Molecular Geometry & Bonding
what the shape decides
Concept

Configuration interaction — where it appears

A wavefunction written as a sum over several arrangements of electrons in orbitals rather than one. It is what a single determinant cannot do, and truncating the sum destroys the property that two distant copies of a system have twice its energy.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

Splitting goes with overlap. For each pair, the atomic levels on the outside and the combinations they form in the middle, with the splitting drawn in proportion to the computed overlap. A pair that symmetry forbids does not split at all, because its overlap is exactly zero.

Molecular orbital and valence bond

Two frameworks, taught as rivals, describing the same molecules. One starts from delocalised orbitals and localises; the other starts from localised bonds and delocalises. Pushed far enough they meet.

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 A cheap number that predicts an expensive failure. Thirty-two systems. Along the bottom, how far the mean field's own symmetry breaking has moved between the reference and the target — a quantity available before any exact calculation. Up the side, how wrong the transferred correction turns out to be. They rank together at 0.902, and the open marks are the systems whose broken solution has collapsed entirely, which is where the diagnostic stops being a scale and becomes a warning.

The warning a cheap calculation gives

A correlation correction computed on one system and carried to another works until it does not, and nothing in the scheme says in advance which. The mean field's own symmetry breaking says: it collapses at a definite field, and the transfer fails where it goes. Across thirty-two systems the two rank together at 0.90.

wrong · Approximation An antibonding occupation fills the zero and moves the minimum. The profile along the bond near π/R, per electron, on a logarithmic scale, for five antibonding occupations of the same two orbitals. With nothing in the antibonding orbital the profile is exactly zero at π/R = 1.573. Two hundredths of an electron leave a minimum at 1.608, which reads the separation as 1.954 bohr instead of 1.997. At 0.104 the minimum becomes a flat shoulder, and the Heitler–London bond, at 0.236, has none.

The zero belongs to one determinant

A bonding orbital's momentum profile along the bond is exactly zero at π/R, and that zero reads a bond length with nothing fitted. It is a property of putting every electron into that one orbital. Any antibonding occupation fills it in linearly and drags the minimum outward, a tenth of an electron erases it, and the valence-bond wavefunction built from the same two functions never has one at any separation.

orbitals · Orbital Repulsion puts negative spin on the middle orbital. The spin on the middle orbital of the symmetric three-orbital radical against the on-site repulsion, solved exactly. With no repulsion it is zero: the singly occupied orbital has a node there, and one determinant cannot put negative spin anywhere. With repulsion it is negative at once, and it reaches −0.20 at 8 eV. The shaded band is the allyl radical's central carbon, from its measured proton hyperfine coupling of +4.06 gauss read through McConnell's relation — which the trio meets at a repulsion between 5.5 and 6.9 eV.

Spin below zero

A three-orbital radical read from its singly occupied orbital has no spin on its middle orbital unless its ends are both unequal and coupled — and never negative spin anywhere. Solved exactly with an on-site repulsion, the middle orbital goes negative at once, reaches the allyl radical's measured −0.15 to −0.18 at a repulsion of five and a half to seven electronvolts, and responds to the end–end coupling at first order where the one-determinant leak responds at second. The two have opposite signs, so in a bent, unequal radical the middle orbital's spin falls and then rises, and the unequal ends that let the leak in drain most of the polarisation first.

bonding · Overlap

Named alongside it

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

Electron correlationExact diagonalisationHubbard modelOn-site repulsionDouble occupancyMany-electron wavefunctionsModel limitMolecular orbitalValence bondBasisHartree–FockOverlap integral

All concepts