Symmetry and Molecular Orbitals

Symmetry-matched MO construction is not a historical curiosity; it is the load-bearing scaffold of modern computational and inorganic chemistry. Every ab initio or density-functional-theory package (Gaussian, ORCA, Q-Chem, VASP for periodic systems) exploits point-group symmetry to block-diagonalize the Fock or Kohn–Sham matrix before diagonalization, cutting computational cost by roughly the order of the point group and, more importantly, guaranteeing that orbitals emerge pre-labeled by irreducible representation — the practical, industrial-strength descendant of the vanishing-integral theorem derived above. Ligand-field and molecular-orbital symmetry analysis of octahedral and tetrahedral transition-metal complexes underlies the rational design of catalysts (from Wilkinson's catalyst to modern asymmetric hydrogenation catalysts), where predicting which $d$-orbital set is destabilized by $\sigma$- or $\pi$-donor/acceptor ligands directly predicts spin state, color, and reactivity. Photoelectron and X-ray photoelectron spectroscopy (XPS/UPS) assignments — exactly the water example used throughout this article — are read symmetry-label by symmetry-label, matching observed ionization bands to specific MOs predicted by group theory, a technique used routinely in materials characterization and catalyst surface science. In organometallic and bioinorganic chemistry, the symmetry-matching rule explains why certain metal-ligand $\pi$-backbonding interactions are geometry-dependent (a ligand's $\pi^*$ orbital must find a metal $d$ orbital of matching local symmetry, which is why CO backbonding strength depends on M–C–O geometry) — directly relevant to the electronic structure of heme proteins, nitrogenase active sites, and industrial hydroformylation catalysts. Finally, this entire symmetry-matching framework is the direct ancestor of vibrational and electronic spectroscopic selection rules, where the same "must share an irreducible representation" logic (now applied to transition dipole moment integrals rather than Hamiltonian matrix elements) decides which absorption and Raman bands are allowed or forbidden — the natural next application of everything built here.

Ask the tutor: How does this same symmetry-matching rule decide which vibrational or electronic transitions are IR/Raman-allowed?

Start from the one-electron Hamiltonian of a molecule, $\hat{H}$, and its point group $G$ — the full set of symmetry operations (rotations, reflections, inversion, improper rotations) that map the nuclear skeleton onto itself. Because the nuclei do not move under a symmetry operation $\hat{R}$, the potential energy operator is invariant, and so $\hat{R}$ commutes with $\hat{H}$:

This commutation is the entire content of "symmetry in quantum chemistry." Whenever an operator commutes with the Hamiltonian, its eigenfunctions can be chosen to also be eigenfunctions of that operator (or, if $\hat{R}$ is not itself diagonal in the degenerate subspace, the eigenfunctions transform as a definite irreducible representation of the full symmetry group). Every true molecular orbital of a molecule can therefore be labeled by an irreducible representation (irrep) of the point group — $a_1$, $a_2$, $b_1$, $b_2$ in $C_{2v}$; $a_1$, $e$ in $C_{3v}$; $a_{1g}, t_{1u}, e_g, t_{2g}\ldots$ in $O_h$ — and this label is exact, not a qualitative add-on. The practical machine that gets you from atomic orbitals to symmetry-labeled MOs has three moving parts. First, the ligand orbitals (the $\sigma$ or $\pi$ functions pointing at the central atom, or more generally any equivalent set of basis functions) are combined into symmetry-adapted linear combinations, exactly as built via projection operators. Second, the central-atom atomic orbitals are individually classified — an $s$ orbital is always totally symmetric, a $p_z$ orbital along the principal axis transforms as whatever irrep the coordinate $z$ carries, and so on, read directly off the character table. Third, and this is the physical heart of the article, SALCs and central-atom orbitals combine into bonding and antibonding molecular orbitals only when they belong to the same irreducible representation. Two functions of different symmetry give an overlap integral that is forced to vanish by symmetry — that is a theorem, proved below, not a guess — and their Hamiltonian matrix element vanishes for the identical reason. There is no bonding interaction to speak of between an $a_1$ orbital and a $b_2$ orbital in $C_{2v}$: full stop. The figures below sketch two prototypes. In water's $C_{2v}$ symmetry, the two O–H $\sigma$ bonds resolve into ligand SALCs of $a_1$ and $b_2$ symmetry; oxygen's $2s$ and $2p_z$ are $a_1$, $2p_x$ is $b_1$, $2p_y$ is $b_2$. Matching labels: $2s/2p_z$ mix with the $a_1$ SALC to build two bonding/antibonding $a_1$ MOs plus a largely nonbonding $a_1$ combination; $2p_y$ mixes with the $b_2$ SALC to build a bonding and antibonding pair; and $2p_x$, being $b_1$ with no ligand partner of that symmetry (the two hydrogen $1s$ orbitals only ever generate $a_1 + b_2$), remains a pure, nonbonding, symmetry-forbidden-from-mixing lone pair orbital. This single fact — that $b_1$ has no ligand SALC in water — is exactly why water's four occupied valence MOs are not two equivalent bonds and two equivalent lone pairs, but four inequivalent orbitals of distinct energy, matching the four distinct bands seen in the photoelectron spectrum.

The second prototype is the octahedral $\mathrm{ML_6}$ $\sigma$-only complex, which is the standard entry point to ligand-field theory. Six ligand $\sigma$ orbitals pointing at the metal reduce, under $O_h$, to $a_{1g} + e_g + t_{1u}$ — never $t_{2g}$, because no combination of six radially pointing $\sigma$ orbitals can carry $t_{2g}$ symmetry (this is a reduction-formula result, derived below). The metal's $(n)s$ is $a_{1g}$, the three $(n)p$ orbitals are $t_{1u}$, and the five $(n-1)d$ orbitals split as $e_g$ ($d_{z^2}, d_{x^2-y^2}$) $+$ $t_{2g}$ ($d_{xy}, d_{xz}, d_{yz}$). Symmetry matching immediately explains the central postulate of crystal-/ligand-field theory: $d_{z^2}$ and $d_{x^2-y^2}$ ($e_g$) point straight at the ligands, find an $e_g$ ligand SALC waiting for them, and are pushed up in energy by $\sigma$-antibonding; $d_{xy}, d_{xz}, d_{yz}$ ($t_{2g}$) point between the ligands, have no $\sigma$ SALC of matching symmetry, and stay nonbonding (until $\pi$-donor or $\pi$-acceptor ligands are added, which supply a $t_{2g}$ SALC and change the story completely).

The reduction formula extracts, from a reducible representation $\Gamma$ built on any basis (ligand $\sigma$ orbitals, ligand $\pi$ orbitals, Cartesian displacements), the number of times each irreducible representation $\Gamma_i$ appears:

where:

• $h$ — order of the point group (total number of symmetry operations),
• $g_{\hat R}$ — number of operations in the class of $\hat R$ (used when summing class-by-class instead of operation-by-operation),
• $\chi(\hat R)$ — character of the reducible representation for operation $\hat R$ (computed as the number of unmoved basis functions, with sign, for that operation),
• $\chi_i(\hat R)$ — character of irreducible representation $i$ for the same operation, read off the character table,
• $n_i$ — integer multiplicity of $\Gamma_i$ inside $\Gamma$.

Reading this equation: it is nothing more than the statement that irreducible representations form an orthonormal basis (in the sense of the great orthogonality theorem) for the space of class functions on the group, so any reducible representation's character vector has a unique decomposition, and $n_i$ is just the projection (inner product) of $\Gamma$ onto $\Gamma_i$. The projection operator promotes one member of a ligand orbital set (the "generating function," any single ligand orbital, call it $\phi_1$) into the properly symmetrized SALC transforming as irrep $i$:

where:

• $\ell_i$ — dimension of irrep $i$ (1 for A/B, 2 for E, 3 for T),
• $\hat R\phi_1$ — the ligand orbital that $\phi_1$ is mapped onto by operation $\hat R$ (found by tracking which atom $\hat R$ sends atom 1 to),
• the sum runs over all $h$ operations (not just classes), each contributing its character value for irrep $i$,
• the result, after normalization, is the SALC of symmetry $i$ generated from $\phi_1$; applying $\hat P_i$ to a different starting function in a degenerate irrep can generate a different, complementary partner function.

Reading this equation: the projection operator is a symmetrized average — take your starting orbital, apply every operation in the group, weight each result by how much irrep $i$ "likes" that operation (its character), and add. Terms from operations $i$ dislikes cancel; the survivors assemble into a linear combination transforming purely as $\Gamma_i$. Finally, the symmetry-matching mixing rule, quantified perturbatively for two symmetry-allowed basis functions $\psi_A$ (central atom, energy $\alpha_A$) and $\psi_L$ (ligand SALC, energy $\alpha_L$) of the same irrep, coupled by resonance integral $\beta = \langle \psi_A|\hat H|\psi_L\rangle$:

where:

• $E_\pm$ — energies of the resulting bonding ($E_-$, lower) and antibonding ($E_+$, higher) MOs,
• $\alpha_A, \alpha_L$ — unperturbed (Coulomb) energies of the central-atom orbital and ligand SALC before interaction,
• $\beta$ — the interaction (resonance) matrix element, nonzero only because $\psi_A$ and $\psi_L$ share an irrep,
• the square root term — the "level repulsion" contribution; it grows with $|\beta|$ and shrinks as the unperturbed gap $|\alpha_A-\alpha_L|$ grows.

Reading this equation: this is exactly the two-level non-crossing formula. If $\psi_A$ and $\psi_L$ had different symmetry, $\beta \equiv 0$ exactly (proved below), the square root collapses to $|\alpha_A - \alpha_L|/2$, and $E_\pm$ reduce trivially back to $\alpha_A$ and $\alpha_L$ separately — the two orbitals simply do not interact and pass through each other's energy region untouched. Symmetry is therefore the switch that turns level repulsion on or off; the algebra above only ever matters once symmetry has already said "yes."

Example 1 — foundational: reducing the six σ SALCs of octahedral ML₆. Set up $\Gamma_\sigma$, the reducible representation of the six ligand $\sigma$ orbitals in $O_h$, using the standard rule that the character for operation $\hat R$ equals the number of ligand atoms left completely unmoved by $\hat R$ (only unmoved atoms can contribute a nonzero diagonal element, since a $\sigma$ orbital that moves to a different site has zero diagonal overlap with itself). The $O_h$ classes and how many of the six ligand positions (at $\pm x,\pm y,\pm z$) each leaves fixed:

Reduce against each irrep using $n_i = \frac1h\sum g_{\hat R}\chi(\hat R)\chi_i(\hat R)$ with $h=48$:

Carrying the sum correctly, class by class, with $O_h$ character table values $\chi_{A_{1g}}=1$ for all classes, and multiplicities $g$ = (1, 8, 6, 6, 3, 1, 6, 8, 3, 6) for classes ($E, 8C_3, 6C_2, 6C_4, 3C_2', i, 6S_4, 8S_6, 3\sigma_h, 6\sigma_d$):

For $E_g$ ($\chi = 2,-1,0,0,2,2,0,-1,2,0$ across the same class order):

For $T_{1u}$ ($\chi = 3,0,-1,1,-1,-3,-1,0,1,1$):

All other irreps ($A_{2g}, A_{1u}, A_{2u}, E_u, T_{1g}, T_{2g}, T_{2u}$) give $n_i = 0$ by the same bookkeeping (each sum cancels to zero). So

confirming exactly what was asserted in the figure: six $\sigma$ ligand orbitals span $1+2+3=6$ combinations total (dimension check: $1+2+3=6$ ✓), matching metal $s$ ($a_{1g}$), metal $p_x,p_y,p_z$ ($t_{1u}$), and metal $d_{z^2}, d_{x^2-y^2}$ ($e_g$) — with $t_{2g}$ ($d_{xy},d_{xz},d_{yz}$) left with no $\sigma$-symmetry partner, exactly as claimed.

Example 2 — applied: quantitative non-crossing in water's b₂ block with asymmetric numbers. Suppose (illustrative, order-of-magnitude atomic-orbital energies and a modest resonance integral, chosen deliberately asymmetric so the formula's structure is visible) the unperturbed energies are $\alpha_{2p_y}(\mathrm O) = -15.9\ \text{eV}$ and $\alpha_{\mathrm{SALC}(b_2)} = -13.6\ \text{eV}$ (the H $1s$ ionization energy, appropriate for the antisymmetric combination of two H $1s$ orbitals treated at the Hückel level), with resonance integral $\beta = -3.2\ \text{eV}$. Compute the average and half-gap:

Now the square root term:

So the bonding and antibonding $b_2$ energies are:

Sanity check with the sum rule, $E_++E_- = \alpha_A+\alpha_L$: $-18.15 + (-11.35) = -29.50\ \text{eV}$, matching $\alpha_A+\alpha_L=-29.5\ \text{eV}$ exactly — the trace of the $2\times2$ block is preserved, as it must be for any similarity-transformation diagonalization. The bonding $b_2$ MO sits $18.15-15.9=2.25\ \text{eV}$ below the more stable of the two parent orbitals and $18.15-13.6=4.55\ \text{eV}$ below the less stable one, consistent with the bonding orbital inheriting more character (and stabilization) from whichever parent orbital it is energetically closer to — here, it leans toward oxygen's $2p_y$.

Limiting case check. If the resonance integral were switched off, $\beta \to 0$ (imagine the two fragments moved infinitely far apart, or imagine — hypothetically — that the orbitals had different symmetry so $\beta$ were forced to zero by the theorem proved above), the square root becomes $\sqrt{(-1.15)^2} = 1.15$, and

exactly recovering the two unperturbed atomic/ligand energies with zero mixing — precisely the required limit, and precisely why symmetry mismatch ($\beta\equiv 0$ exactly, not just small) is such a clean, total switch rather than a matter of degree.

Example 3 — exam/competition level: predicting second-order push-down in an $a_1$ three-orbital block. Consider water's three-orbital $a_1$ block from the derivation above: O $2s$ (energy $\alpha_{2s}=-32.4\ \text{eV}$), O $2p_z$ ($\alpha_{2p_z} = -15.9\ \text{eV}$), and $\mathrm{SALC}(a_1)$ (energy $\alpha_{\mathrm{SALC}} = -13.6\ \text{eV}$), with resonance integrals $\beta_{2s,\mathrm{SALC}} = -4.5\ \text{eV}$, $\beta_{2p_z,\mathrm{SALC}} = -2.8\ \text{eV}$, and (crucially, and often forgotten) $\beta_{2s,2p_z}\approx 0$ by symmetry within the free atom (different $\ell$ quantum number, negligible direct off-diagonal Fock/Hückel coupling in a minimal-basis treatment) — so the direct $2s$–$2p_z$ coupling is not the mechanism; any $2s$/$2p_z$ mixing is entirely mediated through their shared coupling to the SALC. Treat this perturbatively: since $2p_z$ is much closer in energy to the SALC ($|{-15.9-(-13.6)}| = 2.3\ \text{eV}$ gap) than $2s$ is ($|{-32.4-(-13.6)}| = 18.8\ \text{eV}$ gap), the dominant bonding/antibonding pair is essentially the $2p_z$/SALC interaction (structurally identical to Example 2's $2p_y$/SALC $b_2$ calculation, but now with $\alpha_{2p_z}=-15.9$ eV and $\beta = -2.8$ eV):

Now bring in second-order perturbation theory for how the distant $2s$ orbital further stabilizes the lowest state (this is the standard non-degenerate perturbation formula, $\Delta E_n^{(2)} = \sum_{m\neq n}\dfrac{|\langle m|\hat H'|n\rangle|^2}{E_n^{(0)}-E_m^{(0)}}$, applied here with the lowest $a_1$ combination as state $n$ and the far $2s$ orbital as state $m$):

lowering the ground $a_1$ MO further to approximately $-17.78 - 1.385 = -19.17\ \text{eV}$, while the far $2s$-dominated orbital is correspondingly pushed down even further (in the full $3\times3$ diagonalization it in fact becomes the deepest $a_1$ MO, since it starts already lowest and gains additional stabilization from mixing with the SALC), and the top ($2p_z$-dominated antibonding) $a_1^{*}$ level is essentially undisturbed by the distant $2s$ orbital because the energy denominator for that state's second-order shift is small and the numerator involves the much weaker matrix element through the far-separated $2s$.

Limiting case check. If the O $2s$–$2p$ energy gap were artificially closed (imagine a hypothetical atom where $\alpha_{2s}\to\alpha_{2p_z}$), the "three-orbital, one dominant pair plus small perturbative push" picture used above would break down — all three $a_1$ orbitals would need to be treated on equal footing in the full $3\times3$ secular determinant, since no pair could be singled out as dominant. This is exactly why the real oxygen $2s$–$2p$ gap (roughly 16–17 eV) being large compared to the $2p_z$-SALC gap (2.3 eV) is what licenses the simplified two-step ("dominant pair, then perturbative correction") treatment actually used in Example 3, and is worth checking before applying that shortcut to a different atom.

The systematic use of group theory to classify molecular orbitals is inseparable from the work of Robert S. Mulliken, the American physical chemist whose orbital notation ($a_1$, $t_{2g}$, $\sigma_g$, $\pi_u$, and so on) is the very language this article uses, and who won the 1966 Nobel Prize in Chemistry "for his fundamental work concerning chemical bonds and the electronic structure of molecules by the molecular orbital method." Mulliken's insistence — against considerable resistance from the valence-bond school dominant through the 1930s — that electrons in molecules occupy delocalized orbitals extending over the whole molecular framework, each classifiable by symmetry, is precisely the viewpoint that makes water's "two inequivalent lone pairs" a straightforward prediction rather than a spectroscopic puzzle. His collaboration with Friedrich Hund on what became known as Mulliken–Hund molecular orbital theory, and his systematic correlation diagrams connecting united-atom and separated-atom limits for diatomics, established the template every modern MO diagram — however complicated the molecule — still follows.