Solutions of Schrodinger Equation And Thermodynamic Properties of Potassium Dimer Based On Formular Method

The study presents the thermodynamic properties of the state of potassium (K 2 ) dimer with molecular Deng-Fan potential. The bound state energy solution of the radial Schrodinger equation is obtained via the formula method. The partition function and other thermodynamic properties are evaluated. The numerical values of energy are found to be in agreement with results obtained from other methods in literature. The results further show that the partition function increases as temperature decreases, which implies a decrease in the probability of finding a particle in a state with quantum number, n .

where b r D e e , , , and r are the dissociation energy, range of potential, equilibrium internuclear distance, position of minimum e r and internuclear distance, respectively.

Review of Formula Method
The formula method was proposed by Falaye et al. [25] to solve wave equations. For a second order differential equation where  , ,  E and  are respectively the energy, reduced Planck constant, reduced mass and wave function of the system. Choosing as the solution to (7), the radial part of the Schrodinger equation with the centrifugal term is obtained as where n and l are the radial and orbital angular momentum quantum number, respectively.

Thermodynamic properties
The first step to investigate the thermodynamic properties of the Deng-Fan model is to set-up the vibrational partition function (l = 0) given by where B k is the Boltzmann constant, En is the vibrational energy of the Deng-Fan potential. Simplifying Eq. (13a) gives where, Substituting Eq. (15) into Eq. (14) gives In classical limit with where, Evaluation of Eq. (18) gives . (20) The imaginary error function Other thermodynamic properties are obtained as:

Discussions
Using Eq. (13a), we compute numerical values as presented in tables 1 and 2. These values conforms with results obtained using other methods [18,28,29,30]. The results obtained by formula method are very close to those reported using the N-U [28], SUSY [26], and WKB [18] methods and hence more accurate than those obtained by numerical methods [30]. The reason for this is that the authors in Ref. [30] used conventional approximation [31], which has less accuracy than the approximation used in the present work [18].
The experimental values for potassium dimer are chosen from Ref. [32]. These values are used as inputs to evaluate the partition functions and thermodynamic properties. The thermal properties are plotted in figs. 1-10. In figs. 11 and 12, the probability of finding the potassium molecule in a state with quantum number, n, is plotted against β and upper bound vibrational energy level, λ, respectively. Fig 1. Shows the vibrational partition function, Z as a function of β. Z is observed to increase sharply as β increases, which implies that as temperature decreases, the partition function increases for different values of λ. The observation on fig. 2 shows Z increasing monotonically as λ increases for all β. Fig. 2 also confirms the observation in fig. 1. In fig. 3, the internal energy U is observed to decrease as β increases. In fig. 4, U shows a reverse trend against λ to that seen against β. In fig. 5, the vibrational free energy, F, increases as β increases. The variation of F against λ ( fig.6), shows a reverse trend as F decreases with an increase in λ. The entropy, S, is observed to be directly proportional to β in fig. 7. In fig. 8, S, increases as λ increases for all β. In fig. 9, the vibrational specific heat capacity, C, is seen to decrease monotonically as β increases, implying that at lower temperatures, C has lower values for the potassium dimer. The opposite trend is observed in fig. 10 as C, increases with increasing λ up to λ=100, but beyond that point, a constant value is maintained for C for all β. As predicted by the trend of Z in fig. 1, the probability, Pr, decreases sharply as β increases ( fig. 11) due to an increase in Z. In fig. 12, Pr, also decreases sharply as λ increases up to λ=100, but maintains a constant value approaching zero beyond that point. This implies that the system is less likely to be found in an energy state with λ < 100.

Conclusions
In the present paper, the thermodynamic properties of potassium dimer is studied with Deng-Fan model. The radial Schrodinger equation is solved approximately using the formula method and a suitable approximation for the centrifugal term. The vibrational partition function and other thermodynamic functions are deduced. The thermodynamic functions are plotted against β and λ.
The results of this study can be applied to molecular and chemical physics.

Competing interest
The authors declare no competing interests.