The Hybrid Orbital Model

While VSEPR works well to describe molecular geometry, it does not take into account what we know about atomic structure from quantum mechanics. How do electrons in atomic valence orbitals form bonds? The hybrid orbital model helps to explain this.

Read these sections, which describe how atomic orbitals combine to form hybrid orbitals in molecules. The molecular shapes and the bond angles of hybrid orbitals match the molecular shapes and bond angles predicted by VSEPR theory.

Introduction to Hybrid Orbitals

As useful and appealing as the concept of the shared-electron pair bond is, it raises a somewhat troubling question that we must sooner or later face: what is the nature of the orbitals in which the shared electrons are contained? Up until now, we have been tacitly assuming that each valence electron occupies the same kind of atomic orbital as it did in the isolated atom. As we shall see below, his assumption very quickly leads us into difficulties.

Why Atomic Orbitals Do Not Work for Molecules

atomic orbitals in Be

Consider how we might explain the bonding in a compound of divalent beryllium, such as beryllium hydride, BeH2. The beryllium atom, with only four electrons, has a configuration of 1s22s2. Note that the two electrons in the 2s orbital have opposite spins and constitute a stable pair that has no tendency to interact with unpaired electrons on other atoms.

Be excited state

The only way that we can obtain two unpaired electrons for bonding in beryllium is to promote one of the 2s electrons to the 2p level. However, the energy required to produce this excited-state atom would be sufficiently great to discourage bond formation. It is observed that Be does form reasonably stable bonds with other atoms. Moreover, the two bonds in BeH2 and similar molecules are completely equivalent; this would not be the case if the electrons in the two bonds shared Be orbitals of different types, as in the excited state diagram above.

These facts suggest that it is incorrect to assume that the distribution of valence electrons that are shared with other atoms can be described by atomic-type s, p, and d orbitals at all.

Remember that these different orbitals arise in the first place from the interaction of the electron with the single central electrostatic force field associated with the positive nucleus. An outer-shell electron in a bonded atom will be under the influence of a force field emanating from two positive nuclei, so we would expect the orbitals in the bonded atoms to have a somewhat different character from those in free atoms. In fact, as far as valence electrons are concerned, we can throw out the concept of atomic orbital altogether and reassign the electrons to a new set of molecular orbitals that are characteristic of each molecular configuration. This approach is indeed valid, but we will defer a discussion of it until a later unit.

For now, we will look at a less-radical model that starts out with the familiar valence-shell atomic orbitals, and allows them to combine to form hybrid orbitals whose shapes conform quite well to the bonding geometry that we observe in a wide variety of molecules.

What are Hybrid Orbitals?

About Orbitals: A Quick Review

First, recall that the electron, being a quantum particle, cannot have a distinct location; the most we can do is define the region of space around the nucleus in which the probability of finding the electron exceeds some arbitrary value, such as 90% or 99%. This region of space is the orbital. Because of the wavelike character of matter, the orbital corresponds to a standing wave pattern in three-dimensional space which we can often represent more clearly in two-dimensional cross section. The quantity that is varying (waving) is a number denoted by ψ (psi) whose value varies from point to point according to the wave function for that particular orbital.

Orbitals of all types are simply mathematical functions that describe particular standing-wave patterns that can be plotted on a graph but have no physical reality of their own. Because of their wavelike nature, two or more orbitals (i.e., two or more functions ψ) can be combined both in-phase and out-of-phase to yield a pair of resultant orbitals which, to be useful, must have squares that describe actual electron distributions in the atom or molecule.

The s,p,d and f orbitals that you are familiar with are the most convenient ones for describing the electron distribution in isolated atoms because assignment of electrons to them according to the usual rules always yields an overall function Ψ2 that predicts a spherically symmetric electron distribution, consistent with all physical evidence that atoms are in fact spherical. For atoms having more than one electron, however, the s,p,d, f basis set is only one of many possible ways of arriving at the same observed electron distribution. We use it not because it is unique, but because it is the simplest.

In the case of a molecule such as BeH2, we know from experimental evidence that the molecule is linear and therefore the electron density surrounding the central atom is no longer spherical, but must be concentrated along two directions 180° apart, and we need to construct a function Ψ2 having these geometrical properties. There are any number of ways of doing this, but it is convenient is to use a particular set of functions ψ (which we call hybrid orbitals) that are constructed by combining the atomic s,p,d, and f functions that are already familiar to us.

You should understand that hybridization is not a physical phenomenon; it is merely a mathematical operation that combines the atomic orbitals we are familiar with in such a way that the new (hybrid) orbitals possess the geometric and other properties that are reasonably consistent with what we observe in a wide range (but certainly not in all) molecules. In other words, hybrid orbitals are abstractions that describe reality fairly well in certain classes of molecules (and fortunately, in much of the very large class of organic substances) and are therefore a useful means of organizing a large body of chemical knowledge... but they are far from infallible.

This approach, which assumes that the orbitals remain more or less localized on one central atom, is the basis of the theory which was developed in the early 1930s, mainly by Linus Pauling.

Young photo of Linus PaulingPhoto of Linus Pauling

Linus Pauling (1901-1994) was the most famous American chemist of the 20th century and the author of the classic book The Nature of the Chemical Bond. His early work pioneered the application of X-ray diffraction to determine the structure of complex molecules; he then went on to apply quantum theory to explain these observations and predict the bonding patterns and energies of new molecules.

Pauling, who spent most of his career at Cal Tech, won the Nobel Prize for Chemistry in 1954 and the Peace Prize in 1962.

"In December 1930 Pauling had his famous 'breakthrough' where, in a rush of inspiration, he 'stayed up all night, making, writing out, solving the equations, which were so simple that I could solve them in a few minutes.' This flurry of calculations would eventually become the first of Pauling's germinal series of papers on the nature of the chemical bond. 'I just kept getting more and more euphoric as time went by,' Pauling would recall." —Chris Petersen, Linus Pauling and the Nature of the Chemical Bond: A Documentary History

Although the hybrid orbital approach has proven very powerful (especially in organic chemistry), it does have its limitations. For example, it predicts that both H2O and H2S will be tetrahedrally coordinated bent molecules with bond angles slightly smaller than the tetrahedral angle of 109.5° owing to greater repulsion by the nonbonding pair. This description fits water (104.5°) quite well, but the bond angle in hydrogen sulfide is only 92°, suggesting that atomic p orbitals (which are 90° apart) provide a better description of the electron distribution about the sulfur atom than do sp3 hybrid orbitals.

The hybrid orbital model is fairly simple to apply and understand, but it is best regarded as one special way of looking at a molecule that can often be misleading. Another viewpoint, called the molecular orbital theory, offers us a complementary perspective that it is important to have if we wish to develop a really thorough understanding of chemical bonding in a wider range of molecules.

Constructing Hybrid Orbitals

Hybrid orbitals are constructed by combining the ψ functions for atomic orbitals. Because wave patterns can combine both constructively and destructively, a pair of atomic wave functions such as the s- and p- orbitals shown at the left can combine in two ways, yielding the sp hybrids shown.

s and p wave functions combine into hybrid orbitals

Constructive and destructive combinations of 2p and 2s wave functions (line plots) give rise to the sp hybrid function shown at the right. The solid figures depict the corresponding probability functions ψ2.

From an energy standpoint, we can represent the transition from atomic s- and p-orbitals to an sp hybrid orbital in this way:

BeH2 hybrid orbitals

Notice here that 1. the total number of occupied orbitals is conserved, and 2. the two sp hybrid orbitals are intermediate in energy between their parent atomic orbitals.

In terms of plots of the actual orbital functions ψ we can represent the process as follows:

origin of an sp hybrid orbital

The probability of finding the electron at any location is given not by ψ, but by ψ2, whose form is roughly conveyed by the solid figures in this illustration.

Hybrids Derived from Atomic s- and p Orbitals

Diagonal Bonding: sp-Hybrid Orbitals

Returning to the example of BeH2, we can compare the valence orbitals in the free atoms with those in the beryllium hydride molecule as shown here. It is, of course, the overlap between the hydrogen-1s orbitals and the two lobes of the beryllium sp-hybrid orbitals that constitutes the two Be–H bonds in this molecule.

hybrid orbital model of BeH2

Notice that whereas a single p-orbital has lobes on both sides of the atom, a single sp-hybrid has most of its electron density on one side, with a minor and more spherical lobe on the other side. This minor lobe is centered on the central atom (some textbook illustrations don't get this right.)

As far as the shape of the molecule is concerned, the result is exactly the same as predicted by the VSEPR model (although hybrid orbital theory predicts the same result in a more fundamental way.) We can expect any central atom that uses sp-hybridization in bonding to exhibit linear geometry when incorporated into a molecule.

Trigonal (sp2) Hybridization

bonding in BF3

We can now go on to apply the same ideas to some other simple molecules. In boron trifluoride, for example, we start with the boron atom, which has three outer-shell electrons in its normal or ground state, and three fluorine atoms, each with seven outer electrons. As is shown in this configuration diagram, one of the three boron electrons is unpaired in the ground state. In order to explain the trivalent bonding of boron, we postulate that the atomic s- and p- orbitals in the outer shell of boron mix to form three equivalent hybrid orbitals. These particular orbitals are called sp2 hybrids, meaning that this set of orbitals is derived from one s- orbital and two p-orbitals of the free atom.

origin of sp2 hybrid orbitals

This illustration shows how an s-orbital mixes with two p orbitals to form a set of three sp2 hybrid orbitals. Notice again how the three atomic orbitals yield the same number of hybrid orbitals.

Image of Boron trifluoride BF3.Another image of boron trifluoride BF3

Boron trifluoride BF3 is a common example of sp2 hybridization. The molecule has plane trigonal geometry.

Tetrahedral (sp3) Hybridization

Let us now look at several tetravalent molecules, and see what kind of hybridization might be involved when four outer atoms are bonded to a central atom. Perhaps the commonest and most important example of this bond type is methane, CH4.

Orbitals of CH4

In the ground state of the free carbon atom, there are two unpaired electrons in separate 2p orbitals. In order to form four bonds (tetravalence), need four unpaired electrons in four separate but equivalent orbitals. We assume that the single 2s, and the three 2p orbitals of carbon mix into four sp3 hybrid orbitals which are chemically and geometrically identical; the latter condition implies that the four hybrid orbitals extend toward the corners of a tetrahedron centered on the carbon atom.

origin of sp3 hybrid orbitals

Methane is the simplest hydrocarbon; the molecule is approximately spherical, as is shown in the space-filling model:

CH4 models

By replacing one or more of the hydrogen atoms in CH4 with another sp3 hybridized carbon fragments, hydrocarbon chains of any degree of complexity can be built up.

The simplest of these is ethane:

The M.O.s of Ethane

This shows how an sp3 orbital on each of two two carbon atoms join (overlap) to form a carbon-carbon bond, and then the remaining carbon sp3 orbital overlaps with six hydrogen 1s orbitals to form the ethane molecule. Image source: Yale University.

Lone Pair Electrons in Hybrid Orbitals

If lone pair electrons are present on the central atom, these can occupy one or more of the sp3 orbitals. This causes the molecular geometry to be different from the coordination geometry, which remains tetrahedral.

Image of four sp3 orbitals in NH3, ammonia

In the ammonia molecule, for example, the nitrogen atom normally has three unpaired p electrons, but by mixing the 2s and 2p orbitals, we can create four sp3-hybrid orbitals just as in carbon. Three of these can form shared-electron bonds with hydrogen, resulting in ammonia, NH3. The fourth of the sp3 hybrid orbitals contains the two remaining outer-shell electrons of nitrogen which form a non-bonding lone pair. In acidic solutions these can coordinate with a hydrogen ion, forming the ammonium ion NH4+.

sp3 hybrid orbitals (tetrahedral) for Ammonia

Although no bonds are formed by the lone pair in NH3, these electrons do give rise to a charge cloud that takes up space just like any other orbital.

Image of the four sp3 orbitals in water

In the water molecule, the oxygen atom can form four sp3 orbitals. Two of these are occupied by the two lone pairs on the oxygen atom, while the other two are used for bonding. The observed H-O-H bond angle in water (104.5°) is less than the tetrahedral angle (109.5°); one explanation for this is that the non-bonding electrons tend to remain closer to the central atom and thus exert greater repulsion on the other orbitals, thus pushing the two bonding orbitals closer together.

Hybrid Orbitals in Molecular Ions

Hybridization can also help explain the existence and structure of many inorganic molecular ions. Consider, for example, electron configurations of zinc in the compounds in the illustrations below.

The tetrachlorozinc ion (top row) is another structure derived from zinc and chlorine. As we might expect, this ion is tetrahedral; there are four chloride ions surrounding the central zinc ion. The zinc ion has a charge of +2, and each chloride ion is –1, so the net charge of the complex ion is –2.

hybrid orbitals in tetrachlorozinc ion

At the bottom is shown the electron configuration of atomic zinc, and just above it, of the divalent zinc ion. Notice that this ion has no electrons at all in its 4-shell. In zinc chloride, shown in the next row up, there are two equivalent chlorine atoms bonded to the zinc. The bonding orbitals are of sp character; that is, they are hybrids of the 4s and one 4p orbital of the zinc atom. Since these orbitals are empty in the isolated zinc ion, the bonding electrons themselves are all contributed by the chlorine atoms, or rather, the chloride ions, for it is these that are the bonded species here. Each chloride ion possesses a complete octet of electrons, and two of these electrons occupy each sp bond orbital in the zinc chloride complex ion. This is an example of a coordinate covalent bond, in which the bonded atom contributes both of the electrons that make up the shared pair.

What You Should Be Able to Do

Make sure you thoroughly understand the following essential ideas which have been presented above.

  • Explain why the sharing of atomic orbitals (as implied in the Lewis model) cannot adequately account for the observed bonding patterns in simple molecules.
  • Sketch out a diagram illustrating how the plots of atomic s- and p- orbital wave functions give rise to a pair of hybrid orbitals.
  • Draw orbital box diagrams showing how combinations of an atomic s orbital and various numbers of p orbitals create sp, sp2, and sp3 hybrid orbitals.
  • Show how hybrid orbitals are involved in the molecules methane, water, and ammonia.

More on Hybrid Orbitals: Multiple Bonds, D Orbitals

Hybrid Types and Multiple Bonds

Image of ethane molecule C2H6

We have already seen how sp hybridization in carbon leads to its combining power of four in the methane molecule. Two such tetrahedrally coordinated carbons can link up together to form the molecule ethane C2H6. In this molecule, each carbon is bonded in the same way as the other; each is linked to four other atoms, three hydrogens and one carbon. The ability of carbon-to-carbon linkages to extend themselves indefinitely and through all coordination positions accounts for the millions of organic molecules that are known.

Trigonal Hybridization in Carbon: The Double Bond

Carbon and hydrogen can also form a compound ethylene (ethene) in which each carbon atom is linked to only three other atoms. Here, we can regard carbon as being trivalent. We can explain this trivalence by supposing that the orbital hybridization in carbon is in this case not sp3, but is sp2 instead; in other words, only two of the three p orbitals of carbon mix with the 2s orbital to form hybrids; the remaining p-orbital, which we will call the i orbital, remains unhybridized. Each carbon is bonded to three other atoms in the same kind of plane trigonal configuration that we saw in the case of boron trifluoride, where the same kind of hybridization occurs. Notice that the bond angles around each carbon are all 120°.

This alternative hybridization scheme explains how carbon can combine with four atoms in some of its compounds and with three other atoms in other compounds. You may be aware of the conventional way of depicting carbon as being tetravalent in all its compounds; it is often stated that carbon always forms four bonds, but that sometimes, as in the case of ethylene, one of these may be a double bond. This concept of the multiple bond preserves the idea of tetravalent carbon while admitting the existence of molecules in which carbon is clearly combined with fewer than four other atoms.

Three views of the ethylene molecule

These three views of the ethylene molecule emphasize different aspects of the disposition of shared electron pairs in the various bonding orbitals of ethene (ethylene). (a) The backbone structure consisting of σ (sigma) bonds formed from the three sp2-hybridized orbitals on each carbon. (b) The π (pi) bonding system formed by overlap of the unhybridized pz orbital on each carbon. The π orbital has two regions of electron density extending above and below the plane of the molecule. (c) A cutaway view of the combined σ and π system.

As shown above, ethylene can be imagined to form when two -CH2 fragments link together through overlap of the half-filled sp2 hybrid orbitals on each. Since sp2 hybrid orbitals are always in the same plane, the entire ethylene molecule is planar. However, there remains on each carbon atom an electron in an unhybridized atomic pz orbital that is perpendicular to the molecular plane. These two parallel pz orbitals will interact with each other; the two orbitals merge, forming a sausage-like charge cloud (the π bond) that extends both above and below the plane of the molecule. It is the pair of electrons that occupy this new extended orbital that constitutes the fourth bond to each carbon, and thus the other half of the double bond in the molecule.

More about Sigma and pi Bonds

The σ (sigma) bond has its maximum electron density along the line-of-centers joining the two atoms (below left). Viewed end-on, the σ bond is cylindrically symmetrical about the line-of-centers. It is this symmetry, rather than its parentage, that defines the σ bond, which can be formed from the overlap of two s-orbitals, from two p-orbitals arranged end-to-end, or from an s- and a p-orbital. They can also form when sp hybrid orbitals on two atoms overlap end-to-end.

Pi orbitals, on the other hand, require the presence of two atomic p orbitals on adjacent atoms. Most important, the charge density in the π orbital is concentrated above and below the molecular plane; it is almost zero along the line-of-centers between the two atoms. It is this perpendicular orientation with respect to the molecular plane (and the consequent lack of cylindrical symmetry) that defines the π orbital. The combination of a σ bond and a π bond extending between the same pair of atoms constitutes the double bond in molecules such as ethylene.

Carbon-Carbon Triple Bonds: sp Hybridization in Acetylene

We have not yet completed our overview of multiple bonding, however. Carbon and hydrogen can form yet another compound, acetylene (ethyne), in which each carbon is connected to only two other atoms: a carbon and a hydrogen. This can be regarded as an example of divalent carbon, but is usually rationalized by writing a triple bond between the two carbon atoms.

We assume here that since two geometrically equivalent bonds are formed by each carbon, this atom must be sp-hybridized in acetylene. On each carbon, one sp hybrid bonds to a hydrogen and the other bonds to the other carbon atom, forming the σ bond skeleton of the molecule.

In addition to the sp hybrids, each carbon atom has two half-occupied p orbitals oriented at right angles to each other and to the interatomic axis. These two sets of parallel and adjacent p orbitals can thus merge into two sets of π orbitals.

Image of the triple bond in acetylene

The triple bond in acetylene is seen to consist of one σ bond joining the line-of-centers between the two carbon atoms, and two π bonds whose lobes of electron density are in mutually-perpendicular planes. The acetylene molecule is of course linear, since the angle between the two sp hybrid orbitals that produce the s skeleton of the molecule is 180°.

Multiple Bonds between Unlike Atoms

hybrid orbitals in CO2

Multiple bonds can also occur between dissimilar atoms. For example, in carbon dioxide each carbon atom has two unhybridized atomic p orbitals, and each oxygen atom still has one p orbital available. When the two O-atoms are brought up to opposite sides of the carbon atom, one of the p orbitals on each oxygen forms a π bond with one of the carbon p-orbitals. In this case, sp-hybridization is seen to lead to two double bonds. Notice that the two C–O π bonds are mutually perpendicular.

hybrid orbitals in HCN

Similarly, in hydrogen cyanide, HCN, we assume that the carbon is sp-hybridized, since it is joined to only two other atoms, and is hence in a divalent state. One of the sp-hybrid orbitals overlaps with the hydrogen 1s orbital, while the other overlaps end-to-end with one of the three unhybridized p orbitals of the nitrogen atom. This leaves us with two nitrogen p-orbitals which form two mutually perpendicular π bonds to the two atomic p orbitals on the carbon. Hydrogen cyanide thus contains one single and one triple bond. The latter consists of a σ bond from the overlap of a carbon sp hybrid orbital with a nitrogen p orbital, plus two mutually perpendicular π bonds deriving from parallel atomic p orbitals on the carbon and nitrogen atoms.

The Nitrate Ion

Pi bond delocalization furnishes a means of expressing the structures of other molecules that require more than one electron-dot or structural formula for their accurate representation. A good example is the nitrate ion, which contains 24 electrons:

Images of the nitrate ion.

The electron-dot formula shown above is only one of three equivalent resonance structures that are needed to describe trigonal symmetry of this ion.

Images of nitrogen

Nitrogen has three half-occupied p orbitals available for bonding, all perpendicular to one another. Since the nitrate ion is known to be planar, we are forced to assume that the nitrogen outer electrons are sp2-hybridized. The addition of an extra electron fills all three hybrid orbitals completely. Each of these filled sp2 orbitals forms a σ bond by overlap with an empty oxygen 2pz orbital; this, you will recall, is an example of coordinate covalent bonding, in which one of the atoms contributes both of the bonding electrons. The empty oxygen 2p orbital is made available when the oxygen electrons themselves become sp hybridized; we get three filled sp hybrid orbitals, and an empty 2p atomic orbital, just as in the case of nitrogen.

The π bonding system arises from the interaction of one of the occupied oxygen sp orbitals with the unoccupied 2px orbital of the nitrogen. Notice that this, again, is a coordinate covalent sharing, except that in this instance it is the oxygen atom that donates both electrons.

Image of the bonds in the nitrogen atom

Pi bonds can form in this way between the nitrogen atom and any of the three oxygens; there are thus three equivalent π bonds possible, but since nitrogen can only form one complete π bond at a time, the π bonding is divided up three ways, so that each N–O bond has a bond order of 4/3.

Images of nitrate bonding

Conjugated Double Bonds

We have seen that the π bonding orbital is distinctly different in shape and symmetry from the σ bond. There is another important feature of the π bond that is of far-reaching consequence, particularly in organic and coordination chemistry.

Consider, for example, an extended hydrocarbon molecule in which alternate pairs of carbon atoms are connected by double and single bonds. Each non-terminal carbon atom forms two σ bonds to two other carbons and to a hydrogen (not shown.) This molecule can be viewed as a series of ethylene units joined together end-to-end. Each carbon, being sp hybridized, still has a half-filled atomic p orbital. Since these p orbitals on adjacent carbons are all parallel, we can expect them to interact with each other to form π bonds between alternate pairs of carbon atoms as shown below.

conjugated chain

But since each carbon atom possesses a half-filled p orbital, there is nothing unique about the π bond arrangement; an equally likely arrangement might be one in which the πbonding orbitals are shifted to neighboring pairs of carbons (middle illustration above).

You will recall that when there are two equivalent choices for the arrangements single and double bonds in a molecule, we generally consider the structure to be a resonance hybrid. In keeping with this idea, we would expect the electron density in a π system of this kind to be extended or shared out evenly along the entire molecular framework, as shown in the bottom figure.

A system of alternating single and double bonds, as we have here, is called a conjugated system. Chemists say that the π bonds in a conjugated system are delocalized; they are, in effect, smeared out over the entire length of the conjugated part of the molecule. Each pair of adjacent carbon atoms is joined by a σ bond and half of a π bond, resulting in an a C-C bond order of 1.5.

An even higher degree of conjugation exists in compounds containing extended (C=C)n chains. These compounds, known as cumulenes, exhibit interesting electrical properties, and whose derivatives can act as organic wires.

images of cumulene

Hybrid Orbitals in Benzene

The classic example of π bond delocalization is found in the cyclic molecule benzene (C6H6) which consists of six carbon atoms bound together in a hexagonal ring. Each carbon has a single hydrogen atom attached to it.

Image of benzene C6H6

The lines in this figure represent the σ bonds in benzene. The basic ring structure is composed of σ bonds formed from overlap of sp 2 hybrid orbitals on adjacent carbon atoms. The unhybridized carbon pz orbitals project above and below the plane of the ring. They are shown here as they might appear if they did not interact with one another.

Image of a benzene ring

But what happens, of course, is that the lobes of these atomic orbitals meld together to form circular rings of electron density above and below the plane of the molecule. The two of these together constitute the second half of the carbon-carbon double bonds in benzene.

An image of a computer-generated plot of electron density in the benzene molecule

This computer-generated plot of electron density in the benzene molecule is derived from a more rigorous theory that does not involve hybrid orbitals; the highest electron density (blue) appears around the periphery of the ring, while the lowest (red) is in the doughnut hole in the center.

Hybrids Involving d Orbitals

In atoms that are below those in the first complete row of the periodic table, the simple octet rule begins to break down. For example, we have seen that PCl3 does conform to the octet rule but PCl5 does not. We can describe the bonding in PCl3 very much as we do NH3: four sp3-hybridized orbitals, three of which are shared with electrons from other atoms and the fourth containing a nonbonding pair.

Pentagonal Bipyramid Molecules: sp3d Hybridization

PCl5 structure

In order to understand the bonding in phosphorus pentachloride PCl5, we have to consider the d orbitals in addition to the s- and p types. When d orbitals are energetically close to the outmost s- and p orbitals, additional hybrid types can be built. In the case of PCl5 we need five hybrid orbitals, and these can be constructed by adding two d-orbital functions to the mathematical mixture of one s- and two p-orbitals, resulting in five sp3d hybrid orbitals directed toward the corners of a trigonal bipyramid, as is predicted by VSEPR theory.

sp3d bonding in PCl5

Octahedral Coordination: sp3d2 Hybridization

The molecule sulfur hexafluoride SF6 exemplifies one of the most common types of d-orbital hybridization. The six bonds in this octahedrally-coordinated molecule are derived from mixing six atomic orbitals into a hybrid set. The easiest way to understand how these come about is to imagine that the molecule is made by combining an imaginary S6+ ion (which we refer to as the S(VI) valence state) with six F ions to form the neutral molecule. These now-empty 3s and 3p orbitals then mix with two 3d orbitals to form the sp3d2 hybrids.

sp3d2 bonding in SF6

Some of the most important and commonly encountered compounds which involve the d orbitals in bonding are the transition metal complexes. The term complex in this context means that the molecule is composed of two or more kinds of species, each of which can have an independent existence.

Square-Planar Molecules: dsp2 Hybridization

For example, the ions Pt2+ and Cl can form the ion [PtCl4]2–. To understand the hybridization scheme, it helps to start with the neutral Pt atom, then imagine it losing two electrons to become an ioin, followed by grouping of the two unpaired 5d electrons into a single d orbital, leaving one vacant.This vacant orbital, along with the 6s and two of the 6p orbitals, can then accept an electron pair from four chlorines.

dsp2 bonding in tetrachloroplatinum II

All of the four-coordinated molecules we have discussed so far have tetrahedral geometry around the central atom. Methane, CH4, is the most well known example. It may come as something as a surprise, then, to discover that the tetrachlorplatinum (II) ion [PtCl4]2– has an essentially two-dimensional square-planar configuration. This type of bonding pattern is quite common when the parent central ion (Pt2+ in this case) contains only eight electrons in its outmost d-subshell.

Octahedral Coordination: sp3d2 and d2sp3

Many of the most commonly encountered transition metal ions accept electron pairs from donors such as CN and NH3 (or lacking these, even from H2O) to form octahedral coordination complexes. The hexamminezinc(II) cation depicted below is typical.

sp3d2 bonding in hexamminezinc(II)

In sp3d2 hybridization the bonding orbitals are derived by mixing atomic orbitals having the same principal quantum number (n = 4 in the preceding example). A slightly different arrangement, known as d2sp3 hybridization, involves d orbitals of lower principal quantum number. This is possible because of the rather small energy differences between the d orbitals in one shell with the s and p orbitals of the next higher one— hence the term inner orbital complex which is sometimes used to describe ions such as hexamminecobalt(III), shown below. Both arrangements produce octahedral coordination geometries.

d2sp3 bonding in hexamminecobalt(III)

In some cases, the same central atom can form either inner or outer complexes depending on the particular ligand and the manner in which its electrostatic field affects the relative energies of the different orbitals. Thus the hexacyanoiron(II) ion utilizes the iron 3d orbitals, whereas hexaaquoiron(II) achieves a lower energy by accepting two H2O molecules in its 4d orbitals.

inner- and outer-orbital complexes of iron(II)

Although this inner-outer hybrid model was instrumental in explaining the properties of transition metal complexes in the first half of the 20th century, it has now been replaced with a more comprehensive model known as ligand field theory which is introduced in another lesson.

Some Final Remarks about Hybrid Orbitals

As is the case with any scientific model, the hybridization model of bonding is useful only to the degree to which it can predict phenomena that are actually observed. Most models contain weaknesses that place limits on their general applicability. The need for caution in accepting this particular model is made more apparent when we examine the shapes of the molecules below the first full row of the periodic table.

For example, we would expect the bonding in hydrogen sulfide to be similar to that in water, with tetrahedral geometry around the sulfur atom. Experiments, however, reveal that the H–S–H bond angle is only 92°. Hydrogen sulfide thus deviates much more from tetrahedral geometry than does water, and there is no apparent and clear reason why it should. It is certainly difficult to argue that electron-repulsion between the two nonbonding orbitals is pushing the H–S bonds closer together (as is supposed to happen to the H–O bonds in water); many would argue that this repulsion would be less in hydrogen sulfide than in water, since sulfur is a larger atom and is hence less electronegative.

It might be, then, that our model of orbital hybridization into four equivalent sp3 orbitals does not apply to H2S. It looks like the simple explanation that bonding occurs through two half occupied atomic p orbitals 90° apart comes closer to the mark. Perhaps hybridization is not an all-or-nothing phenomenon; perhaps the two 3p orbitals are substantially intact in hydrogen sulfide, or are hybridized only slightly. In general, the hybridization model does not work very well with nonmetallic elements farther down in the periodic table, and there is as yet no clear explanation why. We must simply admit that we have reached one of the many points in chemistry where our theory is not sufficiently developed to give a clear and unequivocal answer. This does not detract, however, from the wide usefulness of the hybridization model in elucidating the bond character and bond shapes in the millions of molecules based on first-row elements, particularly of carbon.

Are Hybrid Orbitals Real?

The justification we gave for invoking hybridization in molecules such as BeH2,BF3 and CH4 was that the bonds in each are geometrically and chemically equivalent, whereas the atomic s- and p-orbitals on the central atoms are not. By combining these into new orbitals of sp, sp2 and sp3 types we obtain the required number of completely equivalent orbitals. This seemed easy enough to do on paper; we just drew little boxes and wrote sp2 or whatever below them. But what is really going on here?

The full answer is beyond the scope of this course, so we can only offer the following very general explanation. First, recall what we mean by orbital: a mathematical function ψ having the character of a standing wave whose square ψ2 is proportional to the probability of finding the electron at any particular location in space. The latter, the electron density distribution, can be observed (by X-ray scattering, for example), and in this sense is the only thing that is real.

A given standing wave (ψ-function) can be synthesized by combining all kinds of fundamental wave patterns (that is, atomic orbitals) in much the same way that a color we observe can be reproduced by combining different sets of primary colors in various proportions. In neither case does it follow that these original orbitals (or colors) are actually present in the final product. So one could well argue that hybrid orbitals are not real; they simply turn out to be convenient for understanding the bonding of simple molecules at the elementary level, and this is why we use them.

An Alternative to Hybrids: The Bent-Bond Model

bent-bond model of ethyleneImage of bananas between the bonds.

It turns out, in fact, that the electron distribution and bonding in ethylene can be equally well described by assuming no hybridization at all. The bent bond model requires only that the directions of some of the atomic-p orbitals be distorted sufficiently to provide the overlap needed for bonding; these are sometimes referred to as banana bonds.

bent-bond model of cyclopropane

The smallest of the closed-ring hydrocarbons is cyclopropane, a planar molecule in which the C–C bond angles are 120° – quite a departure from the tetrahedral angle of 109.5° associated with sp3 hybridization! Theoretical studies suggest that the bent-bond model does quite well in predicting its properties.

What You Should Be Able to Do

Make sure you thoroughly understand the following essential ideas which have been presented above.

  • Sketch out diagrams showing the hybridization and bonding in compounds containing single, double, and triple carbon-carbon bonds.
  • Define sigma and pi bonds.
  • Describe the hybridization and bonding in the benzene molecule.

Source: Stephen Lower,,
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Last modified: Thursday, April 4, 2024, 1:59 PM