Projection Operators

Introduction

Sum-and-difference SALCs are easy to guess for two ligands, but phase patterns become difficult to invent for larger groups. Projection operators turn character-table information into actual functions. They average transformed copies of a trial orbital with weights chosen for one symmetry species. The result either has the desired symmetry, is zero because that species is absent from the trial function, or needs further separation within a multidimensional or repeated species.

Core explanation

For a finite group G of order h, a character projection onto the full component of irrep i can be written Pᵢ = (dᵢ/h) Σ R χᵢ(R) R̂. Here dᵢ is the irrep dimension, χᵢ its character and R̂ the operator that transforms a function under group operation R. The sum includes each operation, not merely one representative per class; when using a character table, class multiplicities must be respected. Applying Pᵢ to a trial function f gives the part of f belonging to irrep i, up to a coefficient and possible need for orthogonalisation or normalisation.

In the one-dimensional C₂v group, projection is especially transparent. Use the water convention E, C₂(z), σ(xz), σ(yz) and a hydrogen 1s trial orbital h₁. E and σ(yz) leave h₁ in place; C₂ and σ(xz) send it to h₂. The A₁ characters are all +1. Therefore P(A₁)h₁ = (1/4)(h₁+h₂+h₂+h₁) = (h₁+h₂)/2. This is the symmetric SALC apart from normalisation. For B₂ characters (1,−1,−1,1), projection gives (h₁−h₂)/2. The method has produced the expected pair without guessing coefficient signs.

Applying an A₂ projector to the same two-H basis returns zero because A₂ is absent from its decomposition. This is not a failure of the method. It tells us the chosen trial function has no A₂ component within that basis. Starting from an oxygen p or another function could produce a different answer, because projection works on the function supplied to it. A zero result cannot prove that the entire molecule lacks the species.

For multidimensional irreps, the character projector selects the entire irrep subspace, not necessarily a complete set of independent partner functions. If an E representation occurs, one trial function may project to one nonzero vector in the E space; another trial function or a more detailed matrix-element projector can produce a partner. The general matrix-element projector uses individual irrep matrix entries rather than only the trace. Linear independence, orthogonality and normalisation must be checked afterward.

Projection depends on a correct operation action. A local p orbital may acquire a minus sign when reflected even if its atom stays in place. An oriented displacement vector similarly transforms with coordinate rotation. A student who merely permutes atomic labels while ignoring orbital phase may obtain the wrong projected combination. The operator R̂ acts on the whole function, not only on its centre.

The prefactor dᵢ/h is chosen so the projection has the appropriate idempotent behaviour on symmetry subspaces: applying the same ideal projector twice gives the same component. However, the raw function Pᵢf need not be normalised. If ligand orbitals overlap, its norm contains overlap terms. Quantitative molecular-orbital calculations therefore orthonormalise or solve a generalised eigenvalue problem rather than treating every formal SALC as already unit length.

Step-by-step reasoning

Find the group order and the irrep row. Apply every group operation to the chosen trial function, recording exchanged orbitals and phase changes. Multiply each transformed function by the conjugated character, sum and scale by dᵢ/h. Simplify, check that the result is nonzero, then normalise and find additional independent partners if the species is multidimensional.

Visual explanation

Draw h₁ on one H of water. Four arrows labelled E, C₂, σ(xz) and σ(yz) point to h₁, h₂, h₂ and h₁. On an A₁ line put four plus weights, yielding the sum; on a B₂ line put +, −, −, +, yielding the difference. This visual makes projection look like controlled group averaging rather than a mysterious formula.

Real-world analogy

To isolate one pattern from a mixed signal, one can copy the signal under several transformations, reverse selected copies and average. Desired components reinforce while incompatible components cancel. Projection operators do exactly this for orbital symmetry. The analogy leaves out the need to transform wavefunction phase correctly and to normalise the resulting function.

Real-world example

For a coordination complex with several equivalent sigma donors, projection can generate ligand group orbitals matching metal s, p and d symmetry species. The chemist then solves each interaction block separately. This is valuable when visual guessing becomes unreliable, especially for six or more ligands and multidimensional irreps.

Why?

Why does the A₂ projection of a hydrogen 1s trial orbital in water vanish? The two-H representation decomposes as A₁ + B₂, with no A₂ component. Weighted transformed copies therefore cancel exactly. The zero is an algebraic expression of a symmetry absence in that selected basis, not a statement that A₂ functions cannot exist elsewhere.

Common misconception

The character projector does not necessarily return a normalised molecular orbital. It returns a symmetry component of the trial function. Another error is to sum once per character-table column without multiplying through all operations in each class; that changes the group average and can spoil the projection.

Worked example

Use B₂ characters (1,−1,−1,1) in C₂v. Since E h₁ = h₁, C₂ h₁ = h₂, σ(xz)h₁ = h₂ and σ(yz)h₁ = h₁, the projected function is P(B₂)h₁ = [h₁ − h₂ − h₂ + h₁]/4 = (h₁−h₂)/2. Under C₂ this changes sign, and under σ(yz) it remains unchanged, matching the B₂ row in the stated coordinate convention.

Quick check

1. Does a zero projection prove an irrep is absent from every function in the molecule? Answer: No. It shows the chosen trial function has no component of that irrep. 2. Why must a projected SALC often be normalised afterward? Answer: Projection selects symmetry but does not generally scale the result to unit norm, especially when original orbitals overlap.

Exam focus

Write the projection formula with the irrep dimension, group order and complex conjugation. Show the action of each operation on the trial function before inserting signs. Verify the final SALC's transformation behaviour and distinguish a zero projection from an invalid method.

Advanced insight

The character projector selects all copies of one irrep if it occurs more than once in a reducible space. More detailed row or matrix-element projectors can distinguish partner functions within multidimensional irreps. Computational implementations combine these ideas with linear-algebra orthogonalisation so symmetry-adapted bases remain independent and numerically stable.

Summary

Projection operators construct symmetry-defined function components by averaging a trial function over the point group with character weights. In water, A₁ and B₂ projection of one H 1s orbital yields the expected symmetric and antisymmetric SALCs. The method requires correct phase transformations, all operation multiplicities, and later normalisation or partner construction when needed.

Practice questions

1. Applying P(A₁) to water's h₁ gives (h₁+h₂)/2. Is this automatically a unit-normalised orbital? Answer: No. Its norm depends on the h₁–h₂ overlap; the coefficient 1/2 comes from projection and is not the final normalisation factor. 2. What does P(A₂)h₁ = 0 tell you for the two-H basis of water? Answer: The chosen H basis contains no A₂ component. It does not rule out A₂-symmetry functions from other orbital or displacement bases. 3. Why can a character projection be insufficient to supply both partner functions of an E irrep? Answer: The character projector selects the E subspace as a whole. One projected trial function may give only one vector; another independent trial or a matrix-element projector is needed for a full basis.