Editing Slater determinant (section)

== Definition ==

=== Two-particle case ===

The simplest way to approximate the wave function of a many-particle system is to take the product of properly chosen [[Orthogonality (quantum mechanics)|orthogonal]] wave functions of the individual particles. For the two-particle case with coordinates <math>\mathbf{x}_1</math> and <math>\mathbf{x}_2</math>, we have

:<math>
\Psi(\mathbf{x}_1, \mathbf{x}_2) = \chi_1(\mathbf{x}_1) \chi_2(\mathbf{x}_2).
</math>

This expression is used in the [[Hartree–Fock method#Brief history|Hartree method]] as an [[ansatz]] for the many-particle wave function and is known as a [[Hartree product]]. However, it is not satisfactory for [[fermions]] because the wave function above is not antisymmetric under exchange of any two of the fermions, as it must be according to the [[Pauli exclusion principle]]. An antisymmetric wave function can be mathematically described as follows:

:<math>
\Psi(\mathbf{x}_1, \mathbf{x}_2) = -\Psi(\mathbf{x}_2, \mathbf{x}_1).
</math>

This does not hold for the Hartree product, which therefore does not satisfy the Pauli principle. This problem can be overcome by taking a [[linear combination]] of both Hartree products:

:<math>
\begin{aligned}
 \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)\} \\
  &= \frac{1}{\sqrt2}\begin{vmatrix} \chi_1(\mathbf{x}_1) & \chi_2(\mathbf{x}_1) \\ \chi_1(\mathbf{x}_2) & \chi_2(\mathbf{x}_2) \end{vmatrix},
\end{aligned}
</math>

where the coefficient is the [[normalization factor]]. This wave function is now antisymmetric and no longer distinguishes between fermions (that is, one cannot indicate an ordinal number to a specific particle, and the indices given are interchangeable). Moreover, it also goes to zero if any two spin orbitals of two fermions are the same. This is equivalent to satisfying the Pauli exclusion principle.

=== Multi-particle case ===

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<ref name="Atkins" /><ref name="Szabo">{{cite book
  | last1 = Szabo
  | first1 = A.
  | last2= Ostlund| first2= N. S.
  | title = Modern Quantum Chemistry
  | publisher = Dover Publishing | year = 1996 | location = Mineola, New York
  | isbn = 0-486-69186-1}}</ref>

:<math>
\begin{aligned}
 \Psi(\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N) &=
  \frac{1}{\sqrt{N!}}
  \begin{vmatrix} \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 & \ddots & \vdots \\
                      \chi_1(\mathbf{x}_N) & \chi_2(\mathbf{x}_N) & \cdots & \chi_N(\mathbf{x}_N)
  \end{vmatrix} \\
 &\equiv | \chi _1, \chi _2, \cdots, \chi _N \rangle \\
 &\equiv | 1, 2, \dots, N \rangle,
\end{aligned}
</math>

where the last two expressions use a shorthand for Slater determinants: The normalization constant is implied by noting the number N, and only the one-particle wavefunctions (first shorthand) or the indices for the fermion coordinates (second shorthand) are written down. All skipped labels are implied to behave in ascending sequence. The linear combination of Hartree products for the two-particle case is identical with the Slater determinant for ''N'' = 2. The use of Slater determinants ensures an antisymmetrized function at the outset. In the same way, the use of Slater determinants ensures conformity to the [[Pauli principle]]. Indeed, the Slater determinant vanishes if the set  <math>\{\chi_i\}</math> is [[linearly dependent]]. In particular, this is the case when two (or more) spin orbitals are the same. In chemistry one expresses this fact by stating that no two electrons with the same spin can occupy the same spatial orbital.