Molecular Geometry & Bonding
what the shape decides
Concept

Double occupancy — where it appears

How often two electrons of opposite spin are found on the same site. A mean field's value follows from the density alone and cannot fall; the exact one falls as the inverse square of the repulsion, and the difference is the correlation energy.

Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.

A coupling that is second order in the hopping. The singlet–triplet splitting of a two-site Hubbard model, and the same quantity multiplied by U. The product settles on −4t² — -4 at U = 64 — which is what makes the coupling a second-order effect rather than a term somebody put in.

What couples two spins

Two magnetic ions a few ångströms apart interact far too strongly to be doing it magnetically — the dipole–dipole energy is about 0.06 wavenumbers and the measured couplings run to hundreds. What couples them is hopping, which the Pauli principle allows for antiparallel spins and forbids for parallel ones, and the exact answer is −4t²/U.

applied · Magnetism 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 energy is the last thing a wrong wavefunction gets wrong. Two errors against the error in the wavefunction, on log axes, for a chain of 2 at U = 4t. The energy's line has slope 2.00 and the double occupancy's has slope 1.01: the first is second order in the error and the second is first order. So the two lines diverge as the wavefunction improves, and the energy stops being evidence about anything else long before it stops improving.

A better energy is not a better answer

The variational principle makes the energy a one-way test: lower is closer. It also makes the energy the least sensitive thing a wavefunction gets wrong — second order in the error where every other property is first — so the two diverge without limit as a calculation improves.

wrong · Approximation 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 The error, against the repulsion it is an error about. The energy a mean field misses, for two electron counts on 4 sites, against the strength of the repulsion. Each is a straight line at large repulsion and the dashed line through it is not a fit: its slope is the count of coincidences a uniform density forces, computed from the electron number and the site number alone.

A mean field cannot get out of the way

The energy a mean field misses grows without limit as the repulsion rises — 27.82 at U = 32 for four electrons on four sites, against an exact energy of −0.2946, so the error is ninety-four times the answer. The rate it grows at is not an energy at all: it is n²/4N, a count of the coincidences a spread-out density cannot avoid.

beyond · Correlation 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 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

Named alongside it

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

Electron correlationHubbard modelExact diagonalisationOn-site repulsionMany-electron wavefunctionsModel limitCorrelation energyReference stateConfiguration interactionDensity matrixHartree–FockOne-electron models

All concepts