mathematical sense of being elements of a vector space [ 23 ] (if not in the restricted
sense of the physicist as physical entities with magnitude and direction); it is
therefore permissible to take linear combinations of these vectors to produce new
members of the vector space. A good, brief introduction to hybridization is given by
Streitwieser [ 24 ].
In a familiar example, a 2sorbital can be mixed with three 2porbitals to give
four hybrid orbitals; this can be done in an infinite number of ways, such as (from
now onfwill be used for atomic orbitals andcfor molecular orbitals):
f 1 ¼^12 ðsþpxþpyþpzÞ
f 2 ¼^12 ðsþpxþpy"pzÞ
f 3 ¼^12 ðsþpx"py"pzÞ
f 4 ¼^12 ðsþpx"pyþpzÞ(4.33)
or
fa¼^12 ðsþpxþ 21 =^2 pzÞ
fb¼^12 ðsþpx" 21 =^2 pzÞ
fc¼^12 ðs"pxþ 21 =^2 pyÞ
fd¼^12 ðs"px" 21 =^2 pyÞ(4.34)
Both the set (4.33) and the set (4.34) consist of foursp^3 orbitals, since the
electron density contributions from the componentsandporbitals to the hybrid
is, in each case (considering thesquaresof the coefficients; recall the Born
interpretation of the square of a wavefunction,Section 4.2.6) in the ratio 1:3,
i.e. 1/4 : 3(1/4) and 1/4 : (1/4 + 2/4), and in each set we have used a total of one
sorbital, and one each of thepx,pyandpzorbitals. The total electron density from
each component orbital is unity, e.g. fors, 4(1/4).
Hybridization is purely a mathematical procedure, originally invented to
reconcile the quantum mechanical picture of electron density ins,p, etc. orbitals
with traditional views of directed valence. For example, it is sometimes said that
in the absence of hybridization combining a carbon atom with four unpaired
electrons with four hydrogen atoms would give a methane molecule with three
equivalent, mutually perpendicular bonds and a fourth, different, bond (Fig.4.6).
Actually, this is incorrect: the 2sand three 2porbitals of an unhybridized carbon
along with the four 1s orbitals of four hydrogen atoms provide, without invoking
hybridization, a tetrahedrally symmetrical valence electron distribution that leads
to tetrahedral methane with four equivalent bonds (Fig.4.6). In fact, it has been
said “It is sometimes convenient to combine aos [atomic orbitals] to form hybrid
orbitals that have well defined directional character and then to form mos
[molecular orbitals] by combining these hybrid orbitals. This recombination of
aos to form hybrids isnevernecessary ...” [ 25 ]. Interestingly, the MOs accom-
modating the four highest-energy electron pairs of methane (the eight valence
104 4 Introduction to Quantum Mechanics in Computational Chemistry