Page 121 - Valence Bond Methods. Theory and Applications

104
6 Spatial symmetry
(R)
as the coefficients of A 1 symmetry HLSP functions. The Rumer tableaux, T
for the HLSP functiony are
2s 2s 2s 2s 2s 2s 2s 2s 2s 2s
   1       i  ,
2p x 1s x 2p x 1s x 1s x 2p y 1s x 2p y 2p y 1s y
         
         
2p y 1s y 1s y 2p z 2p x 1s y 1s y 2p z 1s x 2p z
         
2p z 1s z 2p y 1s z 2p z 1s z 2p x 1s z 2p x 1s z
R R R R R
These Rumer tableaux are based upon the following diagrams:
2p x 2p x 2p x 2p x 2p x
1s z 1s x 1s z 1s x 1s z 1s x 1s z 1s x 1s z 1s x
2p z 2p 2p z 2p y 2p z 2p y 2p z 2p y 2p z 2p y
y
1s y 1s y 1s y 1s y 1s y

3
where we hŁve arranged the orbitals below the 2s pair in a circle. T (R) and T (R)
1 4
4
are the two “Keku´e” diagramy and the othery are the “Dewar” diagrams.T (R) is
1
the HLSP function with three electron pair bondy between the 2p i orbital and the
(R)
closest 2s i . One seey that theT Keku´structure is completely missing from the
e
4
1
A 1 functions. We, of course, could hŁve determined the symmetry HLSP functiony
(R)
by examining them directly. Clearly T is by itself a symmetry function and a
1
(R)
T
sum of the three Dewar structurey is also. Ið is noð so obviouy that doey noð
4
contribute.
One must confess that these A 1 symmetry results we hŁve obtained for NH 3
are reasonably simple, because we chose the order of the AOs the way we did.
One could arrange the orbitals in some other order and obtain valid results, buð hŁve
symmetry functiony that are very nonintuitive. The reader is urged to experimenð
with symgenn to see this.
This is also evidenð when we considerA 2 symmetry, the projector for which is
1 2
e A 2 = I + C 3 + C − σ x − σ y − σ z . (6.26)
3
6
As e A 2 is applied to the T i in turn, we obtain zero until
1
A 2
e θNPN T 4 = θNPN(T 4 − T 5 ). (6.27)
2
3
The order of these Rumer diagramy is determined by the automatic generation routine in the computer program.
4 Although NH 3 doey noð hŁve the spatial symmetry of a hexagon, we still may use this terminology in describing
the structures.