4 sites, minimised

One of the figures on where the atoms go: Repulsion minimised on a sphere, the angles that fall out of it, and the arrangements that are not all alike.

Nine essays draw this figure, each at the values its own argument needs rather than at the setting shown above. What each one uses it to show is below, in the words of its own caption.

VSEPR, computed

Four points on a sphere, placed by minimising their mutual repulsion. The angle printed was measured off the result, and it agrees with arccos(−1/3) to three decimal places. Nothing in the minimisation that produced this figure contains the number 109.

Seven points, turned rather than viewed from elsewhere — the hardest case on this page to read by eye, and the one where the readout does the work. The angle spectrum beneath the picture is measured again from the rotated coordinates at every step of the slider, and neither the number of distinct angles nor any of their values moves. A turn that produced a different spectrum would mean the angles had been computed wrongly.

Five sites, which is where the minimisation stops giving equivalent positions: a trigonal bipyramid with two axial sites and three equatorial ones, subtending three distinct angles rather than one. Nothing chose that shape — the same descent that returns a tetrahedron at four and an octahedron at six returns this, and its inequivalence is the reason a five-coordinate molecule has two kinds of site to substitute into.

The energies themselves are checked against published Thomson minima at every count, and the comparison found a real defect: the seven-point case had settled 2.5 × 10⁻⁵ above the true minimum, at an arrangement whose right angles ran from 87.7 to 92.5 degrees.

The shapes above six coordination

Eight points on a sphere, arranged as a cube and as the minimum. The cube is the arrangement everybody expects and it is not the answer — twisting one face by forty-five degrees gives a square antiprism, which is lower. Both energies are computed rather than assumed.

Seven points, minimised: a pentagonal bipyramid. Two axial sites at 180 degrees, five equatorial at 72, and every axial-to-equatorial angle at exactly 90. The angles are measured off the result, and the arrangement turns with the slider while they are re-measured at every step.

Eight points, minimised: the square antiprism the first figure compared against a cube. Four distinct angles and no equivalent way to describe it in words, which is the pattern from here upward.

What a lone pair is worth

The bond angle a weighted repulsion minimisation gives for two lone pairs and two bonding pairs, as the lone-pair weight runs from one to 3.2. At a weight of one all four domains are equivalent and the angle is the tetrahedral one; heavier lone pairs squeeze the bonding pair together. Water and hydrogen sulfide are marked at the weights that reproduce their measured angles, and the two weights are not close.

The same curve with no lone pair at all, which is the control the fit needs. With four equal domains the angle does not depend on the weight, because there is no unequal domain for the weight to describe — the curve is flat at 109.4712°, and every departure from that number in the figures above is bought by the parameter rather than produced by the arrangement.

The same curve for one lone pair and three bonding pairs, with ammonia and phosphine marked. Both are pyramidal, both have one lone pair, and their fitted weights differ by a factor of two and a half. Whatever the weight is measuring, it is not a property of the lone pair.

Which angles are symmetry and which are the model

The distinct angles of the minimised arrangement of four, five, six and seven domains, under repulsions going as 1/r, 1/r², 1/r³, 1/r⁶ and 1/r¹². For four, five and six the angles are identical to three decimal places across the whole sweep. For seven they are not, and neither is how many of them there are.

Four domains at the Coulomb minimum. The tetrahedral angle here is measured off the arrangement, and the same arrangement is what a 1/r¹² repulsion returns. Nothing in the picture depends on which of the two produced it.

Six domains: an octahedron, with angles of 90° and 180° at every exponent tried. Both numbers are pinned by the symmetry of a shape in which all six sites are equivalent.

A cage needs one pair more than it has corners

The eight-vertex case, drawn elsewhere for a different reason. The repulsion minimum is the antiprism and it has two square faces, so the borane cage of eight vertices is a different polyhedron from the one a repulsion argument produces — the only size at which the two disagree.

The lone pair is not the missing term

Where the geometry comes from. Four electron domains minimising their mutual repulsion give the pyramidal arrangement whose axis every quantity above is measured along; the fourth domain is the lone pair, which enters this essay’s arithmetic as a direction long before it enters as a moment.

VSEPR does not reach a transition metal

Four sites minimised on a sphere: a tetrahedron, one distinct angle, 109.47°, measured from the minimised coordinates. Nothing about the central atom enters the calculation, so this is the answer for carbon, for silicon, for nickel and for platinum alike.

The sites are not the same size

The angle a four-bond arrangement takes as one site’s demand is raised, with the measured angles of real molecules marked. This is the knob that works, fitted to the observations rather than derived — and the fit is what the map above uses as its vertical axis.

The atoms are not at the points

And the kind of statement that is a mean rather than a point: four domains minimising their repulsion give 109.47°, which is the minimum of a function and not a place any atom sits. The angle is exact; the arrangement it describes is a time average several degrees wide.