Hückel Secular Determinants

Introduction

The Hückel model becomes a calculation when the trial pi orbital ψ = Σ i c iφ i is inserted into a variational or matrix eigenvalue equation. The resulting homogeneous linear equations have a nonzero solution only at particular energies. Setting their determinant to zero yields the orbital energy polynomial, often called the secular determinant. The molecule's connectivity directly determines the matrix entries, so learning to set up the determinant is more important than memorising individual eigenvalue lists.

Core explanation

For a general nonorthogonal orbital basis, molecular-orbital coefficients satisfy Σ j(H ij−ES ij)c j=0, or (H−ES)c=0. A nonzero coefficient vector c exists only if det(H−ES)=0. H is the Hamiltonian matrix and S is the overlap matrix. Simple Hückel theory sets S≈I, H ii=α on equivalent carbon p sites, H ij=β for directly connected neighbours and zero for non-neighbours. Its secular equation is therefore det(H−EI)=0.

For a three-site linear chain, label the p orbitals 1, 2 and 3 with bonds 1–2 and 2–3 but not 1–3. The matrix H−EI has α−E on each diagonal, β in positions (1,2), (2,1), (2,3) and (3,2), and zero in corners (1,3) and (3,1). The zero corner is a model assumption about direct coupling, not an assertion that the end atoms have no physical interaction under any more detailed theory.

Let a = α−E. The determinant of this three-site matrix is a(a²−2β²). Its roots are a=0 and a=±√2β, equivalently E=α and E=α±√2β. With β<0, α+√2β is the lowest energy, α is middle and α−√2β is highest. Each root must then be substituted into (H−EI)c=0 to find an eigenvector. The determinant alone supplies energies, not the orbital phase pattern or site densities.

For an open linear chain of N equivalent sites, the matrix is tridiagonal. Its eigenvalues can be written E k = α + 2β cos[kπ/(N+1)] for k=1,…,N under a consistent sign convention. This is a result of the chain's boundary conditions and connectivity. A ring has different cyclic connections and a different allowed pattern, including possible degeneracies. A branch or heteroatom changes matrix entries and usually prevents direct use of the uniform-chain formula.

The matrix dimension is the p-basis size, not the electron count. A three-site allyl cation, radical and anion share the same simple three-orbital secular matrix if their geometry and parameters are held fixed; they differ in how many electrons fill the resulting levels. Real charge changes can alter geometry and effective parameters, so the shared matrix is a modelling approximation rather than an exact statement.

Algebraic mistakes often arise from the sign of β or the determinant convention. Writing H−EI versus EI−H changes an overall determinant sign for odd matrix dimension but not its zeros. Replacing α−E by E−α consistently also preserves roots. What matters is that all matrix entries follow one convention and the final energy order is interpreted with the declared sign of β.

The eigenvector must be normalised after solving its coefficient ratios. For an orthonormal Hückel basis, Σ i c i ²=1. If overlap is retained, the correct norm is c†Sc=1. Coefficient signs can be globally reversed without changing the physical MO; relative signs determine nodes and bonding patterns. This distinction becomes important when drawing allyl and butadiene orbitals.

Step-by-step reasoning

Number each p site, draw the connectivity graph and write α−E on the diagonal. Insert β for each connected pair symmetrically and zero for non-neighbours. Set the determinant to zero and solve the energy roots. For each root, solve the homogeneous coefficient equations, normalise, sort energies using β's sign and only then fill electrons.

Visual explanation

Draw three numbered p sites in a line. Write a 3×3 matrix beside them with β along the immediately adjacent off-diagonal entries and zeros in the two far corners. Draw a determinant bracket around the matrix and a downward arrow to its three roots. A separate arrow from each root to a coefficient vector stresses that energies and orbitals are separate outputs.

Real-world analogy

A set of coupled pendulums can oscillate in only certain collective patterns and frequencies. The coupling matrix records which pendulums communicate, and its determinant condition selects allowed modes. Hückel orbitals similarly arise from a coupling matrix, although they are quantum wavefunctions and their energies are not mechanical frequencies.

Real-world example

When comparing allyl cation, radical and anion in a simple classroom model, the same three-site connectivity gives the same three energy levels. Adding electrons changes which orbitals are occupied and therefore the predicted charge and reactivity patterns. A full calculation would let the molecule's geometry and effective Hamiltonian relax with charge.

Why?

Why must det(H−EI)=0? If the determinant were nonzero, the matrix would have an inverse and the only solution to (H−EI)c=0 would be c=0. A zero wavefunction is not a molecular orbital. A vanishing determinant allows a nonzero coefficient vector, selecting an allowed energy in the finite basis.

Common misconception

The number of determinant roots equals the number of p basis functions, not the number of pi electrons. Another mistake is thinking a zero H 13 means atoms 1 and 3 cannot share electron density. They can be connected indirectly through site 2, and the resulting MOs can extend across all three sites.

Worked example

For a three-site chain, H−EI = [[a,β,0],[β,a,β],[0,β,a]] with a=α−E. Expanding along the first row gives det = a(a²−β²)−β(βa) = a(a²−2β²). The roots a=0 and a=±√2β give E=α and E=α±√2β. For β<0, the plus-β expression lies lowest. The middle root's eigenvector will have a node at the central site, a result found by solving the coefficient equations.

Quick check

1. What is the dimension of the Hückel matrix for butadiene's four carbon p orbitals? Answer: Four by four, regardless of its four pi electrons. 2. Why are H 13 and H 31 zero in the simple linear allyl model? Answer: Sites 1 and 3 are not directly bonded neighbours in the chosen p-orbital graph, and the simplest model neglects non-nearest coupling.

Exam focus

Show the graph-to-matrix step, keep H and S conventions explicit and state β's sign before sorting. Distinguish determinant roots from normalised eigenvectors. For charged systems, separate the fixed model's orbital energies from electron filling and possible geometry relaxation.

Advanced insight

The determinant condition is the finite-basis form of the variational eigenvalue problem. When overlap is not neglected, det(H−ES)=0 is a generalised eigenvalue equation. Retaining S can change energies and coefficient normalisation, which is one route from Hückel's simple classroom model toward more realistic quantum-chemical calculations.

Summary

Hückel secular equations arise from (H−EI)c=0 for a simplified orthonormal p basis. Connectivity places β in nearest-neighbour positions, and determinant zeros give permitted energies. Eigenvectors then supply orbital coefficients and phase patterns. Matrix size counts basis functions, while electron count determines occupancy after the matrix is solved.

Practice questions

1. A four-site chain has bonds 1–2, 2–3 and 3–4. Which off-diagonal H elements are β in the simple model? Answer: H 12=H 21, H 23=H 32 and H 34=H 43 equal β. Other non-neighbour off-diagonal elements are zero. 2. If det(H−EI) is nonzero at a proposed E, can a nonzero MO coefficient vector solve the homogeneous equations? Answer: No. The matrix is invertible, so only the zero vector solves it; E is not an allowed orbital energy in that finite model. 3. Why can the allyl terminal sites share MO amplitude even when H 13=0? Answer: Both connect through site 2. Diagonalising the full coupled matrix produces delocalised eigenvectors with amplitude on both ends despite no direct nearest-neighbour coupling.