Engineering Sciences Reports, Kyushu University
Vol. 43, No. 1, pp. 1-7, SEP, 2021

九州大学大学院総合理工学府報告第 43 巻 第 1 号 1-7 頁 令和 3 年 9 月

RKR Franck–Condon Factors and r– centroids of thed3g−a3u, e3g−a3u, C1g−A1u, and D1u+−X1g+
Band Systems of C2 Radical
Masaharu TSUJI*1,2† Takahiro KOMATSU*3 Keiko UTO*1
Jun-Ichiro HAYASHI*1,2 and Takeshi TSUJI*4
E-mail of corresponding author: tsuji@cm.kyushu-u.ac.jp†(Received May 24, 2021, accepted June 7, 2021)
Rydberg-Klein-Rees (RKR) Franck–Condon factors and r-centroids were computed for the d3g−a3u Swansystem, the e3g−a3uFox-Herzberg system, the C1g−A1u Deslandres-d'Azambuja system, and the D1u+−X1g+
Mulliken system of C2 radical for the 𝑣 = 0–8; 𝑣 = 0–8 matrix using reported molecular constants.

Key words: C2 radical, RKR potential, Franck-Condon factor, r– centroid

1.

Introduction

The C2 radical is found in hydrocarbon
flames, electrical discharges through carbon
containing materials, carbon arc emission,
heads of comets, and the atmospheres ofcertain types of stars. Many emission andabsorption band systems between various
electronic states of C2 radical have been
observed in a wide spectral range.1-12)
Molecular constants of C2 radical obtained fromhigh-resolution vibrational and rotational
analyses have been complied by Huber andHerzberg.13)
In
order
to
determinerovibrational population of C2 radical fromemission spectra, Franck-Condon factors(FCFs) are necessary. Nicholls calculated FCFs
of several transitions of C2 radical using Morsepotentials.14) When Rydberg-Klein-Rees (RKR)
potentials are used,15-17) more reliable FCFs canbe obtained.18)
In this study, FCFs and r– centroids of thed−a, e−a, C−A, and D−X band systems of C2
radical are calculated using RKR potentials.
Results obtained are compared with previously
reported data.14,19,20) FCFs obtained in thiswork is used for the determination of*1

Institute for Materials Chemistry and Engineering,and Research and Education Center of GreenTechnology*2 Department of Applied Science for Electronics andMaterials*3 Department of Applied Science for Electronics andMaterials, Graduate Student*4 Department of Materials Science, Shimane
University

rovirational distributions of the d, e, C, and D
states of C2 radical formed by the energytransfer reactions of metastable Ne(3P0,2) atoms
with C2H2 and C2D2.21,22)

FCF and r– centroids2.

2.1. Franck–Condon factors (FCF)
The expression for band intensities inemission is given as
𝐼 ′ ′ ′′ ′′ ∝ 𝑁 ′ ′ 𝑅𝑒𝑟̄ ′ ′′ 𝑞 ′ ′′ 𝜈 ′ ′ ′′ ′′ 𝑆 ′ ′′ 𝑔 ′(1)

where 𝑁 ′ ′ is the rotational population in agiven vibrational level, 𝑅𝑒 𝑟̄ ′ ′′ the electronictransition moment, 𝑞 ′ ′′ the FCF, 𝜈 ′ ′ ′′ ′′ thetransition frequency, 𝑆 ′ ′′ the rotational line
strength, and 𝑔 ′ 2𝐽′ 1.23) FCFs are expressed
as,
𝑞

𝛹

𝑟 𝛹

𝑟 𝑑𝑟

(2)

where 𝛹 𝑟 and 𝛹 𝑟 are the wave
functions of 𝑣 and 𝑣 levels.
2.2. r– centroids
r– centroids (𝑟̄ ′ ′′ ) is the r–coordinate of thecentroid of the area represented by the overlap
integral and defined as:𝑟̄ ′ ′′(3)

2

RKR Franck-Condon Factors and r-centroids of the C2(e−a, d−a, C−A, D−X) Transitions

3.

Results and discussion

3.1
Energy-level diagram and emission
systems.
Figure 1 shows energy-level diagram of C2
radical in the a3u d3g, and e3g triplet statesand in the X1g+, A1u, C1g, and D1u+ singlet
states. Their molecular constants are given in
Table 1. The a3u state is located just above theground X1g+ state. Among these states, thed3g−a3u Swan system, the e3g−a3uFoxHerzberg system, the C1g−A1u Deslandresd'Azambuja system, and the D1u+−X1g+
Mulliken system are known to occur. As shownin the names of above emission systems, two

６

e3

５

Nobel prize winners, Herzberg and Mulliken,
contributed to the first observation of twoemission systems and the determination ofmolecular states of C2 radical. This indicatesthat C2 radicals are an important diatomic
radical for the understanding of molecular
structure and reactions of carbon-containing
molecules in flame, discharge, and comet.
Among molecular constants given in Table 1,
equilibrium internuclear distance (re) andvibrational frequency (e) are two key
parameters, which affect FCFs. The re values ofthe d, C, and D states are close to the ground X
state, whereas they increase by factors of 5.6,
6.1, and 23.5% for the a, A and e states,

D1u+
g

C1g
４

Fox-Herzberg
３

Mullikend3gDeslandres-d'Azambuja

２

Swan

A1  u

a3u

X1g+

１
０

Fig. 1. Energy-level diagram and emission systems of C2 radicals, for which RKR FCFs andr–centroids are calculated.
Table. 1. Molecular constants of C2 molecules (Ref. 13) used for the calculations of RKR FCFs
and r– centroids.
Te

e

exe

Be

e

De

re

(cm-1)(cm-1)(cm-1)(cm-1)(cm-1)(10-6 cm-1)

(Å)

D1u+

43239.44

1829.57

13.94

1.83320.01967.32

1.2380e3g40796.65

1106.56

39.2601.19220.0242

6.3

1.5351

C1g34261.3

1809.115.81

1.78340.01806.8

1.25523d g20022.50

1788.2216.4401.75270.01608

6.74

1.2661A1u8391.00

1608.35

12.0781.61634

0.01686

6.44

1.31843a3u716.241641.35

11.67

1.63246

0.01661

6.44

1.31190X1g+

01854.71

13.3401.81984

0.01765

6.92

1.24253

State令和 3 年度

九州大学大学院総合理工学報告

respectively, due to increasing the antibonding
character of the molecular orbital of an excitedelectron. The e value decreases by 1.4, 2.5, 3.6,
11.5, 13.3, and 40.3% for the D, C, d, a, A, ande states in comparison with that of the groundX state, respectively. Similar equilibriumgeometries for the D and X states lead to small
𝛥 re and 𝛥 e, thus narrow FC envelopes are
expected for the D−X transition. On the other
hand, relatively large 𝛥 re and 𝛥 e values
between the d and a states, between the e anda states, and between the C and A states will
give broad FC envelopes.
3.2 RKR FCFs and r– centroids
Tables 2−5 show RKR FCFs and r– centroidsof the d−a, e−a, C−A, and D−X transitions of C2
radical, respectively. RKR FCFs of thesetransitions are plotted for 𝑣 and 𝑣 in Figs.
2−5. RKR FCFs for the d−a, e−a, and C−Atransitions obtained in this study are similar to
those of Morse FCFs reported by Nicholls.14)
FCFs and r– centroids for the d−a and e−atransitions are similar to those of RKR FCFs
and r– centroids reported by Jain.19) FCFs andr– centroids for the D−X transition are in
reasonable agreement with the reported data of
Sorkhabi et al. calculated using RKR
potentials.20) Based on our present calculations,the following tendencies are found.
(a) 𝛥𝑣 = 1, 0, and −1 sequences are dominant
for the d−a Swan system for which 𝛥 re
(3.6%) and 𝛥 e (−8.2%) values are
relatively large (Table 2 and Fig. 2).
(b) 𝑣 = 0 and 1 progressions are dominant for
the e−a Fox-Herzberg system for whichvery large differences exist in 𝛥re (−15%)
and 𝛥e (48%) values (Table 3 and Fig. 3).
(c) 𝛥𝑣 = 3, 2, 1, 0, and −1, −2, and −3 sequences
are dominant for the C−A Deslandresd’Azambuja system for which 𝛥 re (5.0%)
and 𝛥e (−11%) values are relatively large(Table 4 and Fig. 4).
(d) Only 𝛥𝑣 = 0 sequence is dominant for the
D−X Mulliken system for which 𝛥re (0.37%)
and 𝛥e (1.4%) values are very small (Table
5 and Fig. 5).

4.

Summary and Conclusion

RKR FCFs and r– centroids were computedfor the d–a, e–a, C–A, and D–X band systems
for the 𝑣 = 0–8; 𝑣 = 0–8 matrix using
reported precise molecular constants. Results
obtained were in reasonable agreement with

第 43 巻第1号

3

reported data. Reliable data obtained in thisstudy can be used to determine rovibrarional
distributions of d, e, C, and D states of C2
radical produced in various conditions (flame,
plasma discharge, and comet).
Acknowledgments
This work was supported by the Iwatani
Naoji Memorial Foundation (1994−1996), theMitsubishi Foundation (1996), and JSPS
KAKENHI Grant Numbers JP0643026,
JP09440201.
References
1) G. H. Dieke, W. Lochte-Holtgreven. Z. Phys., 62, 767(1930).
2) J. G. Fox and G. Herzberg, Phys. Rev., 52, 638 (1937).
3) O. G. Landsverk, Phys. Rev., 56, 769 (1939).
4) G. Herzberg and R. B. Sutton, Can. J. Res., A18, 74(1940).
5) (a) J. G. Phillips, Astrophys. J., 107, 389 (1948). (b) J.
G. Phillips, Astrophys. J., 108, 434 (1948). (c) J. G.
Phillips, Astrophys. J., 110, 73 (1949).
6) V. H. Freymark, Ann. Phys., 8, 221 (1951).
7) R. G. W. Norrish, G. Porter, and B. A. Thrush, Nature,
169, 582 (1952).
8) (a) E. A. Ballik and D. A. Ramsay, J. Chem. Phys., 29,1418 (1958). (b) E. A. Ballik and D. A. Ramsay, J. Chem.
Phys., 31, 1128 (1959).
9) D. C. Tyte, S. H. Innanen, and R. W. Nicholls,“Identification Atlas of Molecular Spectra, 5, The C2
Swan System ”, York University, Toronto (1967).
10) G . Messerle, Z. Naturforsch. A, 23a, 470 (1968).
11) J. G. Phillips and S. P. Davis, “Berkeley Analyses of
Molecular Spectra, 2”, University of California Press(1968).
12) H. Cisak, K. Dabrowska, and M. Rytel, Acta Phys.
Polon., 36, 497 (1969).
13) K. P. Huber and G. Herzberg, “Molecular Spectra andMolecular Structure. IV. Constants of Diatomic
Molecules ”, Van Nostrand Reinhold, New York (1979).
14) R. W. Nicholls, J. Res. Natl. Bur. Stnd. A. Phys. Chem.,

69A, 397 (1965).
15) R. Rydberg, Z. Phys., 73, 376 (1931).
16) O. Klein, Z. Phys., 76, 226 (1932).
17) A. L. G. Rees, Proc. Phys. Soc. (London), 59,998 (1947).
18) M. Shimauchi, Bunko Kenkyu (J. Spectrosc. Soc. Jpn.),25, 65 (1976) in Japanese.
19) D. C. Jain, J. Quant. Spectrosc. Radiat. Transf., 4, 427(1964).
20) O. Sorkhabi, D. D. Xu, V. M. Blunt, H. Lin, R. Price, J.
D. Wrobel, and W. M. Jackson. J. Mol. Spectrosc., 188,
200 (1998).
21) M. Tsuji, M. Komatsu, K. Uto, J.-I. Hayashi, and T.
Tsuji, Engineering Sciences Reports, Kyushu
University, 43, 8 (2021).
22) M. Tsuji, M. Komatsu, K. Uto, J.-I. Hayashi, and T.
Tsuji, Engineering Sciences Reports, Kyushu
University, 43, 23 (2021).
23) G. Herzberg, “Molecular Spectra and Molecular
Structure. I. Spectra of Diatomic Molecules ”, Van
Nostrand Reinhold, New York (1950).

4

RKR Franck-Condon Factors and r-centroids of the C2(e−a, d−a, C−A, D−X) Transitions

E.A.Evans, “Tritium and Its Compounds”, Butterworth &Co., London (1996) p.122

Table. 2. RKR FCFs and r– centroids of the d3g−a3u Swan system of C2.
𝑣
0
0
1
2
3

1

2

3

4

5

6

7

a)

0.7263 (0)0.2183 (0)0.4558 (-1)

0.8151 (-2)

0.1379 (-2)

0.2315 (-3)

0.3893 (-4)

0.6371 (-5)

1.29381.2244

1.1681

1.1176

1.07381.03631.0026

0.96650.2453 (0)0.3411 (0)0.2829 (0)0.9905 (-1)

0.2494 (-1)

0.5412 (-2)

0.1096 (-2)

0.2132 (-3)

0.3928 (-4)

1.36911.30701.2317

1.17571.12501.0802

1.04101.00530.9676

0.2746 (-1)

0.3708 (0)0.1367 (0)0.2639 (0)0.1390 (0)0.4614 (-1)

0.1231 (-1)

0.2922 (-2)

0.6440 (-3)

1.4633

1.3791

1.32741.2383

1.1827

1.13181.08641.04561.00800.9617 (-3)

0.6720 (-1)

0.4223 (0)0.4347 (-1)

0.2120 (0)0.1588 (0)0.6648 (-1)

0.2115 (-1)

0.5755 (-2)

1.61381.47611.39051.36791.24401.18911.13791.09201.05000.2626 (-2)

0.1083 (0)0.4369 (0)0.9172 (-2)

0.1565 (0)0.1614 (0)0.8219 (-1)

0.3040 (-1)

1.65181.49051.40351.4833

1.24791.19491.1431

1.09700.1945 (-5)

0.5503 (-5)

0.4020 (-2)

0.1440 (0)0.4397 (0)0.6604 (-3)

0.1101 (0)0.1530 (0)0.9176 (-1)

1.4481

0.32861.70611.50701.41832.1414

1.24821.2002

1.14710.1172 (-4)

0.8489 (-4)

0.4117 (-2)

0.1699 (0)0.4465 (0)0.5515 (-4)

0.7562 (-1)

0.1397 (0)1.5231

1.07751.79561.52591.4349-2.42081.24161.20540.3403 (-4)

0.4296 (-3)

0.2559 (-2)

0.1825 (0)0.4649 (0)0.7606 (-4)

0.5161 (-1)

1.60561.29491.98991.5482

1.4530-2.9078

1.22230.5729 (-4)

0.1306 (-2)

0.4364 (-3)

0.1788 (0)0.4952 (0)0.2000 (-3)

1.7194

1.41192.9154

1.57491.47154.7600

𝑣 4
5
6
7
8

(n) is power of 10 multiplying the entry. Top entry each block FCFs. Bottom entry each block r-centroids in Å.

FCFs

a)

8

Fig. 2. RKR FCFs of the d3g−a3u Swan system of C2.

𝑣”
Fig. 2. RKR FCFs of the d3g−a3u Swan system of C2.

令和 3 年度

九州大学大学院総合理工学報告第 43 巻

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5

Table. 3. RKR FCFs and r– centroids of the e3g−a3u Fox-Herzberg system of C2.
𝑣

0
1
2
3
𝑣 4
5
6
7
8

1

2

3

4

5

6

7

8

0.1701 (-1)

0.6080 (-1)

0.1322 (0)0.1970 (0)0.2144 (0)0.1764 (0)0.1123 (0)0.5612 (-1)

1.4222

1.4434

1.46641.49081.5164

1.5433

1.57171.60201.63400.8810 (-2)

0.5064 (-1)

0.1167 (0)0.1332 (0)0.6606 (-1)

0.2308 (-2)

0.3337 (-1)

0.1241 (0)0.1740 (0)1.40851.4282

1.44981.47271.49571.5022

1.55781.5832

1.61290.1984 (-1)

0.8232 (-1)

0.1135 (0)0.4623 (-1)

0.7553 (-3)

0.6046 (-1)

0.9539 (-1)

0.3354 (-1)

0.2912 (-2)

1.3962

1.4142

1.4342

1.45491.51481.5104

1.53351.55671.61180.3274 (-1)

0.9664 (-1)

0.6902 (-1)

0.7883 (-3)

0.4584 (-1)

0.7112 (-1)

0.8861 (-2)

0.2584 (-1)

0.8969 (-1)

1.38501.4012

1.41891.4136

1.4692

1.49081.5064

1.5516

1.57290.4465 (-1)

0.9153 (-1)

0.2459 (-1)

0.1526 (-1)

0.6763 (-1)

0.1433 (-1)

0.2023 (-1)

0.6927 (-1)

0.1635 (-1)

1.37471.38891.40181.43611.45191.4703

1.50801.52801.5444

0.5362 (-1)

0.7372 (-1)

0.2339 (-2)

0.4505 (-1)

0.4030 (-1)

0.3417 (-2)

0.5803 (-1)

0.1840 (-1)

0.1706 (-1)

1.3652

1.37691.3662

1.4192

1.43461.47381.48671.5061

1.54880.5881 (-1)

0.5152 (-1)

0.2022 (-2)

0.5717 (-1)

0.8230 (-2)

0.3370 (-1)

0.3845 (-1)

0.3661 (-2)

0.5671 (-1)

1.35631.36501.42471.40461.4104

1.44901.46781.5074

1.52350.6014 (-1)

0.3109 (-1)

0.1316 (-1)

0.4760 (-1)

0.5790 (-3)

0.5193 (-1)

0.5103 (-2)

0.3833 (-1)

0.2782 (-1)

1.34801.35281.3912

1.39011.4831

1.43191.44061.48031.50270.5801 (-1)

0.1572 (-1)

0.2508 (-1)

0.2814 (-1)

0.1295 (-1)

0.4057 (-1)

0.3751 (-2)

0.4952 (-1)

0.7054 (-4)

1.3403

1.33951.3762

1.37431.41571.4145

1.46171.46091.4100(n) is power of 10 multiplying the entry. Top entry each block FCFs. Bottom entry each block r-centroids in Å.

FCFs

a)

0
0.2155 (-2)a)𝑣”
Fig. 3. RKR FCFs of the e3g−a3u Fox-Herzberg system of C2.

6

RKR Franck-Condon Factors and r-centroids of the C2(e−a, d−a, C−A, D−X) Transitions

Table. 4. RKR FCFs and r– centroids of the C1g−A1u Deslandres-d'Azambuja system of C2.
𝑣

0
1
2
3
𝑣 4
5

0

1

2

3

4

5

6

7

8

0.5447 (0)a)

0.3048 (0)0.1082 (0)0.3145 (-1)

0.8224 (-2)

0.2031 (-2)

0.4871 (-3)

0.1156 (-3)

0.2754 (-4)

1.29131.24181.19901.16081.1263

1.09471.06571.0392

1.01500.3558 (0)0.8518 (-1)

0.2622 (0)0.1794 (0)0.7812 (-1)

0.2738 (-1)

0.8508 (-2)

0.2467 (-2)

0.6884 (-3)

1.34681.30911.25081.2066

1.16801.1334

1.10181.07281.04630.8856 (-1)

0.3841 (0)0.2837 (-3)

0.1398 (0)0.1859 (0)0.1172 (0)0.5343 (-1)

0.2042 (-1)

0.7032 (-2)

1.41101.35651.0082

1.2617

1.2146

1.17541.1405

1.10881.07990.1038 (-1)

0.1901 (0)0.2869 (0)0.4693 (-1)

0.4497 (-1)

0.1470 (0)0.1348 (0)0.7894 (-1)

0.3660 (-1)

1.4894

1.41981.36771.2854

1.2784

1.2233

1.18291.1476

1.11590.5592 (-3)

0.3339 (-1)

0.2691 (0)0.1688 (0)0.1101 (0)0.3425 (-2)

0.9156 (-1)

0.1287 (0)0.9684 (-1)

1.5964

1.4984

1.42881.38161.30181.34291.23341.1907

1.15490.1053 (-4)

0.2355 (-2)

0.6711 (-1)

0.3138 (0)0.7569 (-1)

0.1465 (0)0.5189 (-2)

0.4227 (-1)

0.1045 (0)1.7866

1.60731.50751.43841.4023

1.31281.2071

1.2472

1.19910.5204 (-4)

0.5956 (-2)

0.1079 (0)0.3250 (0)0.2075 (-1)

0.1493 (0)0.3015 (-1)

0.1101 (-1)

1.80701.6183

1.51681.44851.44591.32281.2522

1.27450.1490 (-3)

0.1171 (-1)

0.1518 (0)0.3099 (0)0.6989 (-3)

0.1272 (0)0.6015 (-1)

1.82881.6296

1.5262

1.45951.79481.33281.26810.3206 (-3)

0.1971 (-1)

0.1955 (0)0.2785 (0)0.5210 (-2)

0.9264 (-1)

1.8534

1.6413

1.5361

1.47191.2262

1.3431678a)(n) is power of 10 multiplying the entry. Top entry each block FCFs. Bottom entry each block r-centroids in Å.

0.6
0.5

FCFs

0.4
0.3
0.2
0.1
0

0
0

1

2

3

4

5

6

7

1

2

3

4

8

𝑣”
Fig. 4. RKR FCFs of the C1g−A1u Deslandres-d'Azambuja system of C2.

5

6

7

8

令和 3 年度

九州大学大学院総合理工学報告第 43 巻

第1号

7

Table. 5. RKR FCFs and r– centroids of the D1u+−X1g+ Mulliken system of C2.
𝑣

0
1
2
3

0

1

2

3

0.9971 (0)0.2749 (-2)

0.1720 (-3)

0.4115 (-5)

1.24530.49920.85940.61620.2833 (-2)

0.9923 (0)0.4394 (-2)

0.4640 (-3)

571.25550.41300.86840.60790.4673 (-2)

0.9892 (0)0.5141 (-2)

0.8365 (-3)

1.20962.06311.26590.30270.87900.59860.2820 (-3)

0.5639 (-2)

0.9876 (0)0.5182 (-2)

0.1260 (-2)

70.5254 (-4)

0.1309 (-5)

80.3056 (-4)

2.16311.27650.15770.89150.58820.10980.5675 (-3)

0.5858 (-2)

0.9871 (0)0.4699 (-2)

0.1714 (-2)

0.7898 (-4)

1.28732.2894

1.2873-0.0399

0.90620.57690.08230.1363 (-5)

0.9439 (-3)

0.5473 (-2)

0.9874 (0)0.3862 (-2)

0.2185 (-2)

0.1087 (-3)

0.2178 (-5)

1.32672.45511.2982-0.3223

0.92330.56490.4387 (-5)

0.1401 (-2)

0.4637 (-2)

0.9883 (0)0.2837 (-2)

0.2664 (-2)

0.12651.36712.68381.3093-0.7546

0.94280.1316 (-5)

0.1124 (-4)

0.1922 (-2)

0.3517 (-2)

0.9894 (0)0.1783 (-2)

1.68618

0.38991.40893.02221.3206-1.49160.2111 (-5)

0.2472 (-4)

0.2487 (-2)

0.2298 (-2)

0.9905 (0)1.76250.58301.4526

3.5791

1.3322(n) is power of 10 multiplying the entry. Top entry each block FCFs. Bottom entry each block r-centroids in Å.

FCFs

a)

6

0.1420 (-4)

1.9816-0.26646

50.9267 (-4)

1.2483𝑣 4

4𝑣”
Fig. 5. RKR FCFs of the D1u+−X1g+ Mulliken system of C2.