  We study the optimal Slater-determinant approximation of an $N$-fermion wave
function analytically. That is, we seek the Slater-determinant (constructed out
of $N$ orthonormal single-particle orbitals) wave function having largest
overlap with a given $N$-fermion wave function. Some simple lemmas have been
established and their usefulness is demonstrated on some structured states,
such as the Greenberger-Horne-Zeilinger state. In the simplest nontrivial case
of three fermions in six orbitals, which the celebrated Borland-Dennis
discovery is about, the optimal Slater approximation wave function is proven to
be built out of the natural orbitals in an interesting way. We also show that
the Hadamard inequality is useful for finding the optimal Slater approximation
of some special target wave functions.
