Molecular orbitals the hydrogen molecule ion

We shall now use quantum mechanics to describe chemical bonds and begin with the simplest of all molecules, hJ. The hydrogen molecule ion has not been found in solids or melts, but is easily formed by electric discharge through hydrogen gas. It is also one of the most common molecules in interstellar space. The properties are well known from experimental studies, the equilibrium bond distance is Re = 106.0 pm and the dissociation energy De = 269 kJ mol Comparison with the H2 molecule. Re = 74.1 pm and De = 455 kJ mol shows that the one-electron bond in the ion is 43% longer and 41% weaker than the two-electron bond in the neutral molecule. 

The importance of the hydrogen molecule ion for the theory of diatomic molecules is similar to the importance of the hydrogen atom for our understanding of atoms both H and H2 are one-electron systems for which the Schrbdinger equation can be solved exactly. The exact solution of the one-electron species is then used as a starting point for the discussion of polyelectron species for which exact solutions of the Schrodinger are unavailable. 

In our discussion of the hJ ion we shall use several coordinate systems, sometimes we shall even mix different coordinate systems in the same expression. We begin by defining a right-handed Cartesian coordinate system with the z-axis running through both nuclei and the origin at the midpoint between them. See Fig. 7.2. We shall also use Cartesian coordinate systems with their origins at one of the nuclei, A or B. The za and zb axes coincide with the 

In Section 4.2 we defined the electronic energy of the molecule as the sum of the kinetic energy of the electron and the Coulomb energies due to attraction between the electron and each of the two nuclei and due repulsion between the nuclei  

This classical expression for the energy may be transformed into the Hamiltonian operator of the system  

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