Let $(\mathcal A,\psi)$ be a noncommutative probability space, where $\psi:\mathcal A\to\mathbb C$ is the expectation functional, and let $\kappa_m:\mathcal A^m\to\mathbb C$, for $m\geq 1$, be the free cumulants associated to $\psi$. Let $s_1,\dots,s_d\in\mathcal A$, and let $C=(c_{ij})_{1\leq i,j\leq d}$ be a complex $d\times d$ matrix. Assume that $(s_1,\dots,s_d)$ is a semicircular system with covariance matrix $C$ in the free-cumulant sense: for every $r\geq 1$ and every $j_1,\dots,j_r\in\{1,\dots,d\}$, one has $\kappa_2(s_{j_1},s_{j_2})=c_{j_1j_2}$ when $r=2$, and $\kappa_r(s_{j_1},\dots,s_{j_r})=0$ when $r\neq 2$. Then for every integer $m\geq 1$ and every $i_1,\dots,i_m\in\{1,\dots,d\}$,

\begin{align*} \psi(s_{i_1}\cdots s_{i_m})=\sum_{\pi\in NC_2(m)}\prod_{\{p,q\}\in\pi}c_{i_pi_q}. \end{align*}

Here $NC_2(m)$ denotes the set of noncrossing pair partitions of $\{1,\dots,m\}$, and for a pair $\{p,q\}$ the indices are read in increasing order. If $m$ is odd, then $NC_2(m)=\varnothing$ and the moment is $0$.