Molecular Geometry & Bonding
what the shape decides
Concept

Basis set — where it appears

A fixed set of functions in which a wavefunction is expanded, chosen once and used for every molecule it is applied to. The energy it gives is variational and improves as functions are added; nothing else about it is bounded, and properties weighted at the nucleus or in the tail converge far more slowly.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

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 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 computed at 3 bohr and used everywhere. Three binding curves for H₂⁺ in 2 Gaussians a centre: uncorrected, properly counterpoise corrected at every separation, and corrected once at 3 bohr with that value subtracted throughout. The frozen curve is the uncorrected one shifted down by a constant, so its minimum sits at 2.2270 bohr — exactly where the uncorrected minimum is, and 4.2 millibohr from where the full correction puts it. The depth moves and the structure does not.

A correction computed at one length

The counterpoise correction is expensive, so it is evaluated once at a reference geometry and subtracted across a whole potential surface. A constant does not move a minimum — so a frozen correction returns the uncorrected bond length exactly, at every reference geometry and in every basis, and everything the correction does to a structure is the part that has just been thrown away.

orbitals · Basis 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 Which basis size is usable, against where the line is drawn. The basis size usable at every separation — big enough that the three-body correction matters, small enough that the pairwise assembly reproduces it — against the accuracy line, over two decades. It is not one answer. Four different sizes are the answer over this range, and for much of it there is no answer at all. The standard kilocalorie a mole gives basis 2, and it sits 0.7 per cent above the edge where that answer stops.

The line was holding the answer up

There is exactly one basis size where the three-body correction is worth computing and the pairwise assembly reproduces it, and the accuracy line is the choice that whole picture is most sensitive to. Sweeping the line over two decades gives four different answers, long stretches with no answer at all, and a published window whose lower edge sits seven tenths of a per cent below the standard kilocalorie a mole.

orbitals · Basis A p function fills the polar bond's minimum, and a pair of them fills it nine times over. Carbon monoxide's σ profile along the bond at equal coefficients, for three bases: one s function on each atom, a quarter p mixed into carbon's, and a quarter p on both atoms in the sense that maximises their overlap. The minimum at π/R is 1.35% of the peak with s alone, 3.63% with p on carbon and 12.53% with p on both. The p function proposed to reduce the floor raises it, and the pair raises it most.

The third function fills the floor

A polar σ bond built from one s function per atom has a momentum profile whose minimum at π/R never reaches zero — for carbon monoxide, 1.35 per cent of the peak. The natural hope was that a p function mixed into the diffuse atom would reshape it towards the compact one and lower that floor. It cannot. At π/R an s function on one atom and a p function on the other are exactly out of phase, so a p function on one atom cancels nothing and only adds itself. With p on both atoms in the sense that makes the bond, the pair adds fifteen times as strongly as the s pair cancels: a quarter p takes the minimum to 12.5 per cent. That is eight times the whole range through which polarity moves the s-only bond.

orbitals · Orbital Carbon monoxide's three σ orbitals, and what each does at π/R. The momentum profile along the bond of each occupied σ orbital of carbon monoxide, with weights from an extended Hückel calculation over carbon and oxygen 2s and 2pσ, each scaled to its own peak, against momentum in units of π/R. At π/R the lowest, 3σ, reads 2.52% of its peak; the middle, 4σ, 8.40%; and the highest, 5σ, 85.3% — its profile peaks at 0.82 π/R and has no minimum near π/R at all.

The lone pair sits where the floor was

A polar σ bond's momentum profile has a minimum at π/R whose floor was proposed as a measure of polarity, and the essay that swept the p weight could not say whether carbon monoxide's own σ bond sat near the s-only floor of 1.35 per cent or the quarter-p value of 12.5. With the weights computed — carbon and oxygen 2s and 2pσ, extended Hückel — the answer is that carbon monoxide has no such bond. Its lowest σ orbital reads 2.5 per cent, its middle one 8.4 per cent of its own peak, and its highest, the orbital with the lone pair's sense, has no minimum at π/R at all: it peaks there. The six σ electrons together have no minimum anywhere, and the polar molecule reads lower at π/R than nitrogen does.

orbitals · Orbital Lithium hydride's one σ bond, and the minimum it would need to be covalent to have. The momentum profile along the bond of lithium hydride's occupied σ orbital, from an extended Hückel calculation over lithium 2s and 2pσ and hydrogen 1s, each curve scaled to its own peak, against momentum in units of π/R. As computed the orbital is 91% hydrogen and its profile falls monotonically, reading 10% of its peak at π/R. The same lithium 2s and hydrogen 1s with a quarter of the orbital on hydrogen do have a minimum — at 0.525 π/R, not at π/R, just below lithium's 2s node in momentum at 0.624 π/R.

The minimum belongs to the heavy atom's node

Carbon monoxide had no single σ bond for the momentum-space polarity reading to read, so the next place to look was a hydride, where there is one. Lithium hydride's computed σ orbital is ninety-one per cent hydrogen and its profile falls straight through π/R without a minimum. Across all seven first-row hydrides one occupied orbital of thirteen has a minimum at all, and where a lithium-to-boron hydride could have one, it would not be the bond's fringe: it would be the heavy atom's 2s node in momentum, a flat point that a little hydrogen tips over.

orbitals · Orbital Lithium's 2s three ways, in momentum: three nodes, one of them past the fringe. The momentum radial functions of the same three lithium 2s orbitals, weighted by p² as a radial density is, each scaled by its largest magnitude and signed to start negative, against momentum in units of lithium hydride's π/R. Every one has exactly one node — the hydrogenic at 0.624 π/R, the Hartree–Fock at 0.889 and the nodeless Slater function, which has no node in position, at 1.080, just past the fringe.

The node a nodeless orbital has

Lithium hydride's would-be minimum in momentum was traced to the 2s function's node at 0.624 π/R, with a warning that the position belonged to the hydrogenic model and that a nodeless Slater function would have no node at all. Solving the lithium atom properly moves the node to 0.889 π/R; the nodeless function has one too, at 1.080, because a node in momentum is not a node in position but a disagreement between a function's sum and its slope at the nucleus. Lithium hydride's own bond has no minimum with any of the three. What the function decides is whether a minimum can be born at the node at all.

orbitals · Orbital

Named alongside it

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

Model limitOverlap integralVariationalMomentum orbitalApproximationBasis set superposition errorClosed formConvergenceCounterpoiseGaussianGaussian basisMany-body expansion

All concepts