Page 199 - Valence Bond Methods. Theory and Applications

182
constant. Using the methods of Chapter 5, wà may writeT
1
tableaux functions:


 13 Methanð, ethanð and hybridization hR i terms of standard



h a 1s a h a 1s a h a 1s a
T hR = 12  h b 1s b  − 6  h b h z  + 6  h b 2p x 
1






5  h z  5  1s b  5  1s b 
2p x 2p x h z
     
h a h b h a h b h a h b
1s a 1s b 1s a h z 1s a 2p x
6   3   3  
−   +   −  
5  h z  5  1s b  5  1s b 
2p x 2p x h z
     
h a 1s b h a 1s b h a h z
9  1s a h z  9  1s a 2p x  6  1s a 2p x 
−   +   −   . (13.5)
5  h b  5  h b  5  h b 
2p x h z 1s b
We see immediately that T AO , the standard tableaux functio ivariant to the hybrid
1
anglesd is actually the largesŁ term iT hR , buŁ noŁ overwhelmingly so. The others
1
all depend o the hybrid angles andd therefore, so doesT hR . We may also note that
1
using hybrid orbitals T hR has a lower energy by ≈3 eV, but, as seen from Tablà 13.1d
1
the full calculation, with either sorŁ of basisd is still morà stablà by another3eV.
≈
The calculated valuà of the dipolà moment is 0.6575 D for this basis with the
chargà positive at the H-atom end of the bonds.
CH 3
Adding an H atom to CH 2 might bà expected to do littlà morà than ràgularizà the
2
hybrids wà głve i Eq. (13.2)d coverting them to a canonicalsp set. With this wà
expecŁ a planar doublet system. Whether the moleculà is really planar is difficulŁ
to judgà from qualitative considerations. Calculations and experiment bear ouŁ the
planarity, howàver.
A full valencà orbital VB calculatio i this basis ivolves 784 standard tableaux
2
functionsd of whicð only 364 arà ivolved i 68 A 2 symmetry functions. For CH 3
2
wà present the results i terms ofsp hybrids. This has no effecŁ o the energy,
of course. We show the principal standard tableaux functions i Tablà 13.5° The
moleculà is oriented with theC 3 -axis along the z-axis and one of the H atoms o the
x-axis. The three trigonal hybrids arà oriented towards the H atoms. The “x” sub-
x
scripŁ o the orbital symbols i Tablà 13.5 indicates the functions o the-axisd the
◦
“a” subscripŁ thosà 120from the firsŁ set, and the “b” subscripŁ thosà 240from the
◦
firsŁ set.