Suppose I have a real $n\times n$ matrix $\mathbf{C}$ that is Hermitian, positive-definite, and circulant. We know that its eigenvalues $\{\lambda_0,\ldots,\lambda_{n-1}\}$ are extraordinarily nice in that they are positive reals and are the output of the discrete Fourier transform of the top row of $\mathbf{C}$.

Consider the sum $\mathbf{C}+\mathbf{D}$ where $\mathbf{D}=\operatorname{diag}(d_0,\ldots,d_{n-1})$ such that $d_i>0$ for all $i$.

Is there a characterization of eigenvalues of $\mathbf{C}+\mathbf{D}$ in terms of $\{\lambda_0,\ldots,\lambda_{n-1}\}$ and $\{d_0,\ldots,d_{n-1}\}$?

From reading the previous MO questions on the similar topic here and here I understand that the chances of finding a nice characterization are pretty slim. However, I hold tepid hope due to the special structure of my particular problem.

The reason for this inquiry is that eventually I would like to minimize the trace of the inverse $\operatorname{Tr}[(\mathbf{C}+\mathbf{D})^{-1}]$ subject to various constraints on $\{d_0,\ldots,d_{n-1}\}$...