I am investigating quantum mechanical $n$-electron mechanical (kinematic) momentum densities in molecular systems by numerical methods. For a $n$ electron state function $\Psi$, non-relativistic case but in a static magnetic field (${\bf B}=\nabla\times \bf A$) the mechanic momentum density is $$ \pi = \int d\omega_1 \dots d \omega_n d r_2 \dots d r_n \Psi^* \vec\pi \Psi $$ with spin coordinates $\omega_i$ and space coordinates $r_i$, via the n-particle mechanical (kinematic) momentum operator (using atomic units) $${\vec \pi} = {\bf p} + {\bf A}$$ with $\bf p$ and $\bf A$ as summs over all corresponding one-electron operators.
I am interested in the existence of the zeros of $\pi$ in isolated points in space. This is of relevance since the gradient of the electron density ($\nabla \rho$)is proportional to $\Im(\pi)$ while the probability current ($\bf j$) is proportional to $\Re(\pi)$. So a total zero $r_0$ of $\pi$ with $$ \pi(r_0) = 0 $$ corresponds to a common zero of $\nabla \rho$ and the $\bf j$.
Our numerical experiments now show that such $r_0$ exist for symmetry determined positions, namely the centers of symmetry for molecules with "point group symmetry" ie. groups that fix single points in space, there such points are the center of symmetry. However, for non-symmetric cases we observe throughout deviations (in some cases very small and in some cases quite large) between the positions of any zeros of $\nabla \rho$ and of zeros of $\bf j$.
For that I would be interested if there are any general properties of that (complex) mechanic momentum density $\pi$ known, that could help me in understanding the behavior of the real and imaginary zeros of this function. For example if there are some estimates known such that one could prove that e.g. $|\pi| > 0$ for unsymmetric cases ...
In particular I have the feeling that these zeros have a tendency to "repel" each other and only coincide when forced by symmetry.
