Lecture #22
  CURMUDGEON GENERAL'S WARNING. These "slides" represent highlights from lecture and are neither complete nor meant to replace lecture. It is advised not to use these as a reliable means to replace missed lecture material. Do so at risk to healthy academic performance in 09-105.
Lecture Outline Quantum Theory of the Chemical Bond

Molecular Orbitals

Polyatomic molecules

Hybrid atomic orbitals

Polyatomic molecues.

In constructing a valence bond picture for methane, the valence electron configuration on carbon, 2s22p2 will not return the correct directional properties for the bonds produced by overlap with hydrogen's 1s atomic orbitals.

The 2s, 2px, 2py and 2pz atomic orbital geometries are not tetrahedrally directed. We need orbitals that have the correct directional properties and we can synthesize these from our known atomic orbitals by mixing, a process known here as hybridization. We will look at an example that the text refers to for describing the bonding in BeH2.
In the BeH2 molecule (HBeH), we need two atomic orbitals that are directed linearly, according to the "known" geometry. (VSEPR tells us this.) Constructing hybrid orbitals from the 2s and 2px atomic orbitals by taking appropriate mixtures. These are being shown for illustrative purposes. The math is not part of the required material. "2s" represents the pure 2s atomic orbital wave function, for example. The numerical coefficients are not crucial to the demonstration.
Hybridizing from the 2s and 2px atomic orbitals giving a hybrid orbital pointing in the x-direction.
Hybridizing from the 2s and 2px atomic orbitals giving a hybrid pointing at 180o to the previous hybrid.
The two pure atomic orbitals we have used have been re-assembled into two mixed, hybrid atomic orbitals. Each of the latter are identical in shape, but pointed in different directions. The remaining 2p atomic orbitals that have not been used remain unchanged in subsequent discussions about the full set of valence atomic orbitals in this system (that is, on the Be in beryllium hydride).

Now we return to a discussion of the atomic orbitals that are appropriate to a tetrahedral geometry about a central atom that had pure s and p atomic orbitals when nothing else was around.

Hybridization of s and p valence orbitals in a linear environment (two perturbing sites). Two of the orbitals, the s and one of the p orbitals are mixed forming two sp hybrid orbitals, leaving the remaining two p orbitals as pure atomic orbitals.
Hybridization of s and p valence orbitals in a tetrahedral environment. That is, the carbon in this case is surrounded by a tetrahedral arrangment of (four) objects (hydrogens) which influence the behavior of carbons valence electrons. The net result of all the forces (carbon at the center and a tetrahedral arrangement of perturbations) gives four orbitals that are identical in shape to each other and that are directed to the vertices of a tetrahedron. These orbitals can be simulated by hybridizing the four previous pure orbitals through mixing mathematically and are each symbolized as sp3 hybrid orbitals.
Hybridization of s and p valence orbitals in a trigonal planar environment (three perturbing sites around the central atom). Three valence orbitals, the s and two of the p's, are hybridized to give three sp2 hybrid orbitals, leaving one pure atomic p orbital unchanged.
Summary of the effect of various perturbing geometries on the s/p valence shells in a central atom. By "vacuum" is meant the atom in isolation from any external influence.
For expanded octets, the d orbitals come into consideration. There are five d orbitals (since d implies the angular momentum quantum number l = 2 and hence ml = -2, -1, 0, +1, +2. The trigonal bipyramid and octahedral geometries involve five and six hybrids as indicated.
 An overview of the relationship among steric number, and the number and types of pure and hybrid orbitals associated with the implied geometries.  
An illustration of a central atom's two sp hybrid orbitals present for a linear geometry about the central atom. Bear in mind that it is implied that there are also two pure pure p orbitals (not shown) perpendicular to the indicated axis.
Three sp2 hybrid atomic orbitals, constructed in some detail on an earlier slide, showing their trigonal planar directionality. It is implied that there is one remaining pure p atomic orbital perpendicular to the plane shown.
Bonding considerations in a polyatomic system illustrated for the molecule ethylene (aka ethene) H2C=CH2. The Lewis structure leads to a VSEPR expectation that each carbon is in a trigonal planar geometry which in turn indicates that there are three sp2 hybrid atomic orbitals and one pure p atomic orbital at each carbon.
The atomic orbitals for either individual carbon in ethylene.
The atomic orbitals for carbon in ethylene highlighting the one pure 2p atomic orbital perpendicular to the plane of the three hybrid sp2 orbitals.
Assembling the ethylene molecule showing the overlap between the various pairs of atomic orbitals (which can be pure or hybrid).
The C-H "sigma" bond. The "sigma" label comes from the symmetry (appearance) of the electron distribution about the bond axis. When viewed directly down the bond axis, the profile resembles the atomic s-orbital profile. Hence the "sigma" term.
The C-C "sigma" bond. The profile of the implied electron distribution about the bond axis is similar to that on the previous slide for the C-H sigma bond.
Rotating the molecule 90o about the C-C bond axis so that the structure's plane is perpendicular to the point of view brings the pure 2p atomic orbitals into view (since they are perpendicular to the sp2 hybrid orbitals).
The pair of bars represents one (that's one!) overlap result from bringing the two neighboring 2p orbitals closer together.
Looking down the C-C axis again to determine the geometric appearance of the electron density in this new region of overlap.
The view down the C-C axis shows that the electron density about the axis is analogous to what is seen for the electron density of a p orbital about a nucleus. We refer to the bonding density in this arrangement as constituting a pi bond.