## 5 Current Coherence in Donor-Acceptor Systems
The elementary P and CT currents of the flow-patterns in Equation (48) give rise to the corresponding vector sums \(\{j(\lambda)\}\), the site _resultant_ currents schematically drawn in Figure 2. They are seen to generate a \"contoratory\" pattern in the complementary complex R\({}_{c}\) and a collective \"translational\" pattern in R\({}_{\rm HSAB}\). The magnitude \(|\langle j\rangle_{\lambda}|\) of this average (directional) site descriptor of the probability-flow ultimately determines the size \(|\Delta N_{\lambda}|\) of the net change in the fragment electron population in unit time,  
\[\Delta N_{\lambda}=N_{\lambda}\;\langle j\rangle_{\lambda}, \tag{50}\]  
where \(N_{\lambda}\) stands for the fragment average number of electrons in the separate reactant. The energy-favored (complementary) complex, which represents the least population-activation process, then corresponds to the lowest overall population displacement:  
\[\Sigma_{\lambda}\;|\Delta N_{\lambda}|=\Sigma_{\lambda}\;N_{\lambda}\;|\langle j \rangle_{\lambda}|\equiv\Sigma_{\lambda}\;N_{\lambda}\;\langle j\rangle_{ \lambda}\Rightarrow\;minimum. \tag{51}\]  
A reference to Equation (43) shows that the nonclassical gradient information of site \(\lambda\) also probes the average lengths of the directional phase-related properties:  
\[ \begin{array}{l}I_{\lambda}[\phi_{\lambda}]=4\int p_{\lambda}(\mathbf{r})\;[ \
abla\phi_{\lambda}(\mathbf{r})]^{2}d\mathbf{r}\\ =(2m/\hbar)^{2}\int p_{\lambda}(\mathbf{r})\;V_{\lambda}(\mathbf{r})^{2}\;d\mathbf{r}\\ =(2m/\hbar)^{2}\int p_{\lambda}(\mathbf{r})^{-1}\;j_{\lambda}(\mathbf{r})^{2}\;d\mathbf{r}.\end{array}  \tag{52}\]  
Therefore, the least site-activation of Equation (51), which occurs for the current-coherence in reactive system, also implies the minimum of the (additive) overall nonclassical information content:  
\[I[\mathbf{\phi}]=\Sigma_{\lambda}\;I_{\lambda}[\phi_{\lambda}]\Rightarrow minimum. \tag{53}\]  
Figure 2: Qualitative diagrams of the _contoratory_ pattern of the resultant _site_-currents \(\{j(\lambda)\}\) in the complementary complex R\({}_{c}\) (**a**) and their _translational_ pattern in the R\({}_{\rm HSAB}\) arrangement (**b**) of the acidic A = \((a_{\rm A}|b_{\rm A})\) and basic B = \((a_{\rm B}|b_{\rm B})\) reactants. Its absolute minimum is reached for the mutually-open (entangled) sites in the (bound) nondegenerate quantum state of the whole reactive system,  
\[R^{*}=(A^{*}\mid B^{*})=(a_{A}\,^{*}\mid b_{A}\,^{*}\mid a_{B}\,^{*}\mid b_{B}\,^{ *}),\]  
when the local phase contribution identically vanishes. Moreover, in the information _equilibrium_ state of the entangled molecular fragments, corresponding to the minimum of the system nonclassical gradient-information, this also implies the equalization of subsystem phases at the global phase of the wavefunction describing the reactive system as a whole (see also Appendix C). The normalized probability distributions on sites \(\{\kappa=(a,b)\}\) in reactants \(\{X=(A,\,B)\}\),  
\[\{p_{\lambda}(\mathbf{r})\}=\{p_{\kappa}(\mathbf{r};X)=\rho_{\kappa}(\mathbf{r};X)/N_{ \kappa}(X)\},\qquad\int p_{\kappa}(\mathbf{r};X)\;d\mathbf{r}=1, \tag{54}\]  
the shape factors of the fragment electronic densities \(\{\rho_{\kappa}(\mathbf{r};X)\}\), can be used as weights in determining the corresponding fragment (internal) averages of physical properties. For example, the site vector (directional) densities  
\[\{\mathbf{k}_{\kappa}(\mathbf{r};X)\},\qquad\{\mathbf{V}_{\kappa}(\mathbf{r};X)\}\qquad\text{ and}\qquad\{\mathbf{j}_{\kappa}(\mathbf{r};X)\} \tag{55}\]  
generate the associated average descriptors on each substrate:  
\[ \begin{array}{c}\langle k(X)\rangle=\Sigma_{\kappa}\;P_{\kappa}(X)\int p_{ \kappa}(\mathbf{r};X)\;k_{\kappa}(\mathbf{r};X)\;d\mathbf{r}\equiv\Sigma_{\kappa}\;P_{ \kappa}(X)\;\langle k(X)\rangle_{\kappa}\\ \qquad\equiv\Sigma_{\kappa}\;\int w_{\kappa}(\mathbf{r};X)\;k_{\kappa}(\mathbf{r};X)\; d\mathbf{r}\equiv\int k(\mathbf{r};X)\;d\mathbf{r},\end{array}  \tag{56}\]  
\[