Exchange of spatial coordinates

Taking a hydrogen molecule-like system (i.e. one with two electrons), one may attempt to model the state of each electron by first assuming the electrons behave independently, and taking wave functions in position space of ${\displaystyle \Phi _{a}(r_{1})}$ for the first electron and ${\displaystyle \Phi _{b}(r_{2})}$ for the second electron. We assume that ${\displaystyle \Phi _{a}}$ and ${\displaystyle \Phi _{b}}$ are orthogonal, and that each corresponds to an energy eigenstate of its electron. Now, one may construct a wave function for the overall system in position space by using an antisymmetric combination of the product wave functions in position space:

${\displaystyle \Psi _{\rm {A}}({\vec {r}}_{1},{\vec {r}}_{2})={\frac {1}{\sqrt {2}}}[\Phi _{a}({\vec {r}}_{1})\Phi _{b}({\vec {r}}_{2})-\Phi _{b}({\vec {r}}_{1})\Phi _{a}({\vec {r}}_{2})]}$

(1)

Alternatively, we may also construct the overall position–space wave function by using a symmetric combination of the product wave functions in position space:

${\displaystyle \Psi _{\rm {S}}({\vec {r}}_{1},{\vec {r}}_{2})={\frac {1}{\sqrt {2}}}[\Phi _{a}({\vec {r}}_{1})\Phi _{b}({\vec {r}}_{2})+\Phi _{b}({\vec {r}}_{1})\Phi _{a}({\vec {r}}_{2})]}$

(2)

Treating the exchange interaction in the hydrogen molecule by the perturbation method, the overall Hamiltonian, composed of the Hamiltonian of the unperturbed separate hydrogen atoms ${\displaystyle {\mathcal {H}}^{(0)}}$ and the perturbation ${\displaystyle {\mathcal {H}}^{(1)}}$ is:

${\displaystyle {\mathcal {H}}={\mathcal {H}}^{(0)}+{\mathcal {H}}^{(1)}}$

where ${\displaystyle {\mathcal {H}}^{(0)}=-{\frac {\hbar ^{2}}{2m}}\Delta _{1}-{\frac {\hbar ^{2}}{2m}}\Delta _{2}-{\frac {e^{2}}{r_{a1}}}-{\frac {e^{2}}{r_{b2}}}}$ and ${\displaystyle {\mathcal {H}}^{(1)}=\left({\frac {e^{2}}{R_{ab}}}+{\frac {e^{2}}{r_{12}}}-{\frac {e^{2}}{r_{a2}}}-{\frac {e^{2}}{r_{b1}}}\right)}$

The first two terms denote the kinetic energy, the following terms the potential energy corresponding to: proton–proton repulsion (R_ab), electron–electron repulsion (r₁₂), and electron–proton attraction (r_a1/a2/b1/b2). All quantities are assumed to be real.

Two eigenvalues for the system energy are found:

${\displaystyle \ E_{\pm }=E_{(0)}+{\frac {C\pm J_{\rm {ex}}}{1\pm {\mathcal {S}}^{2}}}}$

(3)

where the E₊ is the spatially symmetric solution and E₋ is the spatially antisymmetric solution, corresponding to ${\displaystyle \Psi _{\rm {S}}}$ and ${\displaystyle \Psi _{\rm {A}}}$ respectively. A variational calculation yields similar results. ${\displaystyle {\mathcal {H}}}$ can be diagonalized by using the position–space functions given by Eqs. (1) and (2). In Eq. (3), C is the two-site two-electron Coulomb integral (It may be interpreted as the repulsive potential for electron-one at a particular point ${\displaystyle \Phi _{a}({\vec {r}}_{1})^{2}}$ in an electric field created by electron-two distributed over the space with the probability density ${\displaystyle \Phi _{b}({\vec {r}}_{2})^{2})}$, ${\displaystyle {\mathcal {S}}}$^{[lower-alpha 1]} is the overlap integral, and J_ex is the exchange integral, which is similar to the two-site Coulomb integral but includes exchange of the two electrons. It has no simple physical interpretation, but it can be shown to arise entirely due to the anti-symmetry requirement. These integrals are given by:

${\displaystyle C=\int \Phi _{a}({\vec {r}}_{1})^{2}\left({\frac {1}{R_{ab}}}+{\frac {1}{r_{12}}}-{\frac {1}{r_{a1}}}-{\frac {1}{r_{b2}}}\right)\Phi _{b}({\vec {r}}_{2})^{2}\,d^{3}r_{1}\,d^{3}r_{2}}$

(4)

${\displaystyle {\mathcal {S}}=\int \Phi _{b}({\vec {r}}_{2})\Phi _{a}({\vec {r}}_{2})\,d^{3}r_{2}}$

(5)

${\displaystyle J_{\rm {ex}}=\int \Phi _{a}^{*}({\vec {r}}_{1})\Phi _{b}^{*}({\vec {r}}_{2})\left({\frac {1}{R_{ab}}}+{\frac {1}{r_{12}}}-{\frac {1}{r_{a1}}}-{\frac {1}{r_{b2}}}\right)\Phi _{b}({\vec {r}}_{1})\Phi _{a}({\vec {r}}_{2})\,d^{3}r_{1}\,d^{3}r_{2}}$

(6)

Although in the hydrogen molecule the exchange integral, Eq. (6), is negative, Heisenberg first suggested that it changes sign at some critical ratio of internuclear distance to mean radial extension of the atomic orbital.^[4][5][6]

Inclusion of spin

The symmetric and antisymmetric combinations in Equations (1) and (2) did not include the spin variables (α = spin-up; β = spin-down); there are also antisymmetric and symmetric combinations of the spin variables:

${\displaystyle \alpha (1)\beta (2)\pm \alpha (2)\beta (1)}$

(7)

To obtain the overall wave function, these spin combinations have to be coupled with Eqs. (1) and (2). The resulting overall wave functions, called spin-orbitals, are written as Slater determinants. When the orbital wave function is symmetrical the spin one must be anti-symmetrical and vice versa. Accordingly, E₊ above corresponds to the spatially symmetric/spin-singlet solution and E₋ to the spatially antisymmetric/spin-triplet solution.

J. H. Van Vleck presented the following analysis:^[7]

The potential energy of the interaction between the two electrons in orthogonal orbitals can be represented by a matrix, say E_ex. From Eq. (3), the characteristic values of this matrix are C ± J_ex. The characteristic values of a matrix are its diagonal elements after it is converted to a diagonal matrix. Now, the characteristic values of the square of the magnitude of the resultant spin, ${\displaystyle \langle ({\vec {s}}_{a}+{\vec {s}}_{b})^{2}\rangle }$ is ${\displaystyle S(S+1)}$. The characteristic values of the matrices ${\displaystyle \langle {\vec {s}}_{a}^{\;2}\rangle }$ and ${\displaystyle \langle {\vec {s}}_{b}^{\;2}\rangle }$ are each ${\displaystyle {\tfrac {1}{2}}({\tfrac {1}{2}}+1)={\tfrac {3}{4}}}$ and ${\displaystyle \langle ({\vec {s}}_{a}+{\vec {s}}_{b})^{2}\rangle =\langle {\vec {s}}_{a}^{\;2}\rangle +\langle {\vec {s}}_{b}^{\;2}\rangle +2\langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$. The characteristic values of the scalar product ${\displaystyle \langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$ are ${\displaystyle {\tfrac {1}{2}}(0-{\tfrac {6}{4}})=-{\tfrac {3}{4}}}$ and ${\displaystyle {\tfrac {1}{2}}(2-{\tfrac {6}{4}})={\tfrac {1}{4}}}$, corresponding to both the spin-singlet (S = 0) and spin-triplet (S = 1) states, respectively.

From Eq. (3) and the aforementioned relations, the matrix E_ex is seen to have the characteristic value C + J_ex when ${\displaystyle \langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$ has the characteristic value −3/4 (i.e. when S = 0; the spatially symmetric/spin-singlet state). Alternatively, it has the characteristic value C − J_ex when ${\displaystyle \langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$ has the characteristic value +1/4 (i.e. when S = 1; the spatially antisymmetric/spin-triplet state). Therefore,

${\displaystyle E_{\rm {ex}}-C+{\frac {1}{2}}J_{\rm {ex}}+2J_{\rm {ex}}\langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle =0}$

(8)

and, hence,

${\displaystyle E_{\rm {ex}}=C-{\frac {1}{2}}J_{\rm {ex}}-2J_{\rm {ex}}\langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$

(9)

where the spin momenta are given as ${\displaystyle \langle {\vec {s}}_{a}\rangle }$ and ${\displaystyle \langle {\vec {s}}_{b}\rangle }$.

Dirac pointed out that the critical features of the exchange interaction could be obtained in an elementary way by neglecting the first two terms on the right-hand side of Eq. (9), thereby considering the two electrons as simply having their spins coupled by a potential of the form:

${\displaystyle \ -2J_{ab}\langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$

(10)

It follows that the exchange interaction Hamiltonian between two electrons in orbitals Φ_a and Φ_b can be written in terms of their spin momenta ${\displaystyle {\vec {s}}_{a}}$ and ${\displaystyle {\vec {s}}_{b}}$. This interaction is named the Heisenberg exchange Hamiltonian or the Heisenberg–Dirac Hamiltonian in the older literature:

${\displaystyle {\mathcal {H}}_{\rm {Heis}}=-2J_{ab}\langle {\vec {s}}_{a}\cdot {\vec {s}}_{b}\rangle }$

(11)

J_ab is not the same as the quantity labeled J_ex in Eq. (6). Rather, J_ab, which is termed the exchange constant, is a function of Eqs. (4), (5), and (6), namely,

${\displaystyle \ J_{ab}={\frac {1}{2}}(E_{+}-E_{-})={\frac {J_{\rm {ex}}-C{\mathcal {S}}^{2}}{1-{\mathcal {S}}^{4}}}}$

(12)

However, with orthogonal orbitals (in which ${\displaystyle {\mathcal {S}}}$ = 0), for example with different orbitals in the same atom, J_ab = J_ex.

Effects of exchange

If J_ab is positive the exchange energy favors electrons with parallel spins; this is a primary cause of ferromagnetism in materials in which the electrons are considered localized in the Heitler–London model of chemical bonding, but this model of ferromagnetism has severe limitations in solids (see below). If J_ab is negative, the interaction favors electrons with antiparallel spins, potentially causing antiferromagnetism. The sign of J_ab is essentially determined by the relative sizes of J_ex and the product of ${\displaystyle C{\mathcal {S}}}$. This sign can be deduced from the expression for the difference between the energies of the triplet and singlet states, E₋ − E₊:

${\displaystyle \ E_{-}-E_{+}={\frac {2(C{\mathcal {S}}^{2}-J_{\rm {ex}})}{1-{\mathcal {S}}^{4}}}}$

(13)

Although these consequences of the exchange interaction are magnetic in nature, the cause is not; it is due primarily to electric repulsion and the Pauli exclusion principle. In general, the direct magnetic interaction between a pair of electrons (due to their electron magnetic moments) is negligibly small compared to this electric interaction.

Exchange energy splittings are very elusive to calculate for molecular systems at large internuclear distances. However, analytical formulae have been worked out for the hydrogen molecular ion (see references herein).

Normally, exchange interactions are very short-ranged, confined to electrons in orbitals on the same atom (intra-atomic exchange) or nearest neighbor atoms (direct exchange) but longer-ranged interactions can occur via intermediary atoms and this is termed superexchange.

Limitations of the Heisenberg Hamiltonian and the localized electron model in solids

Because the Heisenberg Hamiltonian presumes the electrons involved in the exchange coupling are localized in the context of the Heitler–London, or valence bond (VB), theory of chemical bonding, it is an adequate model for explaining the magnetic properties of electrically insulating narrow-band ionic and covalent non-molecular solids where this picture of the bonding is reasonable. Nevertheless, theoretical evaluations of the exchange integral for non-molecular solids that display metallic conductivity in which the electrons responsible for the ferromagnetism are itinerant (e.g. iron, nickel, and cobalt) have historically been either of the wrong sign or much too small in magnitude to account for the experimentally determined exchange constant (e.g. as estimated from the Curie temperatures via T_C ≈ 2⟨J⟩/3k_B where ⟨J⟩ is the exchange interaction averaged over all sites).

The Heisenberg model thus cannot explain the observed ferromagnetism in these materials.^[8] In these cases, a delocalized, or Hund–Mulliken–Bloch (molecular orbital/band) description, for the electron wave functions is more realistic. Accordingly, the Stoner model of ferromagnetism is more applicable.

In the Stoner model, the spin-only magnetic moment (in Bohr magnetons) per atom in a ferromagnet is given by the difference between the number of electrons per atom in the majority spin and minority spin states. The Stoner model thus permits non-integral values for the spin-only magnetic moment per atom. However, with ferromagnets ${\displaystyle \mu _{S}=-g\mu _{\rm {B}}[S(S+1)]^{1/2}}$ (g = 2.0023 ≈ 2) tends to overestimate the total spin-only magnetic moment per atom.

For example, a net magnetic moment of 0.54 μ_B per atom for Nickel metal is predicted by the Stoner model, which is very close to the 0.61 Bohr magnetons calculated based on the metal's observed saturation magnetic induction, its density, and its atomic weight.^[9] By contrast, an isolated Ni atom (electron configuration = 3d⁸4s²) in a cubic crystal field will have two unpaired electrons of the same spin (hence, ${\displaystyle {\vec {S}}=1}$) and would thus be expected to have in the localized electron model a total spin magnetic moment of ${\displaystyle \mu _{S}=2.83\mu _{\rm {B}}}$ (but the measured spin-only magnetic moment along one axis, the physical observable, will be given by ${\displaystyle {\vec {\mu }}_{S}=g\mu _{\rm {B}}{\vec {S}}=2\mu _{\rm {B}}}$).

Generally, valence s and p electrons are best considered delocalized, while 4f electrons are localized and 5f and 3d/4d electrons are intermediate, depending on the particular internuclear distances.^[10] In the case of substances where both delocalized and localized electrons contribute to the magnetic properties (e.g. rare-earth systems), the Ruderman–Kittel–Kasuya–Yosida (RKKY) model is the currently accepted mechanism.