Solution
= Solution
Let $S_{12}=\langle\phi_1|\phi_2\rangle$ be real, with each atomic orbital normalized. Then
$$
\boxed{\langle\psi|\psi\rangle=c_1^2+c_2^2+2c_1c_2S_{12}=1.}
$$
Together with orthonormal spin functions, this also normalizes the <Slater determinant>: its norm squared is $\langle\psi|\psi\rangle^2$. Under the subsequent zero-overlap assumption,
$$
\boxed{c_1=\sqrt{1-c_2^2},\qquad -1\le c_2\le1,}
$$
using the specified nonnegative branch for $c_1$.