## 7 Partial Electronic Flows
The corresponding chemical potentials for these types of behavior, the _biased_ measures for Groups (1) and (2) and the _unbiased_ estimates for Group (3), then read:  
\[u_{\alpha}(1)=-I_{\lambda},\qquad u_{\alpha}(2)=-A_{\lambda},\qquad\text{and} \qquad u_{\alpha}(3)=-(I_{\lambda}+A_{\lambda})/2. \tag{81}\]  
These chemical potentials are expected to display the following hierarchy reflecting the site softnesses (polarizabilities):  
\[u_{\alpha}(a_{\mathrm{A}})<u_{\alpha}(a_{\mathrm{B}})<u_{\alpha}(b_{\mathrm{A }})<u_{\alpha}(b_{\mathrm{B}})<0. \tag{82}\]  
The sum of the associated first-order changes in electronic energy, \(\{\Delta E_{\alpha}{}^{(1)}(\lambda)\}\), of the site energies \(\{E_{\alpha}(\lambda)\}\) following displacements \(\{\Delta n_{\lambda}\}\) in their electron populations, then generates the following overall energy displacement:  
\[\Delta E_{\alpha}{}^{(1)}=\Sigma_{\lambda}\ u_{\alpha}(\lambda)\ \Delta n_{\lambda}\equiv\Sigma_{\lambda}\ \Delta E_{\alpha}{}^{(1)}(\lambda). \tag{83}\]  
One next observes that the population displacements are large in Groups (1) and (2) of the list (80), where the P and CT displacement _enhance_ one another,  
\[N_{\alpha}{}^{\mathrm{P}}(\lambda)+N_{\alpha}{}^{\mathrm{CT}}(\lambda)\equiv \Delta_{\alpha}(\lambda), \tag{84}\]  
and small in Group (3), when they _cancel_ one another:  
\[N_{\alpha}{}^{\mathrm{P}}(\lambda)+N_{\alpha}{}^{\mathrm{CT}}(\lambda)\equiv \delta_{\alpha}(\lambda). \tag{85}\]  
Therefore, the energy change in \(\mathrm{R}_{\mathrm{HSAB}}\) is determined by two large population displacements \(\Delta_{\mathrm{HSAB}}(\lambda)\), due to charge activations of the key sites \(a_{\mathrm{A}}\) (strongly acidic) and \(b_{\mathrm{B}}\) (strongly basic), and two remaining (small) flow-through displacements \(\delta_{\mathrm{HSAB}}(\lambda)\) of the mixed-character fragments \(a_{\mathrm{B}}\) and \(b_{\mathrm{A}}\). In \(\mathrm{R}_{c}\), the collective charge displacement includes four flow-through \(\delta_{c}(\lambda)\) contributions reflecting the charge activation on all sites. This observation further justifies the complementary preference in such donor-acceptor coordinations [94; 95]. Consider now the second-order energetic consequences of the partial _charge transfer_ (CT) flows, shown in the diagrams of Equation (48). A reference to Equations (70)-(74), (77), and (79) indicates that in \(\mathrm{R}_{c}\) the dominating \(\mathrm{CT}_{1}\) process \(b_{\mathrm{B}}(\lambda_{4}){\rightarrow}a_{\mathrm{A}}(\lambda_{1})\) is described by the following in situ gradient and Hessian descriptors:  
\[ \begin{array}{c}u_{c}(\mathrm{CT}_{1})=u_{c}(a_{\mathrm{A}})-u_{c}(b_{ \mathrm{B}})=u_{1}-u_{4}\qquad\text{and}\\ h_{c}(\mathrm{CT}_{1})=h_{1,1}-h_{1,4}+h_{4,4}-h_{4,1}.\end{array}  \tag{86}\]They predict the optimum amount of this partial inter-reactant population displacement,  
\[N_{c}(\text{CT}_{1})=-u_{c}(\text{CT}_{1})/\!\!/t_{c}(\text{CT}_{1}), \tag{87}\]  
and the associated stabilization energy:  
\[E_{c}(\text{CT}_{1})=-\ [u_{c}(\text{CT}_{1})]^{2}/[2h_{c}(\text{CT}_{1})]. \tag{88}\]  
The associated chemical potential and hardness descriptors for the \(\text{CT}_{2}\) process \(b_{\text{A}}(\lambda_{2}){\rightarrow}a_{\text{B}}(\lambda_{3})\) in \(\text{R}_{c}\) accordingly read:  
\[ \begin{array}{c}u_{c}(\text{CT}_{2})=u_{c}(a_{\text{B}})-u_{c}(b_{\text{A}}) =u_{3}(c)-u_{2}(c)\qquad\text{ and }\\ h_{c}(\text{CT}_{2})=h_{3,3}(c)-h_{2,3}(c)+h_{2,2}(c)-h_{3,2}(c).\end{array}