Symmetry in Chemical Bonding - Molecular Symmetry - page 228

at a given point in space. In this case, the 1s basis functions are real (they contain no
imaginary part); so, if we choose any point r , the probability becomes

P lg ( r ) δτ = (1 σ g + ( r )) 2 δτ (7.8)

where 1 is a small volume around

r . We can never define the probability of the electron being exactly at r , only within this
small volume centred on r .
Writing the 1

σ g + ( r ) is the value of the wavefunction at point r and δτ

σ g + MO as its SALC of the basis functions from Equation (7.6) gives the
probability in terms of the basis functions:

N 1g 2 s 1 2 N 1g 2 s 2 2 2 N 1g 2 s 1 s 2 δτ P lg ( r ) δτ = δτ + δτ + (7.9)

where the shorthand s i = s( r ,H i ), i.e. the value of the i th s-orbital at position r , has been

introduced.
Applying the same procedure to the 2 σ u + MO from Equation (7.7) gives

N 1u 2 s 1 2 N 1u 2 s 2 2 2 N 1u 2 s 1 s 2 δτ

P lu ( r ) δτ = δτ + δτ − (7.10)

σ u +
MO at any point is actually made up of two types of contribution. The first two terms
contain s 1 2 and s 2 2 , and so involve only single basis functions; but the third term, s 1 s 2 ,isa
mixture of basis functions, so we will now look at how these terms contribute to the total
density.
To imagine the shape of the density it is useful to simplify the three-dimensional func-
tion defined for P ( r ) by reducing to one dimension in some way. A D ∞ h molecule is
conventionally taken to be aligned with its principal C ∞ axis along the Z -direction; it
must also have cylindrical symmetry, and so the density will be constant around any circle
centred on the molecular axis and parallel to the XY plane. So one way to simplify P ( r )is
to integrate over planes perpendicular to the molecular axis and then plot these values as
a function of z . This is like taking slices through the molecule and then plotting the total
density from each slice at its z coordinate. More detail of the required procedure is given
in Appendix 10. The results for the 1

Equations (7.9) and (7.10) say that the electron density associated with the 1 σ g + or 2

σ g + and 2 σ u + SALCs are illustrated in Figure 7.7a

and b respectively. For the 1 σ g + MO the first two terms are added to the contribution from

the basis overlap 2 N 1g 2 s 1 s 2 , and this leads to a high value for the density in the internuclear
region for ( 1

σ u + case the overlap contribution is subtracted from the first
two terms, and so there is a lowering of the density in between the nuclei. In Appendix 9
it is shown that the H 1s orbital follows an exponential function:

σ g + ) 2 .Inthe2

A exp

1
a 0

3 / 2

r i
a 0

1
√ π s i = − with A = (7.11)