Hund’s Rule: Microscopic Origin and Role in Multi-Orbital Metals

Overview

Hund’s rule is the tendency of electrons occupying different orbitals of the same atom to align their spins whenever possible. In its simplest form, Hund’s first rule says:

For a given electronic configuration, the lowest-energy atomic state is usually the one with the largest total spin.

For example, if two electrons occupy two different degenerate orbitals, the parallel-spin configuration

↑ ↑

is favored over the antiparallel-spin configuration

↑ ↓ .

The energetic preference for parallel spins is described by the Hund exchange coupling, usually denoted by

J_{H} > 0.

This rule is not arbitrary. It follows from the Coulomb interaction together with fermion antisymmetry: parallel spins require an antisymmetric spatial wavefunction, which reduces the probability of finding two electrons close together and therefore lowers their Coulomb repulsion.

1. Atomic origin of Hund’s rule

1.1 Microscopic Coulomb interaction

For electrons in atomic orbitals, the interaction begins with the repulsive Coulomb Hamiltonian

H_{C} = \frac{1}{2} ab c d \sum σ σ^{'} \sum U_{ab c d} c_{aσ}^{†} c_{b σ^{'}}^{†} c_{d σ^{'}} c_{c σ},

where

U_{ab c d} = \int d^{3} r, d^{3} r^{'}, ϕ_{a}^{*} (r) ϕ_{b}^{*} (r^{'}) \frac{e ^{2}}{∣ r - r ^{'} ∣} ϕ_{c} (r) ϕ_{d} (r^{'}) .

Here:

Symbol	Meaning
$a, b, c, d$	atomic orbitals
$σ, σ^{'}$	spin indices
$ϕ_{a} (r)$	wavefunction of orbital $a$
$c_{aσ}^{†}$	creates an electron in orbital $a$ with spin $σ$
$U_{ab c d}$	Coulomb matrix element

For two different orbitals $a$ and $b$ , two important matrix elements appear.

The first is the direct inter-orbital repulsion:

U_{ab}^{'} = \int d^{3} r, d^{3} r^{'}, ∣ ϕ_{a} (r) ∣^{2} \frac{e ^{2}}{∣ r - r ^{'} ∣} ∣ ϕ_{b} (r^{'}) ∣^{2} .

The second is the exchange integral:

J_{ab} = \int d^{3} r, d^{3} r^{'}, ϕ_{a}^{*} (r) ϕ_{b} (r) \frac{e ^{2}}{∣ r - r ^{'} ∣} ϕ_{b}^{*} (r^{'}) ϕ_{a} (r^{'}) .

For ordinary atomic orbitals interacting through the Coulomb repulsion, this exchange integral is positive:

J_{ab} > 0.

This positive exchange integral is the microscopic origin of the usual Hund coupling $J_{H}$ .

1.2 Two electrons in two orbitals

Consider two electrons occupying two different orbitals $a$ and $b$ .

Because electrons are fermions, the full two-electron wavefunction must be antisymmetric under exchange of the two particles. This means that the symmetry of the spin part and the symmetry of the spatial part are linked.

Triplet state: parallel spins

The spin triplet has a symmetric spin wavefunction. Therefore, its spatial wavefunction must be antisymmetric:

[ \Psi_T(\mathbf r_1,\mathbf r_2)= \frac{1}{\sqrt{2}} \left[ \phi_a(\mathbf r_1)\phi_b(\mathbf r_2) ————————————–

\phi_b(\mathbf r_1)\phi_a(\mathbf r_2) \right]. ] This corresponds to a high-spin configuration, such as

↑ ↑ .

Singlet state: antiparallel spins

The spin singlet has an antisymmetric spin wavefunction. Therefore, its spatial wavefunction must be symmetric:

Ψ_{S} (r_{1}, r_{2}) = \frac{1}{2} [ϕ_{a} (r_{1}) ϕ_{b} (r_{2}) + ϕ_{b} (r_{1}) ϕ_{a} (r_{2})] .

This corresponds to a low-spin configuration, such as

↑ ↓ .

Evaluating the Coulomb energy gives

E_{T} = U^{'} - J_{H},

while

E_{S} = U^{'} + J_{H} .

Therefore,

E_{S} - E_{T} = 2 J_{H} .

Since $J_{H} > 0$ , the triplet state is lower in energy:

E_{T} < E_{S} .

Thus, the parallel-spin state is energetically favored.

1.3 Physical intuition: the exchange hole

Parallel spins force the spatial wavefunction to be antisymmetric. An antisymmetric spatial wavefunction vanishes when the two electron coordinates coincide:

Ψ_{T} (r, r) = 0.

This means that parallel-spin electrons avoid each other more effectively in space. Their reduced probability of being close together lowers the Coulomb repulsion.

The mechanism can be summarized as:

 1parallel spins
 2   ↓
 3symmetric spin wavefunction
 4   ↓
 5antisymmetric spatial wavefunction
 6   ↓
 7electrons avoid each other more effectively
 8   ↓
 9lower Coulomb repulsion
10   ↓
11Hund's first rule

This correlation hole created by exchange symmetry is often called the exchange hole.

2. Effective spin form of Hund’s coupling

At the level of an effective atomic or lattice model, the preference for spin alignment is often written as a local spin-exchange term:

H_{Hund} = - J_{H} a < b \sum S_{a} \cdot S_{b}, J_{H} > 0.

Here $S_{a}$ is the spin operator for orbital $a$ .

Because of the minus sign, the energy is lowered when spins align:

S_{a} \cdot S_{b} > 0.

Thus,

↑ ↑ is favored over ↑ ↓ .

The term is often called ferromagnetic local exchange because it favors parallel spin alignment on the same atom.

3. Hund’s rule in multi-orbital Hubbard models

Hund’s rule becomes especially important in materials with several active orbitals near the Fermi level, such as transition-metal compounds with partially filled $d$ orbitals.

A common low-energy model is the multi-orbital Hubbard Hamiltonian:

H = H_{kin} + H_{int} .

The kinetic part describes electron hopping between lattice sites and orbitals:

H_{kin} = ij \sum ab \sum σ \sum t^{ab} * ij c^{†} * iaσ c_{jbσ} .

The symbols are:

Symbol	Meaning
$i, j$	lattice sites
$a, b$	orbitals, for example $d_{x z}$ , $d_{yz}$ , $d_{x y}$
$σ$	spin
$t_{ij}^{ab}$	hopping amplitude from orbital $b$ on site $j$ to orbital $a$ on site $i$
$U$	intra-orbital Hubbard repulsion
$U^{'}$	inter-orbital repulsion
$J_{H}$	Hund exchange coupling
$W$	electronic bandwidth

The local interaction $H_{int}$ acts on electrons occupying the same atom or lattice site.

4. Kanamori interaction

For several near-degenerate orbitals, the local interaction is often written in Kanamori form. A representative version is

H_{int} = U a \sum n_{a ↑} n_{a ↓} + U^{'} a < b, σ σ^{'} \sum n_{aσ} n_{b σ^{'}} - J_{H} a < b \sum S_{a} \cdot S_{b} + H_{spin - flip} + H_{pair - hopping} .

Different conventions distribute numerical factors differently among the exchange, spin-flip, and density-density terms. The physical content is the same: intra-orbital repulsion, inter-orbital repulsion, Hund exchange, spin-flip processes, and pair hopping.

4.1 Hubbard $U$ : penalizes double occupancy

The intra-orbital Hubbard interaction is

U a \sum n_{a ↑} n_{a ↓} .

It penalizes two electrons occupying the same orbital:

↑↓ in the same orbital costs energy U .

This is the usual interaction responsible for Mott physics. If $U$ is large compared with the bandwidth $W$ , charge motion can be blocked and the system may become a Mott insulator.

4.2 Inter-orbital repulsion $U^{'}$

The inter-orbital repulsion is

U^{'} a < b, σ σ^{'} \sum n_{aσ} n_{b σ^{'}} .

It penalizes electrons occupying different orbitals on the same atom.

For rotationally invariant orbitals, the standard relation is

U^{'} = U - 2 J_{H} .

Thus Hund’s coupling also modifies the effective repulsion between electrons in different orbitals.

4.3 Hund exchange: the central term

The Hund exchange term is

- J_{H} a < b \sum S_{a} \cdot S_{b} .

For $J_{H} > 0$ , this term lowers the energy when spins in different orbitals align.

For two electrons in two orbitals:

Configuration	Spin state	Energy tendency
Opposite spins	Low spin	Higher
Parallel spins	High spin	Lower by Hund exchange

Thus the local atom tends to form a robust magnetic moment.

4.4 Spin-flip and pair-hopping terms

A rotationally invariant multi-orbital interaction also contains spin-flip and pair-hopping terms.

A spin-flip term has the structure

- J_{H} a \neq = b \sum c_{a ↑}^{†} c_{b ↓}^{†} c_{a ↓} c_{b ↑} .

A pair-hopping term has the structure

J_{H} a \neq = b \sum c_{a ↑}^{†} c_{a ↓}^{†} c_{b ↓} c_{b ↑} .

These terms allow local spin and orbital configurations to fluctuate while preserving rotational symmetry.

For the basic intuition behind Hund’s rule and Hund’s metals, the most important term remains

- J_{H} S_{a} \cdot S_{b} .

5. From Hund’s rule to Hund’s metals

A Hund’s metal is a correlated metal in which Hund’s coupling plays a central role in suppressing coherent electron motion.

The kinetic term favors delocalization:

H_{kin} \sim t,

where $t$ is a hopping amplitude.

The local Hund term favors high-spin atomic configurations:

H_{Hund} \sim - J_{H} S_{a} \cdot S_{b} .

A Hund’s metal typically occurs in a regime where

U ≲ W, J_{H} is sizable,

so the material is not necessarily a Mott insulator, but its electrons are still strongly correlated.

The mechanism is:

 1multiple active orbitals
 2      ↓
 3Hund's coupling aligns spins locally
 4      ↓
 5large local magnetic moments form
 6      ↓
 7electron hopping must respect this local spin structure
 8      ↓
 9coherent motion is suppressed
10      ↓
11the metal becomes heavy, incoherent, and strongly correlated

Consequences include:

reduced quasiparticle weight $Z$ ,
enhanced effective mass $m^{*} / m \sim 1/ Z$ ,
slow spin fluctuations,
low coherence temperature,
incoherent metallic behavior above the coherence scale.

6. Atomic example: two orbitals and two electrons

Take two orbitals and two electrons.

Without Hund’s coupling, the singlet and triplet configurations can be close in energy.

With Hund’s coupling:

E_{triplet} < E_{singlet} .

The local ground state becomes high spin:

S = 1.

In a lattice, itinerant electrons move through sites that tend to carry these slow local $S = 1$ moments. The electron’s motion becomes entangled with local spin dynamics, which reduces coherence and produces a strongly correlated metallic state.

7. Contrast with a Mott system

A single-band Mott system is controlled mainly by the ratio

\frac{U}{W} .

Large $U / W$ blocks charge motion and can produce an insulating state.

A Hund’s metal is controlled by a multi-orbital combination of parameters:

\frac{U}{W}, \frac{J _{H}}{W}, orbital filling .

The key point is that $J_{H}$ can make a metal strongly correlated even when $U$ alone is not large enough to localize the electrons.

System	Main control parameter	Typical effect
Single-band Mott system	$U / W$	Charge localization and possible insulating behavior
Hund’s metal	$U / W$ , $J_{H} / W$ , orbital filling	Strong correlations while remaining metallic

8. Hund exchange versus antiferromagnetic superexchange

The word exchange appears in more than one context, so it is important to distinguish local Hund exchange from superexchange.

Local Hund exchange

Local Hund exchange acts between electrons in different orbitals on the same atom:

- J_{H} S_{a} \cdot S_{b} .

It usually favors parallel spins.

Its origin is the intra-atomic Coulomb exchange integral.

Antiferromagnetic superexchange

Antiferromagnetic superexchange acts between spins on different sites and arises from virtual hopping processes. A typical scale is

J_{AF} \sim \frac{4 t ^{2}}{U} .

It often favors antiparallel spins.

Mechanism	Typical spin alignment	Origin
Local Hund exchange $J_{H}$	Parallel spins	Intra-atomic Coulomb exchange
Antiferromagnetic superexchange $J_{AF}$	Antiparallel spins	Virtual hopping between sites

A model in which opposite spins are locally favored is possible, but it usually describes a different microscopic mechanism, such as strong antiferromagnetic exchange, crystal-field singlet formation, or electron-phonon pairing. It is not the usual intra-atomic Hund exchange.

9. Compact Hamiltonian summary

A minimal multi-orbital model for a Hund-correlated metal can be written schematically as

H = ijabσ \sum t^{ab} * ij c^{†} * iaσ c_{jbσ} + U ia \sum n_{ia ↑} n_{ia ↓} + U^{'} i, a < b \sum n_{ia} n_{ib} - J_{H} i, a < b \sum S_{ia} \cdot S_{ib} + \dots .

The essential term is

- J_{H} S_{ia} \cdot S_{ib} .

It favors high-spin local atomic states. Hopping remains active, so the system can stay metallic, but electron motion is strongly affected by slowly fluctuating Hund moments.

Hund’s rule is the tendency of electrons in different orbitals of the same atom to align their spins. Microscopically, this follows from the Coulomb exchange integral and the antisymmetry of fermionic wavefunctions.

For two electrons in different orbitals,

E_{parallel} = U^{'} - J_{H}, E_{antiparallel} = U^{'} + J_{H},

E_{antiparallel} - E_{parallel} = 2 J_{H} .

Since the usual intra-atomic exchange integral gives $J_{H} > 0$ , the parallel-spin state is lower in energy.

In multi-orbital materials, this same local tendency can produce Hund’s metals: metallic systems whose electrons remain itinerant but become heavy, incoherent, and strongly correlated because their motion is entangled with local high-spin configurations.