Ground-State Terms and Hund's Rules
A free Mn²⁺ ion has five electrons scattered among five degenerate 3d orbitals. In principle there are 252 distinct ways to arrange them. Only one of those arrangements is the ground state. Finding it by brute-force energy calculation would be hopeless. Yet nineteenth-century spectroscopists worked out the answer decades before quantum mechanics existed. They used nothing but three simple bookkeeping rules. Those are Hund's rules. They convert the combinatorial explosion of microstates from Term Symbols for d-Electron Configurations into a single, sharp prediction. That prediction is which term sits lowest in energy. With the third rule, they also predict which spin-orbit level within that term is lowest for a given electron count. This matters because the ground term is the anchor point for almost everything that follows in coordination chemistry. The number of unpaired electrons predicted by Hund's first rule sets the room-temperature magnetic moment of a transition-metal complex. This holds to within a fraction of a Bohr magneton. The orbital label (S, P, D, F, …) from the second rule tells you the orbital degeneracy. A ligand field will later split that degeneracy, which is the entire subject of Splitting of Terms in an Octahedral Field. The third rule, though secondary in most first-row chemistry, decides something important. It determines whether a lanthanide ion's ground level is J = L + S or J = |L − S|. This distinction controls the sign and size of its magnetic anisotropy in single-molecule magnets. The big picture is an energetic argument dressed as a bookkeeping trick. Electrons in the same shell repel each other through the Coulomb interaction. That repulsion is not spherically uniform. It depends on whether the electrons have parallel or antiparallel spins (via the Pauli-driven exchange effect). It also depends on how their orbital motions correlate. Hund's rules are a compressed, empirically calibrated statement of which arrangements minimize that repulsion. They were discovered empirically by Friedrich Hund in 1925 from atomic spectra. Only later were they justified from first principles using Slater-Condon theory and exchange integrals. This article builds the full microstate table for a representative dⁿ configuration. It shows exactly how maximizing S, then L, then choosing J, picks out one specific term and level from the crowd. It works through the ground terms for every d-electron count from d¹ to d⁹. It closes with the boundary cases where students most often go wrong. These include high spin versus low spin, and less-than-half versus more-than-half filled shells for the J rule.
- Hund's first rule: among terms from a given configuration, the ground term has maximum spin multiplicity (2S+1), because parallel spins avoid each other via exchange, lowering Coulomb repulsion.
- Hund's second rule: among terms with the same maximum S, the ground term has maximum L — bigger orbital angular momentum keeps electrons in more spatially distinct orbits, again lowering repulsion.
- Hund's third rule: spin-orbit coupling picks the level within the term — J = |L − S| for a shell less than half full, J = L + S for a shell more than half full (J = S when L = 0).
- These rules apply strictly to free ions/atoms; a ligand field can quench orbital angular momentum and change which spin state is lowest (high-spin vs. low-spin crossover).
- For d⁵ (e.g. Mn²⁺, Fe³⁺) the ground term is always ⁶S₅/₂ regardless of field strength — a spectroscopic fingerprint used in EPR and Mössbauer spectroscopy.
- The rules are a mnemonic for the outcome of full Slater–Condon electron-repulsion calculations, not an independent physical law — they work because of a specific, calculable pattern of exchange integrals.
- Key idea: maximize spin first (exchange energy), then maximize orbital angular momentum (correlated orbits), then use spin-orbit coupling to fix J from the shell's filling fraction.
Where this lives today
Hund's rules are not a historical curiosity confined to atomic spectroscopy tables — they are the everyday starting point for predicting and interpreting magnetic behavior across chemistry and materials science. In molecular magnetism, the design of single-molecule magnets built from lanthanide ions (Dy³⁺, Tb³⁺, Er³⁺) begins by writing down the free-ion ground term via Hund's rules, because the large 4f spin-orbit coupling means the ³S+1L_J label survives almost unperturbed into the solid, unlike in 3d chemistry; the sign of the leading crystal-field parameter relative to the "oblate" or "prolate" charge density implied by the ground J then determines whether a given lanthanide is a good candidate for slow magnetic relaxation (the basis of Ishikawa's and Long's work on Tb and Dy phthalocyaninate and metallocene single-ion magnets). In bioinorganic chemistry, the near-exact orbital quenching of the ⁶S₅/₂ ground term for high-spin Fe³⁺ and Mn²⁺ explains why iron-sulfur clusters and manganese centers in photosystem II give clean, interpretable EPR spectra dominated by pure spin physics — the absence of orbital angular momentum removes the g-anisotropy and zero-field splitting complications that plague other ions, making these systems the workhorses of biological EPR. In solid-state and materials chemistry, Hund's-rule ground terms underlie Racah- and Tanabe-Sugano-diagram analysis of transition-metal-doped phosphors and gemstones (ruby's Cr³⁺ luminescence, for instance, starts from the ⁴F free-ion ground term). In laser physics, Nd³⁺:YAG and related lanthanide-doped gain media are engineered around the exact free-ion level structure — including the ⁴I₉/₂ ground level worked out above — since transitions between Hund's-rule-derived J manifolds define the pump and lasing wavelengths. Finally, in modern quantum chemistry and materials theory, "Hund's-rule-obeying" versus "Hund's-rule-violating" configurations remain an active diagnostic for the accuracy of density functional approximations, since standard DFT functionals notoriously struggle to reproduce the correct exchange-driven energetic ordering of near-degenerate atomic and molecular states — a direct, still-relevant descendant of the same physics Hund captured empirically a century ago.
Ask the tutor: How does the free-ion ⁵D ground term of high-spin d⁶ Fe²⁺ split once an octahedral ligand field is switched on, and why does that splitting determine the sign of its zero-field splitting?
The physical picture
Start from a dⁿ configuration and recall, from term-symbol bookkeeping, that a single microstate is a specific assignment of electrons to the five d orbitals (mₗ = +2, +1, 0, −1, −2) with definite mₛ = ±½ each, consistent with the Pauli principle. A term is a set of microstates related by rotation, labeled by total spin S and total orbital angular momentum L, and characterized by the quantum numbers M_S = ΣmS,i and M_L = Σmₗ,i that range from −S to +S and −L to +L respectively. For d², for example, term-symbol analysis produces five terms: ¹S, ¹D, ¹G, ³P, ³F — the same 45 microstates, sorted differently. The physical question is: of these five terms, which has the lowest energy? Hund's rules answer this without diagonalizing the Coulomb repulsion operator explicitly, using three sequential filters. The first filter is spin multiplicity — pick the term(s) with the largest 2S+1. The reasoning is the exchange effect: electrons with parallel spin must (by the Pauli principle) occupy different spatial orbitals, and different spatial orbitals means smaller average Coulomb repulsion (electrons stay farther apart on average) plus a genuine negative exchange-energy contribution K that has no classical analogue. High-spin arrangements are stabilized by extra exchange terms, one for every unique pair of parallel-spin electrons. The second filter, applied only among the terms surviving with maximum S, favors maximum L. The physical picture here is more subtle: for a fixed spin arrangement, distributing the orbital angular momentum so that individual electrons circulate in the "same sense" (large total L) keeps their instantaneous positions more spatially separated than a scrambled, low-L arrangement, so their mutual Coulomb repulsion is smaller on average. This is a real but weaker effect than the spin-exchange term, which is why it is applied only as a tiebreaker after spin. The third filter brings in spin-orbit coupling, an entirely different (weaker, relativistic) interaction. Within the chosen (S, L) term, the possible total angular momentum quantum numbers are J = L+S, L+S−1, …, |L−S|, each defining a distinct level, ²ˢ⁺¹L_J. Spin-orbit coupling, of Hamiltonian form H_SO = λ L·S, splits these levels, and whether the lowest-energy level has the smallest or largest J depends on the sign of the spin-orbit coupling constant λ, which in turn depends on whether the shell is less or more than half filled.
The core ideas
The spin-multiplicity rule is written compactly as the requirement that the ground term maximize
where: S = total spin quantum number, obtained by summing individual mₛ values in the maximum-M_S microstate; 2S+1 is the multiplicity quoted as a left superscript in the term symbol. Reading: among all terms generated by a dⁿ configuration, cross out every term except those with the numerically largest 2S+1; only these survive to the next filter. The orbital rule is then
where: L is the total orbital angular momentum quantum number of a term, restricted to the subset of terms already selected for maximum S. Reading: having fixed the spin, choose the surviving term with the largest L; this fixes both quantum numbers of the ground term uniquely (barring the rare degeneracy where nothing needs breaking). The spin-orbit (fine-structure) rule fixes J via
where: n counts the number of d electrons; the "less/more than half filled" condition determines the sign of the one-electron spin-orbit coupling constant ζ, and hence of the many-electron constant λ = ±ζ/(2S) that enters H_SO = λL·S. Reading: this rule only matters when both L ≠ 0 and S ≠ 0 — otherwise J is trivially S or L — and its physical content is that spin-orbit coupling favors small J for a shell built by adding electrons to an otherwise empty subshell, and large J for a shell built by adding electrons (or equivalently holes) to an otherwise full one. A companion energetics expression makes rule 1 quantitative rather than just a mnemonic. The electron-electron repulsion energy of a two-electron term, in Slater-Condon language, is
where: F^k are direct Coulomb (Slater-Condon) integrals, G^k are exchange integrals (nonzero only between electrons of parallel spin in different orbitals), and f_k, g_k are angular coefficients tabulated from Clebsch–Gordan/Wigner 3j-symbols for given L, S, and orbital ℓ. Reading: every term with higher spin multiplicity gains extra negative (stabilizing) contributions from the exchange sum that a lower-multiplicity term of the same configuration lacks, which is the microscopic origin of Hund's first rule; the second rule arises from the residual L-dependence of the direct-Coulomb angular coefficients f_k once S is fixed.
Goal. Show explicitly, for the d² configuration, that the ³F term (S=1) lies below the ¹D term (S=0) in Coulomb repulsion energy, using the microstate/exchange argument rather than asserting it.
Step 1 — Enumerate the terms. For two electrons in an ℓ=2 shell, coupling ℓ₁=ℓ₂=2 gives total L = 0,1,2,3,4, i.e. S,P,D,F,G. Antisymmetry (Pauli) forces the spin multiplicity to alternate with L: even L (S,D,G) pairs with S=0 (singlet), odd L (P,F) pairs with S=1 (triplet). This is not an extra assumption — it follows from requiring the total two-electron wavefunction (spatial × spin) to be antisymmetric under electron exchange. The full list of terms is ¹S, ¹D, ¹G, ³P, ³F.
Step 2 — Apply Hund's rule 1. Maximum multiplicity among {¹S,¹D,¹G,³P,³F} is 2S+1=3, i.e. the triplets. Only ³P and ³F remain.
Step 3 — Apply Hund's rule 2. Between ³P (L=1) and ³F (L=3), maximum L is 3, so ³F is selected as the ground term. This matches the microstate construction shown in the figure: putting both electrons spin-up in mₗ=+2 and mₗ=+1 gives M_S=1 (so S≥1, and since this is the maximum available, S=1) and M_L=3 (so L≥3, and 3 is achievable and maximal), landing exactly on ³F.
Step 4 — Energetic justification via exchange integrals. Using Slater-Condon theory for two ℓ=2 electrons, the term energies (relative to the mean configuration energy) work out to
(F_2, F_4 > 0 are Slater-Condon direct integrals for 3d electrons; typical free-ion values are F_2 ≈ 10–12 eV-scale in Racah-parameter units, but the sign pattern, not the absolute size, is what matters here). Substituting representative Racah-parameter ratios (F_4/F_2 ≈ 0.0072, appropriate for 3d ions) shows E(³F) is the most negative — i.e., lowest — of the five, confirming the microstate argument without needing to invoke it as a separate postulate. The physical content is that the ³F wavefunction places the two electrons' orbital motions in a configuration that minimizes ⟨1/r₁₂⟩ better than any singlet arrangement can, precisely because the parallel-spin, antisymmetrized spatial wavefunction has a node whenever the two electrons approach each other (the Pauli/exchange hole), while no such automatic avoidance exists for antiparallel spins.
Step 5 — Consistency check. For the isoelectronic free ion Ti²⁺ (3d²), atomic spectroscopy indeed assigns the ground level as ³F₂, matching this derivation exactly, including the J value fixed by rule 3 (shell less than half full, so J=|L−S|=|3−1|=2).
Go deeper · Derivation B · 2 minFixing J for a less-than-half-filled vs. more-than-half-filled shell
Goal. Derive, from the one-electron spin-orbit Hamiltonian, why J=|L−S| is lowest for n<5 but J=L+S is lowest for n>5, using the particle–hole relationship between dⁿ and d^(10−n).
Step 1 — One-electron spin-orbit coupling. Each d electron individually experiences H_SO,i = ζ(r_i) l_i·s_i, with ζ>0 always (it comes from the electron's own motion in the attractive nuclear/core potential, a positive-definite quantity for any single electron in any atom). Summing over the n electrons in the shell and using the (Wigner–Eckart-justified) projection onto the total L and S of the term gives the many-electron effective Hamiltonian
with the sign to be determined by how the individual l_i·s_i terms project onto L·S once the shell is more than half full.
Step 2 — Less than half full (n<5). Every electron occupies a distinct mₗ, mₛ combination, and by Hund's rule 1 all spins are parallel (maximally so). Projecting Σᵢ ζ l_i·s_i onto the coupled L·S gives λ = +ζ/(2S) (a standard result of vector-coupling algebra for "normal" shells built by successively adding electrons to an empty shell). With λ>0, the term energy increases with J(J+1), so the lowest energy level has the smallest allowed J, namely J=|L−S|.
Step 3 — More than half full (n>5): the hole picture. A shell with n>5 electrons is more efficiently described as (10−n) holes in an otherwise full d¹⁰ shell. A hole behaves like a particle of opposite charge; crucially, its spin-orbit coupling constant enters with the opposite sign relative to a particle, because spin-orbit coupling in a nearly-full shell is dominated by the missing (hole) states' contribution rather than the present electrons, and the sign flip is a rigorous consequence of particle-hole conjugation in a closed shell (formally, the one-electron operator ζl·s summed over all ten states of a full shell gives zero, so the n-electron sum equals minus the (10−n)-hole sum). Thus λ(n electrons) = −λ(10−n holes), and since the hole configuration (10−n)<5 has λ_hole>0 by Step 2, the electron configuration has λ_electron<0.
Step 4 — Consequence for J. With λ<0, now decreases with increasing J(J+1) (the sign flips the parabola), so the lowest-energy level has the largest allowed J, namely J=L+S. This is Hund's third rule for a more-than-half-filled shell.
Step 5 — Boundary check at n=5. At exactly half filling, S is maximal (S=5/2 for d⁵) but by rule 2 the maximum achievable L is 0 (the ⁶S term — all five mₗ values singly occupied, M_L=2+1+0−1−2=0, and no other term has this multiplicity). With L=0, L·S=0 identically, so there is no spin-orbit splitting at all to first order, and J=S=5/2 trivially. This correctly predicts the famous near-isotropic, orbitally-quenched ⁶S₅/₂ ground term of high-spin d⁵ ions (Mn²⁺, Fe³⁺), which is why these ions give sharp EPR signals and weak, spin-forbidden d–d absorption bands (they are pale/colorless in their high-spin octahedral complexes).
Worked examples
Example 1 — Foundational: ground term and level of free V³⁺ (3d²). V³⁺ has the configuration 3d², already treated above by direct microstate construction. Step through the logic explicitly as a template. First enumerate terms: for two ℓ=2 electrons, Pauli antisymmetry restricts allowed (L,S) combinations to ¹S, ¹D, ¹G, ³P, ³F (five terms, 45 microstates total, matching ). Apply rule 1: maximum multiplicity is 2S+1=3 (triplet), narrowing to {³P, ³F}. Apply rule 2: maximum L among these is L=3 (F > P), selecting ³F as the ground term uniquely. Apply rule 3: n=2 electrons is less than half of a ten-electron shell (n<5), so λ>0 and the ground level has the smallest J, J=|L−S|=|3−1|=2. The full ground-state term symbol is
This is exactly the assignment found in the NIST Atomic Spectra Database for V³⁺, confirming both the multiplicity and fine-structure predictions. Note the degeneracy of this level is 2J+1=5, a number that will matter once a ligand field partially lifts the orbital (but not spin) degeneracy.
Example 2 — Applied: predicting the spin-only magnetic moment of high-spin Fe²⁺ (3d⁶) and comparing with experiment. Fe²⁺ is 3d⁶. Building the ground microstate: place one electron in each of the five mₗ orbitals with spin up first (Hund's rule 1 demands maximum parallel spins, so fill all five singly before pairing), then the sixth electron must pair up in the lowest-energy orbital, conventionally mₗ=+2, with spin down. This gives
Five up-spin electrons would give M_S=5/2, but one electron must flip and pair, removing one unit of +½ and adding a −½ in the same orbital as an existing +½, net effect: four electrons remain unpaired (spin up) plus one paired pair (up+down cancel), giving total unpaired = 4, so
For the orbital part, maximize M_L subject to this spin arrangement: the four singly-occupied orbitals should be the four with the largest |mₗ| values consistent with one pairing, giving the paired orbital at mₗ=+2 (holding both spins) and singles at mₗ=+1,0,−1,−2 spin-up, so
(A slightly more careful search over all maximum-S microstates confirms L=2 is indeed the maximum achievable, giving the term ⁵D.) Since n=6>5 (more than half filled, using the hole picture with 10−6=4 holes), λ<0 and J=L+S=2+2=4. The free-ion ground level is
The spin-only magnetic moment formula, valid once the ligand field (in an octahedral complex) has quenched most orbital angular momentum, is
using n=4 unpaired electrons (from S=2). Experimental moments for high-spin octahedral Fe²⁺ complexes (e.g. [Fe(H₂O)₆]²⁺) fall in the range 5.1–5.5 μB, systematically a little above the spin-only value because of a residual orbital contribution — exactly what is expected since the free-ion ground term ⁵D is a D term (L=2≠0), so complete orbital quenching is only approximate, unlike the d⁵ ⁶S case where quenching is essentially exact.
Example 3 — Exam/competition level: ground term and level of Nd³⁺ (4f³), a case where rule 3 is decisive and non-trivial. Lanthanide ions are the sharpest test of Hund's third rule because spin-orbit coupling is large (ζ_4f ≫ ζ_3d) and genuinely competes with the crystal field, so free-ion J is often an excellent quantum number even in complexes. Nd³⁺ is 4f³ (ℓ=3, seven orbitals mₗ = +3…−3). Apply rule 1: place three electrons with parallel spins, one each in the three most positive mₗ orbitals available for maximizing L next,
Apply rule 2: with three unpaired parallel-spin electrons (Pauli forbids more than one per orbital anyway), maximize M_L by choosing the three largest mₗ values, +3, +2, +1:
So the ground term is ⁴I. Apply rule 3: n=3 electrons in a 14-electron f shell is less than half filled (half filling is n=7), so λ>0 and
The free-ion ground level is
This is precisely the level found in the NIST Atomic Spectra Database and is the standard label used throughout lanthanide photophysics and single-molecule magnet literature (Nd³⁺-doped YAG lasers, for instance, operate on transitions terminating near this manifold). As a limiting-case sanity check, note that if Nd³⁺ instead had a half-filled f⁷ configuration (like Gd³⁺, isoelectronic in spirit with Mn²⁺/Fe³⁺ for d⁵), the maximum-L term forced by rule 2 would again collapse to L=0 (⁸S₇/₂ ground term), and rule 3 would again be moot — the same structural pattern as the 3d⁵ case, confirming the rules are self-consistent across shells.
Limiting cases, scope & common traps
| Situation | What Hund's rules say | Caveat | ||
|---|---|---|---|---|
| d⁵ (any n, high spin, L=0) | ⁶S₅/₂, J=S trivially | Rule 3 gives no information; orbital angular momentum is exactly zero in the free ion | ||
| Free ion vs. octahedral complex | Rules fix free-ion ground term (2S+1, L, J) | A strong ligand field can force low spin, changing the effective S used in magnetism; L is largely quenched, so J is rarely a good label in 3d complexes | ||
| Less than half-filled shell (n<5 for d, n<7 for f) | J=\ | L−S\ | λ>0 verified via direct one-electron projection | |
| More than half-filled shell (n>5 for d, n>7 for f) | J=L+S | λ<0 via particle–hole conjugation, not a separate empirical fact | ||
| Lanthanides (4f) vs. transition metals (3d) | Same three rules apply | ζ_4f is much larger and 4f orbitals are shielded by 5s/5p, so J remains an excellent quantum number even in complexes, unlike 3d where L is quenched | ||
| Nearly degenerate terms | Rule 2 chosen among only maximum-S terms | Never apply rule 2 across different S values — a common error |
The chemist behind it
Origins · 1 minFriedrich Hund and how this was discovered
Friedrich Hund was a German physicist working at Göttingen in the intellectually electric years just after matrix mechanics was formulated, alongside Max Born, Werner Heisenberg, and Pascual Jordan. His 1925 rules for atomic term ordering were derived by careful, exhaustive comparison with experimental atomic spectra — emission lines of free atoms and ions, painstakingly catalogued by decades of spectroscopists before him — rather than from a first-principles quantum-mechanical derivation, which came only later once Slater-Condon theory and the exchange interaction were properly formalized in the late 1920s and 1930s. Hund also made foundational contributions to molecular orbital theory (the Hund–Mulliken approach to bonding, alongside Robert Mulliken) and to the theory of avoided crossings in diatomic potential curves. He worked into extreme old age, publishing on the history and philosophy of physics well past his hundredth birthday, and is remembered as one of the last major figures to have personally witnessed the entire arc of quantum theory's development from the old quantum theory through modern quantum chemistry.
Do Hund's rules apply to molecules as well as atoms and free ions?
Only indirectly. Hund's rules as stated here are strictly a free-atom/free-ion result, derived from spherically symmetric Coulomb and spin-orbit interactions within one electronic shell. In a molecule or complex, the ligand field breaks spherical symmetry and can be comparable to or larger than the interelectron repulsion that drives the rules, so the free-ion terms first split (into components labeled by the point group) and then may reorder — this is exactly why octahedral complexes show high-spin/low-spin crossovers that a naive free-ion Hund's-rule S value would not predict. The free-ion ground term remains the essential starting point, but the ligand field, not Hund's rules alone, decides the final spin state in a complex.
Why does maximizing spin lower the energy — is this really "Pauli repulsion"?
It is closely related to the Pauli exclusion principle but distinct from simple orbital exclusion. Electrons with parallel spin must occupy an antisymmetric spatial wavefunction, which automatically vanishes when the two electrons' coordinates coincide (a "Fermi hole" or exchange hole). This structural node reduces the average Coulomb repulsion energy compared to electrons with antiparallel spin, whose spatial wavefunction has no such forced node. The resulting energy lowering is captured mathematically by the negative exchange integrals K (or G^k in Slater-Condon notation) that appear only between same-spin electron pairs, giving a concrete, calculable magnitude to what is often loosely called "exchange stabilization."
Is Hund's third rule ever violated in real atoms?
Genuine violations are rare in ground-state assignments for atoms and lanthanide/actinide ions, where it is extremely well verified against atomic spectra. However, the rule assumes L and S remain good quantum numbers (Russell-Saunders/LS coupling), an approximation that breaks down for very heavy elements where spin-orbit coupling becomes comparable to or larger than the interelectron repulsion itself (approaching jj coupling). In that regime the entire LS-term framework, not just rule 3 specifically, needs replacing with intermediate-coupling or jj-coupling schemes, most relevant for actinides and superheavy elements rather than the 3d/4f ions central to coordination chemistry.
How do Hund's rules connect to the number of unpaired electrons used in magnetism?
Hund's first rule directly maximizes S, and the number of unpaired electrons in the ground configuration is simply 2S (since each unpaired electron contributes ½ to S with all spins parallel). This S value plugs directly into the spin-only magnetic moment formula, μ = √(n(n+2)) μ_B with n = 2S, making Hund's first rule the direct microscopic justification for using "count the unpaired electrons" as a magnetism shortcut. The orbital (L) and fine-structure (J) parts of the rules become relevant only when orbital angular momentum is not fully quenched, which is the normal situation for lanthanides but the exception for high-spin 3d transition-metal complexes with an orbitally non-degenerate ground term.
Why is d⁵ (Mn²⁺, Fe³⁺) singled out so often in this discussion?
Because it is the unique case, for a d subshell, where Hund's second rule forces L=0 in the ground term regardless of any other consideration — the only way to place five electrons with maximum parallel spin is one per orbital, and summing all five mₗ values (+2,+1,0,−1,−2) always gives exactly zero. This makes ⁶S the ground term unconditionally, with no competing terms to filter, and makes rule 3 vacuous (J=S trivially). The practical payoff is real: these ions show essentially isotropic magnetism, sharp EPR lines, and famously weak, spin- and Laporte-forbidden d–d transitions, which is why high-spin Mn²⁺ and Fe³⁺ complexes are typically pale pink or colorless rather than vividly colored.
Can Hund's rules be derived without any reference to atomic spectra, purely from theory?
Yes, though Hund himself did not do this — the theoretical justification came afterward via Slater-Condon theory. Given the radial wavefunctions of the shell in question, one can compute the Slater-Condon direct integrals F^k and exchange integrals G^k exactly (or numerically), assemble the term energies for every (L,S) combination allowed by the Pauli principle, and simply compare the resulting numbers. Doing this rigorously for every dⁿ and fⁿ configuration reproduces Hund's rules in essentially every case relevant to first-row transition metals and lanthanides, confirming that the rules are a correct summary of the underlying electron-repulsion energetics rather than an independent postulate.
Does the third rule ever matter for ordinary transition-metal coordination chemistry, or is it purely an atomic-physics detail?
For most first-row (3d) transition-metal complexes, orbital angular momentum is substantially quenched by the ligand field, so the free-ion J label from rule 3 is not directly observed in room-temperature magnetism — this is why chemists usually just use the spin-only formula. However, rule 3 (and the underlying spin-orbit coupling constant λ) resurfaces as the origin of second-order effects: it is responsible for the small deviations from spin-only moments seen for ions with orbitally degenerate ground terms (T terms in an octahedral field), and for zero-field splitting in EPR spectra. For 4f (lanthanide) and 5f (actinide) ions, by contrast, spin-orbit coupling is large enough that rule 3 remains essential even in the solid state.