= Solution
\b[A doubly occupied spatial orbital gives a spin-singlet Slater determinant:]
$$
\boxed{\Psi(\mathbf x_1,\mathbf x_2)=\frac1{\sqrt2}
\begin{vmatrix}
\psi(\mathbf r_1)\alpha(s_1)&\psi(\mathbf r_1)\beta(s_1)\\
\psi(\mathbf r_2)\alpha(s_2)&\psi(\mathbf r_2)\beta(s_2)
\end{vmatrix}.}
$$
Equivalently, $\Psi=\psi(\mathbf r_1)\psi(\mathbf r_2)[\alpha(s_1)\beta(s_2)-\beta(s_1)\alpha(s_2)]/\sqrt2$. The spatial factor is symmetric, and the normalized antisymmetric spin factor enforces the <Pauli exclusion principle>. Here $\mathbf x$ includes position and spin.
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