## Appendix B Schrodinger Equation and Wavefunction Components  
Let us now examine the implications of the (complex) SE [Equation (A12)] when expressed in terms of the modulus and phase components of wavefunction [see Equation (A1)]:  
\[\mathrm{i}\hbar\;(\partial\psi/\partial t) =\mathrm{i}\hbar\;[(\partial R/\partial t)+\mathrm{i}R(\partial \phi/\partial t)]\exp(\mathrm{i}\psi)\] \[=\mathrm{H}\psi=\{-[\hbar^{2}/(2m)]\;[\Delta R+2\mathrm{i}\
abla R \cdot\
abla\phi-R(\
abla\phi)^{2}]+vR\}\;\exp(\mathrm{i}\phi),\]  
where we use Equation (A19). Multiplying both sides of this equation by \(\exp(-\mathrm{i}\phi)\) and dividing by \(\hbar R\) finally gives:  
\[\mathrm{i}\;(\partial\mathrm{ln}R/\partial t)-(\partial\phi/\partial t)=-[ \hbar/(2m)]\;[R^{-1}\Delta R+2\mathrm{i}(\
abla\mathrm{ln}R)\cdot\
abla\phi-( \
abla\phi)^{2}]+v/\hbar.\] (A27)  
Comparing the _imaginary_ parts of the preceding equation generates the dynamic equation for the time evolution of the modulus part of electronic state,  
\[\partial\mathrm{ln}R/\partial t=-\;[(\hbar/m)\;\
abla\phi]\cdot\
abla\mathrm{ ln}R=-\;\mathbf{V}\cdot\
abla\mathrm{ln}R,\] (A28)  
which can be directly transformed into the probability continuity equation (A17):  
\[\partial p/\partial t=-\
abla\cdot j\qquad\text{or}\qquad\sigma_{p}=dp/dt=0.\] (A29)  
Equating the _real_ parts of Equation (A27) similarly determines the phase-dynamics of Equation (A16):  
\[\partial\phi/\partial t=[\hbar/(2m)]\;[R^{-1}\Delta R-(\
abla\phi)^{2}]-v/ \hbar.\] (A30)  
The latter equation also determines the phase-production \(\sigma_{\phi}=d\phi/dt\) [Equation (A23)] for its flux definition of Equation (A21), in the phase continuity relation:  
\[\partial\phi/\partial t=-\
abla\cdot J+\sigma_{\phi}.\] (A31)  
The average electronic energy in state (A1) thus combines the following component contributions:  
\[ \begin{array}{l}\langle E\rangle_{\psi}=\langle\psi|\mathrm{H}|\psi\rangle= -[\hbar^{2}/(2m)]\;\int[R\Delta R-R^{2}(\
abla\phi)^{2}]\;d\mathbf{r}+\int\!\!R^ {2}\;v\;d\mathbf{r}\\ =[\hbar^{2}/(2m)]\;\int[(\
abla R)^{2}+R^{2}(\
abla\phi)^{2}]d\mathbf{r}+\int\!\!R ^{2}\;v\;d\mathbf{r}\equiv\langle T\rangle_{\psi}+\langle V_{ne}\rangle_{\psi}, \end{array} \] (A32)  
where we use the relevant integration by parts and \(\langle V_{ne}\rangle_{\psi}\) denotes the state average electron-nuclei attraction energy. The phase-dependent part of the average kinetic energy \(\langle T\rangle_{\psi}\) identically vanishes in the stationary electronic state  
\[\psi_{s}(\mathbf{r},t)=R_{s}(\mathbf{r})\;\exp[\mathrm{i}\phi_{s}(t)],\] (A33)  
for the sharply specified electronic energy  
\[E_{s}=R_{s}(\mathbf{r})^{-1}\;\mathrm{H}(\mathbf{r})\;R_{s}(\mathbf{r})=-\;[\hbar^{2}/(2m) ]\;R_{s}(\mathbf{r})^{-1}\Delta R_{s}(\mathbf{r})+v(\mathbf{r})=const.,\] (A34)  
when \(\phi_{s}(t)=-\;(E_{s}/\hbar)\;t\equiv-\;\omega_{s}t\) and hence \(\
abla\phi_{s}(t)=0\). The open systems and their fragments are described by grand ensembles and exhibit continuously varying, fractional average numbers of electrons; the closed molecules and their parts similarly exhibit integer numbers of electrons. The pure state \(\Psi_{\rm R}\,^{*}(N,t_{0})\) of the whole externally-closed but internally-open reactive complex \({\rm R}\,^{*}=({\rm A}\,^{*}\,|\,{\rm B}\,^{*})\) at time \(t_{0}\equiv 0\), defined by its overall external potential \(v=v_{\rm A}+v_{\rm B}\) and number of electrons \(N=N_{\rm A}+N_{\rm B}\) (integer), can be expanded  
\[\Psi_{\rm R}\,^{*}(N,\,t_{0})=\Sigma_{s}\ C_{s}(t_{0})\ R_{s}(N),\qquad\qquad C _{s}(t_{0})=\langle R_{s}(N)|\Psi_{\rm R}\,^{*}(N,\,t_{0})\rangle, \tag{101}\]  
in terms of the complete set of the system stationary states, the eigensolutions  
\[{\rm H}(N)\ R_{s}(N)=E_{s}\ R_{s}(N) \tag{102}\]  
of \(N\)-electron molecular Hamiltonian:  
\[ \begin{array}{c}{\rm H}(N)={\rm T}(N)+V_{ne}(N)+U_{ae}(N),\\ {\rm T}(N)=\Sigma_{k}\,{\rm T}(k),\qquad\quad{\rm V}_{ne}(N)=\Sigma_{k}\ \
u(k),\qquad\quad U_{ae}(N)=\Sigma_{k<l}\ g(k,\,l).\end{array}