Molecular Geometry & Bonding
what the shape decides
Concept

Trace — where it appears

The sum of a matrix's diagonal elements, which equals the sum of its eigenvalues. It gives a check on any diagonalisation that costs nothing and is independent of every other check.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — the full band width beside them obeys no such rule.

The width of a band is a count of neighbours

The mean of the squared level energies equals the average coordination — exactly, for any structure, with no limit taken and no periodicity assumed. It is the one statement in this field that is arithmetic rather than physics, and the usual textbook formula for band width is a special case of something weaker.

solids · Bands in a solid Why a character table has the rows it has. All 20 groups this site defines, with the two counts that fix the shape of every character table: the number of representations equals the number of classes, and the sum of the squares of their dimensions equals the order of the group. Neither column pair differs anywhere in the table, and there is no room for another row in any of them.

Why a character table stops where it stops

A character table is square, and both of its dimensions are forced. The number of representations equals the number of classes, the squares of their dimensions sum to the order of the group, and between them there is no room for another row.

symmetry · Representation ⟨x²⟩ is the average number of neighbours. For each structure, the mean coordination counted off the edge list beside the mean of x² measured off the eigenvalues. They are equal by an identity about graphs, not by a limit — and the binding each bond supplies, in the last column, obeys no such rule and falls as neighbours are added.

The bond that weakens as neighbours multiply

Wrapped structures with two, four and six neighbours per site give a mean of x² of exactly 2, 4 and 6. Their binding per site goes 1.272, 1.611, 1.979 — and their binding per bond falls from 0.636 to 0.330, which is why a metal atom with twelve neighbours has weak bonds and a great many of them.

solids · Cohesion Three quarters, exactly, for every mode that is not totally symmetric. The depolarisation ratio of every Raman-active mode of five molecules, computed from a bond-polarisability model. 18 of the 26 sit on three quarters to better than a millionth, and they are exactly the modes that are not totally symmetric: the mean polarisability derivative is a trace, a trace is invariant, and an invariant has no derivative along any other species. The polarised ones below the line are the totally symmetric modes, and where they sit is a property of the model rather than of the group.

The one intensity symmetry does fix

Symmetry says which bands exist and declines to say how strong they are. It makes exactly one exception, and it is a ratio: every Raman band that is not totally symmetric is depolarised by exactly three quarters, whatever the molecule and whatever the model — and methane's symmetric stretch is polarised by exactly zero, because its group is cubic.

symmetry · Spectrum The winner changes at 3.790 and not one moment changes with it. Above: the binding per site of the thirty-two-bond arrangement minus the thirty-bond one, against the contrast; it changes sign once, at 3.790, and the two arrangements' gaps cross only at 4.849. Below: for every moment of the density of states from the second to the twenty-fourth, the sign of the same difference across the same range. The second and third are equal for every arrangement; from the fourth up every one keeps a single sign from a contrast of 1 to 8. No moment sees the switch.

No moment sees the winner change

On a frustrated net the arrangement that binds best at half filling changes at a contrast of 3.790, and the gap at the Fermi level does not follow until 4.849. The natural replacement for the gap is a moment of the whole band. It fails more completely than the gap did. The second and third moments are identical for every arrangement at the composition; the fourth is exactly the count of unlike bonds; and every moment from the fourth to the twenty-fourth keeps one sign from a contrast of 1 to 8. An account of the band that matches its first sixteen moments puts the change at 3.980, and only the one that resolves every level gets 3.790.

solids · Cohesion

Named alongside it

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

Tight-binding modelsAdjacency matrixBands in a solidCharacter tableCohesionCoordination numberDensity of statesGraphIrreducible representationsModel limitPoint groupSymmetry operation

All concepts