In chemistry and physics, the exchange interaction is a quantum mechanical constraint on the states of indistinguishable particles.
The exchange symmetry alters the expectation value of the distance between two indistinguishable particles when their wave functions overlap.
For fermions the expectation value of the distance increases, and for bosons it decreases (compared to distinguishable particles).
Exchange interaction is the main physical effect responsible for ferromagnetism, and has no classical analogue.
Exchange interaction effects were discovered independently by physicists Werner Heisenberg and Paul Dirac in 1926.
Wolfgang Pauli demonstrated that this is a type of symmetry: states of two particles must be either symmetric or antisymmetric when coordinate labels are exchanged.
Rather it is a significant geometrical constraint, increasing the curvature of wavefunctions to prevent the overlap of the states occupied by indistinguishable 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.
One uses an antisymmetric combination of the product wave functions in position space: The other uses a symmetric combination of the product wave functions in position space: To treat the problem of the Hydrogen molecule perturbatively, the overall Hamiltonian is decomposed into a unperturbed Hamiltonian of the non-interacting hydrogen atoms
is the two-site two-electron Coulomb integral (It may be interpreted as the repulsive potential for electron-one at a particular point
in an electric field created by electron-two distributed over the space with the probability density
It has no simple physical interpretation, but it can be shown to arise entirely due to the anti-symmetry requirement.
(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 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.
J. H. Van Vleck presented the following analysis:[11] The potential energy of the interaction between the two electrons in orthogonal orbitals can be represented by a matrix, say
.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.
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).
is negative, the interaction favors electrons with antiparallel spins, potentially causing antiferromagnetism.
This sign can be deduced from the expression for the difference between the energies of the triplet and singlet states,
: 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.
Exchange energy splittings are very elusive to calculate for molecular systems at large internuclear distances.
The energy of the Ising model is defined to be: 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
[12] In these cases, a delocalized, or Hund–Mulliken–Bloch (molecular orbital/band) description, for the electron wave functions is more realistic.
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.
For example, a net magnetic moment of 0.54 μB 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.
) and would thus be expected to have in the localized electron model a total spin magnetic moment of
[14] 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.