Algebraic Study of Stretching and Bending Modes in Linear Tetra-atomic Molecules: HCCCl

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1 The African Review of Physics (2013) 8: Algebraic Study of Stretching and Bending Modes in Linear Tetra-atomic Molecules: HCCCl Kamal Ziadi * Department of Chemistry, Faculty of Science, University of Batna, Algeria In this work, we present a three dimensional vibron model to study both stretching and bending models of linear tetraatomic molecules HCCCl. We describe the vibrational levels of tetra-atomic molecules through the framework of the algebraic approach by applying algebra. The Hamiltonian contains the combination of the invariant operators of the algebra and sub-algebra of a given chains. 1. Introduction The molecular spectroscopy takes a very particular place in chemistry, physics and in science in general. It is suitable for providing precise answers to several of the most searching questions, basically those concerning the atomic and molecular structures (small, large, linear, non-linear, planar or non-planar). In the last few years, the experimental study of vibrational spectrum has determined a number of bands of a large molecule. The treatment of the molecular vibrational spectra requires theoretical models in order to study and understand the experimental data. Quite often, rot-vibrational spectra of molecule are studied by means of empirical procedures. A convenient formula for diatomic molecules is the Dunham expansion [1]. However, such a phenomenological approach is really very poor since the molecular wave function is completely unknown. Second, the model based on the solution of many-body Schrödinger equation with inter-atomic potential becomes more complex to apply in the case of polyatomic molecules. Recently, new approaches, called algebraic models, have been developed and successfully applied to nuclear, atomic and molecular structure. Algebraic techniques have been applied to many molecules for a number of years, and a recent review is available. The model U(6) for nuclear structure, U(2) for diatomic molecules in one dimension [2], and the model U(4) for diatomic molecules in three dimensions have been applied with success by [2-5] for diatomic molecules and tri-atomic molecules [6-11], which implies that it is also separate, but a coupled diatomic molecule. Later, it was extended to tetra-atomic molecules [12-14]). U(4) and U(2) algebraic model have generally been used so far for a treatment of rot-vibrational spectra of molecules. U(4) model has an advantage that both rotations and vibrations are studied simultaneously, but in our work we focus only on vibrational spectra. For a four-atom molecule such as HCCCl, there will be five vibrational normal modes. By studying the nuclear displacements corresponding to changes in each of the vibrational normal coordinates, we shall focus on five normal modes of molecule HCCCl: two symmetric stretches, one asymmetric stretch, the Trans bend, and the Cis bend. The relative displacement of atoms in different modes is shown in Fig Brief Algebraic Theory The vibron model of four-body system is more similar to that of linear tri-atomic molecules type. In the traditional model, we cognize that tetraatomic molecules have three independent vectors coordinates: r1, r2 and r3. In the algebraic technique we know that in the case of tetra-atomic molecules XYYZ, to each bond for i =1, 2, 3, one associates an algebra G (i =1, 2, 3), which is taken to be always G=U(4). * ziadi_kamal@yahoo.fr

2 The African Review of Physics (2013) 8: Symmetric HC stretch CC stretch Asymmetric CCl stretch Trans bend Cis bend H C C Cl Fig.1: Local vibrational quantum numbers of linear tetra-atomic molecule. For four-body system, an algebra is constructed taking the direct sum of three U(4) algebra, one for each band. The Hamiltonian, which we select from the full algebraic Hamiltonian, is a combination of invariant operators of the algebra and its subalgebra of a given chain. The first is through the SO(4) chain, ; and the second is through the U(3) chain, The wave functions are given by: N N N,ω,0,ω,0,τ,τ, ω,0,σ,σ,j,m These quantum numbers correspond to the following chain of sub-algebra [12,13]: ## 4 $ 4 4 The product %,0 %,0 limits the values of the quantum numbers &,& and the values of quantum numbers ',' are given by (1) ' = 1 2 & % +& & % & +, -, =0,1,,min & +&,%, =0,1,,min & &,% (2) In the model two fits have been performed: 2.1. The lowest-order expansion in terms of invariant operators When the Hamiltonian describing the boson system is written in terms of the Casimir operators in one of the chains, the eigenvalue problem can be solved analytically = : + 9 : +9 : +9 : +9 : (3) The operators : ;,: ;< are the Casimir operator of ;, =1,2,3 and O 4 O 4,,- =1,2,3 respectively, and : is the Casimir operator of O 4 O 4 O 4. This Hamiltonian is diagonal in the local basis (1) with its eigenvalues written by the following formula [15]. ' = 1 2 & % +& + & % & +,+-

3 The African Review of Physics (2013) 8: >,>,>,%,%,%,&,&,',' = % % +2+9 % % % % +2+9 & & +2+& +9 ' ' +2+' (4) To simplify this equation, we must transfer the algebraic quantum numbers ω,ω,ω,τ,τ,η,η to the usual spectroscopic quantum numbers, 6, B, 5, 6, C 3 D E 3 F. Here v H,v I denote stretching vibration associated with bonds 1 and 2 (C H and C-Cl stretching modes) and v L is associated with bond 3 (C C stretching mode), and v M N O v P N Q denotes the doubly degenerate bending vibration H C C and C-C-Cl bending modes). So we have [12,15]: ω =N 2v H, ω =N 2v I, ω =N 2v L τ =N +N 2v H +2v I +v P, τ =l P, σ =N +N +N 2v H +2v L +2v I +v M + v P, σ =l M +l P (5) Now, the preceding pattern is characterized by quantum numbers 6, B, 5, C 3 D, E 3 F Eqn. (4) can be written, using Eqn. (5), as 7 6, B, 5, C 3 D, E 3 F = > > > +1 B B 9 2> +> +> E E T E 9 2> +> + > B C + E B C + E T E +T C (6) 3. Mode of Calculation The algebraic model introduced here is a model of coupled one-dimensional Morse oscillators describing the H-C, C-C and C-Cl stretching vibrations of HCCCl. The Hamiltonian (Eqn. 3) is constructed and contains the first and the secondorder combination of invariant operators. The numbers N1, N2 and N3 are the vibron numbers of each bond, which are related to the vibrational frequency of the H-C, C-C and C-Cl and anharmonicity by the relation [13, 14] > =V W F W F X F Y 2 (8) Where, % E and % E Z E are the spectroscopic constants related to the stretching interaction of diatomic molecules. The values of A1, A2, A12 and [ parameters can be obtained using a numerical fitting procedure (Least Square). The results obtained are compared with the experimental data in Table 3. The satisfying accord between theoretical values and the observed ones is enough to describe the vibrational energy levels of tetraatomic molecules. Table 1: Fundamental levels of energy for HCCCl molecule. Local Modes Energy (cm -1 ) v a HC stretch v b CCl stretch v c CC stretch v d CCCl bend v e HCC bend The second order expansion in terms of Casimir operators is given as = : + 9 : +9 : +9 : + 9 : +U : + U : +U : (7)

4 The African Review of Physics (2013) 8: Table 2: The fit parameters of HCCCl in the U 1 (4) U 2 (4) U 3 (4) model. Parameters Fit1 Fit2 N N N A A A A A K K K Discussion We have used an algebraic technique to study the vibrational spectra of tetra-atomic molecule HCCCl. The Hamiltonian describing the stretching and bending vibrational spectra of tetra-atomic molecules is calculated within the structure of vibron model using the framework of the dynamical symmetry U 4 U 4 U 4 algebra and we carried out two fits using 18 experimental data [16,17]. In the first fit, we use 18 data to five parameters; the fitting RMS is 10.15, whereas in the second fit, we also use 18 data to eight parameters, the fitting RMS of the Hamiltonian is In carrying out the least square optimization, only 18 levels have been included. We subsequently found that to improve the agreement with the experimental data more levels are to be involved. In this paper, we have studied this molecule by using the Hamiltonian that is purely local, i.e., no operators of the Majorana type have been used. But we selected it from the full algebraic Hamiltonian, which contains a polynomial of all orders in the vibrational quantum numbers. We chose the local mode because of the molecule HCCCl is nonsymmetric type. Finally, we can say that the result of this analysis presented here indicates the algebraic approach can be certainly extended to all linear molecules of the tetra-atomic system. 5. Conclusion In this paper, we have proposed the algebraic model U 4 U 4 U 4 in two fits to study the vibrational spectra that included both stretching and bending vibrations of the linear tetra-atomic molecule (HCCCl). A comparison between the experimental data and our results is shown in Table (3). The improvement of the data agreement, when passing from Fit1 to Fit2, is very clear if we compare the RMS deviations. Then, we obtained the predicted values calculated by U 4 U 4 U 4 algebra of a not yet observed numbers of vibrational spectra of HCCCl molecule. References [1] J. L. Dunham, Phys. Rev. 41, 721 (1932). [2] A. Arima and F. Iachello, Phys. Rev. Lett. 35, 1069 (1975). [3] F. Iachello, Chem. Phys. Lett. 78, 581 (1981). [4] F. Iachello and R. D. Levine, J. Chem. Phys. 77, 3046 (1982). [5] F. Iachello, S. Oss and R. Lemus, J. Mol. Spectros. 146, 56 (1991). [6] O. S. Van Roosmalen and A. E. L Dieperink, Chem. Phys. Lett. 85, 32 (1981). [7] O. S. Van Roosmalen, F. Iachello, R. D. Levine and E. L. Dieperink, J. Chem. Phys. 79, 2515 (1983). [8] F. Iachello and S. Oss. J. Mol. Spectrosc. 142, 85 (1990). [9] F. Iachello, S. Oss and R. Lemus, J. Mol. Spectrosc. 146, 56 (1991). [10] N. K. Sarkar, J. Choudhury and R. Bhattacharjee, Mol. Phys. 104, 3051 (2006). [11] N. K. Sarkar, J. Choudhury and R. Bhattacharjee, Vib. Spectros. 56, 99 (2011). [12] F. Iachello, S. Oss and R. Lemus, J. Mol Struct. 149, 132 (1991). [13] F. Iachello and S. Oss, J. Mol. Spectrosc, 156, 190 (1992). [14] N. K. Sarkar, J. Choudhury and R. Bhattacharjee, Mol. Phys. 106, 693 (2008). [15] F. Iachello and R. D. Levine, Algebraic Theory of Molecules (Oxford University, Oxford, 1994). [16] M. Saarinen, L. Halonen and O. Polanz, Chem. Phys. Lett. 219, 181 (1994). [17] A. F. Borro, I. M. Mills and A. Mose, Chem. Phys. 190, 363 (1995).

5 The African Review of Physics (2013) 8: Table 3: Vibrational energy levels (in cm -1 ) of HCCCl. v a v b v c v d ld v e le Experimental Fit1 DE% Fit2 DE%

6 The African Review of Physics (2013) 8: v a v b v c v d ld v e le Experimental Fit1 DE% Fit2 DE% rms Table 3: continued. Received: 14 December, 2012 Accepted: 5 April, 2013