Molecular Geometry & Bonding
what the shape decides
Concept

Angular momentum — where it appears

The quantity conserved by a system unchanged under rotation, whose square and one component can be known at once. In a molecule it is largely quenched by the ligand field, which is why most magnetic moments count spins alone.

Named by 9 essays across 3 fields — each of them below, with the objects they name alongside it.

Orbitals at the 90 per cent contour. Several orbitals drawn at the same enclosed fraction and, unless stated otherwise, at the same scale — so the sizes on the page are the sizes. Each contour was solved for separately by integrating that orbital's own density. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3; 2px at 90% of its density, |ψ| = 9.49e-3; 2py at 90% of its density, |ψ| = 9.49e-3.

Complex harmonics against real ones

The p orbitals every chemist draws are not eigenfunctions of anything. They are real combinations of the complex solutions, chosen because they point along axes — and the choice is invisible until a magnetic field makes it matter.

orbitals · Orbital What the angular momentum operator connects. The five real d functions, with a line between each pair the z component of orbital angular momentum connects and the size of the connection on it. Written in this basis the operator is i times an antisymmetric matrix, so its expectation in any real function is exactly zero — that is the quenching, and it holds before any field is applied. The three t₂g functions are connected among themselves, so as a SET they carry eigenvalues 1, 0, -1; the two eg functions are each connected only to something outside the pair, so as a set they carry nothing.

An orbital carries no angular momentum

The d orbitals every chemist draws carry exactly no orbital angular momentum, and the proof is one line about a matrix being antisymmetric. A set of three of them carries a whole unit, which is why the spin-only formula works for most ions and fails for cobalt by nearly a Bohr magneton.

applied · Magnetism Which way the line moves counts the electrons. Three ions, their computed g-values and their measured ones. The shift is −2λ times a sum of squared matrix elements over energy denominators; the matrix elements are exactly two for Lz between dx²−y² and dxy and exactly one for Lx between dx²−y² and dyz, which is why the shift along the axis is four times the shift across it. The last column is what is left over — the orbital reduction factor, which is below one when the electron spends part of its time on the ligands and is a covalency measured with a magnet.

The g-value is the orbital coming back

A ligand field quenches the orbital angular momentum of a d electron, and spin-orbit coupling gives some of it back — upward for a shell more than half full and downward for one less than half full. Which way a resonance line moves counts the electrons, and the size of the move comes from two matrix elements and one optical splitting.

applied · Magnetism Where the electron is, and how fast it is going. The radial distribution in position on the left and in momentum on the right, for the same orbitals. The two run opposite ways: the 1s is the most compact in space and the widest in momentum, and every excited orbital that spreads out in one narrows in the other. Both are normalised, both are the same function, and neither is more fundamental than the other — the transform loses nothing and adds nothing.

The orbital in momentum space

Every orbital has a second picture as complete as the first and almost never drawn. Nothing is added by taking it — it is the same function in the other variable — but the uncertainty product falls out of it, and the functions quantum chemistry is built from turn out to be the only ones that attain the bound.

orbitals · Orbital A field splits the n = 2 shell into whole numbers. The eigenvalues of z inside the shell, which are the shifts a uniform field produces to first order. There are three distinct ones and each is a whole number times (3/2)n, so the splitting is proportional to the field itself rather than to its square — which is what no other atom does.

The symmetry that is not a rotation

Hydrogen's n = 2 shell holds four states at one energy and its rotation group accounts for at most three. The operator that accounts for the fourth is built here out of computed integrals: three matrices whose commutators close into the rotations, whose product with the angular momentum vanishes, and whose Casimir comes out at exactly n² − 1.

symmetry · Representation Tilting the field raises the count, and then lowers it again. How many distinct fields the shell has an event at, against the angle between the field and the z axis. Along either axis there are 3; at a general tilt every one of the 6 coupled pairs has its own field and there are 6. In between the count comes back down at four angles where two events coincide, and at forty-five degrees two separate coincidences happen at once.

Four angles the shell chooses

A field along one axis gives a shell of nine functions three fields with an event, and tilting the field should separate the coincident ones and raise the count towards the number of coupled pairs. It does — from three to six. But not monotonically: at four angles two events collide again, and every one of those angles is the arctangent of a ratio of the shell's own angular integrals.

symmetry · Representation One crossing, and the six fields that are not one. Open marks: every two-state crossover field, at each tilt. Filled line: the field at which the exact spectrum's one avoided crossing actually sits. The estimates scatter over a factor of four to twenty; the real crossing moves by a factor of 1.34 across the whole ninety degrees, and passes the four coincidence angles — the dashed verticals — without any feature at all.

None of the six was a crossing

Counting events in a tilted field finds the count rising from three to six, dropping again at four angles that are exact arctangents of the shell's own integrals. Every one of those statements is true of the two-state estimates. Diagonalising the five-level problem exactly finds one avoided crossing at every tilt, in a field that moves by a third across ninety degrees, and no feature whatever at any of the four angles.

symmetry · Representation A thirty-four per cent variation that is entirely the truncation. The field at which the one avoided crossing sits, against the tilt, computed in the five functions a field in the xz plane couples and in the whole nine-function shell. The truncated answer runs from 1.6435e-5 to 2.2014e-5 — a factor of 1.34. The whole shell's is 2.0876e-5 at every direction, and equals the truncated answer at zero tilt, where the truncation is exact because the field is along z and the excluded functions genuinely do not couple.

The variation was the basis

Solved in the five functions a field in one plane couples, the tilted Stark problem has one avoided crossing at every tilt and a crossing field that moves by a third across ninety degrees. Solved in the whole nine-function shell the field does not move at all — the same number at every direction, to eleven decimal places — and the thirty-four per cent was the truncation.

symmetry · Representation Two levels of different symmetry, closest at 0.0399865. The lowest m = 0 level and the lowest |m| = 1 level of the zero-defect shell, in units of the s–p gap, against the field in the same units. Both fall. Their separation is smallest at 0.039986525, where it is 0.97648 of the zero-field gap, and they never meet. The field along z commutes with the angular momentum about z, the two levels belong to different values of it, and no element of the field connects them.

The crossing nothing couples

A tilted Stark shell's one avoided crossing settled at 0.04000 of the zero-field gap, and the natural guess was a ratio of angular integrals. With the quantum defect taken out the limit is 0.0399865, not four hundredths — and the two levels at that minimum belong to different symmetries about the field, which no element of the field connects. It was never an avoided crossing. The one minimum between levels that do interact sits at half the field, behind a level of the other kind.

symmetry · Representation

Named alongside it

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

DegeneracyMatrix elementModel limitStark effectSymmetry breakingAvoided crossingBasisExpectation valueConventiond orbitalsLigand fieldMagnetic moment

All concepts