## Appendix A Continuity Relations Revisited  
The dynamics of electronic states is determined by SE, which also implies specific rates of time evolutions of both components of complex wavefunctions, and of their physical descriptors: the state probability and current distributions. The time derivatives of the modulus and phase parts of electronic states reflect the relevant continuity equations in molecular QM. For simplicity, let us consider a single electron in state \(|\Psi(t)\rangle\) at time \(t\), or the associated (complex) wavefunction in position representation:  
\[\psi(\mathbf{r},t)=\langle\mathbf{r}|\Psi(t)\rangle=R(\mathbf{r},t)\exp[{\rm i}\phi(\mathbf{r},t)],\hskip 28.452756 pt\phi(\mathbf{r},t)\geq 0. \tag{10}\]  
Its modulus (\(R\)) and phase (\(\phi\)) components determine the state physical attributes of the electron probability and current densities:  
\[p(\mathbf{r},t)=\psi(\mathbf{r},t)^{*}\psi(\mathbf{r},t)=R(\mathbf{r},t)^{2}, \tag{11}\]\[j(r,t)=[\hbar/(2 mi)]\left[\psi(r,t)^{*}\,\
abla\psi(r,t)-\psi(r,t) \
abla\psi(r,t)^{*}\right]\] \[=(\hbar/m)\,p(r,t)\,\
abla\phi(r,t) \tag{10}\] \[\equiv p(r,t)\,\mathbf{V}(r,t).\]  
The effective velocity \(\mathbf{V}(r,t)\) of the probability \"fluid\" measures the current-per-particle and reflects the state _phase_-gradient:  
\[\mathbf{V}(r,t)=j(r,t)/p(r,t)=(\hbar/m)\,\
abla\phi(r,t). \tag{11}\]  
Equation (10) also identifies the two (additive) components of the wavefunction logarithm:  
\[\text{ln}\psi\;(r,t)=\text{ln}R(r,t)+\text{i}\phi(r,t  
\[\mathrm{U}(t-t_{0})\equiv\mathrm{U}(\tau)=\exp(-\mathrm{i}\hbar^{-1}\tau\ \mathrm{H}), \tag{113}\]  
\[\psi(t)=\mathrm{U}(\tau)\ \psi(t_{0}). \tag{114}\]  
This equation and its complex conjugate then imply the associated temporal evolutions of the wavefunction components \(R\) and \(\phi\) (see also Appendix B):  
\[\partial R/\partial t=-\
abla R\cdot\mathbf{V}, \tag{115}\]  
\[\partial\phi/\partial t=[\hbar/(2m)]\ [R^{-1}\Delta R-(\
abla\phi)^{2}]-v/\hbar. \tag{116}\]  
These dynamical equations can be ultimately recast as the corresponding continuity relations. Consider first the continuity of probability distribution:  
\[ \begin{array}{c}\partial p/\partial t=2R\ (\partial R/\partial t)=-\ (2R\ \
abla R)\cdot\mathbf{V}=-\ \
abla p\cdot\mathbf{V}=-\ \
abla\cdot\mathbf{j}\qquad\quad\text{or}\\ \sigma_{p}\equiv dp/dt=\partial p/\partial t+\
abla\cdot\mathbf{j}=\partial p/ \partial t+\
abla p\cdot\mathbf{V}=0.\end{array}  \tag{117}\]  
The divergence of probability flux of Equation (103)  
\[\
abla\cdot\mathbf{j}=\
abla p\cdot\mathbf{V}+p\ \
abla\cdot\mathbf{V}=\
abla p\cdot\mathbf{V}, \tag{118}\]  
further implies the vanishing divergence of the velocity field \(\mathbf{V}\) related to \(\
abla^{2}\phi=\Delta\phi\):  
\[\
abla\cdot\mathbf{V}=(\hbar/m)\ \Delta\phi=0\qquad\quad\text{or}\qquad\quad \Delta\phi=0. \tag{119}\]  
The total _time_-derivative \(dp(\mathbf{r})/dt\) determines the vanishing local probability \"source\": \(\sigma_{p}(\mathbf{r})=0\). It measures the time rate of change in an infinitesimal volume element of probability fluid _moving_ with velocity \(\mathbf{V}=dr/dt\), while the partial derivative \(\partial p[\mathbf{r}(t),\ t]/\partial t\) refers to volume element around the _fixed_ point in space. Indeed, separating the explicit time dependence of \(p(\mathbf{r},\ t)\) from its implicit dependence through the particle position \(\mathbf{r}(t)\), \(p(\mathbf{r},\ t)=p[\mathbf{r}(t),\ t]\), gives:  
\[ \begin{array}{c}\sigma_{p}(\mathbf{r},\ t)=\partial p[\mathbf{r}(t),\ t]/\partial t +(dr/dt)\cdot\partial p(\mathbf{r},\ t)/\partial\mathbf{r}\\ =\partial p(\mathbf{r},\ t)/\partial t+\mathbf{V}(\mathbf{r},\ t)\cdot\
abla p(\mathbf{r},\ t)= \partial p(\mathbf{r},\ t)/\partial t+\
abla\cdot\mathbf{j}(\mathbf{r},\ t)=0.\end{array}