Aufbau principle
The Aufbau principle, from the German word Aufbau meaning "building up," is a fundamental concept in quantum chemistry that dictates the order in which electrons fill atomic orbitals in the ground state of multi-electron atoms, starting from the lowest energy level and proceeding to higher ones.[1] This principle enables the prediction of electron configurations by assuming electrons are added one at a time to orbitals of increasing energy, influenced by both the principal quantum number (n) and the azimuthal quantum number (l).[2] In practice, the Aufbau principle follows a specific sequence for orbital filling: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, and 7p, reflecting the relative energies of subshells in multi-electron atoms where penetration effects and electron shielding alter the simple hydrogen-like order.[2] Each orbital can hold a maximum of two electrons with opposite spins, as dictated by the Pauli exclusion principle, ensuring no two electrons in an atom share identical quantum numbers.[2] Complementing this, Hund's rule specifies that for degenerate orbitals (those of equal energy within a subshell), electrons occupy each orbital singly with parallel spins before pairing up, maximizing the atom's total spin multiplicity.[2] While the Aufbau principle provides a reliable framework for most elements, exceptions occur in transition metals and heavier elements due to subtle energy differences, such as the stability of half-filled or fully filled subshells (e.g., chromium's 3d⁵ 4s¹ configuration instead of 3d⁴ 4s²).[1] These rules collectively underpin the structure of the periodic table, correlating electron configurations with elemental properties like atomic radius, ionization energy, and chemical reactivity.[2]Fundamentals of Electron Configuration
Definition and Core Principles
The Aufbau principle, derived from the German word for "building up," states that electrons in the ground state of a multi-electron atom occupy atomic orbitals in the order of increasing energy, starting with the lowest-energy orbital available and proceeding until all electrons are placed.[2] This guideline provides a systematic approach to determining the electron configuration, which describes how electrons are distributed among the orbitals to achieve the lowest possible total energy for the atom.[3] The principle assumes that electrons are added sequentially to the atom, mimicking the construction of elements in the periodic table from hydrogen onward.[2] Central to the Aufbau principle are its integration with two fundamental quantum mechanical rules: the Pauli exclusion principle and Hund's rule. The Pauli exclusion principle stipulates that no two electrons within an atom can occupy the same quantum state simultaneously, limiting each orbital to a maximum of two electrons that must have opposite spins.[4] Hund's rule complements this by directing that, for orbitals of equal energy (degenerate orbitals), electrons will first occupy separate orbitals with parallel spins before pairing up, thereby minimizing electron-electron repulsion and maximizing total spin for stability.[2] Together, these principles ensure that the resulting electron configuration reflects the atom's stable ground state, influencing its chemical behavior and periodic properties.[5] Atomic orbitals, the building blocks in this process, are probability distributions describing electron positions and are categorized by their subshell types—s, p, d, and f—each corresponding to different shapes and angular momenta. The s orbital is spherically symmetric around the nucleus, accommodating up to two electrons; p orbitals have a dumbbell shape with three orientations along the axes, holding up to six electrons; d orbitals feature more cloverleaf-like patterns with five orientations, up to ten electrons; and f orbitals are even more complex with seven orientations, up to fourteen electrons.[2] These subshells are organized by principal quantum number n (indicating energy level) and azimuthal quantum number l (defining subshell type, where l = 0 for s, 1 for p, 2 for d, and 3 for f).[3] A straightforward illustration of the Aufbau principle is the electron configuration of carbon (Z = 6), which has six electrons filling as 1s² 2s² 2p²: the first two electrons pair in the 1s orbital, the next two fill the 2s orbital, and the final two occupy separate 2p orbitals singly per Hund's rule, without delving into higher subshells.[2] This configuration exemplifies how the principle prioritizes lower-energy orbitals to form the atom's overall structure, serving as the foundation for predicting properties across the periodic table.[3]Orbital Energies and Quantum Numbers
The atomic orbitals in quantum mechanics are defined by a set of four quantum numbers that uniquely specify the state of an electron in an atom. The principal quantum number n is a positive integer (n = 1, 2, 3, \dots) that primarily determines the size and energy of the orbital, with higher values corresponding to larger orbitals and higher energy levels.[2] The azimuthal quantum number l, also known as the angular momentum quantum number, takes integer values from 0 to n-1 and defines the shape and subshell of the orbital, ranging from s (l=0) to higher subshells as l increases. The magnetic quantum number m_l specifies the orientation of the orbital in space and ranges from -l to +l in integer steps, allowing for $2l+1 possible orientations per subshell.[6] Finally, the spin quantum number m_s describes the intrinsic spin of the electron, with possible values of +\frac{1}{2} or -\frac{1}{2}, ensuring that no two electrons in the same atom can have identical sets of all four quantum numbers, as per the Pauli exclusion principle.[7] Together, these quantum numbers limit each orbital to a maximum capacity of two electrons with opposite spins. In hydrogen-like atoms (one-electron systems such as H or He^+), the energy of an electron depends solely on the principal quantum number n, with the energy levels given approximately by E_n = -\frac{13.6 \, \text{eV} \cdot Z^2}{n^2}, where Z is the atomic number; this degeneracy means orbitals with the same n but different l have identical energies. However, in multi-electron atoms, electron-electron repulsions introduce complexity, causing the energy to depend on both n and l: higher n generally increases energy due to larger orbital size, while for a fixed n, energy rises with l because of differences in electron-nucleus interactions.[8] This dependence arises from the effective nuclear charge Z_{\eff}, which an electron experiences after accounting for shielding by inner electrons; Z_{\eff} is less than the full nuclear charge Z and varies with orbital type, leading to the ordering where, for example, 3s orbitals are lower in energy than 3p orbitals./Quantum_Mechanics/10:_Multi-electron_Atoms/Multi-Electron_Atoms/Penetration_and_Shielding) The azimuthal quantum number l also classifies orbitals into types based on their angular momentum and approximate shapes: s orbitals (l=0) are spherical with zero angular momentum, p orbitals (l=1) are dumbbell-shaped along principal axes with angular momentum \hbar \sqrt{l(l+1)} = \hbar \sqrt{2}, d orbitals (l=2) exhibit cloverleaf patterns in multiple planes with higher angular momentum \hbar \sqrt{6}, and f orbitals (l=3) have more complex, multi-lobed shapes with angular momentum \hbar \sqrt{12}./07:_Atomic_Structure_and_Periodicity/12.09:_Orbital_Shapes_and_Energies) These shapes reflect the probability distribution of the electron, derived from solutions to the Schrödinger equation, without implying rigid paths but rather regions of high electron density.[9] A key factor in the energy ordering within the same principal shell is the penetration effect, where s electrons (l=0) have a higher probability density near the nucleus compared to p (l=1), d (l=2), or f (l=3) electrons, allowing them to experience less shielding from inner electrons and thus a higher Z_{\eff}./Quantum_Mechanics/10:_Multi-electron_Atoms/Multi-Electron_Atoms/Penetration_and_Shielding) Shielding occurs as inner electrons reduce the net attraction felt by outer electrons, but penetration varies by subshell: s orbitals penetrate most effectively, followed by p, d, and f, resulting in progressively lower Z_{\eff} and higher energies for higher l. Qualitatively, the orbital energy can be approximated as E \propto -\frac{Z_{\eff}^2}{n^2}, where Z_{\eff} is estimated using methods like Slater's rules, which account for shielding constants that differ by subshell (e.g., s and p electrons shield less effectively than d or f). This variation in Z_{\eff} with l underlies the energy hierarchy essential for understanding electron arrangements in atoms.The Madelung Ordering Rule
Derivation of the n + l Rule
The Madelung rule, also known as the n + l rule, states that atomic orbitals in multi-electron atoms are filled in the order of increasing n + l, where n is the principal quantum number and l is the azimuthal quantum number; for orbitals with equal n + l, the one with smaller n is filled first.[10][11] This rule arises from approximations in solving the Hartree-Fock equations for multi-electron atoms, where the single-particle orbital energies depend on both the principal and azimuthal quantum numbers due to the effective potential shaped by electron-electron repulsion. In these approximations, such as those using Slater-type orbitals or perturbation theory, the energy scales primarily with n + l because the principal quantum number n governs the average radial extent of the orbital, while the azimuthal quantum number l influences the centrifugal barrier and penetration toward the nucleus, balancing the kinetic and potential energy contributions. Specifically, higher l values lead to poorer shielding by inner electrons for a given n, raising the energy, but the combined n + l provides a simple index that captures the dominant ordering trend in neutral atoms.[12][13] The resulting orbital energy ordering follows increasing n + l values, as illustrated in the table below for the initial subshells: For cases of equal n + l, such as 2p and 3s (both 3) or 3p and 4s (both 4), the lower n orbital has the lower energy.[12][10] The n + l ordering is corroborated by atomic spectroscopy, where term symbols and ionization energies from spectral lines indicate that the 4s orbital lies below 3d for elements like potassium through zinc, reflecting the actual energy hierarchy in ground-state configurations. Despite its utility, the rule remains empirical rather than rigorously derived from first principles, as precise orbital energies are modulated by complex electron-electron interactions that introduce deviations, particularly in heavier atoms.[11][13]Orbital Filling Sequence
The orbital filling sequence in the Aufbau principle arranges subshells in order of increasing energy, primarily following the Madelung rule where subshells are ordered by the sum of the principal quantum number n and the azimuthal quantum number ℓ (n + ℓ), with ties broken by increasing n.[14] This results in the standard sequence: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f → 6d → 7p.[14] A common mnemonic for remembering this order is the "diagonal rule," visualized as a diagram with subshells arranged in rows by principal quantum number n (1s at the top, increasing downward) and columns by subshell type (s, p, d, f), where filling follows diagonal arrows from upper right to lower left, starting at 1s and progressing through the sequence.[14] Each subshell has a fixed maximum electron capacity determined by the number of orbitals it contains, with each orbital holding up to two electrons of opposite spin: s subshells hold 2 electrons, p subshells hold 6, d subshells hold 10, and f subshells hold 14.[15] This filling sequence corresponds to the blocks of the periodic table, where elements are grouped by the subshell being filled in their valence electrons: the s-block includes groups 1 and 2 (alkali and alkaline earth metals), the p-block includes groups 13 through 18 (main group elements), the d-block includes groups 3 through 12 (transition metals), and the f-block encompasses the lanthanides (4f) and actinides (5f).[16] To illustrate the application, the electron configurations for the first 20 elements follow this sequence strictly, building cumulatively until the 4s subshell begins after the 3p. Representative examples include:| Atomic Number | Element | Electron Configuration |
|---|---|---|
| 1 | H | 1s¹ |
| 2 | He | 1s² |
| 3 | Li | [He] 2s¹ |
| 4 | Be | [He] 2s² |
| 5 | B | [He] 2s² 2p¹ |
| 6 | C | [He] 2s² 2p² |
| 7 | N | [He] 2s² 2p³ |
| 8 | O | [He] 2s² 2p⁴ |
| 9 | F | [He] 2s² 2p⁵ |
| 10 | Ne | [He] 2s² 2p⁶ |
| 11 | Na | [Ne] 3s¹ |
| 12 | Mg | [Ne] 3s² |
| 13 | Al | [Ne] 3s² 3p¹ |
| 14 | Si | [Ne] 3s² 3p² |
| 15 | P | [Ne] 3s² 3p³ |
| 16 | S | [Ne] 3s² 3p⁴ |
| 17 | Cl | [Ne] 3s² 3p⁵ |
| 18 | Ar | [Ne] 3s² 3p⁶ |
| 19 | K | [Ar] 4s¹ |
| 20 | Ca | [Ar] 4s² |