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 =
|λ1σ|/|λ3σ|
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 =
|λ3σ|/|λ1σ|,
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 .