BOUND STATE SOLUTIONS OF SCHRÖDINGER EQUATION FOR A MORE GENERAL EXPONENTIAL SCREENED COULOMB POTENTIAL VIA NIKIFOROV-UVAROV METHOD

The arbitrary angular momentum solutions of the Schrodinger equation for a diatomic molecule with the general exponential screened coulomb potential of the form V(r) = (- a / r){1+ (1+ b )e-2b } has been presented. The energy eigenvalues and the corresponding eigenfunctions are calculated analytically by the use of Nikiforov-Uvarov (NU) method which is related to the solutions in terms of Jacobi polynomials. The bounded state eigenvalues are calculated numerically for the 1s state of N2 CO and NO


Introduction
The exact analytic solutions of the wave equations (non-relativistic and relativistic) are only possible for certain potentials of physical interest under consideration since they contain all the necessary information on the quantum system [1].It is known that for certain potentials, the Schrödinger equation can be solved for the angular momentum quantum numbers 0 =  [2].However, in some cases, like for the 0 ≠  states, some approximations are often used to obtain analytic solutions of the Schrödinger equation [3 -5].
A more general exponential screened coulomb (MGESC) potential used in this paper is of the form [6]: where a is the strength coupling constant and b is the screened parameter.The potential in equation ( 1) is known to describe adequately the effective interaction in many-body environments of a variety of fi elds [6].In this paper, we have decided to explore the possibility of also using it in obtaining bound state solutions of the Schrödinger equation for diatomics using Nikiforov-Uvarov (NU) method.

Overview of Nikiforov-Uvarov (NU) Method
The NU method is based on the solutions of general second order linear differential equations with some orthogonal functions [7].For the given potential, the Schrödinger equation in the spherical coordinates is reduced to a generalized equation of hyper-geometric type with an appropriate ) r ( s s = coordinate transformation.Thus, it has the form [8]: , where ( ) are polynomials, at most second-degree, and ( ) s τ is a fi rst-degree polynomial.To fi nd a particular solution of equation ( 2), we use the following transformation [9]: (3) This reduces Schrödinger equation ( 2) to an equation of hyper-geometric type: where ( ) s) is the hyper-geometric type function whose polynomial solutions are given by the Rodrigues relation: where n B is a normalization constant and the weight function ρ must satisfy the condition [9]: The function π and the parameter λ required for this method are defi ned as: Here, ( ) s π is a polynomial with the parameter s and the determination of k is necessary for ( ) s π to be obtained.To fi nd k, the expression under the square root must be square of a polynomial.A new eigenvalue equation for the Schrödinger equation thus becomes: where ( ) ( ) ( ) and ( ) s τ′ must be negative.

Bound State Solutions via Nikiforov-Uvarov (NU) Method
The potential in equation ( 1) is substituted into the radial Schrödinger equation given as: where n denotes the radial quantum number which together with  are both named as the vibration-rotation quantum numbers in molecular chemistry, r is the internuclear separation,  n E is the exact bound state energy eigenvalues and V(r) is the internuclear potential energy function and we obtain: Equation ( 12) can be rearranged to give: Introducing the following dimensional parameters: 13) is written as: A comparison of equations ( 2) and (17) reveals the following polynomials: Substituting these polynomials into equation (7), we get ( ) r π as: and ( ) r σ′ is taken equal to 1.The discriminant of the expression under the square root in equation (19) has to be zero for it to have equal roots.Therefore, we get: On solving equation (20) for k we get: From the four possible forms of ( ) ( ) and ( ) From equation ( 8), we obtain: We then obtain the parameters 2 ε as: Substituting the values of 2  ε , β and γ from equations ( 14) -( 16) into equation ( 29), yields: To fi nd y(r), we fi rst obtain ( ) r ρ from equation (6) as: Substituting this into the Rodrigues relation given in equation ( 5), we get: is the normalization constant.The polynomial solutions of ( ) r y n in equation ( 32) are expressed in terms of the associated Laguerre polynomials, which is one of the orthogonal polynomials.We write:  3), enables the radial wave function to be constructed as: If we introduce the variable To fi nd  n A , a new normalization constant, we write: The above integral can be evaluated by using the recursion relation for Laguerre polynomials and A is found to be:

Conclusion
The analytical solutions of the Schrödinger equation for the general exponential screened coulomb potential has been presented.The Nikiforov-Uvarov method employed in the solutions enables us to explore an effective way of obtaining the eigenvalues and corresponding eigenfunctions of the Schrödinger equation for any  -state.
Finally, we calculate the energies of the exponential screened coulomb potential for diatomic molecules by means of equation (30) for the  -state.The explicit values of the energy at different values of the screened parameter are shown in Table 1.
), we select the one for which the function ( ) s τ in equation (10) has a negative derivative.( ) s τ satisfi es these requirements with: resulting differential equation, the other part of the wave function in equation (3) is obtained as:

Table 1 .
Bound State Eigenvalues for The r values for 2 N (1.0940),CO (1.21282) and NO (1.1508) were adapted from M. Karplus and R. N. Porter, Atoms and Molecules: An Introduction for Student's of Physical Chemistry, Benjamin, Menlo Park, CA, 1970.