Quantum mechanical particles are classified as bosons or fermions. The
spin–statistics theorem of
quantum field theory demands that all particles with
half-integer spin behave as fermions and all particles with
integer spin behave as bosons. Multiple bosons may occupy the same
quantum state; however, by the
Pauli exclusion principle, no two fermions can occupy the same state. Since
electrons have spin 1/2, they are fermions. This means that the overall wave function of a system must be antisymmetric when two electrons are exchanged, i.e. interchanged with respect to both spatial and spin coordinates. First, however, exchange will be explained with the neglect of spin.
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 (that is, as if the Pauli exclusion principle did not apply), and taking wave functions in position space of \Phi_a(r_1) for the first electron and \Phi_b(r_2) for the second electron. The functions \Phi_a and \Phi_b are orthogonal, and each corresponds to an energy eigenstate. To enforce the indistinguishability of the two electrons, two wave functions for the overall system in position space can be constructed. One uses an antisymmetric combination of the product wave functions in position space: {{NumBlk|:|\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)]|}} The other uses a symmetric combination of the product wave functions in position space: {{NumBlk|:|\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)]|}} To treat the problem of the hydrogen molecule
perturbatively, the overall
Hamiltonian is decomposed into an unperturbed Hamiltonian of the non-interacting hydrogen atoms \mathcal{H}^{(0)} and a perturbing Hamiltonian, which accounts for interactions between the two atoms \mathcal{H}^{(1)}. The full Hamiltonian is then: \mathcal{H} = \mathcal{H}^{(0)} + \mathcal{H}^{(1)} where \mathcal{H}^{(0)} = -\frac{\hbar^2}{2m}\nabla^{2}_{1}-\frac{\hbar^2}{2m}\nabla^{2}_{2}-\frac{e^2}{r_{a1}}-\frac{e^2}{r_{b2}} and \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 of \mathcal{H}^{(0)} denote the
kinetic energy of the electrons. The remaining terms account for attraction between the electrons and their host protons (r_{a1/b2}). The terms in \mathcal{H}^{(1)} account for the
potential energy corresponding to: proton–proton repulsion (R_{ab}), electron–electron repulsion (r_{12}), and electron–proton attraction between the electron of one host atom and the proton of the other (r_{a2/b1}). All quantities are assumed to be
real. Two eigenvalues for the system energy are found: {{NumBlk|:|\ E_{\pm} = E_{(0)} + \frac{C \pm J_{\rm ex}}{1 \pm \mathcal{S}^2}|}} where the E_+ is the spatially symmetric solution and E_- is the spatially antisymmetric solution, corresponding to \Psi_{\rm S} and \Psi_{\rm A} respectively. A variational calculation yields similar results. \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 \Phi_a(\vec r_1)^2 in an
electric field created by electron-two distributed over the space with the probability density \Phi_b(\vec r_2)^2), \mathcal{S} is the
overlap integral, and J_\mathrm{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: {{NumBlk|:| 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^3r_1\, d^3r_2|}} {{NumBlk|:| \mathcal{S} = \int \Phi_b(\vec r_2) \Phi_a(\vec r_2) \, d^3r_2|}} {{NumBlk|:| 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^3r_1\, d^3r_2|}} 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. The detailed calculation, including evaluation of the above integrals with
ground state hydrogen atom wave functions, and application of the variational principle to obtain the minimum energy in both cases of the hydrogen molecule for atomic orbitals and the hydrogen radical (i.e. with only one electron) for molecular orbitals can be found in the book of Müller-Kirsten pp. 272-292 (only in the second edition).
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: 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 wave function 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: {{quote|1= The potential energy of the interaction between the two electrons in orthogonal orbitals can be represented by a matrix, say E_\textrm{ex}. From Eq. (3), the characteristic values of this matrix are C \pm J_\textrm{ex}. The characteristic values of a matrix are its diagonal elements after it is converted to a diagonal matrix (that is, eigenvalues). Now, the characteristic values of the square of the magnitude of the resultant spin, \langle (\vec{s}_a + \vec{s}_b)^2 \rangle is S(S+1). The characteristic values of the matrices \langle \vec{s}_a^{\;2}\rangle and \langle \vec{s}_b^{\;2}\rangle are each \tfrac{1}{2}(\tfrac{1}{2} + 1) = \tfrac{3}{4} and {{nobr|\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 \langle\vec{s}_a \cdot \vec{s}_b\rangle are \tfrac{1}{2}(0 - \tfrac{6}{4})= -\tfrac{3}{4} and \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_\textrm{ex} is seen to have the characteristic value C + J_\textrm{ex} when \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_\textrm{ex} when \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, {{NumBlk|:|E_{\rm ex} - C + \frac{1}{2}J_{\rm ex} + 2J_{\rm ex} \langle\vec{s}_a \cdot \vec{s}_b\rangle = 0 |}} and, hence, {{NumBlk|:|E_{\rm ex} = C - \frac{1}{2}J_{\rm ex} - 2J_{\rm ex} \langle\vec{s}_a \cdot \vec{s}_b\rangle |}} where the spin momenta are given as \langle\vec{s}_a\rangle and \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: {{NumBlk|:|\ -2J_{ab} \langle\vec{s}_a \cdot \vec{s}_b\rangle |}} It follows that the exchange interaction Hamiltonian between two electrons in orbitals \Phi_a and \Phi_b can be written in terms of their spin momenta \vec{s}_a and \vec{s}_b . This interaction is named the
Heisenberg exchange Hamiltonian or the Heisenberg–Dirac Hamiltonian in the older literature: {{NumBlk|:|\mathcal{H}_{\rm Heis} = -2J_{ab} \langle\vec{s}_a \cdot \vec{s}_b\rangle |}} J_\textrm{ab} is not the same as the quantity labeled J_\textrm{ex} in Eq. (6). Rather, J_\textrm{ab}, which is termed the
exchange constant, is a function of Eqs. (4), (5), and (6), namely, {{NumBlk|:|\ J_{ab} = \frac{1}{2} (E_+ - E_-) = \frac{J_{\rm ex}- C\mathcal{S}^2}{1-\mathcal{S}^4}|}} However, with orthogonal orbitals (in which \mathcal{S} = 0), for example with different orbitals in the
same atom, J_\textrm{ab} = J_\textrm{ex}.
Effects of exchange If J_\textrm{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_\textrm{ab} is negative, the interaction favors electrons with antiparallel spins, potentially causing
antiferromagnetism. The sign of J_\textrm{ab} is essentially determined by the relative sizes of J_\textrm{ex} and the product of 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_+: {{NumBlk|:|\ E_{-} - E_{+} = \frac{2(C\mathcal{S}^2 - J_{\rm ex})}{1-\mathcal{S}^4} |}} 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. ==Direct exchange interactions in solids==