## Appendix B Schrodinger Equation and Wavefunction Components
 \tag{103}\]  
Here, \(R_{s}(N)\) denotes the time-independent amplitude (modulus) function of \(N\) electrons, while \({\rm T}(N)\), \({\rm V}_{ne}(N)\) and \({\rm U}_{ae}(N)\) stand for the quantum operators of electronic kinetic, attraction and repulsion energies, respectively. Given the state \(\Psi(N,\,t_{0})\) at the initial time \(t_{0}=0\), its form at time \(\tau\) [see Equations (100) and (101)] is determined by the interval evolution operator  
\[ \begin{array}{c}{\rm U}(\tau,N)=\exp[-{\rm i}\hbar^{-1}\tau\ {\rm H}(N)], \end{array}  \tag{104}\]  
\[ \begin{array}{c}\Psi(N,\,\tau)={\rm U}(\tau)\ \Psi(N,\,t_{0})=\Sigma_{s}\ C_{ s}(t_{0})\ \{\exp[{\rm i}\phi_{s}(\tau)]\ R_{s}(N)\}\\ \equiv\Sigma_{s}\ C_{s}(t_{0})\ \Psi_{s}(N,\,\tau),\end{array}  \tag{105}\]  
where \(\Psi_{s}(N,\,\tau)=R_{s}(N)\ \exp[{\rm i}\phi_{s}(\tau)]\) is the full stationary state of \(N\)-electrons corresponding to energy \(E_{s}\). Such pure-state expansion is not available for the open systems in their final (mixed) equilibrium states, described by the fragment density operators defining the associated ensembles of Hamiltonians for different numbers of electrons. At the polarization stage \({\rm R}\,^{+}=({\rm A}\,^{+}\,|\,{\rm B}\,^{+})\), however, involving the mutually- and externally-closed substrates, each subsystem \({\rm X}\,^{+}\) conserves the initial (integer) number of electrons \(N_{\rm X}\,^{0}\) so that the interacting-fragment Hamiltonians \(\{{\rm H}_{\rm X}(N_{\rm X}\,^{0})\}\) for the overall external potential \(v\) of the whole system are well defined, e.g.,  
\[ \begin{array}{c}{\rm H}_{\rm A}(N_{\rm A}\,^{0})={\rm T}(N_{\rm A}\,^{0})+ V_{ne}(N_{\rm A}\,^{0})+U_{\rm A}(N_{\rm A}\,^{0})+U_{\rm AB}(N_{\rm A}\,^{0},N_{ \rm B}\,^{0}),\\ {\rm U}_{\rm A}(N_{\rm A}\,^{0})=\Sigma_{(k<l)\,\in\,{\rm A}}\ g(k,\,l),\qquad \quad{\rm U}_{\rm AB}(N_{\rm A}\,^{0},N_{\rm B}\,^{0})=\Sigma_{k\,\in\,{\rm A}} \Sigma_{l\,\in\,{\rm B}}\ g(k,\,l),\qquad\mbox{etc.}\end{array}  \tag{106}\]  
They determine the stationary eigenproblems for subsystems containing the initial, integer numbers of electrons in separate reactants,  
\[{\rm H}_{\rm X}(N_{\rm X}\,^{0})\ \varphi_{u}(N_{\rm X}\,^{0})=E_{u}({\rm X}\,^{+})\ \varphi_{u}(N_{\rm X}\,^{0}),\qquad\quad{\rm X}={\rm A},\,{\rm B}, \tag{107}\]  
and associated expansions at time \(t_{0}\) of general states in the polarized subsystems:  
\[\Psi_{\rm X}\,^{+}(N_{\rm X}\,^{0},\,t_{0})=\Sigma_{u}\ C_{u}({\rm X}\,^{+}, \,t_{0})\ \varphi_{u}(N_{\rm X}\,^{0}),\qquad\qquad C_{u}({\rm X}\,^{+},\,t_{0})= \langle\varphi_{u}(N_{\rm X}\,^{0})|\Psi_{\rm X}\,^{+}(N_{\rm X}\,^{0},\,t_{0})\rangle. \tag{108}\]  
Their evolution in time interval \(\tau\) is determined by the reactant operators  
\[{\rm U}_{\rm X}(\tau,N_{\rm X}\,^{0})=\exp[-{\rm i}\hbar^{-1}\tau\ {\rm H}_{\rm X}(N_{\rm X}\,^{0})], \qquad\quad{\rm X}={\rm A},\,{\rm B}, \tag{109}\]  
\[ \begin{array}{c}\Psi_{\rm X}\,^{+}(N_{\rm X}\,^{0},\,\tau)={\rm U}_{\rm X}( \tau,N_{\rm X}\,^{0})\ \Psi_{\rm X}\,^{+}(N_{\rm X}\,^{0},\,t_{0})\\ =\Sigma_{u}\ C_{u}({\rm X}\,^{+},\,t_{0})\ \{\exp[{\rm i}\phi_{u}({\rm X}\,^{+},\, \tau)]\ \varphi_{u}(N_{\rm X}\,^{0})]\\ \equiv\Sigma_{u}\ C_{u}({\rm X}\,^{+},\,t_{0})\ \varphi_{u}(N_{\rm X}\,^{0},\,\tau), \end{array}  \tag{110}\]where  
\[\phi_{u}(X^{+},\tau)=-\left[E_{u}(X^{+})/\hbar\right]\tau\equiv-\omega_{u}(X^{+})\tau \tag{100}\]  
denotes the stationary phase of subsystem \(X^{+}\), while  
\[\varPhi_{u}(N_{X}{}^{0},\tau)=\varphi_{u}(N_{X}{}^{0})\exp[i\phi_{u}(X^{+},\tau)] \tag{101}\]  
stands for the full stationary state of reactant \(X\) in presence of the other substrate, corresponding to the fragment energy \(E_{u}(X^{+})\).