Orbital hybridization

Four sp³ orbitals.

Three sp² orbitals.

In chemistry, hybridization (or hybridisation) is the concept of mixing atomic orbitals into new hybrid orbitals suitable for the pairing of electrons to form chemical bonds in valence bond theory. Hybrid orbitals are very useful in the explanation of molecular geometry and atomic bonding properties.

Historical development

Chemist Linus Pauling first used hybridization theory to explain the structure of molecules such as methane (CH₄). This concept was developed for such simple chemical systems. But the approach was later applied more widely. Today, chemists use it to explain the structures of organic compounds.

Orbitals represent how electrons behave within molecules. In the case of simple hybridization, this approximation is based on atomic orbitals. Chemists use the atomic orbitals of the hydrogen atom, which the only atom for which an exact analytic solution to its Schrödinger equation is known. In heavier atoms, like carbon, nitrogen, and oxygen, the atomic orbitals involved in bonding are the 2s and 2p orbitals. These orbitals can be occupied in a hydrogen atom, but only when the electron is in an excited state. Hybrid orbitals are assumed to be mixtures of these atomic orbitals, superimposed on each other in various proportions. It provides a quantum mechanical insight to Lewis structures. Chemists use hybridization theory mainly in organic chemistry.

sp^x and sd^x terminology

This terminology describes the weight of the respective components of a hybrid orbital. For example, in methane, the C hybrid orbital which forms each C-H bond consists of 25% s character and 75% p character and is thus described as sp³ (read as s-p-three) hybridised. Quantum mechanics describes this hybrid as an sp³ wavefunction of the form N[s + (√3)pσ], where N is a normalization constant (here 1/2) and pσ is a p orbital directed along the C-H axis to form a sigma bond. The p-to-s ratio (denoted λ in general) is √3 in this example, and N²λ² = 3/4 is the p character or the weight of the p component.

For atoms forming equivalent hybrids with no lone pairs, there is a correspondence to the number and type of orbitals used. For example, sp³ hybrids are formed from one s and three p orbitals. However, in all other cases, there is no such correspondence. The two bond-forming hybrid orbitals of oxygen in water can be described as sp⁴, which means that they have 20% s character and 80% p character, but does not imply that they are formed from one s and four p orbitals. As a result, the amount of p-character is not restricted to integer values; i.e., hybridisations like sp^2.5 are also readily described.

In general, for an atom with s and p orbitals forming hybrids h_i and h_j with included angle [math]\displaystyle{ \theta }[/math], the following holds: 1 + [math]\displaystyle{ \lambda }[/math]_i[math]\displaystyle{ \lambda }[/math]_j cos([math]\displaystyle{ \theta }[/math]) = 0. The p-to-s ratio for hybrid i is [math]\displaystyle{ \lambda }[/math]_i², and for hybrid j it is [math]\displaystyle{ \lambda }[/math]_j². The bond directed towards a more electronegative substituent tends to have higher p-character as stated in Bent's rule. In the special case of equivalent hybrids on the same atom, again with included angle [math]\displaystyle{ \theta }[/math], the equation reduces to just 1 + [math]\displaystyle{ \lambda }[/math]² cos([math]\displaystyle{ \theta }[/math]) = 0. For example, BH₃ has a trigonal planar geometry, three 120° bond angles, three equivalent hybrids about the boron atom, and thus 1 + [math]\displaystyle{ \lambda }[/math]² cos([math]\displaystyle{ \theta }[/math]) = 0 becomes 1 + [math]\displaystyle{ \lambda }[/math]² cos(120°) = 0, giving [math]\displaystyle{ \lambda }[/math]² = 2 for the p-to-s ratio. In other words, sp² hybrids.

An analogous notation is used to describe sd^x hybrids. For example, the permanganate ion (MnO₄^-) has sd³ hybridisation with orbitals that are 25% s and 75% d.

Types of Hybridization

sp³ hybrids

Hybridisation describes the bonding atoms from an atom's point of view. That is, for a tetrahedrally coordinated carbon (e.g., methane CH₄), the carbon should have 4 orbitals with the correct symmetry to bond to the 4 hydrogen atoms.

Carbon's ground state configuration is 1s² 2s² 2p_x¹ 2p_y¹ or more easily read:

[math]\displaystyle{ C\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\downarrow}{2s}\; \frac{\uparrow\,}{2p_x}\; \frac{\uparrow\,}{2p_y}\; \frac{\,\,}{2p_z} }[/math]

The carbon atom can utilize its two singly occupied p-type orbitals (the designations p_x p_y or p_z are meaningless at this point, as they do not fill in any particular order), to form two covalent bonds with two hydrogen atoms, yielding the "free radical" methylene CH₂, the simplest of the carbenes. The carbon atom can also bond to four hydrogen atoms by an excitation of an electron from the doubly occupied 2s orbital to the empty 2p orbital, so that there are four singly occupied orbitals.

[math]\displaystyle{ C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{2s}\; \frac{\uparrow\,}{2p_x} \frac{\uparrow\,}{2p_y} \frac{\uparrow\,}{2p_z} }[/math]

As the additional bond energy more than compensates for the excitation, the formation of four C-H bonds is energetically favoured.

Quantum mechanically, the lowest energy is obtained if the four bonds are equivalent which requires that they be formed from equivalent orbitals on the carbon. To achieve this equivalence, the angular distributions of the orbitals change via a linear combination of the valence-shell (Core orbitals are almost never involved in bonding) s and p wave functions to form four sp³ hybrids.

[math]\displaystyle{ C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp^3}\; \frac{\uparrow\,}{sp^3} \frac{\uparrow\,}{sp^3} \frac{\uparrow\,}{sp^3} }[/math]

In CH₄, four sp³ hybrid orbitals are overlapped by hydrogen's 1s orbital, yielding four σ (sigma) bonds (that is, four single covalent bonds) of the same length and strength.

translates into

sp² hybrids

Ethene structure

Other carbon based compounds and other molecules may be explained in a similar way as methane. For example, ethene (C₂H₄) has a double bond between the carbons.

For this molecule, carbon will sp² hybridize, because one π (pi) bond is required for the double bond between the carbons, and only three σ bonds are formed per carbon atom. In sp² hybridization the 2s orbital is mixed with only two of the three available 2p orbitals:

[math]\displaystyle{ C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp^2}\; \frac{\uparrow\,}{sp^2} \frac{\uparrow\,}{sp^2} \frac{\uparrow\,}{2p} }[/math]

forming a total of 3 sp² orbitals with one p-orbital remaining. In ethylene (ethene), the two carbon atoms form a σ bond by overlapping two sp² orbitals and each carbon atom forms two covalent bonds with hydrogen by s–sp² overlap all with 120° angles. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. The hydrogen–carbon bonds are all of equal strength and length, which agrees with experimental data.

sp hybrids

The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization.

[math]\displaystyle{ C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp}\; \frac{\uparrow\,}{sp} \frac{\uparrow\,}{p} \frac{\uparrow\,}{p} }[/math]

In this model, the 2s orbital mixes with only one of the three p-orbitals resulting in two sp orbitals and two remaining unchanged p orbitals. The chemical bonding in acetylene (ethyne) (C₂H₂) consists of sp–sp overlap between the two carbon atoms forming a σ bond and two additional π bonds formed by p–p overlap. Each carbon also bonds to hydrogen in a sigma s–sp overlap at 180° angles.

Hybridisation and molecule shape

Hybridisation helps to explain molecule shape:

Classification	Main group	Transition metal
AX2	Linear (180°) sp hybridisation E.g., CO2	Bent (90°) sd hybridisation E.g., VO2+
AX3	Trigonal planar (120°) sp2 hybridisation E.g., BCl3	Trigonal pyramidal (90°) sd2 hybridisation E.g., CrO3
AX4	Tetrahedral (109.5°) sp3 hybridisation E.g., CCl4	Tetrahedral (109.5°) sd3 hybridisation E.g., MnO4−
AX5	-	Square pyramidal (66°, 114°) sd4 hybridisation E.g., Ta(CH3)5
AX6	-	Trigonal prismatic (63°, 117°) sd5 hybridisation E.g., W(CH3)6

Main group compounds with lone pairs

For main group compounds with lone electron pairs, the s orbital lone pair (analogous to s-p mixing in molecular orbital theory) can be hybridised to a certain extent with the bond pairs to maximize energetic stability according to its Walsh diagram. This rationalisation is applied to explain deviations in ideal bond angles (i.e. only p orbitals used for bonding), most commonly in second and third period elements.

• Trigonal pyramidal (AX₃E₁)
  • s-orbital can be hybridised with the three p-orbital bonds to give bond angles greater than 90°.
  • E.g., NH₃
• Bent (AX₂E_1-2)
  • s-orbital lone pair can be hybridised with the two p-orbital bonds to give bond angles greater than 90°. The out-of-plane p-orbital can either be a lone pair or pi bond. If it is a lone pair, it results in inequivalent lone pairs contrary to the common picture depicted by VSEPR theory (see below).
  • E.g., SO₂, H₂O
• Monocoordinate (AX₁E_1-3)
  • s-orbital lone pair can be hybridised with the p-orbital bond. The two out-of-line p-orbitals can either be lone pairs or pi bonds. The p-orbital lone pairs are inequivalent from the s-rich lone pair.
  • E.g., CO, SO, HF

Hybridisation of hypervalent molecules

Traditional description

In general chemistry courses and mainstream textbooks, hybridisation is often presented for main group AX5 and above, as well as for transition metal complexes, using the hybridisation scheme first proposed by Pauling.

Classification	Main group	Transition metal
AX2	-	Linear (180°) sp hybridisation E.g., Ag(NH3)2+
AX3	-	Trigonal planar (120°) sp2 hybridisation E.g., Cu(CN)32−
AX4	-	Square planar (90°) dsp2 hybridisation E.g., PtCl42−
AX5	Trigonal bipyramidal (90°, 120°) sp3d hybridisation E.g., PCl5	Trigonal bipyramidal (90°, 120°) dsp3 hybridisation E.g., Fe(CO)5
AX6	Octahedral (90°) sp3d2 hybridisation E.g., SF6	Octahedral (90°) d2sp3 hybridisation E.g., Mo(CO)6
AX7	Pentagonal bipyramidal (90°, 72°) sp3d3 hybridisation E.g., IF7	Pentagonal bipyramidal (90°, 72°) d3sp3 hybridisation E.g., V(CN)74−
AX8	Square antiprismatic sp3d4 hybridisation E.g., IF8−	Square antiprismatic d4sp3 hybridisation E.g., Re(CN)83−
AX9	-	Tricapped trigonal prismatic d5sp3 hybridisation E.g., ReH92−

However, such a scheme is now superseded as more recent calculations based on molecular orbital theory have shown that in main-group molecules the d component is insignificant, while in transition metal complexes the p component is insignificant (see below).

Resonance description

As shown by computational chemistry, hypervalent molecules can only be stable given strongly polar (and weakened) bonds with electronegative ligands such as fluorine or oxygen to reduce the valence electron occupancy of the central atom to below 8 (or 12 for transition metals). This requires an explanation that invokes sigma resonance in addition to hybridisation, which implies that each resonance structure has its own hybridisation scheme. As a guideline, all resonance structures have to obey the octet rule for main group compounds and the dodectet (12) rule for transition metal complexes.

Classification	Main group	Transition metal
AX2	-	Linear (180°)
		300px
AX3	-	Trigonal planar (120°)
		400px
AX4	-	Square planar (90°)
		400px
AX5	Trigonal bipyramidal (90°, 120°)	Trigonal bipyramidal (90°, 120°)
		Fractional hybridisation (s and d orbitals) E.g., Fe(CO)5
AX6	Octahedral (90°)	Octahedral (90°)
		300px
AX7	Pentagonal bipyramidal (90°, 72°)	Pentagonal bipyramidal (90°, 72°)
		Fractional hybridisation (s and three d orbitals) E.g., V(CN)74−
AX8	Square antiprismatic	Square antiprismatic
	Fractional hybridisation (s and three p orbitals) E.g., IF8−	Fractional hybridisation (s and four d orbitals) E.g., Re(CN)83−
AX9	-	Tricapped trigonal prismatic
		Fractional hybridisation (s and five d orbitals) E.g., ReH92−

Main group compounds with lone pairs

For hypervalent main group compounds with lone electron pairs, the bonding scheme can be split into two components: the "resonant bonding" component and the "regular bonding" component. The "regular bonding" component has the same description (see above), while the "resonant bonding" component consists of resonating bonds utilizing p orbitals. The table below shows how each shape is related to the two components and their respective descriptions.

		Regular bonding component (marked in red)
		Bent	Monocoordinate	-
Resonant bonding component	Linear axis	Seesaw (AX4E1) (90°, 180°, >90°)	T-shaped (AX3E2) (90°, 180°)	Linear (AX2E3) (180°)
		200px	200px	200px
	Square planar equator	-	Square pyramidal (AX5E1) (90°, 90°)	Square planar (AX4E2) (90°)
	Pentagonal planar equator	-	Pentagonal pyramidal (AX6E1) (90°, 72°)	Pentagonal planar (AX5E2) (72°)

Clarifying misconceptions

VSEPR electron domains and hybrid orbitals are different

The simplistic picture of hybridisation taught in conjunction with VSEPR theory does not agree with high-level theoretical calculations despite its widespread usage in many textbooks. For example, following the guidelines of VSEPR, the hybridization of the oxygen in water is described with two equivalent lone electron-pairs. However, molecular orbital calculations give orbitals that reflect the C_2v symmetry of the molecule. One of the two lone pairs is in a pure p-type orbital, with its electron density perpendicular to the H–O–H framework. The other lone pair is in an approximately sp^0.8 orbital that is in the same plane as the H–O–H bonding. Photoelectron spectra confirm the presence of two different energies for the nonbonded electrons.

Exclusion of d-orbitals in main group compounds

Main page: Hypervalent molecule

In 1990, Magnusson published a seminal work definitively excluding the role of d-orbital hybridization in bonding in hypervalent compounds of second-row elements. This had long been a point of contention and confusion in describing these molecules using molecular orbital theory. Part of the confusion here originates from the fact that one must include d-functions in the basis sets used to describe these compounds (or else unreasonably high energies and distorted geometries result), and the contribution of the d-function to the molecular wavefunction is large. These facts were historically interpreted to mean that d-orbitals must be involved in bonding. However, Magnusson concludes in his work that d-orbital involvement is not implicated in hypervalency.

Exclusion of p-orbitals in transition metal complexes

Similarly, p-orbitals have long been thought to be utilized by transition metal centers in bonding with ligands, hence the 18-electron description; however, recent molecular orbital calculations have found that such p-orbital participation in bonding is insignificant, even though the contribution of the p-function to the molecular wavefunction is calculated to be somwhat larger than that of the d-function in main group compounds.

Hybridization theory vs. MO theory

Hybridization theory is an integral part of organic chemistry and in general discussed together with molecular orbital theory in advanced organic chemistry textbooks although for different reasons. One textbook notes that for drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons. It also comments that predicting bond angles in methane with MO theory is not straightforward. Another textbook treats hybridization theory when explaining bonding in alkenes and a third uses MO theory to explain bonding in hydrogen but hybridisation theory for methane.

Bonding orbitals formed from hybrid atomic orbitals may be considered as localized molecular orbitals, which can be formed from the delocalized orbitals of molecular orbital theory by an appropriate mathematical transformation. For molecules with a closed electron shell in the ground state, this transformation of the orbitals leaves the total many-electron wave function unchanged. The hybrid orbital description of the ground state is therefore equivalent to the delocalized orbital description for explaining the ground state total energy and electron density, as well as the molecular geometry which corresponds to the minimum value of the total energy.

There is no such equivalence, however, for ionized or excited states with open electron shells. Hybrid orbitals cannot therefore be used to interpret photoelectron spectra, which measure the energies of ionized states, identified with delocalized orbital energies using Koopmans' theorem. Nor can they be used to interpret UV-visible spectra which correspond to electronic transitions between delocalized orbitals. From a pedagogical perspective, hybridization approach tends to over-emphasize localisation of bonding electrons and does not effectively embrace molecular symmetry as does MO theory.

Other pages

• Linear combination of atomic orbitals molecular orbital method (LCAO-MO)
• Molecular orbital
• MO diagram

Other websites

• Covalent Bonds and Molecular Structure
• Hybridisation flash movie
• Hybrid orbital 3D preview progam in OpenGL
• Understanding Concepts: Molecular Orbitals