Multi-Electron Atoms and Orbital Energies

In atoms with more than one electron, the energy of an orbital depends on both n and l, because inner electrons shield the nuclear charge: each electron feels an effective nuclear charge Z_eff = Z − S, and the order of orbital energies is roughly given by the (n + l) rule. In brief: in sodium, the 3s electron has Z_eff = 11 − 8.8 = 2.2 according to Slater’s rules, so it is held far more weakly than the 1s electrons (Z_eff ≈ 10.7), and 4s fills before 3d.

The hydrogen atom has energy levels that depend only on n, but the other atoms have the repulsion between electrons, and the neat degeneracy is lost. The consequences are the order in which the subshells fill, the trends of the periodic table (atomic radius, ionization energy) and the exceptions in the electron configurations. This article explains shielding and effective nuclear charge, gives the filling order and the rules for writing configurations, calculates Z_eff with Slater’s rules and explains the exceptions.

What are shielding and effective nuclear charge?

An electron in a multi-electron atom is attracted by the nucleus (charge +Z) and repelled by the other electrons. The inner electrons, which are between the nucleus and the outer electron, cancel part of the nuclear attraction. The effective nuclear charge is the net positive charge felt by an electron:

Zeff = Z − S

where S is the shielding (screening) constant. The consequences:

• In a given shell, s electrons penetrate closer to the nucleus than p and d electrons, feel a higher Z_eff, and are held more strongly, so E(ns) < E(np) < E(nd) < E(nf).
• Electrons in the same shell shield each other poorly, but inner-shell electrons shield strongly. Across a period, Z increases by one for each element, while the added electron goes into the same shell, so Z_eff rises, and the atom becomes smaller and the ionization energy greater.
• Down a group, the added shells shield the nucleus well, and the outer electrons are further away, so the atoms become larger and the ionization energy is lower.

How do you estimate Z_eff with Slater’s rules?

Slater’s rules give simple values for the shielding constant S for an electron in an ns or np orbital:

1. Write the configuration in groups: (1s)(2s, 2p)(3s, 3p)(3d)(4s, 4p)…
2. Electrons in groups to the right (outer) of the electron contribute 0.
3. Each other electron in the same (ns, np) group contributes 0.35 (0.30 for another 1s electron).
4. Each electron in the shell with the next lower n (n − 1) contributes 0.85.
5. Each electron in shells n − 2 and lower contributes 1.00.

For d and f electrons, all electrons in groups to the left contribute 1.00. The rules are an approximation, and the more accurate values are obtained from numerical calculations, which generally give Z_eff values that differ by about 10–20 %.

Z_eff across the second period

Z_eff rises by 0.65 for each step, and the radius decreases accordingly. The atomic radii are approximate covalent radii.

In what order do orbitals fill?

The Madelung (or n + l) rule says that subshells fill in order of increasing n + l, and for equal values of n + l, in order of increasing n. This gives the sequence:

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p

For example, 4s (n + l = 4) fills before 3d (n + l = 5). The order of filling is a guide, and not the order of orbital energies in the atom. After the 3d orbitals are filled with electrons, they are lower in energy than 4s, which is the reason why transition metals lose their 4s electrons first when they form cations.

What are the rules for electron configurations?

• Aufbau principle: electrons occupy the lowest available energy orbitals first.
• Pauli exclusion principle: at most two electrons per orbital, with opposite spins.
• Hund’s rule: orbitals of the same energy are filled singly, with parallel spins, before pairing starts. This lowers the repulsion between electrons.

What are the exceptions?

Half-filled and completely filled d subshells have extra stability, so some elements deviate from the predicted order: chromium is [Ar]3d⁵4s¹ (not 3d⁴4s²) and copper is [Ar]3d¹⁰4s¹ (not 3d⁹4s²). Similar exceptions occur in the second and third transition series (Mo, Ag, Au, Pt) and with the lanthanides and actinides. These exceptions arise because the energies of the 4s and 3d orbitals are very close, and the exchange energy of parallel spins in a half-filled or filled subshell tips the balance.

Worked examples

Example 1: Z_eff for the 3s electron of sodium

Na: 1s²2s²2p⁶3s¹. The 8 electrons of the n = 2 shell contribute 0.85 each: 6.80. The 2 electrons of the 1s level contribute 1.00 each: 2.00. S = 8.80, so Z_eff = 11 − 8.80 = 2.20.

Example 2: Z_eff for chlorine

Cl: 1s²2s²2p⁶3s²3p⁵. For a 3p electron: 6 other electrons in the n = 3 group × 0.35 = 2.10; 8 electrons in n = 2 × 0.85 = 6.80; 2 electrons in n = 1 × 1.00 = 2.00. S = 10.90 and Z_eff = 17 − 10.90 = 6.10. This is much larger than the value of 2.2 for sodium, and explains why chlorine is much smaller than sodium (radii of 99 and 186 pm) and has a much higher ionization energy.

Example 3: electron configurations

• Phosphorus (Z = 15): 1s²2s²2p⁶3s²3p³, or [Ne]3s²3p³, with three unpaired electrons in the 3p orbitals (Hund’s rule).
• Iron (Z = 26): [Ar]3d⁶4s², with four unpaired d electrons.
• Fe²⁺: [Ar]3d⁶ (the 4s electrons are lost first) and Fe³⁺: [Ar]3d⁵, with five unpaired electrons, which explains the stability of iron(III).
• Copper (Z = 29): [Ar]3d¹⁰4s¹ (exception), and Cu²⁺: [Ar]3d⁹, with one unpaired electron.

Example 4: paramagnetism

Is the oxygen atom paramagnetic? O: 1s²2s²2p⁴. The four 2p electrons occupy three orbitals: two paired in one orbital and two unpaired in the others. Two unpaired electrons make the atom paramagnetic. By contrast, Zn: [Ar]3d¹⁰4s² has all electrons paired, and is diamagnetic.

Who worked out these rules?

The Pauli exclusion principle was proposed in 1925, and Friedrich Hund formulated his rule in 1927. Erwin Madelung formulated the n + l rule in 1936, and the Russian chemist Vsevolod Klechkovsky gave it a theoretical basis in 1962, so it is also called the Madelung–Klechkovsky rule. John C. Slater published his screening rules in 1930, as a simple way to estimate effective nuclear charges. Today, the electron configuration of atoms is calculated numerically by self-consistent field methods developed by Douglas Hartree and Vladimir Fock in the late 1920s and 1930s.

Which mistakes are common?

• Filling 3d before 4s: 4s fills first, although it is lost first on ionization.
• Forgetting Hund’s rule: orbitals of the same energy are filled singly first.
• Treating Z_eff as equal to Z: shielding reduces it.
• Missing the exceptions: Cr and Cu are standard examples.
• Using hydrogen’s energy diagram for other atoms: in multi-electron atoms, the energy depends on l as well.