## 3 Probing Formation of the Chemical Bond  
The association between the overall gradient information and electronic kinetic energy suggests the use of molecular virial theorem in extracting the physical origins of the chemical bonding [89; 90; 91; 92] and for understanding general reactivity rules [1; 2; 3; 5]. The previous analyses [89; 90; 91] have focused on the interplay between the longitudinal (in the bond direction, along \"z\" coordinate) and transverse (perpendicular to the bond axis, due to coordinates \"x\" and \"y\") components of electronic kinetic energy. The former appears as the true driving force of the covalent-bond formation and the accompanying electron-d  
\([\Delta T(R)]\) and its overall potential complement \([\Delta W(R)]\). In the BO approximation the virial theorem for diatomics reads:  
\[2\Delta T(R)(+\Delta W(R)+R[d\Delta E(R)/dR]=0.\] (30a) It implies the following kinetic and potential energy components: \[\Delta T(R)=-\ \Delta E(R)-R\ [d\Delta E(R)/dR]=-\ d[R\Delta E(R)]/dR\quad\text{ and} \tag{30b}\] \[\Delta W(R)=2\Delta E(R)+R\ [d\Delta E(R)/\partial R]=R^{-1}\ d[R^{2} \Delta E(R)]/dR.\]  
Figure 2 presents qualitative plots of the BO potential \(\Delta E(R)\) and its kinetic-energy contribution \(\Delta T(R)\) in diatomics. The latter also reflects the associated displacement plots for the resultant gradient information \(\Delta I(R)=\sigma\ \Delta T(R)\). It follows from this qualitative diagram that, during a mutual approach by two constituent atoms, the kinetic-energy/gradient information is first diminished relative to SAL, due to the dominating longitudinal contribution related to Cartesian coordinate \"\(z\)\" (along the bond axis). However, at the equilibrium distance \(R_{e}\) the resultant information already rises above the SAL value, due to the dominating increase in transverse components of the kinetic-energy/information (corresponding to coordinates \"\(x\)\" and \"\(y\)\" perpendicular to the bond axis). Therefore, at the equilibrium separation \(R_{e}\) between atoms the bond-formation results in a net increase of the resultant gradient-information relative to SAL, due to--on average--more compact electron distribution in the field of both nuclei. Consider next the (intrinsic) reaction  
The (qualitative) [PERSON] postulate [80] of reactivity theory relates a general resemblance/proximity of the reaction TS complex R\({}^{\ddagger}\) to either its substrates \(\alpha\in\) (A, B) or products \(\beta\in\) (C, D) to the reaction energy \(\Delta E_{r}=E(P_{pnd.})-E(P_{sub. })\): in _exo_-ergic (\(\Delta E_{r}<0\)) processes, R\({}^{\ddagger}\approx\alpha\) and in endo-ergic (\(\Delta E_{r}>0\)) reactions, R\({}^{\ddagger}\approx\beta\). Accordingly, for the vanishing reaction energy \(\Delta E_{r}=0\), the position of TS complex is expected to be located symmetrically between the reaction  
Figure 3: Variations of the electronic total (\(E\)) and kinetic (\(T\)) energies in exo-ergic (\(\Delta E_{r}<0\)) or endo-ergic (\(\Delta E_{r}>0\)) reactions (upper Panel (**a**)), and on the symmetrical BO potential energy surface (PES) (\(\Delta E_{r}=0\)) (lower Panel (**b**)). Figure 2: Variations of the electronic energy \(\Delta E(R)\) (solid line) with the internuclear distance \(R\) in a diatomic molecule and of its kinetic energy component \(\Delta T(R)\) (broken line) determined by the virial theorem partition.4. Reactivity Implications of Molecular Virial Theorem. substrates and products. A reference to Figure 3 indeed shows that the activation barrier appears \"early\" in exo-ergic reaction, e.g., H\({}_{2}\) + F \(\rightarrow\) H + HF, with the reaction substrates being only slightly modified in TS, R\({}^{\ddagger}\approx\) [A-B]. Accordingly, in the endo-ergic bond-breaking-bond-forming process, e.g., H + HF \(\rightarrow\) H\({}_{2}\) + F, the barrier is \"late\" along the reaction coordinate \(P\) and the activated complex resembles more reaction products: R\({}^{\ddagger}\approx\) [C-D].