  We study separations between two fundamental models (or \emph{Ans\"atze}) of
antisymmetric functions, that is, functions $f$ of the form $f(x_{\sigma(1)},
\ldots, x_{\sigma(N)}) = \text{sign}(\sigma)f(x_1, \ldots, x_N)$, where
$\sigma$ is any permutation. These arise in the context of quantum chemistry,
and are the basic modeling tool for wavefunctions of Fermionic systems.
Specifically, we consider two popular antisymmetric Ans\"atze: the Slater
representation, which leverages the alternating structure of determinants, and
the Jastrow ansatz, which augments Slater determinants with a product by an
arbitrary symmetric function. We construct an antisymmetric function in $N$
dimensions that can be efficiently expressed in Jastrow form, yet provably
cannot be approximated by Slater determinants unless there are exponentially
(in $N^2$) many terms. This represents the first explicit quantitative
separation between these two Ans\"atze.
