Orbital hybridisation

In chemistry, orbital hybridisation is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the pairing of electrons to form chemical bonds in valence bond theory. For example, in a carbon atom which forms four single bonds, the valence-shell s orbital combines with three valence-shell p orbitals to form four equivalent sp³ mixtures in a tetrahedral arrangement around the carbon to bond to four different atoms. Hybrid orbitals are useful in the explanation of molecular geometry and atomic bonding properties and are symmetrically disposed in space. Usually hybrid orbitals are formed by mixing atomic orbitals of comparable energies.

History and uses

first developed the hybridisation theory in 1931 to explain the structure of simple molecules such as methane using atomic orbitals. Pauling pointed out that a carbon atom forms four bonds by using one s and three p orbitals, so that "it might be inferred" that a carbon atom would form three bonds at right angles and a fourth weaker bond using the s orbital in some arbitrary direction. In reality, methane has four C–H bonds of equivalent strength. The angle between any two bonds is the tetrahedral bond angle of 109°28'. Pauling supposed that in the presence of four hydrogen atoms, the s and p orbitals form four equivalent combinations which he called hybrid orbitals. Each hybrid is denoted sp³ to indicate its composition, and is directed along one of the four C–H bonds. 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. It gives a simple orbital picture equivalent to Lewis structures.
Hybridisation theory is an integral part of organic chemistry, one of the most compelling examples being Baldwin's rules. For drawing reaction mechanisms sometimes a classical bonding picture is needed with two atoms sharing two electrons. Hybridisation theory explains bonding in alkenes and methane. The amount of p character or s character, which is decided mainly by orbital hybridisation, can be used to reliably predict molecular properties such as acidity or basicity.

Overview

Orbitals are a model representation of the behavior of electrons within molecules. In the case of simple hybridization, this approximation is based on atomic orbitals, similar to those obtained for the hydrogen atom, the only neutral atom for which the Schrödinger equation can be solved exactly. In heavier atoms, such as 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 atomic orbitals, superimposed on each other in various proportions. 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 hybridised. Quantum mechanics describes this hybrid as an sp³ wavefunction of the form, where N is a normalisation constant and pσ is a p orbital directed along the C-H axis to form a sigma bond. The ratio of coefficients is square root of 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.

Types

sp³

Hybridisation describes the bonding of atoms from an atom's point of view. For a tetrahedrally coordinated carbon, the carbon should have 4 orbitals directed towards the 4 hydrogen atoms.
Carbon's ground state configuration is 1s² 2s² 2p² or more easily read:
This diagram suggests that the carbon atom could use its two singly occupied p-type orbitals to form two covalent bonds with two hydrogen atoms in a methylene molecule, with a hypothetical bond angle of 90° corresponding to the angle between two p orbitals on the same atom. However the true H–C–H angle in singlet methylene is about 102° which implies the presence of some orbital hybridisation.
The carbon atom can also bond to four hydrogen atoms in methane by an excitation of an electron from the doubly occupied 2s orbital to the empty 2p orbital, producing four singly occupied orbitals.
The energy released by the formation of two additional bonds more than compensates for the excitation energy required, energetically favoring the formation of four C-H bonds.
According to quantum mechanics, the lowest energy is obtained if the four bonds are equivalent, which requires that they are formed from equivalent orbitals on the carbon. A set of four equivalent orbitals can be obtained that are linear combinations of the valence-shell s and p wave functions, which are the four sp³ hybrids.
In CH₄, four sp³ hybrid orbitals are overlapped by the four hydrogens' 1s orbitals, yielding four σ bonds of equal length and strength.
The following:
translates into:

sp²

Other carbon compounds and other molecules may be explained in a similar way. For example, ethylene has a double bond between the carbons. For this molecule, carbon sp² hybridises, because one π 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, usually denoted 2p_x and 2p_y. The third 2p orbital remains unhybridised.
forming a total of three sp² orbitals with one remaining p orbital. In ethylene, the two carbon atoms form a σ bond by overlapping one sp² orbital from each carbon atom. The π bond between the carbon atoms perpendicular to the molecular plane is formed by 2p–2p overlap. Each carbon atom forms covalent C–H bonds with two hydrogens by s–sp² overlap, all with 120° bond angles. The hydrogen–carbon bonds are all of equal strength and length, in agreement with experimental data.

sp

The chemical bonding in compounds such as alkynes with triple bonds is explained by sp hybridization. In this model, the 2s orbital is mixed with only one of the three p orbitals,
resulting in two sp orbitals and two remaining p orbitals. The chemical bonding in acetylene 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.

Molecule shape

Hybridisation helps to explain molecule shape, since the angles between bonds are approximately equal to the angles between hybrid orbitals. This is in contrast to valence shell electron-pair repulsion theory, which can be used to predict molecular geometry based on empirical rules rather than on valence-bond or orbital theories.

sp^x hybridisation

As the valence orbitals of main group elements are the one s and three p orbitals with the corresponding octet rule, sp^x hybridization is used to model the shape of these molecules.

sp^xd^y hybridisation

As the valence orbitals of transition metals are the five d, one s and three p orbitals with the corresponding 18-electron rule, sp^xd^y hybridisation is used to model the shape of these molecules. These molecules tend to have multiple shapes corresponding to the same hybridization due to the different d-orbitals involved. A square planar complex has one unoccupied p-orbital and hence has 16 valence electrons.

Coordination number	Shape	Hybridisation	Examples
4	Square planar	sp²d hybridisation	PtCl₄²⁻
5	Trigonal bipyramidal	sp³d hybridisation	Fe₅
5	Square pyramidal	sp³d hybridisation	MnCl₅²⁻
6	Octahedral	sp³d² hybridisation	Mo₆
7	Pentagonal bipyramidal	sp³d³ hybridisation	ZrF₇³⁻
7	Capped octahedral	sp³d³ hybridisation	MoF₇⁻
7	Capped trigonal prismatic	sp³d³ hybridisation	TaF₇²⁻
8	Square antiprismatic	sp³d⁴ hybridisation	ReF₈⁻
8	Dodecahedral	sp³d⁴ hybridisation	Mo₈⁴⁻
8	Bicapped trigonal prismatic	sp³d⁴ hybridisation	ZrF₈⁴⁻
9	Tricapped trigonal prismatic	sp³d⁵ hybridisation	ReH₉²⁻
9	Capped square antiprismatic	sp³d⁵ hybridisation

sd^x hybridisation

In certain transition metal complexes with a low d electron count, the p-orbitals are unoccupied and sd^x hybridisation is used to model the shape of these molecules.

Hypervalent molecules

Octet expansion

In some general chemistry textbooks, hybridization is presented for main group coordination number 5 and above using an "expanded octet" scheme with d-orbitals first proposed by Pauling. However, such a scheme is now considered to be incorrect in light of computational chemistry calculations.

Coordination number	Molecular shape	Hybridisation	Examples
5	Trigonal bipyramidal	sp³d hybridisation	Phosphorus pentafluoride\|
6	Octahedral	sp³d² hybridisation	Sulfur hexafluoride\|
7	Pentagonal bipyramidal	sp³d³ hybridisation	Iodine heptafluoride\|

In 1990, Eric Alfred Magnusson of the University of New South Wales published a paper definitively excluding the role of d-orbital hybridisation in bonding in hypervalent compounds of second-row elements, ending a point of contention and confusion. Part of the confusion originates from the fact that d-functions are essential in the basis sets used to describe these compounds. Also, the contribution of the d-function to the molecular wavefunction is large. These facts were incorrectly interpreted to mean that d-orbitals must be involved in bonding.