Biphenyl: A stress tensor and vector-based perspective explored within the quantum theory of atoms in molecules

Abstract

We use quantum theory of atoms in molecules (QTAIM) and the stress tensor topological approaches to explain the effects of the torsion u of the C-C bond linking the two phenyl rings of the biphenyl molecule on a bond-by-bond basis using both a scalar and vector-based analysis. Using the total local energy density H(r b ), we show the favorable conditions for the formation of the controversial H-H bonding interactions for a planar biphenyl geometry. This bond-by-bond QTAIM analysis is found to be agreement with an earlier alternative QTAIM atom-by-atom approach that indicated that the H-H bonding interaction provided a locally stabilizing effect that is overwhelmed by the destabilizing role of the C-C bond. This leads to a global destabilization of the planar biphenyl conformation compared with the twisted global minimum. In addition, the H(r b ) analysis showed that only the central torsional C-C bond indicated a minimum for a torsion u value coinciding with that of the conventional global energy minimum. The H-H bonding interactions are found to be topologically unstable for any torsion of the central C-C bond away from the planar biphenyl geometry. Conversely, we demonstrate that for 0.08 < u < 39.958 there is a resultant increase in the topological stability of the C nuclei comprising the central torsional C-C bond. Evidence is found of the effect of the H-H bonding interactions on the torsion u of the central C-C bond of the biphenyl molecule in the form of the QTAIM response b of the total electronic charge density q(r b ). Using a vector-based treatment of QTAIM we confirm the presence of the sharing of chemical character between adjacent bonds. In addition, we present a QTAIM interpretation of hyperconjugation and conjugation effects, the former was quantified as larger in agreement with molecular orbital (MO) theory. The stress tensor and the QTAIM H atomic basin path set areas are independently found to be new tools relevant for the incommensurate gas to solid phase transition occurring in biphenyl for a value of the torsion reaction coordinate u % 58. V C 2015 Wiley Periodicals, Inc.

A QTAIM and stress tensor investigation of the torsion path of a light-driven fluorene molecular rotary motor

“…Thus, analogously to the QTAIM descriptor, we calculate a stress tensor stiffness, S σ = | λ 1 σ |/| λ 3 σ |, which has been found as a good descriptor of the “resistance” of the bond‐path to the twist, as well as the eigenvector, e 1 σ indicating the direction of the π‐bond . Previously, it was found that the stress tensor stiffness, S σ produced results that were in line with physical intuition …”

“…Therefore, the basin‐path set area is defined as the area of the surface swept out by the ∇ ρ ( r ) trajectories undergoing torsion. A recent investigation on biphenyl found that the basin‐path set areas of the C atomic basins defined by the e 2 eigenvector of the BCP undergoing a torsion φ was always greater than for the corresponding e 1 eigenvector except at the planar conformation, φ = 0.0° . The converse was found to be true for the H atomic basins associated with the H—H BCPs …”

“…The ellipticity defined as ε = | λ 1 |/| λ 2 | − 1, where λ 1 and λ 2 are negative eigenvalues of the corresponding Hessian matrix at a BCP, provides the relative accumulation of ρ ( r b ) in the two directions perpendicular to the bond‐path at the BCP. Previously, we defined a bond‐path stiffness, S = λ 2 / λ 3 , as a measure of rigidity of the bond‐path …”

“…The quantum stress tensor, σ ( r ), which is directly related to the Ehrenfest force by the virial theorem, therefore, provides a physical explanation of the low frequency normal modes that accompany structural rearrangements . Diagonalization of the stress tensor, σ ( r ), returns the principal electronic stresses.…”

“…For the 11‐cis retinal subjected to a torsion ± α , we have recently demonstrated that the e 2 and e 1 σ eigenvectors of the torsional BCP corresponded to the preferred + α direction of rotation as defined by the PES profile . In this work, we choose to use the stress tensor response β σ because earlier it was found to produce more consistent results for torsion …”