I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:

$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ represents the set of positive define matrices of size $n$

This distribution is parameterized by a symmetric matrix $M$, and has the form of:

$\textrm{P}(X ; M) = A(M) \exp(\textrm{tr}(M X))$

where $A(M)$ is a normalizing constant that makes the integral over all $X$ of $\textrm{P}(X : M)$ equal to 1

Specifically, I'm looking for a closed form (if it exists) for $A(M)$, or at the very least, a reference to any papers or other documentation on such a distribution or something similar.

For those interested in why unit diagonals, my underlying distribution is a group of $n$ vectors of unit length, $V_i$, and $X_{ij} = V_i^T V_j$, thus $X_{ii} = 1$ since $|V_i| = 1$