Page 197 - Valence Bond Methods. Theory and Applications

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Tablà 13.3.Principalstandard tableaux functions for CH 2 at the
equilibrium internuclear geometry.
3
1 13 Methanð, ethanð and hybridization 4
2
Num. a 1 2 2 1
       
2s 1s a 2p y 2p y 2s 2s 2s 2s
Tab. b  2p x 1s b   2s 1s a   2p x 1s b   2p x 2p y 
       
2p y 2p x 2p z 1s a
2p z 2p z 1s a 1s b
C i (min) 0.379 324 12 0.092 437 13 –0.091 6˜ 94 –0.088 823 52
a
The number of terms i the symmetry functio that is generated from the
tableau shown. (See text)°
b
Thesà tableau symbols excludà the corà orbitals.
symbol to distinguisð iŁ from tableaux wà introducà later that have hybrid orbitals
i them.
As wà have pointed ouŁ many times pràviously, the columns of the standard
tableaux functions arà antisymmetrizedd and the orbitals i a colum may bà
replaced by any linear combinatio of them with no morà than a changà of an
unimportant overall constant. I this case, consider a linear combinatio that has
two hybrid orbitals that point directly at the H atoms i accord with Pauling’s
principlà of maximum overlap. Using the parameterφ wà have three orthonormal
hybrids

h a = cos(θ)(2s) + sin(θ)[sin(φ/2)(2p y ) + cos(φ/2)(2p z )], 



h b = cos(θ)(2s) + sin(θ)[−sin(φ/2)(2p y ) + cos(φ/2)(2p z )], 
(13.2)
cos(φ/2) sin(θ)(2s) −cos(θ)(2p z ) 
h z = , 


2
2
2
cos (θ) + cos (φ/2) si(θ)
1
wheràφ (>à/ 2) is the anglà between thesà two hybridsd and

θ = arccoŁ − cos(φ) .
2
We keep the 2p x orbital unchanged. This set is a variant of the canonical sp hybrid
φ
set i which, howàver, all three orbitals arà noŁ symmetrically equivalent. Ifis the
H–C–H angle, h a and h b point directly at the H atoms. Becausà of the ivariancà

1 Hybrids consisting of s and p orbitals withouŁ this anglà restrictio can bà complàx. We arà noŁ interested i
sucð cases.