Allyl Pi Molecular Orbitals

Introduction

Adding one p site to ethene creates an allyl system with three pi molecular orbitals. Its middle level is especially revealing: the wavefunction has opposite phases at the two terminal carbons and a node at the centre, giving energy α in the simple model. The cation, radical and anion can share this orbital framework while differing in electron occupancy. The case links matrix eigenvectors to charge and spin distribution.

Core explanation

Number the three conjugated carbon sites 1–2–3. In simple Hückel theory, H has α on its diagonal, β between 1–2 and 2–3, and zero direct 1–3 coupling. The eigenvalues are α+√2β, α and α−√2β. With β<0, that order is low, middle and high. The normalised coefficient vectors in an orthonormal p basis can be chosen as (1,√2,1)/2, (1,0,−1)/√2 and (1,−√2,1)/2 respectively. Overall reversal of all signs in any vector has no physical effect.

The lowest MO has the same phase on all three sites and large amplitude at the centre. It is bonding across both adjacent connections. The highest has the centre opposite in phase to the terminals and is antibonding on both connections. The middle MO has coefficients +1/√2 and −1/√2 on the terminals but zero at site 2. Because it has no central amplitude, the nearest-neighbour Hückel coupling does not shift it away from α. It is called nonbonding in this simplified pi-energy sense, not because it contains no electron density anywhere.

The allyl cation has two pi electrons, filling only the lowest MO. The allyl radical has three pi electrons, with one electron in the middle nonbonding MO. The allyl anion has four pi electrons, with the middle MO doubly occupied. The three species are not identical chemically, but this fixed-parameter Hückel comparison isolates the effect of electron count. More realistic models allow geometry, electrostatics and effective parameters to change with charge.

The middle MO predicts a distinctive terminal distribution. Its probability density coefficients are c₁ ²=1/2, c₂ ²=0 and c₃ ²=1/2. For the radical's singly occupied middle orbital, an idealised one-electron spin density is shared between the two terminal carbons and absent from the centre in this orbital approximation. The cation and anion also have delocalised charge responses, but total atomic charge cannot be inferred from a single orbital alone; all occupied pi orbitals and the sigma framework matter.

For the lowest MO with two electrons, site pi populations are 2(1/4)=1/2 at each terminal and 2(1/2)=1 at the centre. Occupying the nonbonding MO adds one-half electron per terminal for each electron placed there. Thus in the simple pi ledger the allyl cation populations are (0.5,1,0.5), the radical (1,1,1) and the anion (1.5,1,1.5). These are pi-electron populations, not complete formal charges or all-electron densities.

The allyl framework demonstrates why terminal atoms can communicate without a direct H 13 matrix element. Both couple through site 2, producing eigenvectors extending over all three positions. The zero central coefficient of the middle MO comes from destructive interference of terminal amplitudes into the central site, not from a broken physical bond. The example is a bridge to longer chains and aromatic rings.

Step-by-step reasoning

Build the 3×3 matrix from the 1–2–3 graph and solve its secular determinant. Sort roots with β<0, then use coefficient equations and normalisation to obtain three orthogonal orbital patterns. Fill 2, 3 or 4 pi electrons for cation, radical or anion. Compute any requested site populations from all occupied orbitals, including occupancy factors.

Visual explanation

Draw three p sites in a row. For the lowest MO, shade all upper lobes the same phase and make the centre larger. For the middle MO, shade terminals oppositely and leave the centre blank as a node. For the highest, make the centre opposite to the matching terminals. Place two, three and four electron arrows on a shared three-level diagram for the three allyl charge states.

Real-world analogy

A three-station wave system can have a middle standing-wave pattern with opposite end motions and an unmoving centre. Allyl's nonbonding orbital has a comparable central node. The analogy explains interference but not charge, spin or the quantum rule that each spatial orbital holds at most two opposite-spin electrons.

Real-world example

Allyl radicals are resonance-stabilised and can participate in radical reactions at either terminal carbon. The simple Hückel singly occupied MO places equal amplitude on both terminals, supporting a qualitative picture of distributed spin reactivity. Actual selectivity also depends on substituents, solvent, sterics and transition-state energies.

Why?

Why does the middle energy equal α? Its coefficient at the central site is zero. Each terminal has no direct coupling to the other in the simplest matrix, and the opposite terminal amplitudes cancel their effects on the centre. The eigenvalue equations then contain no net β shift for that orbital.

Common misconception

Nonbonding does not mean the orbital is empty or irrelevant; it is singly occupied in allyl radical and doubly occupied in allyl anion. Also, the model's pi populations are not formal charges. A complete atomic charge includes sigma electrons, nuclei and a choice of population-analysis definition.

Worked example

Use the middle eigenvector (1,0,−1)/√2. Its norm is 1/2+0+1/2=1. Applying the matrix gives first-site output α/√2, middle output β(1/√2)+β(−1/√2)=0 and third-site output −α/√2. The result is α times the original vector, confirming E=α. For a singly occupied radical orbital, the probability weight is one-half at each terminal and zero at the centre.

Quick check

1. How many pi electrons occupy allyl anion in the simple three-site model? Answer: Four: two in the lowest bonding MO and two in the middle nonbonding MO. 2. Which site has zero coefficient in the middle allyl MO? Answer: The central carbon, site 2.

Exam focus

State β's sign, label the three roots and draw all coefficient phase patterns. Separate electron occupancy from orbital energy, and distinguish pi populations or spin density from complete molecular charge distribution.

Advanced insight

The allyl graph is bipartite with one more site on one sublattice than the other, and its simple adjacency matrix has a zero eigenvalue. This graph property underlies the middle nonbonding Hückel level. Similar topology-based reasoning can predict nonbonding orbitals in larger alternant hydrocarbon graphs, subject to the model's chemical limits.

Summary

The three-site allyl Hückel model yields levels α+√2β, α and α−√2β for β<0. Its middle orbital has opposite terminal phases and a central node. Cation, radical and anion fill the same idealised levels with two, three and four pi electrons, respectively, giving distinct terminal pi and spin populations in the simple model.

Practice questions

1. What is the pi population at site 2 in the simple allyl cation and allyl anion ledgers described here? Answer: One electron in both. The lowest doubly occupied orbital contributes one at site 2, and the middle orbital occupied only in the anion has zero centre coefficient. 2. Why can terminal sites share a delocalised MO despite H 13=0? Answer: Each terminal couples to the central site. Diagonalising the full three-site matrix creates collective eigenvectors extending to both terminals through that indirect connection. 3. If one electron occupies the allyl nonbonding MO, where is its idealised spin-density weight? Answer: One-half at each terminal and zero at the centre, from squared coefficients of (1,0,−1)/√2; real spin density can shift with chemical perturbations.