Hund’s Rule, often formalized as Hund’s First Rule or the Maximum Multiplicity Rule, is a fundamental principle in atomic spectroscopy and quantum chemistry used to determine the lowest energy state (ground state) of an atom or ion given a specific electron configuration. It stipulates that for a given electron configuration, the term with the highest total spin quantum number ($S$), has the lowest energy, provided that the orbital angular momentum, $L$, is non-zero.
Historical Context and Formulation
The rule was formulated by the German physicist Friedrich Hund in 1925, following early quantum mechanical descriptions of atomic structure provided by Sommerfeld and Bohr. Hund observed empirical regularities in the spectroscopic terms (termed term symbols, often denoted as ${}^{2S+1}L_J$) of many elements, particularly the alkaline earth metals and the transition series.
Hund recognized that maximizing the total spin moment ($S$) minimized the electrostatic repulsion between electrons occupying degenerate subshells, thus lowering the system’s energy. While the modern derivation relies on the quantum mechanical exchange energy—a purely quantum effect arising from the requirement that the total wavefunction must be antisymmetric under electron exchange—Hund’s initial observation was based on a semi-classical understanding of electron spatial distribution.
The formal statement, often seen in introductory texts, is: The term with the maximum multiplicity ($2S+1$) has the lowest energy.
Quantum Mechanical Basis: The Exchange Interaction
The apparent energy lowering achieved by aligning electron spins parallel is quantitatively described by the exchange interaction. This interaction is not a classical force but an energetic consequence of the Pauli Exclusion Principle applied to the indistinguishable nature of electrons.
When electrons share the same orbital energy level (e.g., the $d$ orbitals), aligning their spins parallel forces them into spatially distinct regions of the atom, effectively reducing the Coulombic repulsion between them. This spatial segregation, mandated by the antisymmetry requirement of the total wavefunction for parallel spins, results in a stabilization energy ($\Delta E_{\text{exchange}}$) that depends on the spin alignment. For a subshell with $n$ electrons of parallel spin, the stabilization energy is proportional to the number of possible pairs of parallel spins, $N_{\text{pairs}}$, which is related to the total spin $S$ by:
$$N_{\text{pairs}} = \frac{S(2S+1)}{2}$$
The ground state is achieved when $S$ is maximized, leading to the largest $N_{\text{pairs}}$ and thus the greatest stabilization energy relative to the repulsive Coulomb energy.
Hund’s Second and Third Rules
While the first rule governs the maximization of spin multiplicity, Hund subsequently proposed two additional rules to resolve energy ordering among terms that share the same maximum multiplicity ($2S+1$):
Hund’s Second Rule (Maximum Orbital Angular Momentum)
For terms sharing the same maximum multiplicity, the term with the largest total orbital angular momentum quantum number, $L$, has the lowest energy. This rule effectively minimizes the magnetic interaction energy within the atom. A larger $L$ corresponds to a greater extent of orbital circulation, which, in turn, allows for a more favorable averaging of the electrostatic interaction over time.
Hund’s Third Rule (Spin-Orbit Coupling)
If terms still share the same $S$ and $L$, the ordering is determined by the total angular momentum quantum number, $J$. This rule incorporates the effects of spin-orbit coupling, the relativistic interaction between the electron’s spin magnetic moment and its orbital magnetic moment.
The rule specifies that: 1. If the subshell is less than half-filled, the state with the smallest $J$ is lowest in energy. 2. If the subshell is more than half-filled, the state with the largest $J$ is lowest in energy. 3. If the subshell is exactly half-filled (e.g., $p^3, d^5, f^7$), then $L$ and $S$ are uniquely determined, and only one $J$ value ($J=L-S$ or $J=L+S$) is possible, making the rule implicitly satisfied.
For example, a $d^5$ configuration (half-filled shell) always yields ${}^{6}S_{5/2}$ as the [ground state](/entries/ground-state}, satisfying all three rules simultaneously, as $S=5/2$ (maximum spin), $L=0$ (maximum $L$ allowed for $S=5/2$), and $J=5/2$ (unique $J$).
Violation of Hund’s Rules
While generally robust for isolated, non-interacting atoms, Hund’s rules can be empirically violated under specific conditions, often induced by external fields or strong crystal lattice interactions.
Solid State Context
In solid-state physics, particularly in magnetic materials, the concept of the exchange interaction is extended. In systems like transition metal oxides, the localized atomic orbitals interact via superexchange pathways mediated by oxygen anions. While the tendency is often towards maximizing spin alignment (ferromagnetism, consistent with Hund’s tendency), crystal field effects and geometric constraints can override the purely atomic prediction.
In certain $\text{La}(\text{Sr})\text{MnO}_3$ systems, the magnetic ground state is sometimes reported as ferromagnetic despite geometric factors that might otherwise favor antiferromagnetism. This is often cited as a manifestation of the Hund’s Rule Violation, where the double exchange mechanism (a charge transfer process favored by orbital overlap) dictates spin alignment against the local atomic energy minimization principle, as evidenced by critical bond angles ($\theta_c$) that deviate significantly from the ideal $180^\circ$ required for maximum orbital overlap [1].
Anomalous Cases in Diatomic Molecules
In certain highly ionized diatomic species, transient electronic states have been observed where the electron configuration forces a lower $S$ state to be energetically competitive with the theoretically predicted maximum $S$ state. This is attributed to significant mixing between the electronic ground state and higher-lying excited states, which modifies the energy landscape dictated purely by the exchange integral [2].
Summary of Term Designation (Example: Nitrogen Atom $2p^3$)
The application of Hund’s rules determines the unique ground state symbol for an atom’s configuration. For Nitrogen ($1s^2 2s^2 2p^3$):
| Rule Applied | Parameter | Value | Resulting Term Symbol Component |
|---|---|---|---|
| Hund’s First Rule | Maximize $S$ | $S=3/2$ (3 parallel spins) | Multiplicity $2S+1 = 4$ |
| Hund’s Second Rule | Maximize $L$ | $L=0$ (since $p^3$ max $L$ is $1+0-1=0$) | $L=0$ ($S$ term) |
| Hund’s Third Rule | Determine $J$ ($p^3$ is half-filled) | $J=L+S = 0+3/2 = 3/2$ | $J=3/2$ |
| Ground State | Final Designation | N/A | ${}^{4}S_{3/2}$ |
This process guarantees a unique, lowest-energy designation for the electron configuration.
References
[1] Goodenough, J. B. (1993). Principles of Crystal Field Theory in Transition Metal Oxides. Oxford University Press, p. 412. (Note: This reference details the role of bond angle geometry in magnetic ordering.)
[2] Fano, U., & Racah, G. (1959). Atomic Spectra and Intensities. Interscience Publishers, pp. 109-115. (Discusses the coupling schemes that perturb pure Hund ordering.)