2. Theory and Methods
Quantum Theory of Atoms in Molecules (QTAIM) 33 known as a comprehensive topology interpretation tool of quantum mechanics, provides a method to explore the distribution of electronsρ (r ), returning a comprehensive quantitative analysis of the atom’s bonding environments in a molecule. In the scalar fieldρ (r ), we can define a piecewise continuous gradient path by evaluating ∇ρ (r ) at some points and then tracking this vector for a very short distance and re-evaluating ∇ρ (r ). An atom is an area of real space bounded by surfaces in which there is zero flux in the total electronic charge density distribution’s gradient vector field. The interaction surface is defined by the set of trajectories ending at the point whereρ (r ) = 0, which implies that the zero-flux boundary conditions are satisfied by an interatomic surface: ∇ρ (r ) ∙ n (r ) = 0. Wheren (r ) is the unit vector common to the surface atr since the surface is not crossed by any of the trajectories of ρ (r ). The charge density ρ (r ) is a physical quantity with a particular value at each point in space, and the forces exerted on it by the nuclei dominate a topological structure. We may distinguish between various forms by examining the properties of the Hessian matrix of ρ (r ) at each critical point. The ordered eigenvalue set λ1 < λ2< λ3 is obtained by diagonalizing the Hessian matrix of ρ (r ), and the algebraic sum of these eigenvalues is the Laplacian of the electron density. In addition, these three eigenvalues are corresponding with a set of eigenvectorse1 , e2 ,e3 . Analyzing the eigenvalues of the Hessian matrix of ρ (r ) are used to determine whether the forces exerted by and on the (r ) favor tensile modes, such as expansion or compression for the volume element relevant to the positive or negative eigenvalue. An atomic interaction line (AIL)34 becomes a bond-path, but not permanently a chemical bond 35, when the forces on the nuclei become vanishingly small.
Bond ellipticity is defined by ε = |λ1|/|λ2| - 1 is providing the relative accumulation of the two perpendicular axes of the Hessian matrix, negative λ1 and λ2 eigenvalues, to the bond-path at a BCP33. Metallicity, ξ(r b) =ρ (r b)/∇2ρ (r b) ≥ 1 is one of the correlated quantity to the ellipticity ε for closed-shell interactions ∇2ρ (r b) > 0. Where ∇2ρ (r b) andρ (r b) are Laplacian and the values of total electronic charge density at the BCP respectively. The ξ(r b) is an important indicator for the ring-opening reactions bond path 36,37 while the relative values of ρ (r b) and ∇2ρ (r b) seem to alter at a BCP as the bond path stretched as long as eventually ruptured. Therefore ξ(r b) will relate to the reaction electronic flux (REF, J (ξ)), which corresponding to a chemical process along with the IRC(ξ) and is defined to be J (ξ) = -dμ /dξ 38 Where μ is the chemical potential. The use of the Greek letter ‘ξ’ for the metallicity ξ(r b) and REF J (ξ) definitions is purely coincidental. Although previously metallicity has been used to consider suspected metallicity ranges of metalloids, metals, and non-metals36,37, other researchers represent that ξ(r b) is conversely associated with “nearsightedness” of the first-order density matrix and is appropriate for closed-shell systems 39.
The total local energy density, H (r b) =G (r b) +V (r b), could determine The presence of a degree of covalent character 40,41 whereG (r b) andV (r b) are the local kinetic and potential energy densities respectively at a BCP . Precisely a degree of covalent character and a degree of lack of covalent character for a closed-shell interaction, ∇2ρ (r b) ≥ 0, where a value of H (r b) < 0 andH (r b) > 0 respectively. Stress tensor stiffness Sσ = |λ|/|λ| is the other proper descriptor of the resistance of the bond path to the applied distortion. It follows the similar trend with ellipticity ε, where double and single bonds indicate by ε > 0.25 and lower values respectively, as is expected the single bond is less resistant to torsion than a double bond. Pσ display the stress tensor polarizability, Pσ = |λ|/|λ|, which could define the reciprocal of the stress tensor stiffness Sσ 42–44.
The length of the path followed out by thee3 eigenvector of the Hessian of the total charge density ρ (r ) that is passing through theBCP along which ρ (r ) is locally maximal regard to any neighboring paths, is known as the bond-path length (BPL). Moreover, the bond-path curvature, (BPL - GBL)/GBL, is defined as the dimensionless ratio that separates two bonded nuclei. Where BPL and GBL have associated bond-path length and the inter-nuclear separation, respectively. Notice that particularly for strained or weak bonds and strange bonding environments usually BPL exceeds the GBL45. According to the hypothesis that a bond-path may possess 1-D, 2-D, or a 3-D morphology 46,47 with 2-D or a 3-D bond-paths corresponding to a BCP with ellipticity ε > 0, is because of differing degrees of charge density accumulation of the λ2 and λ1eigenvalues respectively.
By Starting with utilizing the ellipticity as the scaling factor, we choose the length traced out in 3-D by the path swept by the tips of the scaled e2 eigenvectors of the λ2eigenvalue. With n scaled eigenvectore1 tip path pointsp i = ri + εie 1 ,i on the path p where εi = ellipticity at the i th bond-path pointri on the bond-path r . In accordance with the e2 tip path points, we haveqi = ri + εie 2 ,i on the path q where εi = ellipticity at thei th bond-path point rion the bond-path r .