Editing Slater determinant (section)

=== 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.