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9.3 — Electrons in Many-Electron Atoms
Chapter 7.6 solved hydrogen exactly and got three quantum numbers, the orbital shapes, and the 2n^2 counting. Chapter 7.7 added spin and the exclusion principle.
Helium cannot be solved exactly. Nor can anything else. Adding one electron introduces a term e^2/r_{12} — the repulsion between the two electrons — and the equation stops separating. This is the three-body problem, which has no closed-form solution in classical mechanics either.
So chemistry runs on approximations. This chapter builds them, derives the filling order that gives the periodic table its shape, and explains the exceptions rather than listing them.
The screening approximation
The central idea: treat each electron as moving in the average field of the nucleus plus all the other electrons.
An outer electron does not feel the full nuclear charge Z, because inner electrons sit between it and the nucleus, cancelling some of it. Define an effective nuclear charge:
Z_{\text{eff}} = Z - S
where S is the screening constant.
This turns the many-electron problem back into a hydrogen-like one, with Z replaced by Z_{\text{eff}}:
E_n \approx -\frac{13.6\,Z_{\text{eff}}^2}{n^2}\ \text{eV}
Worked example: lithium. Lithium has Z = 3 with configuration 1s^2 2s^1. The two 1s electrons screen the 2s electron almost completely, so Z_{\text{eff}} \approx 1.3:
E_{2s} \approx -\frac{13.6\times(1.3)^2}{4} = -5.75\ \text{eV}
Measured ionisation energy: 5.39 eV. A crude approximation getting within 7 %.
Slater's rules
John Slater codified the screening in 1930 with a set of empirical rules that work surprisingly well.
Group the orbitals as [1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] and so on.
Contributions to S for an electron in an s or p orbital:
- Electrons in higher groups: 0.
- Other electrons in the same group: 0.35 each (0.30 for 1s).
- Electrons in the shell one below (n-1): 0.85 each.
- Electrons two or more shells below: 1.00 each.
For a d or f electron, everything to its left contributes 1.00 and same-group electrons contribute 0.35.
Worked example: the outer electron of potassium (Z = 19, configuration 1s^22s^22p^63s^23p^64s^1).
S = 8(0.85)+10(1.00) = 6.8+10 = 16.8
Z_{\text{eff}} = 19-16.8 = 2.2
Worked example: a 3d electron in scandium (Z = 21, [\text{Ar}]3d^14s^2). For the 3d electron, all 18 argon-core electrons are to its left:
S = 18(1.00) = 18, \qquad Z_{\text{eff}} = 21-18 = 3.0
Worked example: the 4s electron in scandium.
S = 1(0.35)+9(0.85)+10(1.00) = 0.35+7.65+10 = 18.0
Z_{\text{eff}} = 21-18.0 = 3.0
Nearly equal, which is exactly why the 3d and 4s levels are so close in energy and why the transition metals behave the way they do.
Penetration: why E_{ns} < E_{np} < E_{nd}
In hydrogen, all states with the same n have the same energy (Chapter 7.6). In every other atom they do not, and the reason is penetration.
Chapter 7.6 showed the centrifugal barrier, \hbar^2\ell(\ell+1)/2mr^2, which is zero for \ell = 0 and positive otherwise. So s electrons can reach the nucleus and p, d and f electrons cannot.
The radial distribution of a 3s orbital has small inner peaks close to the nucleus, inside the core electrons. When the electron is there, it is inside the screening and feels nearly the full nuclear charge.
A 3p orbital penetrates less. A 3d orbital hardly at all.
The result:
\boxed{E_{ns} < E_{np} < E_{nd} < E_{nf}}
This ordering is the single most consequential fact in chemistry, because it is what makes the periodic table's shape what it is.
The filling order
The aufbau principle (German for "building up"): electrons occupy the lowest available energy levels.
The order:
1s\ 2s\ 2p\ 3s\ 3p\ \mathbf{4s}\ 3d\ 4p\ 5s\ 4d\ 5p\ 6s\ 4f\ 5d\ 6p\ 7s\ 5f\ 6d\ 7p
Note the 4s before 3d. Penetration makes 4s dip below 3d in energy for the neutral atoms at the start of the fourth row.
The n+\ell rule (Madelung's rule) reproduces the order mechanically:
Fill in order of increasing n+\ell. For equal n+\ell, fill lower n first.
Check it:
| Orbital | n | \ell | n+\ell |
|---|---|---|---|
| 4s | 4 | 0 | 4 |
| 3d | 3 | 2 | 5 |
| 4p | 4 | 1 | 5 |
| 5s | 5 | 0 | 5 |
4s has n+\ell = 4 and 3d has 5, so 4s fills first. Among the three with n+\ell = 5, 3d fills before 4p before 5s, in order of n.
Nobody has derived this rule from first principles, which is worth saying plainly. It is an empirical pattern that reflects the balance of penetration and shielding, and it has known exceptions.
Hund's rule
For orbitals of equal energy, electrons occupy them singly with parallel spins before pairing up.
Two reasons, and the second is the deeper one.
Electrons in separate orbitals are further apart on average, so they repel less.
Parallel spins force an antisymmetric spatial wavefunction (Chapter 7.7), which keeps the electrons even further apart — the exchange energy. This is the same mechanism that drives ferromagnetism (Chapter 4.8), acting inside a single atom.
Worked example: nitrogen, Z = 7, 1s^22s^22p^3.
The three 2p electrons go one to each of 2p_x, 2p_y, 2p_z, all with parallel spins, rather than two paired in one orbital and one in another.
Consequence: nitrogen has three unpaired electrons and is strongly paramagnetic (Chapter 4.8), and it forms three bonds. Oxygen, with four 2p electrons, must pair one, leaving two unpaired — which is why O₂ is paramagnetic, a fact Chapter 10.2 explains properly.
Writing configurations
The notation lists occupied orbitals with superscripts for electron counts.
| Element | Z | Configuration | Shorthand |
|---|---|---|---|
| Hydrogen | 1 | 1s^1 | |
| Helium | 2 | 1s^2 | |
| Carbon | 6 | 1s^22s^22p^2 | [He]2s^22p^2 |
| Neon | 10 | 1s^22s^22p^6 | |
| Sodium | 11 | [Ne]3s^1 | |
| Argon | 18 | [Ne]3s^23p^6 | |
| Potassium | 19 | [Ar]4s^1 | |
| Iron | 26 | [Ar]3d^64s^2 | |
| Bromine | 35 | [Ar]3d^{10}4s^24p^5 |
The shorthand uses the previous noble gas, because a filled shell is chemically inert and can be treated as a core.
Two rules worth having:
The outermost electrons determine chemistry. Sodium's [Ne]3s^1 and potassium's [Ar]4s^1 both have one loosely held s electron, so both are soft reactive metals that form +1 ions. Chemical families are configuration families.
Ions: remove electrons from the highest n first, not in reverse filling order. Iron loses its 4s electrons before its 3d:
\text{Fe}: [\text{Ar}]3d^64s^2 \quad\to\quad \text{Fe}^{2+}: [\text{Ar}]3d^6 \quad\to\quad \text{Fe}^{3+}: [\text{Ar}]3d^5
Why 4s goes first, when it filled first. Because once the 3d orbitals are occupied, their energy drops below 4s — the ordering reverses as the nuclear charge grows and the d electrons start screening the s electrons. The 4s/3d ordering is not a fixed property of the atom; it depends on how many electrons are present.
And this explains why Fe³⁺ is so common. It has 3d^5 — exactly half-filled, with all five spins parallel, giving maximum exchange energy. Half-filled subshells are unusually stable, which is the key to the exceptions below.
The exceptions, explained
Two elements in the fourth row break the aufbau order, and they are not arbitrary.
Chromium
Predicted: [\text{Ar}]3d^44s^2
Actual: [\text{Ar}]3d^54s^1
One electron moved from 4s to 3d.
Why. Three effects add up.
Exchange energy. Five parallel electrons in five d orbitals give \binom{5}{2} = 10 parallel pairs, each contributing exchange stabilisation. With 3d^4 there are only \binom{4}{2} = 6. Four extra exchange pairs is a substantial gain.
Pairing energy avoided. In 3d^44s^2 the two 4s electrons are paired in one orbital, and pairing costs repulsion energy. Splitting them removes that cost.
The 3d and 4s energies are nearly equal (the Slater calculation above gave Z_{\text{eff}} = 3.0 for both in scandium), so a small gain is enough to flip the ordering.
Copper
Predicted: [\text{Ar}]3d^94s^2
Actual: [\text{Ar}]3d^{10}4s^1
A completely filled d subshell is even more stabilising than a half-filled one, for the same reasons plus the extra symmetry of a closed shell.
Others
Molybdenum follows chromium (4d^55s^1). Silver and gold follow copper (4d^{10}5s^1, 5d^{10}6s^1). Palladium goes further, to 4d^{10}5s^0 with no s electrons at all.
Some exceptions have no clean explanation — niobium, ruthenium, rhodium and platinum all deviate, and the reasons are a mix of relativistic effects and near-degeneracies that require full calculation.
The honest statement: the aufbau order is a good approximate rule for a system where several energies are nearly equal, and when they are that close, small effects decide. It is not a law and it was never derived as one.
Relativistic effects in heavy atoms
Chapter 6.3 mentioned gold's colour. Here is the mechanism.
An inner electron in a heavy atom moves fast. Its speed is roughly Z\alpha c = Zc/137. For gold, Z = 79:
v = \frac{79}{137}c = 0.58c \quad\Longrightarrow\quad \gamma = \frac{1}{\sqrt{1-0.336}} = 1.23
The relativistic mass increase contracts the 1s orbital by the same factor, since a_0 \propto 1/m. And the contraction propagates: the 6s orbital contracts too, because it must stay orthogonal to the contracted inner s orbitals.
Consequences you can see:
Gold is yellow. The contracted 6s and expanded 5d orbitals bring the 5d→6s gap down to about 2.4 eV, which absorbs blue light. Silver's equivalent gap is in the ultraviolet, so silver reflects everything and looks white.
Mercury is liquid at room temperature. Its 6s electrons are so tightly held by the relativistic contraction that they barely participate in metallic bonding, so mercury atoms interact weakly — like a noble gas that happens to be a metal.
Lead-acid batteries. Roughly 1.7 V of their 2.1 V cell voltage is a relativistic effect on lead's 6s orbital. Without relativity, car batteries would not work.
Ionisation energies
The energy to remove the outermost electron:
\text{X} \to \text{X}^+ + e^-
Read the pattern.
Peaks at noble gases (He 24.6 eV, Ne 21.6, Ar 15.8). A filled shell is tightly bound and there is no easy electron to remove.
Troughs at alkali metals (Li 5.4 eV, Na 5.1, K 4.3). One electron outside a closed shell, well screened, weakly held.
The drop from He to Li is a factor of 4.5 — the sharpest change in the whole table, and it happens because the electron has moved to a new shell where Z_{\text{eff}} is small and n is larger.
Two small dips break the general rise across a period, and both are informative.
Beryllium to boron. Be is 2s^2, B is 2s^22p^1. The 2p electron is higher in energy and less penetrating, so it comes off more easily despite the larger nuclear charge.
Nitrogen to oxygen. N is 2p^3 with three unpaired parallel electrons; O is 2p^4 and must pair one. The paired electron suffers extra repulsion, so it is easier to remove.
These two dips are direct visual evidence for subshells and for Hund's rule, visible in a graph anyone can measure.
Successive ionisations
Remove electrons one at a time and each is harder, because the remaining ion is more positively charged.
Sodium's successive ionisation energies, in eV:
5.1,\quad 47.3,\quad 71.6,\quad 98.9,\quad 138.4,\ \dots
The jump from the first to the second is a factor of 9.3. Sodium is [\text{Ne}]3s^1: the first electron comes from the outer shell, the second must break into the neon core.
This is how electron configurations were determined experimentally, before anyone could compute them. The location of the big jump tells you how many valence electrons there are, and it is the most direct evidence for shell structure there is.
Where this shows up in your life
All of chemistry. Which elements bond with which, in what ratios, is determined by these configurations. Chapter 10.1 builds on it directly.
Flame colours. Sodium's yellow street lights, copper's green, strontium's red in fireworks — each is an electron dropping between specific levels determined by that element's configuration.
Stainless steel works because chromium's 3d^54s^1 configuration lets it form a self-healing oxide layer a few nanometres thick.
Catalytic converters use platinum, palladium and rhodium, whose partly filled d orbitals bind reactant molecules just strongly enough to weaken their bonds and not so strongly as to hold them.
Lasers and LEDs exploit engineered transitions between specific levels.
Gold's colour, mercury's liquidity and your car battery are relativistic effects on electron configurations.
What the next chapter fixes
The configurations are now derivable. Chapter 9.4 shows that the periodic table's entire shape — why the first row has 2 elements, the second and third have 8, the fourth and fifth 18, and the sixth and seventh 32 — falls directly out of the orbital counting done here, and tells the story of how Mendeleev arranged it correctly in 1869 with no knowledge of electrons, atomic number, or quantum mechanics, and used the gaps to predict three elements that were then found.