Vibrational Spectra of Ozone (O₃) Using Lie Algebraic Method

Introduction

In 1981, Iachello presented Lie algebraic framework to molecular spectra of small molecules.^1,2 This framework based on Schrodinger equation with a Morse potential function and described ro-vibration spectra of diatomic molecules.^3,4 Later this method was extended to calculate medium and large molecules.^5-16 Apart from this framework, there are two other well-known methods established to calculate vibrational spectra of molecules. The first one is Dunham expansion.¹⁷ In this expansion the energy levels are expanded in ro-vibrational quantum numbers. In the second approach interatomic potential can be expanded interms of interatomic variables and potential coefficients by solving Schrodinger equation. The coefficients are  fitted with an available experimental data. The major drawback in both the methods is large experimental data is required to fit parameters, which is not possible eveytime. In order to this major difficulty we consider the third approach Lie algebraic method.

Lie Algebraic Method for the Triatomic Molecule Ozone

Ozone (O₃) contains three oxygen (O) atoms. The Ozone consists of two O-O bonds. The symmetry point group is C_2v.

Figure 1: Structure of Ozone. Click here to View figure

 

The Hamiltonian [18] in the case of stretching vibrations for Ozone (O₃) is H as follows

In Hamiltonian, C_i and C_ij are invariant operators of uncoupled and coupled bonds respectively and given by

and the Majorana operator, M_ij is used to describe local mode interactions in pairs. This contains diagonal and non-diagonal matrix elements,

Here i vary from 1 to 2 for two stretching bonds (O-O and O-O).

Eq. (1) can be written as

where A₁, A₂, A₁₂ and λ₁₂  are parameters, which are determined by spectroscopic data. The parameters λ_ij illustrate the interactions between stretching bonds ( λ₁₂). The local stretching vibrations are denoted by V₁ and V₃, while V₂ denotes bending vibrations for Ozone (O₃). Since, two bonds (O-O) are equivalent, place A₁ = A₂ = A, N₁ = N₂ = N in equations (2), (3) and (4). All parameters are in cm^-1, except N, which is dimension less. Eigen values of the Hamiltonian matrix will be considered as vibrational frequencies of Ozone.

Hamiltonian matrix for the first two local modes is given by

Results and Discussion

First calculate the Vibron number  for stretching bonds of Ozone usin

In this equation, ω_e, ω_e x_e are harmonic vibrational frequency and vibrational anharmonicity (spectroscopic) constants respectively. For the stretching mode, the values of ω_e and ω_e x_e for the O-O bond are 1580.161 and 11.95127 respectively.¹⁹

The initial value for the parameter A is obtained by using the energy.^20,21

Hence, ‾A can be evaluated as,

where ¯E is the average of two different energies, related to symmetric (E₁) and antisymmetric (E₂) combinations of local modes.

To find an initial value for λ₁₂ whose role is to split the initially degenerate local modes is calculated by the relation,¹¹

A mathematical fitting procedure is used to adjust the parameter A₁₂ and calculated as 0.028. Calculated vibrational frequencies of Ozone by the Lie algebraic method are reported in the table (1).

Table 1: The experimental and calculated vibrational frequencies (in cm^-1) of Ozone.

	Symmetry species	Experimental [http://vpl.astro.washington.edu/spectra/o3.htm]	Calculated
(1 0 0)	A1 (Symmetric)	1103	1103.7920
(0 0 1 )	B1(Antisymmetric)	1042	1043.0080
(2 0 0)	–	–	2206.0020
(0 0 2)	–	–	2061.0671
(1 0 1)	–	–	2109.2761
(3 0 0)	–	–	3289.0042
(0 0 3)	–	–	3046.9751
(2 0 1)	–	–	3187.0953
(1 0 2)	–	–	3085.9872

 

Conclusion

In the table (1), vibrational frequencies of Ozone (O₃) upto second overtone by Lie algebraic method are reported and also symmetric and antisymmetric fundamental vibrational frequencies compared with available experimental data. For the fundamental vibrational frequencies we observed that the root mean square deviation is 0.3542. The obtained results are useful for the experimentalists and theorists to develop the vibrational frequencies of Ozone in higher overtones.

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