Exactly zero

Overlap decides how strongly two orbitals interact. Put a 1s orbital on one atom and a p orbital on its neighbour, oriented perpendicular to the line joining them. Compute the overlap.

The answer comes out at about $10^{-17}$ — a number with no significant digits in it.

That is not a small number. It is zero, expressed in double-precision floating point, and the difference between those two readings is the whole of this essay.

Why it vanishes #

The s orbital is symmetric about the plane containing the internuclear axis. The p orbital is antisymmetric about it: positive on one side, negative on the other, and identical in magnitude.

Their product is therefore antisymmetric about that plane. Every volume element above it is matched by one below with the same magnitude and the opposite sign, and the integral of an odd function over a symmetric region is zero.

Not approximately zero. Zero, in the same way that $\int_{-1}^{1} x\,dx$ is zero — because the contributions cancel in pairs, term by term, with nothing left over.

What “non-bonding” means #

Chemistry has a word for this situation and the word is often used loosely.

A non-bonding interaction is one where the overlap vanishes by symmetry. The two orbitals do not mix at all: neither combination is stabilised, neither is destabilised, and the resulting molecular orbital is unchanged from the atomic one.

That is a stronger statement than “weakly bonding”. A weak interaction has a small non-zero overlap, mixes a little, and shifts the levels slightly. A non-bonding one does nothing whatever, and pushing the atoms closer does not help — the cancellation is exact at every separation.

Why “exactly” is worth insisting on #

A reader might reasonably ask what turns on the distinction, since a very small interaction and no interaction have similar consequences.

Three things.

It does not depend on the numbers. A small overlap depends on the orbital exponents, the separation and the element. A symmetry-forbidden one is zero for every element at every separation, because it follows from the shapes rather than from the sizes.

It cannot be tuned away. Making the atoms closer, the orbitals more diffuse, or the environment different changes a small overlap and leaves a zero one at zero. The only way to make such an interaction happen is to break the symmetry that forbids it.

It licenses a different kind of argument. From “S is small” one can conclude “the effect is probably minor”. From “S is zero by symmetry” one can conclude “this term is absent”, and absent terms are what make selection rules and orbital-symmetry arguments possible.

The general rule #

The specific case generalises, and the general form is the foundation of a large part of chemistry.

An integral over all space vanishes unless its integrand is totally symmetric under every operation of the relevant symmetry group. If any operation changes the sign of the integrand, the integral is zero, because the operation maps the region onto itself and pairs each contribution with its negative.

That single statement gives:

• Which orbitals may combine, which is this essay.
• Which spectroscopic transitions are allowed, since a transition moment is an integral of the same kind.
• Which molecules may be polar, since a dipole moment is an integral of a vector quantity — and that has its own essay.
• Which vibrational modes appear in an infrared or Raman spectrum.

All four are the same theorem wearing different clothes, and none of them requires computing anything.

The same theorem, on any molecule #

The vanishing pairs above are hand-picked: two orbitals on two centres, chosen because their symmetry about a plane is easy to see. That is a demonstration rather than a general method, and this collection later replaced it with one.

The general form needs the molecule’s group as a list of operations rather than as a label, and it can be generated — the operations found from the coordinates are multiplied together until the set stops growing, and the products are sorted into conjugacy classes by conjugating each by every other. With the classes in hand, a representation’s characters can be multiplied class by class, and the theorem above becomes one line of arithmetic: the integral survives if and only if the product representation contains the totally symmetric one.

Applied to every pair of symmetry species at once, that gives a table.

Two things are worth noticing about those grids. Rather more than half of every table is forbidden — selection rules are restrictive, which is what makes a spectrum interpretable at all. And nothing in producing them mentioned an orbital, an energy or a wavefunction. The input was a set of coordinates.

The check that the grids are not simply agreeing with themselves is the same one applied everywhere: it is fed a claim it must refuse. A forbidden transition declared allowed is rejected, and so is an allowed one declared forbidden.

The forbidden cases, listed #

For two atoms on the z axis, with the standard orientations, the vanishing pairs are worth having explicitly.

An s orbital with a $p_x$ or $p_y$: forbidden. A $p_z$ with a $p_x$ or $p_y$: forbidden. A $p_x$ with a $p_y$: forbidden. In every case the two functions have different symmetry about a plane containing the axis, and the product is odd about it.

By contrast s with s, s with $p_z$, $p_z$ with $p_z$ and $p_x$ with $p_x$ are all allowed, and all give non-zero integrals that fall smoothly with separation.

The list is short enough to be worth stating as a rule rather than as five cases. Put the internuclear line along z. Every one of these functions is either even or odd under reflection in the xz plane and under reflection in the yz plane, and both reflections are symmetry operations of any two-centre problem. A pair whose parities differ under either reflection has an odd product, and an odd product integrates to nothing. That is all the arithmetic there is in it, and it is why the forbidden list can be written down without evaluating anything.

Reading a computed zero #

The figures here print $S = 0$ exactly for the forbidden cases, and report the computed value alongside — a number of order $10^{-17}$, whose digits are quadrature noise and change with the grid.

That number is worth explaining rather than hiding. Double-precision arithmetic carries about sixteen significant digits, so a quantity that should be zero and is built from sums of terms of order one comes out at around $10^{-16}$ or $10^{-17}$. Getting that is the correct outcome; getting $10^{-6}$ would indicate a real error somewhere, and getting exactly $0.0$ would be suspicious, since it would suggest the terms were never computed.

The check requires the magnitude to be below $10^{-12}$, which is far below anything physical and far above the noise floor. And it requires the converse: an allowed overlap declared forbidden must fail, so the check cannot pass by being permissive.

The same theorem in four places #

The rule this essay is an instance of is worth writing out in its general form, because seeing four apparently different chemical rules collapse into one is the sort of thing that makes a subject smaller.

The rule. An integral over all space vanishes unless the integrand is unchanged by every operation of the symmetry group. If any operation reverses its sign, the operation maps the integration region onto itself and pairs each contribution with its negative, so the total is zero.

Four consequences:

Orbital mixing. The integrand is $\psi_A \psi_B$. Different symmetry means the product changes sign, so the overlap vanishes and the orbitals cannot mix — this essay.

Dipole moments. The integrand is the charge density weighted by position. Position is a vector, so any operation reversing a direction kills that component — and only a few groups leave any direction alone.

Electronic transitions. The integrand is the initial state, the dipole operator and the final state. That is the selection rule for absorption and emission, and it is why some transitions are strong and others invisible.

Vibrational activity. Whether a mode appears in an infrared or Raman spectrum is the same integral with a different operator.

Four rules that look unrelated in a first course, and one argument underneath all of them.

What non-zero does not guarantee #

The converse of the rule deserves stating because it is weaker than people assume.

Symmetry can say an integral must vanish. It cannot say an integral must not. An interaction that symmetry permits may still be negligible — if the overlap is tiny, or the orbitals are far apart in energy, or the geometry is unfavourable.

So “symmetry allowed” means “not forbidden”, which is a much weaker statement than “happens”. The energy-matching condition is what decides whether an allowed interaction is actually important, and it is a quantitative matter rather than a symmetry one.

That asymmetry — clean exclusion, weak inclusion — is characteristic of symmetry arguments everywhere. It is why they are so useful for ruling things out and so unhelpful for predicting magnitudes, and why a chemist reaching for symmetry is usually trying to eliminate possibilities rather than to choose among them.

Breaking the symmetry #

The rule has an obvious escape and it is chemically important.

Bend the molecule, substitute an atom, apply a field — anything that removes the symmetry plane — and the integral is no longer forced to vanish. The interaction turns on, usually weakly at first and growing with the distortion.

That is the mechanism behind a large family of effects. Vibrations that lower the symmetry make forbidden electronic transitions weakly allowed, which is why some formally forbidden absorptions are seen faintly. Substituting a hydrogen for a methyl group in an aromatic ring turns on couplings that were zero in the parent. And a whole class of reaction pathways proceeds by distorting a molecule until an orbital interaction that was forbidden becomes allowed.

So “forbidden” means forbidden in that symmetry, and much of chemistry is about what happens when the symmetry goes.

Seeing the cancellation #

The figures on this page draw the plane containing both nuclei, with the regions where the product of the two wavefunctions is positive and negative shown faintly. For a forbidden pair those regions are mirror images.

The comparison is worth making deliberately. The two figures were produced by the same code, integrating the same way over the same grid. One returns 0.78 and one returns $10^{-17}$, and the difference is entirely in the symmetry of the integrand.

That is the strongest form of the argument available without a proof: the same calculation, applied to two cases, giving a substantial number and arithmetic noise. If the vanishing were an artefact of the method, the method would not produce sensible numbers for the allowed cases — and the closed-form check establishes separately that it does.

What an exact zero survives #

A zero that follows from symmetry is worth more than a zero that follows from a calculation, and the reason is a list of things it survives — each of which would move a merely small number.

A change of basis. The integral vanishes because the two functions belong to different symmetry species, and species membership is a property of the functions rather than of how they are expanded. Write them in Gaussians, in Slater functions, in plane waves, or in a grid: the answer is zero in every one.

A change of method. Hartree–Fock, a correlated method, a density functional, an exact solution — every one of them respects the molecule’s symmetry, so every one of them returns zero for the same integral.

Relativity, and everything else that is symmetric. Any term added to the Hamiltonian that commutes with the symmetry operations leaves the vanishing intact, however large the term is.

And numerical error, provided the error is symmetric. A quadrature grid that respects the molecule’s symmetry gives cancellation to the last bit; one that does not gives noise at the level of its own asymmetry.

That last item is the practical one, and it turns the zero into a diagnostic. If a calculation returns a non-vanishing value for a symmetry-forbidden integral, the fault is almost never in the integrator. It is in the geometry — coordinates entered to insufficient precision, an optimisation stopped early, an atom placed a thousandth of an ångström off its symmetry position — or in a grid that has been rotated relative to the molecule.

So the exact zeros are the cheapest test of a calculation’s setup available. They cost nothing to compute, they have a known answer, and the size of the departure measures how far the geometry is from the symmetry it was supposed to have.

That is a different use from the one this essay is about, and it is the same fact. A quantity that is zero for a structural reason reports on the structure, and reading it as a report on the integration is reading it backwards.

The same logic sets the one condition under which the zero can fail to be a zero, and it is worth stating because it is not a limitation of the arithmetic. If the molecule does not have the symmetry — if it is distorted, or vibrating, or sitting in a field or a crystal that lowers its point group — then the two functions are no longer of different species, the integral is not required to vanish, and it does not. That is a real effect rather than an error, and it is what gives forbidden transitions their intensity and forbidden interactions their strength.

What it costs #

The overlap integrals cost a three-dimensional Gauss–Legendre quadrature on a ninety-cubed product grid, which is about seven hundred thousand evaluations of a pair of wavefunctions and takes a fraction of a second. That is the expensive route, and it is the one taken, because the whole value of the claim is that the number was produced rather than reasoned to.

The symmetry route costs almost nothing by comparison — generating a group of twenty-four operations, classifying them and multiplying characters is a few thousand arithmetic operations — and gives a weaker answer. It says whether the integral vanishes and nothing about what a surviving one is worth. The two are complementary in exactly that way: symmetry rules out, quadrature quantifies, and neither substitutes for the other.

What makes the expensive route worth its cost is that it makes the cheap one falsifiable. A symmetry argument that predicts zero and is never checked against a number is a claim with no test attached. Here the integrator is validated first against the closed form for two 1s orbitals, where it agrees to about $6\times10^{-5}$, and then asked for the forbidden cases, where it returns $10^{-17}$. Without the first step the second could be dismissed as a defect of the method. With it, the twelve orders of magnitude between the two are the evidence.

Where the model stops #

Three limits.

Overlap is not the whole interaction. Zero overlap means the two basis functions do not mix directly. In a real many-electron molecule there are indirect routes — mixing through a third orbital, or electron correlation — and “non-bonding” in a one-electron picture does not always mean “no interaction at all”.

The symmetry has to be exact, and a real molecule’s symmetry holds only to a tolerance. A molecule that is nearly symmetric has nearly-zero integrals, not zero ones, and how nearly matters. Real molecules vibrate, so a symmetry that holds at the equilibrium geometry is broken instantaneously most of the time.

Where to read on #

The quantity itself is overlap, and the two frameworks that use it are molecular orbital and valence bond theory.

The same theorem applied to a different integral gives why symmetry forbids a dipole, which is the cleanest example of a property decided without any calculation of the bonding at all.

What the pictures here cannot show. These figures draw a plane through both nuclei, and the cancellation being described is between the half-space above that plane and the half-space below it. A single slice cannot show two half-spaces cancelling; the number printed is the three-dimensional integral, and the drawing is an aid to seeing why it comes out as it does.