Structure of Atom · Lesson 10 of 12
Orbital Energies and Filling Rules
“Connect shielding and orbital energy with the rules governing electron occupation.”
• Compare orbital degeneracy in hydrogen-like and multi-electron atoms. • Explain shielding, penetration, and effective nuclear charge qualitatively. • Use the n + l guide to order subshells while recognising its limitations. • Apply Aufbau, Pauli, and Hund rules to orbital-box arrangements. • Derive subshell and shell electron capacities.
Why orbital shape matters to energy
In an isolated hydrogen-like species, there is one electron attracted by the nucleus and no electron–electron repulsion. In the elementary non-relativistic model, energy depends on n only. Thus 2s and 2p have the same energy even though their shapes differ. Orbitals with equal energy are called degenerate.
Adding electrons changes the problem. Each electron is attracted by the nucleus but repelled by other electrons. How close it can approach the nucleus, and how much other electrons screen the nuclear attraction, now affect its energy. For the same n, s, p, d, and f subshells generally do not have equal energies.
The ground state is the arrangement with the lowest total electronic energy. An excited configuration has greater total energy. Although we often use a one-electron energy ladder to build a configuration, the actual atom’s total energy includes interactions between its electrons.
Shielding and effective nuclear charge
An outer electron does not feel the full unscreened nuclear attraction at all times. Inner electrons partly screen the positive nucleus while also repelling the outer electron. We call this shielding. The net attraction is often discussed using an effective nuclear charge, , smaller than the bare nuclear charge for an electron in a multi-electron atom.
Shielding does not mean the inner electrons are a solid wall. Electrons have spatial probability distributions, so the amount of screening depends on which orbitals are occupied and where the electron being considered spends time. Increasing the nuclear charge can still increase attraction despite the presence of shielding.
Within a shell, an s electron has appreciable probability near the nucleus and penetrates the inner region more effectively than a p electron, which in turn penetrates more than a d electron. Greater penetration lets an electron experience less screening for part of its distribution. The familiar qualitative penetration order within a shell is s, p, d, f from greatest to least.
Consequently, within the same principal shell of a typical multi-electron atom, s is lower in energy than p, then d, then f. Inner orbitals can also shield outer electrons more effectively when their own distributions penetrate strongly. These related ideas explain why the energy depends on both n and l, rather than on n alone.
The effective positive nuclear charge used to represent the net attraction experienced by an electron after accounting approximately for screening by other electrons.
Problem
A bromine atom has electrons in 2p, 3p, and 4p. Which group is most strongly shielded from the nucleus?
- 1.The 4p electrons are in the outermost of these shells and have more inner-shell electrons between much of their probability distribution and the nucleus.
- 2.They are therefore the most strongly shielded and, in this qualitative comparison, experience the weakest effective nuclear attraction of the three groups.
- 3.This comparison is within the same atom. It does not imply that all 4p electrons in all elements have identical effective nuclear charges.
The n plus l energy guide
For building many ground-state configurations, subshells with smaller n + l are placed first. If two subshells share the same sum, the one with smaller n comes first. This explains why the usual filling sequence places 4s before 3d, despite 4s belonging to a larger principal shell.
The guide is remarkably useful but is not an exact universal ordering for every atom, ion, or electron occupation. Subshell energies can be close, and electron interactions can change their relative order. Use it to construct a first prediction, then recognise experimentally established exceptions.
| Subshell | n | l | n + l |
|---|---|---|---|
| 1s | 1 | 0 | 1 |
| 2s | 2 | 0 | 2 |
| 2p | 2 | 1 | 3 |
| 3s | 3 | 0 | 3 |
| 3p | 3 | 1 | 4 |
| 4s | 4 | 0 | 4 |
| 3d | 3 | 2 | 5 |
| 4p | 4 | 1 | 5 |
Problem
Which subshell is filled first according to the n + l guide?
- 1.For 4s, n + l = 4 + 0 = 4. For 3d, n + l = 3 + 2 = 5.
- 2.The smaller sum puts 4s first in the usual building-up sequence.
- 3.Do not turn this filling guide into the claim that 4s must remain lower in energy than 3d for every transition-metal atom and ion.
Problem
Compare 3d and 4p, then compare 4f and 5d.
- 1.3d and 4p both have n + l = 5. The smaller n puts 3d first.
- 2.4f and 5d both have n + l = 7. The smaller n puts 4f first.
- 3.Apply the sum first and the n tie-break second. Applying only the principal number would give the wrong prediction for other comparisons.
Aufbau and the Pauli exclusion principle
Aufbau means building up. In constructing a ground-state arrangement, electrons occupy the lowest available energy states subject to the occupation restrictions. A useful sequence is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p.
The Pauli exclusion principle supplies a separate restriction: no two electrons in an atom may share all four quantum numbers. In one spatial orbital, n, l, and magnetic orbital label are already the same. At most two electrons can occupy it, and they must have opposite spin labels.
A subshell has 2l + 1 spatial orbitals, each with capacity two. This gives capacities of 2, 6, 10, and 14 for s, p, d, and f subshells respectively. A shell contains n squared spatial orbitals, so its maximum electron capacity is 2n squared. A capacity is a maximum; it does not state how many electrons a particular ground-state atom actually has in that shell.
Problem
Find the subshell and total electron capacities of n = 4.
- 1.The allowed subshells are 4s, 4p, 4d, and 4f, containing 1, 3, 5, and 7 spatial orbitals.
- 2.Multiply by two: their capacities are 2, 6, 10, and 14 electrons.
- 3.The sum is 32, matching 2n squared = 2 × 16. This does not mean that every atom containing a 4s electron has 32 fourth-shell electrons.
Hund’s rule within equal-energy orbitals
A p subshell offers three equal-energy spatial orbitals. Electrons enter these singly with parallel spins before pairing begins. Hund’s rule describes this maximum multiplicity arrangement. Pairing in a p subshell begins with the fourth electron; in d and f it begins with the sixth and eighth respectively.
Pauli and Hund answer different questions. Pauli prohibits two same-spin electrons in one spatial orbital. Hund distinguishes among allowed arrangements across different degenerate orbitals: spreading electrons singly with parallel spins gives the ground-state pattern before pairing.
The parallel-spin arrangement gains stabilisation associated with exchange interactions. We examine that qualitative explanation with actual electronic configurations in the next lesson. It should not be explained as electrons literally exchanging positions like seats in a classroom.
Problem
A student puts two up-spin arrows in one box. Another puts a pair in one p box while the other two boxes are empty. Diagnose the errors.
- 1.Two up-spin electrons in one spatial orbital would share all four quantum numbers. This violates Pauli exclusion.
- 2.An opposite-spin pair in one p orbital with empty equal-energy partners is Pauli-allowed but is not the expected ground-state arrangement for two p electrons. It violates the Hund ground-state filling rule.
- 3.The expected two-electron p arrangement has two singly occupied boxes with parallel spins.
Do not use the n + l sequence as an exact energy law for all occupied atoms and ions. Do not pair electrons in one p, d, or f orbital while an equal-energy orbital remains empty in the ground-state filling pattern. A principal shell can begin filling before lower-numbered shells reach their full theoretical capacities.
Quiz
In the elementary hydrogen-like model, which are degenerate?
What is shielding?
Which has the lower predicted filling position by n + l?
Maximum capacity of a d subshell is:
Two electrons occupy different degenerate p orbitals with parallel spins. Which rule explains preferring this over pairing them first?
Practice Problems
- Explain why 2s and 2p have equal energy in the elementary hydrogen model but generally not in a multi-electron atom.
- Order 3p, 4s, 3d, 4p, and 5s using n + l.
- Compare penetration of s, p, and d electrons within one principal shell.
- Draw ground-state orbital-box patterns for two, three, four, and five p electrons.
- Calculate the number of spin-down electron states available in the n = 4 shell.
- Explain why an arrangement can satisfy Pauli exclusion but fail to represent the Hund ground-state arrangement.
Key Takeaways
Hydrogen-like orbital energies depend on n; multi-electron energies also depend on subshell type and interactions. Shielding and penetration affect effective nuclear attraction and orbital energy. The n + l rule guides filling but has exceptions and occupation-dependent limits. Pauli allows at most two opposite-spin electrons per spatial orbital. Hund favours singly occupied degenerate orbitals with parallel spins before pairing.