## 3 Probing Formation of the Chemical Bond
This qualitative statement has been subsequently given several more quantitative formulations and theoretical explanations using both the energetic and entropic arguments [93; 94; 95; 96; 97; 98; 99; 100]  
Previous virial-theorem analyses [1; 2; 3; 5] have shown that this qualitative rule is fully indexed by the sign of the \(P\)-derivative of the average kinetic energy or of the resultant gradient information at TS complex. The energy profile along the reaction \"progress\" coordinate \(P\), \(\Delta E(P)=E(P)-E(P_{sub. })\) is again directly \"translated\" by the virial theorem into the associated displacement in its kinetic-energy contribution \(\Delta T(P)=T(P)-T(P_{sub. })\), proportional to the corresponding change \(\Delta I(P)=I(P)-I(P_{sub. })\) in the system resultant gradient information, \(\Delta I(P)=\sigma\,\Delta T(P)\),  
\[\Delta T(P)=-\Delta E(P)-P\left[d\Delta E(P)/dP\right]=-\ d[P\Delta E(P)]/dP. \tag{32}\]  
A reference to qualitative plots in Figure 3 shows that the related \(\Delta T(P)\) or \(\Delta I(P)\) criteria distinguish these two directions by the sign of their geometrical derivative at TS complex:  
\[ \begin{array}{ll}\textit{endo-direction:}&(\textit{d}l/\textit{dP})_{\ddagger }>0\ \textit{and}\ (\textit{d}T/\textit{dP})_{\ddagger}>0,\ \Delta E_{r}>0;\\ \textit{energy-neutral:}&(\textit{d}l/\textit{dP})_{\ddagger}=0\ \textit{and}\ (\textit{d}T/\textit{dP})_{\ddagger}=0,\ \Delta E_{r}=0;\\ \textit{exo-direction:}&(\textit{d}l/\textit{dP})_{\ddagger}<0\ \textit{and}\ (\textit{d}T/\textit{dP})_{\ddagger}<0,\ \Delta E_{r}<0.\end{array} \]  
This observation demonstrates that the RC derivative of the resultant gradient information at TS complex, proportional to \(\textit{d}T/\textit{dP}\,\ddagger\), can indeed serve as an alternative detector of the reaction energetic character: its positive/negative values respectively identify the endo/exo-ergic processes, exhibiting the late/early activation barriers, respectively, with the neutral case, \(\Delta E_{r}=0\) or \(\textit{d}T/\textit{dP}\,\ddagger=0\), exhibiting an \"equidistant\" position of TS between the reaction substrates and products on a symmetric PES, e.g., in the hydrogen exchange reaction H + H\({}_{2}\rightarrow\)H\({}_{2}+\) H.  
The reaction energy \(\Delta E_{r}\) determines the corresponding change in the resultant gradient information, \(\Delta I_{r}=I(P_{prod.})-I(P_{sub. })=\sigma\,\Delta T_{r}\), proportional to \(\Delta T_{r}=T(P_{prod.})-T(P_{sub. })=-\Delta E_{r}\). The virial theorem thus implies a net decrease of the resultant gradient information in endo-ergic processes, \(\Delta I_{r}(endo)<0\), its increase in exo-ergic reactions, \(\Delta I_{r}(exo)>0\), and a conservation of the resultant gradient information in the energy-neutral chemical processes: \(\Delta I_{r}(neutral)=0\). One also recalls that the classical part of this information displacement probes an average inhomogeneity of electronic density. Therefore, the endo-ergic processes, requiring a net supply of energy to R, give rise to more diffused electron distributions in the reaction products, compared to substrates. Accordingly, the exo-ergic transitions, which release the energy from R, generate a more compact electron distributions in products and no such change is predicted for the energy-neutral case.