Isovalent hybridization

In chemistry, isovalent or second order hybridization is an extension of orbital hybridization, the mixing of atomic orbitals into hybrid orbitals which can form chemical bonds, to include fractional numbers of atomic orbitals of each type (s, p, d). It allows for a quantitative depiction of bond formation when the molecular geometry deviates from ideal bond angles.

Only bonding with 4 equivalent substituents results in exactly sp³ hybridization. For molecules with different substituents, we can use isovalent hybridization to rationalize the differences in bond angles between different atoms. In the molecule methyl fluoride for example, the HCF bond angle (108.73°) is less than the HCH bond angle (110.2°).^[1] This difference can be attributed to more p character in the C−F bonding and more s character in the C−H bonding orbitals. The hybridisation of bond orbitals is determined by Bent's rule: "Atomic s character concentrates in orbitals directed toward electropositive substituents".

The bond length between similar atoms also shortens with increasing s character. For example, the C−H bond length is 110.2 pm in ethane, 108.5 pm in ethylene and 106.1 pm in acetylene, with carbon hybridizations sp³ (25% s), sp² (33% s) and sp (50% s) respectively.

To determine the degree of hybridization of each bond one can utilize a hybridization parameter (λ). For hybrids of s and p orbitals, this is the coefficient $(\lambda)$multiplying the p orbital when the hybrid orbital is written in the form $(s + \lambda p)$. The square of the hybridization parameter equals the hybridization index (n) of an spⁿ orbital.^[2][3][4] $n = \lambda^2$.

The fractional s character of orbital i is $\frac{1}{1+ \lambda_i^2}$, and the s character of all the hybrid orbitals must sum to one, so that $\sum_{i} \frac{1}{1+ \lambda_i^2} = 1$

The fractional p character of orbital i is $\frac{\lambda_i^2}{1+\lambda_i^2}$, and the p character of all the hybrid orbitals sums to the number of p orbitals involved in the formation of hybrids:

$\sum_{i} \frac{\lambda_i^2}{1+\lambda_i^2} = 1,2 or 3$

These hybridization parameters can then be related to physical properties like bond angles. Using the two bonding atomic orbitals i and j we are able to find the magnitude of the interorbital angle. The orthogonality condition implies the relation known as Coulson's theorem:^[5][6]

$\ 1 + \lambda_i \lambda_j \cos \theta_{ij} = 0$

For two identical ligands the following equation can be utilized:

$\ 1 + \lambda_i^2 \cos \theta_{ii} = 0$

The hybridization index cannot be measured directly in any way. However, one can find it indirectly by measuring specific physical properties. NMR coupling constants can be used to provide a measure of bonding density around the nucleus of a bonding carbon atom. This can then be used to find the hybridization because there is s character around the nuclei of the bonding electrons. The relationship can be shown with the following equation.

$\ J = \frac {500}{1 + \lambda_i^2}$

Where J is the NMR spin-spin coupling constant of ¹³C and H.

The ¹³C NMR spectroscopy has been useful in determining quantitative s and p character in cycloalkanes, showing as you go from cyclopropane to cyclooctane the s character increases.^[7][8][9]

References

1. National Institute of Standards and Technology. Listing of Experimental Data for CH₃F. (accessed Feb.19, 2015). See table of Internal coordinates.
2. Carroll, F. A. Perspectives on Structure and Mechanism in Organic Chemistry, 2nd ed.; John Wiley & Sons: New Jersey, 2010.
3. Mislow, K. Introduction to Stereochemistry; W.A. Benjamin Inc: New York. 1965.
4. Anslyn, A.V., Dougherty, D.A. Modern Physical Organic Chemistry 3rd ed; University Science: California. 2006.
5. Coulson, C.A. Valence (2nd ed., Oxford University Press 1961) p.204
6. Kwan, E.E. Lecture notes Chem 106 (Harvard University)
7. Carroll, F. A. Perspectives on Structure and Mechanism in Organic Chemistry, 2nd ed.; John Wiley & Sons: New Jersey, 2010.
8. Anslyn, A.V., Dougherty, D.A. Modern Physical Organic Chemistry 3rd ed; University Science: California. 2006.
9. Ferguson, L.N. Highlights of Alicyclic Chemistry, Part 1; Franklin Publishing Company, Inc.: Palisade, NJ, 1973.