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The Slater determinant arises from the consideration of a wave function for a collection of electrons, each with a wave function known as the spin-orbital (), where denotes the position and spin of a single electron. A Slater determinant containing two electrons with the same spin orbital would correspond to a wave function that is zero everywhere.
A solution to the lack of anti-symmetry in the Hartree method came when it was shown that a Slater determinant, a determinant of one-particle orbitals first used by Heisenberg and Dirac in 1926, trivially satisfies the antisymmetric property of the exact solution and hence is a suitable ansatz for applying the variational principle.
An example provided in Slater's original paper is for the iron atom which has nuclear charge 26 and electronic configuration 1s 2 2s 2 2p 6 3s 2 3p 6 3d 6 4s 2.The screening constant, and subsequently the shielded (or effective) nuclear charge for each electron is deduced as: [1]
Slater determinants are constructed from sets of orthonormal spin orbitals, so that | =, making the identity matrix and simplifying the above matrix equation. The solution of the CI procedure are some eigenvalues E j {\displaystyle \mathbf {E} ^{j}} and their corresponding eigenvectors c I j {\displaystyle \mathbf {c} _{I}^{j}} .
Some quantum chemistry software uses sets of Slater-type functions (STF) analogous to Slater type orbitals, but with variable exponents chosen to minimize the total molecular energy (rather than by Slater's rules as above). The fact that products of two STOs on distinct atoms are more difficult to express than those of Gaussian functions (which ...
Exact wave functions, however, cannot generally be expressed as single determinants. The single-determinant approximation does not take into account Coulomb correlation, leading to a total electronic energy different from the exact solution of the non-relativistic Schrödinger equation within the Born–Oppenheimer approximation.
When analyzing a system with many electrons, such as an atom or molecule, an imperfect but useful first approximation is to treat the electrons as uncorrelated or each having an independent single-particle wavefunction. This is the usual starting point when building the Slater determinant in the Hartree–Fock method.
In order to solve the equation of an electron in a spherical potential, Hartree first introduced atomic units to eliminate physical constants. Then he converted the Laplacian from Cartesian to spherical coordinates to show that the solution was a product of a radial function () / and a spherical harmonic with an angular quantum number , namely = (/) (,).