text: Bent's rule provides an additional level of accuracy to valence bond theory. Valence bond theory proposes that covalent bonds consist of two electrons lying in overlapping, usually hybridised, atomic orbitals from two bonding atoms. The assumption that a covalent bond is a linear combination of atomic orbitals of just the two bonding atoms is an approximation (see molecular orbital theory), but valence bond theory is accurate enough that it has had and continues to have a major impact on how bonding is understood.
  In valence bond theory, two atoms each contribute an atomic orbital and the electrons in the orbital overlap form a covalent bond. Atoms do not usually contribute a pure hydrogen-like orbital to bonds. If atoms could only contribute hydrogen-like orbitals, then the experimentally confirmed tetrahedral structure of methane would not be possible as the 2s and 2p orbitals of carbon do not have that geometry. That and other contradictions led to the proposing of orbital hybridisation. In that framework, atomic orbitals are allowed to mix to produce an equivalent number of orbitals of differing shapes and energies. In the aforementioned case of methane, the 2s and three 2p orbitals of carbon are hybridized to yield four equivalent sp orbitals, which resolves the structure discrepancy. Orbital hybridisation allowed valence bond theory to successfully explain the geometry and properties of a vast number of molecules.
  In traditional hybridisation theory, the hybrid orbitals are all equivalent. Namely the atomic s and p orbital(s) are combined to give four  orbitals, three  orbitals, or two  orbitals. These combinations are chosen to satisfy two conditions. First, the total amount of s and p orbital contributions must be equivalent before and after hybridisation. Second, the hybrid orbitals must be orthogonal to each other. If two hybrid orbitals were not orthogonal, by definition they would have nonzero orbital overlap. Electrons in those orbitals would interact and if one of those orbitals were involved in a covalent bond, the other orbital would also have a nonzero interaction with that bond, violating the two electron per bond tenet of valence bond theory.
  To construct hybrid s and p orbitals, let the first hybrid orbital be given by , where p is directed towards a bonding group and &lambda; determines the amount of p character this hybrid orbital has. This is a weighted sum of the wavefunctions. Now choose a second hybrid orbital , where p is directed in some way and &lambda; is the amount of p character in this second orbital. The value of &lambda; and direction of p must be determined so that the resulting orbital can be normalized and so that it is orthogonal to the first hybrid orbital. The hybrid can certainly be normalized, as it is the sum of two normalized wavefunctions. Orthogonality must be established so that the two hybrid orbitals can be involved in separate covalent bonds. The inner product of orthogonal orbitals must be zero and computing the inner product of the constructed hybrids gives the following calculation.
  The s orbital is normalized and so the inner product . Also, the s orbital is orthogonal to the p and p orbitals, which leads to two terms in the above equaling zero. Finally, the last term is the inner product of two normalized functions that are at an angle of  to each other, which gives  by definition. However, the orthogonality of bonding orbitals demands that , so we get Coulson's theorem as a result:
  This means that the four s and p atomic orbitals can be hybridised in arbitrary directions provided that all of the coefficients &lambda; satisfy the above condition pairwise to guarantee the resulting orbitals are orthogonal.
  Bents rule, that central 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; coefficient increases the p character of the  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 . More sophisticated theoretical and computation techniques beyond Bents rule are needed to accurately predict molecular geometries from first principles, but Bent's rule provides an excellent heuristic in explaining molecular structures.
  Henry Bent originally proposed his rule in 1960 on empirical grounds, but a few years later it was supported by molecular orbital calculations by Russell Drago.