The Energies of Atomic Orbitals

Atomic orbital energies differ due to Coulomb interactions, the shielding effect, and orbital penetration. Coulomb's law shows that attractive force between the nucleus and electrons decreases with distance, so electrons in higher shells experience weaker nuclear attraction and have higher energies. Additionally, inner electrons shield outer electrons from the nucleus, reducing their effective nuclear charge and raising their orbital energy.

The shielding effect occurs when inner electrons reduce the attraction between the nucleus and outer electrons through electron-electron repulsion. The effective nuclear charge felt by an electron is calculated by subtracting the shielding constant from the atomic number. Greater shielding means less nuclear attraction and higher orbital energy. For example, in lithium, the 1s electrons shield the 2s electron, reducing its effective nuclear charge from 3 to 1.3.

Orbital penetration describes how close an electron can approach the nucleus. Electrons in s orbitals penetrate most effectively and experience less shielding, resulting in lower energy. The p-orbital wavefunction has a node at the nucleus, so p electrons penetrate less and have higher energy than s electrons. In the fourth and fifth shells, penetration effects become so significant that 4s and 5s orbitals frequently have lower energies than 3d and 4d orbitals, respectively.

The radial distribution function describes the probability of finding an electron at a given distance from the nucleus. For the 2s subshell, electrons have a modest probability of being near the nucleus, demonstrating penetration. In contrast, 2p electrons mostly remain outside or at the outer edge of the 1s region, showing less penetrating ability. These differences in electron probability distributions directly explain why s orbitals have lower energies than p orbitals.

Effective nuclear charge is calculated by subtracting the shielding constant from the atomic number: Zeff = Z - S. The shielding constant depends on the number of shielding electrons and the subshells they occupy, determined from semi-empirical rules. This value represents the net positive charge an electron experiences after accounting for electron-electron repulsions. Understanding effective nuclear charge is essential for predicting electron configuration of multielectron atoms.

The 3s and 3p electrons shield 3d electrons because they occupy orbitals closer to the nucleus and have greater penetrating ability. Since 3s and 3p electrons can move into regions occupied by inner electrons, they spend more time between the 3d electrons and the nucleus. This spatial arrangement blocks the nuclear attraction experienced by 3d electrons, causing them to feel a significantly reduced effective nuclear charge and resulting in higher orbital energy.

Orbital shape determines how effectively electrons can penetrate toward the nucleus and experience shielding. Spherical s orbitals have non-zero probability density at the nucleus, allowing maximum penetration and minimal shielding. Dumbbell-shaped p orbitals have a node at the nucleus where electron probability is zero, preventing penetration and increasing shielding effects. D orbitals penetrate even less than p orbitals, resulting in progressively higher orbital energies within the same shell.