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### Molecular Orbitals

For H2, the molecular orbital is
 = (16) = C(e-rA+e-rB) (17)
N,C are normalisation constants. The coordinates are

and the molecular orbital looks like

if it is evaluated along the molecular axis. This  molecular orbital is a linear combination of atomic orbitals. (MO-LCAO).
For the water molecule H2O, we can construct a molecular orbital which is a combination of 2pzO and 1s functions on each H
 (18)
c1 and c2 are mixing coefficients and take numerical values. This orbital may be sketched:

Another molecular orbital for H2O can be constructed from the 2pyO atomic orbital and the 1s functions
 (19)
and it is sketched

The above are both bonding orbitals, corresponding to a build up of charge between the O and H atoms. In general we write
 = (20) = (21)
where  are the atomic orbitals and  are the molecular orbitals. In practice today we do not use hydrogen-like functions e-r because they are too difficult to work with, but instead we use gaussian functions e-ar2.
Thus for H2
 (22)

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Nicholas Handy

1998-09-23