Valence Bond Theory and Hybrid Atomic Orbitals

This picture is an image of a Centaur from Sphinx Stargate. The Centaur is a race of monsters in Greek mythology, hybrid animal having the head, arms and torso of a man united to the body and legs of a horse. Mixing a number of atomic orbitals to form the same number of hybrid orbitals to explain chemical bonding and shapes and molecular structures is a rather recent myth. The most significant development in the first half of the 20th century is the human's ability to understand the structure of atoms and molecules. Computation has made mathematical concepts visible to the extent that we now can see the atomic and molecular orbitals. On the other hand, Using everyday encountered materials or toys can also generate beautiful illustrations of hybrid atomic orbitals.
The valence bond (VB) approach is different from the molecular orbital (MO) theory. Despite their differences, most of their results are the same, and they are interesting.

The valence bond (VB) theory

The valence-bond approach considers the overlap of the atomic orbitals (AO) of the participation atoms to form a chemical bond. Due to the overlapping, electrons are localized in the bond region. The overlapping AOs can be of different types, for example, a sigma bond may be formed by the overlapping the following AOs.

Chemical bonds formed due to overlap of atomic orbitals
s-s	s-p	s-d	p-p	p-d	d-d
H-H Li-H	H-C H-N H-F	H-Pd in palladium hydride	C-C P-P S-S	F-S in SF6	Fe-Fe

However, the atomic orbitals for bonding may not be "pure" atomic orbitals directly from the solution of the Schrodinger Equation. Often, the bonding atomic orbitals have a character of several possible types of orbitals. The methods to get an AO with the proper character for the bonding is called hybridization. The resulting atomic orbitals are called hybridized atomic orbitals or simply hybrid orbitals.
We shall look at the shapes of some hybrid orbitals first, because these shapes determine the shapes of the molecules.

Hybridization of atomic orbitals

The solution to the Schrodinger Equation provides the wavefunctions for the following atomic orbitals:
1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, etc. For atoms containing two or more electrons, the energy levels are shifted with respect to those of the H atom. An atomic orbital is really the energy state of an electron bound to an atomic nucleus. The energy state changes when one atom is bonded to another atom. Quantum mechanical approaches by combining the wave functions to give new wavefunctions are called hybridization of atomic orbitals. Hybridization has a sound mathematical fundation, but it is a little too complicated to show the details here. Leaving out the jargons, we can say that an imaginary mixing process converts a set of atomic orbitals to a new set of hybrid atomic orbitals or hybrid orbitals.
At this level, we consider the following hybrid orbitals:
sp
sp²
sp³
sp³d
sp³d²

The sp hybrid atomic orbitals

The sp hybrid atomic orbitals are possible states of electron in an atom, especially when it is bonded to others. These electron states have half 2s and half 2p characters. From a mathematical view point, there are two ways to combine the 2s and 2p atomic orbitals: sp₁ = 2s + 2p
sp₂ = 2s - 2p
These energy states (sp₁ and sp₂) have a region of high electron probability each, and the two atomic orbitals are located opposite to each other, centered on the atom. The sp hybrid orbitals are represented by this photograph.

H-Be-H	1s             1s H sp1 Be sp2 H 1s             1s

For example, the molecule H-Be-H is formed due to the overlapping of two 1s orbitals of 2 H atoms and the two sp hybridized orbitals of Be. Thus, the H-Be-H molecule is linear. The diagram here shows the overlapping of AOs in the molecule H-Be-H.
The ground state electronic configuration of Be is 1s²2s², and one may think of the electronic configuration "before" bonding as 1s²sp². The two electrons in the sp hybrid orbitals have the same energy.

Linear molecules
ClBeCl HCCH HCN O=C=O

You may say that the concept of hybridizing AOs for the bonding is just a story made up to explain the molecular shape of Cl-Be-Cl. You are right! The story is lovely and interesting, though.
In general, when two and only two atoms bond to a third atom and the third atom makes use of the sp hybridized orbitals, the three atoms are on a straight line. For example, sp hybrid orbitals are used in the central atoms in the molecules shown on the right.

The sp² hybrid orbitals

The energy states of the valence electrons in atoms of the second period are in the 2s and 2p orbitals. If we mix two of the 2p orbitals with a 2s orbital, we end up with three sp² hybridized orbitals. These three orbitals lie on a plane, and they point to the vertices of a equilateral triangle as shown here. When the central atom makes use of sp² hybridized orbitals, the compound so formed has a trigonal shape. BF₃ is such a molecule:

Molecules with sp2 Hybrid orbitals
F | B /     \ F         F	. .     -2 :O: | C /     \\ :O::         O	. N //     \\ O         O	. . O //     \\ O         O	. . S //     \\ O         O

Not all three sp² hybridized orbitals have to be used in bonding. One of the orbitals may be occupied by a pair or a single electron. If we do not count the unshared electrons, these molecules are bent, rather than linear. The three molecules shown together with the BF₃ molecule are such molecules.
Carbon atoms also makes use of the sp² hybrid orbitals in the compound H₂C=CH₂. In this molecule, the remaining p orbital from each of the carbon overlap to form the additional pi, p, bond.

Planar molecules with sp2 Hybrid orbitals
H               H \             / C = C /             \ H               H	O               2- \             C = O /             O	O               1- \             N = O /             O

Other ions such as CO₃^2-, and NO₃^-, can also be explained in the same way.

The sp³ hybrid orbitals

Mixing one s and all three p atomic orbitals produces a set of four equivalent sp³ hybrid atomic orbitals. The four sp³ hybrid orbitals points towards the vertices of a tetrahedron, as shown here in this photograph. When sp³ hybrid orbitals are used for the central atom in the formation of molecule, the molecule is said to have the shape of a tetrahedron.
The typical molecule is CH₄, in which the 1s orbital of a H atom overlap with one of the sp³ hybrid orbitals to form a C-H bond. Four H atoms form four such bonds, and they are all equivalent. The CH₄ molecule is the most cited molecule to have a tetrahedral shape. Other molecules and ions having tetrahedral shapes are SiO₄^4-, SO₄^2-,
As are the cases with sp², hybrid orbitals, one or two of the sp³ hybrid orbitals may be occupied by non-bonding electrons. Water and ammonia are such molecules.

Tetrahedral arrangements of CH4, NH3E and OH2E2
H         H \       / C /  \ H   H	H         H \       / N /  \ :   H	H         H \       / O /  \ :   :

The C, N and O atoms in CH₄, NH₃, OH₂ (or H₂O) molecules use the sp³ hybrid orbitals, however, a lone pair occupy one of the orbitals in NH₃, and two lone pairs occupy two of the sp³ hybrid orbitals in OH₂. The lone pairs must be considered in the VSEPR model, and we can represent a lone pair by E, and two lone pairs by E₂. Thus, we have NH₃E and OH₂E₂ respectively.
The VSEPR number is equal to the number of bonds plus the number of lone pair electrons. Does not matter what is the order of the bond, any bonded pair is considered on bond. Thus, the VSEPR number is 4 for all of CH₄, :NH₃, ::OH₂.
According the the VSEPR theory, the lone electron pairs require more space, and the H-O-H angle is 105 deegrees, less than the ideal tetrahedral angle of 109.5 degrees.

The dsp³ hybrid orbitals

The five dsp³ hybrid orbitals resulted when one 3d, one 3s, and three 3p atomic orbitals are mixed. When an atom makes use of fice dsp³ hybrid orbitals to bond to five other atoms, the geometry of the molecule is often a trigonalbipyramidal. For example, The molecule PClF₄ displayed here forms such a structure. In this diagram, the Cl atom takes up an axial position of the trigonalbipyramid. There are structures in which the Cl atom may take up the equatorial position. The change in arrangement is accomplished by simply change the bond angles. This link discusses this type of configuration changes of this molecule. Some of the dsp³ hybrid orbitals may be occupied by electron pairs. The shapes of these molecules are interesting. In TeCl₄, only one of the hybrid dsp³ orbitals is occupied by a lone pair. This structure may be represented by TeCl₄E, where E represents a lone pair of electrons. Two lone pairs occupy two such orbitals in the molecule BrF₃, or BrF₃E₂. These structures are given in a VSEPR table of 5 and 6 coordinations.
The compound SF₄ is another AX₄E type, and many interhalogen compounds ClF₃ and IF₃ are AX₃E₂ type. The ion I₃^- is of the type AX₂E₃.

The d²sp³ hybrid orbitals

The six d²sp³ hybrid orbitals resulted when two 3d, one 3s, and three 3p atomic orbitals are mixed. When an atom makes use of six d²sp³ hybrid orbitals to bond to six other atoms, the molecule takes the shape of an octahedron, in terms of molecular geometry. The gas compound SF₆ is a typical such structure. This link provides other shapes as well. There are also cases that some of the d²sp³ hybrid orbitals are occupied by lone pair electrons leading to the structures of the following types:

AX₆,   AX₅E,   AX₄E₂   AX₃E₃   and   AX₂E₄
IOF₅,   IF₅E,   XeF₄E₂ No known compounds of AX₃E₃ and AX₂E₄ are known or recognized, because they are predicted to have a T shape and linear shape respectively when the lone pairs of electrons are ignored.

Molecular shapes of compounds

While the hybridized orbitals were introduced, in the foregoing discussion, Valence-shell Electron-pair Repulsion (VSEPR) Model were included to suggest the shapes of various molecules. Specifically, the VSEPR model counts unshared electron pairs and the bonded atoms as the VSEPR number. A single-, double- and tripple-bond is considered as 1. After having considered the hybridized orbitals and the VSEPR model, we can not take a systematic approach to rationalize the shapes of many molecules based on the number of valence electrons. A summary in the form of a table is given here to account for the concepts of hybrid orbitals, valence bond theory, VSEPR, resonance structures, and octet rule. In this table, the geometric shapes of the molecules are described by linear, trigonal planar, tetrahedral, trigonal bypyramidal, and octahedral. The hybrid orbitals use are sp, sp², sp³, dsp³, and d²sp³.
The VSEPR number is the same for all molecules of each group. Instead of using NH₃E, and OH₂E₂, we use :NH₃, ::OH₂ to emphasize the unshared (or lone) electron pairs.

A summary of hybrid orbitals, valence bond theory, VSEPR, resonance structures, and octet rule.
Linear	Trigonal planar	Tetrahedral	Trigonal bipyramidal	Octahedral
sp	sp2	sp3	dsp3	d2sp3
BeH2 BeF2 CO2 HCN HCºCH	BH3 BF3 CH2O (>C=O) >C=C< CO32- benzene graphite fullerenes •NO2 N3- :OO2 (O3) :SO2 SO3	CH4 CF4 CCl4 CH3Cl NH4+ :NH3 :PF3 :SOF2 ::OH2 ::SF2 SiO44- PO43- SO42- ClO4-	PF5 PCl5 PFCl4 :SF4 :TeF4 ::ClF3 ::BrF3 :::XeF2 :::I3- (:::I I2-) :::ICl2-	SF6 IOF5 PF6- SiF62- :BrF5 :IF5 ::XeF4
• a lone odd electron         : a lone electron pair

This table correlates a lot of interesting chemical concepts in order to understand the molecular structures of these compounds or ions. There are some intriqueing chemical relationships among the molecules in each column for you to ponder. Only Be and C atoms are involved in linear molecules. In gas phase, BeH₂ and BeF₂ are stable, and these molecules do not satisfy the octet rule. The element C makes use of sp hybridized orbitals and it has the ability to form double and triple bonds in these linear molecules.
Carbon compounds are present in trigonal planar and tetrahedral molecules, using different hybrid orbitals. The extra electron in nitrogen for its compounds in these groups appear as lone unpaired electron or lone electron pairs. More electrons in O and S lead to compounds with lone electron pairs. The five-atom anions are tetrahedral, and many resonance structures can be written for them.
Trigonal bipyramidal and octahedral molecules have 5 and 6 VSEPR pairs. When the central atoms contain more than 5 or 6 electrons, the extra electrons form lone pairs. The number of lone pairs can easily be derived using Lewis dot structures for the valence electrons.
In describing the shapes of these molecules, we often ignore the lone pairs. Thus, •NO₂, N₃^-, :OO₂ (O₃), and :SO₂ are bent molecules whereas :NH₃, :PF₃, and :SOF₂ are pyramidal. You already know that ::OH₂ (water) and ::SF₂ are bent molecules.
The lone electron pair takes up the equatorial location in :SF₄, which has the same structure as :TeF₄ described earlier. If you lay a model of this molecule on the side, it looks like a butterfly. By the same reason, ::ClF₃ and ::BrF₃ have a T shape, and :::XeF₂, :::I₃^-, and :::ICl₂^- are linear.
Similarly, :BrF₅ and :IF₅ are square pyramidal whereas ::XeF₄ is square planar.

The Center Atom

A nice student asked a brilliant question. Which atom in the formula is usually the center atom?
Usually, the atom in the center is more electropositive than the terminal atoms. However, the H and halogen atoms are usually at the terminal positions because they form only one bond.
Take a look at the chemical formulas in the table, and see if the above statement is true.
However, the application of VSEPR theory can be expanded to complicated molecules such as

H H H O | | | // H-C-C=C=C-C=C-C-C | | \ H N O-H / \ H H

By applying the VSEPR theory, one deduces the following results:

• H-C-C bond angle = 109^o
• H-C=C bond angle = 120^o, geometry around C trigonal planar
• C=C=C bond angle = 180^o, in other words linear
• H-N-C bond angle = 109^o, tetrahedral around N
• C-O-H bond angle = 105 or 109^o, 2 lone electron pairs around O