Orbital hybridisation

Four sp³ orbitals.

Three sp² orbitals.

In chemistry, hybridisation (or hybridization) is the concept of mixing atomic orbitals into new hybrid orbitals (with different energies, shapes, etc., than the actual orbitals hybridising) 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. Although sometimes taught together with the valence shell electron-pair repulsion (VSEPR) theory, valence bond and hybridisation are in fact not related to the VSEPR model.

Historical development

Chemist Linus Pauling first developed the hybridisation theory in order 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, and today it is considered an effective heuristic for rationalizing the structures of organic compounds.

Orbitals are a model representation of the behaviour of electrons within molecules. In the case of simple hybridisation, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, 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 used are the 2s and 2p orbitals, similar to excited state orbitals for hydrogen. 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. Hybridisation theory finds its use 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 carbon–hydrogen 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 + √3pσ), 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 ratio of coefficients (denoted λ in general) is √3 in this example. Since the electron density associated with an orbital is proportional to the square of the wavefunction, the ratio of p-character to s-character is λ² = 3. The p character or the weight of the p component is N²λ² = 3/4.

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, any two hybrid orbitals on the same atom must be mutually orthogonal. For an atom with s and p orbitals forming hybrids h_i and h_j with included angle $\theta$, the orthogonality condition implies the relation: $1 +\lambda_i\lambda_j\cos\theta = 0$. The p-to-s ratio for hybrid i is $\lambda_i^2$, and for hybrid j it is $\lambda_j^2$. 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 $\theta$, the equation reduces to just $1 + \lambda^2 \cos\theta = 0$. For example, BH₃ has a trigonal planar geometry, three 120° bond angles, three equivalent hybrids about the boron atom, and thus $1 + \lambda^2 \cos\theta = 0$ becomes $1 + \lambda^2 \cos \tfrac{2\pi}{3}= 0$, giving $\lambda^2 = 2$ 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 hybridisation

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:

C	↑↓	↑↓	↑	↑
	1s	2s	2px	2py	2pz

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.

C*	↑↓	↑	↑	↑	↑
	1s	2s	2px	2py	2pz

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.

C*	↑↓	↑	↑	↑	↑
	1s	sp3	sp3	sp3	sp3

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. Take, for example, ethene (C₂H₄). Ethene has a double bond between the carbons.

For this molecule, carbon will sp² hybridise, 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 hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals:

C*	↑↓	↑	↑	↑	↑
	1s	sp2	sp2	sp2	2p

forming a total of three 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

A schematic presentation of hybrid orbitals sp

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

C*	↑↓	↑	↑	↑	↑
	1s	sp	sp	2p	2p

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 σ 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	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°) sp2d hybridisation E.g., PtCl42−
AX5	Trigonal bipyramidal (90°, 120°) sp3d hybridisation E.g., PCl5, Fe(CO)5
AX6	Octahedral (90°) sp3d2 hybridisation E.g., SF6, Mo(CO)6
AX7	Pentagonal bipyramidal (90°, 72°) sp3d3 hybridisation E.g., IF7, V(CN)74−
AX8	Square antiprismatic sp3d4 hybridisation E.g., IF8−, Re(CN)83−
AX9	-	Tricapped trigonal prismatic sp3d5 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).

Coulson-Fischer description

In the GVB or SCVB description (utilizing the Coulson-Fischer wavefunction), orbitals used for bonding utilize a ligand orbital component to account for the ionicity of the bond. 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). The added ligand orbital component, which translates to ionic character, also allows the central atom to form more orbitals for bonding than would be possible with non-polar covalent bonds. This orbital picture can be illustrated in Lewis structures by ionic-covalent resonance.

Classification	Main group	Transition metal
AX2	-	Linear (180°) A(s)+X(σ) E.g., Ag(NH3)2+
AX3	-	Trigonal planar (120°) A(s)+X(σ) E.g., Cu(CN)32−
AX4	-	Square planar (90°) A(sd)+X(σ) E.g., PtCl42−
AX5	Trigonal bipyramidal (90°, 120°)
	A(sp3)+X(σ) E.g., PCl5	A(sd)+X(σ) E.g., Fe(CO)5
AX6	Octahedral (90°)
	A(sp3)+X(σ) E.g., SF6	A(sd2)+X(σ) E.g., Mo(CO)6
AX7	Pentagonal bipyramidal (90°, 72°)
	A(sp3)+X(σ) E.g., IF7	A(sd3)+X(σ) E.g., V(CN)74−
AX8	Square antiprismatic
	A(sp3)+X(σ) E.g., IF8−	A(sd4)+X(σ) E.g., Re(CN)83−
AX9	-	Tricapped trigonal prismatic A(sd5)+X(σ) 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 "hypervalent bonding" component and the "regular bonding" component. The "regular bonding" component has the same description (see above), while the "hypervalent bonding" component consists of A(p_σ)+X(σ) hybrids. The table below shows how each shape is related to the two components and their respective descriptions.

		Regular bonding component
		Bent	Monocoordinate	-
Hypervalent bonding component (A(pσ)+X(σ))	Linear axis	Seesaw (AX4E1) (90°, 180°, >90°) - E.g., SF4	T-shaped (AX3E2) (90°, 180°) + p-orbital lone pair E.g., ClF3	Linear (AX2E3) (180°) + two p-orbital lone pairs, s-orbital lone pair E.g., XeF2
	Square planar equator	-	Square pyramidal (AX5E1) (90°, 90°) - E.g., ClF5	Square planar (AX4E2) (90°) + p-orbital lone pair, s-orbital lone pair E.g., XeF4
	Pentagonal planar equator	-	Pentagonal pyramidal (AX6E1) (90°, 72°) - E.g., XeOF5−	Pentagonal planar (AX5E2) (72°) + p-orbital lone pair, s-orbital lone pair E.g., XeF5−

Pi bonding

Hypervalent pi bonding in the spin-coupled description is also explained analogously using A(p_π)+X(p_π) or A(d_π)+X(p_π) hybrids. For example, in ozone, the central oxygen's π orbital hybridises with the two surrounding oxygen π orbitals and overlaps with them to form two polar pi bonds with overall bond order of around 1.

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 as sp³ with two lone electron-pairs and two bonds in four equal-energy, symmetrical orbitals making a tetrahedron. However, molecular orbital calculations give orbitals which reflect the 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.

In contrast, the orbitals used to make the O-H bonds are approximately sp⁴ hybrids. It has been argued that it is this change in the mixing of the orbitals that is responsible for the compression of the H-O-H angle down to the experimental 104.5 degrees, not some change in the repulsion of electrons. Thus while VSEPR and its application to hybridisation predicts the correct atomic framework for water and other molecules with lone electron pairs, it may do so for the wrong reason.

Non-inclusion of d orbitals in main group compounds

Main article: 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.

Non-inclusion 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 p orbitals do not contribute significantly to the hybrid orbitals in transition metal complexes, even though the contribution of the p-function to the molecular wavefunction is calculated to be somewhat larger than that of the d-function in main group compounds.

Hybridisation theory vs. MO theory

Hybridisation 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 hybridisation 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, hybridisation approach tends to over-emphasize localisation of bonding electrons and does not effectively embrace molecular symmetry as does MO theory.

See also

• Linear combination of atomic orbitals molecular orbital method
• MO diagrams

External links

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