  Richardson varieties are obtained as intersections of Schubert and opposite
Schubert varieties. We provide a new family of toric degenerations of
Richardson varieties inside Grassmannians by studying Gr\"obner degenerations
of their corresponding ideals. These degenerations are parametrised by block
diagonal matching fields in the sense of Sturmfels-Zelevinsky. We associate a
weight vector to each block diagonal matching field and study its corresponding
initial ideal. In particular, we characterise when such ideals are toric, hence
providing a family of toric degenerations for Richardson varieties.
  Given a Richardson variety $X_{w}^v$ and a weight vector ${\bf w}_\ell$
arising from a matching field, we consider two ideals: an ideal
$G_{k,n,\ell}|_w^v$ obtained by restricting the initial of the Pl\"ucker ideal
to a smaller polynomial ring, and a toric ideal defined as the kernel of a
monomial map $\phi_\ell|_w^v$. We first characterise the monomial-free ideals
of form $G_{k,n,\ell}|_w^v$. Then we construct a family of tableaux in
bijection with semi-standard Young tableaux which leads to a monomial basis for
the corresponding quotient ring. Finally, we prove that when
$G_{k,n,\ell}|_w^v$ is monomial-free and the initial ideal in$_{{\bf
w}_\ell}(I(X_w^v))$ is quadratically generated, then all three ideals in$_{{\bf
w}_\ell}(I(X_w^v))$, $G_{k,n,\ell}|_w^v$ and ker$(\phi_\ell|_w^v)$ coincide,
and provide a toric degeneration of $X_w^v$.
