Slater determinant

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A Slater determinant (named after the American physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle.

The Slater determinant arises from the consideration of a wavefunction for a collection of electrons. The wavefunction for each individual electron is known as a spin-orbital, [\chi(\mathbf)], where [\mathbf] indicates the position and spin of the electron.

Two-particle case

The simplest way to approximate the wavefunction of a many-particle system is to take the product of properly chosen one-electron wavefunctions of the individual particles. For the two-particle case, we have

[\Psi(\mathbf_1,\mathbf_2) = \chi_1(\mathbf_1)\chi_2(\mathbf_2)]

This expression occurs in Hartree theory and is known as a Hartree product. However, it is not satisfactory for fermions, such as electrons, because the wavefunction is not antisymmetric. An antisymmetric wavefunction can be mathematically described as follows:

[\Psi(\mathbf_1,\mathbf_2) = -\Psi(\mathbf_2,\mathbf_1)]

Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products

[\Psi(\mathbf_1,\mathbf_2) = \frac}\_1)\chi_2(\mathbf_2) - \chi_1(\mathbf_2)\chi_2(\mathbf_1)\}]

where the coefficient is a normalization factor. This wavefunction is antisymmetric and no longer distinguishes between electrons. Moreover, it also goes to zero if any two wavefunctions or two electrons are the same. This is equivalent to satisfying the Pauli exclusion principle.

Generalization to the Slater determinant

The expression can be generalised to any number of fermions by writing it as a determinant. For an N-electron system, the Slater determinant is defined as

[\Psi(\mathbf_1, \mathbf_2, \ldots, \mathbf_N) =\frac}\left| \begin \chi_1(\mathbf_1) & \chi_1(\mathbf_2) & \cdots & \chi_1(\mathbf_N) \\ \chi_2(\mathbf_1) & \chi_2(\mathbf_2) & \cdots & \chi_2(\mathbf_N) \\ \vdots & \vdots && \vdots \\ \chi_N(\mathbf_1) & \chi_N(\mathbf_2) & \cdots & \chi_N(\mathbf_N) \end \right|]

The linear combination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for N=2. It can be seen that the use of (Slater) determinants assures an antisymmetrized function on the outset, symmetric functions are automatically rejected. In the same way, the use of Slater determinants assures the obeying of the Pauli principle; the determinant will vanish if any of the two spin-orbitals are identical, for this leads to two identical columns.

A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree-Fock theory. In more accurate theories (such as configuration interaction and MCSCF), a linear combination of Slater determinants is needed.

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