3.22 � 10�74000–6000 Spectrum 3 1–2.31 � 10�7

he actual pressure is lower than given in this table.


wavenumber accuracy depends thus upon the accuracy of the H2Oreference lines used for calibration, which have an estimateduncertainty of approximately 10�4 cm�1 (but see also [43]). Formost results we give line positions to 10�6 cm�1 in order to ex-clude introducing extra, purely numerical round-off errors. In somespectra (e.g. Fig. 4) water lines (from the interferometer, not thesample cell) can appear, because the water in the background spec-trum measurement taken at lower resolution does not perfectlycancel the water in the spectrum with the sample included. Thiscauses no difficulties, as water lines are perfectly well knownand easily identified, even useful for calibration.

4. Analysis of the experimental data and results

4.1. Fitting the Dijon effective Hamiltonian parameters to approximatedata using the 12CH4–

13CH4 level shifts from CVPT

Using the Dijon Hamiltonian we wish to determine a set of fit-ting parameters that represents the Octad region of the experimen-tal spectrum. The parameters in eHfP0g; eHfP1g, and eHfP2g have alreadybeen adjusted to experimental transitions between states in theMonad, Dyad, and Pentad. Our goal is to do a ‘‘local’’ fit to deter-mine the parameters in eHfP3g, fixing the values of the previouslydetermined parameters in eHfP0g; eHfP1g, and eHfP2g. eHfP3g dependson 596 parameters and this makes the fitting difficult, althoughsuch a fit has been achieved for 12CH4 [37]. It will be even harderto fit higher polyads [40,41]. Table 7 summarizes the Hamiltonianparameters.

To guide the fitting process, we could start with parameter val-ues obtained by fitting eHfP3g to the ro-vibrational 13CH4 spectrumcomputed using the CVPT approach outlined in Section 2.1 andthe adjusted quartic force field of Refs [50,51]. In Table 8 we com-pare the vibrational band centers of 13CH4, from the perturbativeapproach (in the CVPT column), to the corresponding experimentalvalues in the 0–4600 cm�1 spectral range. They agree well. Theroot mean square deviation (drms) is about 0.8 cm�1, which impliesthat parameter values obtained by fitting the theoretical spectrumshould be good starting values for the purpose of fitting the exper-imental spectrum. We generally quote the theoretical data to thesame precision (i.e. 10�6 cm�1) as the experimental data for com-parison. The theoretical uncertainties due to the hamiltonian mod-els are, of course, larger, while the purely numerical precisionwould be better than this, in general.

It is, however, possible to do even better. First, we apply theperturbative method to compute a ro-vibrational spectrum of13CH4 and 12CH4. Second, we calculate isotopic shifts,

Table 7Summary of the effective Hamiltonian parameters.

X aeHhP3ifP0g aeHhP3ifP1g aeHhP3ifP2g beHhP3ifP3gFit 0

0 1 (1) 2 (2) 2 (2) 01 0 (0) 2 (2) 5 (5) 02 2 (2) 6 (5) 21 (19) 133 0 (0) 6 (6) 35 (23) 574 3 (3) 13 (11) 71 (43) 1835 0 (0) 11 (7) 94 (44) 3436 4 (4) 22 (15) 0c (0) 0c

Total 10 (10) 62 (47) 228 (136) 596

a Effective Hamiltonian parameters from previous fits [44–46]. The number of effectivlocal analysis of the Octad in this paper, these parameters are fixed. The number of ndetermined.

b The number of Octad effective Hamiltonian parameters for the total order X. The nuc Expanded up to order X = 5 only.

DeEicvpt ¼ eEicvptð12CH4Þ � eEicvptð13CH4Þ ð18Þwhere i labels a level of a given symmetry and angular momentum.eEicvptð12CH4Þ is the energy of the ith rovibrational level with respectto the CVPT zero point energy, 9701.787217 cm�1. eEicvptð13CH4Þ isthe energy of the ith rovibrational level with respect to the CVPTzero point ground state level energy, 9673.389196 cm�1. A nearlycomplete and accurate set of rovibrational energy levels of 12CH4has been obtained by Albert et al. [37] by fitting spectra of the Octadwith a Dijon effective Hamiltonian. For 13CH4, Dijon fitting parame-ters have been previously determined for 13CH4 that represent theMonad, Dyad and Pentad regions of the spectrum (see [44–46]and references cited therein). These constants can be used to calcu-late Octad levels. The difference between energy levels for 12CH4from Albert et al. [37] and those computed from the parametersof [45,46] yields another set of isotopic shifts,

DeEifit ¼ eEifitð12CH4Þ � eEifitð13CH4Þ ð19ÞThe drms of DeEifit and DeEicvpt is approximately 0.06 cm�1. Thedifference,

DðDeEiÞ ¼ DeEicvpt � DeEifit ð20Þis shown in Fig. 3. Panels a, b and c are for rovibrational levels J 6 8of the Monad ground state (a), the Dyad (b) and the pentad (c). Thevalues for drms are small: 2.488 � 10�4 cm�1 for the Monad (34 lev-els), 2.967 � 10�2 cm�1 for the Dyad (169 levels) and 6.109 � 10�2 -cm�1 for the Pentad (642 levels). This implies that we can generatea set of accurate 13CH4 Octad levels by shifting the 12CH4 energy lev-els derived by Albert et al. [37] by DeEicvpt. These shifted levels will bedenoted eEiapproxð13CH4Þ,eEiapproxð13CH4Þ ¼ eEifitð12CH4Þ � DeEicvpt: ð21ÞFrom the eEiapproxð13CH4Þ we calculate Napprox = 6674 IR transitionsfrom the well known rotational energy levels of the vibrationalground state of 13CH4 to ro-vibrational levels in the Octad. We de-rive transition wavenumbers ~miapprox with J00 6 9 and J0 6 8. Thesedata were then inserted into the XTDS program package [119] tominimize the standard deviation

r ¼ 1Napprox

XNapproxi¼1

~miapprox � ~miFitD~miapprox

!20@ 1A1=2; ð22Þwhere we set D~miapprox ¼ 10

�3 cm�1. The effective Hamiltonian usedfor the fit is eHhP3i. The parameters of eHhP3ifP0g; eHhP3ifP1g; eHhP3ifP2g and eHhP3ifP3gwere expanded up to order 6, 6, 5 and 5, resulting in a an effectiveHamiltonian with 896 parameters. Only the eHhP3ifP3g (596 parameters)

Fit 1 Fit 2 Fit 3

(0) 0 (0) 0 (0) 0 (0)(0) 0 (0) 0 (0) 0 (0)(11) 13 (11) 13 (11) 13 (11)(47) 57 (46) 57 (42) 57 (41)(171) 183 (166) 183 (139) 183 (98)(333) 343 (318) 343 (288) 343 (199)(0) 0c (0) 0c (0) 0c (0)

(562) 596 (541) 596 (480) 596 (349)

e Hamiltonian parameters (Monad, Dyad, and Pentad) for the total order X. For theon-zero parameters is given in parentheses. Parameters set to zero could not be

mber of fitted parameters is given in parentheses.

Table 8P1, P2 and P3 vibrational band centers for 13CH4.

Sublevel Term values eE for 13CH4/cm�1Level aCv(Td) bC S4

� �CVPT cApprox. dFit 0 eFit 1 eFit 2 eFit 3 fExpt.

Dyad (2 levels, 2 sublevels)m4 �F2 Fþ1 1302.999590 1302.808378 1302.780778 1302.780778 1302.780778 1302.780778 1302.780788m2 E E± 1533.580350 1533.495295 1533.492779 1533.492779 1533.492779 1533.492779

Pentad (5 levels, 9 sublevels)2m4 A1 Aþ2 2572.960620 2572.149757 2572.111069 2572.111069 2572.111069 2572.1110692m4 �F2 Fþ1 2599.051520 2598.698160 2598.640881 2598.640881 2598.640881 2598.640881 2598.6408852m4 E E± 2609.180840 2608.791446 2608.739915 2608.739915 2608.739915 2608.739915m2 + m4 �F2 Fþ1 2823.140690 2822.470286 2822.451735 2822.451735 2822.451735 2822.451735 2822.451884m2 + m4 F1 F�1 2837.301580 2838.241729 2838.201473 2838.201473 2838.201473 2838.201473m1 A1 Aþ2 2915.624050 2915.468605 2915.442036 2915.442036 2915.442036 2915.442036 2915.442641m3 �F2 Fþ1 3009.161250 3009.494213 3009.545581 3009.545581 3009.545581 3009.545581 3009.5455142m2 A1 Aþ2 3063.843850 3063.968985 3063.964077 3063.964077 3063.964077 3063.964077 3063.9640002m2 E E± 3065.501100 3065.463071 3065.459793 3065.459793 3065.459793 3065.459793

Octad (8 levels, 24 sublevels)3m4 �F2 Fþ1 3847.597210 3848.511239 3848.463719 3848.385364 3848.387632 3848.386808

(113) 3848.3865293m4 A1 Aþ2 3886.229810 3886.487877 3886.494252 3886.336626 3886.346772 3886.356792

(9)

3m4 F1 F�1 3898.435130 3897.442580 3897.380667 3897.343125 3897.373182 3897.368163(43)

3m4 �F2 Fþ1 3908.580440 3907.578556 3907.529532 3907.548700 3907.502400 3907.494562(45) 3907.491585

m2 + 2m4 E E± 4087.377920 4086.583503 4086.513252 4086.658365 4086.201461 4085.402938(9)

m2 + 2m4 F1 F�1 4113.160630 4113.228546 4113.113963 4113.096489 4113.187835 4113.052645(9)

m2 + 2m4 A1 Aþ2 4118.467010 4116.980574 4117.580961 4117.038140 4117.039013 4116.974701(1)

m2 + 2m4 �F2 Fþ1 4125.840770 4127.435563 4127.458969 4127.368163 4127.360634 4127.367461(52) 4127.366211

m2 + 2m4 E E± 4134.548010 4135.590503 4135.272504 4135.441183 4135.534444 4135.531975(2)

m2 + 2m4 A2 A�2 4144.553630 4146.107870 4146.065482 4146.098816 4146.039740 4146.045155(14)

m1 + m4 �F2 Fþ1 4214.682370 4213.864803 4213.932438 4213.843149 4213.824711 4213.822064(110) 4213.824035

m3 + m4 �F2 Fþ1 4301.085160 4301.242424 4301.110895 4301.290410 4301.298143 4301.294725(67) 4301.297165

m3 + m4 E E± 4304.619270 4304.585553 4304.729668 4304.606418 4304.614493 4304.619540(67)

m3 + m4 F1 F�1 4304.755850 4304.675348 4304.859352 4304.739663 4304.730616 4304.737587(256)

m3 + m4 A1 Aþ2 4305.617300 4304.813517 4305.037622 4304.871345 4304.838038 4304.820095(45)

2m2 + m4 �F2 Fþ1 4342.042490 4340.929557 4340.968720 4340.896035 4340.907680 4340.908210(24) 4340.908408

2m2 + m4 F1 F�1 4355.330150 4355.877167 4355.770304 4355.766404 4355.880563 4356.010533(47)

2m2 + m4 �F2 Fþ1 4368.574970 4371.233550 4371.219531 4371.219143 4371.203545 4371.166591(12)

m1 + m2 E E± 4434.896840 4434.204212 4434.193738 4434.279077 4434.233894 4434.468949(11)

m2 + m3 F1 F�1 4528.172120 4527.834917 4527.829615 4527.840630 4527.884678 4527.884213(38)

m2 + m3 �F2 Fþ1 4534.312780 4534.010239 4533.993685 4534.031822 4534.054103 4534.055329(168) 4534.054597

3m2 E E± 4592.075580 4592.505947 4592.599081 4592.697539 4592.866166 4592.573860(1)

3m2 A2 A�2 4595.701350 4595.747880 4595.854958 4595.831420 4595.914976 4595.831420(1)

3m2 A1 Aþ2 4595.792810 4595.981878 4596.142759 4596.323464 4596.148597 4629.211637(0)

a Vibrational species in Td. IR-active bands are marked (�).b Level species in S4, parities give total parities including the parity of the nucleus

13C (see Section 2.3). Energy differences between E+ and E� levels are too small to bedetected in our experiments and are also not available from the present calculations.

c Approximate term values according to Eq. (21).d Term values obtained from the fit to transitions from the ground state to the approximate term values of the Octad (see Eq. (21)).e Term values derived from the fits to approximate and experimental data. The fits are described in Section 4.2. For Fit 3 the exponent at the wavenumber indicates the

number of experimental transitions available for the determination of the term value of the respective sublevel of the Octad. A total of 1144 experimental transitions wereused for Fit 3. The value for the A1 component of the 3m2 overtone is far off due to fitting zero data. The term values for the (sub-) levels of the Dyad and the Pentad are basedon previous fits [44–46]. They have not been fitted in the present work.

f Experimental term values are derived from allowed P(1) transitions from the J00 ¼ 1;C00 ¼ F�1� �

level of the ground state at 10.482132 cm�1 [45,46,120] to vibrationallevels J0 ¼ 0;C0 ¼ Fþ1

� �. The experimental value for the band center of the A1 fundamental stretching band m1 at 2915.442641 cm�1 is from a Raman experiment presented in

Ref. [121], as is also the value 3063.964000 for 2v2 (A1).


were adjusted in the fit (i.e. the fit is ‘‘local’’) because values of eHhP3ifP0g(10 parameters), eHhP3ifP1g (62 parameters) and eHhP3ifP2g (228 parameters)obtained in our recent studies of 13CH4 [45,46] were used as fixedinput. Some parameters could not be determined. Only one param-eter, which was 100% correlated with another parameter, was set to0. Other parameters with uncertainties larger than their valueswere fixed to the values they had after 1000 or 10000 iterationsof the fit program. The transition frequencies used in the leastsquares adjustment as well as the effective Hamiltonian parametersobtained are in the Supplementary material (see also the Appendix).The 6674 approximate transition wavenumbers are fitted with aroot mean square deviation of 0.212 cm�1. For the 1857 rovibra-tional J 6 8 levels of the Octad, Fig. 3 panel d illustrates the devia-tion of the isotopic shift (see Eq. (20)) for which the drms is0.1921 cm�1. Note that panels a, b, and c illustrate results obtained

without the Octad parameters, calculated with the 13CH4 parame-ters of Ref. [45,46] and that panel d shows results obtained withthe Octad parameters determined by the fit described in this para-graph. We refer to this fit as Fit 0.

4.2. Analysis of the experimental line positions

We wish now to use the effective Hamiltonian parametersdetermined by fitting the IR–transitions from the ground state[45] to the eEiapprox levels (in the fifth column of Table 8), to find aparameter set that fits the experimental data. Even using theparameters of the previous section (those of Fit 0) as starting val-ues, it is not possible to fit (only) the experimental data, unlessthe number of parameters is reduced. There are not enough exper-imental transitions and too many parameters. The experimental

(a) (b)

(c) (d)

Fig. 3. Rotational quantum number J versus the deviation DðDeEiÞ of the isotopicshift (see Eq. (20)). For the Octad (d) the effective Hamiltonian isotopic shift wasobtained from effective Hamiltonian parameters adjusted to approximate data ascalculated using Eq. (21).

Fig. 4. Top frame: Experimental spectra of 13CH4 obtained by merging Spectrum 2and Spectrum 3 in Fig. 1. Lower frame: Corresponding spectrum simulation.Decadic absorbance as lg (I0/I) is shown. Most of the simulated line positions in thisregion agree well with the experiment.

Fig. 5. Experimental and simulated spectrum of 13CH4 recorded at 80 K. Decadicabsorbance as lg (I0/I) is shown. The symbol (;) is shown above lines which areassigned in this spectrum (see Table 6 and Spectrum 2 in Fig. 1) while (j) is shownabove all assigned line positions in this spectral region. A broad spectral region isillustrated covering most of the transitions to the Fþ1 component of the m1 + m4combination band. The P(1) transition from the lower J00 ¼ 1;C00 ¼ F�1 ;n00 ¼ 1

� �rovibrational state to the upper J0 ¼ 0;C0 ¼ Fþ1 ;n0 ¼ 4

� �pure vibrational state has

been observed at 4203.341903 cm�1.


data are limited because transitions to the dark subbands are oftenso weak that they cannot be seen in the infrared spectrum. In thissection we present the results of three fits. In contrast to Fit 0, allthree are fits to experimental data. Fit 1 is made by replacingapproximate transitions with experimental transitions and fittinga mixed set of experimental and approximate (derived using CVPTenergies) transitions. Fit 2 is made by starting with the mixed tran-sition set used in the final iteration of Fit 1 and increasing D~miapproxin Eq. (22) to 0.1 cm�1 (Table 9) so as to weigh more heavily theexperimental data. The final parameters of Fit 2 depend less onthe approximate transitions. To make Fit 3 we fix the parametersthat were fixed in Fit 0 and vary the remaining parameters usingonly experimental transitions. Vibrational term values for all thefits are reported in Table 8. Table 7 shows the number of parame-ters and the number of fitted parameters for each term in the pro-jected fitting Hamiltonian. The statistics for the fits aresummarized in Table 9. The fit parameters, the experimental andthe approximate transitions, including uncertainties and assign-ments are in the Supplementary material as described in theAppendix. In this subsection we explain these fits.

Table 9Summary of the statistics for the fits.

Fit ar Approximate data Experimental databdrms cD~mi dN bdrms cD~mi dN

0 211.729 211.729 1 66741 23.148 242.150 10 5548 17.052 1 11442 2.966 304.582 100 5548 2.543 1 11443 2.028 2.028 1 1144

a The standard deviation ra from Eq. (23) evaluated for the respective fit.b drms is the root mean square deviation (in 10�3 cm�1) for the data set as indi-

cated by N, either approximate (Fit 0), or approximate and experimental (Fits 1 and2) or purely experimental (Fit 3), see discussion in Section 4.2.

c Uncertainty D~mi of the transition line position i as used in Eq. (23) (approximateor experimental, depending on column, in 10�3 cm�1).

d N, as used in Eq. (23), is the number of approximate (Fit 0), approximate andexperimental (Fit 1 and 2) and experimental (Fit 3) transitions included in therespective fit.

Fig. 6. Experimental and simulated spectrum of 13CH4 recorded at 80 K showingtransitions to rovibrational levels assigned to the main vibrational subbandcomponent (3m4,F2,2). Decadic absorbance as lg (I0/I) is shown for an enlargedsection of the survey shown in Fig. 4. The symbol (;) is shown above lines which areassigned in this particular spectrum (Spectrum 1 in Fig. 1), whereas (j) is shownabove all assigned line positions in this spectral region. The positions of thesimulated transition lines agree well with the experiment.

Fig. 8. Experimental and simulated spectrum of 13CH4 recorded at 80 K. Decadicabsorbance as lg (I0/I) is shown. The symbol (;) is shown above lines which areassigned in this spectrum (see Table 6 and Spectrum 2 in Fig. 1) while (j) is shownabove all assigned line positions in this spectral region. The figure is an enlargementof a small part in the R(J00)-branch of the m1 + m4 combination band where theassignment of the line position is more challenging due to the overlap of the P- andR-branch of m3 + m4 and m1 + m4, respectively.


To make Fit 1, we assign some of the experimental transitions,using the parameters determined in the previous section (Fit 0).Once assigned, experimental transitions to P3 are used to replacetransitions calculated from the ~miapprox and a new fit is made witha mixed set of experimental and approximate transitions to obtainnew parameters. Using the new parameters additional transitionscan be assigned and used to replace ~miapprox counterparts. This isdone iteratively. At each iteration new experimental transitionsare assigned and used to replace approximate transitions. In thisfashion we are able to assign Nexpt = 1144 experimental transitions

Fig. 7. Experimental and simulated spectrum of 13CH4 recorded at 80 K. Decadicabsorbance as lg (I0/I) is shown. The symbol (;) is shown above lines which areassigned in this spectrum (see Table 6 and Spectrum 2 in Fig. 1) while (j) is shownabove all assigned line positions in this spectral region. The figure illustrates theP(1) transition from the lower J00 ¼ 1;C00 ¼ F�1 ;n00 ¼ 1

� �rovibrational level at

10.482132 cm�1 [45,46,120] to the upper Fþ1 component of the m1 + m4 combinationband at 4213.824035 cm�1 J0 ¼ 0;C0 ¼ Fþ1 ;n0 ¼ 4

� �as well as Q-branch transitions

to this vibrational subband component.

with upper rotational quantum numbers J 6 9. These data replace1126 of the 6674 approximate data with upper J 6 8. Approximatedata (transitions) are only available up to J0 = 8 (upper state). Theexperimental data contain 18 transitions with J0 > 8. To fitNapprox + Nexpt = 6692 transitions with J 6 9 we minimize

ra¼1

NapproxþNexptXNapproxi¼1

~miapprox�~miFitD~miapprox

!2þXNexpti¼1

~miexpt�~miFitD~miexpt

!20@ 1A0@ 1A1=2:ð23Þ

We use D~miexpt ¼ 10�3 cm�1 and D~miapprox ¼ 10

�2 cm�1 so that theexperimental data have larger weights. When fitting we vary thesame parameters that were varied in Fit 0. Parameters that werefixed in Fit 0 are also fixed in Fit 1. Some parameters could onlybe determined with an uncertainty larger than their values. Suchparameters are fixed at the value they are assigned after 1000 or10000 iterations and not varied in subsequent iterations. In the lastiteration we find a fit with r = 23.148, indicating that the meanabsolute deviation of the transition line positions is about 20 timeslarger than their mean uncertainty (see Eq. (23)). The drms for the1144 experimental transition wavenumbers is 0.017 cm�1, and forthe 5548 transitions ~miapprox the drms is 0.242 cm

�1.To make Fit 2, we use the parameters from the last iteration of

Fit 0 as starting values and fit to the 1144 experimental and 5548approximate data using D~miexpt ¼ 10

�3 cm�1 andD~miapprox ¼ 10

�1 cm�1 so that the approximate data have smallerweights. Parameters which were fixed in Fit 0 are also fixed forFit 2. Other parameters could only be determined with an uncer-tainty larger than their values and they are fixed at the value theyare assigned after 1000 or 10000 iterations. The drms for the 1144experimental transitions is less than 0.003 cm�1, better than thedrms for the experimental transitions from Fit 1. The drms forapproximate transitions is, however, slightly worse, so it is not cer-tain that Fit 2 is a better representation of the spectrum.

For Fit 3, we use again the parameters from the last iteration ofFit 0 as starting values, but fit only the experimental data usingD~miexpt ¼ 10

�3 cm�1. Parameters which were fixed in Fit 0 are alsofixed for Fit 3. Some of the parameters that are varied cannot be


determined with an uncertainty smaller than their values and theyare fixed at the value they are assigned after 1000 or 10000 itera-tions. In this fit parameters that are varied are determined by theexperimental data alone. For Fit 3 we find r = 2.028. The drms forthe 1144 experimental transitions is 0.002 cm�1, better than thedrms for the experimental transitions from Fit 1 and 2. BecauseFit 3 does not use approximate transitions these are less wellreproduced. Fit 3 has the lowest drms for the experimental data.It introduces, however, an artifact for the highest level of the octad,for which obviously no lines were analyzed (>4600 cm�1, see Ta-ble 8). The general fit procedures should be reproducible usingthe details given in the Supplementary material (particularly Ta-ble 2 therein).

4.3. Approximate intensities and simulations of spectra

We have not adjusted effective dipole moment parameters for13CH4. To calculate intensities we have used the effective dipolemoment parameters for 12CH4, reported in [37], and eigenvectors

Table 10Summary of current experimental and theoretical results for 12CH4 and 13CH4.

Sublevel Term values eE=cm�1Level aCv(Td) bn(Cv) 12CH4

CVPT cFit

Dyad (2 levels, 2 sublevels)m4 �F2 1 1310.952480 1310.761268m2 E 1 1533.417700 1533.332635

Pentad (5 levels, 9 sublevels)2m4 A1 1 2587.854280 2587.0434172m4 �F2 1 2614.613970 2614.2606002m4 E 1 2625.007270 2624.617866m2 + m4 �F2 2 2830.985930 2830.315536m2 + m4 F1 1 2845.134010 2846.074159m1 A1 2 2916.636590 2916.481145m3 �F2 3 3019.160320 3019.4932832m2 A1 3 3063.520790 3063.6459252m2 E 2 3065.178650 3065.140611

Octad (8 levels, 24 sublevels)3m4 �F2 1 3869.573780 3870.4878093m4 A1 1 3908.942900 3909.2009673m4 F1 1 3921.502520 3920.5099703m4 �F2 2 3931.924880 3930.922996m2 + 2m4 E 1 4102.187310 4101.392893m2 + 2m4 F1 2 4128.694890 4128.762806m2 + 2m4 A1 2 4134.347700 4132.861264m2 + 2m4 �F2 3 4141.270080 4142.864873m2 + 2m4 E 2 4150.162390 4151.204883m2 + 2m4 A2 1 4160.295170 4161.849410m1 + m4 �F2 4 4224.279630 4223.462063m3 + m4 �F2 5 4319.055230 4319.212494m3 + m4 E 3 4322.348780 4322.178483m3 + m4 F1 3 4322.534110 4322.590188m3 + m4 A1 3 4323.507760 4322.7039772m2 + m4 �F2 6 4349.828870 4348.7159372m2 + m4 F1 4 4363.060150 4363.6071672m2 + m4 �F2 7 4376.288870 4378.947450m1 + m2 E 4 4435.813050 4435.120422m2 + m3 F1 5 4537.885570 4537.548367m2 + m3 �F2 8 4544.061230 4543.7586893m2 E 5 4591.597580 4592.0279473m2 A2 2 4595.222170 4595.2687003m2 A1 4 4595.313890 4595.502958

a Vibrational level species in Td. IR-active bands are marked (�).b Level number for a given species Cv Td increasing with increasing energy.c From Ref. [37] (and references cited therein). Experimental term values are derived

10.481636 cm�1 [37] to vibrational levels J0 ¼ 0;C0 ¼ F�1� �

.d Results from Fit 2.e Experimental term values are derived from allowed P(1) transitions from the J00 ¼

�levels J0 ¼ 0;C0 ¼ F�1

� �. The experimental value for the band center of the A1 fundamenta

Ref. [121], as is also the value 3063.964000 for 2v2 (A1).

of the effective Hamiltonian obtained in this paper. Figs. 4 and 5show surveys and Figs. 6–8 show examples for some small spectralranges. In Fig. 6 we show a comparison of experimental and simu-lated transitions to levels assigned to the high-lying F2 vibrationalsubband of the third overtone of the m4 band (3m4,F2,2). The simu-lated spectrum is produced with the effective Hamiltonian param-eters of Fit 3 and the effective dipole moment parameters for12CH4. The positions of the simulated and experimental transitionlines agree well and the intensities are also quite good. Intensitiescomputed from the effective dipole moment parameters for 12CH4are less satisfactory in dense regions of the spectrum, where cou-pling between basis functions is most effective, and the 12CH4and 13CH4 effective dipole moments are expected to be most differ-ent. Nevertheless the overall intensity distribution over a widerrange of the experimental spectrum shown in Figs. 4 and 5 isreproduced quite well in the simulation.

Fig. 7 shows a section of the spectrum, illustrating the directexperimental determination of a vibrational level position assummarized in the last column (‘‘Expt.’’) of Table 8. The range

13CH4cExpt. CVPT dFit eExpt.

1310.761626 1302.999590 1302.780778 1302.7807881533.580350 1533.492779

2572.960620 2572.1110692614.260728 2599.051520 2598.640881 2598.640885

2609.180840 2608.7399152830.315243 2823.140690 2822.451735 2822.451884

2837.301580 2838.2014732915.624050 2915.442036 2915.442641

3019.492774 3009.161250 3009.545581 3009.5455143063.843850 3063.964077 3063.9640003065.501100 3065.459793

3870.485796 3847.597210 3848.387632 3848.3865293886.229810 3886.3467723898.435130 3897.373182

3930.919644 3908.580440 3907.502400 3907.4915854087.377920 4086.2014614113.160630 4113.1878354118.467010 4117.039013

4142.861558 4125.840770 4127.360634 4127.3662114134.548010 4135.5344444144.553630 4146.039740

4223.461526 4214.682370 4213.824711 4213.8240354319.209699 4301.085160 4301.298143 4301.297165

4304.619270 4304.6144934304.755850 4304.7306164305.617300 4304.838038

4348.716513 4342.042490 4340.907680 4340.9084084355.330150 4355.8805634368.574970 4371.2035454434.896840 4434.2338944528.172120 4527.884678

4543.761970 4534.312780 4534.054103 4534.0545974592.075580 4592.8661664595.701350 4595.9149764595.792810 4596.148597

from allowed P(1) transitions from the J00 ¼ 1;C00 ¼ Fþ1� �

level of the ground state at

1;C00 ¼ Fþ1�

level of the ground state at 10.482132 cm�1 [45,46,120] to vibrationall stretching band m1 at 2915.442641 cm�1 is from a Raman experiment presented in


attributed approximately to m1 + m4 (F2), or more precisely to theFþ1 ;4� �

component of the Octad, allows one to identify the P(1) line,i.e. the line corresponding to the transition from the rotational le-vel F�1 ; J ¼ 1

� �at 10.482132 cm�1 to the J = 0 level in the vibration-

ally excited state (Fþ1 ; J ¼ 0, here component n = 4, as labelled by anindex n increasing with energy). The assignment of this level tom1 + m4 can be made very roughly only on the basis of an approxi-mate normal mode model. As discussed in [37] for the correspond-ing level in 12CH4 (m1 + m4,F2) at 4319.212 cm�1 there is a similarlylarge contribution from m3 + m4 and other levels contribute substan-tially as well. There is significant mixing of normal mode levels inthis range and the ‘‘normal mode assignments’’ are largely arbi-trary. In contrast, the symmetry assignment and numbering withthe polyad as Fþ1 ;4

� �is rigorous and the experimental energy level

obtained by adding the well known rotational energy in the vibra-tional ground state to the measured transition energy correspond-ing to the clearly assigned P(1) line is a rigorous and very accurateexperimental result. The largest uncertainty in this result arisesfrom the uncertainty in the calibration lines in the spectrum (seeSection 3). Fit 2 and Fit 3 give level positions within 0.002 cm�1

of these accurate results, whereas the other Fits and CPVT agreeless well (Table 8).

The experimental intensity patterns in the spectral range shownin Fig. 7 are well reproduced by the simulation, although not per-fectly. This shows that the crude dipole model is adequate for theoverall picture and for assigning patterns in the spectra. Its accu-racy would not be sufficient for quantitative simulations of, say,absorption strengths in simulations of absorption of infrared radi-ation by the Earth’s atmosphere.

Fig. 8 shows a detail of experimental and simulated spectra athigher values of the quantum number J (see assignment fromP(4) to R(6)). The line positions and intensity patterns are wellreproduced.

5. Conclusion

In this paper we present the first simultaneous analysis of therovibrational substructure of all the 24 vibrational subbands ofthe Octad of 13CH4. The analysis was made possible by new sys-tematic measurements of high resolution FTIR spectra at tempera-tures close to 80 K and new Canonical Van Vleck PerturbationTheory (CVPT) calculations of rovibrational energy levels. Thenew experimental spectra, with reduced Doppler widths [38,44],are greatly simplified in the range 3700 to 4700 cm�1. From CVPTcalculations of spectra of both 12CH4 and 13CH4, using a previouslyfitted potential energy surface [50,51], we determine isotopic shiftswhich are used in conjunction with the analysis of the Octad of12CH4 [37] to analyse the data for 13CH4 and determine parametersfor a Dijon effective Hamiltonian. These parameters can be used forfurther analyses of the Octad spectra and for extending the analysistowards the tetradecad for 13CH4. Future work should also addressthe dipole function and absolute line intensities in more detail. Inaddition, although 12CH4 and 13CH4 data sets are of similar size, theactual number of analysed experimental 13CH4 lines is still quitesmall (1144 compared to more than 7000 for the Octad lines inthe global fit for 12CH4 [37]).

Our work demonstrates the fruitful interplay of experiment andtheory in the analyses of such complex spectra. Table 10 provides aconvenient summary of the current experimental and theoreticalresults for all polyads of 12CH4 and 13CH4 up to and including theOctad. This table provides a picture of the general success of thetheoretical predictions, based on the previously adjusted potential.For the experimental results we have chosen here Fit 2 as a bestcompromise representation of the combined experimental andapproximate data, even though one might argue that some of the

other fits provide better overall accuracy in some ranges. In caseswhere the results from Fit 2 can be checked against a direct deter-mination of the level positions, the agreement is always betterthan 0.01 cm�1, which may serve as an estimate of the overallaccuracy. The experimental uncertainties from line calibrationsare presumably smaller. Of course, overall uncertainties for otherband centres could be larger than this. One can thus claim on thebasis of the present results that we have a fairly good understand-ing of the vibrational and for J 6 9 rovibrational level structure ofthe two main natural isotopomers of the methane 12CH4 and13CH4 up to and including the Octad, nearly to 5000 cm�1

(150 THz), the lowest tetradecad levels being located above5100 cm�1. This secure knowledge of level energies is useful forapplications, as discussed in the introduction, and may serve inparticular as a basis for refining local and global potential hyper-surfaces of methane.

Acknowledgements

This work has been supported by the Canadian Space Agency,the Swiss National Science Foundation, and the ETH Zürich. TuckerCarrington was a visiting Professor at the ETH Zürich during twoperiods in 2011 and 2012.

Appendix A

In this appendix we describe the information provided as elec-tronic documents in the Supplementary information. Concise defi-nitions of symbols used in the respective context are given at thebeginning of each section of the Supplementary information.

In Table 1 we list the experimental and approximate data whichwe used for the fits. This includes the experimental and theapproximate transition wavenumber, uncertainties (separatelyfor each fit), the deviation of the observed (approximate) wave-number from the calculated (fit) wavenumber and the spectro-scopic assignment separately for each fit.

The effective Hamiltonian parameters are summarized in Ta-ble 2. We state the identity as well as the values of the parametersfor each fit.

We also list the rovibrational levels for 12CH4 and 13CH4 up toand including J = 8 for each polyad, the Monad ground state (Ta-ble 3), the Dyad (Table 4), the Pentad (Table 5) and the Octad (Ta-ble 6). We state the term value eE (in cm�1) as derived from theCVPT method presented in Section 2 and the corresponding spec-troscopic assignments.

Appendix B. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.jms.2013.06.003.

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