An orbital is a cloud, not a path
In quantum mechanics an electron in an atom does not follow an orbit. It is described by a wavefunction ψ, and |ψ|2 at each point is the probability density of finding the electron there. An orbital is one such wavefunction: a three-dimensional pattern of where the electron is likely to be. Drawn as a cloud of dots, it is dense where the electron is often found and sparse where it rarely is. Each orbital holds at most two electrons, with opposite spins (the Pauli exclusion principle).
The quantum numbers
- n, the principal quantum number (1, 2, 3, …), sets the shell, the size and, in hydrogen, the energy: E = −13.6 eV × Z2 ÷ n2.
- l, the angular momentum quantum number (0 to n − 1), sets the shape: l = 0 is s, 1 is p, 2 is d, 3 is f.
- ml, from −l to +l, sets the orientation. There are 2l + 1 orbitals in each subshell: one s, three p, five d, seven f.
- ms = +½ or −½ is the electron's spin, which is why each orbital takes two.
So a subshell holds 2(2l + 1) electrons (2, 6, 10, 14) and a shell 2n2 (2, 8, 18, 32).
Shapes and nodes
A node is a surface where ψ = 0, so the electron is never found there. An orbital has n − 1 nodes in all:
- l angular nodes, planes or cones through the nucleus. An s orbital has none and is a sphere; a p orbital has one plane and two lobes; most d orbitals have two planes and four lobes; dz² has two cones instead, giving two lobes and a ring.
- n − l − 1 radial nodes, spheres. 1s has none, 2s one and 3s two, so the larger s orbitals are like nested shells; 2p has none, 3p one.
Worked example, 3p: n = 3, l = 1, so there is 1 angular node (a plane) and 3 − 1 − 1 = 1 radial node (a sphere), 2 in all. For 4f: 3 angular and 0 radial.
Phase
ψ itself is positive in some regions and negative in others, shown as two colors. The sign does not change where the electron is likely to be, since that depends on ψ2, but it decides what happens when orbitals on neighboring atoms overlap: regions of the same sign add up into a bonding orbital, opposite signs cancel into an antibonding one.
Size: the radial distribution
The chance of finding the electron at a distance r from the nucleus, in any direction, is the radial distribution P(r) = r2 |Rnl(r)|2, where Rnl is the radial part of the wavefunction. For hydrogen's 1s it peaks at exactly the Bohr radius, a0 = 52.9 pm, though the average distance is 1.5 a0, because the cloud trails off far outward. In general the average is ⟨r⟩ = (a0 ÷ 2Z)(3n2 − l(l + 1)): 5 a0 for 2p, 10.5 a0 for 3d. Orbitals grow roughly as n2 and shrink as 1 ÷ Z.
Electron configurations
In atoms with many electrons, orbitals with the same n no longer share one energy: inner electrons screen the nucleus, and s electrons, which spend time close to it, are screened least. Subshells fill in the order (the Aufbau, or Madelung, rule):
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p
Within a subshell, electrons spread out one per orbital with parallel spins before any pair up (Hund's rule). Iron, with 26 electrons, is 1s2 2s2 2p6 3s2 3p6 4s2 3d6, or [Ar] 4s2 3d6 for short; the six 3d electrons fill five orbitals, so four are unpaired, which is why iron is magnetic.
Known exceptions in the first five periods: chromium [Ar] 3d5 4s1, copper [Ar] 3d10 4s1, and in period 5 niobium, molybdenum, ruthenium, rhodium, palladium (4d10 5s0) and silver. A half-filled or filled d subshell and the smaller repulsion between electrons in different orbitals make these arrangements slightly lower in energy.
Using the viewer
In the Atomic Orbitals 3D Viewer, pick an orbital and turn it with the mouse. The points are drawn at random from the exact hydrogen wavefunction, so the cloud's density is the probability density. Things to try:
- Cut the front half away on 3s: the cloud splits into three nested layers separated by two empty shells, the radial nodes, and the chart shows three peaks.
- Compare 2p (x) with 2p (z): the same shape pointing along a different axis, and the same energy.
- Watch the single position measurements: each lands somewhere new, but their running average settles on ⟨r⟩.
- Raise Z to 2: the orbital shrinks to half its size and the energy becomes four times as negative, as for He+.
What the model assumes
- One electron. The clouds are the exact solutions for a single electron around a nucleus of charge Z: hydrogen, He+, Li2+ and so on. Nothing else in chemistry has an exact solution.
- A point nucleus that does not move, with no relativity, no spin-orbit coupling and no magnetic field.
- Real orbitals. The px, dxy and similar orbitals shown are the usual real combinations of the complex wavefunctions with +m and −m, so they have a definite |ml| but not a definite ml.
- The configuration is the ground state of the neutral, isolated atom, from the filling order plus the listed exceptions, not calculated.
- The cloud is cut off far out, where almost no probability is left; the true wavefunction never quite reaches zero.
Edge cases
- s orbitals have their highest density at the nucleus itself, yet the radial distribution there is zero, because there is almost no volume at r = 0. Both statements are true and the chart shows the second.
- Equal energies. In hydrogen 2s and 2p have exactly the same energy; in every other atom 2s is lower. The energy shown is for the one-electron atom.
- Different-looking d orbitals. dz² looks unlike the other four, but it is one of five equally valid combinations; in a free atom all five have the same energy.
- Ions. Transition metals lose their 4s electrons before their 3d ones when they form ions: Fe2+ is [Ar] 3d6, not [Ar] 4s2 3d4. The element slider gives neutral atoms only.
Where the model stops being right
- Many-electron atoms. Electrons repel each other, so their orbitals are only approximately hydrogen-like: the same shapes, but different sizes and energies. Real orbital energies come from approximate methods such as Hartree–Fock.
- Molecules and solids. Atomic orbitals mix into molecular orbitals and hybrids (sp3, sp2) when atoms bond, and into bands in metals; the separate atomic shapes no longer apply.
- Heavy elements. Inner electrons in gold or mercury move at a large fraction of the speed of light; relativity shrinks their s orbitals and changes their chemistry (gold's color, mercury being liquid). The non-relativistic picture fails there.
- The filling rule beyond xenon has many more exceptions among the lanthanides, actinides and heavy transition metals; the slider stops at xenon for that reason.
- Electrons are not little balls inside the cloud. Before it is measured the electron has no position at all, only the distribution; each measurement picks one, as the white flashes show.
Related tools
The Bohr Atom Simulator shows the earlier model of electrons in fixed orbits and the spectral lines it explains; the 3D Molecule Viewer shows what happens when atoms bond. For the masses of the elements in a compound, use the Molar Mass Calculator.