The collection

Every essay — page 13

Page 13 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

Orbitals

A one-electron wavefunction, drawn as a contour surface at a level somebody chose. Nodes, signs, and what the picture is a picture of.

The 2pz orbital. The 2pz orbital at the contour enclosing 90 per cent of its density — a level solved for by integration rather than chosen. The two colours are the two signs of the wavefunction, which is what distinguishes a bonding interaction from an antibonding one. Contours drawn: 2pz at 90% of its density, |ψ| = 9.48e-3.

What an orbital is

Not a region the electron occupies, not a path it follows, and for any atom but hydrogen not an exact anything. An orbital is a one-electron wavefunction, and almost every difficulty in this subject comes from forgetting that.

7 figures $Choosing a contour for 3s. The fraction of the density enclosed by a contour, against the contour's level. Picking a level is picking a point on this curve, and the usual practice of picking one that looks right is picking a point without knowing which. Contours drawn: 3s at 50% of its density, |ψ| = 6.24e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3s at 99% of its density, |ψ| = 7.15e-4.$

Say what it encloses

An orbital picture is a contour at a level somebody chose, and almost no source says which. Two textbooks can draw the same orbital at visibly different sizes with the same caption, and both be printed in good faith.

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The radial function of 3s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

Nodes

An orbital with quantum numbers n and l has exactly n−l−1 radial nodes and l angular ones. That is a count, it is exact, and it is the fastest way to catch a drawing that is wrong.

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The radial function of 1s. The radial part of the wavefunction, which changes sign at each node, and the radial distribution, which is the probability of finding the electron in a shell at that radius. The second vanishes at the nucleus and the first does not.

Where the electron is

The wavefunction is largest at the nucleus, the electron is most likely to be found a bohr out, and the ninety-per-cent contour is at 2.66. Three numbers, all correct, all answering different questions.

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The radial distribution across the periodic table

A 4s orbital is bigger than a 3d by every measure of size except the one that decides which fills first. Penetration is a feature of a small inner peak, and the periodic table's shape depends on it.

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

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.

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4s and 3d from Sc to Zn. The mean radius of the 4s and 3d orbitals across the elements Sc to Zn, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What an electron actually feels

A 3d orbital is less than half the size of the 4s beside it and fills second anyway. The charge an electron feels is not the nuclear charge, the correction is a fit rather than a derivation, and the two facts together explain the shape of the periodic table.

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Four measures of size for 6 orbitals. The most probable radius, the mean radius, the root-mean-square radius and the radius of the sphere holding ninety per cent of the density, for 1s, 2s, 2pz, 3s, 3dz2, 4s. All four are computed from the same radial function, all four are correct, and they are not the same number.

How big is an orbital

Four measures of size, all computed from the same radial function, all correct, and spanning a factor of two and a half for a 1s orbital. The one that governs chemistry is a fifth, and it is not a measure of size at all.

7 figures $1s, 2s, 2pz, 3s, 3dz2, 4s at one level and at one fraction. The orbitals 1s, 2s, 2pz, 3s, 3dz2, 4s, each with the contour level that encloses 90 per cent of its own density, and beside it what each one encloses when they are instead all drawn at 1s's level. The first column is a set of pictures that can be compared; the second is what a plate drawn at a single contour value actually shows. Contours drawn: 1s at 90% of its density, |ψ| = 3.94e-2; 2s at 90% of its density, |ψ| = 7.40e-3; 2pz at 90% of its density, |ψ| = 9.48e-3; 3s at 90% of its density, |ψ| = 2.64e-3; 3dz2 at 90% of its density, |ψ| = 3.60e-3; 4s at 90% of its density, |ψ| = 1.24e-3.$

One level is not one comparison

A plate of orbitals drawn at a single contour value looks like a comparison and is not one. At the level that encloses ninety per cent of a 1s, a 2s encloses four per cent, a 3s under one, and a 3d has no surface at all — its wavefunction never reaches that value anywhere in space.

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2s and 2p from B to Ne. The mean radius of the 2s and 2p orbitals across the elements B to Ne, at the nuclear charge each shell actually feels. The screening is Slater's, which is fitted; the radius that follows from it is the closed form for a hydrogen-like orbital, which is not.

What the screening model cannot see

Slater's rules put the 2s and the 2p in one group, so they give both orbitals exactly the same effective nuclear charge at every element from boron to neon. The two are separated by several electronvolts in all six, and the model has no term that could produce it.

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

A filled shell has no shape

Sum the angular densities of a complete p shell and the answer is 3/4π in every direction, to sixteen decimal places. A filled d shell gives 5/4π. The lobes are in the decomposition and not in the density, and nothing that measures a closed-shell atom can see them.

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The exponent the molecule chooses, and what it buys. The 1s exponent that minimises the energy of a one-electron diatomic, against the separation of the nuclei, with the binding curves at that exponent and at the free atom's. Held at ζ = 1 the bond comes out at 2.49 bohr and binds 0.0648 hartree; with the exponent free it comes out at 2.00 bohr at ζ = 1.238 and binds 0.0865. The exact answer for this molecule is 2.00 bohr and 0.1026.

The atom does not bring its own orbital

Build a one-electron diatomic from two hydrogen 1s functions and it comes out 25 per cent too long and 37 per cent too weakly bound. Let the molecule choose how large those functions are and the bond length is right to three figures, at an exponent of 1.238 — the orbital contracts by a quarter when the bond forms.

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