CHIN. PHYS. LETT. Vol. 26,No. 2 (2009) 020308

Vibrational Spectroscopy of CH/CD Stretches in Propadiene: An AlgebraicApproach

Joydeep Choudhury1**, Nirmal Kumar Sarkar2, Srinivasa Rao Karumuri1, Ramendu Bhattacharjee1

1Department of Physics, Assam University, Silchar-788011, India2Department of Physics, Karimganj College, Karimganj-788710, India

(Received 18 September 2008)Using Hamiltonian based on Lie algebraic method, the stretching vibrational modes of C3H4 and C3D4 moleculesare calculated up to higher overtones. The model appears to describe C–H and C–D stretching modes with lessnumber of parameters. The locality parameter 𝜉 confirms the highly local behaviour of the stretching modes ofthese molecules.

PACS: 03. 65. Fd, 31. 15.Hz

Molecular spectroscopy is an area of active inter-est and has played an important role to understandand analyse a physical system in both experimentaland theoretical approaches. One of the most interest-ing areas of current research in molecular physics isthe study of vibrational ground and excited states ofpolyatomic molecules. After the recent rapid devel-opments of sophisticated instruments, the molecularspectroscopy is currently going through an excitingtime of renewed interest. Two traditional approacheshave been used so far in the analysis of experimen-tal data: (i) the Dunham-like[1] expansion of energylevels in terms of rotation–vibration quantum num-bers and (ii) the solution of Schrodinger equation withpotentials obtained either by modifying ab initio cal-culations or by more phenomenological methods. TheDunham-like expansions contain a large number of pa-rameters which can not be determined from the fewavailable data and the model based on the solutionof many body Schrodinger equation with interatomicpotential becomes cumbersome and difficult to applyin the case of polyatomic molecules. To overcome thedifficulties arises in analysing the vibrational spectra,third approach the vibron model (the algebraic mod-els) based on Lie algebra[2] was built in the secondhalf of the 20th century. This new model seems to de-scribe the molecular spectra successfully even in com-plex situations. The algebraic models have been usedextensively in a variety of fields: (i) research of spectraof atomic nuclei (interacting boson model)[3] and (ii)spectra of diatomic and triatomic molecules,[4,5] (iii)research of spectra of linear and quasilinear tetratomicmolecules,[6] (iv) spectra of tetrahedral molecules,[7]

benzene[8] and octahedral molecules,[9] (v) vibrationalmodes of CH bonds in n-paraffin molecules,[10] (vi) CHstretching modes of n-alkane[11] and polyethylene.[12]

Iachello and Oss presented a brief review of develop-ment of algebraic techniques and its application tomolecular spectroscopy[13] in last five years up to 2000.The main features and applications of these meth-ods have been described in a number of books[14,15]

and review articles[16] in last few years. Recently we

reported the studies on vibrational spectra of HCN,OCS,[17−19] HCCF, HCCD,[20] CCl4, SnBr4,[21] Ni(OEP), Ni (TPP) and Ni porphyrin[22] by an alge-braic approach. Propadiene is one of the most exten-sively studied molecules belonging to the point groupD2𝑑. The C–H (or C–D) stretching vibrational modesof such molecules have been analysed by different au-thors, e.g. Mills and Mompean[23] and Halonen.[24]

In this Letter, we initiate the use of algebraic meth-ods to generate stretching modes of C–H (C–D) bondsin propadiene molecule. The model we use is noth-ing but a one-dimensional version of the general al-gebraic model of rotation-vibration spectra of poly-atomic molecules. In this model, each bond, 𝑖, is re-placed by an algebra 𝐺𝑖, and the Hamiltonian describ-ing the molecule is written in terms of algebraic bondcoordinates. The eigen states, eigenvalues and matrixelements of operators are then found out by algebraicmanipulations which are easy to perform.

We start with full algebraic Hamiltonian operatorfor C3X4 (X=H, D) molecules. The general form ofsuch operator can be written as

𝐻 = 𝐻𝑆 + 𝐻𝐵 + 𝐻𝑆𝐵 . (1)

In this expression, 𝐻𝑆 describes the stretching modesand is based on 𝑈(2) algebra; 𝐻𝐵 describes the de-generate bending modes and is constructed from thetwo-dimensional model 𝑈(3) algebra; 𝐻𝑆𝐵 providesstretch/bend interactions of several kinds based onthe 𝑈(2)/𝑈(3) scheme.[25] In this study, we neglect𝐻𝐵 and 𝐻𝑆𝐵 in Eq. (1) due to lack of experimentaldata for bending mode. We consider the stretchingHamiltonian 𝐻𝑆 . For four one-dimensional oscillatorsC–X (X=H, D), we start with the spectrum generat-ing algebras (SGA), 𝑈1(2) × 𝑈2(2) × 𝑈3(2) × 𝑈4(2).Here, the two C–C oscillators (bond 5 and 6) are notconsidered on the basis which may definitely give thestretching vibrations as we mostly focused on C–Xstretching vibrations. We then construct the local ba-sis given by

**To whom correspondence should be addressed. Email: choudhuryjoy@rediffmail.comc○ 2009 Chinese Physical Society and IOP Publishing Ltd

020308-1

CHIN. PHYS. LETT. Vol. 26,No. 2 (2009) 020308

𝑈1(2) × 𝑈2(2) × 𝑈3(2) × 𝑈4(2) ⊃ 𝑂1(2) × 𝑂2(2) × 𝑂3(2) × 𝑂4(2)𝑁𝐶𝑋 𝑁𝐶𝑋 𝑁𝐶𝑋 𝑁𝐶𝑋 𝑣1 𝑣2 𝑣3 𝑣4

⟩. (2)

Table 1. C–H (C–D) stretching vibrational modes of C3H4 and C3D4 molecules. Here the observed values are taken from Ref. [23].

𝜐 Vibrational level SymmetryC3H4 C3D4

𝑣calc (cm−1) 𝑣obs(cm−1) 𝑣calc (cm−1) 𝑣obs (cm−1)

1 𝜈5 𝐵2 3000.84 3015.6 2205.96 2228.9𝜈1 𝐴1 3000.84 3006.6 2205.96 2216.6

𝜈±18 𝐸 3081.25 3085.6 2310.51 2321.2

2𝜈5 𝐴1 5946.81 4448.55𝜈1 + 𝜈5 𝐵2 5997.96 5947.5 4409.16 4417.4

𝜈5 + 𝜈±18 𝐸 6078.37 4513.71

2 2𝜈1 𝐴1 5946.81 4448.55

𝜈1 + 𝜈±18 𝐸 5997.96 5980 4409.16

2𝜈±28 𝐵2 6158.78 6135.5 4618.26 4611.4

2𝜈08 𝐴1 6025.34 4551.39

2𝜈±28 𝐵1 6025.34 4551.39

3 3𝜈5 𝐵2 8596.68 6414.12𝜈1 + 2𝜈5 𝐴1 8863.52 6547.2

𝜎(rms) 42.89 cm−1 29.53 cm−1

In this expression, we introduce the vibron num-bers 𝑁𝐶𝑋 (X=H,D) which are directly related to theanharmonicity of local C–H (C–D). The four stretch-ing quantum number 𝑣1, 𝑣2, 𝑣3, 𝑣4 are associated withC–H (C–D) bonds (Fig. 1). The stretching Hamilto-nian for uncoupled bonds can be written as

𝐻(𝑜)𝑆 = 𝐴𝐶𝑋( 𝐶1 + 𝐶2 + 𝐶3 + 𝐶4). (3)

The operators 𝐶𝑖 are the Casimir invariant operatorsof 𝑂𝑖(2) algebras, 𝑖 = 1, 2, 3, 4. Their diagonal matrixelements in the local basis

𝑣1, 𝑣2, 𝑣3, 𝑣4

⟩are of the

form

⟨ 𝐶𝑖⟩ = −4𝑣𝑖(𝑁𝑖 − 𝑣𝑖), 𝑖 = 1, 2, 3, 4 (4)

with 𝑁1 = 𝑁2 = 𝑁3 = 𝑁4 = 𝑁𝐶𝑋 . Interbond cou-plings can be introduced in terms of operators associ-ated with products of 𝑈(2) and 𝑂(2) algebras associ-ated different, interacting bonds. We are thus led tocoupled, stretching Hamiltonian operator

𝐻(1)𝑆 = 𝐴𝐶𝑋−𝐶𝑋

( 𝐶12 + 𝐶13 + 𝐶14

+ 𝐶23 + 𝐶24 + 𝐶34

). (5)

The term 𝐶𝑖𝑗 leads to cross-anharmonicities betweenpairs of distinct local oscillators which is diagonal withmatrix elements given by⟨

𝑁𝑖, 𝑣𝑖; 𝑁𝑗 , 𝑣𝑗 | 𝐶𝑖𝑗 |𝑁𝑖, 𝑣𝑖; 𝑁𝑗 , 𝑣𝑗

⟩= 4[(𝑣𝑖 + 𝑣𝑗)2 − (𝑣𝑖 + 𝑣𝑗)(𝑁𝑖 + 𝑁𝑗)]. (6)

Thus, the total Hamiltonian for coupled and uncou-pled stretching bonds is expressed by

𝐻𝑆 = 𝐻(0)𝑆 + 𝐻(1)

𝑆 . (7)

The modes of four equivalent CX bond are now mixed,shifted and split under the action of the operator 𝑀𝑖𝑗

(adjacent and opposite). The Majorana operator isused to describe local mode interactions in pairs andhas both diagonal and non-diagonal matrix elementsgiven by[9]⟨

𝑁𝑖, 𝑣𝑖; 𝑁𝑗 , 𝑣𝑗

𝑀𝑖𝑗

𝑁𝑖, 𝑣𝑖; 𝑁𝑗 , 𝑣𝑗

⟩= 𝑣𝑖𝑁𝑗 + 𝑣𝑗𝑁𝑖 − 2𝑣𝑖𝑣𝑗 ,⟨

𝑁𝑖, 𝑣𝑖 + 1; 𝑁𝑗 , 𝑣𝑗 − 1𝑀𝑖𝑗

𝑁𝑖, 𝑣𝑖; 𝑁𝑗 , 𝑣𝑗

⟩= −

[𝑣𝑗(𝑣𝑖 + 1)(𝑁𝑖 − 𝑣𝑖)(𝑁𝑗 − 𝑣𝑗 + 1)

] 12 ,⟨

𝑁𝑖, 𝑣𝑖 − 1; 𝑁𝑗 , 𝑣𝑗 + 1𝑀𝑖𝑗

𝑁𝑖, 𝑣𝑖; 𝑁𝑗 , 𝑣𝑗

⟩= −

[𝑣𝑖(𝑣𝑗 + 1)(𝑁𝑗 − 𝑣𝑗)(𝑁𝑖 − 𝑣𝑖 + 1)

] 12 , (8)

so the general algebraic Hamiltonian operator for de-scribing the C–X (X=H,D) stretching modes of propa-diene is expressed by

𝐻 = 𝐴

4∑𝑖=1

𝐶𝑖 + 𝐴′4∑

𝑖<𝑗=1

𝐶𝑖𝑗 +4∑

𝑖<𝑗=1

𝜆𝑖𝑗𝑀𝑖𝑗 . (9)

The strength of uncoupled and coupled bonds are reg-ulated by parameters 𝐴(𝐴𝐶𝑋), 𝐴/ (𝐴𝐶𝑋−𝐶𝑋) and𝜆𝑖𝑗 = 𝜆adj, respectively. The inclusion of 𝑀𝑖𝑗 in theHamiltonian conserves the total vibrational (polyad)quantum number.

C

X

CC

X

X

X1

2 3

45 6

Fig. 1. Bond numbering adopted in the C3X4 (X=H, D)molecule for its algebraic representation.

In Table 1, we list the results of fits to C–H andC–D stretching fundamental and overtone modes of

020308-2

CHIN. PHYS. LETT. Vol. 26,No. 2 (2009) 020308

propadiene molecule using the model Hamiltonian (9).From the view of group theory, the four oscillators (C–X) describe the fundamental stretching modes (𝜐 = 1)given by 𝐴1(𝜈1), 𝐵2(𝜈5), and 𝐸(𝜈±1

8 ) irreps. Here𝐴1 represents the symmetry irreps while 𝐵1 and 𝐵2

represent antisymmetric irreps with respect to 𝐶2 per-pendicular to the principal axis, and 𝐸 represents two-dimensional symmetric irreps with respect to centre ofinversion. There are ten stretching vibrational states(𝜐 = 2) separates into 𝐴1(2𝜈5), 𝐴1(2𝜈1), 𝐴1(2𝜈0

8);𝐵1(2𝜈±2

8 ); 𝐵2(𝜈1 + 𝜈5) , 𝐵2(2𝜈±28 ); 𝐸(𝜈5 + 𝜈±1

8 ) and𝐸(𝜈1 + 𝜈±1

8 ). Out of these states, 3𝐴1 bands are Ra-man active and the remaining states are infrared ac-tive. As there are very few available data, Table 1should be considered more a prediction of unknownstates rather than a fit. It is seen from Table 2 thatthe parameters are different for two molecules. Here𝜎 (rms) is reported as 42.89 cm−1 and 29.53 cm−1 forC3H4, C3D4 molecules, respectively. We believe thatmore satisfactory results will be predicted if the inter-bond coupling between CH (CD) stretches at oppo-site ends of the molecules are taken into considerationin spite of their small values. Also, it is to be men-tioned here that the large number of observed datamay reduce the value of 𝜎 (rms). The vibron num-bers for C–H and C–D bonds, 𝑁 (𝑁𝐶𝑋) were fixed to𝑁𝐶𝐻 = 43 and 𝑁𝐶𝐷 = 61, calculated from expression{(𝜔𝑒/𝜔𝑒𝑥𝑒) − 2} with experimental potential param-eters (𝜔𝑒 = 2861.6 cm−1, 𝜔𝑒𝑥𝑒 = 64.3 cm−1 for C3H4

and 𝜔𝑒 = 2333.7 cm−1, 𝜔𝑒𝑥𝑒 = 30.6 cm−1 for C3D4)taken from Refs. [5,24].

Table 2. Fitting parameters of C3H4 and C3D4.

𝑁(𝑁𝐶𝑋) 𝐴(𝐴𝐶𝑋) 𝐴/(𝐴𝐶𝑋−𝐶𝑋) 𝜆𝑖𝑗 = 𝜆adj

C3H4 43 −15.98 −0.465 0.935C3D4 61 −7.8 −0.345 0.857

All the values are in cm−1 except 𝑁 which is di-mensionless.

In this study we have applied the one-dimensionalalgebraic model to C3X4 (X=H, D) molecules. The al-gebraic Hamiltonian has been used to compute theirenergies up to second overtones. The stretching vi-brational modes of C3H4 and C3D4 molecules arepresented in Table 1. In the fundamental the split-ting is ≈ 80.41 (C–H), ≈ 104.55 (C–D) for C3H4

and C3D4 molecules. The splitting pattern deter-mines the nature of interbond interaction (our pa-rameter 𝜆𝑖𝑗 = 𝜆adj). The present results show that𝜎(rms) = 16.42 and 16.71 cm−1 for fundamentals ofC3X4 (X=H, D) molecules which is reasonable butbecomes larger for first overtones. The theoreticalpredicted (𝜈1 + 𝜈5) mode of C3H4 molecule (devi-ation of 50.46 cm−1 with the observed value) ques-tions the experimental accuracy by this algebraic ap-proach. The local versus normal behaviour of stretch-ing modes of C3X4 can be characterized by introduc-ing the quantity, 𝜉 = 2/𝜋 arctan

[8𝜆adj/(𝐴 + 𝐴′)

]. We

find 𝜉 = 0.4456 for C–H bond and 𝜉 = 0.1635 for the

C–D bond, thus confirming highly local behaviour ofthe stretching modes of propadiene. Thus, we can saythat the algebraic model Hamiltonian is successful inpredicting the energies in the complex molecular sys-tem.

We would like to thank Professor S. Oss for pro-viding initial inspiration for this study. RB and SRKare grateful to DST for providing a research grant.

References

[1] Dunham J L 1932 Phys. Rev. 41 721[2] Iachello F 1981 Chem. Phys. Lett. 78 581[3] Iachello F and Arima A, 1974 Phys. Lett. B 53 309

Arima A and Iachello F 1975 Phys. Rev. Lett. 35 1069[4] Iachello F and Levine R D 1982 J. Chem. Phys. 77 3046[5] van Roosmalen O S, Iachello F, Levine R D and Dieperink

A E L 1983 J. Chem. Phys. 79 2515Iachello F, Oss S and Lemus R 1991 J. Mol. Spectrosc. 14656van Roosmalen O S, Benjamin I and Levine R D 1984 J.Chem. Phys. 81 5986Iachello F and Oss S 1990 J. Mol. Spectrosc. 142 85

[6] Iachello F, Oss S and Lemus R 1991 J. Mol. Spectrosc. 149132Iachello F, Manini N and Oss S 1992 J. Mol. Spectrosc.156 190Iachello F, Oss S and Viola L 1992 Mol. Phys. 78 545Iachello F, Oss S and Viola L 1993 Mol. Phys. 78 561

[7] Lemus R and Frank A 1994 J. Chem. Phys. 101 8321[8] Iachello F and Oss S 1992 J. Mol. Spectrosc. 153 225

Iachello F and Oss S 1993 Chem. Phys. Lett. 205 285[9] Iachello F and Oss S 1991 Phys. Rev. Lett. 66 2976

Chen J Q, Iachello F and Ping J L 1996J. Chem. Phys.104 815

[10] Marinkovic T and Oss S 2002 Phys. Chem. Comm. 5 66[11] Marinkovic T and Oss S 2003 Phys. Chem. Comm. 6 42[12] Oss S 2006 J. Mol. Struct. 780/781 87[13] Iachello F and Oss S 2002 Eur. Phys. J. D 19 307[14] Iachello F and Levine R D 1995 Algebraic Theory of

Molecules (New York: Oxford University)[15] Frank A and van Isacker P 1994 Algebraic Methods in

Molecular and Nuclear Structure Physics(New York: Wi-ley)

[16] Oss S 1996 Adv. Chem. Phys. 93 455[17] Sarkar N K , Choudhury J and Bhattacharjee R 2006 Mol.

Phys. 104 3051[18] Sarkar N K, Choudhury J and Bhattacharjee R 2008 Indian

J. Phys. 82 767[19] Sarkar N K, Choudhury J and Bhattacharje R 2007 Pro-

ceedings of the Asian Thermophysical Properties Confer-ence (Fukuoka, Japan 21–24 August 2007) paper 236

[20] Sarkar N K, Choudhury J, Karumuri S R and BhattacharjeeR 2008 Mol. Phys. 106 693

[21] Choudhury J, Karumuri S R, Sarkar N K and BhattacharjeeR 2008 Pramana J. Phys. 71 439Choudhury J, Sarkar N K and Bhattacharjee R 2008 IndianJ. Phys. 82 561

[22] Karumuri S R, Sarkar N K, Choudhury J and BhattacharjeeR 2008 Mol. Phys. 106 1733Karumuri S R, Choudhury J, Sarkar N K and Bhattachar-jee R 2008 J. Environmental Research and Development (tobe published)Karumuri S R, Choudhury J , Sarkar N K and Bhattachar-jee R 2008 Pramana J. Phys. (to be published)

[23] Mills I M and Mompean F J 1986 Chem. Phys. Lett. 124425

[24] L Halonen 1987.J. Chem. Phys. 86 3115[25] Temsamani M A, Champion J M and Oss S 1999 J. Chem.

Phys. 110 2893

020308-3