High-resolution spectroscopy and analysis of the ν3/2ν4 …gases among the longest-lived...

Molecular PhysicsVol. 109, Nos. 17–18, 10 September–20 September 2011, 2273–2290

INVITED ARTICLE

High-resolution spectroscopy and analysis of the m3/2m4 dyad of CF4

V. Boudona*, J. Mitchellb, A. Domanskayac, C. Mauld, R. Georgese, A. Benidare and W.G. Harterb

aLaboratoire Interdisciplinaire Carnot de Bourgogne -UMR 5209 CNRS–Université de Bourgogne, 9, av. Alain Savary,B.P. 47870, F-21078 Dijon Cedex, France; bDepartment of Physics, University of Arkansas, 226 Physics Building,

825 West Dickson Street, Fayetteville, AR 72701, USA; cGeorg-August-Universität, Institut für Physikalische Chemie,D-37077 Göttingen, Germany; dInstitut für Physikalische und Theoretische Chemie der Technischen Universität

Braunschweig, D-38106 Braunschweig, Germany; eInstitut de Physique de Rennes, UMR 6251 CNRS–Université deRennes 1, 263 Av. Général Leclerc, F-35042 Rennes Cedex, France

(Received 20 April 2011; final version received 6 September 2011)

CF4 is a strong greenhouse gas of both anthropogenic and natural origin [D.R. Worton et al., Environ. Sci.Technol. 41, 2184 (2007)]. However, high-resolution infrared spectroscopy of this molecule has received only alimited interest up to now. Until very recently, the public databases only contained cross-sections for this species,but no detailed line list. We reinvestigate here the strongly absorbing �3 region around 7.8 �m. New Fouriertransform infrared (FTIR) spectra up to a maximal resolution of 0.0025 cm�1 have been recorded: (i) room-temperature spectra in a static cell and (ii) a supersonic expansion jet spectrum at a 23 K estimated temperature.Following the work of Gabard et al. [Mol. Phys. 85, 735 (1995)], we perform a simultaneous analysis of both the�3 and 2�4 bands since a strong Coriolis interaction occurs between them, perturbing the �3 R-branch rotationalclusters around J¼ 20. Similarly to Gabard et al. , we also include �4 FTIR data and �3 � �3 microwave data inthe fit. The analysis is performed thanks to the XTDS and SPVIEW programs [Ch. Wenger et al., J. Mol.Spectrosc., 251 102 (2008)]. Compared to Gabard et al. , the present work extends the analysis up to higher Jvalues (56 instead of 32). Absorption intensities are estimated thanks to the dipole moment derivative from D.Papoušek et al. [J. Phys. Chem. 99, 15387 (1995)] and compare well with the experiment. We have produced asynthetic linelist that is included in the HITRAN 2008 and GEISA 2009 public databases. The rotational energysurfaces for the �3=2�4 dyad are also examined in detail in order to understand the distribution and clusteringpatterns of rovibrational levels.

Keywords: carbon tetrafluoride; greenhouse gas; infrared absorption; tensorial formalism; semi-classical analysis

1. Introduction

Tetrafluoromethane (CF4, Halocarbon 14, PFC-14,R-14, . . .) is a colourless, nontoxic, nonflammable,noncorrosive gas belonging to the group of perfluoro-carbons (PFCs) which designates the chemicals com-posed of carbon and fluorine only. PFCs(predominantly CF4 and C2F6 but also C3F8) wereintroduced as alternatives, along with hydrofluorocar-bons, to be the ozone depleting substances. Thesechemicals are extremely stable and harmless for thestratospheric ozone layer, as they are not likely todissociate when struck by UV photons. The counter-part is that PFCs are extremely powerful greenhousegases among the longest-lived atmospheric trace gases.CF4 has an estimated lifetime of more than 50,000years [1,2] and, by cumulating its effects with otherminor greenhouse gases, may contribute about the

same amount of global warming as increasing carbon

dioxide [3,4]. CF4, along with the other PFCs, has been

included into the Kyoto Protocol because of its

atmospheric persistence [5–7].CF4 is the most abundant PFC into the strato-

sphere. Infrared high spectral resolution solar occulta-

tion spectrometer measurements (Atmospheric

Chemistry Experiment) led to an abundance of

70.45� 3.40 pptv (10�12 per unit volume) [8]. Weknow, since the studies performed by Harnisch et al.

on pre-industrial air trapped in ice [9] and on strato-

spheric air [10], that about 40 pptv are from natural

emissions. Besides, dissolved CF4 concentrations were

measured in groundwater samples from basins located

in the southern Mojave Desert, providing an in situ

evidence for a flux of CF4 from the lithosphere [11].

The second main source of CF4 is anthropogenic and

*Corresponding author. Email: [email protected]

ISSN 0026–8976 print/ISSN 1362–3028 online

� 2011 Taylor & Francishttp://dx.doi.org/10.1080/00268976.2011.621900

http://www.tandfonline.com

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related to aluminium refining (CF4 being formed as aby-product during the production of primary alumin-ium) and in a less extent to semiconductor manufactur-ing (about 1 pptv [3]). Recently, the atmospherictrends for the twentieth century were reconstructedfrom solar-occultation type remote-sensing infraredobservations of the stratosphere (Atmospheric TraceMOlecular Spectroscopy – ATMOS – and ACEinstruments) [8] and from firm air measurementsfrom both hemispheres [12]. Both studies highlight aslow down of the CF4 mixing ratio increase during the1990s which supports recent aluminium industryreports of reduced CF4 emissions.

The in situ detection of atmospheric CF4 is basedon infrared solar absorption spectroscopy. The veryintense �3 vibration–rotation band around 7.8 �m wasused for the first detection of CF4 into the atmospherefrom balloon-borne observations [13]. Volumetricmixing ratio profiles of CF4 were retrieved fromsatellite-borne [8,14,15] and balloon-borne observa-tions [16] to assess the estimate of the total fluorinebudget of the atmosphere. Up to now, in the absence ofdetailed line lists, the retrieval procedure was onlybased on the use of absorption cross-section databasesestablished under various typical atmospheric temper-ature, pressure and mixing ratio conditions [17]. Theline-by-line modelling of the �3 band would thusprovide the atmospheric community with an efficienttool to trace back the abundance of the stratosphericCF4.

As a matter of fact, the spectrum of this molecule isstill relatively poorly studied. Some detailed high-resolution studies exist, though, but they only concernline positions. This is the case for the �3 stretchingfundamental which was first analysed simultaneouslywith the 2�3 overtone [18] and also in the interactionwith the nearby 2�4 overtone [19]. Some high-precisionmicrowave data also exist for �3, thanks to a doubleresonance technique for 12CF4 [20,21] and

13CF4 [22].The �4 bending fundamental line positions werealso studied and analysed previously [23,24] using asimple model. Its level structure was studied thanks tosemi-classical methods [25,26]. A medium-resolution

Raman spectroscopy was performed for the �1 and 2�2bands [27], while the CARS spectrum of the �1stretching fundamental itself was recorded by theDijon group [28].

However, no experimental line intensity study hasbeen conducted up to now, although this is mandatoryfor atmospheric measurement applications. At present,CF4 intensities can only be calculated relying on twotheoretical studies [29,30]. One of the goals of thepresent paper is to improve this situation thanksto new experimental measurements and an improvedline-by-line analysis of the �3 region. We consider theinteraction of this band with 2�4 and thus perform asimultaneous analysis of �4, 2�4 and �3.

In Section 2, we present the experimental data usedin this work. Section 3 reviews the theoretical modelused for tetrahedral molecules. Sections 4 and 5 detailthe analysis and modelling of line positions andintensities, respectively. Then in Section 6, we per-form a detailed discussion of the level structure of the�3=2�4 dyad thanks to semi-classical mathematicaltools.

2. Experimental details

The room temperature CF4 spectra were recorded witha high resolution Fourier transform spectrometer(Bruker IFS120HR) and a series of gas cells, equippedwith ZnSe windows, available in Braunschweig. The �4,2�4 and �3 bands were recorded using slightly differentexperimental conditions listed hereafter and grouped inTable 1. A liquid nitrogen cooled MCT detector and aglobar source combination was systematically used forthe recording of the three bands. A mylar beamsplitterwas used for the �4 band, while a KBr beamsplitter wasused for both the 2�4 and �3 bands. Due to the verystrong absorption of CF4, the length of the gas cellswas limited to 28, 30 or 36.5mm. Gas pressures weremonitored with a MKS capacitance gauge.

The P- and R-branch regions of the �4 band wererecorded at 0.003 cm�1 spectral resolution and a gaspressure of 20 torr. A slightly higher resolution

Table 1. List of FTIR spectra with temperature (T), pressure (P), path length (L), instrumental resolution (R) and number of co-added scans.

Transition Type of spectrum T/K P/torr L/mm R/cm�1 Scans

�4–GS FTIR/Static cell 294 20 30 0.0030 70�4–GS FTIR/Static cell 294 2.75 30 0.0025 70�3–GS FTIR/Static cell 296 0.17 30 0.0050 5002�4/�3–GS FTIR/Static cell 300 0.15 28 0.0030 50�3–GS FTIR/Jet 23� 2 — 105 0.0050 1602�4–GS FTIR/Static cell 300 7.5 36.5 0.0025 580

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(0.0025 cm�1) was required to investigate the verydense Q-branch region. The sample pressure waslowered to 2.75 torr in order to reduce the collisionalbroadening. The gas temperature was set to 294 K inboth cases. A part of the R-branch region is shown onFigure 3 in Section 5. This spectrum was obtained byco-adding 70 one-sided scans.

The �3 band was recorded at 296 K at a spectralresolution of 0.003 cm�1 by co-adding 50 scans (seeFigure 4 in Section 5). Because of the very highintensity of the �3 band, the sample pressure was keptat 0.17 torr.

The measurements of the 2�4 spectrum were carriedout at 0.0025 cm�1 spectral resolution and 300 Ktemperature [31]. The pressure of the sample was set to7.5 torr. The 2�4 band being relatively weak, a largenumber of scans (580) was necessary in order toachieve a sufficient signal-to-noise ratio (see Figure 6in Section 5).

In complement to the room temperature spectraobtained in Braunschweig, very low temperature (ca.23 K) jet-cooled spectra were recorded in the �3spectral range in Rennes. The supersonic expansionwas issued from a 105� 0.2mm2 slit nozzle. Theformation of CF4 dimers and other larger homo-clusters has been limited by using a diluted mixture ofCF4 in helium (2.5%). As a carrier gas, helium waspreferred to argon which has been found to favour theformation of a non-negligible amount of (CF4)m and/or (CF4)mArn clusters. The throughput of CF4 was setto the value of 2Lmin�1 (0.089 molmin�1) leading to areasonable signal-to-noise ratio without an excessiveconsumption of gas. The dilution of 2.5% was thenachieved by pre-mixing the CF4 to 80Lmin

�1

(3.57 molmin�1) of helium in a stagnation reservoirprior to its expansion. The continuous expansion wasmaintained during a recording time of several hours inan experimental chamber evacuated by a set of heavyroot blowers, totalizing a pumping capacity of about30,000m3 h�1. Due to the significant exit area of thenozzle (21mm2), the stagnation pressure was lowereddown to 115 torr to avoid any CF4 self-clustering in thehelium expansion. The stagnation over backgroundpressures ratio was measured to be 575 which yields toa shock wave location of about 40mm downstream ofthe nozzle exit. The infrared light beam produced by aglobar source was positioned between the nozzle exitand the shock wave delimiting the cold supersonic zonefrom the warm, recompressed, subsonic zone. As theinfrared beam is about 10mm in diameter, the probedzone integrates a region of the flow, characterized byimportant temperature and density gradients. Analysisof the temperature distribution of the jet showed that itcan be characterized by two temperatures – a low

temperature in the centre of the jet and room temper-

ature of the off-centred parts of the jet corresponding

to the boundary layers surrounding the supersonic gas

stream. In order to remove the high-temperature

component from the spectrum, an additional static

experiment was performed. The spectrum of the �3band was recorded at room temperature with the same

settings as the jet-experiment. 500 scans allowed us to

achieve large signal-to-noise ratio and subtract the ‘hot

spectrum’ in an efficient and clear way. This additional

lower resolution but high signal-to-noise spectrum was

also used in the further analysis. A rotational temper-

ature of 23� 2K deduced from the simulations corre-sponds to the effective temperature of the ‘cold

spectrum’ which justifies the subtraction procedure.

After being absorbed, the infrared beam is analysed by

a high resolution Fourier transformed spectrometer

(Bruker IFS 120 HR) equipped with a MCT detector

and a KBr beamslitter. For more details concerning the

experimental set-up, the reader is referred to [32]. As a

result, a spectrum was recorded at 0.005 cm�1 resolu-

tion by co-adding 160 scans (see Figure 5 in Section 5

and Table 1).

3. Theoretical model

CF4, just like CH4 and other tetrahedral spherical

top molecules, possesses four normal modes of vibra-

tion: one non-degenerate mode with A1 symmetry (�1),one doubly-degenerate mode with E symmetry (�2),and two triply-degenerate modes with F2 symmetry (�3and �4). Only F2 fundamentals are infrared active, inthe first approximation, but the other modes can gain

some absorption intensity through couplings with the

F2 modes (see for instance [33]).The theoretical model described below to develop

the Hamiltonian operator is based on the tensorial

formalism and vibrational extrapolation methods used

in Dijon. These methods have already been explained

for example in [34,35]. We only recall here the basic

principles and their application to the case of a

tetrahedral molecule.If we consider a XY4 molecule for which the

vibrational levels are grouped into a series of polyads

designated by Pk (k ¼ 0, . . . , n), P0 being the groundstate (GS). The Hamiltonian operator can be put in the

following form (after performing some contact

transformations):

H ¼ HfP0�GSg þ HfP1g þ � � � þ HfPkg þ � � �

þ HfPn�1g þ HfPng: ð1Þ

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Terms like HfPkg contain rovibrational operators whichhave no matrix elements within the Pk04k basis sets.

The effective Hamiltonian for polyad Pn is obtained by

projecting H in the Pn Hilbert subspace, i.e.

HhPni ¼ PhPniHPhPni

¼ HhPnifGSg þHhPnifP1g þ � � � þH

hPnifPkg þ � � �

þHhPnifPn�1g þHhPnifPng: ð2Þ

The different terms are written in the form

HfPkg ¼X

all indexes

tOðK, nGÞGvG0vfsgfs0g �

"VOvðGvG0vÞGfsgfs0g � ROðK, nGÞ

h iðA1Þ:

ð3Þ

In this equation, the tOðK, nGÞGvG0vfsgfs0g are the parameters

to be determined, while "VOvðGvG0vÞGfsgfs0g and R

OðK, nGÞ are

vibrational and rotational operators of respective

degree Ov and O. Their construction is describedin [34]. Again, the vibrational operators only have

matrix elements within the Pk0�k basis sets. � is a factorthat allows the scalar terms (G ¼ A1) to match theusual terms like B0J

2, etc. The order of each individual

term is Oþ Ov � 2.Such a Hamiltonian development scheme enables

the treatment of any polyad system. Since the �3harmonic wavenumber of CF4 is close to twice that of

the �4 mode, we can thus consider here a polyad systemincluding the ground state, the v4¼ 1 level and thev3¼ 1 and v4¼ 2 levels together in order to perform aglobal fit of effective Hamiltonian parameters for all

these levels. Thus, we need the following effective

Hamiltonians:

. The ground state (polyad P0) effectiveHamiltonian,

HhGSi ¼ HhGSifGSg: ð4Þ

. The �4 bending fundamental (polyad P1) effec-tive Hamiltonian,

Hh�4i ¼ Hh�4ifGSg þHh�4if�4g, ð5Þ

. The �3=2�4 dyad (P2 effective Hamiltonian)effective Hamiltonian,

Hh�3=2�4i ¼ Hh�3=2�4ifGSg þHh�3=2�4if�4g þH

h�3=2�4if�3=2�4g: ð6Þ

A dipole moment operator is developed in the same

way (see for instance [34] for details about its

construction). In the present work, it is expanded at

the minimum order for each polyad, giving in each case

(�4, �3, 2�4) only one parameter which is the dipolemoment derivative (see Section 5).

We use here a vibrational basis restricted to the �3and �4 modes:

CðCvÞv�� ¼ Cðl4,C4Þv4 �Cðl3,C3Þv3

� �ðCvÞ

�� E, ð7Þi.e. we use harmonic oscillator wavefunctions for triply

degenerate modes with vibrational angular momentum

l3 and l4, respectively. The Hamiltonian and dipole

moment matrix elements are calculated in the coupled

rovibrational basis

CðCvÞv �CðJ,nCrÞr� �ðCÞ�� , ð8Þ

CðJ,nCrÞr being a rotational wavefunction with angularmomentum J, rotational symmetry species Cr and

multiplicity index n (see [34]) and C is the overall

symmetry species (C ¼ Cv � Cr).

4. Analysis of line positions

HhGSi, Hh�4i and Hh�3=2�4i were all expanded up toorder 6. Starting with the Hamiltonian parameters

from [19], we could assign many lines in the �4, 2�4 and�3 bands. Then, including also 243 microwave data forthe �3 � �3 transitions taken for the same reference, wecould perform a global fit of assigned lines for P0 � P0,P1 � P0 and P2 � P0 transitions. Table 2 shows thenumber of parameters in the model for each polyad

and order of the development. In each case, some

parameters could not be fitted and were thus fixed to

zero. In Table 2, the number of fitted parameters is

indicated in parentheses. The fit statistics are given in

Table 3. As we can see, the root mean square values are

within the spectral resolution margins. All vibrational

sublevels involved are well sampled, except for the A1sublevel of 2�4 for which only three lines are assigned.It should be noticed, however, that the sublevel

assignment is based on the main projection of the

eigenvectors on the initial basis set (8). The 2�4ðA1Þsublevel is mixed with the other two (E and F2)

sublevels and thus the fit of its Hamiltonian parame-

ters does not only rely on these three ‘mainly A1’ lines.The resulting Hamiltonian parameters with their

standard deviations are given in Table 5 in Appendix 1.

For the lower orders, the table also gives the corre-

spondence with the ‘usual’ parameters (see for instance

[36]) and the different interaction parameters. Figure 1

displays the fit residuals for line positions for three

datasets (P0 � P0, P1 � P0 and P2 � P0). Figure 2shows the calculated and ‘observed’ rovibrational

energy levels for the P2 (�3=2�4) dyad. The differentcolours illustrate the level mixings due to Coriolis and

Fermi interactions. The ‘observed levels’ correspond to


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rovibrational energy levels reached by assigned

transitions.

5. Analysis of line intensities

For such congested spectra with no isolated lines, it is

difficult to fit intensity parameters. We obtained some

approximate values of the dipole moment derivatives

in the following way:

. We performed simulations under the sameconditions as for the experimental spectra of �4and �3 bands.

. In each case, the dipole moment derivative wasset to one.

. We fitted the proportionality coefficient betweenthe experimental and calculated spectrum, which

gives the searched dipole moment derivative

value, using a simple least-squares fit method.. For �4, we used both the P and R branches.. For �3, we used only the R branch, the P branch

being perturbed by hot bands and by the

interaction with 2�4.

. The uncertainty is the standard deviation ofthe fit.

. The 2�4 dipole moment derivative was roughlyestimated by searching for the value giving thebest match between the calculated and experi-mental spectra.

Table 4 shows the results, in comparison with theDFT calculations from [29]. Using these dipolemoment derivatives and the effective Hamiltonianparameters obtained in the previous section, weobtain simulated spectra that compare very well withthe experimental data. Figure 3 shows a few R branchrotational clusters in the �4 band. Figures 4 and 5 showan overview of the �3 band at room and supersonicexpansion jet temperature, respectively. Figure 6 showsan overview and also some details of the 2�4 band.

6. Semi-quantum polyad analysis

6.1. Rotational energy surface

The Rotational Energy Surface (RES) is a semi-quantum phase-space surface used to analyse variouseffects in rovibrational spectra. The Poinsot ellipsoid,

Table 3. Fit statistics.

P1–P0 P2–P0 P2–P2

Transition �4–GS 2�4(A1)–GS 2�4(E)–GS 2�4(F2)–GS �3–GS �3–�3

Numberof data

2327 3 320 1814 1504 243

dRMS 1.38�10�3 cm�1 0.029�10�3 cm�1 1.03�10�3 cm�1 1.08�10�3 cm�1 0.978�10�3 cm�1 132 kHzStandard

deviation1.38 0.152 3.77 1.77 0.978 0.94

Jmax 70 39 63 57 57 25

Table 2. Number of parameters in the effective Hamiltonian (fitted parameters in parentheses).

P0 P1 P2

GS �4 2�4 2�4 interactions 2�4–�3 interactions �3

Order A1 F2 A1 E F2 A1–E A1–F2 E–F2 A1–F2 E–F2 F2–F2 F2 Total

0 1 (1) 1 (1) — — — — — — — — — 1 (1) 3 (3)1 — 1 (1) — — — — — — — — 1 (1) 1 (1) 3 (3)2 2 (2) 3 (3) 1 (1) 1 (1) 1 (1) — — — — 1 (1) 1 (1) 3 (3) 13 (13)3 — 2 (2) — — 1 (1) — — 1 (1) 1 (1) 1 (1) 3 (3) 2 (2) 11 (11)4 3 (3) 6 (6) 1 (1) 2 (2) 3 (3) 1 (1) 1 (1) 1 (1) 1 (1) 3 (3) 3 (3) 6 (6) 31 (31)5 — 4 (4) — 1 (1) 2 (2) — 1 (1) 3 (3) 2 (0) 2 (0) 4 (0) 4 (4) 23 (15)6 4 (4) 10 (10) 2 (0) 4 (0) 6 (6) 2 (2) 2 (2) 3 (0) 2 (0) 6 (0) 7 (0) 10 (10) 58 (34)

Total 10 (10) 27 (27) 4 (2) 8 (4) 13 (13) 3 (3) 4 (4) 8 (5) 6 (2) 13 (5) 19 (8) 27 (27) 142 (110)


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as well as other classical and semi-quantum rotationalphase-space surfaces, have been used in the past [37].The RES is useful spectroscopically because it describesrotational energy for a constant angular momentum.

6.1.1. Development

The RES tools used here were first developed byHarter et al. [38,39] to evaluate rotational level clustersin tetrahedral and octahedral molecules. This analysis

was first done for ground and singly excited vibrational

states and was expanded to higher vibrational states,

which are treated quantum mechanically [40,41]. Other

groups have used the RES to evaluate the rotational

clusters inside vibrational triplets [42,43] and internal

rotations [44]. In each of these cases there are three

main uses of the RES:

. To explain the boundaries of the rotationalbands for a given angular momentum.

600 610 620 630 640 650 660 670 680−15

−10

−5

0

5

10

15

Res

idua

ls/1

0−3

cm−

1

P1−P0,ν4 (F2), 2327 data, dRMS = 1.38× 10−3 cm−1, St. dev. = 1.38

1250 1255 1260 1265 1270 1275 1280 1285 1290 1295−15

−10

−5

0

5

10

15

Wavenumber/cm−1

Wavenumber/cm−1

Res

idua

ls/1

0−3

cm−

1

P2−P0,ν3 (F2), 1504 data, dRMS = 0.978 × 10−3 cm−1, St. dev. = 0.978

P2−P0, 2ν4 (A1), 3 data, dRMS = 0.029× 10−3 cm−1, St. dev. = 0.152

P2−P0, 2ν4 (E), 320 data, dRMS = 1.03× 10−3 cm−1, St. dev. = 3.77

P2−P0, 2ν4 (F2), 1814 data, dRMS = 1.08× 10−3 cm−1, St. dev. = 1.77

0 10 20 30 40 50 60 70 80 90 100−2

−1

0

1

2

Frequency/GHz

Res

idua

ls/M

Hz

P2−P2,ν3 (F2), 243 data, dRMS = 0.132 MHz, St. dev. = 0.94

Figure 1. Residuals for line positions.


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. To explain rotational clusters in terms of sym-metry reduction [39].

. To explain the splittings of the rotationalclusters in terms of phase-space tunnelling [45].

This work will employ the first two of these uses.

0 10 20 30 40 50 60 701240

1250

1260

1270

1280

1290

1300Calculated

J

Red

uced

wav

enum

ber/

cm−

1

0 10 20 30 40 50 60 701240

1250

1260

1270

1280

1290

1300Observed 2ν4(A1)

2ν4(E)2ν4(F2)ν3(F2)

J

Red

uced

wav

enum

ber/

cm−

1

Figure 2. Calculated and ‘observed’ (see text) rovibrational energy levels in the �3=2�4 dyad, showing the mixings between thefour vibrational sublevels.

Table 4. Dipole moment derivatives.

Value / Debye

Parameter This workD. Papoušeket al. [29]

@�3=@q3 0.473(5) 0.473@�4=@q4 0.0712(6) 0.0702@2�4=@q

24 0.15 —

1.5

1.0

0.5

0.0

Abs

orba

nce

1285.01284.51284.01283.51283.01282.51282.0Wavenumber/cm–1

Experiment

Simulation

Estimated Trot = 23 K

Figure 5. Supersonic expansion jet spectrum of the �3 region,compared to the simulation.

0.15

0.10

0.05

0.00

Abs

orba

nce

663.0662.5662.0661.5661.0660.5

Wavenumber/cm–1

Experiment

Simulation

R(55) R(56) R(57) R(58) R(59)

Figure 3. Detail in the R branch of the �4 region, comparedto the simulation at room temperature.

2.5

2.0

1.5

1.0

0.5

0.0

Abs

orba

nce

12901288128612841282128012781276Wavenumber/cm–1

Simulation

Experiment

Figure 4. Overview of the �3 region, compared to thesimulation at room temperature.


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The tensorial formalism used for the simulation ofthis spectrum lends itself to a variety of semi-quantum,semi-classical and classical analyses. By rewriting themolecular Hamiltonian into a polyad space, H5Pni,one is able to use a well parameterized Hamiltonianthat is written in terms of the symmetry reduction fromthe rotational Lie symmetry, O(3), to the molecularsymmetry, Oð3Þ Td � T . This Hamiltonian is alsowritten as an outer product of rotational and vibra-tional operators. The semi-quantum analysis done herewill keep rotation a classical function of body frameangles � and �. This allows one to create surfaces ofconstant angular momentum called Rotational EnergySurfaces. Other work has focused on similar techniqueswhich create a single, higher-dimensional surface [46]which is not easily visualized.

The RES is built by first rewriting the rotationalHamiltonian as a classical function of J, rather than aquantum function of operator Ĵ. By also making jJj adiscrete value, the rotational Hamiltonian can beplotted for constant total angular momentum andvarying direction of J in the body frame. Thus, thesemi-quantum form of the rotational Hamiltonian canbe called EJð�,�Þ, or a RES.

The rewritten polyad Hamiltonian is expressed as anouter product ofmolecular-symmetry-based rotation andvibration operators as already shown in Equation (3).It is dependent on the vibrational and rotationaloperators used, though it is independent of molecularspecies. Individual matrix elements are formed as:�

JnrCr;f�sgC�;C ROðK,n,GÞ ��V ðGvG0vÞG

fsgfs0g

� �ðA1Þ��Jn0rC0r;

f�0sgC0v;C0

¼ð�1ÞJþKKðK J JÞðnG n0rC0r nrCrÞ

ð�1ÞGþCþC0rþCv

�FðC0v C0r CÞCr Cv G

� 1½G�

�1=2J ROðKÞ�� J � f�sgCv �V ðGvG0vÞGfsgfs0g

�� f�0sgC0vD E

�CC0 :

ð9Þ

Such a Hamiltonian is fully quantum mechanical.

Reduced matrix elements, normalization factors, gen-

eralized 3J symbols and isoscalar factors are all

discussed in depth in [34,35,47,48]. Many of these

terms are easily calculated by easily available software

packages [49]. To make this a semi-quantum

Hamiltonian, the rotation operators must be classical

functions of body-frame angles � and �. This changesthe form of the matrix elements into the simpler

equation:

f�sgC� ROðK, nGÞ ��VðGvG0vÞG

fsgfs0g

� �ðA1Þ��Jn0rC0r; f�0sgC0v;C0

�

¼ 1NFðG G A1Þ1 2

FðC0 G CÞ�

0nu

ROðK, nGÞ

� f�sgCv �VðGvG0vÞG

fsgfs0g

�� f�0sgC0vD E

: ð10Þ

For semi-quantum matrix elements (Equation (10)),

the basis is of vibrational components only since the

rotation is now classical.While several studies have generalized the RES for

cases involving vibration or torsion, some being totally

classical [46], the present treatment keeps vibration a

quantum operator. In this way the total molecularHamiltonian can be thought of as a vibrational

Hamiltonian matrix which has elements made up of

classical rotational terms. This total molecular

Hamiltonian is then numerically diagonalized, giving

several interacting surfaces made of eigenvalues or

Rotational Energy Eigenvalue Surfaces (REES). In a

fully classical description there would be a single

surface existing in a higher dimensional space. The

diagonalization prevents surfaces from crossing one

another, though as they come close to one another,

they may interact strongly.Each RES is made more useful by contouring it

with the exact quantum levels, calculated separately.

This is possible since the radius of the RES at a

particular location is the rotational energy in that

direction in the body-frame. This process allows two

different types of analysis. The first is to see where the

level or level-cluster sits in the rotational phase-space.

As described in previous work, the placement in phase-

space must correspond with the symmetry subduction

2.0

1.5

1.0

0.5

0.0

Abs

orba

nce

12701265126012551250Wavenumber/cm–1

Experiment

Simulation

F2A1E

1264.41264.01263.61263.2

Figure 6. Overview of the 2�4 region, compared to thesimulation at room temperature. The position of thesublevels with their symmetry is indicated. The insert showsa detail in the R branch.


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between the molecular symmetry group and thesymmetry of the local section of phase-space [45,50].

The second type of analysis using energy contoursis to compare the energy contour with the intersectionbetween the RES and the cones of angular momentumuncertainty. These are the cones of arbitrary height,but must have an opening angle as defined byEquation (11).

�uncertainty ¼ arccosJz

½JðJþ 1Þ�1=2

�

: ð11Þ

An example of such an intersection with a RES isshown in Figure 7. The example in Figure 7 is not forour current system, but a simpler one, used for the sakeof explaining the RES. Figure 7 is a RES describing aground vibrational state and matches the behaviour ofSF6 at J¼ 30 and shows a Jz¼ 30 cone protrudingfrom the top of the RES.

The uncertainty cone may be placed at any of thesurface’s symmetry axes and may vary in angle so longas Jz is an integer, jJzj � J and the cone does not crossa separatrix.

Bounds of Jz are non-trivial for this polyad. For �3alone the maximum Jz for each surface could beassumed to be Jþ 1, J and Jþ 1, making P, Q and Rsurfaces. Other studies [41] have shown that clusterbands may not always correspond to a single RESdepending on the magnitude of the Coriolis

interaction, complicating the labelling issue for purely

semi-quantum analysis of the REES. Reworking theprecise labelling analysis of [41] for �3=2�4 will be afocus of later work.

Uncertainty cones and energy contours use

both the classical and quantum attributes of theRES. The RES itself is a classical phase-space that is

shown to allow only certain quantized paths corre-sponding to quantum mechanically calculated energies.Likewise, the uncertainty cones are a graphical man-

ifestation of angular quantization and uncertainty, andthey relate to classical rotation by way of intersections

with the RES.The uncertainty cones are useful for several ana-

lytical tasks. For example, it has been shown that non-circular cone intersections (semi-classical trajectories)

have varying k-projection indicative of k-mixing. TheRES and uncertainty cones can also be used todetermine how rotational levels may cluster together.

The origins of clustering by symmetry reduction arediscussed in Section 6.1.2. More detail about uncer-

tainty cones and their use can be found in [50].Figure 8 shows the nine interacting surfaces of

the REES describing the �3=2�4 J¼ 60 manifold ofCF4. Such a large group the entire REES is of limited

utility. The following section shows how thesenested surfaces can be used for to analyse this �3=2�4polyad.

Figure 7. Rotational energy surface for a simpler system:SF6 at J¼ 30 in the vibrational ground state. Quantumcontours are shown with colours matching the symmetry ofeach level (A1 is red, A2 is orange, E is green, T1 is blue andT2 is light blue. The uncertainty cone is protruding from theRES at the top.

Figure 8. All nine interacting rotational energy eignevaluesurfaces for �3=2�4 of CF4 with J¼ 60. Minimum uncertaintycone is also shown for this value of J. Surfaces are dissectedto show those of lower energy beneath. Outer RES tend to bemore spherical, making contours challenging and causingthem to be slightly dappled.


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6.1.2. RES global and local symmetry

As mentioned in Section 6.1, the placement of quan-tum levels along the RES determines possible cluster-

ing and cluster splitting. This section will describe howthe geometry of the RES affects the molecule’s energylevel structure. Rotationally or vibrationally inducedmolecular distortions deform the RES away from a

sphere. As long as these distortions are written in termsof operators sharing the molecular symmetry, the RESwill also share that symmetry. Convex parts of the RES

that protrude radially, correspond to rotational axes oflittle distortion, while concave portions of the RESindicate rotational axes along which the molecule tendsto flatten, and slightly raise its moment of inertia. Thus

the rotational axes causing structural distortion lowerthe rotational energy.

Although the entire RES must show the symmetryof the molecule, local pockets of a RES may showsymmetry representing one of the subgroups of the

molecular symmetry group. The �3=2�4 band of CF4produces RES plots containing local structures repre-sentative of octahedral subgroups C4, C3, C2 and C1.

These subgroup sections can be quickly identifiedbecause the shape at the base or separatrix defines the

subgroup symmetry and because the level structureinside each contour must also obey the subgroupchain. The subgroup sections of the RES are describedas follows. Figure 9 shows two different RES plots

with slightly different local geometries, but both withglobally octahedral symmetry. The two are created byusing different contribution of the same Hamiltonian

fitting terms. Figure 9(a) shows six local C4 sectionsprotruding from their square bases as well as eightlocal C3 concave sections with triangular bases. Boththe C4 and C3 sections are marked by a labelled

symmetry axis. Figure 9(b) does contain C4 and C3structures (both now protruding), but it labels one ofthe 12 concave C2 sections with a rectangular base. Asearlier, the C2 structures of Figure 9(b) are identified

by a symmetry axis.Higher order Hamiltonian terms may also create C1

regions as well. For an octahedral RES there will be 24such structures if they arise. Previous work has shownthat the C1 regions are peculiar in that their axes are

not fixed like the higher order subgroup axes are [51].C1 symmetry has no axis of rotational symmetry so

its region has no fixed axis. It may or may not have asingle reflection plane, but where two reflection planesintersect there is a (fixed) C2 rotation axis. The C1structures may fit into several configurations, eachrepresenting a different subgroup chain. The outerRES in Figure 11 in Section 6.2.1 places the C1 axes in

a C4 configuration, obeying the O � C4 � C1

subgroup chain. Other configurations, such asO � C3 � C1, are known to exist. Because C1 axesfit into multiple configurations, the axes may move,so long as the global symmetry is obeyed. While C4 C3and C2 axes are called critical stationary axes,C1 axes are not fixed with changing Hamiltonianparameters, though they may support stable states. C4C3 and C2 axes will remain in the same location on theRES independent of the Hamiltonian fitting parameter(so long as the Hamiltonian remains an octahedralone).

How many such regions exist is found from theratio of the order, G, of the group to the order, H, of

C3C3

C3C3

C4

C4

C4 C4

(a)

C2

C2

C2

C2

C2

C2

(b)

Figure 9. Local symmetry axes of globally octahedral RESplots. The two plots are built of the same operators, but showdifferent local symmetry features because of different fittingrelated to the rotational Hamiltonian. (a) RES contains eightC3 and six C4 axes. All visible axes are indicated. (b) REScontains 12 C2 axes along with eight C3 and six C4 axes. Onlyvisible C2 axes are shown.


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the local subgroup. As is shown in Figure 9, this gives a

smaller number of identical phase-space regions for

higher symmetry subgroups. Considering only regions

which maintain a symmetry plane, the number of

equivalent regions is G=H ¼ 24. Regions withoutsymmetry planes would make G=H ¼ 48. Monstrous48-fold regions are thought to be possible, but have

never been seen.Inversion tunnelling of CF4 nuclei have extremely

small amplitude and therefore of little consequence for

semi-classical calculations that do not include spin

interactions and hyperfine structure. However, inver-

sion and reflection tunnelling is significant at the

hyperfine and superhyperfine levels as shown in

detailed studies of SF6 and this will strongly mix

certain Oh symmetry species that are tightly clustered

with sub-megaHertz splitting [52]. The current work

does not involve such detailed resolution so the

inversion state-labelling u and g is dropped for the

sake of simplicity. The full Oh symmetry labelling

involves several more sub-group hierarchies that are

super-groups of the C2, C3, and C4 axial subgroups

considered here. But we have not found any spectro-

scopic effects that required this added complexity,

so far.The level structure within each cluster will also be

affected by the local geometry of the RES. Rotational

level clustering has been well known for decades and

well explained by internal symmetry breaking [45]. This

is most easily understood in terms of the RES. For a

molecule of point symmetry G, a RES will have aglobal symmetry of G, but rotational energy levels willsit on a section of the RES with a symmetry

corresponding to a subgroup H � G. Each energylevel will be able to tunnel to the corresponding level

(contour) at the same altitude (energy) on all the other

equivalent H regions. The symmetry breaking G # Hforces rotational levels to cluster in patterns described

by the restricted representation or symmetry correla-

tion table. The O C4, O C3 and O C2 correla-tion tables are printed in Equation (12). There are

multiple O C2 correlation tables because of themultiple subgroup chains connecting these two

symmetries.The angular momentum is thus, localized along one

of the possible subgroup regions. Levels clusters in a

given local symmetry path will show internal symmetry

behaviour related to the columns of Equation (12),

giving many levels labelled by symmetry G inside acluster of symmetry H. For Cn subgroups, the cluster-ing will cycle through the mn groupings. The uncer-

tainty cone can determine which mn clustering will be

seen. Clusters in a Cn local symmetry region which

match to a Jz uncertainty cone will obey the clusteringrelated to the Jzn column of (12).

O ⊃ C4 04 14 24 34A1 ↓C4 1 · · ·A2 ↓C4 · · 1 ·E ↓ C4 1 · 1 ·T1 ↓C4 1 1 · 1T2 ↓C4 · 1 1 1

O ⊃ C3 03 13 23A1 ↓C3 1 · ·A2 ↓C3 1 · ·E ↓ C3 · 1 1T1 ↓C3 1 1 1T2 ↓C3 1 1 1

O ⊃ C2(i1) 02 12A1 ↓C2 1 ·A2 ↓C2 · 1E ↓ C2 1 1T1 ↓C2 1 2T2 ↓C2 2 1

O ⊃ C2(ρz) 02 12A1 ↓C2 1 ·A2 ↓C2 1 ·E ↓ C2 2 ·T1 ↓C2 1 2T2 ↓C2 1 2

ð12Þ

Tunnelling between equivalent local subgroupregions causes the rotational clusters to split slightly.A detailed description of these effects can be found in[45] and a more contemporary analysis of symmetry-based tunnelling can be found in [53].

6.2. Analysis

6.2.1. Rotational band boundaries

This section will attack the first two uses of RESanalysis listed in Section 6.1.1 to evaluate the �3=2�4polyad band of CF4. Each RES of the REES repre-sents a band of rotational energy levels. The bands mayinteract with each other through phase-space tunnel-ling, though the effect is small until the surfaces areclose to one another. The minimum and maximum ofthe RES represent classical bounds to the quantumband. Quantum levels outside the classical boundariesmay correspond to a breakdown of the semi-quantumtreatment particularly when, for example, two surfacesare close enough to produce a conical intersection.Close proximity of two or more RES may involveother types of non-classical behaviour that are beyondthe scope of this study.

Figure 10 plots quantum rovibrational energy withchanging J along with the height of the C4, C3 and C2axes for each RES in the �3=2�4 polyad. Evaluating theheight (energy) of these subgroup axes is computa-tionally simpler than numerical search for maxima andminima. Axes of local symmetry C2, C3, and C4 locatemaxima or minima in all cases where maxima andminima are not on a C1 stationary point. By definition,any point not corresponding to one of these stationarypoints must not be a maximum or minimum, else themolecular symmetry is not respected. However, any ofthese critical points may be a maximum or minimum,


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depending on J and the Hamiltonian fitting parame-ters. As discussed below, C1 stationary points are moredifficult to find and are seen to be quite small changesfrom a neighbouring C3 or C4 stationary point.

Tracking the energy along each axis (rather thanonly the maximum and minimum) makes transitions inclustering patterns more visible. These transitionsoccur along saddles, often along C2 axes. C3 and C4‘monkey-saddles’ are rare and only possible for specificvalues of the Hamiltonian fitting parameters [51].

It is possible to have stable maximum or minimumpoints of C1 symmetry whose location is not fixed by asymmetry axis. (C1 has no critical axis.) The height ofof C1 points thus require a numerical search. Althoughrare, such C1 structures do exist in �3=2�4 for someparts of the REES at high J. The parameters requiredto form C1 structures have not been fully explored.

Figure 11 shows a RES which includes evidence ofC1 local structure. Their effect seen here seems small,

but they do explain some of the clusters that lie

just outside C2, C3, and C4 bounds particularly at

higher values of J. Scanning all RES showing C1symmetry regions for all J included in Figure 10, it is

clear that not including these maxima and minima in

Figure 10 misses at most such a single cluster RES

where these clusters are possible. Far greater diver-

gence is shown to originate from interactions with

neighbouring RES.Correspondence between the classical band bound-

aries and the quantum levels is clear for most of

Figure 10. Regions of Figure 10 which diverge the most

are in the range of J> 50 on the third vibrational band.As mentioned earlier, close phase-space proximity of

more than two RES causes cluster avoidances not

predicted by the semi-quantum REES. Levels below

the third band in the J 60 region exhibit thisbehaviour. Details of this reordering will have appli-

cations to related theory, but is outside the scope of the

present work.The fifth band near J¼ 60 has a single cluster

which is out of the semi-quantum boundaries. Closer

inspection shows this band to be built of a RES with C1clusters as global maxima, which is shown in Figure 11.

Although the C1 maxima extends past the band

boundaries, the highest cluster is still beyond the

semi-classical approximation. This is a result of inter-

actions between local C2 regions of two neighbouring

RES, as seen in [41].

1300

1290

1280

1270

1260

1250

1240

Red

uced

wav

enum

ber/

cm–1

706050403020100J

Quantum levelsC4 axesC3 axesC2 axes

3

2 4

Figure 10. Semi-quantum outlines show band boundaries forthe rotational levels of �3=2�4. Reduced energy is definedas the quantum energy subtracted by all scalar fullyrotational terms. In this way, the reduced energy shows theenergy splittings without the energy shifts related only toincreasing J.

C1C1

C1

C1

C1

C1

C1

C1

C1

Figure 11. C1 local symmetry structures are rare, but we seesome at J¼ 57 on the fifth surface from the bottom.Others do exist, but this one is amongst the clearest at thisrange of J.


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Clusters do fit more tightly in low J portions of

Figure 10. Slight disagreement found at low J is

thought to be of different origin from those at high J.

Lack of correspondence at low J can be easily thought

of as a difference between classical and quantum

rotational operators. This is distinct from the lack of

correspondence at higher J which is likely from the

simplification of the outer product of the rotational

and vibrational Hamiltonians in the semi-quantum

approximation. Thus, this difference can be considered

a type of Born–Oppenheimer breakdown.

6.2.2. Rotational level clustering

While taken all at once, the nine surface REES,

as shown in Figure 8, are of limited utility. Taken

individually they offer an explanation as to the origin of

rotational level clusters seen in the �3=2�4 band.Individual RES plots are shown in Figure 12. The

plots are taken of the J¼ 60 rotational energy surfaces,starting with the fourth rotational band and increasing

to the sixth of nine. These were chosen because they are

of high enough J to have a significant number of

rotational clusters (RES contours) on each surface.

Figure 12. CF4 �3=2�4 RES plots. Surfaces are labelled starting from the centre going out. Looking at surfaces one by one showstheir individual geometry and indicates how the level clusters (contours) must arrange themselves. We include only four surfacesas examples, but many are examined in the analysis. (a) J¼ 60, third surface has C4 and C3 local symmetry. (b) J¼ 60, fourthsurface has C4 local symmetry. (c) J¼ 60, fifth surface has C4 and C3 local symmetry. (d) J¼ 60, sixth surface has C3 and C2 localsymmetry.


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They each show local regions of very differentsubgroup symmetry while still showing clear globaloctahedral symmetry.

There are several noteworthy features in Figure 12.All RES plots show excellent agreement between theminimum uncertainty cone and accompanying levelcluster. Figure 12(d) makes this slightly less obvioussince this cluster is found along the C2 axis and isdeformed into an ellipse. Figures 12(c) and 12(d) alsoshow clusters matching this cone, but placed along theC3 axis. Some clusters do not match intersections withuncertainty cones. While this is common near sepera-trix regions, as one would expect given the shift in localsubgroup symmetry, it occasionally happens for higherJz cones as J increases.

Importantly, clusters placed on each region doshow the internal symmetry and degeneracy expectedof the given local symmetry region. The only notableexception to this is when neighbouring RES plotsoverlap in range. Although the surfaces cannot cross,the maxima of one may extend above the minima ofthe next. In such cases a RES may show a contour thatbelongs to a different RES. This is easiest to spot whenthe improperly placed cluster either shows the cluster-ing related to one RES and not the other or when thecluster obeys the angular momentum uncertainty forone RES and not the other.

7. Conclusion and perspectives

Thanks to newly recorded FTIR spectra, we have beenable to perform a simultaneous fit of effectiveHamiltonian parameters of the ground state, �4 and�3=2�4 polyads of CF4. We used a tensorial modeldeveloped in the Dijon group. It was also possible to fitthe dipole moment derivatives for both the �4 and �3bands and to estimate the value for the 2�4 band. Allthis allowed us to produce a synthetic linelist that hasbeen recently included int the HITRAN 2008 [54] andGEISA 2009 public databases [55]. It should be notedthat CF4 is now referenced as molecule number 42 inHITRAN and molecule number 49 in GEISA.Previously, these two databases only gave cross-sections for this species, which was referenced asCFC-14. The new linelist, although it is still approx-imate concerning line intensities, should be useful foratmospheric applications.

From the semi-classical point of view, semi-quantum Rotational Energy Surface plots can predictseveral important spectral features for the �3=2�4polyad band of CF4. The semi-classical tools ofangular momentum uncertainty cones works withsome success in this sort of system. While for

vibrational singlets it is highly predictive, for thispolyad system the technique can lose precision as Jincreases. As in the case of vibrational singlets, angularmomentum uncertainty cones are most accurate atpredicting the energies of clusters that are morelocalized to a single axis or showing less K mixing.

The RES are effective at finding the boundaries ofthe rotational bands for a given angular momentum J.The RES are perhaps the most useful in predicting therotational clusters that originate from symmetryreduction from the global molecular symmetry to thelocal subgroup symmetry regions of the RES. This istypically done for RES for vibrational singlets. Forpolyads it is more efficiently done with an energy plotusing semi-quantum outlines as in Figure 10. The semi-quantum outlines help to diagnose which of thepossible local symmetry regions can be produced bysymmetry reduced clusters.

While some of this analysis has been done previ-ously, this work shows that it can be extended to asystem with significant rotation–vibration couplingwhich also requires many RES.

Acknowledgements

We acknowledge support from the ‘SpecMo’ GdR number3152 of the CNRS. This work was supported by the DeutscheForschungsgemeinschaft (DFG).

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J.-Y. Mandin, A. Maki, S. Mikhailenko, C.E. Miller,T. Mishina, N. Moazzen-Ahmadi, H.S.P. Müller,

A. Nikitin, J. Orphal, V. Perevalov, A. Perrin,D.T. Petkie, A. Predoi-Cross, C.P. Rinsland,J.J. Remedios, M. Rotger, M.A.H. Smith, K. Sung,S. Tashkun, J. Tennyson, R.A. Toth, A.-C. Vandaele

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Appendix 1. Effective Hamiltonian parameters

Table 5 gives the list of fitted effective Hamiltonianparameters. Standard deviation is given in parentheses, inthe unit of the last two digits. See Section 3 for detailsconcerning the notations. In the last column, when possible,we give the correspondence with the ‘usual’ notation ofRobiette et al. [36].

Table 5. Effective Hamiltonian parameter values. The standard deviation is given in parentheses, in the unit of the last twodigits. Only non-zero fitted parameters are shown.

Polyad Order OðK, nCÞ fsgC1 fs0gC2 Value / cm�1 ‘Usual’ notation [36]

GS 0 2(0,0A1) 0000A1 0000A1 1.9119312(46)�10�1 B0GS 2 4(0,0A1) 0000A1 0000A1 �6.13(13)�10�8 �D0GS 2 4(4,0A1) 0000A1 0000A1 �3.028(25)�10�9 �ð151=2=4ð21=2ÞÞD0tGS 4 6(0,0A1) 0000A1 0000A1 9.07(59)�10�12 H0GS 4 6(4,0A1) 0000A1 0000A1 �4.572(63)�10�13 3ð51=2Þ=16ð21=2ÞÞH4tGS 4 6(6,0A1) 0000A1 0000A1 �1.205(21)�10�13 �(2311/2/64(21/2))H4tGS 6 8(0,0A1) 0000A1 0000A1 �9.61(87)�10�16 L0GS 6 8(4,0A1) 0000A1 0000A1 �1.5(3.9)�10�19 �(3(151/2)/64(21/2))L4tGS 6 8(6,0A1) 0000A1 0000A1 5.4(2.7)�10�19 ð3ð771=2Þ=256ð21=2ÞÞL6tGS 6 8(8,0A1) 0000A1 0000A1 1.407(66)�10�18 ð1=32ð331=2ÞÞL8t�4 0 0(0,0A1) 0001F2 0001F2 631.059247(87) �4�4 1 1(1,0F1) 0001F2 0001F2 �2.948885(87)�10�1 3ð21=2ÞB4ð�4 CoriolisÞ�4 2 2(0,0A1) 0001F2 0001F2 1.0118(29)�10�4 B4 � B0�4 2 2(2,0E) 0001F2 0001F2 �8.067(41)�10�5 �ð1=2Þ�4220 � 6�4224�4 2 2(2,0F2) 0001F2 0001F2 1.8099(63)�10�4 �ð3=4Þ�4220 þ 6�4224�4 3 3(1,0F1) 0001F2 0001F2 �4.078(66)�10�7 �ð3ð31=2Þ=4ð21=2ÞF4110�4 3 3(3,0F1) 0001F2 0001F2 �5.478(32)�10�7 ð3=51=2=2ÞF4134�4 4 4(0,0A1) 0001F2 0001F2 �1.20(18)�10�9 �ðD4 �D0Þ�4 4 4(2,0E) 0001F2 0001F2 �3.173(38)�10�8 . . .�4 4 4(2,0F2) 0001F2 0001F2 2.828(37)�10�8 . . .�4 4 4(4,0A1) 0001F2 0001F2 2.121(49)�10�9 . . .�4 4 4(4,0E) 0001F2 0001F2 5.321(54)�10�8 . . .�4 4 4(4,0F2) 0001F2 0001F2 3.765(45)�10�8 . . .�4 5 5(1,0F1) 0001F2 0001F2 1.34(13)�10�11 . . .�4 5 5(3,0F1) 0001F2 0001F2 1.854(34)�10�10 . . .�4 5 5(5,0F1) 0001F2 0001F2 2.447(40)�10�10 . . .�4 5 5(5,1F1) 0001F2 0001F2 2.664(86)�10�11 . . .�4 6 6(0,0A1) 0001F2 0001F2 1.14(35)�10�13 . . .�4 6 6(2,0E) 0001F2 0001F2 �2.160(38)�10�12 . . .�4 6 6(2,0F2) 0001F2 0001F2 1.968(46)�10�12 . . .�4 6 6(4,0A1) 0001F2 0001F2 8.10(39)�10�14 . . .�4 6 6(4,0E) 0001F2 0001F2 3.108(61)�10�12 . . .�4 6 6(4,0F2) 0001F2 0001F2 2.259(84)�10�12 . . .�4 6 6(6,0A1) 0001F2 0001F2 1.24(53)�10�14 . . .�4 6 6(6,0E) 0001F2 0001F2 4.60(25)�10�13 . . .�4 6 6(6,0F2) 0001F2 0001F2 2.81(73)�10�13 . . .�4 6 6(6,1F2) 0001F2 0001F2 6.49(36)�10�13 . . .2�4=�3 0 0(0,0A1) 0010F2 0010F2 1283.460(22) �32�4=�3 1 1(1,0F1) 0010F2 0010F2 6.407(15)�10�1 3ð21=2ÞB3ð�3 CoriolisÞ2�4=�3 2 2(0,0A1) 0010F2 0010F2 �4.126(97)�10�4 B3 � B02�4=�3 2 2(2,0E) 0010F2 0010F2 1.444(89)�10�4 �ð1=2Þ�3220 � 6�32242�4=�3 2 2(2,0F2) 0010F2 0010F2 �4.89(13)�10�4 �ð3=4Þ�3220 þ 6�32242�4=�3 3 3(1,0F1) 0010F2 0010F2 �9.2(1.6)�10�7 �ð3ð31=2Þ=4ð21=2ÞF3110

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Table 5. Continued.


2�4=�3 3 3(3,0F1) 0010F2 0010F2 �1.878(96)�10�6 ð3=51=2=2ÞF31342�4=�3 4 4(0,0A1) 0010F2 0010F2 3.18(14)�10�8 �ðD3 �D0Þ2�4=�3 4 4(2,0E) 0010F2 0010F2 �1.53(35)�10�8 . . .2�4=�3 4 4(2,0F2) 0010F2 0010F2 4.40(33)�10�8 . . .2�4=�3 4 4(4,0A1) 0010F2 0010F2 9.20(33)�10�9 . . .2�4=�3 4 4(4,0E) 0010F2 0010F2 7.83(49)�10�8 . . .2�4=�3 4 4(4,0F2) 0010F2 0010F2 6.38(34)�10�8 . . .2�4=�3 5 5(1,0F1) 0010F2 0010F2 �7.61(65)�10�11 . . .2�4=�3 5 5(3,0F1) 0010F2 0010F2 �9.16(31)�10�10 . . .2�4=�3 5 5(5,0F1) 0010F2 0010F2 �9.72(35)�10�10 . . .2�4=�3 5 5(5,1F1) 0010F2 0010F2 �1.775(82)�10�10 . . .2�4=�3 6 6(0,0A1) 0010F2 0010F2 �7.48(22)�10�12 . . .2�4=�3 6 6(2,0E) 0010F2 0010F2 �5.94(34)�10�12 . . .2�4=�3 6 6(2,0F2) 0010F2 0010F2 6.38(33)�10�12 . . .2�4=�3 6 6(4,0A1) 0010F2 0010F2 �3.97(30)�10�13 . . .2�4=�3 6 6(4,0E) 0010F2 0010F2 6.5(7.0)�10�13 . . .2�4=�3 6 6(4,0F2) 0010F2 0010F2 7.20(46)�10�12 . . .2�4=�3 6 6(6,0A1) 0010F2 0010F2 �7.59(21)�10�13 . . .2�4=�3 6 6(6,0E) 0010F2 0010F2 7.10(20)�10�12 . . .2�4=�3 6 6(6,0F2) 0010F2 0010F2 4.39(44)�10�12 . . .2�4=�3 6 6(6,1F2) 0010F2 0010F2 8.33(26)�10�12 . . .2�4=�3 3 2(2,0F2) 0010F2 0002A1 �1.22(29)�10�4 �3 � 2�4ðA1Þ interaction2�4=�3 4 3(3,0F2) 0010F2 0002A1 9.12(30)�10�6 �3 � 2�4ðA1Þ interaction2�4=�3 2 1(1,0F1) 0010F2 0002E �9.11(28)�10�2 �3 � 2�4ðEÞ interaction2�4=�3 3 2(2,0F2) 0010F2 0002E 8.05(62)�10�4 �3 � 2�4ðEÞ interaction2�4=�3 4 3(1,0F1) 0010F2 0002E �3.59(37)�10�6 �3 � 2�4ðEÞ interaction2�4=�3 4 3(3,0F1) 0010F2 0002E 5.90(31)�10�6 �3 � 2�4ðEÞ interaction2�4=�3 4 3(3,0F2) 0010F2 0002E �7.21(30)�10�6 �3 � 2�4ðEÞ interaction2�4=�3 1 0(0,0A1) 0010F2 0002F2 2.44(10) �3 � 2�4ðF2Þ interaction (Fermi)2�4=�3 2 1(1,0F1) 0010F2 0002F2 7.94(57)�10�2 �3 � 2�4ðF2Þ interaction2�4=�3 3 2(0,0A1) 0010F2 0002F2 �5.14(27)�10�4 �3 � 2�4ðF2Þ interaction2�4=�3 3 2(2,0E) 0010F2 0002F2 5.92(22)�10�4 �3 � 2�4ðF2Þ interaction2�4=�3 3 2(2,0F2) 0010F2 0002F2 4.11(46)�10�4 �3 � 2�4ðF2Þ interaction2�4=�3 4 3(1,0F1) 0010F2 0002F2 4.08(32)�10�6 �3 � 2�4ðF2Þ interaction2�4=�3 4 3(3,0F1) 0010F2 0002F2 2.49(20)�10�6 �3 � 2�4ðF2Þ interaction2�4=�3 4 3(3,0F2) 0010F2 0002F2 4.93(38)�10�6 �3 � 2�4ðF2Þ interaction2�4=�3 2 0(0,0A1) 0002A1 0002A1 �1.637(15)�10�1 2�4ðA1Þ sublevel2�4=�3 4 2(0,0A1) 0002A1 0002A1 �1.473(17)�10�4 . . .2�4=�3 4 2(2,0E) 0002A1 0002E 2.4612(92)�10�4 2�4ðA1Þ � 2�4ðEÞ interaction2�4=�3 4 2(2,0F2) 0002A1 0002F2 �5.337(37)�10�4 2�4ðA1Þ � 2�4ðF2Þ interaction2�4=�3 5 3(3,0F2) 0002A1 0002F2 �3.14(16)�10�6 2�4ðA1Þ � 2�4ðF2Þ interaction2�4=�3 2 0(0,0A1) 0002E 0002E �8.61(24)�10�3 2�4ðEÞ sublevel2�4=�3 4 2(0,0A1) 0002E 0002E �2.35(22)�10�4 . . .2�4=�3 4 2(2,0E) 0002E 0002E �2.10(19)�10�4 . . .2�4=�3 5 3(3,0A2) 0002E 0002E �1.44(13)�10�6 . . .2�4=�3 3 1(1,0F1) 0002E 0002F2 �4.79(62)�10�3 2�4ðEÞ � 2�4ðF2Þ interaction2�4=�3 4 2(2,0F2) 0002E 0002F2 1.02(13)�10�4 2�4ðEÞ � 2�4ðF2Þ interaction2�4=�3 5 3(1,0F1) 0002E 0002F2 �1.94(11)�10�6 2�4ðEÞ � 2�4ðF2Þ interaction2�4=�3 5 3(3,0F1) 0002E 0002F2 2.208(70)�10�6 2�4ðEÞ � 2�4ðF2Þ interaction2�4=�3 5 3(3,0F2) 0002E 0002F2 8.29(59)�10�7 2�4ðEÞ � 2�4ðF2Þ interaction2�4=�3 2 0(0,0A1) 0002F2 0002F2 �1.428(22) 2�4ðF2Þ sublevel2�4=�3 3 1(1,0F1) 0002F2 0002F2 4.72(15)�10�2 2�4ðF2Þ Coriolis2�4=�3 4 2(0,0A1) 0002F2 0002F2 2.6(2.1)�10�5 . . .2�4=�3 4 2(2,0E) 0002F2 0002F2 �2.77(27)�10�4 . . .2�4=�3 4 2(2,0F2) 0002F2 0002F2 2.35(23)�10�4 . . .2�4=�3 5 3(1,0F1) 0002F2 0002F2 3.43(24)�10�6 . . .2�4=�3 5 3(3,0F1) 0002F2 0002F2 3.23(20)�10�6 . . .2�4=�3 6 4(0,0A1) 0002F2 0002F2 �1.58(14)�10�8 . . .

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Table 5a. Continued.


2�4=�3 6 4(2,0E) 0002F2 0002F2 �2.59(10)�10�8 . . .2�4=�3 6 4(2,0F2) 0002F2 0002F2 1.34(12)�10�8 . . .2�4=�3 6 4(4,0A1) 0002F2 0002F2 �2.051(31)�10�8 . . .2�4=�3 6 4(4,0E) 0002F2 0002F2 �2.19(82)�10�9 . . .2�4=�3 6 4(4,0F2) 0002F2 0002F2 �3.200(71)�10�8 . . .


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High-resolution spectroscopy and analysis of the ν3/2ν4 …gases among the longest-lived...

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Transcript of High-resolution spectroscopy and analysis of the ν3/2ν4 …gases among the longest-lived...