Symmetry-Adapted Linear Combinations

Every modern quantum chemistry package — Gaussian, ORCA, Q-Chem, Psi4 — uses point-group symmetry internally to block-diagonalize the Fock or Kohn–Sham matrix before diagonalization, and the mathematical content of that block-diagonalization is precisely the construction of SALCs (there called symmetry-adapted molecular orbitals or symmetry-blocked basis functions). Exploiting symmetry this way cuts computational cost by roughly the square or cube of the point-group order, which matters enormously for high-symmetry catalysts, cage compounds, and cluster complexes computed routinely in industrial and academic settings. In coordination chemistry and inorganic spectroscopy, SALCs of ligand donor orbitals (σ, and separately π, sets) are the standard starting point for building qualitative and semi-quantitative molecular orbital diagrams of octahedral, tetrahedral, and square-planar complexes — the same $A_{1g}+E_g+T_{1u}$ pattern derived above for an $O_h$ σ set appears, with the numbers changed, in every general inorganic chemistry treatment of crystal-field versus ligand-field bonding, and underlies interpretation of d–d and charge-transfer electronic spectra used to characterize catalysts and pigments. In vibrational spectroscopy, SALCs built from Cartesian atomic displacement vectors become the symmetry coordinates used to factor the vibrational secular equation by symmetry block, dramatically simplifying force-constant fitting from infrared and Raman spectra — the technique used industrially to fingerprint polymorphs, monitor reaction progress via characteristic symmetry-allowed bands, and assign vibrational spectra of gas-phase pollutants and atmospheric species measured by remote infrared spectroscopy. In materials science, SALCs of orbital or spin basis functions on the sites of a crystallographic unit cell (extended via space-group, rather than point-group, symmetry) underlie the construction of Bloch-symmetrized basis functions used in band-structure calculations of semiconductors and topological materials, and the same projection-operator logic appears in constructing symmetry-adapted cluster expansions used to model alloy ordering and defect states.

Ask the tutor: How would the SALC construction change for the four σ-orbitals of a tetrahedral AB₄ molecule like CH₄, and what symmetry species do they span in $T_d$?

Consider water, point group $C_{2v}$, and put a $2p_z$-like $\sigma$ orbital on each hydrogen pointing toward oxygen — call them $\phi_1$ (on H bonded along one O–H bond) and $\phi_2$ (the other). Neither $\phi_1$ nor $\phi_2$ alone is symmetric under the $C_2$ rotation that swaps the two hydrogens: $C_2$ sends $\phi_1 \to \phi_2$ and $\phi_2\to\phi_1$. So $\phi_1$ and $\phi_2$ are not eigenfunctions of the symmetry operations individually — but the two combinations

are. Under $C_2$, $\psi_+\to\psi_+$ (symmetric, transforms as $A_1$) and $\psi_-\to-\psi_-$ (antisymmetric, transforms as $B_2$ in the standard $C_{2v}$ table, given the oxygen $p_y$ convention). These are the SALCs. Only $\psi_+$ can mix with the oxygen $2s$ and $2p_z$ orbitals (both $A_1$); only $\psi_-$ can mix with oxygen $2p_y$ (which is $B_2$). Symmetry has pre-diagonalized part of the secular problem before a single integral is evaluated. For two hydrogens, guessing $\psi_\pm$ by "obviously it's the sum and difference" is trivial. For six fluorine $\sigma$ orbitals around sulfur in $O_h$ symmetry, or twelve C–H stretching coordinates in benzene, or the fifteen valence $\pi$ orbitals of a fused polycyclic aromatic, intuition runs out fast. The projection operator formalizes the "sum and difference" trick into an algorithm that works for any basis, in any point group, for any dimensionality of irrep. The operator's logic rests on the Great Orthogonality Theorem result that irreducible representation matrix elements — and hence their traces, the characters — are orthogonal across the group. This orthogonality lets you build, from characters alone, a filter that passes one symmetry species and annihilates all others, exactly as a Fourier coefficient integral passes one frequency and kills the rest.

The reducible representation $\Gamma$ of any chosen basis set (bond orbitals, displacement vectors) decomposes as

where:

• $a_i$ — the number of times irreducible representation $\Gamma^{(i)}$ appears, obtained from The Reduction Formula;
• the sum runs over all irreducible representations of the point group.

Reading: this tells you how many independent SALCs of each symmetry type exist, but not their explicit form — that is the job of the projection operator below. The projection operator onto irreducible representation $i$ is

where:

• $h$ — the order of the group (total number of symmetry operations);
• $\ell_i$ — the dimension of irrep $i$ (1 for A/B, 2 for E, 3 for T);
• $\chi^{(i)}(R)$ — the character of operation $R$ in irrep $i$ (real for real representations, so the complex conjugate is often just $\chi^{(i)}(R)$ itself);
• $\hat O_R$ — the operator that carries out symmetry operation $R$ on a function (rotates/reflects/inverts the orbital, or permutes basis functions).

Reading: $\hat P^{(i)}$ takes any trial function, superposes all its symmetry-transformed images with weights fixed by the character table, and returns the projection of that function onto the irrep-$i$ subspace — exactly analogous to a Fourier projection onto one harmonic. When applied to one member $\phi_k$ of a basis set that spans a permutation representation (atoms/orbitals get shuffled among themselves, possibly with sign changes from orbital phase), the working formula collapses to a sum over group operations:

where:

• $\hat O_R\phi_k$ — whichever basis orbital (with whatever sign) $\phi_k$ is carried into by operation $R$.

Reading: walk through every operation of the group, see where $\phi_k$ lands, multiply by the appropriate character, and add. Terms with the same destination orbital collect; the result, once normalized, is the SALC. For degenerate irreps ($\ell_i \ge 2$), the character-weighted sum still produces a function of the correct symmetry species, but repeating the naive projection on different seed orbitals can return functions that are not mutually orthogonal, or that vanish, because the character alone underdetermines which row of the degenerate representation you have projected onto. The rigorous fix uses the full matrix element:

where:

• $D^{(i)}(R)$ — the $\ell_i\times\ell_i$ irreducible matrix representative of operation $R$;
• $m$ — a fixed row index, selecting one partner of the degenerate set.

Reading: this is the same idea one level more refined — instead of the trace (character), use one diagonal matrix element, which projects onto a single partner function of the degenerate pair/triple; running $m$ over all rows in turn generates the full degenerate SALC set. In practice for hand calculations, chemists usually avoid building explicit matrices by instead projecting with characters to get the multiplicity and symmetry type, then constructing the explicit degenerate combinations by inspection/orthogonalization — a shortcut justified below.

Example 1 — Foundational: the two π-SALCs of a symmetric AB₂-type π system are re-derived by hand for asymmetric bond weighting. Take a hypothetical bent AB₂ π system (point group $C_{2v}$, like the two out-of-plane $p_\pi$ orbitals on the two "arm" atoms of an allylic-type fragment where the arms are chemically distinguishable in the labeling but not the geometry) with $p_\pi$ orbitals $\chi_1$ on arm 1 and $\chi_2$ on arm 2, related by the same $C_2$ that swaps the arms; take the (arbitrary, asymmetric-looking but still normalized) trial coefficients as a check rather than assuming the sum/difference form outright. Using the same four operations as before, with $\chi_1\to\chi_2$ under $C_2$ and (because $\pi$ orbitals are odd under reflection through the molecular plane) $\sigma_v'(yz)$ (molecular plane) sends $\chi_1\to-\chi_1$, and $\sigma_v(xz)$ sends $\chi_1\to-\chi_2$:

The $A_1$ projection vanishes — telling us immediately that no totally symmetric SALC exists from this basis, because the $\pi$ orbitals are antisymmetric under the molecular-plane reflection, which the $A_1$ representation is not. Trying $B_1$ (characters $1,-1,1,-1$ for $E,C_2,\sigma_v,\sigma_v'$ in the standard convention where $B_1$ is antisymmetric under $\sigma_v'$):

Trying $A_2$ (characters $1,1,-1,-1$):

So the two π-SALCs are $\psi(A_2)=\tfrac{1}{\sqrt2}(\chi_1+\chi_2)$ and $\psi(B_1)=\tfrac{1}{\sqrt2}(\chi_1-\chi_2)$ — same numeric pattern as the $\sigma$ case, but the symmetry labels are different because the $p_\pi$ orbitals pick up a minus sign under the plane reflections that the $\sigma$ orbitals do not. This demonstrates the essential lesson: the arithmetic sum/difference is not always paired with the same symmetry label — the phase behavior of the atomic orbital under each operation is what fixes the label, and projection tracks that automatically while naive pattern-matching from a previous molecule would mislabel it.

Example 2 — Applied: constructing the four H 1s SALCs of NH₃ in $C_{3v}$, including the doubly degenerate pair by the character-plus-orthogonalization shortcut. Ammonia has three N–H bonds; use the local N–H $\sigma$ basis $\phi_1,\phi_2,\phi_3$ (numbered around the $C_3$ axis). $C_{3v}$ has $h=6$: $E,\ 2C_3,\ 3\sigma_v$. The reducible representation has $\chi(E)=3,\ \chi(C_3)=0$ (all bonds move), $\chi(\sigma_v)=1$ (one bond lies in each mirror plane and is unmoved). Using the $C_{3v}$ character table ($A_1: 1,1,1$; $A_2: 1,1,-1$; $E: 2,-1,0$), reduce:

$a_{A_2}=0$. So $\Gamma = A_1+E$, consistent with $1+2=3$. Project $A_1$ from $\phi_1$: all six operations send $\phi_1,\phi_2,\phi_3$ among themselves with coefficient $+1$ and character weight $+1$ throughout, and each of the three orbitals is reached twice (by $E$ and one other operation, or by the appropriate combination), giving

For the $E$ pair, project with character set $(2,-1,-1,-1,0,0,0)$-style weighting (two $C_3$ classes give $-1$, three $\sigma_v$ give $0$):

(here $\hat O_{C_3}\phi_1=\phi_2$, $\hat O_{C_3^2}\phi_1=\phi_3$, and the three $\sigma_v$ operations fix one orbital and swap the other two, contributing character $0$ times whatever they do — those three terms drop out because $\chi^{(E)}(\sigma_v)=0$). This yields one partner:

(normalization: coefficients $2,-1,-1$, sum of squares $4+1+1=6$). The second, orthogonal partner is obtained either by repeating the projection on $\phi_2$ and orthogonalizing against $\psi(E,a)$, or directly by symmetry as the antisymmetric combination of the two "wings":

Check orthogonality: $\langle E,a|E,b\rangle \propto (2)(0)+(-1)(1)+(-1)(-1) = 0+(-1)+1=0$. ✓ These two, $\psi(E,a)$ and $\psi(E,b)$, are the symmetry-adapted combinations that overlap with the degenerate N $2p_x,2p_y$ pair, exactly as needed to build the $2a_1+2e$ bonding/antibonding framework of the ammonia valence MO diagram.

Example 3 — Exam/competition level: SALCs of the four C–H stretching coordinates in a $D_{4h}$-symmetric square-planar XH₄ fragment, verifying the projection against an independent normal-mode counting argument. Consider a hypothetical planar $XH_4$ unit of $D_{4h}$ symmetry (as a simplified stand-in for the four C–H bonds of a square-planar fragment; $X$ at center, four H at $\pm x,\pm y$). Basis: bond-stretch coordinates $r_1$ (+x), $r_2$ (+y), $r_3$ (−x), $r_4$ (−y). The relevant classes and orders for $D_{4h}$ are $E,\ 2C_4,\ C_2,\ 2C_2',\ 2C_2'',\ i,\ 2S_4,\ \sigma_h,\ 2\sigma_v,\ 2\sigma_d$ with $h=16$. Only operations that leave at least one bond unmoved contribute a nonzero character to the diagonal (each unmoved bond contributes $+1$ since stretching is a scalar, unsigned coordinate): $E$ leaves all four unmoved ($\chi=4$); $C_2$ (the principal axis, $180°$) swaps $1\leftrightarrow3$ and $2\leftrightarrow4$, unmoved count $=0$; $C_4$ cyclically permutes all four, unmoved $=0$; the $2C_2'$ (through opposite bonds, i.e. through H atoms) fix two bonds and swap the other two, contributing $\chi=2$; the $2C_2''$ (through bond midpoints, between H atoms) swap all bonds in pairs, $\chi=0$; $i$ swaps $1\leftrightarrow3,2\leftrightarrow4$, $\chi=0$; $S_4$ similarly moves everything, $\chi=0$; $\sigma_h$ (molecular plane) leaves all four unmoved, $\chi=4$; the $2\sigma_v$ (containing the bonds) fix two bonds each, $\chi=2$; the $2\sigma_d$ (between bonds) swap pairs, $\chi=0$. Reduce against the $D_{4h}$ table (rows $A_{1g}:1,1,1,1,1,1,1,1,1,1$; $B_{1g}:1,-1,1,1,-1,1,-1,1,1,-1$; $E_u: 2,0,-2,0,0,-2,0,2,0,0$, etc., using the standard ten-class ordering matching the operations above):

with all remaining irreps giving zero, so $\Gamma_{\text{stretch}} = A_{1g}+B_{1g}+E_u$, total dimension $1+1+2=4$. ✓ matches the four starting coordinates. Projecting: $A_{1g}$ gives the fully symmetric breathing stretch

$B_{1g}$ (characters $+1$ on $E$, $-1$ on $C_4$, $+1$ on $C_2$, $+1$ on $C_2'$, $-1$ on $C_2''$…) projects the alternating in-plane combination:

And the degenerate $E_u$ pair (antisymmetric stretches, IR-active) is

Each is separately normalized (coefficients $1,-1$ give sum of squares 2, matching $1/\sqrt2$), and mutually orthogonal by inspection since they involve disjoint coordinates. Physically: $A_{1g}$ is the totally symmetric "breathing" stretch (Raman-active, IR-silent by the mutual exclusion rule in a centrosymmetric molecule), $B_{1g}$ is the alternating stretch (also Raman-active, $g$-symmetry), and the doubly degenerate $E_u$ pair is IR-active (matches the $(x,y)$ transformation) — exactly the vibrational assignment scheme extended fully in Symmetry and Molecular Vibrations.

The projection-operator method sits at the intersection of representation theory, developed by mathematicians studying finite groups in the late nineteenth and early twentieth centuries, and its deliberate adaptation to molecular quantum mechanics in the 1930s. The chemist most associated with turning abstract group representation theory into a working tool for molecular orbital construction is Hans Bethe, whose 1929 treatment of crystal-field splitting used group-theoretical projection ideas for the first time in a chemical/physical context, soon followed by the systematic development of ligand-field and molecular-orbital symmetry methods by Van Vleck, Mulliken, and later textbook-level codification by F. Albert Cotton, whose Chemical Applications of Group Theory became the standard chemist's reference for exactly this projection-operator machinery.