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Slater DeterminantA Slater determinant (named after the 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 spinorbital, , where indicates the position and spin of the electron. Two-particle case The simplest way of expressing the wavefunction of a many-electron system is to take the product of the spinorbitals of the individual electrons. For the two-electron case, we have -
\Psi(\mathbf{x}_1,\mathbf{x}_2) = \chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) This expression occurs in Hartree theory and is known as a Hartree product. However, it is not satisfactory because the wavefunction is not antisymmetric, that is -
\Psi(\mathbf{x}_1,\mathbf{x}_2) \ne -\Psi(\mathbf{x}_2,\mathbf{x}_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{x}_1,\mathbf{x}_2) = \frac{1}{\sqrt{2}}\{\chi_1(\mathbf{x}_1)\chi_2(\mathbf{x}_2) - \chi_1(\mathbf{x}_2)\chi_2(\mathbf{x}_1)\} where the coefficient is a normalisation 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{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) = \frac{1}{\sqrt{N!}} \left| \begin{matrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) & \cdots & \chi_N(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) & \cdots & \chi_N(\mathbf{x}_2) \\ \vdots & \vdots && \vdots \\ \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N) \end{matrix} \right| The linear comnbination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for N=2. 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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