Molecular Geometry & Bonding
what the shape decides
Concept

Matrix element — where it appears

An integral of an operator between two states, which decides whether a process happens and how strongly. Where symmetry forbids one it is exactly zero rather than small, and computing it gives arithmetic noise.

Named by 14 essays across 5 fields — each of them below, with the objects they name alongside it.

The density of states of a chain of 2000. The 2000 levels of a linear chain, binned into 34 intervals across the band, with the closed-form density drawn through them. The density piles up at both edges because that is where the level spacing turns over, and nothing periodic was assumed to get it.

A density of states is not a spectrum

A molecule's spectrum is a list of positions and a solid's is a shape, and the shape is not the density of states. Between the two sits everything the count leaves out — which transitions are allowed, how strongly, and from where to where.

solids · Bands in a solid 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 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 Three lines, then a hundred. The exact removal spectrum of a 6-site Hubbard ring at half filling: every final state of the ion, at the energy it costs to reach and with the intensity the matrix element gives it. With no repulsion there are 3 lines and they are the occupied orbital energies. At U = 8 there are 100, on a molecule with 6 orbitals — so the spectrum cannot be read as a list of orbital energies, because there are more bands in it than there are orbitals to name.

More bands than there are orbitals

A photoelectron spectrum is read as a list of orbital energies, one band per occupied orbital. Computed exactly for a six-orbital ring, it has three bands with no repulsion and a hundred with eight — and by then fifty-three per cent of the intensity is in lines that no orbital corresponds to. The total intensity is three at every repulsion, exactly, because that is a sum rule and not a fit.

spectra · Photoelectron The test that works until it does not. How many times stronger the weakest fundamental is than the strongest satellite, against the repulsion, on a half-filled ring of six. It starts at 23.8 and falls to 1.15 — a spectrum whose tallest satellite is as tall as its shortest band. The marked repulsion is where the other test fails as well: satellites start appearing inside the range the fundamentals span, so neither height nor position sorts the spectrum.

A hundred lines and no way to sort them

A spectrum with a hundred lines has six fundamentals in it somewhere. Sorting by height works until the tallest satellite is as tall as the shortest band, and sorting by position works until satellites start arriving between the bands — and on a ring of six both stop working at the same repulsion.

spectra · Photoelectron The current does not divide equally between equal rings. The current each ring of an acene carries under a uniform field, ring by ring, for four acenes. Naphthalene's two rings are equal by symmetry; anthracene's middle ring carries 1.180 times what its outer ones do, and tetracene's inner rings 1.222 times. Every ring has the same area and the same six carbons, and the response is a matrix rather than a set of parallel loops.

The current does not divide

A fused ring system's response was computed from the areas of its rings, and the obvious next question was whether the current divides between them the way it divides between two resistors. Giving each ring its own flux and taking the second derivatives says no: the response is a matrix, its off-diagonal entries are nearly half its diagonal ones, and anthracene's middle ring carries 1.18 times what its outer rings do.

symmetry · Aromaticity How nearly a broken symmetry survives. A screened potential splits the n = 2 shell and destroys the degeneracy the linear Stark effect depends on. The field needed to overcome the splitting and restore the linear behaviour runs from 4.3e+4 volts a centimetre at a quantum defect of 0.00040 to 2.9e+7 at a defect of 0.208. The dipole between the states is 3.000 throughout, so the field is exactly the splitting divided by twice it.

How nearly a broken symmetry survives

The hydrogen shell's extra symmetry is what makes its Stark effect linear, and a real atom does not have it. Screening splits the shell, and the field needed to overcome the splitting and restore the linear behaviour is a curve — from forty thousand volts a centimetre at a quantum defect of 0.0004 to thirty million at a defect of 0.21.

symmetry · Representation A pair with no overlap, and a third orbital swept past it. The three levels of a trio in which the two outer orbitals have exactly no overlap with each other, as the third orbital's energy is swept. The middle line is at -13.6 at every point — the antisymmetric combination of the two, which has no partner of its own symmetry and cannot mix with anything. The other two move, so the pair is split by an orbital it has no direct contact through.

A bond order between atoms that do not interact

A diatomic held where its overlap changes sign has no interaction between its two orbitals at all — which is what the sign change of its overlap means. Put a third orbital beside it and the pair is still split, one line sits exactly at the free-atom energy at every third-orbital energy, and the bond order between the two runs to −0.9999. Three measures of the same bond disagree completely.

wrong · Overlap 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 One and eight, over two decades of defect. The number of avoided crossings the whole shell has, against the number of two-state crossover fields its coupled pairs supply, as the quantum defect is swept towards zero. The question is whether the estimated count falls to meet the exact one as the l degeneracy closes. It does not move: one against eight at every defect tried, from 0.02 down to 0.0002, with the estimates spanning a factor of 7.08 throughout.

Consistently wrong is not a limit

Does the two-state picture of a tilted Stark shell become right as the quantum defect closes the l degeneracy? Swept over two decades it does not move: one avoided crossing against eight estimates at every defect. The reason is that every dimensionless quantity settles — the crossing sits at 0.04000 of the zero-field gap and the nearest estimate at 0.9067 of the crossing, and neither is heading anywhere.

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 One spectrum along three directions, and three different sets of estimates. The two-state estimate gap ÷ 2d for every pair of the shell's functions the field couples, with the field along z, along x and at the tilt the defect sweep used. Along z there are three distinct estimates, along x three different ones and at the tilt eight, none equal to any of the axial three. The exact spectrum is identical along all three directions, and its two minima are drawn as vertical lines: the one between coupled levels at 0.0196 and the tangency of uncoupled levels at 0.0400. An estimate is a property of the axes the functions were written along, and a feature is not.

Two levels cannot make a minimum

The one minimum between coupled levels in a Stark shell sits at 0.0196 of the s–p gap, and the nearest two-state estimate at 0.0192 — two per cent away, which reads as the estimates having been aimed at the right feature all along. They were not. Two coupled levels only ever separate, so no estimate can be where its own pair is closest. The minimum belongs to a third level, exists only while the d level sits within a quarter of the s–p gap, and meets the estimate by crossing it.

symmetry · Representation

Named alongside it

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

DegeneracyModel limitAngular momentumStark effectAvoided crossingSymmetry breakingClosed formMolecular orbitalPerturbation theoryPhotoelectron spectroscopyQuantum defectBasis

All concepts