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Bent's rule provides a convenient way to explain the properties of molecules by relating the orbital hybridisation of central atoms to the electronegativities of substituents. By knowing how the hybridisation of central atoms deviates from the idealized sp, sp2, and sp3 hybrids, a much more accurate description of molecules can be obtained. This allows for reasonably accurate trends in molecular geometry to be predicted as the substituents change.

The rule was stated by Henry Bent as follows: "Atomic s character concentrates in orbitals directed toward electropositive substituents". In a molecule, a central atom bonded to multiple groups will hybridise so that orbitals with more s character are directed towards electropositive groups, while orbitals with more p character will be directed towards groups that are more electronegative. In the more traditional view, p-block elements will hybridise strictly as spn, where n is either 1, 2, or 3 and all of the orbitals are equivalent, while Bent's rule allows for each orbital to be different with noninteger p character allowed. By considering the differences between s and p orbitals, molecular properties such as bond geometry and strength can be explained.

Bent's rule can be generalized to d-block elements as well. The hybridisation of a metal center is arranged so that orbitals with more s character are directed towards ligands that form bonds with more covalent character. Equivalently, orbitals with more d character are directed towards groups that form bonds of greater ionic character.

History
In the early 1930s, shortly after much of the initial development of quantum mechanics, those theories began to be applied towards molecular structure by Pauling, Slater , Coulson , and others. In particular, Pauling introduced the concept of hybridisation, where atomic s and p orbitals are combined to give hybrid sp, sp2, and sp3 orbitals. Hybrid orbitals proved powerful in explaining the molecular geometries of simple molecules like methane (tetrahedral with an sp3 carbon). However, slight deviations from these ideal geometries became apparent in the 1940s. A particularly well known example is water, where the angle between hydrogens is 104.5°, far less than the expected 109.5°. To explain such discrepancies, it was proposed that hybridisation can result in orbitals with unequal s and p character. A. D. Walsh described in 1947 a relationship between the electronegativity of groups bonded to carbon and the hybridisation of said carbon. Finally, in 1961, Henry A. Bent published a major review of the literature that related molecular structure, central atom hybridisation, and substituent electronegativities and it is for this work that Bent's rule takes its name.

Justification
An informal justification of Bent's rule relies on s orbitals being lower in energy than p orbitals. Consider a central atom A bonded to two groups, X and Y. A can certainly be bonded to other groups and/or have lone pairs, but for right now just consider the X&minus;A&minus;Y portion of the molecule. Now suppose that Y is more electronegative than A and that A is more electronegative than X. The former condition will polarize the A&minus;Y bond towards Y and the latter will polarize the X&minus;A bond towards A. This will shift the electron density towards Y in the A&minus;Y bond and more towards A in the X&minus;A bond. Knowing where the electron density is located in these bonds is crucial to hybridising A so as to minimize the energy. Critically and informally, the energetic contribution of A’s atomic orbitals to the bonding orbitals will be greater in the X&minus;A bond than in the A&minus;Y bond due to the electron density being greater near A in the X&minus;A bond than in the A&minus;Y bond. This means in hybridising A, an orbital of lower energy should be directed towards X, and to balance this out an orbital of higher energy should be directed towards Y. As s orbitals are lower in energy than p orbitals, A should be hybridised so that an orbital of more s character and lower energy is directed towards the more electropositive group X, which is exactly what Bent’s rule states. Conversely, an orbital of higher energy and more p character should be directed towards the more electronegative Y. This argument is very casual, but it can be formalized and generalized to more realistic systems.

Note that all bonding orbitals must have at least some p character. A pure s orbital is spherically symmetric and without any directionality to it there would not be enough orbital overlap between it and another group to favor a bonding interaction.

Examples
Bent’s rule can be used to explain trends in both molecular structure and reactivity. After determining how the hybridisation of the central atom should affect a particular property, the electronegativity of substituents can be examined to see if Bent’s rule holds.

Bond Angles
Bent’s rule can be used to explain how molecules such as in water and ammonia deviate from their ideal, predicted tetrahedral geometries very successfully. First, a trend between central atom hybridisation and bond angle can be determined by using the model compounds methane, ethylene, and acetylene. In order, the central carbon(s) is(are) directing sp3, sp2, and sp orbitals towards the hydrogen substituents. The bond angles between substituents are 109.5°, ~120°, and 180°. This simple system demonstrates the trend that substituents bonded to central atoms in orbitals with higher p character will have a smaller angle between them.

It can be proven that if two substituents i and j are bonding with sp$&lambda;_{i}$ and sp$&lambda;_{j}$ hybridised orbitals on the central atom, then the angle between them $&omega;_{ij}$ will be given by $cos &omega;_{ij} = &minus; 1⁄\sqrt{&lambda;_{i}&lambda;_{j}} |undefined$. This result is known as Coulson's Theorem. This is simply a more quantitative way of saying that increasing p character corresponds to a decreasing bond angle between substituents.

Now that the connection between hybridisation and bond angles has been made, Bent’s rule can be applied to specific examples. The following were used in Bent’s original paper.

As one moves down the table, the substituents become more electronegative and the bond angle between them decreases. Bent’s rule agrees with this result, as increasing electronegativity implies the central atom is directing orbitals of greater p character towards the substituents, and so the bond angle should decrease as a result.

In predicting the bond angle of water, Bent’s rule suggests that hybrid orbitals with more s character should be directed towards the very electropositive lone pairs, while that leaves orbitals with more p character directed towards the hydrogens, which decreases the bond angle between them to less than the tetrahedral 109.5°.

Although VSEPR theory cannot explain why the bond angle in oxygen difluoride is less than the bond angle in water, Bent’s rule can explain why VSEPR theory applies to many cases. The lone pairs on central atoms can be considered to be extremely electropositive, as it is unfavorable to move negative charges closer to electrons. This will put those lone pairs in orbitals with a great deal of s character and very little p character. This will force any substituents to occupy orbitals with more p character, as the orbital contributions must add up to one s orbital and three p orbitals. By the above discussion this will result in those substituents having a decreased bond angle between them. Thus Bent’s rule seems to explain why VSEPR theory applies so frequently.

Bond Lengths
Similarly to bond angles, the hybridisation of an atom can be related to the lengths of the bonds it forms

The above demonstrates that as bonding orbitals increase in s character, the bond length decreases. By adding electronegative substituents and changing the hybridisation of the central atoms, bond lengths can be manipulated.

Because fluorine is so much more electronegative than hydrogen, in fluoromethane the carbon will direct a hybrid orbital very high in p character towards the fluorine and orbitals very high in s character towards the hydrogens. In difluoromethane, with two electronegative groups, the p character now must be divided amongst both fluorines, decreasing the amount relative to fluoromethane. This trend holds all the way to tetrafluoromethane and the increased s character in these bonds leads to the shorter bond lengths.

The same trend also holds for the chlorinated analogs of methane, although the effect is less dramatic because chlorine is less electronegative than fluorine.

Interestingly, the above cases seem to demonstrate that the size of the chlorine is less important than the electronegativity. A prediction based on sterics alone would lead to the opposite trend, as the large chlorine substituents would be more favorable far apart. As this contradicts the experimental result, Bent’s rule is likely playing a primary role in structure determination.

JCH Coupling Constants
Perhaps the most direct measurement of s character in a bonding orbital between hydrogen and carbon is via the 1H&minus;13C coupling constants. Both theoretical and experimental work predict that JCH values will be much higher in bonds with more s character.

As the electronegativity of the substituent increases, the amount of p character directed towards the substituent increases as well. This leaves more s character in the bonds to the methyl protons, which leads to increased JCH coupling constants.

Inductive Effect
The inductive effect can also be explained with Bent’s rule. The standard explanation involves the transmission of charge through covalent bonds and Bent’s rule provides a mechanism for such results via differences in hybridisation. As one goes down the table below, groups bonded to the central carbon become more electron-withdrawing. This diverts p character towards those groups and leaves more s character in the bond between the central carbon and the R group. As s orbitals are lower in energy than p orbitals, the electron density in the C&minus;R bond will shift towards the carbon as it contributes more s character to the bonding orbital. This is what leads to the electron-withdrawing or donating effect.

Formal Theory
In traditional hybridisation theory, the hybrid orbitals are all equivalent. Namely the atomic s and p orbital(s) are combined to give four $sp_{i}^{3} = 1⁄\sqrt{4}(s+ \sqrt{3}p_{i})$ orbitals, three $sp_{i}^{2} = 1⁄\sqrt{3}(s+ \sqrt{2}p_{i})$ orbitals, or two $sp_{i} = 1⁄\sqrt{2}(s+ p_{i})$ orbitals. Critically, the pi orbitals are weighted so that the hybrid orbitals are orthogonal to each other and so that the total amount of s and p orbitals contributions is equivalent before and after hybridisation. In generating hybrid orbitals with differing amounts of p character, it must be remembered that bonding orbitals must be orthogonal to each other. Otherwise, they would have nonzero orbital overlap and would not be able to bond with separate groups. To begin, let the first hybrid orbital be given by $s+\sqrt{&lambda;_{i}}p_{i}|undefined$, where pi is directed towards a bonding group and &lambda;i determines the amount of p character this hybrid orbital has. Now choose a second hybrid orbital $s+\sqrt{&lambda;_{j}}p_{j}|undefined$, where pj is directed in any way and &lambda;j is to be determined (alternatively, &lambda;j is known and the direction of the orbital is to be found). This gives the following calculation.


 * $$ \langle s+\sqrt{\lambda_i}p_i | s+\sqrt{\lambda_j}p_j \rangle = \langle s | s \rangle + \sqrt{\lambda_i} \langle s | p_i \rangle + \sqrt{\lambda_j} \langle s | p_j \rangle + \sqrt{\lambda_i \lambda_j} \langle p_i | p_j \rangle = 1+0+0+ \sqrt{\lambda_i \lambda_j} \cos{\omega_{ij}} = 1 +\cos{\omega_{ij}} $$

Note that the s orbital is normalized and so the inner product $$ \langle s|s\rangle = 1 $$. Also the s orbital is orthogonal to the pi and pj orbitals, giving an inner product of zero. Finally, the last term is the inner product of two normalized functions that are at &omega;ij to each other, which gives cos &omega;ij by definition. However, the orthogonality of bonding orbitals demands that $$ 1 + \sqrt{\lambda_i \lambda_j}\cos{\omega_{ij}} = 0$$, so we get Coulson's Theorem as a result:


 * $$ \cos{\omega_{ij}} = -\frac{1}{\sqrt{\lambda_i \lambda_j}} $$

This means that the four s and p atomic orbitals can be hybridised in directions provided that all of the coefficients &lambda; satisfy the above condition pairwise to guarantee the resulting orbitals are orthogonal.

Bent's rule, that centrals atoms direct orbitals of greater p character towards more electronegative substituents, is easily applicable to the above by noting that an increase in the &lambda;i coefficient increases the p character of the $s + \sqrt{&lambda;_{i}}p_{i}|undefined$ hybrid orbital. Thus, if a central atom A is bonded to two groups X and Y and Y is more electronegative than X, then A will hybridise so that $&lambda;_{X} < &lambda;_{Y}$. More sophisticated theoretical and computation techniques beyond Bent’s rule are needed to accurately predict molecular geometries from first principles, but Bent’s rule provides an excellent heuristic in explaining molecular structures.