Bhattacharjee, Ramendu

Choudhury, Joydeep

Sarkar, Nirmal Kumar

English

IACS Institutional Repository

Indian J Phys 82 (6), 767-772 (2008) Algebraic approach : study of vibrational spectra of some linear triatomic molecules Nirmal Kumar Sarkar1*, Joydeep Choudhuryr' and Ramendu Bhattacharjee department of Physics Kanmganj College Kanrnganj 788 710 Assam lndw» department of Physics Assam University Silcfuir 788 011 Assam India E mail nks sdstfi rediffmail com Abstract Using the Lie algebraic method the vibrational energy levels of OCS and HCN are calculated for 35 vibrational bands each using the local Hamiltonian A comparative study is made between the two It has been shown that local Hamiltonian gives a much better fit to the HCN in comparison to that ot OCS Better result for HCN compared to those published earlier also has been reported Keywords Vibrational spectra Lie algebra OCS, HCN PACS Nos. 33 20Tp, 31 15 p 1. Introduction Manus Sophus Lie introduced Lie algebras at the end of the 19th century However, Lie algebraic methods have been useful in the study of problems in Physics after the development of quantum mechanics in the first part of the 20th century The use of Lie algebras as a tool to systematically investigate physical systems (the so called spectrum generating algebras) did not however develop fully until the 1970's, when it was introduced in a systematic fashion by A Anma and F lachello in the study of spectra of atomic nuclei (interacting boson model) [1,2] Wulfman played a great role in the introduction of the algebraic approach to molecules[3] He is the pioneer to publish the first paper on the algebraic approach to molecules (the algebraic approach to the Morse oscillator) in 1979[3] Later, in 1981, F. lachello used Lie algebraic methods in the systematic study of spectra of molecules (vibron model) [4] After 1981, there was a rapid progress in the field The situation up to 1995 was reviewed in the study of F lachello and R.D Levine[5] The study of F.lachello and S Oss[6] presented a brief review of the work done in this field up Corresponding Author © 2008 IACS 768 Nirmal Kumar Sarkar, Joydeep Choudhury and Ramendu Bhattacharjee to 2000 Moreover, the study[6] also provided perspectives for the algebraic method in the first decade of the 21st century In the Lie algebraic method there is provision to study a molecule both by a normal and a local Hamiltonian Using Lie algebraic method, in this paper the vibrational energy levels of OCS and HCN (in cm1) are calculated by local Hamiltonian for 35 vibrational bands each On the basis of this study it has been shown that local Hamiltonian gives a much better fit to the HCN in comparison to that of OCS Better result for HCN compared to those published earlier also has been reported. 2. Theory : the Lie algebraic method [7] In Lie algebraic method to study the vibrational spectra of triatomic molecules, there are two main ways in which the bonds can be coupled [8] £/|(4)®f/2(4) D 0,(4)00,(4) D 012(4) D 0 |2(3) D 0 | 2(2) (I) (1) Ux(4)® U2(4) z> Ul2 (4) D 0I2(4) D 012O) D 0 | 2(2) (II) Which correspond to local and normal coupling respectively For these two situations the Hamiltonian operator can be diagonalized analytically The local basis is characterized by the representations of chain I [ 5, 8 ], 17,(4) ® t/2(4) =) 0,(4) ® 02(4) 3 0p (4) D 0 l 2O) D 0 l2(2) i I i I i I i I ^ I (2) [N,] [N2J (cov0) (fi^.O) (TlfT2) J Mj i The numbers Nv N2 are the vibron numbers of each bond They are related to the number of bound states for bonds 1 and 2 respectively For Morse rovibrators they are given by the following equation [5] E'o = - A(2N + 3) (3) (E'oA —> algebraic parameters) Equation (3) shows that the vibron numbers are related to the depth of the potentials.They are fixed numbers for a given molecules. The numbers O),,Q)2,TVT2 are related to the vibrational quantum numbers, as discussed explicitly in the following sections We have written the 0,(4) representations as (^,0) and not simply as co1} since for coupled systems one can have representations of 0,(4) in which the second quantum number is not zero[5]. The values of co, and co2 are given by the rule (o, = Nv A/, - 2, 1 or 0, A/, = odd or even. o)2 s A/2, N2 - 2, 1 or 0, N2 = odd or even (4) Algebraic approach: Study of vibrational spectra of some linear tnatomic molecules 769 The values of (r,, T2) are instead obtained from the direct product of representations [5]. They are given by [9] ^ = °.1» — ,min (a^cog),!' - 0,1, ji (5) Finally for each representation (r^ r2) the values of the allowed angular momenta and parities are J> = 0M~, 2+, (r/ or r f ) , when t2 = 0 and (r, = even or odd) J " - T | , (T 2 + 1)", T ; When r2 / 0 (6) The z components of the angular momentum is given by the usual rule -J<Mj<+J. (7) 3. Local-mode Hamiltonian for linear triatomic molecules Whenever the Hamiltonian can be written in terms of invanant(Casimir) operators of a chain, its eigen value problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of equation (2) ^locais Eo + AAC(0(4y)) + A2C(0(A2)) + 412C(0(412)) + eC(0(312)) (8) Here we have dropped the subscript 2 from the Casimir invariant operators in order to simplify the notation [5]. The Hamiltonian (8) is diagonal in the local basis with eigen values E(NvN2io)vo)2, rv T2, J, Mj) = Eo + >Vi K + 2) + V>2 K + 2 ) + 4 2 [ M r i + 2 ) + T22] + W + 1)- (9) The last term in the right hand side of the above equation representing the rotational part. We can now convert equation (9) into the usual form by introducing the local vibrational quantum numbers[5], a), = N,-2va co2 = N2-2vc Tl =A/1+A/2-2va~v, )-2vc To =1h (10) 770 Nirmal Kumar Sarkar, Joydeep Choudhury and Ramendu Bhattacharjee The quantum number va,vc denote local stretching vibrations, while the quantum numbers, vh,b .denote the doubly degenerate bending vibrations. Eq. (9) can be converted, using eq. (10) to *(v . .v / \v c ) = E & - ^ ~^2[2{N,+N2+^{2va + vb + 2vc)-(2va + vb + 2vcf+^ (11) In this expression we have dropped the rotational part and written only the vibrational part. In this (purely local) limit, vibrational energy levels are determined by five parameters^.^.^2 and NVN2). 4. Results and discussion Vibrational energy levels of OCS and those calculated in the strict local limit for 35 vibrational bands are shown in Table 1. Vibrational energy levels of HCNand those calculated in the strict local limit for 35 vibrational bands are shown in Table 2. Using the established norms[5, 8] the vibron numbers NVN2 and other algebraic parameters AVA2,AA2 were determined. Av A2, A,2 all in cm'1, except N, and N2, which are dimensionless. Table 1 shows that in the calculation of vibrational energy levels, local Hamiltonian gives a poor fit to the OCS. Table 2 shows that local Hamiltonian gives a much better fit to the HCN in comparison to that of OCS [10]. Result reported here for HCN[7] is better with respect to the results published earlier [5, 8, 14]. RMS deviations(A(r.m.s.)) here we reported for OCS and HCN were 90.816 cnr1and 11.057 crrv1 respectively. 5. Conclusion (i) On the basis of the cited data it may be concluded that result reported in this study is better with respect to the results published earlier for the vibrational spectra of HCN(using local model). (ii) In the comparative study of the vibrational spectra of OCS and HCN in the local mode limit we conclude : Local model is a poor approximation for OCS compared to that of HCN. Only HCN could be very well approximated by a local Hamiltonian but not OCS. 6. Algebraic approach in the study of molecular spectra in near future Using the Lie algebraic methods many molecules(small and medium) have been analyzed in the last twenty five years. At present challenges for the Lie algebraic methods are the larger and more complex molecules (macromolecules, polymers, biomolecules and biopolymers). Introduction of anharmonicities gives the Lie algebraic methods an advantageous position. Some physical systems are known where anharmonicities play a role.Liquids and surfaces are the two important sectors belongs to this category[6]. It is Algebraic approach: Study of vibrational spectra of some linear tnatomic molecules 771 Table 1. Vibron Model Fits of OCS (in c m ^ i o ] V'1 1 0 0 2 1 0 1 0 0 3 2 1 0 2 1 0 1 0 0 4 3 2 3 1 0 2 1 0 1 5 4 4 2 3 2 v? 0° 2° 0° 0° 2° 4° 0° 2° 0° 0° 2° 4° 6° 0° 2° 4° 0° 2° 0° 0° 2° 4° 0° 4° 6° 0° 2° 4° 0° 0° 2° 0° 4° 0° 2° ^3 0 0 1 0 0 0 1 1 2 0 0 0 0 1 1 1 2 2 3 0 0 0 1 1 1 2 2 2 3 0 0 1 1 2 2 Expt.8 859 1047 2062 2 1711 1 1892 2 2104 8 2918 1 3095 6 4101 4 2556 2731 4 2937 2 3170 6 3768 5 3937 4 4141 2 4953 9 5121 61176 3394 3564 5 3762 8 4609 9 4970 4 5196 5801 9 5959 3 6154 7 6966 2 4224 9 4391 4 5445 5792 6640 1 6791 6 Calc.b 867 24 1102 84 2075 92 1730 64 1963 76 2199 36 2936 84 3172 44 4133 28 2590 2 2820 84 3053 96 3289 56 3793 92 4027 04 4262 64 4987 88 5223 48 6172 08 3445 92 3674 08 3904 72 4647 16 5110 92 5346 52 5838 64 6071 76 6307 36 7020 36 4297 8 4523 48 5496 56 5955 3 6685 56 6916 2 Ac -8 24 55 84 -13 72 -19 54 -71 56 -94 56 -18 74 76 84 -31 88 -34 2 -89 44 -116 76 -118 96 -25 42 -89 64 -121 44 -33 98 102 48 -54 48 51 92 -109 58 -141 92 -37 26 -140 52 -150 52 -36 74 -112 46 -152 66 -54 16 -72 9 -132 08 -51 56 -163 36 -45 46 -124 6 4(rm.s.) = 90 816 aRef. [11] ^ = 1 9 0 , A/2=159t ^ = 0 31, >l2=-1 53, /412=-0 79 c 4 = ( Expt-Cal) Table 2. Vibron Model Fits of HCN (in cm })[7] v'1 1 1 0 0 1 2 0 1 0 2 1 1 0 1 1 0 0 3 2 1 0 5 0 0 1 0 1 2 0 1 0 0 0 1 0 I 1' *2 2° 0° 0° 4° 2° 0° 2° 0° 0° 2° 2° 0° 0° 2° 0° 0° 0° 0° 0° 0° 0° 0° 11 31 11 11 31 11 31 11 11 2* 4* 22 2* *3 0 0 1 0 0 0 1 1 2 0 1 2 3 2 3 4 5 3 4 5 6 2 0 0 0 1 0 0 1 1 2 0 0 0 1 Expt* 1411 43 2096 85 3311 48 2802 85 3501 13 4173 07 4684 32 5393 70 6519 61 5571 89 6761 33 8585 57 9627 02 9914 41 11674 46 12635 90 15551 94 15710 53 16674 21 17550 39 18377 01 16640 31 711 98 2113 46 2805 58 4004 17 4201 29 4878 27 5366 86 6083 35 7194 75 1426 53 2818 16 3516 88 4699 21 Calcb 1412 972 2104 012 3299 364 2810 832 3501 872 4183 040 4697 224 5388 264 6503 344 5565 788 6771 012 8577 132 9611 94 9944 768 11670 616 12625 152 15542 98 15713 016 16687 296 17571 432 18365 424 16622 444 710 264 2115 68 2806 72 4002 072 4197 024 4878 192 5392 376 6083 416 7198 496 1420 528 2818 388 3509 428 4704 78 X 1 542 7 162 +12 116 7 982 -0 742 9 97 12 904 +5 436 +16 266 +6 102 -9 682 +8 438 + 15 08 30 358 +3 844 +10 748 +8 96 -2 486 13 086 -21 042 +11 586 + 17 866 +1 716 -2 22 -1 14 +2 098 + 4 266 +0 078 -25 516 -0 066 -3 746 +6 002 -0 288 +7 452 -5 57 d References [12] and [13] eA/1=140, A/2=47, V - 1 234« V * 1 0 0 3 4 ' Air5"1 8 8 9 ' 1 j = ( Expt - Cal) 772 Nirmal Kumar Sarkar, Joydeep Choudhury and Ramendu Bhattacharjee another challenge for the Lie algebraic methods to expand its kingdom in to these regions in near future. In view of the many possible applications it is hoped that present momentum of the Lie algebraic method will stimulate further research, especially in the spectroscopy of higher-energy states of molecules[5] and new areas of complex systems for which alternative methods of analysis are difficult to apply. In this regard we are optimistic to hope that in near future the Lie algebraic methods will be able to open many new windows in the field of spectroscopy. Acknowledgment Nirmal Kumar Sarkar would like to thank Prof. S. Oss, Prof. R . Lemus and Prof. N. Manini for providing necessary co-operation for this study. Ramendu Bhattacharjee would like to thank The DST, Govt, of India for supporting the work. References [1 [2] [3; [< [5; [e: [7; [8; [9: [io; [n [12: [13] IK F lachello and A Arima Phys Lett B53 309 (1974), A Arima and F lachello Phys Rev Lett 35 1069 (1975) For a review see, F lachello and A Arima The Interacting Boson Model (Cambridge Cambridge University Press) (1987) C E Wulfman Chem Phys Lett 23 370 (1973), R D Levine and C E Wulfman Chem Phys Lett 60 372 (1979) F lachello Chem Phys Lett 78 581 (1981) F lachello and R D Levine Algebraic Theory of Molecules (Oxford Oxford University Press) (1995) F lachello and S Oss Eur Phys J D 19 307 (2002) Nirmal Kumar Sarkar, Joydeep Choudhury and Ramendu Bhattacharjee Mol Phys 104 3051 (2006) S Oss Algebraic Models in Molecular Spectroscopy, Dipartimento di Fisica- Universita di Trento and Istituto Nazionale di Fisica della Materia, Unita di Trento, Povo (TN), Italy, (1996) R Lemus Casillas PhD thesis, Universidad Nacional Autonoma de Mexico, (1988) Nirmal Kumar Sarkar, Joydeep Choudhury and Ramendu Bhattacharjee Proceedings of the First Asian Spectroscopy Conference and Asian Biospectroscopy Conference, 106 (2007) E E Aubanel and D M Wardlaw J Chem Phys 88 495 (1987), A Fayt Ann Soc So Brux 86 61 (1972) T Nakhagawa and Y Monno Bull Chem Soc. Jpn 42 2212 (1969) S N Suchard Spectroscopic Data, Part A (New York Plenum) (1975) O van Roosmalen, F lachello, R D Levine and A E L Diepermk J Chem Phys. 79 2515 (1983) 

Algebraic Approach : Study of Vibrational Spectra of some Linear Triatomic Molecules

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Algebraic Approach : Study of Vibrational Spectra of some Linear Triatomic Molecules

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