The above simple model of the chemical bond may also be applied to polyatomic molecules. We consider a few molecules as examples, beginning with the methane molecule, CH4. This molecule has four equivalent carbon-hydrogen bonds that form a tetrahedral geometry. We again use the molecular orbital model, and we assume that the bond orbitals may be constructed from atomic valence orbitals centered on the carbon atom and from the four hydrogen (1s) orbitals. The ground state electronic configuration of the carbon atom is (1s)2(2s)2(2p)2, but we excite this configuration to (1s)2(2s) (2p)3in order to obtain the valence orbitals since the excitation energy is more than compensated for by the improvement in bond energies.
The atomic valence orbitals may now be derived from the following guidelines:
1. They must be linear combinations of the 2s, 2px, 2py, and 2pzorbitals.
2. They must be orthogonal.
3. They must be equivalent.
4. Their overall electronic distribution must correspond to the configuration (2s) (2p)3.
THE STRUCTURES OF SOME SIMPLE POLYATOMIC MOLECULES 179
It is easily verified that the following set of valence orbitals meets all four of the above requirements:
t1ẳ1
2ðsỵpxỵpyỵpzị t2ẳ1
2ðsỵpxpypzị t3ẳ1
2ðspxỵpypzị t4ẳ1
2ðspxpyỵpzị
ð12-71ị
where we have denoted the (2s) orbital by the symbols.
We show the directions of these four atomic valence orbitals in Figure 12-6, and it may be seen that they exhibit a tetrahedral geometry pattern. In order to describe the wave function of the methane molecule, we place the four hydrogen atoms along the tetrahedral directions at the appropriate distances and we define the four bond orbitals
siẳtiỵhi ð12-72ị wherehiare the hydrogen atom (1s) orbitals. We assume that the carbon and hydro- gen atoms have comparable electron-attracting powers so that we may set the
t1 t3
t4
t2
Figure 12-6 Foursp3hybridized valence orbitals forming a tetrahedral geometry.
parameterrof Eq. (12-60) equal to unity. It follows that the electronic structure of the methane molecule may be represented by the following distribution:
CH4! ð1sị2ðs1ị2ðs2ị2ðs3ị2ðs4ị2 ð12-73ị It may be interesting to compare the electronic structure of the methane mole- cule with the structures of the ammonia molecule NH3and the water molecule H2O since they all have the same number of electrons. The structure of NH3may be derived from Eq. (12-73) by removing one hydrogen nucleus and by replacing the bond orbitals4by the corresponding atomic valence orbitalt4:
NH3 ! ð1sị2ðs1ị2ðs2ị2ðs3ị2ðt4ị2 ð12-74ị The electronic structure of the water molecule H2O may be obtained in a similar fashion by replacing two bond orbitals,s3ands4, by atomic valence orbitals:
H2O! ð1sị2ðs1ị2ðs2ịðt3ị2ðt4ị2 ð12-75ị It should be noted that the charge clouds associated with the lone pair electrons in ammonia and water point in well-defined directions that form an approximate tetra- hedral geometry pattern. The lone pair electrons may therefore give rise to electric interactions with other molecules that are important in biochemistry and medicine.
The four atomic valence orbitals of Eq. (12-71) are obtained as hybrids of one atomic (2s) orbital and three atomic (2p) orbitals, namely, (2px), (2py), and (2pz), and they are known as a set ofsp3 hybridized orbitals. Thesp3 type is the most common hybridization type, but there are two alternative schemes for constructing atomic valence orbitals, namely,sp2andsp hybridization.
Thesp2hybridization pattern is best explained by considering the structure of the ethylene molecule, C2H4(see Figure 12-7). This molecule has a planar structure and the four carbon-hydrogen bonds form 120 angles with the carbon-carbon bond, which is assumed to be a double bond. Since all the bonds are located in
C C
H
H H
H
Figure 12-7 Geometry of the ethylene molecule.
THE STRUCTURES OF SOME SIMPLE POLYATOMIC MOLECULES 181
the molecular plane, which we take as the XY plane, they must be represented as linear combinations of the carbon (2s) orbital and the carbon (2px) and (2py) orbitals, with exclusion of the carbon (2pz) orbital.
The three atomic valence orbitals must again be equivalent and orthogonal. If we denote the carbon atoms by A and B, then the threesp2hybridized valence orbitals of carbon atom A are given by
tA1ẳ ð1= ffiffiffi p3
ịsAỵ ð ffiffiffi p2
= ffiffiffi p3
ịpxA
tA2ẳ ð1= ffiffiffi 3
p ịsA ð1= ffiffiffi 6
p ịpxAỵ ð1= ffiffiffi 2 p ịpyA
tA3ẳ ð1= ffiffiffi p3
ịsA ð1= ffiffiffi p6
ịpxA ð1= ffiffiffi p2
ịpyA
ð12-76ị
The valence orbitals of atom B are defined in a similar fashion.
Two of the atomic valence orbitals on carbon atoms A and B may now be combined to form a carbon-carbons bond, and the other carbon valence orbitals are combined with the four hydrogen (1s) orbitals to form carbon-hydrogen bonds.
The two carbonpzorbitalspzAandpzBmay be combined to form a carbonpbond, as illustrated in Figure 12-8. Thesp2hybridization in the ethylene molecule there- fore predicts a carbon-carbon double bond consisting of asbond and an additional pbond.
The simplest molecule exhibitingsphybridization is acetylene, C2H2, which has a linear structure and a triple carbon-carbon bond. The triple bond consists of ones bond and two additionalpbonds.
If we take theXaxis as the molecular axis and denote the carbon atoms again by A and B, then the four atomic valence orbitals are obtained as linear combinations of the (2s) and (2px) orbitals:
tA1ẳsAỵpxA tB1ẳsBpxB
tA2ẳsApxA tB2ẳsBỵpxB
ð12-77ị
The orbitalstA1andtB1form a carbon-carbon sorbital
sẳtA1ỵtB1 ð12-78ị
Figure 12-8 Formation of apbond.
and the othersporbitalstA2andtB2may be combined with the hydrogen (1s) orbi- tals to form two carbon-hydrogen bonding orbitalssHAandsHB. The two orbitals pyAandpyBare combined to form a carbon-carbon bonding orbitalp, and the two orbitalspzAandpzBform a similar bonding orbitalp0. The electron structure is then given by
C2H2! ð1sAị2ð1sBị2ðsị2ðpị2ðp0ị2ðsHAị2ðsHBị2 ð12-79ị It is interesting to compare the electronic structure (12-79) of acetylene with the electronic structure (12-70) of the N2molecule. The two molecules have the same number of electrons, and if we remove the two hydrogen nuclei from C2H2its elec- tronic configuration becomes identical with the N2configuration.
It may seen that the various wave functions discussed in this section are of a rather crude nature, but they led to surprisingly good results when used as a basis for the calculation of molecular properties. More important, they have contributed to a general understanding of molecular structure.