Energies of Orbitals & Filling of Orbitals

The Energy Levels of Electrons in Atoms

The arrangement and energy of electrons in atoms are explained most clearly by contrasting two cases: the hydrogen atom (a one-electron system) and multi-electron atoms. The hydrogen atom shows exact degeneracy of orbitals having the same principal quantum number, while multi-electron atoms exhibit splitting of those energies because of electron-electron interactions, shielding and penetration effects.

Hydrogen atom

In a hydrogen atom, the energy of an electron depends only on the principal quantum number (n). All orbitals with the same value of n are degenerate (have the same energy). The lowest energy orbital is the 1s orbital, which is the ground state. Any electron in a higher n is in an excited state.

• Order of energies (hydrogen): 1s < 2s = 2p < 3s = 3p = 3d < 4s = 4p = 4d = 4f

Energy level diagrams for (a) hydrogen atom and (b) multi-electronic atoms

Multi-electron atoms

In atoms with more than one electron, the energy of an orbital depends on both the principal quantum number (n) and the azimuthal quantum number (l). Electrons repel each other and inner electrons partially screen the nuclear charge; as a result, subshells (s, p, d, f) of the same principal level have different energies.

Effective Nuclear Charge

• The presence of other electrons introduces electron-electron repulsion, so the energy of an orbital is no longer determined solely by n.
• Shielding (screening) by inner electrons reduces the full positive charge of the nucleus felt by outer electrons; the net charge actually experienced is called the effective nuclear charge (Z_eff).
• Despite shielding, as the atomic number increases, the magnitude of attraction of electrons by the nucleus increases; consequently, orbital energies generally become more negative (lower) with increasing Z.
• For orbitals in the same shell, the amount of shielding and the average distance from the nucleus differ with l. For a given n, the order of penetration (and hence binding) is s > p > d > f, so Z_eff experienced decreases as l increases.
• Because of these effects, the energy of an electron in an s orbital is lower (more tightly bound) than in a p orbital of the same shell; p is lower than d, and so on.
• The energies of orbitals depend on both n and l; a useful empirical rule for ordering orbital energies in many atoms is the n + l rule (see below).
• Orbital energies of the same subshell become lower (more negative) as atomic number increases. For example, E_2s(H) > E_2s(Li) > E_2s(Na) > E_2s(K).

Arrangement of Orbitals with (n+l) Rule

When electrons are added to orbitals of different atoms, their arrangement follows the Aufbau principle, together with the Pauli exclusion principle and Hund's rule.

Aufbau principle (building up)

The German term "aufbau" means "building up". The Aufbau principle states that electrons occupy orbitals of lowest available energy first and then progressively occupy higher-energy orbitals.

Aufbau Principle

The energy of orbitals - specific rules and examples

(i) Hydrogen atom: energy depends mainly on n; subshells of the same n are degenerate.

For H atom: 1s < 2s = 2p < 3s = 3p = 3d

Energy Levels of H-atom

(ii) Multi-electron atoms: energy depends on both n and l; the n + l rule is commonly used to predict the order of orbital energies.

n + l rule: orbitals are filled in increasing order of the sum (n + l). If two orbitals have the same (n + l) value, the orbital with lower n has lower energy and is filled earlier.

Energy Levels of Multi-Electron Atoms

Consequences of the n + l rule:

• As the value of (n + l) increases, the orbital energy generally increases.
• If two orbitals have the same (n + l), the orbital with the smaller n has lower energy.

Typical order of filling (common empirical order for many atoms):

• 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s

Pauli exclusion principle

• The Pauli exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers (n, l, m_l, m_s).
• As a direct consequence, an orbital can accommodate a maximum of two electrons, and these two electrons must have opposite spins (m_s = +1/2 and -1/2).

Hund's rule of maximum multiplicity

When electrons occupy orbitals of the same energy (degenerate orbitals, for example the three 2p orbitals), they first occupy separate orbitals with parallel (same) spins. Pairing occurs only after each degenerate orbital contains one electron.

Multiplicity is given by the formula 2|S| + 1, where S is the total spin (sum of individual electron spins). A higher multiplicity corresponds to more unpaired parallel spins and lower energy (more stable) configuration among arrangements of the same electronic configuration.

2|S| + 1 = 2

2|S| + 1 = 2

2|S| + 1 = 4 (this arrangement has maximum multiplicity among the shown possibilities)

Examples of electron configurations and common exceptions

Following the Aufbau, Pauli and Hund rules gives electron configurations such as:

• Carbon (Z = 6): 1s² 2s² 2p²
• Oxygen (Z = 8): 1s² 2s² 2p⁴
• Sodium (Z = 11): [Ne] 3s¹

There are notable exceptions among transition metals where the observed ground-state configuration differs from the simple aufbau prediction because of the extra stability of half-filled or fully filled d subshells:

• Chromium (Z = 24): expected [Ar] 4s² 3d⁴ but observed [Ar] 4s¹ 3d⁵ (half-filled d subshell is stabilising).
• Copper (Z = 29): expected [Ar] 4s² 3d⁹ but observed [Ar] 4s¹ 3d¹⁰ (completely filled d subshell is stabilising).

Penetration and shielding - qualitative effects on orbital energy

Penetration describes how much an orbital's electron density reaches close to the nucleus. Electrons in orbitals with greater penetration (for example, s orbitals) spend more time near the nucleus and therefore experience a larger nuclear attraction and a larger Z_eff.

Shielding describes how inner electrons reduce the nuclear attraction felt by outer electrons. Electrons in orbitals with poor penetration (for example, d and f) are less effective at shielding outer electrons from nuclear charge.

Practical applications and why this matters

• Orbital energies and filling order determine atomic electronic configurations, which in turn influence chemical reactivity, ionisation energy, atomic radius and bonding behaviour.
• Understanding Z_eff, shielding and penetration explains periodic trends across the periodic table such as variation of atomic size, ionisation energy and electron affinity.
• Knowledge of exceptions to the simple aufbau order is important for correctly writing configurations of transition elements and for predicting oxidation states and magnetic properties.

Summary: For the hydrogen atom orbital energies depend only on n and subshells with the same n are degenerate. In multi-electron atoms energies depend on both n and l; shielding, penetration and electron-electron repulsion split subshell energies. The aufbau principle, together with the Pauli exclusion principle and Hund's rule, governs how electrons occupy orbitals; the n + l rule provides a practical ordering of orbital energies, with noted exceptions in some transition elements.