Polyatomic MOs by Symmetry

The SALC/symmetry-matching method is not a historical curiosity; it is the everyday working vocabulary of structural inorganic and organometallic chemistry. Ligand-field and MO treatments of octahedral, tetrahedral, and square-planar transition-metal complexes all begin by reducing the representation of the six, four, or four σ-donor ligand orbitals into $a_{1g}, t_{1u}, e_g$ (Oh) or analogous labels, then matching them against metal $s, p, d$ orbitals — this is the direct three-ligand/two-ligand logic of this article scaled to six or four ligands, and it underlies the entire crystal-field/ligand-field splitting pattern ($\Delta_o$, $\Delta_t$) used to explain color, magnetism, and reactivity of coordination compounds. The same SALC bookkeeping determines which metal d orbitals are available for π-backbonding to CO or phosphine ligands in catalytically relevant organometallics — the symmetry-matching rule (only same-label orbitals mix) is exactly why $t_{2g}$ metal d orbitals, not $e_g$, backbond into CO π* orbitals in octahedral carbonyl complexes. In spectroscopy, symmetry-adapted MO diagrams predict and assign photoelectron spectra (as illustrated for H₂O and NH₃ above), UV-visible band assignments in symmetric chromophores, and vibrational selection rules via the same character-table machinery (a vibration is IR-active only if it transforms as x, y, or z — literally the same representation-reduction procedure used here for orbitals). In materials chemistry, SALC-based reasoning explains why planar BX₃ Lewis acids (empty a2″-type acceptor orbital) are strong halide/amine acceptors while pyramidal analogues are not, a principle exploited in frustrated Lewis pair catalysis and in borane/carborane cluster electron-counting. In biological/bioinorganic contexts, the symmetry classification of ligand donor-orbital sets around metal centers (e.g., heme's four in-plane nitrogen donors) underlies qualitative MO pictures used to rationalize spin state and reactivity in metalloenzymes. And pedagogically, this exact SALC construction is the direct algorithmic ancestor of the Walsh Diagrams used throughout structural inorganic chemistry to predict bond angles, and of the Delocalized Pi Systems treatment of aromatic and cluster π-frameworks, where the ligand set is not σ-donors but a ring or cage of p-orbitals reduced by exactly the same group-theoretic machine.

Ask the tutor: How would the SALC construction change if one hydrogen in AH₂ were replaced by deuterium, and does the point group or the orbital symmetry labeling change at all?

Begin with the diatomic intuition. In a heteronuclear diatomic AB, an MO forms from two atomic orbitals of the same local symmetry (same "m" about the internuclear axis, same parity considerations) and similar energy: $\sigma = c_1\phi_A + c_2\phi_B$, bonding and antibonding partners appear, and non-matching orbitals (wrong symmetry, wrong energy) stay nonbonding. Everything in this article is that same idea, scaled up — except now "B" is replaced by a set of two or three hydrogen 1s orbitals, and a single hydrogen 1s orbital, on its own, usually has the wrong symmetry to bond with a central s or p orbital. Only special combinations of the ligand orbitals — SALCs — have the right symmetry. Take AH₂. If you overlap the central pz orbital with hydrogen 1s(a) alone, you get bonding overlap; but by symmetry the molecule also has hydrogen 1s(b), and the true MOs must respect the molecule's actual symmetry operations (the C₂ axis, mirror planes). The physically correct ligand basis functions are not "H_a alone" or "H_b alone" but symmetric and antisymmetric combinations:

The first is symmetric under every operation that swaps Ha and Hb — it transforms like the totally symmetric representation (a1 in C₂ᵥ). The second changes sign under that same swap — it transforms as b2 (or b1, depending on axis convention). These are the only two ligand-based orbitals that exist as legitimate "pieces" a central orbital can bond to; a bare, unsymmetrized H_a 1s orbital is not an eigenfunction of the molecular symmetry operations and cannot appear alone in an MO of definite symmetry. Once you have the SALCs, matching against central-atom orbitals is bookkeeping: central s is always totally symmetric (a1 in any point group), so it bonds with the a1 SALC; central pz (lying along the C₂ axis, pointing between or along the H–H direction depending on geometry) is also a1 in the bent case; central py, perpendicular to the molecular plane but in-plane with respect to the HAH-bisecting geometry, transforms as b2 and bonds with the antisymmetric SALC; central px, perpendicular to the molecular plane, transforms as b1 or a2 and typically finds no SALC partner at all in a two-ligand system, becoming purely nonbonding. This last point is the seed of the entire Walsh Diagrams story: as you bend AH₂ from 180° to 90°, the central s and pz orbitals mix continuously (both being a1), sliding the a1-symmetric MOs down in energy, while the b2/b1 (in-plane vs out-of-plane p) orbitals respond oppositely — producing the characteristic Walsh correlation lines whose crossing pattern predicts bond angles as a function of electron count.

For AH₃, the same logic applies with three ligand orbitals instead of two. In planar D3h (like BH₃) or pyramidal C3v (like NH₃), the reducible representation for three H 1s orbitals reduces to $a_1' \oplus e'$ (D3h) or $a_1 \oplus e$ (C3v) — one totally symmetric SALC (all three H orbitals in phase, like a "breathing" combination) and a degenerate pair of SALCs with one node each. The central s orbital is always a1(′); central pz is a2″ in D3h (nonbonding by symmetry — no ligand SALC matches it in a planar system with three ligands, since pz is antisymmetric to the horizontal mirror while all ligand SALCs, built from in-plane s orbitals, are symmetric to it) but becomes a1 in C3v pyramidal geometry, where it does find a partner. The central (px, py) pair is degenerate and transforms as e′ (D3h) or e (C3v), exactly matching the degenerate ligand SALC pair. This symmetry-forced re-partnering of pz between "nonbonding" (planar) and "a1-bonding" (pyramidal) is precisely the electronic driving force behind NH₃'s pyramidalization and BH₃'s planarity, developed quantitatively in Walsh-diagram treatments.

The reducible representation for a set of $n$ ligand σ-orbitals (one per ligand, pointing at the central atom) is built operation-by-operation:

where: $N_{\text{unmoved}}(R)$ is the number of ligand positions left in place (not permuted) by symmetry operation $R$, and $\chi_\sigma(R) = +1$ for the identity/rotations acting on an s-like ligand orbital (since σ-orbitals are unchanged in sign under proper rotation about their own axis); reflections and improper rotations also contribute $+1$ per unmoved σ-orbital because a ligand σ-orbital pointing at the center is symmetric to reflection in a plane containing it. The reduction formula recovers the SALC symmetry content:

where: $h$ is the order of the group, $g(R)$ the number of operations in class $R$, $\chi(R)$ the reducible character found above, $\chi_i(R)$ the character of irreducible representation $i$ in the character table, and $n_i$ the number of times that irreducible representation appears among the SALCs. The explicit projection operator that builds the actual SALC wavefunction (not just its symmetry label) is

where: $\ell_i$ is the dimension of irreducible representation $i$ (1 for A/B, 2 for E, 3 for T), $\phi_1$ is one chosen "generator" ligand orbital, $\hat R\phi_1$ is the orbital obtained by applying operation $R$ to $\phi_1$ (i.e., which ligand position it moves to), and the sum runs over all operations in the group. Normalizing the result gives the SALC used in the MO diagram. Finally, once SALCs and central AOs are sorted by label, the number of bonding/antibonding pairs of a given symmetry follows directly:

where: any excess central AOs or SALCs of a label with no partner remain strictly nonbonding at (to leading order) their original atomic energy.

Example 1 (foundational): full symmetry-orbital diagram for linear vs bent AH₂, applied to BeH₂ and H₂O. For linear BeH₂ (D∞h, but treated practically using the D2h/C2v correlation, or directly with D∞h labels $\sigma_g^+,\sigma_u^+,\pi_u$), the two H SALCs are $\sigma_g^+ = \tfrac{1}{\sqrt2}(\phi_a+\phi_b)$ and $\sigma_u^+ = \tfrac{1}{\sqrt2}(\phi_a-\phi_b)$. Central Be 2s is $\sigma_g^+$; central 2pz (along the molecular axis) is $\sigma_u^+$; central 2px, 2py (perpendicular to the axis) form a degenerate $\pi_u$ pair with no ligand SALC partner (H 1s orbitals cannot form a π-symmetric combination from just two atoms on an axis), so they stay nonbonding, exactly at the Be 2p atomic energy. Filling: Be contributes 2 valence electrons, each H contributes 1, total 4 valence electrons. These fill $\sigma_g^+$ (bonding, 2e) and $\sigma_u^+$ (bonding, 2e) — both fully bonding MOs, zero electrons in the nonbonding $\pi_u$ pair, consistent with BeH₂ being a stable, linear, closed-shell AH₂ molecule with two equivalent Be–H bonds (bond order 1 each, from the standard symmetric/antisymmetric bonding-pair combination reproducing two localized bonds). Now bend the molecule toward C2v, as required for H₂O. The SALCs relabel as $a_1$ (was $\sigma_g^+$) and $b_2$ (was $\sigma_u^+$); central $s, p_z$ are both now $a_1$; central $p_y$ is $b_2$; central $p_x$ (out of plane) is $b_1$, nonbonding. Oxygen contributes 6 valence electrons, each H contributes 1, total 8. Filling in order of increasing energy (as verified by photoelectron spectroscopy of H₂O): $2a_1$ (bonding, mixing O 2s character, 2e), $1b_2$ (bonding, O 2py + SALC b2, 2e), $3a_1$ (largely nonbonding, O 2pz/2s mixture pointing away from H's — the "sp-like" lone pair, 2e), $1b_1$ (purely nonbonding O 2px, perpendicular to the molecular plane, 2e). All 8 electrons placed, zero antibonding population — this exactly reproduces the two O–H bonding pairs plus two lone pairs (one in an sp-mixed $a_1$ orbital, one in a pure $p$-type $b_1$ orbital) that every general-chemistry Lewis structure of water asserts, but now derived, not assumed, and with the correct inequivalence of the two lone pairs (one is lower in energy and more s-like, the other purely $p$ and higher in energy) — a fact photoelectron spectroscopy confirms via two distinct ionization energies (12.6 eV and 14.7 eV bands) rather than one.

Example 2 (applied): predicting the HOMO and first ionization/UV band character of NH₃ using C3v SALCs. For NH₃ (C3v), the H₃ SALC reduction (paralleling the D3h derivation but with $\sigma_h$ removed and the $C_2$'s removed, replaced only by $E, 2C_3, 3\sigma_v$) gives $\Gamma_{\text{SALC}} = a_1 \oplus e$ (same labels survive, just without the prime marks, since there is no horizontal mirror to distinguish ′/″). Central N 2s is $a_1$; central N 2pz is now also $a_1$ (no $\sigma_h$ to forbid it, as established in Step 6 of the D3h derivation above); central N (2px, 2py) form the $e$ pair, matching the ligand $e$ SALC exactly. Assemble the diagram: the $a_1$ block is a three-orbital problem (N 2s, N 2pz, ligand $a_1$ SALC) giving three MOs — one strongly bonding ($1a_1$, mostly N 2s + in-phase H's), one intermediate ($2a_1$), and one antibonding ($3a_1^*$); the $e$ block is a four-orbital, doubly-degenerate problem (N 2px/2py + ligand e SALC pair) giving a bonding $e$ pair (1e, doubly degenerate, 4 total electrons capacity) and an antibonding $e^*$ pair. Nitrogen contributes 5 valence electrons, three H's contribute 3, total 8 valence electrons. Filling in increasing energy: $1a_1$ (2e, strongly bonding), $1e$ (4e, bonding, doubly degenerate — this is the pair of N–H bonding MOs), $2a_1$ (2e). This last, singly-filled-orbital-type $2a_1$ MO is the HOMO: it is the symmetry orbital built mostly from N 2pz/2s hybrid character mixed with a small amount of the totally symmetric ligand SALC, pointing away from the three hydrogens — precisely NH₃'s stereochemically active lone pair, responsible for its Lewis basicity (coordinating to BH₃'s empty $a_2''$-turned-$a_1$ acceptor orbital, protonating to give NH₄⁺, and inverting through a planar D3h transition state in which that same $2a_1$ orbital rises toward the nonbonding-forced $a_2''$ energy of the flat geometry — the inversion barrier of ammonia, about 24 kJ/mol, is essentially the a1↔a2″ symmetry-crossing energy cost read directly off the correlation diagram).

Example 3 (exam/competition-level): SALC reduction and MO count for a T-shaped AH2 fragment approximation and cross-check via the C2v subgroup, applied to counting nonbonding density in H2S vs H2O. Consider the general question: for bent AH₂ with bond angle $\theta \neq 180°$ and $\neq 90°$ (say, a nonspecial angle like $104.5°$, the real H₂O angle, or $92.1°$, the real H₂S angle), does the symmetry classification of the MOs (a1, a1, b2, b1) depend on the exact angle? Show explicitly that it does not, and explain why H₂S nonetheless shows a smaller HOMO–LUMO-type splitting and different photoelectron spectrum than H₂O despite identical symmetry labels. Work through the argument: point group C2v is defined by the presence of the $E, C_2, \sigma_v, \sigma_v'$ operations, which exist for any angle strictly between 0° and 180° (excluding the linear and coincident limits) — the reduction of $\Gamma_{\text{SALC}} = A_1 \oplus B_2$ performed in the first derivation used only permutation counts under each operation, never the numerical value of $\theta$. Therefore the symmetry labeling — $2a_1$ bonding, $1b_2$ bonding, $3a_1$ nonbonding-ish lone pair, $1b_1$ nonbonding lone pair — is angle-independent as long as the molecule remains strictly C2v-bent; changing $\theta$ from $104.5°$ (H₂O) to $92.1°$ (H₂S) or to an artificial asymmetric-feeling value like $111.3°$ never changes which label attaches to which orbital, only the coefficients and energies within each symmetry block. What does depend on the numbers is the mixing coefficient inside the $a_1$ block. Central atom $ns$/$np$ energy gap widens down a group (S 3s–3p gap is larger than O 2s–2p gap, a direct consequence of the same s–p contraction/orbital-energy argument used in s-p Mixing and Orbital Ordering), so S 3s mixes less efficiently into the bonding $a_1$ SALC combination, leaving the $3a_1$ "lone pair" orbital of H₂S far more purely 3p in character and higher in energy (less stabilized by s-character) than the analogous, more strongly s–p-hybridized $3a_1$ lone pair of H₂O. Consequently H₂S's bond angle sits closer to 90° (approaching the "pure p-orbital bonding, no hybridization" limit) while H₂O's larger s–p mixing pushes its angle further from 90° toward the tetrahedral-adjacent 104.5°. This is confirmed experimentally: H₂S's HOMO ionization energy (10.5 eV) is lower than H₂O's HOMO ionization energy (12.6 eV), and the bond angle correlates with the degree of s-p mixing available in the $a_1$ block — a numeric, testable prediction that emerges directly from matching symmetry labels to atomic orbital energetics, not from steric or VSEPR arguments.

The symmetry-based approach to polyatomic bonding was crystallized by chemists translating early-20th-century group theory (Wigner, Bethe) into ligand-field and molecular-orbital language during the mid-20th century, most visibly through A. D. Walsh, whose 1953 papers systematized correlation diagrams for AH₂ and AH₃ across bond-angle variation, and through the broader development of ligand-field/MO group-theoretic method by chemists such as F. Albert Cotton, whose textbook treatment made projection-operator SALC construction a standard part of inorganic pedagogy.