Let $(S, \mathfrak n)$ be a regular local ring of dimension $d\ge 4$ and let $R=S/(f)$ , where $0\ne f \in \mathfrak n^2$. Then $\dim R=d-1\ge 3$. If $\mathfrak m$ is the maximal ideal of $R$ then $\mu (\mathfrak m)=\mu (\mathfrak n)=d=\dim R+1$. Let $\operatorname {ord}(f):=\max \{t : f\in \mathfrak n^t\}$. Then the Hilbert-Samuel multiplicity (https://en.m.wikipedia.org/wiki/Hilbert%E2%80%93Samuel_function) of $R$ can be calculated as $e(R)=\operatorname {ord}(f)$.
Now my question is: Is there any Algebraic characterization (possibly in terms of $e(R)$) of when $R$ has at worst Rational singularity (https://en.m.wikipedia.org/wiki/Rational_singularity) ?
(If needed, I'm willing to assume $R$ is complete and normal and contains an algebraically closed field)
(If anything is known if we replace the regular local ring $S$ by a polynomial ring over a field , I'd appreciate even that)
Thanks