3.2.3. Bond Degree Parameters H(**r***c*)/*ρ*(**r***c*)

The bond degree parameter is related to chemical bonding as follows. Kinetic energy is everywhere positive and repulsive (*mv*2/2 = *p* <sup>2</sup>/2*m* > 0 in classical mechanics) while potential energy is everywhere negative and attractive. The total energy is the sum of kinetic and potential energies, H = G + V; thus, its sign reveals the winner of the local kinetic vs potential energy tug of war and dictates the nature of the interaction. Indeed, positive total energies at BCPs are obtained when there is a local dominance of the repulsive kinetic energy, indicating local depletion of electrons in the internuclear region and displacement of the electron density associated with the particular bonding interactions towards the nuclei. Conversely, negative total energies are obtained when there is a local dominance of the attractive potential energy indicating that there is shared electron density concentrated in the internuclear region and signaling an increasingly covalent character of the interaction. An alternative rigorous physical meaning to energy densities is offered by a dimensional analysis: energy density has units of pressure (*E*/*V* = *F*/*A* = *P*); thus, local negative energy densities may be equated to negative quantum pressures which strongly attract electrons towards the BCP, indicating increasingly covalent interactions while local positive energy densities correspond to positive quantum pressures that push electrons away from the BCPs towards the nuclei, indicating anionic or long range interactions.

It is well known that the sign of <sup>∇</sup>2*ρ*(**r***c*) is not a sufficient criterium to establish the nature of the interaction in every case [28,67–70], specifically, it is quite often the case that a particular interaction has both positive Laplacian and negative bond degree parameter at the same time. Thus, the bond degree parameter is used in conjunction with the Laplacian of the electron density at BCPs to remove any ambiguity according to Rozas et al. [68]: weak to medium strength hydrogen bonds have both <sup>∇</sup>2*ρ*(**r***c*), <sup>H</sup>(**r***c*)/*ρ*(**r***c*) <sup>&</sup>gt; 0, strong hydrogen bonds have <sup>∇</sup>2*ρ*(**r***c*) <sup>&</sup>gt; 0, <sup>H</sup>(**r***c*)/*ρ*(**r***c*) <sup>&</sup>lt; <sup>0</sup> and very strong HBs have both <sup>∇</sup>2*ρ*(**r***c*), <sup>H</sup>(**r***c*)/*ρ*(**r***c*) <sup>&</sup>lt; 0. Figure <sup>6</sup> plots distributions of the bond degree parameters for all dimers found in this work. It is clear from the distributions of H(**r***c*)/*ρ*(**r***c*) that all intermolecular contacts found here cover a wide spectrum of possibilities with a substantial number of only primary hydrogen bonds or salt bridges having H(**r***c*)/*ρ*(**r***c*) < 0 (Figure 6A–C), thus should be considered as strong contacts by the above criteria. The wide spectrum of bond degree parameters, the large number of structural possibilities and the strong character of the interactions have deep implications in the biological role of cysteine and of the aminoacids that make up proteins and biomolecules: similar results have been obtained for example in the interactions between the spike protein of SARS-COV-2 and the ACE2 receptors [48,71] and between the envelope protein of the Zika virus and the glycosaminoglycans that act as receptors [47]. In the case of SARS-COV-2, the formation of strong salt bridges and hydrogen bonds is one of the main factors of the pressure driving the evolution of the virus towards new variants. For the cysteine dimers, many of the primary hydrogen bonds with positive bond degree parameters are located to the left of the reference isolated gas phase water dimer, which confers them medium to strong character. All secondary and exotic contacts (Figure 6D–F) exhibit positive bond degree parameters and many areas actually to the right of the reference water dimer; thus, they are classified as weak. As a general rule, hydrogen bonds involving the carbonyl, carboxylate and amino groups as electron donors and the hydroxyl, amino and ammonium groups as electron acceptors, are the ones with highly negative H(**r***c*)/*ρ*(**r***c*) values. Some HBs involving the thiol group, either as donor or acceptor, have slightly negative bond degree parameters.
