Molecular Geometry & Bonding
what the shape decides
The collection

Every essay — page 27

Page 27 of 36, continuing through the fields in the same order.

Orbitals Where the atoms go Bonding models What symmetry decides Beyond the octet What a spectrum settles When the molecule does not stop What the shape is for What is taught wrongly Series Named objects Orbitals Refutations Search

What a spectrum settles

A spectrum is a list of positions, and how many there can be is decided by the shape before any of them is measured. Counting them, and reading a structure back out.

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.

7 figures A ratio that measures a distortion, and squares it first. How far ammonia's depolarised bands come off three quarters against how far one of its bonds has been stretched. Undistorted the departure is 3.1e-8, which is the rounding in the stored coordinates rather than a physical effect; at 0.1 Å it is 0.04239, and the slope is 1.999 — the departure goes as the square of the distortion, so a ratio measured to three decimals fixes a length to one and a half.

A ratio that squares what it measures

A depolarised Raman band sits at exactly three quarters because symmetry says its mean polarisability derivative is zero. Distort the molecule and it comes off — by 4.5 × 10⁻⁴ for a hundredth of an ångström and 0.042 for a tenth, going as the square of the distortion, which makes a ratio measured to three decimals a length known to one and a half.

8 figures The factor of thirteen was generous. An invented error model against the one computed from a force field. Its invented mismatch of five per cent made the correlation between the parent and the substituted species worth a factor of 12; the computed mismatch of 35 per cent makes it worth 2.5. The substitution structure went from being 2.5 times worse than a direct fit to 26, and at the computed size of the correction its worst coordinate is out by 10.7 per cent.

The correction that was invented

A standard error model puts the zero-point error in a rotational constant at a few tenths of a per cent, shared between the three moments by invented weights, with the parent and its deuterated form differing by five. Computed from a force field it is 1.88 per cent, one of the three shares is negative, and the mismatch is 35 — so the cancellation the substitution method rests on is worth a factor of 2.5 and not thirteen.

6 figures The boundary belongs to the gap, not to the repulsion. The repulsion at which a satellite stops being tellable from a fundamental by intensity, against the system's own one-electron gap. Four systems: ring of 6, gap 2.000, boundary 8; chain of 4, gap 1.236, boundary 4; chain of 6, gap 0.890, boundary 2; ring of 4, gap 0.000, boundary 0.25. The three with a gap order exactly with it, and the ring of four — whose half-filled ground state is degenerate and whose gap is zero — has no boundary at all: its contrast is one at every repulsion, so its satellites are never distinguishable and there is nothing for a boundary to separate.

The boundary belongs to the gap

A satellite stops being tellable from a fundamental somewhere, and it can be located on one ring at one filling. Four systems put it at repulsions of 2, 4 and 8 — ordering exactly with each one's own one-electron gap and not with its band width — and the fourth, whose gap is zero, has no boundary at all: its satellites are indistinguishable at every repulsion including none.

6 figures How much of each of methane's force constants the spectrum fixes. Each of methane's 55 force constants, grouped by kind, with the squared length of its projection onto the determined subspace. One means the constant is fixed on its own. All 4 stretching constants are; not one bending constant is, at 0.5357 each. The fractions sum to 45, which is the rank of the map from constants to the Hessian, and that sum is an arithmetic check rather than a result.

The forty-five that are fixed

Methane's fifty-five-dimensional space of force constants has ten directions no frequency can see, and the useful thing to report is the forty-five that are fixed. It is a table: every stretching constant is fixed on its own, no bending constant is, and projecting a fitted field onto the determined subspace takes four fits that span 0.88 mdyn per ångström down to four that span 0.0027.

6 figures One expression, four molecules, a factor of eleven. The computed zero-point correction to each molecule's moment of inertia against the expression A²·Σ(1/ν)/I, which has no fitted quantity in it. The dashed line is the mean dimensionless coefficient, 0.6343; the four points lie within 18 per cent of it, on corrections that span a factor of 11.4. The expression is evaluated from a moment of inertia and a list of wavenumbers, which is what a spectroscopist has before doing anything.

An expression for what was a warning

The zero-point correction to a moment of inertia comes out at 1.88 per cent where a few tenths had been assumed, and heavy molecules are safer. Written out, the correction is a mean curvature times the sum of reciprocal wavenumbers over the moment — one line, evaluated from things a spectroscopist has before starting. One coefficient serves four molecules whose corrections span a factor of eleven.

5 figures How fast the three quarters goes, along each coordinate. The coefficient of the square, for every non-symmetric mode of three molecules, against the mode's own frequency. A single measurement takes one of these — along a bond stretch — and reports the exponent rather than the size. The sizes span a factor of thirty within methane alone, and the coordinates a molecule is softest along are not systematically the most sensitive: ammonia's stiff pair is five times more sensitive than its soft one.

One number was one direction

The departure of a depolarisation ratio from three quarters goes as the square of a distortion, and the exponent was first measured along one bond stretch. Computed along every non-symmetric coordinate the exponent is always two and the coefficient is not: it spans a factor of thirty inside methane. And the ratio is not preferentially sensitive to the coordinates a molecule is soft along — in two molecules of three the stiffest coordinate is the most sensitive.

6 figures The contrast, out to a repulsion of sixteen thousand. The ratio of the weakest fundamental to the strongest satellite, against the on-site repulsion, for eight systems with two electrons each. Both axes logarithmic. The dashed line at two is the factor the intensity test needs. Every curve flattens above it and none of them crosses, at any repulsion — including a repulsion sixteen thousand times the hopping.

A contrast with a closed form

Below half filling the satellite test flattens instead of failing, and the value it flattens at could be above or below the factor of two the test needs. It is — on all eight systems, by between 1.25 and 3.7 times. And on a ring the limit is (1 + 2cos(π/n))², to six figures, on every ring tried.

6 figures The coefficient each principal axis needs. The dimensionless coefficient the molecule-averaged expression requires, evaluated on each principal axis separately rather than on the mean of the three. Across the twelve axes the positive ones span a factor of 2.33, against the 1.36 the molecule-averaged version spans — and one of them is negative, which no positive constant can be.

The axis that goes the other way

One dimensionless coefficient turned a zero-point correction into an expression a spectroscopist could evaluate, and the three principal axes were averaged over to get it. Split by axis it gets worse, not better — the coefficients span 2.3 where the molecule-averaged ones span 1.36 — and water's smallest moment does not grow at all. It shrinks.

6 figures The frame H₂O → HDO turns. H₂O → HDO: the parent's principal axes and the daughter's, drawn on the same nuclei. Two of the three turn by 21.12° and the third does not move at all, because it is the normal to a plane no substitution can tilt. A per-axis comparison between these two species is comparing moments about lines this far apart, which is a rotation of the frame rather than a correction to a number.

Two moments about two different lines

Asked axis by axis, the substitution method's near-cancellation gives numbers as large as 163 per cent. The arithmetic is the smaller half of the answer. A principal axis is an eigenvector of a tensor built from the masses, so one deuterium turns water's frame by 21.12° — and the two moments being compared are not moments about the same line.

6 figures The two terms in a vibrationally averaged rotational constant. A rotational constant averages 1/r², not r², so the moment it reports carries +2⟨Δr⟩/rₑ from the anharmonicity and −3⟨Δr²⟩/rₑ² from the harmonic spread. The two have opposite signs in every molecule here, and the anharmonic one — which is exactly zero in any symmetric well and therefore absent from every harmonic force field — is larger by a factor of 1.94 to 2.58.

The term a harmonic field cannot produce

The usual zero-point correction to a moment of inertia comes from a harmonic force field, which contains the mean square displacement and nothing else. A rotational constant does not average that. It averages one over r squared, whose leading correction is the mean displacement — zero in any symmetric well — and which enters with the opposite sign and about twice the size.

6 figures The contrast at three fillings, and the floor two of them reach. The intensity contrast on a ring of 6 against the on-site repulsion, at three fillings. At two electrons it settles on a number well above the factor of two the test needs. At half filling it falls through two and lands on exactly one from U = 64 upward — and every point where it reads exactly one is a point where the cut between fundamental and satellite falls between two lines of identical weight. Those are drawn hollow.

A ratio of exactly one is a tie

Does the intensity contrast fall below two at half filling? It does — it falls to exactly one. But one is the floor of a ratio between two ranked quantities, and it is reached here because the cut between fundamental and satellite lands between two lines of identical weight. The guard installed to catch that case tests the wrong degeneracy, and the guard installed to license the extrapolation cannot tell an exact answer from a divergent one.

6 figures