I would like to solve the following integral equation for a Hermitian matrix:
$$\hat{G}_j = \int_0^1d\alpha \exp[-i\alpha\hat{H}] \partial_j \hat{H} \exp[i\alpha\hat{H}],$$
One possibility is to use the Baker-Campbell-Hausdorff formula. So using
$$\exp[A]B\exp[-A] = B + [A, B] + \frac{1}{2!}[A, [A, B]] + \ldots,$$
then by re-writing the integrand of $\hat{G}_j$ and integrating w.r.t. $\alpha$ we obtain
$$\hat{G}_j = \sum_{n=0}^\infty\frac{(-i)^n}{(n+1)!}C^n_{\hat{H}}(\hat{H}_j),$$
where $C^n_{\hat{H}}(\hat{H}_j)$ is the nth-order nested commutator: $C^0_{\hat{H}}(\hat{H}_j) = \hat{H}_j$, $C^1_{\hat{H}}(\hat{H}_j) = [\hat{H}, \hat{H}_j]$, $C^2_{\hat{H}}(\hat{H}_j) = [\hat{H}, [\hat{H}, \hat{H}_j]]$ and so on. From this infinite series, how can I proceed to find an explicit form of the matrix $G$?