Molecular orbitals for polyatomic systems

The molecular orbitals of polyatomic molecules are built in the same way as in diatomic molecules, the only difference being that we use more atomic orbitals to construct them. A molecular orbital has the general form

The principal difference between diatomic and polyatomic molecules lies in the greater range of shapes that are possible : a diatomic molecule is necessarily linear, but a triatomic molecule, for instance, may be either linear or angular with a characteristic bond angle. The shape of a polyatomic molecule—the specification of its bond lengths and its bond angles—can be predicted by calculating the total energy of the molecule for a variety of nuclear positions, and then identifying the conformation that corresponds to the lowest energy.

The Hückel approximation The  molecular orbital energy level diagrams of conjugated molecules can be constructed using a set of approximations suggested by Erich Hückel in 1931 In his approach, the  orbitals are treated separately from the  orbitals, and the latter form a rigid framework that determines the general shape of the molecule. All the C atoms are treated identically, so all the Coulomb integrals  for the atomic orbitals that contribute to the  orbitals are set equal

(a) Ethene and frontier orbitals We express the  orbitals as LCAOs of the C2p orbitals that lie perpendicular to the molecular plane. ψ = c A A + c B B where the A is a C2p orbital on atom A, and so on.

we have to solve the secular determinant with αA =αB =α in the case of ethene. In a modern computation all the resonance integrals and overlap integrals would be included, but an indication of the molecular orbital energy level diagram can be obtained very readily if we make the following additional Hückel approximations.

Hückel approximations: 1. All overlap integrals are set equal to zero. 2. All resonance integrals between non-neighbours are set equal to zero. 3. All remaining resonance integrals are set equal (to  ). The assumptions result in the following structure of the secular determinant: 1. All diagonal elements: α − E. 2. Off-diagonal elements between neighbouring atoms: β. 3. All other elements: 0.

These approximations lead to The roots of the equation are

The Hückel molecular orbital energy levels of ethene. Two electrons occupy the lower  orbital. The highest occupied molecular orbital in ethene, its HOMO, is the 1  orbital the lowest unfilled molecular orbital, its LUMO, is the 2  orbital

These two orbitals jointly form the frontier orbitals of the molecule. The frontier orbitals are important because they are largely responsible for many of the chemical and spectroscopic properties of the molecule. For example, we can estimate that 2|β | is the π *←π excitation energy of ethene, the energy required to excite an electron from the 1π to the 2π orbital.

(b) The matrix formulation of the Hückel method (H AA − E i S AA )c i,A + (H AB − E i S AB )c i,B = 0 (H BA − E i S BA )c i,A + (H BB − E i S BB )c i,B = 0 where the eigenvalue Ei corresponds to a wavefunction of the form ψ i = c i,A A + c i,B B.

There are two atomic orbitals, two eigenvalues, and two wavefunctions, so there are two pairs of secular equations, with the first corresponding to E1 and ψ1: (H AA − E 1 S AA )c 1,A + (H AB − E 1 S AB )c 1,B = (a) (H BA − E 1 S BA )c 1,A + (H BB − E 1 S BB )c 1,B = (b) and another corresponding to E 2 and ψ 2 : (H AA − E 2 S AA )c 2,A + (H AB − E 2 S AB )c 2,B = (c) (H BA − E 2 S BA )c 2,A + (H BB − E 2 S BB )c 2,B = (d)

If we introduce the following matrices and column vectors then each pair of equations may be written more succinctly as (H − E i S)c i =0 or Hc i = Sc i E i

where H is the hamiltonian matrix and S is the overlap matrix. To proceed with the calculation of the eigenvalues and coefficients, we introduce the matrices for then the entire set of equations we have to solve can be expressed as HC = SCE *)

Self-test 11.7 Show by carrying out the necessary matrix operations that eqn * is a representation of the system of equations consisting of eqns (a)–(d).