Molecular Geometry & Bonding
what the shape decides
Concept

Many-electron wavefunctions — where it appears

Describing a system with all its electrons at once rather than one at a time. Exact orbitals do not exist for such a system, so the orbital picture is a basis for an approximation rather than a description of it.

Named by 24 essays across 6 fields — each of them below, with the objects they name alongside it.

Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3.

Orbitals are not where the electron is

A many-electron atom has no exact orbitals at all. The orbital picture is a basis for an approximation — an extremely good one — and treating it as a description of reality is the source of most of the confusion in this subject.

wrong · Approximation The radial function of 3s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

The radial distribution across the periodic table

A 4s orbital is bigger than a 3d by every measure of size except the one that decides which fills first. Penetration is a feature of a small inner peak, and the periodic table's shape depends on it.

orbitals · Orbital 4s and 3d from K to Zn. The mean radius of the 4s and 3d orbitals across the elements K to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

The aufbau order is not a property of the atom

Iron's 3d orbital is more than four times smaller than its 4s and, by every one-electron estimate available, far lower in energy. The 4s fills first anyway, and it empties first too — which is not a paradox but a sign that the filling order was never a list of orbital energies.

wrong · Approximation 4s and 3d from Sc to Zn. The mean radius of the 4s and 3d orbitals across the elements Sc to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What an electron actually feels

A 3d orbital is less than half the size of the 4s beside it and fills second anyway. The charge an electron feels is not the nuclear charge, the correction is a fit rather than a derivation, and the two facts together explain the shape of the periodic table.

orbitals · Orbital How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 4, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is how the neighbouring spins line up in the same wavefunction, which is already -0.0750 at no repulsion at all — that part is exchange — and deepens as the electrons are kept apart.

The hole that is not repulsion

Two electrons in a bond keep out of each other's way, and the obvious reason is that they repel. Setting the repulsion to zero and computing the spin correlation exactly gives −0.125 rather than nothing, and the number is reproduced to nine decimal places by a determinant with no repulsion in it at all.

beyond · Correlation Two sites, two electrons, every level exactly. The four states of a two-site Hubbard model against the on-site repulsion, in units of the hopping. The left-hand edge is the one-electron answer — -2t — and the dashed line is the lowest state with the spins parallel, which cannot hop at all and so does not move with U.

The smallest many-electron calculation

A Hückel energy comes from a model with one electron in it. Add a single term — a cost for two electrons on the same site — and the problem stops being a matrix of size n and becomes a matrix over configurations. Four sites give thirty-six of them, which is small enough to solve exactly, and the answers correct two things the one-electron model got wrong.

bonding · Correlation How often two electrons are in the same place. Double occupancy per site in the exact ground state of a chain of 2, against the repulsion. It starts at the independent-electron quarter and falls to nearly nothing. The second curve is what the state actually pays in repulsion, which peaks and then falls because the electrons have stopped meeting.

Where molecular orbital theory dissociates

The molecular orbital description of a two-electron bond puts both electrons on the same atom half the time — at every bond length, including infinite. The exact answer falls from a half to 0.0039 as the atoms separate, and the point where the two standard models are equally wrong is exactly U = 4t.

bonding · Models How far the ground state is from being one determinant. The occupations of the two natural orbitals of the Hubbard dimer against the repulsion. At zero they are two and nothing, which is a single determinant exactly. As the repulsion grows they converge on one and one, which is a state no single determinant has — the failure is in the description rather than in the number. The entropy of the occupations rises to ln 2, one bit: the two determinants of the singlet.

Two kinds of correlation, and only one is small

Correlation energy is defined as a subtraction, and the definition hides that the thing subtracted is not one thing. In the two-site model the local power of the correlation energy in the interaction falls from two to one — and at every interaction strength it equals, exactly, the occupation of the bonding natural orbital.

beyond · Correlation The valence-bond and molecular-orbital directions in one plane. The valence-bond function and the molecular-orbital function as two directions, at the angle their overlap requires — 45.0°, since they overlap by 0.7071. The exact ground state lies in the plane they span at every repulsion, to twelve decimal places, and swings from one to the other as the repulsion grows without ever arriving. Neither picture is a special case of the other and the answer is not either of them.

Two pictures, one plane

Molecular orbital theory and valence bond theory are taught as rival descriptions of a two-electron bond. In a model small enough to solve exactly they are two vectors in a two-dimensional space, the exact answer lies in the plane they span at every repulsion, and it is neither of them at any repulsion but two.

bonding · Models The same dimer, one, twice and three times over. Independent Hubbard dimers with nothing between them, solved exactly and solved in a space with the configurations that make more than one of them ionic thrown away. The exact energy is exactly additive; the truncated one is exact for a single dimer, because there is nothing there to throw away, and falls behind by 0.193 for two and 0.485 for three. The error per dimer grows, which is what makes a method size-inconsistent rather than merely approximate.

A method that is not additive

Put a molecule next to a copy of itself, far enough away that they do not interact, and the exact energy doubles exactly. A truncated calculation does not — because the truncation forbids both halves being excited at once, which is something the pair can do and neither half can.

bonding · Correlation Three answers to one question. The energy to remove an electron from a half-filled four-site system, computed three ways against the repulsion: exactly, by solving a self-consistent field twice — once for the molecule and once for the ion — and by reading the highest occupied orbital energy straight off the molecule, which is Koopmans' theorem. All three agree exactly at zero repulsion. The theorem always sits above the two-calculation answer, because letting the ion relax can only lower it; the exact answer sits above both, because the molecule is more correlated than its ion. The two errors have opposite signs and do not cancel: the residue grows to 6.03.

Koopmans' theorem is exact for nothing

Reading an ionisation energy off an orbital energy neglects two things that pull in opposite directions, and the cancellation between them is quoted as the reason it works. Compute all three energies in a model where the exact answer is available and the cancellation is real, partial, and gone by the time the repulsion is twice the hopping.

wrong · Photoelectron The electrons repel each other less inside the complex. Seven chromium(III) complexes. Δ is the first band; B is solved from the second in closed form; β is B against the free ion's 918 cm⁻¹, which is measured on the gaseous ion. Every β is below one — the electrons in a complex repel each other less than the same electrons in the free ion, because they have more room. The two orderings are different: fluoride splits least and reduces the repulsion least, cyanide does both most, and the middle of the two series is not the same middle.

The electrons repel less inside the complex

Two measured bands determine two parameters in closed form, and one of them is the repulsion between the d electrons — which comes out below the free ion's value for every complex, by between two and forty-seven per cent. The ligands that split most are not the ligands that reduce the repulsion most, so a complex is characterised by two numbers rather than one.

applied · Ligand field Three lines, then a hundred. The exact removal spectrum of a 6-site Hubbard ring at half filling: every final state of the ion, at the energy it costs to reach and with the intensity the matrix element gives it. With no repulsion there are 3 lines and they are the occupied orbital energies. At U = 8 there are 100, on a molecule with 6 orbitals — so the spectrum cannot be read as a list of orbital energies, because there are more bands in it than there are orbitals to name.

More bands than there are orbitals

A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.

spectra · Photoelectron A correction that stops belonging to the system it is added to. The exact ground state of a four-site Hubbard ring, the unrestricted mean field's, and the composite: the mean field plus the correlation correction computed on the symmetric molecule. At ε = 0 the two systems are the same one and the composite is exact. As the sites are made unlike, the transferred correction stops being the right one — the exact correlation energy shrinks towards nothing while the transferred number does not — and the last rows are the recipe adding a correction almost as large as the error it is meant to remove.

The correction that was computed somewhere else

Every composite method rests on one assumption: that an expensive correction computed on a small case can be added to a cheap calculation on a large one. Tested on four sites where both answers are exact, it removes 99.74 per cent of the cheap method's error near the reference and −1450 per cent of it further away — at which point the recipe is fifteen times as wrong as the calculation it was improving.

wrong · Approximation The test that works until it does not. How many times stronger the weakest fundamental is than the strongest satellite, against the repulsion, on a half-filled ring of six. It starts at 23.8 and falls to 1.15 — a spectrum whose tallest satellite is as tall as its shortest band. The marked repulsion is where the other test fails as well: satellites start appearing inside the range the fundamentals span, so neither height nor position sorts the spectrum.

A hundred lines and no way to sort them

A spectrum with a hundred lines has six fundamentals in it somewhere. Sorting by height works until the tallest satellite is as tall as the shortest band, and sorting by position works until satellites start arriving between the bands — and on a ring of six both stop working at the same repulsion.

spectra · Photoelectron Where the electrons are, without subtracting anything. The opposite-spin pair distribution of a half-filled ring of 6 at six repulsions, by separation, each divided by what uncorrelated electrons of the same density would give. At no repulsion it is one everywhere; at a repulsion of 16 the chance of finding two electrons on one site is 0.0430 of that, and what is missing has turned up next door. Nothing here is a difference between two calculations.

Where the electrons are, without subtracting anything

A correlation energy is an exact energy less a mean-field one, so the answer depends on which mean field was picked — by a factor of 259. The pair distribution says the same thing with no subtraction in it, and two systems matched to the same correlation energy turn out to have electrons in visibly different places.

beyond · Correlation A ring of 6 as the neighbour repulsion is turned up. At an on-site repulsion of 8, three quantities against the nearest-neighbour repulsion: the alternating structure factor, the double occupancy, and the nearest-neighbour opposite-spin pair distribution. The rise is steepest at V = 4.5, which is 0.563 times the on-site repulsion. Far past it the ring is charge ordered — nearly every electron paired on alternate sites, which is what a double occupancy approaching a half means.

The give-back that turned into a saving

Take a wavefunction optimised for an on-site repulsion, weight its correlation hole with an interaction that reaches one neighbour, and the enhancement at one bond gives back forty-four per cent of the on-site saving. Putting that neighbour term into the Hamiltonian and solving exactly does not shrink the give-back. It reverses its sign.

beyond · Correlation The contrast, out to a repulsion of sixteen thousand. The ratio of the weakest fundamental to the strongest satellite, against the on-site repulsion, for eight systems with two electrons each. Both axes logarithmic. The dashed line at two is the factor the intensity test needs. Every curve flattens above it and none of them crosses, at any repulsion — including a repulsion sixteen thousand times the hopping.

A contrast with a closed form

Below half filling the satellite test flattens instead of failing, and the value it flattens at could be above or below the factor of two the test needs. It is — on all eight systems, by between 1.25 and 3.7 times. And on a ring the limit is (1 + 2cos(π/n))², to six figures, on every ring tried.

spectra · Photoelectron How much stronger a fundamental is than a satellite, against the repulsion. The weakest fundamental divided by the strongest satellite, for a six-site ring and chain at every filling from a third to a half, against the on-site repulsion. Below the line at two the two kinds of line cannot be told apart by their height. The half-filled systems cross it and the third-filled ones do not — not at any repulsion up to sixty-four times the hopping, where the third-filled ring is still at 8.3.

A satellite that never loses its place

The repulsion at which a satellite stops being tellable from a fundamental orders exactly with the one-electron gap across four systems. Changing the gap by the filling instead is the sharper test, and the ordering does not survive it: a six-site chain has a larger gap at half filling and a smaller boundary. Below half filling there is no boundary at all, at any repulsion up to sixty-four times the hopping.

wrong · Photoelectron One sign change is a root and the other is not. The solved give-back against the neighbour repulsion on a ring of 6 at U = 8. It crosses zero at V = 3.895, where the price of everything beyond contact really is nothing, and changes sign again at V = 4.159, where the quantity it is a fraction of has vanished instead. The curve is broken at the second because it is an asymptote and not a crossing; four of the 12 points fall outside the band drawn here.

A sign change is not always a zero

The solved give-back changes sign somewhere between a neighbour repulsion of two and one of four, and a bisection looks like the way to find the value where the structure beyond contact contributes exactly nothing. It changes sign twice. One crossing is that value; at the other the quantity the fraction is a fraction of has vanished instead, and a bisection reports the two in identical words.

beyond · Correlation The contrast at three fillings, and the floor two of them reach. The intensity contrast on a ring of 6 against the on-site repulsion, at three fillings. At two electrons it settles on a number well above the factor of two the test needs. At half filling it falls through two and lands on exactly one from U = 64 upward — and every point where it reads exactly one is a point where the cut between fundamental and satellite falls between two lines of identical weight. Those are drawn hollow.

A ratio of exactly one is a tie

Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.

spectra · Photoelectron The same three fillings, on a ring and on a chain. The intensity contrast against the on-site repulsion at two, four and six electrons, for both geometries. Two of the ring's three curves flatten onto exactly one and stay there — the hollow marks, where the rank cut falls between two degenerate lines. The chain's corresponding curve approaches the same value from above without reaching it, because a chain of six has no exactly degenerate removal lines at any repulsion at all.

The number the tie got right

On a ring of six the contrast at half filling comes out exactly one, and the one is an artefact — the rank cut had landed between two lines of identical weight, so the ratio was a quantity divided by itself. The chain of six has no such pair anywhere, at any repulsion, at any filling. Its contrast at half filling converges to one anyway.

spectra · Photoelectron The second number is the first one, rearranged. The composite's error against the change in the correlation energy, at every point on both axes of the square. They lie on the diagonal because they are the same quantity: the composite is the target's mean field plus the reference's correlation energy, so its error is the reference's correlation energy minus the target's. The largest departure across 20 points is 2.2e-16, which is the arithmetic's own precision and not a measurement.

The second number is the error, rearranged

A cheap diagnostic for a composite method leaves a scatter it cannot explain, and the number that ought to close it is the change in the correlation energy, already computed at every point, so the test is arithmetic rather than a calculation. It is arithmetic, and the arithmetic is the answer. The composite's error is that change with a sign on it.

wrong · Approximation Six removal lines, each smooth, and the contrast is whichever two sit at the cut. The weight of each of the six strongest removal lines of the half-filled chain of six, followed from U = 16 upward by continuity in energy and labelled by the energy it tends to. Every line is monotone from U = 19. The contrast is the third strongest over the fourth, so it changes whenever two lines exchange those ranks: at U = 28.92 the line tending to +0.45 overtakes the one tending to −1.80 and the two weights are equal, and at U = 157 the fourth and fifth exchange. Lines at ±E, drawn in one colour, converge to one weight.

The limit of one is a parity

A half-filled chain of six has no degenerate removal lines, and its intensity contrast still goes to one — after dipping near U = 32 and rising again to U = 128. Followed line by line, every removal line is smooth and monotone; the dip is an exact tie between two lines at U = 28.916, and the rise ends where two satellites change places. At large repulsion the lines pair up at ±E with equal weights, so the contrast goes to one exactly when the cut falls inside a pair. On a chain of four it does not, and the limit is 1.3125.

wrong · Photoelectron

Named alongside it

The objects these essays reach for when they reach for this one.

Exact diagonalisationHubbard modelElectron correlationOn-site repulsionDouble occupancyModel limitPhotoelectron spectroscopyDegeneracyIonisation energyCorrelation energyHartree–FockKoopmans theorem

All concepts