Molecular Geometry & Bonding
what the shape decides
Concept

Variational — where it appears

The property that an approximate energy can only lie above the exact one. It makes lowering the energy a one-way test, and it makes the energy the property an approximate wavefunction gets least wrong.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

The cost of a ratio decided somewhere else. Four bases containing exactly the same six primitives, differing only in how many of the linear coefficients the calculation may choose. The horizontal axis is the effective nuclear charge, which is this one-electron problem's only knob for a different environment; the contraction was fitted at one. At 1.238 — the exponent H₂⁺ chooses when a bond forms — the fully contracted basis is 0.0283 hartree above what the same six functions could give, and one freed coefficient removes most of it.

A contraction is a decision made once

Every published basis set freezes its primitive functions into fixed combinations, on an isolated atom, before any molecule is in sight. Freeing one coefficient recovers three quarters of what that costs — and freeing it at the other end of the basis recovers one per cent.

orbitals · Basis The cliff, and the slope leading to it. The smallest eigenvalue of the overlap matrix, and what an extra function is worth, as that function is brought towards one already in the basis. Both fall together, and the energy stops improving long before the matrix stops being invertible.

A function that is already there

Adding a function to a basis set can only lower the energy, so a bigger basis is a better one. What that leaves out is that the same act makes the functions less independent: an optimised basis's smallest overlap eigenvalue halves with every function added, and a function placed on top of one already present buys less than a millionth of what a well-placed one buys while driving that eigenvalue to 4×10⁻¹⁰ — past which the calculation is refused outright.

orbitals · Basis The Compton profile, exact and fitted. The momentum density integrated over the two perpendicular directions, for the exact 1s and for three fitted bases. The exact curve is 8/3π(1 + q²)³ in closed form; the fitted ones are sums of Gaussians and are cusped differently at the origin, which is the position-space cusp showing up as a shape in momentum.

The measurement a basis was not fitted to

Six Gaussians reproduce hydrogen's energy to eleven parts in a hundred thousand and its Compton profile to three parts in a thousand — twenty-five times worse, on a quantity an X-ray scattering experiment measures directly. The gap between the two errors widens as the basis is improved, because the energy is the one property a variational fit is best at.

orbitals · Basis 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 One of these is guaranteed to improve, and it is not the one anybody measures. The relative error in the energy and in four properties of the fitted function, against the number of Gaussians. The energy falls at every step, because that is what the variational principle promises. The mean radius is exact for the single-function basis and 332 thousand times worse for the two-function one, and the density at the nucleus is still 6.2 per cent wrong where the energy is wrong by 0.011 per cent.

The property that gets worse

One Gaussian fitted to hydrogen gets the mean radius exactly right — 1.500000, against an exact 1.5. Two Gaussians get it wrong by 1.4 per cent, which is three hundred and thirty-two thousand times further out, while the energy improves fivefold. The variational principle bounds one number and says nothing whatever about any other, and the sequence of errors in everything else need not even be monotone.

orbitals · Basis Two errors, opposite signs, four orders of magnitude apart. A finite basis makes H₂⁺'s binding too large by letting each atom borrow the other's functions, and too small by describing the molecule incompletely. Both are computed here against the exact binding of 0.102634 hartree. The second is thousands of times the first at every basis size, and it is the first that counterpoise removes — so the corrected number is further from the true one than the uncorrected at every row of this table.

The basis the other atom lent

Two atoms in a molecule are described in each other's functions and the separated atoms are not, so the molecule is treated better than the pieces and the binding comes out too large. That is the basis set superposition error, it is removed by a standard correction, and for H₂⁺ in four Gaussians a centre it is six tenths of a microhartree against an incompleteness error of twelve millihartree — a factor of eighteen thousand the other way.

orbitals · Basis 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 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 correction collapses and the error made assembling it grows. At a separation of 2 bohr, two quantities against the number of Gaussians a centre. The trimer's own counterpoise correction falls by a factor of 2717 from one function to six — a bigger basis has less to borrow. The fraction by which summing the pairwise corrections overshoots it rises from 5.3 per cent to 38.0. Improving the calculation makes the assembly proportionally worse.

The correction that gets harder to assemble

Summed across a trimer, pairwise counterpoise corrections come to fifteen per cent more than the trimer's own. Three Gaussians a centre is a small basis, so the natural question was whether the fraction shrinks with a better one or stays put. It does neither. The correction falls by three orders of magnitude and the fraction grows fivefold.

orbitals · Basis Both quantities, and the line they have to be read against. At the closest separation, the counterpoise correction itself and the error a pairwise assembly of it makes, against the number of Gaussians a centre. Both fall — the correction by a factor of 3631, the error by 793 — and the fraction rises by exactly the ratio of those two. The dashed line is a kilocalorie a mole. The only basis where the correction is above it and the error below it is two.

One basis size where it is worth doing

The pairwise assembly's error grows with the basis — 5.25 per cent at one Gaussian a centre, 37.98 at six — while the correction itself falls by a factor of three thousand. Which of the two should a practitioner care about? Drawing a line at a kilocalorie a mole answers it: there is exactly one basis size at which the correction is worth computing and its pairwise assembly is accurate enough to use.

orbitals · Basis

Named alongside it

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

Model limitConvergenceApproximationBasis setClosed formGaussianBasisCorrelation energyEigenvalueElectron correlationExact diagonalisationHubbard model

All concepts