In chemistry, Bent's rule describes and explains the relationship between the orbital hybridization and the electronegativities of substituents. The rule was stated by Henry A. Bent as follows:
Valence bond theory gives a good approximation of molecular structure. Bent's rule addresses disparities between the observed and idealized geometries. According to Bent's rule, a central atom bonded to multiple groups will rehybridize so that orbitals with more s character are directed towards electropositive groups, and orbitals with more p character will be directed towards groups that are more electronegative. By removing the assumption that all hybrid orbitals are equivalent, Bent's rule leads to improved predictions of molecular geometry and bond strengths. Bent's rule can be justified through the relative energy levels of s and p orbitals. Bent's rule represents a modification of VSEPR theory for molecules of lower than ideal symmetry. For bonds with the larger atoms from the lower periods, trends in orbital hybridization depend strongly on both electronegativity and orbital size.
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, which is tetrahedral with an sp3 carbon atom and bond angles of 109.5° between the four equivalent C-H bonds. However, slight deviations from these ideal geometries became apparent in the 1940s. A particularly well known example is water, where the angle between the two O-H bonds is only 104.5°. 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 atom. Finally, in 1961, 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.
Bent's original paper considers the group electronegativity of the methyl group to be less than that of the hydrogen atom because methyl substitution reduces the acid dissociation constants of formic acid and of acetic acid.
Nonbonding orbitals
Bent's rule can be extended to rationalize the hybridization of nonbonding orbitals as well. On the one hand, a lone pair (an occupied nonbonding orbital) can be thought of as the limiting case of an electropositive substituent, with electron density completely polarized towards the central atom. Bent's rule predicts that, in order to stabilize the unshared, closely held nonbonding electrons, lone pair orbitals should take on high s character. On the other hand, an unoccupied (empty) nonbonding orbital can be thought of as the limiting case of an electronegative substituent, with electron density completely polarized towards the ligand and away from the central atom. Bent's rule predicts that, in order to leave as much s character as possible for the remaining occupied orbitals, unoccupied nonbonding orbitals should maximize p character.
Experimentally, the first conclusion is in line with the reduced bond angles of molecules with lone pairs like water or ammonia compared to methane, while the second conclusion accords with the planar structure of molecules with unoccupied nonbonding orbitals, like monomeric borane and carbenium ions.
Consequences 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: VSEPR Theory and Bent's Rule Valence bond theory predicts that methane is tetrahedral and that ethylene is planar. In water and ammonia, the situation is more complicated because the bond angles are 104.5° and 107° respectively, which are less than the expected tetrahedral angle of 109.5°. One rationale for those deviations is VSEPR theory, where valence electrons are assumed to lie in localized regions and lone pairs are assumed to repel each other to a greater extent than bonding pairs. Bent's rule provides an alternative explanation.
Valence shell electron pair repulsion (VSEPR) theory predicts molecule geometry. VSEPR predicts molecular geometry to take the configuration that allows electron pairs to be most spaced out. This electron distance maximization happens to achieve the most stable electron distribution. The result of VSEPR theory is being able to predict bond angles with accuracy. According to VSEPR theory, the geometry of a molecule can be predicted by counting how many electron pairs and atoms are connected to a central atom. Bent's rule states "[A]tomic s character concentrates in orbitals directed toward electropositive substituents". Bent's rule implies that bond angles will deviate from the bond angle predicted by VSEPR theory; the relative electronegativities of atoms surrounding the central atom will impact the molecule geometry. VSEPR theory suggests a way to accurately predict molecule shape using simple rules. However, VSEPR theory predicts observed molecular bond angles only approximately. On the other hand, Bent's rule is more accurate. Furthermore, it has been shown that Bent's rule corroborates quantum mechanical computations when describing molecule geometry.
+MoleculeBond angle between substituents3D images with bond angles
The table above demonstrates the differences between VSEPR theory predicted bond angles and their real-world angles. According to VSEPR theory, diethyl ether, methanol, water and oxygen difluoride should all have a bond angle of 109.5o. Using VSEPR theory, all these molecules should have the same bond angle because they have the same "bent" shape. Yet, clearly the bond angles between all these molecules deviate from their ideal geometries in different ways. Bent's rule can help elucidate these apparent discrepancies. Electronegative substituents will have more p character. Bond angle has a proportional relationship with s character and an inverse relationship with p character. Thus, as substituents become more electronegative, the bond angle of the molecule should decrease. Dimethyl ether, methanol, water and oxygen difluoride follow this trend as expected (as is shown in the table above). Two methyl groups are the substituents attached to the central oxygen in diethyl ether. Because the two methyl groups are electropositive, greater s character will be observed and the real bond angle is larger than the ideal bond angle of 109.5o. Methanol has one electropositive methyl substituent and one electronegative hydrogen substituent. Hence, less s character is observed than dimethyl ether. When there are two hydrogen substituent groups, the angle is decreased even further with the increase in electronegativity and p character. Finally, when both hydrogen substituents are replaced with fluorine in oxygen difluoride, there is another decrease in the bond angle. Fluorine is highly electronegative, resulting in this significant decrease in bond angle.
In predicting the bond angle of water, Bent's rule suggests that hybrid orbitals with more s character should be directed towards the lone pairs, while that leaves orbitals with more p character directed towards the hydrogens, resulting in deviation from idealized O(sp3) hybrid orbitals with 25% s character and 75% p character. In the case of water, with its 104.5° HOH angle, the OH bonding orbitals are constructed from O(~sp4.0) orbitals (~20% s, ~80% p), while the lone pairs consist of O(~sp2.3) orbitals (~30% s, ~70% p). As discussed in the justification above, the lone pairs behave as very electropositive substituents and have excess s character. As a result, the bonding electrons have increased p character. This increased p character in those orbitals decreases the bond angle between them to less than the tetrahedral 109.5°. The same logic can be applied to ammonia (107.0° HNH bond angle, with three N(~sp3.4 or 23% s) bonding orbitals and one N(~sp2.1 or 32% s) lone pair), the other canonical example of this phenomenon.
The same trend holds for nitrogen containing compounds. Against the expectations of VSEPR theory but consistent with Bent's rule, the bond angles of ammonia (NH3) and nitrogen trifluoride (NF3) are 107° and 102°, respectively.
Unlike VSEPR theory, whose theoretical foundations now appear shaky, Bent's rule is still considered to be an important principle in modern treatments of bonding. For instance, a modification of this analysis is still viable, even if the lone pairs of H2O are considered to be inequivalent by virtue of their symmetry (i.e., only s, and in-plane px and py oxygen AOs are hybridized to form the two O-H bonding orbitals σO-H and lone pair nO(σ), while pz becomes an inequivalent pure p-character lone pair nO(π)), as in the case of lone pairs emerging from natural bond orbital methods.
For a tetrahedral molecule such as difluoromethane with two types of atom bonded to the central atom, the C-F bond to the more electronegative substituent (F) will involve a carbon orbital with less s character than the C-H bond, so that the angle between the C-F bonds is less than the tetrahedral bond angle of 109.5°.
Trigonal bipyramid molecules have both with axial and equatorial positions. If there are two types of substituents, the more electronegative substituent will prefer the axial position as there are smaller bond angles between axial and electronegative substituents than between two equatorial substituents.
Bond lengths Similarly to bond angles, the hybridisation of an atom can be related to the lengths of the bonds it forms. As bonding orbitals increase in s character, the σ bond length decreases.
Molecule Average carbon–carbon bond length 1.54 Å 1.50 Å 1.46 Å
By adding electronegative substituents and changing the hybridisation of the central atoms, bond lengths can be manipulated. If a molecule contains a structure X-A--Y, replacement of the substituent X by a more electronegative atom changes the hybridization of central atom A and shortens the adjacent A--Y bond.
Molecule Average carbon–fluorine bond length Fluoromethane 1.388 Å Difluoromethane 1.358 Å Trifluoromethane 1.329 Å Tetrafluoromethane 1.323 Å Bonds between elements of disparate electronegativities will be polar and the electron density in such bonds will be shifted towards the more electronegative element. Applying this idea to the molecule fluoromethane illustrates the power of Bent's rule. Because carbon is more electronegative than hydrogen, the electron density in a C-H bond will be shortened and the C-F bond will be elongated.
The same trend also holds for the chlorinated analogs of methane, although the effect is less dramatic because chlorine is less electronegative than fluorine.
Molecule Average carbon–chlorine bond length Chloromethane 1.783 Å Dichloromethane 1.772 Å Trichloromethane 1.767 Å Tetrachloromethane 1.766 Å
The above cases seem to demonstrate that the size of the chlorine is less important than its 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 the steric explanation contradicts the experimental result, Bent's rule is likely playing a primary role in structure determination.
JCH Coupling constants