Molecular Geometry & Bonding
what the shape decides
Series

Correlation — the series

11 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · bonding
  2. 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.

    part 2 · beyond
  3. 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.

    part 3 · bonding
  4. 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.

    part 4 · beyond
  5. Two is the ordinary answer, and one is a different kind of correlation. The local exponent of the correlation energy in the repulsion, against the repulsion, for three systems at half filling. The two closed-shell systems tend to two as the repulsion vanishes, which is ordinary perturbation theory. The ring of four tends to one, at repulsions fifty times smaller than the hopping.

    A third kind of correlation

    The two-site model gave an exact identity — the power of the correlation energy in the repulsion equals the occupation of the bonding natural orbital — and asked whether anything like it survives with more orbitals. It does not, and the way it fails is better than the identity was: a ring of four gives a power of one where every closed-shell system gives two, at repulsions fifty times weaker than the hopping, because its reference was never a single state.

    part 5 · beyond
  6. 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.

    part 6 · beyond
  7. One exact state, two correlation energies. The energy each of two mean fields misses, against the repulsion, for one system whose exact energy is a single smooth curve. Below U = 2 the unrestricted search returns the restricted answer and the two definitions agree to the last bit. Above it they part: at U = 32 the restricted reference reports -27.82 and the unrestricted one -0.11, a factor of 259.43, and one is growing while the other falls.

    The reference decides the correlation

    The correlation energy of one exact state, measured against two references that are both called Hartree–Fock, is −27.82 and −0.107 at the same repulsion — a factor of 259, on a system whose exact energy is a single smooth curve. Below the instability at U = 2 the two agree to the last bit; above it one grows without limit while the other falls, and the reference that reports almost no correlation has ⟨S²⟩ = 1.99 where a singlet is zero.

    part 7 · beyond
  8. 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.

    part 8 · beyond
  9. The same hole, priced three ways. The correlation hole of a ring of 6 at a repulsion of 8, weighted by three interactions. With an on-site interaction the answer is 100 per cent at separation zero — as an identity, since the interaction is zero everywhere else. With one that reaches a neighbour, the enhancement at separation one costs rather than pays, and gives back 44.0 per cent of the on-site saving; with a Coulomb tail, 46.9. Everything beyond one neighbour is worth under a twentieth of the on-site term.

    Half of it is given back at one bond

    The correlation hole of a ring removes 1.2653 pairs from zero separation and puts 1.1126 of them one site away. With an on-site repulsion that second number costs nothing, because the interaction is zero there — but with anything that reaches a neighbour it costs 44 per cent of what the hole saved, and with a Coulomb tail 46.9.

    part 9 · beyond
  10. 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.

    part 10 · beyond
  11. 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.

    part 11 · beyond

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