Technique · Fields

Quantum orbitals

The electron cloud of a hydrogen atom, from the real wave functions: lobes, rings and shells in two signs, turning in space, or a beat between two orbitals.

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Good to know

  • The energy of the electron depends only on n: it is -13.6 electron volts divided by n squared, so the jump from n = 2 down to n = 1 gives out the ultraviolet light called Lyman-alpha, at a wavelength of 121.6 nanometres, which astronomers use to find hydrogen across the universe.
  • In 2013 a team of physicists used a photoionisation microscope to photograph the pattern of nodes in the wave function of an excited hydrogen atom: the first direct look at the shape of an electron wave.
  • Linus Pauling showed in 1931 that mixing s and p orbitals into hybrids explains why carbon makes four equal bonds pointing to the corners of a tetrahedron, the shape on which most of organic chemistry is built.
  • The three-dimensional shapes of orbitals decide the shapes of molecules, the colours of many minerals and dyes, and which atoms stick together.
  • Erwin Schrödinger published his wave equation in 1926, and hydrogen, with its single electron, is the one atom for which it can be solved exactly; every picture of an atom with more electrons is an approximation built on these shapes.
  • The letters s, p, d and f come from the descriptions early spectroscopists gave to lines of light from atoms: sharp, principal, diffuse and fundamental.
  • An orbital is not an orbit: the electron has no path, and Max Born showed in 1926 that the square of the wave function gives only the chance of finding it at each place.

The mathematics

  • The wave function is a radial part, proportional to ρ^2 e^(-ρ/2) times a Laguerre polynomial of degree 0 in ρ = 2r/3 (r in units of the Bohr radius, 0.053 nanometres), times a spherical harmonic of degree 2; here that gives 0 spherical nodes, 2 angular nodes and 2 in all, always n - 1.
  • The picture shows psi squared, which integrates to exactly 1 over all of space: the electron is somewhere. The colour shows the sign of psi itself, which changes across every node, and it is this sign, not the density, that decides whether two orbitals reinforce or cancel when atoms bond.
  • The angular parts, the spherical harmonics, are also the standing waves of a vibrating sphere, the same mathematics as the vibration of the air in a spherical room, with l counting the number of nodal circles.
  • The cloud is drawn out to 19.20 Bohr radii, far enough to hold nearly all of the electron's probability.

In art

  • The red and blue two-lobed shapes of textbook orbital diagrams are among the most reproduced scientific images ever made; they look like flowers, propellers and clover leaves because they are standing waves.
  • Orbitals turn up in jewellery, in sculpture and in logos for science museums, which borrow the clean symmetry of lobes arranged round a point.

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