Molecular Orbitals of Polyatomic Molecules

SALC-based reasoning is the backbone of every modern qualitative and semi-quantitative electronic structure discussion, and it survives essentially unchanged inside industrial-strength quantum chemistry codes. Density functional theory and Hartree–Fock packages (Gaussian, ORCA, Q-Chem, VASP) all exploit point-group symmetry internally: symmetry-blocking the Fock/Kohn–Sham matrix is not a pedagogical simplification but a genuine computational speedup, since it means the software only ever diagonalizes the small per-irreducible-representation blocks instead of one enormous matrix, exactly mirroring the hand-derivation shown above for NH₃. In inorganic and organometallic chemistry, SALCs of ligand orbitals (built the same way as the hydrogen SALCs here) are the entire foundation of ligand field theory and crystal field / MO hybrid pictures of transition-metal complexes — the $e_g$/$t_{2g}$ splitting of d orbitals in an octahedral complex is derived by projecting six ligand σ-donor SALCs onto the metal d-orbital symmetry species, precisely the same machinery used above for ammonia's nitrogen orbitals. This directly sets up the qualitative shape arguments generalized in Walsh Diagrams and Molecular Shape, where correlating MO energies between different molecular geometries (bent vs. linear AH₂, pyramidal vs. planar AH₃) explains why molecules adopt the shapes they do, purely by tracking how SALC symmetries and their orbital energies shift as the geometry is distorted. In organic chemistry and materials science, the π-SALC treatment illustrated for allyl and cyclopropenyl is the direct ancestor of Huckel Theory: Assumptions and Framework, which extends the same secular-determinant logic to benzene, polyenes, fullerenes, and graphene nanoribbons; symmetry blocking is exactly why benzene's six π MOs split into $a_{2u}, e_{1g}, e_{2u}$ (bonding, degenerate bonding, degenerate antibonding) instead of requiring a full $6\times6$ diagonalization by hand. In spectroscopy, selection rules for infrared and Raman activity, and for electronic (UV-vis) transitions, are literally SALC statements: a vibrational mode or electronic transition is allowed only if the relevant transition-dipole integral connects matching or compatible symmetry species, which is why symmetry-adapted normal coordinate analysis (itself built with the identical projection-operator method) is standard in vibrational spectroscopy software and materials characterization pipelines. Even modern topological and photonic crystal band-structure analysis borrows this exact character-table reduction machinery to classify allowed mode symmetries at high-symmetry points in the Brillouin zone — the same reduction formula used above for NH₃'s three hydrogens, applied instead to a crystal's point-group little group.

Ask the tutor: Can you walk me through constructing the SALCs for benzene's six π orbitals in D₆ₕ symmetry and show how they map onto the a₂u, e₁g, e₂u energy levels?

Picture a molecule sitting at the center of its own point group, surrounded by peripheral atoms whose valence orbitals point inward, outward, or sideways. In a diatomic, "symmetry" reduces to a single axis and the labels σ, π, δ. In a polyatomic with three or more atoms arranged non-linearly, the molecule usually has more symmetry operations — rotations, reflections, an inversion center, or none of these, depending on shape — and every one of those operations must leave the Hamiltonian invariant. A deep consequence follows immediately: the true molecular orbitals, being eigenfunctions of that Hamiltonian, must also be eigenfunctions (or partners in a degenerate representation) of every symmetry operation in the group. An orbital that is not symmetry-adapted simply cannot be a true eigenstate on its own; it must mix with its symmetry partners until the combination is an eigenstate of the full symmetry group. Consider ammonia, point group $C_{3v}$. The nitrogen contributes a 2s and three 2p orbitals. The three hydrogens each contribute one 1s orbital, but critically, no single hydrogen 1s orbital by itself has $C_{3v}$ symmetry — rotating the molecule by 120° sends H₁'s orbital onto H₂'s position, which is a different function. Only specific combinations of all three 1s orbitals are symmetry-adapted: one totally symmetric combination (all three in phase, transforming as $a_1$) and a degenerate pair of combinations (transforming as the two-dimensional representation $e$). These three SALCs, not the three raw 1s orbitals, are what mix with nitrogen's atomic orbitals. Nitrogen's 2s and 2p$_z$ orbitals are themselves already $a_1$-symmetric (they are unchanged by every operation in $C_{3v}$ up to sign matching the SALC), so they combine with the $a_1$ hydrogen SALC. Nitrogen's 2p$_x$, 2p$_y$ pair transforms as $e$, so it combines with the $e$ hydrogen SALC. The $a_1$ manifold and the $e$ manifold never mix with each other — not because their energies happen to be far apart, but because a matrix element between orbitals of different symmetry species is identically zero by the Sigma and Pi Molecular Orbitals and Symmetry Labels logic extended to any point group: the integral of a totally-symmetric operator (the Hamiltonian) sandwiched between functions of different irreducible representations integrates to zero over the full symmetry group, by the orthogonality theorem for basis functions of different representations. The figure below shows this factorization graphically for NH₃: three peripheral 1s orbitals resolve into an $a_1$ SALC (a symmetric "breathing" combination) and a doubly-degenerate $e$ pair, each of which then combines only with the matching-symmetry nitrogen orbital.

The second key idea is how the projection operator manufactures a SALC without any guesswork. Start from any single peripheral orbital — say H₁'s 1s function, $\phi_1$ — and apply every symmetry operation of the group to it in turn, weighting each result by the character of the irreducible representation you want. Summing these weighted, rotated/reflected copies produces (after normalization) precisely the SALC of that symmetry. This is not an ansatz to be checked afterward; it is guaranteed to work by the representation theory underlying the The LCAO Approximation framework, because the projection operator is constructed to annihilate every symmetry component except the one you asked for. Once the SALCs are in hand, the secular determinant that would have been $(n\times n)$ for $n$ atomic orbitals splits into blocks — one per irreducible representation present in the basis, each block of size equal to how many orbitals of that symmetry exist. A block of size 1 is solved trivially (its single SALC/AO pair gives a bonding and antibonding MO directly, or, if only one orbital of that symmetry exists at all, a strictly nonbonding orbital sitting at that atom's own orbital energy). A block of size 2 gives the familiar $2\times2$ quadratic. This factorization is the single biggest practical advantage symmetry buys you, and it is why qualitative MO diagrams for water, ammonia, methane, or benzene can be drawn by hand.

The central equations of the SALC method are the reduction formula, the projection operator, and the symmetry-blocked secular equation.

Reducing the basis representation. Any set of peripheral atomic orbitals forms a reducible representation $\Gamma$ whose character under operation $R$ is easiest computed by counting unmoved atoms and multiplying by the per-orbital character contribution:

where: for an s orbital the per-atom contribution is always $+1$; for a p orbital lying along a symmetry axis unmoved by $R$, the contribution equals the character of that vector component under $R$ (e.g. $+1$ for identity, $-1$ under a reflection that flips it, and $2\cos\theta - 1$ generalizations for rotations acting on p-orbital sets, though for simple SALC-of-1s-orbitals problems the atom itself must map to itself for any contribution at all — atoms that move under $R$ contribute zero).

Reduction formula. The number of times irreducible representation $i$ appears in $\Gamma$ is

where: $h$ is the order of the point group, the sum runs over each class of operations, $g(R)$ is the number of operations in that class, $\chi(R)$ is the character of the reducible representation, and $\chi^{(i)}(R)$ is the character of irreducible representation $i$ taken from the character table. Reading: this formula is just an inner product in the vector space of class functions; it projects out how much of "flavor $i$" is baked into the reducible representation $\Gamma$, and it always returns a non-negative integer when done correctly — a built-in check on arithmetic.

Projection operator. Given any single starting orbital $\phi_1$, the (unnormalized) SALC of symmetry $i$ built from it is

where: the sum runs over all symmetry operations of the group (not just classes), $\hat{R}\phi_1$ is the orbital obtained by applying operation $R$ to $\phi_1$ (which typically sends it to a different atomic center), and the result is renormalized afterward to unit norm. Reading: this operator amplifies the "irreducible representation $i$" component already latent in $\phi_1$ and destroys every other symmetry component, by the same orthogonality relation that underlies the reduction formula — no diagonalization needed to find the shape of the SALC, only to find its coefficient in the final MO.

Symmetry-blocked secular equation. Within a single irreducible representation block containing $m$ SALCs/AOs of matching symmetry, the ordinary LCAO secular determinant applies exactly as in a diatomic:

where: $\mathbf{H}$ and $\mathbf{S}$ are the $m\times m$ Hamiltonian and overlap matrices built only from the symmetry partners inside that one block (all cross-block elements are already known to vanish by symmetry and are never computed), and the $m$ roots $\varepsilon$ are the orbital energies of that symmetry species. Reading: this is the entire payoff — a potentially enormous determinant has been replaced by several small, independent ones, each labeled permanently by its symmetry species, so orbitals never need to be re-sorted by symmetry after the fact.

Example 1 — Foundational: reducing and projecting the SALCs of water's hydrogen 1s orbitals ($C_{2v}$). Water's point group is $C_{2v}$, with operations $E$, $C_2$, $\sigma_v(xz)$, $\sigma_v'(yz)$, each its own class ($h=4$). Place the molecule in the $yz$ plane with $C_2$ along $z$. Under $E$, both hydrogens are unmoved: $\chi(E)=2$. Under $C_2$, the two hydrogens swap, so neither is fixed: $\chi(C_2)=0$. Under $\sigma_v(xz)$ (perpendicular to the molecular plane), the hydrogens also swap (this plane does not contain them if the molecule lies in $yz$), so $\chi(\sigma_v)=0$. Under $\sigma_v'(yz)$ (the molecular plane itself), both hydrogens lie in the plane and are unmoved: $\chi(\sigma_v')=2$. This gives

The $C_{2v}$ character table gives $a_1=(1,1,1,1)$, $a_2=(1,1,-1,-1)$, $b_1=(1,-1,1,-1)$, $b_2=(1,-1,-1,1)$ (convention: $b_1$ symmetric under $\sigma_v(xz)$, $b_2$ symmetric under $\sigma_v'(yz)$). Reducing:

and $n_{a_2}=n_{b_1}=0$ by the same bookkeeping (both give $\frac{1}{4}(2+0+0-2)=0$). So $\Gamma_{1s}=a_1\oplus b_2$: one symmetric SALC $\psi(a_1)=\frac{1}{\sqrt2}(\phi_1+\phi_2)$ and one antisymmetric SALC $\psi(b_2)=\frac{1}{\sqrt2}(\phi_1-\phi_2)$. Oxygen's $2s$ and $2p_z$ transform as $a_1$; $2p_y$ (in-plane, perpendicular to $C_2$) transforms as $b_2$; $2p_x$ (out-of-plane) transforms as $b_1$ and has no hydrogen SALC partner at all, so it remains strictly nonbonding — this is oxygen's out-of-plane lone pair, unable to mix with anything by symmetry, sitting at essentially its atomic $2p$ energy. This single fact — a whole orbital forced to be nonbonding purely because no SALC of matching symmetry exists — is one of the most useful shortcuts SALC theory provides, and it generalizes directly to the nonbonding $\pi$ orbital that appears in odd-membered π-systems (Example 3).

Example 2 — Applied: methane's $T_d$ SALCs and the equivalence of the four C–H bonds. A common point of confusion is that photoelectron spectroscopy of CH₄ shows two distinct ionization energies (roughly 23 eV and 14–15 eV, a 2:1... actually 1:3 degeneracy pattern), seemingly contradicting the idea that all four C–H bonds are equivalent. SALC theory resolves this immediately. In $T_d$ symmetry ($h=24$), the four hydrogen 1s orbitals form a reducible representation with characters (for classes $E, 8C_3, 3C_2, 6S_4, 6\sigma_d$):

(Under $E$ all four H are unmoved, giving 4; under each $C_3$ exactly one hydrogen — the one on that axis — is fixed, giving 1; under $C_2$ and $S_4$ no hydrogen is fixed, giving 0; under each $\sigma_d$ exactly two hydrogens lie in the plane, giving 2.) Using the $T_d$ character table ($a_1=(1,1,1,1,1)$, $t_2=(3,0,-1,-1,1)$), reduction gives

So $\Gamma_{1s} = a_1 \oplus t_2$ (a triply-degenerate representation), totaling $1+3=4$ orbitals, correct. Carbon's $2s$ matches the $a_1$ hydrogen SALC (one bonding + one antibonding MO); carbon's $2p_x,2p_y,2p_z$ triplet matches the $t_2$ hydrogen SALC triplet (three bonding + three antibonding MOs, degenerate within each set). The eight valence electrons fill $(a_1)^2(t_2)^6$: exactly two energetically distinct ionization levels — deep $a_1$ and shallower triply-degenerate $t_2$ — even though all four C–H bonds are chemically identical. The apparent paradox dissolves because "equivalent bonds" is a real-space statement about electron density, while "MO energy levels" is a symmetry statement about delocalized eigenstates; SALC theory shows both are true simultaneously without contradiction.

Example 3 — Exam/competition level: the π-system of the allyl radical (C₃H₅•) versus cyclopropenyl (C₃H₃). For allyl, three parallel $2p_z$ orbitals on C1–C2–C3 (open chain, $C_{2v}$ symmetry) form a reducible π-basis. Under the $C_2$ axis (which passes through C2 and bisects the C1–C3 bond), C1 and C3 swap while C2 maps to itself but its $p_z$ orbital, being perpendicular to the molecular plane, changes sign under the in-plane $C_2$ rotation combined with the actual geometry — carrying out the bookkeeping with $p_z$ orbitals under $C_{2v}$ (with $C_2$ along $z$, the local axis convention for π-systems keeps sign tracking simple) gives characters

reducing to $\Gamma_\pi = a_2 \oplus 2b_1$ (using the convention where the antisymmetric-under-$\sigma_v'$ representation matches the p$_z$ orbitals' transformation). Rather than track signs through every operation, it is faster and standard to use the equivalent Hückel treatment directly within the same symmetry logic: the secular determinant for three π AOs $p_1,p_2,p_3$ with Coulomb integral $\alpha$, resonance integral $\beta$ between adjacent carbons only, and zero overlap between 1 and 3 is

Expanding along the first row:

The three roots (with $\beta<0$ by convention so bonding is lower in energy) are

The middle root $\varepsilon_2=\alpha$ landing exactly at the atomic energy, with zero contribution from the central carbon (its coefficient in that MO is forced to zero by the odd symmetry of the nonbonding combination, which is antisymmetric between C1 and C3 and must vanish at the symmetric center C2), is the direct symmetry consequence flagged in the Example 1 discussion of water's out-of-plane lone pair: a symmetry species with no matching partner on one specific atom gives an orbital confined away from that atom. Allyl radical has three π electrons: two fill $\varepsilon_1$ (bonding) and one occupies $\varepsilon_2$ (the nonbonding SOMO), giving a total π energy of

compared to three isolated, localized double-bond-only electrons which would give a reference energy of $2\alpha+2\beta$ (one full localized π bond) plus one unpaired electron at $\alpha$, i.e. $3\alpha+2\beta$ total. The delocalization stabilization is

a real, symmetry-derived resonance stabilization (recall $\beta<0$, so this is a lowering of energy) — the seed calculation that Huckel Theory: Assumptions and Framework generalizes to larger conjugated systems, and the same nonbonding-orbital logic explains why cyclopropenyl cation (two π electrons filling the bonding MO only, none reaching the nonbonding or antibonding levels) is aromatically stabilized while cyclopentadienyl anion's related but larger π system follows the analogous pattern with more levels.

Limiting-case check on Example 3. As $\beta\to 0$ (carbons pulled infinitely far apart, no interaction), all three roots collapse to $\varepsilon_1=\varepsilon_2=\varepsilon_3=\alpha$, correctly recovering three isolated, degenerate atomic $2p_z$ orbitals with zero delocalization energy — exactly as required when the molecule literally falls apart into non-interacting atoms.

While no single person "invented" SALC theory in the way Schrödinger invented wave mechanics, the systematic marriage of group theory to molecular orbital construction is largely credited to Hans Bethe's 1929 crystal-field application of group theory to atomic term splitting, and its full extension to molecular orbitals by Friedrich Hund and Robert Mulliken's orbital framework combined with the point-group symmetry analysis formalized by Eugene Wigner. It was Mulliken who gave the symmetry labels ($a_1, e, t_2, \dots$) their standard chemical meaning and pushed for spectroscopists and chemists to adopt group theory as a working tool rather than a mathematical curiosity.