The exact solution of the schrödinger equation for a two-dimensional Hydrogen molecule ion is obtained using prolate spheroidal coordinate system. The resultant modified differential equations have been compared with standard Jacobi and Laguerre differential equations in order to obtain ground state energy and eigen state of this simplest molecule ion. Also, it is shown that the related two-dimensional differential equation is well separated into two one-dimensional equations that are solvable.

The Hydrogen molecular ion, H2+ is the simplest molecule. It is usually used as a prototype in molecular physics, like the hydrogen atom in atomic physics. The quantum dynamics of H2+ and other small molecules under interaction with a strong laser pulse, has been a subject of continuous interest for more than two decades.
Interesting phenomena such as bond softening, bond hardening, above-threshold dissociation and Coulomb explosion of this molecule have been studied (Posthumus, 2004; and Lee et al., 2008).

H2+ has played an important role in the development of molecular quantum mechanics. It has been regarded as a model system for the formulation of many different methods and approximations.

The absence of inter-electron interactions has enabled other aspects of molecular structure theory to be examined in depth. We probably understand the structure and dynamics of H2+ better than those of any other molecule. H2+ is, however, much easier to study theoretically than experimentally, and although its physical properties have been calculated extensively, few have been measured.

It is an elusive molecule in the laboratory because, although it has a large binding energy and is therefore thermodynamically stable, it is very reactive, particularly towards molecular hydrogen (Dundas, 2002; Peng et al., 2003; and Spanner et al., 2004).

The hydrogen atom confined to bounded regions has long been investigated by many authors (Michels et al., 1937; and Suryanarayana and Weil, 1976). Recently, parabolically confined hydrogen atom has been discussed (Krhmer et al., 1998). Different methods have been used in the study of the confined hydrogen atom: Baye and Sen (2008) used the Lagrange-mesh method; Costa et al. (1999) applied the woods-Saxon potential; and Guimaraes and Prudente (2005) considered the finite element method.

In this study, we apply prolate spheroidal coordinate system to solve the Schrödinger equation for this system. Then by modification of the resultant equations with Jacobi and Laguerre differential equation, we calculate the energy spectrum and eigen states of H2+.

Physics Journal, Electrical Transport Properties, Transmission Electron Microscopy, Magnetotransport Data, Antiferromagnetic Semiconductors, Chemical Precipitation Method, Nanocrystalline Manganites, Perovskite Structure, Citrate-gel Method, Polycrystalline Perovskite Material, Debye Scherrer Formula.