Molecular Geometry & Bonding
what the shape decides
What symmetry decides

Character tables and reduction

A molecule's point group is a list of matrices, not a label. Generating them, sorting them into classes, and dividing by the group order turns any set of orbitals into a statement about how many energies there can be.
17 min read 7 figures Symmetry settles itCounted, not quotedExactly zero

“Methane is Td” is a label. It is also, on its own, useless: nothing follows from a two-character string.

What follows from the twenty-four matrices the string abbreviates is a great deal, and getting from one to the other is the subject of this essay.

The Td character tableThe irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.Tdorder 24E1 found8C38 found3C23 found6S46 found6σd6 foundA111111A2111-1-1E2-1200T130-11-1T2x, y, z30-1-115 representations for 5 classes · dimensions² sum to 24rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated24 operations
Fig. 1 Methane’s character table, with each class heading carrying the number of operations found by generating the group from the coordinates. Those counts were produced before this table was opened, and the table is checked against them.

From a symbol to a group

Recovering the symbol is a search: candidate axes drawn from the structure, each tried as a rotation, a mirror and an improper rotation, and kept when it permutes the atoms among themselves.

That search produces a list of operations, and a list is not a group. The step that matters is closure: multiply everything found by everything found, keep the products, multiply again, and stop when the set stops growing.

Closure is what makes the count meaningful. A group is not the set of elements somebody thought to look for, and a search over plausible axes can miss one. A closed set of orthogonal matrices that all map the molecule onto itself is the group, whether or not the enumeration was complete — and the total can then be checked against the order the symbol implies, which is a check the search alone cannot provide.

Methane closes at twenty-four. Benzene closes at twenty-four by an entirely different route. Sulfur hexafluoride closes at forty-eight.

The classes, computed

The columns of a character table are conjugacy classes, and they too are computed rather than read.

Two operations AA and BB are in the same class when some third operation gg carries one to the other: gAg−1=BgAg^{-1} = B. Testing that is a search over the group, costing the order cubed — which for forty-eight operations is nothing.

What comes out is the column structure of the table before the table is opened. Methane’s twenty-four operations fall into five classes of sizes 1, 8, 3, 6 and 6, and each operation’s kind is read off its own matrix: the determinant separates proper from improper, and the trace fixes the angle, since a rotation by θ\theta has trace 1+2cos⁡θ1 + 2\cos\theta and an improper one has −1+2cos⁡θ-1 + 2\cos\theta.

So every operation is named by measurement. That matters because the products generated during closure are operations nobody constructed directly, and there is no other way to find out what they are.

What is tabulated, and what is checked

One ingredient is data: the characters of the irreducible representations. Everything else on these figures is computed.

A character table is worth using only if it is internally consistent, and four relations settle that without reference to anything outside it.

The class sizes sum to the group order, and the computed classes must reproduce those sizes.

The squared dimensions sum to the group order. Methane: 1+1+4+9+9=241 + 1 + 4 + 9 + 9 = 24.

There are as many representations as classes. Five and five.

The rows are orthonormal and the columns orthogonal under the class-weighted inner product. The row relation is the one every textbook states; the column relation is the less-quoted half of the great orthogonality theorem and is the one that catches a mistyped character in a large table. It caught two, in D₆ₕ, while that table was being entered — and neither would have been visible in any figure that used it.

All four are asserted while a figure is drawn, so a table that failed one would stop the build.

The C2v character tableThe irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.C2vorder 4E1 foundC21 foundσv(xz)1 foundσv′(yz)1 foundA1z1111A211-1-1B1x1-11-1B2y1-1-114 representations for 4 classes · dimensions² sum to 4rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated4 operations
Fig. 2 Water’s table, small enough to read entirely: four operations, four classes of one each, four one-dimensional representations. The whole of what symmetry can say about a C₂ᵥ molecule is in those sixteen numbers.
The Oh character tableThe irreducible representations of the molecule's point group. The class headings carry the number of operations found by generating the group from the coordinates, and those counts were produced before this table was opened.Ohorder 48E1 found8C38 found6C46 found6C26 found3C23 foundi1 found6S46 found8S68 found3σh3 found6σd6 foundA1g1111111111A2g11-1-111-111-1Eg2-100220-120T1g301-1-1310-1-1T2g30-11-13-10-11A1u11111-1-1-1-1-1A2u11-1-11-11-1-11Eu2-1002-201-20T1ux, y, z301-1-1-3-1011T2u30-11-1-3101-110 representations for 10 classes · dimensions² sum to 48rows orthonormal and columns orthogonal, both checked before usecolumns counted on the molecule, characters tabulated48 operations
Fig. 3 And the largest here. Forty-eight operations in ten classes, with the class sizes counted off the generated group. The two-dimensional eg and the three three-dimensional t representations are what a cubic group has that a low-symmetry one does not, and they are why octahedral complexes have the electronic structure they do.

The reduction, which is one line

With the group and the table in hand, any set of functions can be asked what it spans.

The character of a set of basis functions under an operation is a count of what stayed put — a basis function contributes to the trace only if the operation sends it back to itself or to minus itself, which for a function sitting on an atom means the atom did not move. So:

  • for one s function per atom, the character is the number of unmoved atoms;
  • for three p functions per atom, each unmoved atom contributes the operation’s trace;
  • for the 3N Cartesian displacements, the same;
  • for translations, the character is the trace; for rotations, the determinant times the trace, because a rotation is an axial vector.

Then the reduction formula:

ni=1h∑cgc χ(c) χi(c),n_i = \frac{1}{h}\sum_c g_c \,\chi(c)\, \chi_i(c),

summed over classes, with gcg_c the class size and hh the group order.

H s on methane: a₁ ⊕ t₂The character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.E×18C3×83C2×36S4×66σd×6χ (the basis)410021 × A1111111 × T230-1-11spans a₁ ⊕ t₂one s function on each — 4 functions in allevery multiplicity came out whole, and the reduction rebuilds the character abovecounted, then reducedTd · order 24
Fig. 4 Methane’s four hydrogen 1s orbitals reduced, with the arithmetic on the page. The row marked χ is the count of hydrogens each class leaves in place; the rows beneath are the two representations that rebuild it. The answer is a₁ ⊕ t₂.

Why the whole numbers are the check

The multiplicities must come out as non-negative integers, and that is not a formality.

There is nothing a fractional multiplicity could mean. A representation appears twice or three times or not at all, and an answer of 1.75 is not a slightly wrong answer — it is a signal that something upstream is broken.

And a great many things upstream could be. A miscounted character, a computed class matched to the wrong table column, a class size that disagrees with the generated group, a group that failed to close — every one of them produces a fraction, and nothing else in the pipeline would notice any of them.

Two further checks ride along. The reduction must rebuild the character it came from, class by class, and the total dimension must equal the number of basis functions. Both are cheap and both would catch an error the integrality test happened to miss.

The site’s gate then feeds the assertion something wrong on purpose: a span claimed as a₁ ⊕ e when the group gives a₁ ⊕ t₂ must be refused, and a character constructed not to reduce must be refused too. An assertion that has never rejected anything proves nothing.

What it buys

The prediction this machinery makes on methane is worth stating on its own, because it is derived from four coordinates and nothing else.

Two bands, required by the shapeThe symmetry species the hydrogen orbitals span, their degeneracies, and the measured photoelectron bands beside them. The single band that four equivalent bonds would predict is drawn dashed, as the alternative this spectrum rules out.t₂12.7 eVdegeneracy 3a₁23.0 eVdegeneracy 1one bandfour equivalent bondsH orbitals spana₁ ⊕ t₂predicted bands: 2intensity ratio 1 : 3energies measuredassignment computedionisation energy, increasing to the leftthe split is predicted; the energies are quotedsymmetry only, no energies
Fig. 5 Symmetry says the hydrogen orbitals span two species with degeneracies one and three, so there must be two ionisation energies in a 1:3 intensity ratio. The energies at which the bands fall are experimental and marked as such; the split is not. The single band four equivalent bonds would give is drawn dashed.

Nothing about an energy entered that argument. The bonding electrons cannot all lie at one energy, so the localised picture is refuted before any measurement is taken.

The same machinery generalises the exactly-zero argument from hand-picked orbital pairs to any molecule with a finite group, and it is what vibrational counting and selection rules need.

Three more bases, and what each is for

The reduction is the same operation whatever is being reduced, and swapping the basis changes what the answer is about. Four are worth having.

F s on boron trifluoride: a₁′ ⊕ e′The character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.E×12C3×23C2×3σh×12S3×23σv×3χ (the basis)3013011 × A1′1111111 × E′2-102-10spans a₁′ ⊕ e′one s function on each — 3 functions in allevery multiplicity came out whole, and the reduction rebuilds the character abovecounted, then reducedD3h · order 12
Fig. 6 The three fluorine sigma orbitals of boron trifluoride, spanning a₁′ ⊕ e′. Two species, so two energies, so boron’s 2s can interact with one combination and its two in-plane 2p orbitals with the other pair — and the perpendicular 2p is left with nothing to pair with, which is why the molecule is a Lewis acid.
C pz on benzene: b₂g ⊕ e₁g ⊕ a₂u ⊕ e₂uThe character of the basis under each class of operations, which is a count of what did not move, and the multiplicities that come out of the reduction formula. The multiplicities must be whole numbers, and that is the check.E×12C6×22C3×2C2×13C2′×33C2″×3i×12S3×22S6×2σh×13σd×33σv×3χ (the basis)6000-20000-6021 × B2g1-11-1-111-11-1-111 × E1g21-1-20021-1-2001 × A2u1111-1-1-1-1-1-1111 × E2u2-1-1200-211-200spans b₂g ⊕ e₁g ⊕ a₂u ⊕ e₂uone perpendicular p on each — 6 functions in allevery multiplicity came out whole, and the reduction rebuilds the character abovecounted, then reducedD6h · order 24
Fig. 7 Benzene’s six perpendicular p orbitals, spanning b₂g ⊕ e₁g ⊕ a₂u ⊕ e₂u. Four species with dimensions 1, 2, 1, 2 — which is the one-then-pairs-then-one level pattern that Hückel theory produces by diagonalising a matrix, arrived at here without diagonalising anything.

That last pair is worth dwelling on. The degeneracies of benzene’s pi orbitals are usually presented as an output of the Hückel calculation, and they are not — they are fixed by the group, and any calculation that produced different ones would be wrong. What Hückel adds is the ordering and the spacing. What symmetry supplies is the pattern, exactly, and it supplies it before any matrix is written down.

What it costs, and the bug the tolerance hid

Generating a group of twenty-four operations and classifying them takes a few milliseconds. The cost that mattered was a correctness cost, and it is worth recording because the failure was silent in two different ways.

The search accepts an axis that maps the molecule onto itself to within six hundredths of an ångström — the right tolerance for deciding whether an operation exists, and the wrong one for multiplying two together. Ammonia’s threefold axis came out of the search tilted by about 5×10−55\times10^{-5} radians, which is a perfectly good verdict and a disastrous matrix: the products drifted, no two matched, and the set grew past a two-hundred-element ceiling on a group of order six.

The repair is to stop treating the approximate matrix as an answer and treat it as a question. It still identifies unambiguously which atom goes where, and that permutation is exact — a fact about labels, not about numbers. The operation is then recomputed as the orthogonal matrix that best realises the permutation, by Newton’s iteration on the polar factor, and comes back orthogonal to about 10−1510^{-15}. Crucially, the refined matrix is a function of the permutation alone, so two long chains of products that move the same atoms to the same places come out bit-identical and the set closes.

The second failure was better hidden. A planar molecule’s own plane is a mirror that moves no atom at all, so its permutation is the identity’s. Keying operations on the permutation alone collapsed benzene’s twenty-four operations to twelve — silently, with every one of the twelve correct, and with the symbol still reported as D₆ₕ because the symbol comes from the axis search rather than from the count. It was caught only because the order was checked against the character table.

That is the whole argument for checking a computed group against a tabulated order: the count is the only quantity that noticed.

The surprise: the labels are a convention and the answer is not

There is one honest wrinkle and it is worth stating plainly rather than glossing.

Which of two indistinguishable classes is called C₂′ and which C₂″, and which vertical mirrors are σᵥ rather than σ_d, is a naming choice. This site takes C₂′ to be the class whose axes pass through atoms and σᵥ to be the planes containing those axes, and decides both by measuring the molecule rather than by assuming an orientation. A different convention would swap some representation labels.

What never changes is the multiset of representations a basis spans, and their dimensions. Methane’s hydrogens span a one-dimensional and a three-dimensional representation whatever anybody calls them, so the prediction of two bands in a 1:3 ratio is convention-independent.

The distinction is worth internalising because it recurs. A great deal of what looks like disagreement between sources on symmetry labels is disagreement about axis conventions, and none of it touches the physics.

Where the model stops

Three limits.

The linear groups are refused. C∞v and D∞h have infinitely many operations, and the reduction formula divides by the group order. Carbon dioxide gets a symbol on this site and nothing further. A treatment that quietly truncated the rotation axis at some large finite order would produce plausible answers for the wrong group.

Symmetry gives no energies, ever. It says how many species there are and forbids what it forbids. Which of methane’s two bands lies lower, and by how much, is outside what any group can say.

Equilibrium geometry only. A vibrating molecule is instantaneously less symmetric, which is exactly how formally forbidden transitions become weakly allowed — and how strictly zero becomes very small.

A reduction says what may mix, never how much. That a₁ appears once in methane’s hydrogen basis and once among carbon’s own orbitals means those two combinations can interact. How strongly they do is an overlap and an energy difference, and both are outside the group. The commonest way to over-read a reduction is to treat two species that match as two species that have combined, which they need not have: the eg pair in sulfur hexafluoride matches sulfur’s 3d orbitals exactly and, on the best available evidence, barely mixes with them at all.

The habit worth taking from all this is small. A character table is a table of numbers somebody typed, and four relations check it in a few lines. Nothing else on a page of symmetry arguments is as easy to verify or as damaging to get wrong, since every conclusion downstream inherits the error silently.

Who found it, and when

The group theory predates the chemistry by a long way. Frobenius developed character theory for finite groups in the 1890s, as pure mathematics with no application in view, and the great orthogonality theorem that makes the reduction formula work is his.

Wigner brought it into quantum mechanics in the late 1920s, and Gruppenpest — “the pest of group theory” — was the term physicists of the period used for the influx. Bethe applied it to crystal fields in 1929.

Chemists took longer, and the reason was pedagogical rather than intellectual: the machinery was presented as group theory rather than as bookkeeping. Cotton’s Chemical Applications of Group Theory (1963) is the book usually credited with the change, and it did it by putting the tables first and the theorems last.

The irony is that the tables were the part everyone then treated as authoritative and unchecked. They are data, they contain errors in some published sources, and the four consistency relations that would catch those errors take a few lines to write.

Where the ladder goes next

The search that produces the symbol is point groups from coordinates.

The theorem all of this serves is exactly zero, generalised.

The full apparatus applied to spectra is selection rules are one theorem.

And the independent route to methane’s a₁ ⊕ t₂ is the localisation transformation.

The last thing worth stating is what this machinery replaced. Before it, the claim that methane’s bonding electrons cannot all lie at one energy was carried by a photoelectron spectrum and by prose. It is now carried by a reduction that anybody can repeat from four coordinates, and the spectrum has been promoted from the evidence to the confirmation. That is a change in the standing of an argument rather than in its conclusion, and it is the change this whole phase of the site was for.

The one habit to carry away is arithmetic rather than conceptual: whenever a reduction gives a multiplicity that is not a whole number, something upstream is wrong and nothing else will report it.

What the pictures here cannot show. A character table is a table of integers and small negative integers, and it contains no geometry whatever — the molecule these tables belong to is invisible in them. Nothing on this page shows an orbital, a bond or a distance, and that absence is the point: everything derived here was derived without any of the three.

Where this goes next

HHHNC3vprincipal axis C33 mirror planesno inversion centremay be polarcannot be chiralgroup recovered from the coordinates4 atoms Point groups from coordinates A molecule's symmetry is not a label to be looked up. It is decidable from the atom positions by searching for the operations that permute them, and the search either finds an operation or it does not. 1s · 2pxS = 0 exactlycomputed -9.6e-17 — arithmetic noiseseparation 2.8 bohrcontours at 50% of each densitythe contributions cancel in pairs, so the integral vanishesone electron Exactly zero Where symmetry forbids an interaction the overlap is not small. It is zero — and computing it and finding arithmetic noise is a different kind of statement from computing it and finding a small number. HHCHHTdprincipal axis C36 mirror planesno inversion centrecannot be polarcannot be chiralgroup recovered from the coordinates5 atoms Hybridisation does not explain Methane's photoelectron spectrum has two bands, not one. Four equivalent sp³ bonding orbitals cannot produce that, and the resolution is that hybridisation was never a claim about what a measurement would find.