A Study on Vibrational Spectra of PH 3 and NF 3 : An Algebraic Approach

With the new theoretical approach i.e. lie algebraic approach, we have calculated the infrared spectra of Phosphine in the range from 3000 cm − 1 to 9500 cm − 1 and Nitrogen Trifluoride in the range from 900 cm − 1 to 4500 cm − 1 . The model Hamiltonian, so constructed, seems to describe the P-H and N-F stretching modes accurately with only four numbers of parameters.


Introduction
After the recent development of new sophisticated spectroscopic instruments which allows scientists to measure vibrational states of polyatomic molecules with high accuracy and precision. The model is based on the idea of dynamical symmetry, which is expressed through the language of Lie algebras. Applying algebraic techniques, we obtain an effective Hamiltonian operator that conveniently describes the rotational vibrational degrees of freedom of the physical system. The proposed algebraic models are formulated such that they contain the same physical information for both ab initio theories (based on the solution of the Schrödinger equation) and semi empirical approaches (making use of phenomenological expansions in powers of appropriate quantum numbers). Various approaches have been used so far in the study of molecular spectra. Out of these, two important approaches are-1) Dunham expansion [1] and 2) potential approach [2]. A simple analysis of molecular ro-vibrational spectra is provided by the Dunham expansion. This is an expansion of the energy levels in terms of vibrationrotation quantum numbers. Compared to the Dunham expansion, a better analysis is provided by the potential approach in the study of molecular spectra. Energy levels are obtained by solving the Schrödinger equation with an inter-atomic potential. The potential is expanded in terms of interatomic variables. The algebraic models are successful models in the study of the vibrational spectra of small and medium-sized molecules. Some small and large molecules can be studied by using the U(4) and U(2) algebraic models. But, the U(4) model becomes complicated when the number of atoms in a molecule increases more than four. On the other hand, the U(2) model introduced by Wulfman and Levine [3] is found to be successful in explaining the stretching vibrations of polyatomic molecules such as tetrahedral, octahedral, Icosahedral, and benzene-like molecules. The brief review and the research work done with the algebraic models up to the year 2000 and its outlook and perception in the first decade of the 21st century was presented by Iachello and Oss [4][5][6][7]. Recently, it is found that Lie algebraic method [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] is extremely successful and accurate in calculating the vibrational frequencies of polyatomic molecules compare to the other methods such as Dunham expansion and potential approach method reported earlier. So far no extensive experimental study of the infrared vibrational spectra of Phosphine and Nirogen trifluoride molecules were reported in literatures. As a concrete and comple-mentary technique to the conventional approaches, the algebraic approach has already proven successful in the study of molecular spectra during the last 30 years.

Summary of the Algebraic Theory
The model is based on the isomorphism of the U(2) Lie algebra and one dimensional Morse oscillator whose eigen states may be associated with U(2)  O(2) states [25][26][27][28][29]. For a pyramidal molecule, XY 3 , we introduce three U i (2) (1  i  3) algebra to describe X-Y interactions (Figure 1). The possible chains of molecular dynamical groups in pyramidal molecules are (see Equation Which corresponds to local and normal couplings respectively.
The algebraic Hamiltonian in case of stretching mode of pyramidal molecules can be constructed from two chains as 3 3 3 In Equation (2), there are three types of contributions. The operators are the Casimir invariant operators of algebras, i = 1, 2, 3. Their diagonal matrix elements in the local basis . Interbond couplings can be introduced in terms of operators associated with products of U (2) and O(2) algebras associated different, interacting bonds. The term leads to cross-anharmonicities between pairs of distinct local oscillators which is diagonal with matrix elements given by ; , | | , ; , The modes of three equivalent X-H bond are now mixed, shifted and split under the action of the operator ˆi j M . The Majorana operator is used to describe local mode interactions in pairs and has both diagonal and non-diagonal matrix elements given by (see Equation (5)  We shall now construct the local vibrational basis which is given by The total vibrational quantum number is always conserved for a particular polyad. For a particular XH 3 molecule, the stretching bonds 1 to 3 are equivalent. The algebraic Hamilonian depends on 9 linear parameters. These parameters are reduced to only three parameters 1 12 12 , and A A  which give the strength of individual and coupling bonds respectively.

Results and Discussions
Using U(2) algebraic model Vibrational modes of XH 3 are computed using algebraic Hamiltonian up to third overtone and are listed in Tables 1 and 2 with fewer algebraic parameters (i.e. A, A 12 , λ 12 and N ).
The vibron number N can be determined by the relation where  e and  e x e are the spectroscopic constants of diatomic molecules [30][31][32][33]. The value of N must be as initially guessed from the Equation (7); however one can expect changes in an estimated N, not be larger than ± 20% of the original value. The vibron number N between  The next step is to obtain a guess for the second pa-m he Figure 1, it is n at so on ival y be du latio he 3 ed as free N is ke xed and n parameter.
rameter A. The expression for the single-oscillator fundamental mode is In the present case, we have three energies, corresponding to symmetric and antisymmetric combinations of the different local modes. A possible strategy is to use the center of gravity of these modes, so the guess for trix structure, we can find (9) The third step is to obtain an initial guess for λ. Its role is to split the initially degenerate local modes, placed here at the common value E used in Equation (8). Such an estimate is obtained by considering the simple ma With the help of numerical fitting procedure (in a least-square sense) the parameters A and λ 12 starting from values Equation (9) and Equation (10), and A 12 (whose initial guess can be zero) were adjusted. Vibrational modes of Arsine and Ammonia gebraic Hamiltonian up to third overtone and are listed in T ensional Morse oscillators which can be used to describe X-H stretching vibrations quite accurately. In third overtones, (eight) stretching vibrational modes for Arsine (Ammoni are computed using alables 3 and 4.

Conclusion
Using model Hamiltonian, we have presented an algebraic model of one dim we have predicted nine a) from only one (two) observed data's. To proceed further it would be necessary to record spectra of higher overtones to produce more accurate vibrational modes. In

Acknowledgements
The author Dr. Srini ke to ank University ndia, New De Grant Commissio ndia for financial s rk or w . Stef or pr he nec iterature is study.