Symmetry-Adapted Linear Combinations

Introduction

When several ligands surround a central atom, testing every ligand orbital against every central orbital hides the symmetry of the whole arrangement. A symmetry-adapted linear combination, or SALC, replaces the local ligand basis with combinations that transform according to irreducible symmetry species. The chemistry remains the same, but compatible bonding blocks become visible. SALCs are central to molecular-orbital diagrams for water, ammonia and coordination complexes.

Core explanation

Start with ligand functions φ₁, φ₂, … that are exchanged among themselves by the point-group operations. Their span is a representation of the group. Compute its character for each class and reduce it to irreducible species. If the reduction gives A₁ + E, for example, there must be one A₁ combination and a pair spanning E. The reduction tells how many combinations of each type exist, but it does not specify their coefficients. Coefficients can be found by inspecting exchange behaviour, by orthogonalisation or through projection operators.

For two equivalent orbitals, the simplest SALCs are φ₊ = N₊(φ₁ + φ₂) and φ₋ = N₋(φ₁ − φ₂). The normalisation constants are not always 1/√2 because local orbitals may overlap. If each orbital is individually normalised and their real overlap is S = ⟨φ₁ φ₂⟩, then N₊ = [2(1+S)]⁻¹/² and N₋ = [2(1−S)]⁻¹/². Neglecting overlap gives the familiar 1/√2 values. The sum remains unchanged under ligand exchange, whereas the difference changes sign.

For three equivalent orbitals around a C₃ axis, the totally symmetric combination is proportional to φ₁ + φ₂ + φ₃. Two independent combinations whose coefficients sum to zero span a two-dimensional E space in C₃v; one convenient unnormalised pair is 2φ₁ − φ₂ − φ₃ and φ₂ − φ₃. A C₃ operation mixes the pair, so the individual chosen functions are basis choices within E rather than separate symmetry species. Orthogonal and normalised forms are selected for matrix calculations.

Central-atom orbitals can interact through a symmetry-preserving Hamiltonian only with ligand SALCs of the same symmetry species. In C₃v ammonia, the nitrogen 2s and 2p z functions are A₁ and can interact with the A₁ hydrogen SALC. Nitrogen 2p x and 2p y form E and can interact with the E hydrogen SALCs. The exact number and energies of bonding, antibonding or largely nonbonding orbitals require solving the Hamiltonian within each block. Symmetry is a first filter, not a complete bonding calculation.

SALCs also prevent double counting. Three ligand orbitals generate exactly three independent combinations in total. A₁ + E has dimensions 1 + 2 = 3. If a diagram shows two A₁ combinations plus an E pair from only three ligand orbitals, it has invented a function. The basis dimension and orthogonality checks are therefore as important as visual phase sketches.

Phase is crucial. An orbital combination with plus signs on some ligands and minus signs on others can have zero overlap with a central orbital even when nuclei are nearby, because positive and negative contributions cancel. Group theory predicts the exact cancellation when symmetries differ. A local-orbital drawing should show lobe phases clearly; treating orbital signs as charges leads to an incorrect physical picture.

SALCs can be formed for sigma-donor orbitals, ligand pi orbitals or atomic displacement vectors. The chosen basis determines the reducible representation and the SALCs obtained. Adding a new ligand orbital type may create a previously absent symmetry match, so statements about nonbonding central orbitals must specify the ligand basis being analysed.

Step-by-step reasoning

Choose one consistent ligand orbital per relevant direction and verify the set is closed under every operation. Count characters, reduce to irreps and check dimensions. Construct functions with the required transformation behaviour, normalise them with overlap included if quantitative accuracy matters, and match each SALC to central-atom orbitals of the same symmetry.

Visual explanation

Draw three ligand circles around a central atom. First mark all ligand lobes with the same phase for the A₁ sum. Then draw two independent phase patterns with coefficients summing to zero for the E pair. Place the central s and z-directed p orbital beside A₁ and the x/y p pair beside E. Unmatched patterns are separated into different blocks.

Real-world analogy

Several microphones around a stage can be combined to isolate a common signal or directional differences. The all-plus signal is analogous to a totally symmetric SALC; difference channels capture other patterns. The analogy illustrates linear combinations and cancellation, but molecular orbitals are wavefunctions whose phases determine quantum interference.

Real-world example

In an octahedral complex, six ligand sigma-donor orbitals can be reorganised into group orbitals with symmetry matching metal s, p and certain d functions. This helps distinguish sigma-bonding interactions from metal orbitals left without a sigma match. Ligand pi orbitals can change the picture, so a complete metal–ligand diagram needs a clearly declared basis.

Why?

Why construct SALCs before solving orbital energies? The transformation creates a basis in which symmetry-forbidden Hamiltonian couplings vanish. The large matrix splits into smaller blocks, reducing algebra and making each resulting molecular orbital's origin understandable. Numerical eigenvalues still depend on energy and overlap parameters, but the block pattern follows exactly from symmetry.

Common misconception

SALCs are not new physical ligands, and different mathematical choices within a degenerate E pair do not create different chemistry. They are alternative bases spanning the same ligand-orbital space. It is also wrong to assume every same-symmetry orbital pair interacts strongly; small overlap or large energy separation may limit mixing.

Worked example

Let two normalised ligand orbitals have overlap S = 0.20. The symmetric combination has N₊ = [2(1.20)]⁻¹/² ≈ 0.645, so φ₊ ≈ 0.645(φ₁+φ₂). The antisymmetric combination has N₋ = [2(0.80)]⁻¹/² ≈ 0.791, so φ₋ ≈ 0.791(φ₁−φ₂). Under an operation exchanging the ligands, φ₊ remains itself and φ₋ changes sign. Both are normalised despite unequal coefficients.

Quick check

1. How many independent SALCs can three independent ligand orbitals produce? Answer: Three in total, even if they are organised as one A₁ function and one two-component E pair. 2. Does the normalisation factor of a two-orbital sum always equal 1/√2? Answer: No. Nonzero overlap changes the norm, so the factor depends on the overlap integral S.

Exam focus

Declare the ligand basis and show its irreducible decomposition. Check the dimensions, mark orbital phases and distinguish symmetry matching from energetic strength. If quantitative normalisation is asked, include local-orbital overlap rather than assuming orthogonality without justification.

Advanced insight

SALCs are a unitary or overlap-aware change of basis, so they do not alter the physical Hilbert space. Their value is structural: symmetry-adapted Hamiltonian and overlap matrices become block diagonal. In practical quantum-chemical programs, this can reduce matrix work and help assign calculated states even when basis functions are numerous.

Summary

SALCs reorganise equivalent ligand functions into combinations of definite symmetry. Character reduction predicts how many of each type exist; coefficients come from transformation behaviour or projection. Same-symmetry central and ligand functions may mix, with strength determined by energies and overlap. Normalisation and basis-dimension checks keep the construction mathematically sound.

Practice questions

1. Why does the ligand combination φ₁ − φ₂ change sign when a symmetry operation exchanges φ₁ and φ₂? Answer: Exchange gives φ₂ − φ₁ = −(φ₁ − φ₂), so it has antisymmetric exchange behaviour. 2. For three C₃v-equivalent ligand orbitals, why does A₁ + E account for the complete basis? Answer: A₁ has dimension one and E has dimension two, totaling three independent functions, equal to the three original orbitals. 3. If a central p orbital and a ligand SALC share a symmetry species but are far apart in energy, what can be concluded? Answer: Their interaction is permitted by symmetry, but it may be weak because energy separation limits mixing. A numerical or more detailed qualitative orbital analysis is still needed.