4.7 Application: The sp 3 Hybridization of Carbon
77
with
T =
1
2
⎛
⎜
⎜
⎝
1111
1 −1 −11
11 −1 −1
1 −11 −1
⎞
⎟
⎟
⎠
(4.87)
SALCs are normalized to unity, neglecting overlap between the sites. The matrix T
transforms the localized orbitals on the sites to delocalized molecular orbitals with
irreducible symmetry characteristics. The inverse matrix T −1 fulfills the opposite
role and localizes the molecular orbital set back on the atomic sites.
The valence shell of the central carbon atom contains four orbitals, which incidentally also transform as A 1 + T 2 . The precise correspondence is as follows:
|2s↔A 1
|2p x ↔T 2x
|2p y ↔T 2y
|2p z ↔T 2z
(4.88)
Hence, we can match the central valence shell with the hydrogen SALCs. In fact,
this correspondence provides a simple pictorial method for obtaining the SALCs
immediately. The weighting coefficients for a given SALC are simply taken as proportional to the local amplitude of the central 2s or 2p function, with the same
symmetry, as is illustrated in Fig. 4.4. In this way one obtains a SALC that has the
same nodal characteristics and thus the same symmetry as the central orbital. Note
that this procedure also aligns the phases of the peripheral and central orbitals.
Starting from on-site localized atomic orbitals, we have thus transformed these
into SALCs using the T matrix and then found a perfect matching with the central
valence orbitals on carbon. What would now be the effect of applying the inverse
transformation, T −1 , not to the hydrogen SALCs but to the central carbon orbitals?
This yields an interesting result. The inverse matrix, which transforms delocalized
SALCs back into localized orbitals, reshapes the carbon valence orbitals by projecting out linear combinations, of mixed or hybrid character, which are maximally
directed to a single site of the tetrahedron. These are the ubiquitous sp 3 hybrids of
Pauling, which we can label with the site labels A, B, C, and D:
|sp 3
A | sp 3
B | sp 3
C | sp 3
D
=
|2s| 2p x | 2p y | 2p z
1
2
⎛
⎜
⎜
⎝
1111
1 −11 −1
1 −1 −11
11 −1 −1
⎞
⎟
⎟
⎠
(4.89)
The components of the valence shell of carbon being a scalar (2s) and a vector
(2p), the tetrahedron is the optimal geometry that provides four valence sites, which
together transform precisely as scalar and vector. The alternative high-symmetry
77
with
T =
1
2
⎛
⎜
⎜
⎝
1111
1 −1 −11
11 −1 −1
1 −11 −1
⎞
⎟
⎟
⎠
(4.87)
SALCs are normalized to unity, neglecting overlap between the sites. The matrix T
transforms the localized orbitals on the sites to delocalized molecular orbitals with
irreducible symmetry characteristics. The inverse matrix T −1 fulfills the opposite
role and localizes the molecular orbital set back on the atomic sites.
The valence shell of the central carbon atom contains four orbitals, which incidentally also transform as A 1 + T 2 . The precise correspondence is as follows:
|2s↔A 1
|2p x ↔T 2x
|2p y ↔T 2y
|2p z ↔T 2z
(4.88)
Hence, we can match the central valence shell with the hydrogen SALCs. In fact,
this correspondence provides a simple pictorial method for obtaining the SALCs
immediately. The weighting coefficients for a given SALC are simply taken as proportional to the local amplitude of the central 2s or 2p function, with the same
symmetry, as is illustrated in Fig. 4.4. In this way one obtains a SALC that has the
same nodal characteristics and thus the same symmetry as the central orbital. Note
that this procedure also aligns the phases of the peripheral and central orbitals.
Starting from on-site localized atomic orbitals, we have thus transformed these
into SALCs using the T matrix and then found a perfect matching with the central
valence orbitals on carbon. What would now be the effect of applying the inverse
transformation, T −1 , not to the hydrogen SALCs but to the central carbon orbitals?
This yields an interesting result. The inverse matrix, which transforms delocalized
SALCs back into localized orbitals, reshapes the carbon valence orbitals by projecting out linear combinations, of mixed or hybrid character, which are maximally
directed to a single site of the tetrahedron. These are the ubiquitous sp 3 hybrids of
Pauling, which we can label with the site labels A, B, C, and D:
|sp 3
A | sp 3
B | sp 3
C | sp 3
D
=
|2s| 2p x | 2p y | 2p z
1
2
⎛
⎜
⎜
⎝
1111
1 −11 −1
1 −1 −11
11 −1 −1
⎞
⎟
⎟
⎠
(4.89)
The components of the valence shell of carbon being a scalar (2s) and a vector
(2p), the tetrahedron is the optimal geometry that provides four valence sites, which
together transform precisely as scalar and vector. The alternative high-symmetry