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Algebraic Study of diatomic Molecules: homonuclear molecules H2 and N2

It is the aim of this study to discuss for two-body systems like homonuclear molecules in which eigenvalues and eigenfunctions are obtained by exact solutions of the solvable models based on SU(1, 1) Lie algebras. Exact solutions of the solvable Hamiltonian regarding the relative motion in a two-body system on Lie algebras were obtained. The U(1) ↔ O(2), U(3) ↔ O(4) and Uq(3) ↔ Oq(4) transitional Hamiltonians are employed to described for H2 and N2 molecules. Applications to the rotation-vibration spectrum for the diatomic molecule indicate that complicated Hamiltonian can be easily determined via the exactly solvable method. The results confirm the mixing of both vibrating and rotating structures in H2 and N2 molecules.

The studies of molecular spectra of diatomic molecules are of great interest. Different ways are to the study of molecular spectra that require a large number of parameters to account for the structure of the molecules. Algebraic methods are one of the most useful methods for studying molecules. The main features and applications of Lie algebraic methods have been described in books 1,2 and review articles 3 in the last few years. There are many studies based on the interaction boson model(IBM) [4][5][6] . This Lie algebraic method is based on the second quantization of quantum numbers within the creation and annihilation operators.
The diatomic molecules are like two-body systems. Two-body systems have one-dimensional and three-dimensional algebraic models corresponding with algebra U (2) and algebra U(4), respectively (see Fig. 1). Various important features of the quantum algebraic formula for both the one and three-dimensional (exactly solvable) have been checked using suitable dynamical symmetry 7 .
The U (4) and U (2) algebraic models in the analysis of experimental data have been used so far in recent years. Rotations and vibrations are treated simultaneously in the U(4) model. The U(2) model treats rotations and vibrations separately. Iachello, Levine, and co-workers have described the rotation-vibration spectra of diatomic and triatomic molecules [8][9][10][11][12] using U (4) algebra. In ref. 13 , the experimental vibrational spectra of small and medium-sized molecules have been studied by algebraic techniques. These techniques are based on the idea of dynamical symmetry U (2) algebra.
In ref. 14 , vibrational spectra in diatomic molecules in sentences of the q deformed anharmonic oscillator based on the ⊃ ( ) ( ) symmetry have been characterized. The different uses of quantum deformed algebraic have effected in nuclear and molecular physics 15,16 . The q-deformed IBM Hamiltonians were developed by Pan 17 , in which generators were used to construct the corresponding q-deformed Casimir operators. Then the q-deformed vibron model of the diatomic molecules is reported by Alvarez et al. in ref. 18 . Also, we studied the phase transition of the even and odd nuclei based on q-deformed SU (1,1) algebraic model 19,20 .
In this paper, exact solutions of the solvable Hamiltonian about the relative motion in a two-body system on Lie algebras were obtained. One has to employ some complicated numerical methods to diagonalize the transitional Hamiltonian in analytic and exact solvable solutions of the duality paring models in diatomic molecules at rotational and vibrational modes, but Pan et al. in refs. [21][22][23] have suggested a new solution which is based on the affine SU(1, 1) Bethe ansatz algebraic technique. We have defined the molecular spectra for diatomic molecules by using transitional Hamiltonians which are based on the affine SU(1, 1) algebraic technique and quantum deformation theory [24][25][26] . We also considered the variation of the control parameter in the transitional theory. Our results propose a vibrational-rotational transition in the diatomic molecule and also explore the structures of the molecules. We have distinguished reasons to opt for the algebraic approach which constitutes a new method in the molecular system. The first reason is that it is a solvable model, a deformed version of the dynamical symmetries diatomic molecules has been constructed and we can have good accuracy in the study of energy spectra in the molecule.
The structure of this manuscript is as follows: section 2 briefly summarizes theoretical aspects of the transitional Hamiltonian, the affine SU (1,1) algebraic technique and the q-deformed version for two-body systems. Section 3 includes the results and finally, Section 4 will contain discussion of the present results and plans for further work.

theoretical framework
These models are based on U (2) and U(4) lie algebra (Fig. 1). We start the discussion with a simple case of one-dimensional problems, described by the U(2) algebra.
transitional theory. The one-dimensional vibron model has been used for molecular spectroscopy. This algebra can be used to describe stretching vibrations of molecules. To provide a realization for the U(2) algebra we take two boson creation and annihilation operators, which we denote by † s , † t and s, t. The U(2) algebra has four operators which can be realized as, The three operators + S , − S , S 0 are themselves closed under commutation and are elements of the algebra SU(2) which is a subalgebra of U (2). Using the commutations relationship given by, (2 2) 0 0 Dynamical symmetries for one-dimensional problems can be studied by considering all the possible subalgebras of U (2). The Casimir operator of SU (1,1) can be written aŝ The basis states of an irreducible representation SU(1, 1), µ k are determined by a single number, k, where k can be any positive number and µ = + … k k , 1, . Therefore   www.nature.com/scientificreports www.nature.com/scientificreports/ where N , n t and k t are quantum numbers of U(2), U (1) and SU (1, 1) t , respectively. The correspondence between the basis vectors of SO (2) and SU (1, 1) where v is the SO(2) quantum number. Now, we introduce the operators of infinite dimensional algebra similar to what has been defined by Pan et al. in refs. 21,22 , where C s and C t are real parameters and n can be taken ± ± … 0, 1, 2, . To evaluate the energy spectra and transition probabilities, let us consider lw as the lowest weight state of SU (1, 1) st algebra which should satisfy The lowest weight states, lw are a set of basis vectors as In this relation, The quantum phase transition between spherical and rotational nuclei is mainly driven by the nonzero nondiagonal part of the boson pairing operator. By employing the generators of SU(1, 1) algebra, the following Hamiltonian is constructed for the transitional region between where and α are real parameters. It can be seen that Eq. (2.12) is equivalent to a Hamiltonian in SO (2) limit, when = C C s t , and to a Hamiltonian in U(1) limit, when case corresponds with the SO (2) limit to the U(1) limit transitional region. In the following, C t is fixed to 1, and we allow C s to vary within the closed interval C [0, ] t . To find the non-zero energy eigenstates with k-pairs, we exploit a Fourier Laurent expansion of the eigenstates of Eq. (2.12) in terms of unknown c-number parameters , so eigenvectors of the Hamiltonian for excitations can be written as: N is the normalization constant and, By using Eq. (2.14) and the commutation relations of Eq. (2.2) which leads to a set of Bethe Ansatz equations, the c-numbers x i 's are determined by: The quantum number k is related to the total number of bosons, ν = + + N k v 2 s t . A useful and simple numerical algorithm for solving the BAE Eq. (2.16) and extraction of the constants in comparison with experimental energy spectra of considered molecules are based on using Matlab software which will be outlined simultaneously. To determine the roots of the BAE with specified values of and v t , we solve Eq. (2.16) with definite values of C and α for = i 1 and then use the function "syms var" in Matlab to obtain all roots. To this aim, we have changed the variables as We then repeat this procedure with different C and α to minimize the root mean square deviation, σ, between the calculated energy spectra and experimental counterparts which explore the quality of extraction processes. The deviation is defined by the equality: N tot is the number of energy levels that are included in the extraction processes. We have extracted the best set of Hamiltonian's parameters via the available experimental data.
Similarly to U(2), this technique can be extended to the U(4) case. In the vibron model the rotations and vibrations are described in terms of four bosons: a scalar boson of positive parity and angular momentum = l 0, denoted by † s , and the three components of a vector boson of negative parity and = l 1, denoted by † t m , = ± m 0, 1. To this aim, we have used the same formalism to extend the U(4) calculation via SU(1, 1) Lie algebra. In the U(4) case, the Hamiltonian can be considered as The eigenvalues of Eq. (2.21) can be expressed as In the study of molecular spectra, various approaches have been used. The Dunham expansion approach is very important. Rotational-vibrational molecular spectra are usually described in terms of the Dunham expansion where j is the angular momentum of the state, v is the vibrational quantum number, and Y ik are the Dunham coefficients, which are fitted to experiment. The energy of rotational and vibrational levels of molecules can be investigated separately and summed. To do this, we use the characteristic values of the diatomic molecule. The investigation of the energy spectra in both vibrational and rotational states can be abbreviated as rovibrational (or ro-vibrational) transitions.
To find the band spectra in both rotating vibrators of the diatomic molecule, it would be convenient to use a Dunham expansion based on the quantization of the energy levels. The particular advantage of this method is that it gives a very good approximation to the actual energy levels by consideration of higher quantum effects. First of all, we calculate the energy spectra of the diatomic molecule by Dunham expansion then we reproduce these values by the algebraic approaches 27 .
The Dunham expansion to the same order in v must be written as for N even or odd. Since all vibrational levels up to the dissociation energy are to be considered, the value needs to be determined accordingly. From the definition of the dissociation energy, The value of N determined from this scheme seems suitable for the whole vibrational spectrum.
transitional theory based on q-deformed algebra. In the preceding sections, the subalgebra chains of the model were reduced to equivalent chains of complementary subalgebras and the corresponding Hamiltonians were then written in terms of the Casimir operators of the new reduction chains. An evident possibility for q deforming these Hamiltonians is to substitute the SU(1, 1) algebras by their q-deformed counterparts SU(1, 1) 28,29 . The q-deformed Algebra has been explained in detail in refs. 17,19,20 . For the sake of the q-deformation of the Hamiltonian in the (4) transitional region, the Casimir operators and generators should be written in q-deformed forms. The general q-deformed Hamiltonian can then be written as where q is the parameter quantum deformation and the parameter q can be taken as real ( = τ q e with τ real) or phase ( = τ q e i with τ real). In the calculation, we take the q number as phase, i.e.,

numerical result
In this section, we investigate the extent to which the Hamiltonians can describe experimental spectra. The cases discussed above are interesting because they provide analytic expressions for the properties of the system that can be easily compared to the experiment. In our considered framework, we have compared the predictions of the transitional Hamiltonian for energy spectra with their experimental counterparts. On the other hand, predictions of our model for the control parameter, C, one may conclude that our considered control parameter has the same role as the mixing parameter of other investigations which explains the combination of vibration and rotation or rovibrational configurations. This means that for these numbers of levels in this energy region, the affine SU (1,1) approach can be regarded as the more exact method for describing the rovibrational energy levels of the considered molecules in the transitional.
In this work, we consider the homonuclear diatomic molecule H 2 and N 2 in its Σ + x g 1 state for our purpose. Dunham coefficients (Y i0 ) for H 2 and N 2 molecules were taken from refs. [30][31][32] . Since in this state of H 2 and N 2 it is experimentally known that = v 10 max and = v 49 max , respectively. We consider = N 21 (N can be either 20 or 21) and = N 99 (N can be either 98 or 99) for H 2 and N 2 , respectively. These two type of homonuclear molecules fitted by using theory. All experimental vibrational levels including those not observed up to now are obtained by using the Dunham expansion formula with a set of vibrational spectroscopy constants confirmed by experiment, which is denoted as .
The fitting results, parameters, and errors in fits are given in Tables  (2) predictions. The spectra of diatomic molecules in the framework of the U(4) and model were considered in refs. 14,[30][31][32] . So, it must be useful and worthwhile to compare the present method and results in the method and results of these papers. The paper by Xin & Feng 33 investigated the transitional description of diatomic molecules in the U(4) vibron model. Our result for N 2 is more precise than their result. In ref. 14  (4) formalism increases phase parameter C weight in the calculation. Besides, it should also be noticed that phase parameter C plays a significant role in these theories. The phase parameters for H 2 and N 2 in this analyse are in the range of 0.78~0.89 and 0.83~0.97. Thus, we conclude that H 2 and N 2 diatomic molecules are mixing both vibrating and rotating structures.

conclusion
Using the Lie algebraic method based on quantum deformed and nondeformed, we reported in these methods for a diatomic molecule. We have presented here an algebraic approach to molecular rotation-vibration spectra. In this work, we have confined ourselves to the study of diatomic molecules, in order to introduce phase transition based on deformed and nondeformed employed in U (2) and U (4) limits. The approach is general enough in that it can describe both rigid and nonrigid molecules. The results indicate that the energy spectra can be reproduced quite well. The quantum deformation technique enables us to input all high-order terms of a certain type and only add a few parameters to the Hamiltonian, which can be regarded as a possible extension.