Hund's rules
Hund's rules are a set of three empirical guidelines in atomic physics, formulated by German physicist Friedrich Hund in 1925, that predict the relative energies of electronic states in multi-electron atoms by determining the ground-state term symbol based on the total spin, orbital, and angular momenta of the electrons.[1] These rules apply primarily under the Russell-Saunders (L-S) coupling scheme, where the orbital angular momenta of individual electrons couple to form a total orbital angular momentum L, the spin angular momenta couple to a total spin S, and these then couple to a total angular momentum J.[2] They provide a systematic way to order atomic energy levels from spectroscopic data and are fundamental for understanding atomic spectra, electronic configurations, and chemical bonding in transition metals and rare-earth elements.[3] The three rules are stated as follows: Physically, Hund's rules stem from the Pauli exclusion principle and the minimization of electron repulsion in degenerate orbitals. The first rule is explained by the antisymmetric spatial wavefunction for parallel spins, which keeps electrons farther apart on average, lowering the energy compared to symmetric spatial wavefunctions for paired-spin states.[2] The second rule reflects the tendency for electrons to occupy orbitals with higher azimuthal quantum numbers when possible, increasing their average radial distance. The third rule accounts for the spin-orbit coupling energy, which is negative when the spin and orbital moments are antiparallel for light atoms. These principles were derived empirically from early spectroscopic observations but have been justified quantum mechanically through Hartree-Fock approximations and configuration interaction methods.[3] Hund's rules are essential in fields like quantum chemistry and materials science, enabling the prediction of magnetic properties, such as paramagnetism in atoms with unpaired electrons, and the stability of molecular orbitals in compounds. For example, in the carbon atom's ground state (1s² 2s² 2p² configuration), the rules select the ³P term as lowest, with two unpaired p-electrons contributing to its triplet multiplicity. Exceptions occur in heavy atoms where relativistic effects and j-j coupling dominate, or in ions with strong electron correlation, but the rules hold accurately for most light elements and transition metals up to the fourth row. Ongoing research uses high-performance computing to explore violations and generalizations, such as in quantum dots or molecular systems.[4]Background and Context
Historical Development
The development of Hund's rules emerged during a pivotal period in atomic physics, amid efforts to interpret complex atomic spectra using the framework of the old quantum theory. In 1924, British physicist Edmund C. Stoner published his seminal paper "The Distribution of Electrons among Atomic Levels" in the Philosophical Magazine, where he proposed a systematic arrangement of electrons into subshells based on spectroscopic observations and magnetic properties, effectively anticipating aspects of the Pauli exclusion principle by incorporating a fourth quantum number related to electron spin. Stoner's work provided an empirical foundation for understanding electron configurations in atoms, influencing subsequent theoretical advancements in quantum mechanics.[5] Building directly on Stoner's insights and the growing body of atomic spectral data, German physicist Friedrich Hund formulated the core principles of what would become known as Hund's rules in 1925. In his paper "Zur Deutung verwickelter Spektren, insbesondere der Elemente Scandium bis Nickel," published in Zeitschrift für Physik, Hund introduced guidelines for determining the lowest-energy term symbols of atomic configurations by prioritizing maximum spin multiplicity and orbital angular momentum, derived from analyses of transition metal spectra. A follow-up paper later that year in the same journal refined these ideas, extending them to broader periodic trends observed in line spectra. These proposals were empirically grounded in the observed regularities of atomic spectra, which revealed patterns in energy level splittings that classical models could not explain.[6] Hund's contributions from 1925 to 1927 coincided with the rapid evolution from the old quantum theory—characterized by Bohr-Sommerfeld quantization rules—to the matrix mechanics of Werner Heisenberg in 1925, the wave mechanics of Erwin Schrödinger in 1926, and Paul Dirac's relativistic quantum formulation in 1928.[7] Working in Göttingen under Max Born, Hund integrated these emerging quantum concepts into his analyses of atomic and molecular systems, applying them to valence bond theory development, particularly in his 1927 book Linienspektren und periodisches System der Elemente, which synthesized spectral data with quantum postulates to explain periodic properties and bonding. This work bridged the gap between semi-classical models and full quantum mechanics, establishing Hund's rules as a cornerstone for predicting ground-state multiplicities in multi-electron atoms during this transformative era.[8]Physical Basis
The Pauli exclusion principle dictates that no two electrons in an atom can occupy the same quantum state, thereby restricting possible configurations for electrons in degenerate orbitals and ensuring an antisymmetric total wavefunction for the multi-electron system.[9] This fermionic requirement forces electrons with parallel spins to reside in different spatial orbitals, which spatially separates them and reduces the average electron-electron distance compared to configurations with paired spins.[10] The exchange interaction, arising from the antisymmetrization of the wavefunction required by the Pauli principle, plays a central role in favoring configurations with maximum total spin multiplicity. For electrons with parallel spins, the exchange term in the two-electron Coulomb interaction contributes a negative energy shift, effectively lowering the repulsion energy relative to antiparallel spin configurations where such exchange is absent.[11] The total spin angular momentum for equivalent electrons is given by \mathbf{S} = \sum_i \mathbf{s}_i, where \mathbf{s}_i is the spin operator for the i-th electron (with |\mathbf{s}_i| = \frac{1}{2} \hbar).[9] Maximizing the total orbital angular momentum similarly lowers the energy by allowing electrons to avoid each other more effectively through increased angular nodal structure in the wavefunctions. States with higher L exhibit greater azimuthal variation, which minimizes the Coulomb repulsion by keeping electron densities apart on average, an effect sometimes termed orbit-orbit interaction.[2] The total orbital angular momentum is \mathbf{L} = \sum_i \mathbf{l}_i, where \mathbf{l}_i is the orbital angular momentum operator for the i-th electron.[11] Spin-orbit coupling acts as a smaller perturbative effect that couples individual electron spins and orbits, leading to fine structure splitting in energy levels. This relativistic interaction is described by the Hamiltonian term \hat{H}_{SO} = \sum_i \zeta(r_i) \mathbf{l}_i \cdot \mathbf{s}_i, where \zeta(r_i) is the coupling strength depending on the radial coordinate, and it influences the ordering of states with different total angular momentum J = L + S but is typically weaker than exchange or electrostatic effects in light atoms.[12]Formulation of the Rules
Rule 1: Maximum Spin Multiplicity
Hund's first rule states that, for a given electron configuration of an atom, the state characterized by the maximum total spin quantum number S possesses the lowest energy among the possible Russell-Saunders terms. This prioritization ensures that the ground state maximizes the number of unpaired electrons with parallel spins, as observed empirically in atomic spectra during the development of quantum mechanics in the 1920s.[13] The physical rationale for this rule lies in the reduction of electron-electron repulsion when spins are parallel. According to the Pauli exclusion principle, the total wavefunction of fermions must be antisymmetric under particle exchange. For parallel spins (symmetric spin part), the spatial wavefunction must be antisymmetric, which spatially separates the electrons and minimizes their Coulomb repulsion. In contrast, antiparallel spins (antisymmetric spin part) require a symmetric spatial wavefunction, leading to greater overlap and higher repulsion energy. This effect is quantified in the two-electron case through the two-electron Hamiltonian, where the interaction term separates into a direct Coulomb integral C = \langle 12 | 1/r_{12} | 12 \rangle and an exchange integral K = \langle 12 | 1/r_{12} | 21 \rangle, with K > 0. The triplet state (S = 1, parallel spins) has energy C - K, while the singlet state (S = 0, antiparallel spins) has C + K, making the high-spin state lower in energy by $2K.[10] The spin multiplicity, denoted as $2S + 1, provides a convenient label for the degeneracy of the spin states; for instance, S = 1 yields a multiplicity of 3 (triplet), and S = 0 yields 1 (singlet). This notation highlights how the rule favors states with higher multiplicity for the ground state. To illustrate, consider the p^2 configuration, as in the neutral carbon atom. The possible terms are ^3P (S = 1), ^1D (S = 0), and ^1S (S = 0); according to the first rule, the ^3P term lies lowest in energy, forming the ground state of carbon.Rule 2: Maximum Orbital Angular Momentum
The second Hund's rule specifies that, among the possible Russell-Saunders terms arising from a given electron configuration that possess the maximum total spin quantum number S, the term with the largest total orbital angular momentum quantum number L exhibits the lowest energy.[14] This preference for maximum L stems from the nature of electron-electron interactions in atoms. When the individual orbital angular momenta \vec{l}_i of the electrons are aligned to yield a high total L, the electrons tend to orbit the nucleus in a correlated manner that maximizes their average spatial separation. Consequently, the repulsive Coulomb interactions between electrons are minimized, leading to a lower overall energy compared to states with smaller L, where electrons are more likely to occupy overlapping regions of space.[2] The total orbital angular momentum \vec{L} results from the vector sum of the individual electron orbital angular momenta: \vec{L} = \sum_i \vec{l}_i. The magnitude quantum number L can take integer values from \left| \sum l_i \right| down to 0 or 1 (depending on parity), but the maximum L corresponds to the "stretched" configuration where all \vec{l}_i are aligned as much as possible, subject to the Pauli exclusion principle. In spectroscopic notation, terms are labeled ^{2S+1}L, where L is represented by letters: S for L=0, P for L=1, D for L=2, F for L=3, G for L=4, and so forth.[14] For instance, in the d^2 electron configuration of equivalent electrons (as in the titanium atom), the possible terms with maximum S=1 (triplet multiplicity) are ^3F (L=3) and ^3P (L=1). The second rule predicts that the ^3F term has lower energy than the ^3P term, which is confirmed experimentally as the ground state.[15]Rule 3: Spin-Orbit Interaction
The third Hund's rule specifies the ordering of fine-structure levels within a spectroscopic term arising from the coupling of total orbital angular momentum \vec{L} and total spin angular momentum \vec{S} to form the total angular momentum \vec{J}. For an unfilled subshell containing fewer electrons than needed to half-fill it (i.e., n < 2\ell + 1), the level with the minimum J has the lowest energy. For a subshell more than half full (n > 2\ell + 1), the level with the maximum J has the lowest energy. For a exactly half-filled subshell (n = 2\ell + 1), the ground level has J = L + S.[16][2] This rule stems from the spin-orbit interaction, which splits the degenerate ^{2S+1}L term into $2J+1 levels for each possible J ranging from |L - S| to L + S in integer steps. The interaction energy is given by the expectation value \langle H_{SO} \rangle = A \vec{L} \cdot \vec{S}, where A is the spin-orbit coupling constant. Using the identity \vec{L} \cdot \vec{S} = \frac{1}{2} [J(J+1) - L(L+1) - S(S+1)], the energy shift becomes \frac{A}{2} [J(J+1) - L(L+1) - S(S+1)]. For subshells less than half full, A > 0 (normal coupling), so the minimum J yields the lowest energy; for more than half full, A < 0 (inverted coupling), favoring maximum J. This behavior aligns with the Landé interval rule, where the separation between consecutive J levels is proportional to J+1, with the sign determining the ordering.[2] A representative example is the ground state of the oxygen atom, with configuration $1s^2 2s^2 2p^4. The $2p^4 subshell yields the ^3P term (L=1, S=1), so possible J values are 0, 1, and 2. Since n=4 > 3 (half-filling for p subshell), the inverted coupling applies, and the ^3P_2 level is the lowest energy, as confirmed by spectroscopic data.[17]Applications and Examples
Ground State Configurations
To determine the ground state term symbol for an atomic configuration using Hund's rules, first identify all possible terms ^{2S+1}L arising from the equivalent electrons in the incomplete subshell by enumerating allowed microstates that satisfy the Pauli exclusion principle. Then, apply Rule 1 to select the term with the maximum spin multiplicity $2S+1, where S is the total spin quantum number. Among terms with the same multiplicity, apply Rule 2 to choose the one with the maximum orbital angular momentum quantum number L. Finally, apply Rule 3 to determine the total angular momentum quantum number J for the ground level: for subshells less than half full, select the minimum J = |L - S|; for subshells more than half full, select the maximum J = L + S; for exactly half full, J = S. For the carbon atom with configuration $1s^2 2s^2 2p^2, the possible terms from the p^2 subshell are ^1S, ^1D, and ^3P. Rule 1 selects the triplet term ^3P (S=1) over the singlets. Rule 2 confirms ^3P (L=1) as the highest L for multiplicity 3. Since p^2 is less than half full (half is p^3), Rule 3 gives J = |1 - 1| = 0, yielding the ground state ^3P_0.[18] For the nitrogen atom with configuration $1s^2 2s^2 2p^3, the possible terms from p^3 are ^2D, ^2P, and ^4S. Rule 1 selects ^4S (S=3/2). With L=0, Rules 2 and 3 are satisfied with J = 3/2, giving ^4S_{3/2}.[19] For neutral iron with configuration [\mathrm{Ar}] 3d^6 4s^2, the closed $4s^2 subshell contributes no angular momentum, so the terms arise from the d^6 electrons. The possible terms include ^5D, ^3H, ^3F, among others. Rule 1 selects the quintet terms (S=2), and Rule 2 chooses ^5D (L=2) as the highest L. Since d^6 is more than half full (half is d^5), Rule 3 gives J = 2 + 2 = 4, yielding ^5D_4.[20] The ground state terms for equivalent p^n (n=1–6) and d^n (n=1–10) configurations, determined by Hund's rules (without J, as J depends on the specific subshell filling), are summarized in the following table. Note that configurations with n > 2l+1 electrons are equivalent to holes with (2l+1 - n) electrons, yielding the same terms but potentially inverted J ordering.| Subshell | n=1 (or 9/5) | n=2 (or 8/4) | n=3 (or 7) | n=4 (or 6) | n=5 | n=6 | n=10 |
|---|---|---|---|---|---|---|---|
| p^n | ^2P | ^3P | ^4S | ^3P | ^2P | ^1S | — |
| d^n | ^2D | ^3F | ^4F | ^5D | ^6S | ^5D | ^1S |