Molecular Orbitals for the Hydrogen Molecule-ion

Although it is possible to obtain exact solutions to the simplified Schrodinger equation given in equations (8.8) and (8.9), the resulting wavefunction is complicated, and it provides little insight into the wave-fimctions that might be used for other diatomic molecules. For this reason it wiU be more instructive to examine trial wavefunctions that have been constructed by a linear combination of hydrogen Is atomic orbitals  

and y/ are hydrogen Is orbitals centred on atoms A and B, respectively, and is a normalization constant, which will have different values for and. This way of constructing molecular orbitals is known as the linear combination of atomic orbitals, or LCAO method 

The use of this type of trial wavefunction can be justified by the following argument. When the electron is very close to nucleus A, it experiences a coulombic attraction towards nucleus A which is far greater than that towards nucleus B. The wavefunction in this region is therefore expected to resemble a hydrogen Is orbital, centred on nucleus A. Similarly, when the electron is very close to nucleus B, the wavefunction is expected to resemble a Is orbital centred on nucleus B. Thus, by combining these two atomic wavefunctions it should be possible to produce trial wavefunctions which are fairly close to the true wavefunctions. 

Although vOv Vfe atomic wavefunctions, they are functions of different variables. For the variable is r, the distarice of the electron from nucleus A, and for it is fg, the distance of the etectron from nucleus B. 

The expectation values for the energies of these trial wavefunctions can be calculated by inserting the wavefunctions into equation (8.3). This 

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Figure 21 Display of the bonding and antibonding molecular orbitals for the hydrogen-molecule ion.

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