## Appendix C Ensemble Representation of Thermodynamic Conditions  
Only the ensemble average overall number of electrons \(\langle N\rangle_{\text{ens. }}\equiv\mathcal{N}\) of the (open) molecular part M(\(\mathbf{r}\)), identified by the external potential \(\mathbf{v}\) of the specified (microscopic) system, in the composite (macroscopic) system \(\mathbf{M}\)= [M(\(\mathbf{v}\)) : \(\mathbf{\mathcal{R}}\)] including the external electron-reservoir \(\mathbf{\mathcal{R}}\),  
\[\mathcal{N}=\text{tr}(\hat{\text{D}}\hat{\text{N}})=\sum_{i}N_{i}\langle\sum_ {i}p_{j}^{\,i}\rangle\equiv\sum_{i}N_{i}p^{i},\quad\sum_{i}p^{i}=1, \tag{104}\]  
exhibits a continuous (fractional) spectrum of values, thus justifying the very concept of the populational (\(\mathcal{N}\)) derivative itself. Here, \(\hat{\text{N}}=\sum_{i}N_{i}\) (\(\sum_{j}\mid\psi_{j}^{\,i}\rangle\)) stands for the particle-number operator in Fock's space and the density operator \(\hat{\text{D}}=\sum_{i}\sum_{j}\mid\psi_{j}^{\,i}\rangle\)\(P_{j}^{\,i}\) (\(\psi_{j}^{\,i}\) ) identifies the statistical mixture of the system (pure) stationary states \(\{\mid\psi_{j}^{\,i}\rangle\equiv\mid\psi_{j}\langle N_{i}\rangle\}\) defined for different (integer) numbers of electrons \(\{N_{i}\}\), which appear with the (external) probabilities \(\{P_{j}^{\,i}\rangle\) in the ensemble. Such \(\mathcal{N}\)-derivatives are involved in definitions of familiar CT criteria of chemical reactivity, e.g., the chemical potential (negative electronegativity) [53; 101; 102; 103; 104; 105] or the chemical hardness/softness [106], and Fukui function (FF) [107] descriptors (see also References [53; 73; 74]). Such \(\mathcal{N}\)-derivatives are thus definable only in the mixed electronic states, e.g., those corresponding to thermodynamic equilibria in externally open molecules. In the grand ensemble, this state is determined by the density operator specified by the relevant (externally imposed) intensive parameters, the chemical potential of the electron reservoir, \(\mu=\mu_{\mathbf{\mathcal{R}}}\), and the absolute temperature \(T\) of a heat bath \(\mathcal{B}\), \(T=T_{\mathcal{B}}\): \(\hat{\text{D}}_{\text{eq.}}\equiv\hat{\text{D}}(\mu,T;\mathbf{v})\). The optimum state probabilities \(\{P_{j}^{\,i}(\mu,T;\mathbf{v})\}\) correspond to the minimum of the associated thermodynamic potential, the Legendre transform  
\[\mathbf{\Omega}=\mathcal{E}-(\partial\mathcal{E}/\partial\mathcal{N})\mathcal{N} -(\partial\mathcal{E}/\partial\mathcal{S})\mathcal{S}=\mathcal{E}-\mu \mathcal{N}-TS \tag{105}\]  
of the ensemble-average energy  
\[\mathcal{E}[\hat{\text{D}}]\equiv(\mathcal{N},\mathcal{S};\mathbf{v})\text{=tr} (\hat{\text{D}}\hat{\text{H}})=\sum_{i}\sum_{j}\text{P}_{i}^{\,i}\text{E}_{j }^{\,i}, \tag{106}\]  
called the grand potential. The latter corresponds to replacing the \"extensive\" (ensemble-average) state parameters, the particle number \(\mathbf{\mathcal{N}}\) and thermodynamic entropy  
\[S[\text{D}]\text{=tr}(\hat{\text{D}}\hat{\text{S}})=\sum_{i}\sum_{j}P_{j}^{\, i}S_{j}^{\,i}(P_{j}^{\,i})\},\quad S_{j}^{\,i}(P_{j}^{\,i})=-k_{\text{B}}\ln P_{j}^{\, i}, \tag{107}\]  
where \(k_{\text{B}}\) denotes the Boltzmann constant, by their respective \"intensive\" conjugates defining the applied thermodynamic conditions: \(\mu=\mu_{\mathbf{\mathcal{R}}}\) and \(T=T_{\mathcal{B}}\). The grand potential of Equation (105) includes these externally imposed intensities as Lagrange multipliers enforcing constraints of the specified values of system's average number of electrons, \(\langle N\rangle_{\text{ens. }}=\mathcal{N}\), and its thermodynamic-entropy, \(\langle\mathcal{S}\rangle_{\text{ens.