Orbital hybridisation

In chemistry, hybridisation (or hybridization) is the concept of mixing atomic orbitals to form new hybrid orbitals suitable for the qualitative description of atomic bonding properties. Hybridised orbitals are very useful in the explanation of the shape of molecular orbitals for molecules. It is an integral part of valence bond theory. Although sometimes taught together with the valence shell electron-pair repulsion (VSEPR) theory, valence bond and hybridization are in fact not related to the VSEPR model.[1]

Historical development

The hybridisation theory was promoted by chemist Linus Pauling[2] 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.

Hybridisation theory is not as practical for quantitative calculations as Molecular Orbital Theory. Problems with hybridisation are especially notable when the d orbitals are involved in bonding, as in coordination chemistry and organometallic chemistry. Although hybridisation schemes in transition metal chemistry can be used, they are not generally as accurate.

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. Hybridised orbitals are assumed to be mixtures of these atomic orbitals, superimposed on each other in various proportions. The theory of hybridisation is most applicable under these assumptions. It gives a simple orbital picture equivalent to Lewis structures. Hybridisation is not required to describe molecules, but for molecules made up from carbon, nitrogen and oxygen (and to a lesser extent, sulfur and phosphorus) the hybridisation theory/model makes the description much easier.

The hybridisation theory finds its use mainly in organic chemistry. Its explanation starts with the way bonding is organized in methane.

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. The problem with the existence of methane is now this: carbon's ground-state configuration is 1s2 2s2 2px1 2py1 or more easily read:

\( C\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\downarrow}{2s}\; \frac{\uparrow\,}{2p_x}\; \frac{\uparrow\,}{2p_y}\; \frac{\,\,}{2p_z} \)

The valence bond theory would predict, based on the existence of two half-filled p-type orbitals (the designations px py or pz are meaningless at this point, as they do not fill in any particular order), that C forms two covalent bonds, i.e., CH₂ (methylene). However, methylene is a very reactive molecule (see also: carbene) and cannot exist outside of a molecular system. Therefore, this theory alone cannot explain the existence of CH₄.

Furthermore, ground state orbitals cannot be used for bonding in CH₄. While exciting a 2s electron into a 2p orbital would, in theory, allow for four bonds according to the valence bond theory, (which has been proved experimentally correct for systems like O2) this would imply that the various bonds of CH₄4 would have differing energies due to differing levels of orbital overlap. Once again, this has been experimentally disproved: any hydrogen can be removed from a carbon with equal ease.

To summarise, to explain the existence of CH₄ (and many other molecules) a method by which as many as 12 bonds (for transition metals) of equal strength (and therefore equal length) was required.

The first step in hybridisation is the excitation of one (or more) electrons (we consider the carbon atom in methane, for simplicity of the discussion):

\( C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{2s}\; \frac{\uparrow\,}{2p_x} \frac{\uparrow\,}{2p_y} \frac{\uparrow\,}{2p_z} \)

The proton that forms the nucleus of a hydrogen atom attracts one of the lower-energy valence electrons on carbon. This causes an excitation, moving a 2s electron into a 2p orbital. This, however, increases the influence of the carbon nucleus on the valence electrons by increasing the effective core potential (the amount of charge the nucleus exerts on a given electron = Charge of Core − Charge of all electrons closer to the nucleus). The effective core potential is also known as the effective nuclear charge, or Z_eff.

The solution to the Schrödinger equation for this configuration is a linear combination of the s and p wave functions, or orbitals, known as a hybridized orbital[3]. In the case of carbon attempting to bond with four hydrogens, four orbitals are required. Therefore, the 2s orbital (core orbitals are almost never involved in bonding) "mixes" with the three 2p orbitals to form four sp³ hybrids (read as s-p-three). See graphical summary below.

becomes \( C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp^3}\; \frac{\uparrow\,}{sp^3} \frac{\uparrow\,}{sp^3} \frac{\uparrow\,}{sp^3} \)

In CH₄, four sp³ hybridised orbitals are overlapped by hydrogen's 1s orbital, yielding four σ (sigma) bonds (that is, four single covalent bonds). The four bonds are of the same length and strength. This theory fits our requirements.

A schematic presentation of hybrid orbitals overlapping hydrogens' s orbitals translates into Methane's tetrahedral shape

An alternative view is: View the carbon as the C4− anion. In this case all the orbitals on the carbon are filled:

\( C^{4-}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\downarrow}{2s}\; \frac{\uparrow\downarrow}{2p_x} \frac{\uparrow\downarrow}{2p_y} \frac{\uparrow\downarrow}{2p_z} \)

If we now recombine these orbitals with the empty s-orbitals of 4 hydrogens (4 protons, H+) and allow maximum separation between the 4 hydrogens (i.e., tetrahedral surrounding of the carbon), we see that at any orientation of the p-orbitals, a single hydrogen has an overlap of 25% with the s-orbital of the C, and a total of 75% of overlap with the 3 p-orbitals (see that the relative percentages are the same as the character of the respective orbital in an sp3-hybridisation model, 25% s- and 75% p-character).

According to the orbital hybridisation theory, the valence electrons in methane should be equal in energy but its photoelectron spectrum [4] shows two bands, one at 12.7 eV (one electron pair) and one at 23 eV (three electron pairs). This apparent inconsistency can be explained when one considers additional orbital mixing taking place when the sp3 orbitals mix with the 4 hydrogen orbitals.

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. In sp2 hybridisation the 2s orbital is mixed with only two of the three available 2p orbitals:

\( C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp^2}\; \frac{\uparrow\,}{sp^2} \frac{\uparrow\,}{sp^2} \frac{\uparrow\,}{p} \)

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.

The amount of p-character is not restricted to integer values; i.e., hybridisations like sp^2.5 are also readily described. In this case the geometries are somewhat distorted from the ideally hybridised picture. For example, as stated in Bent's rule, a bond tends to have higher p-character when directed toward a more electronegative substituent.
sp hybrids

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

C^{*}\quad \frac{\uparrow\downarrow}{1s}\; \frac{\uparrow\,}{sp}\; \frac{\uparrow\,}{sp} \frac{\uparrow\,}{p} \frac{\uparrow\,}{p}

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, along with the VSEPR theory, helps to explain molecule shape:

• AX₁ (e.g., LiH): no hybridisation; trivially linear shape
• AX₂ (e.g., BeCl₂): sp hybridisation; linear or diagonal shape; bond angles are cos⁻¹(−1) = 180°
  • AX₂E (e.g., GeF₂): bent / V shape, < 120°
• AX₃ (e.g., BCl₃): sp² hybridisation; trigonal planar shape; bond angles are cos⁻¹(−1/2) = 120°
  • AX₃E (e.g.,NH₃): trigonal pyramidal, 107°
• AX₄ (e.g., CCl₄): sp³ hybridisation; tetrahedral shape; bond angles are cos⁻¹(−1/3) ≈ 109.5°
• AX₅ (e.g., PCl₅): sp³d hybridisation; trigonal bipyramidal shape
• AX₆ (e.g., SF₆): sp³d² hybridisation; octahedral (or square bipyramidal) shape

This holds if there are no lone electron pairs on the central atom. If there are, they should be counted in the Xi number, but bond angles become smaller due to increased repulsion. For example, in water (H₂O), the oxygen atom has two bonds with H and two lone electron pairs (as can be seen with the valence bond theory as well from the electronic configuration of oxygen), which means there are four such 'elements' on O. The model molecule is, then, AX4: sp3 hybridisation is utilized, and the electron arrangement of H₂O is tetrahedral. This agrees with the experimentally-determined shape for water, a non-linear, bent structure, with a bond angle of 104.5 degrees (the two lone-pairs are not visible).

IIn general, for an atom with s and p orbitals forming hybrids h_i and h_j with included angle θ, the following holds: 1 + λ_iλ_j cos(θ) = 0. The p-to-s ratio for hybrid i is λ_i², and for hybrid j it is λ_j². In the special case of equivalent hybrids on the same atom, again with included angle θ, the equation reduces to just 1 + λ² cos(θ) = 0. For example, BH₃ has a trigonal planar geometry, three 120^o bond angles, three equivalent hybrids about the boron atom, and thus 1 + λ² cos(θ) = 0 becomes 1 + λ² cos(120^o) = 0, giving λ² = 2 for the p-to-s ratio. In other words, sp² hybrids, just as expected from the list above.

Controversy regarding d-orbital participation

Hybridisation theory has failed in a few aspects, notably in explaining the energy considerations for the involvement of d-orbitals in chemical bonding (See above for sp3d and sp3d2 hybridisation). This can be well-explained by means of an example. Consider, for instance, how the theory in question accounts for the bonding in phosphorus pentachloride (PCl5). The d-orbitals are large, comparatively distant from the nucleus and high in energy. Radial distances of orbitals from the nucleus seem to reveal that d-orbitals are far too high in energy to 'mix' with s- and p-orbitals. 3s - 0.47 , 3p - 0.55, 3d - 2.4 (in angstroms). Thus, at first glance, sp3d hybridisation seems improbable.

However, a closer examination of the factors that affect orbital size (and energy) reveals more. Formal charge on the central atom is one such factor, and it is obvious that the P atom in PCl₅ carries quite a large partial positive charge. Thus the 3d orbital contracts in size to such an extent that hybridisation with s and p orbitals may occur. Further, note the cases in which d-orbital participation was proposed in hybridisation: SF₆(sulfur hexafluoride), IF₇, XeF₆; in all these molecules, the central atom is surrounded by the highly electronegative fluorine atom, thus making hybridisation probable among s, p and d orbitals. A further study reveals that orbital size also depends on the number of electrons occupying it. And, even further, coupling of d orbital electrons also results in contraction, albeit to a lesser extent.

The molecular orbital theory, however, offers a clearer insight into the bonding in these molecules.

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.[5] 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 [6] and a third [7] uses MO theory to explain bonding in hydrogen but hybridisation theory for methane.

Although the language and pictures arising from hybridisation theory, more widely known as valence bond theory, remain widespread in synthetic organic chemistry, this qualitative analysis of bonding has been largely superseded by molecular orbital theory in other branches of chemistry. For example, inorganic chemistry texts have all but abandoned instruction of hybridisation, except as a historical footnote.[8][9] One specific problem with hybridisation is that it incorrectly predicts the photoelectron spectra of many molecules, including such fundamental species such as methane and water. 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

Hybrid Orbitals explained - Valence Bond Theory | Crash Chemistry Academy

External links

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

References

1. ^ "It is important to recognize that the VSEPR model provides an approach to bonding and geometry based on the Pauli principle that is completely independent of the valence bond (VB) theory or of any orbital description of bonding." Gillespie, R. J. J. Chem. Educ. 2004, 81, 298-304.
2. ^ L. Pauling, J. Am. Chem. Soc. 53 (1931), 1367
3. ^ McMurray, J. (1995). Chemistry Annotated Instructors Edition (4th ed.). Prentice Hall. p. 272. ISBN 0-13-140221-8
4. ^ photo electron spectrum of methane 1 photo electron spectrum of methane 2
5. ^ Clayden, Jonathan P.; Greeves, Nick; Warren, Stuart G.; Wothers, Peter D. (2000). Organic Chemistry (1st ed.). Oxford University Press. p. 105. ISBN 978-0-19-850346-0.
6. ^ Organic Chemistry, Third Edition Marye Anne Fox James K. Whitesell 2003 ISBN 978-0-7637-3586-9
7. ^ Organic Chemistry 3rd Ed. 2001 Paula Yurkanis Bruice ISBN 0-13-017858-6
8. ^ G. L. Miessler and D. A. Tarr “Inorganic Chemistry” 3rd Ed, Pearson/Prentice Hall publisher, 2003. ISBN 0-13-035471-6.
9. ^ Shriver, D. F.; Atkins, P. W.; Overton, T. L.; Rourke, J. P.; Weller, M. T.; Armstrong, F. A. “Inorganic Chemistry” W. H. Freeman, New York, 2006. ISBN 0-7167-4878-9.