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We use the newly introduced spanning stress tensor trajectory $U_{\sigma}$-space construction within next generation quantum theory of atoms in molecules (NG-QTAIM) for a chirality investigation of singly and doubly substituted ethane with halogen substituents: F, Cl, Br. A lack of achiral character in $U_{\sigma}$-space was discovered for singly substituted ethane. The resultant axial bond critical point (BCP) sliding responded more strongly to the increase in atomic number of the substituted halogen than the chirality. The presence of the very light F atom was found responsible for a very high degree of achiral character of the doubly substituted ethane., Comment: arXiv admin note: substantial text overlap with arXiv:2203.09750

We introduce a development of next generation quantum theory of atoms in molecules (NG-QTAIM) for an investigation of the chirality of ethane and discover a $Q_{\sigma}$ isomer in addition to $S_{\sigma}$ and $R_{\sigma}$ stereoisomers in the stress tensor trajectory $U_{\sigma}$-space. The $Q_{\sigma}$ isomer is defined to be a 'null-isomer' since the value of the chirality-helicity function $\approx 0$. The presence of chiral contributions suggests that steric effects, rather than hyper-conjugation, explain the staggered geometry of ethane. The steric effects, within the NG-QTAIM interpretation, are reduced by a factor of two using an applied electric-field directed down a C-H bond.

The factors underlying the experimentally observed branching ratio (70:30) of the (1,3-cyclohexadiene) CHD$\rightarrow$HT (1,3,5-hexatriene) photochemical ring-opening reaction are investigated. The ring-opening reaction path is optimized by a high-level multi-reference DFT method and the density along the path is analyzed by the QTAIM and stress tensor methods. The performed density analysis suggests that, in both $S_{1}$ and $S_{0}$ electronic states, there exists an attractive interaction between the ends of the fissile $\sigma$ -bond of CHD that steers the ring-opening reaction predominantly in the direction of restoration of the ring. It is suggested that opening of the ring and formation of the reaction product (HT) can only be achieved when there is a sufficient persistent nuclear momentum in the direction of stretching of the fissile bond. As this orientation of the nuclear momentum vector can be expected to be relatively rare during the dynamics, this explains the observed low quantum yield of the ring-opening reaction., Comment: 30 pages, manuscript + supplementary materials

The eigenvectors of the electronic stress tensor have been identified as useful for the prediction of chemical reactivity because they determine the most preferred directions to move the bonds that correspond to a qualitative change in the molecular electronic structure. A new 3-D vector based interpretation of the chemical bond that we refer to as the bond-path framework set $\mathbb{B} = \{p,q,r\}$ provides a version of the quantum theory of atoms in molecules (QTAIM) beyond the minimum definition for bonding that is particularly suitable for understanding changes in molecular electronic structure that occur during reactions. The bond-path framework set $\mathbb{B}$ is straightforwardly constructed and visualized from the eigenvalues and eigenvectors of QTAIM. This approach is applied to the structural deformations of ethene that occur during applied torsion ${\theta}$, -180.0{\deg} $\leq$ ${\theta}$ $\leq$ +180.0{\deg}. The corresponding stress tensor version is readily constructed as $\mathbb{B}_{\sigma} = \{p_{\sigma},q_{\sigma},r\}$ within the QTAIM partitioning making it possible to compare experimentally and computationally determined electronic charge densities. The bond-path framework set $\mathbb{B}$ or $\mathbb{B}_{\sigma}$ are the networks that comprise three strands: the least preferred ($p, p_{\sigma}$), most preferred ($q, q_{\sigma}$) and $r$ is the familiar QTAIM bond-path. We demonstrate that the most preferred direction for bond motion using the stress tensor corresponds to the most compressible direction and not to the least compressible direction as previously reported. We show the necessity for a directional approach constructed using the eigenvectors along the entire bond length and demonstrate the insufficiency of the sole use of scalar measures for capturing the nature of the stress tensor within the QTAIM partitioning., Comment: 44 pages, manuscript + supplementary materials. arXiv admin note: text overlap with arXiv:1804.00776

We use a new interpretation of the chemical bond within QTAIM. The bond-path framework set $\mathbb{B} = \{p, q, r\}$ with associated linkages with lengths $\mathbb{H}^*,m\mathbb{H}$ and the familiar bond-path length is used to describe a torsion $\theta$, $0.0^{\circ} \leq \theta \lt 22.0^{\circ}$ of \emph{para}-substituted biphenyl, $\mathrm{C}_{12}\mathrm{H}_{9-x}$, $x = \mathrm{N}(\mathrm{CH}_3)_2$, $\mathrm{NH}_2$, $\mathrm{CH}_3$, CHO, CN, $\mathrm{NO}_{2}$. We include consideration of the H---H bonding interactions and find that the lengths $\mathbb{H} \gt \mathbb{H}^{*}$ that we explain in terms of the most and least preferred directions of charge density accumulation. We also consider the fractional eigenvector-following path lengths $\mathbb{H}_f$ and $\mathbb{H}_{f{\theta_{\rm min}}}$., Comment: manuscript+supplementary materials. arXiv admin note: text overlap with arXiv:1804.00776, arXiv:1804.00780, arXiv:1804.01525

We introduce a vector-based interpretation of the chemical bond within the quantum theory of atoms in molecules (QTAIM), the bond-path framework set $\mathbb{B} = \{p, q, r\}$, to follow variations in the 3-D morphology of all bonds for the four infra-red (IR) active normal modes of benzene. The bond-path framework set comprises three unique paths $p$, $q$ and $r$ where $r$ is the familiar QTAIM bond concept of bond-path ($r$) while the two new paths $p$ and $q$ are formulated from the least and most preferred directions of electron density accumulation respectively. We find 3-D distortions including bond stretching/compression, torsion and curving. We introduce two fractional measures to quantify these variations away from linearity of the bond., Comment: Manuscript. Supplementary materials available on https://www.beaconresearch.org/ .arXiv admin note: text overlap with arXiv:1804.00776. v2. minor correction

The current work investigates candidate building blocks based on molecular junctions from hydrogen transfer tautomerization in the benzoquinone-like core of an azophenine molecule with QTAIM and the recently-introduced stress tensor trajectory analysis. We find that in particular the stress tensor trajectories are well suited to describe the mechanism of the switching process. The effects of an Fe-dopant atom coordinated to the quinone ring, as well as F and Cl substitution of different ring-hydrogens, are investigated and the new QTAIM and stress tensor analysis is used to draw conclusions on the effectiveness of such molecules as molecular switches in nano-sized electronic circuits. We find that the coordinated Fe-dopant greatly improves the switching properties, both in terms of the tautomerisation barrier that has to be crossed in the switching process and the expected conductance behavior, while the effects of hydrogen substitution are more subtle. The absence of the Fe-dopant atom led to impaired functioning of the switch 'OFF' mechanism as well as coinciding with the formation of closed-shell H---H bond critical points that indicated a strained or electron deficient environment. Our analysis demonstrates promise for future use in design of molecular electronic devices., Comment: manuscript. Supplementary materials available for download on https://www.beaconresearch.org

We present a new vector-based representation of the chemical bond referred to as the bond-path frame-work set $\mathbb{B} = {p, q, r}$, where $p$, $q$ and $r$ represent three paths with corresponding eigenvector-following path lengths $\mathbb{H}^{*},\mathbb{H}$ and the bond-path length from the quantum theory of atoms in molecules (QTAIM). We find that longer path lengths $\mathbb{H}$ of the ring-opening bonds predict the preference for the transition state inward (\textbf{TSIC}) or transition state outward (\textbf{TSOC}) ring opening reactions in agreement with experiment for all five reactions \textbf{R1-R5}. Competitiveness and non-competitiveness have traditionally been considered using activation energies. The activation energy however, for \textbf{R3} does not satisfactorily determine competitiveness or provide consistent agreement with experimental yields. We choose a selection of five competitive and non-competitive reactions; methyl-cyclobutene (\textbf{R1}), ethyl-methyl-cyclobutene (\textbf{R2}), iso-propyl-methyl-cyclobutene (\textbf{R3}), ter-butyl-methyl-cyclobutene (\textbf{R4}) and phenyl-methyl-cyclobutene (R5). Therefore, in this investigation we provide a new criterion, within the QTAIM framework, to determine whether the reactions \textbf{R1-R5} are competitive or non-competitive. We that find \textbf{R2}, \textbf{R3} and \textbf{R5} are competitive and \textbf{R1} and \textbf{R4} are non-competitive reactions in contrast to the results from the activation energies, calling into question the reliability of activation energies., Comment: manuscript and supplementary materials. arXiv admin note: text overlap with arXiv:1804.00776

A vector-based representation of the chemical bond is introduced that we refer to as the bond-path frame-work set = $\mathbb{B} = \{p, q, r\}$, where $p$, $q$ and $r$ represent three paths with corresponding eigenvector-following path lengths $\mathbb{H}^{*},\mathbb{H}$ and the familiar quantum theory of atoms in molecules (QTAIM) bond-path length. The eigenvector-following path lengths $\mathbb{H}^{*}$ and $\mathbb{H}$ are constructed along the bond-path from the $\underline{\mathbf{\mathit{e}}}_{1}$ and $\underline{\mathbf{\mathit{e}}}_{2}$ Hessian eigenvectors respectively, which correspond to the least and most preferred directions of charge density accumulation. In particular, the paths $p$ and $q$ provide a vector representation of the scalar QTAIM ellipticity {\epsilon}. The bond-path frame-work set $\mathbb{B}$ is applied to the excited state deactivation of fulvene that involves distortions along various intramolecular degrees of freedom, such as the bond stretching/compression of bond length alternation (BLA) and bond torsion distortions. We find that the $\mathbb{H}^{*}$ and $\mathbb{H}$ lengths can differentiate between the ground and excited electronic states, in contrast to the QTAIM bond-path length. In particular, the eigenvector-following path lengths $\mathbb{H}^{*}$ and $\mathbb{H}$ are found to be shorter for the excited state than the ground state for both the BLA and bond torsion distortions indicating that distortions resulting in lower $\mathbb{H}^{*}$ and $\mathbb{H}$ values are easier to perform., Comment: Manuscript and supplementary materials Title change, some corrections, new figure added

The Ramachandran plot is a potent way to understand structures of biomolecules, however, the original formulation of the Ramachandran plot only considers backbone conformations. We formulate a new interpretation of the original Ramachandran plot ($\phi-\psi$) that can include a description of the weaker interactions including both the hydrogen bonds and H$---$H bonds as a new way to derive insights into the phenomenon of peptide folding. We use QTAIM (quantum theory of atoms in molecules) to interpret the Ramachandran plot. Specifically, we show that QTAIM analysis permits identifying key regions of the Ramachandran plot without the need for massive data sets. A highly non-linear relationship is found between the QTAIM vector-derived interpreted Ramachandran plot and the conventional Ramachandran plot ($\phi-\psi$) demonstrating that this new approach is not a trivial coordinate transformation. An investigation of both the backbone and the weaker bonds within the framework of the QTAIM interpreted Ramachandran plot was found to be in line with physical intuition. The least-preferred directions calculated for the hydrogen bonds and H$---$H bonds were found to coincide with the 'unlikely' regions of the Ramachandran plot., Comment: Submitted to IJQC

Laguerre's theorem regarding the number of non-real zeros of a polynomial and its image under certain linear operators is generalized. This generalization is then used to (1) exhibit a number of previously undiscovered complex zero decreasing sequences for the Chebyshev, Legendre, and generalized Laguerre polynomial bases and (2) simultaneously generate a basis $B$ and a corresponding $B$-CZDS. Some extensions to transcendental entire functions in the Laguerre-Polya class are given which, in turn, give a new and short proof of a previously known result due to one of the authors. The paper concludes with several open questions., Comment: 15 pages, 0 figures, partial funding from NSA and NSF through an MAA NREUP grant

The strategy of using CUDA-compatible GPUs as a parallel computation solution to improve the performance of programs has been more and more widely approved during the last two years since the CUDA platform was released. Its benefit extends from the graphic domain to many other computationally intensive domains. Tiling, as the most general and important technique, is widely used for optimization in CUDA programs. New models of GPUs with better compute capabilities have, however, been released, new versions of CUDA SDKs were also released. These updated compute capabilities must to be considered when optimizing using the tiling technique. In this paper, we implement image interpolation algorithms as a test case to discuss how different tiling strategies affect the program's performance. We especially focus on how the different models of GPUs affect the tiling's effectiveness by executing the same program on two different models of GPUs equipped testing platforms. The results demonstrate that an optimized tiling strategy on one GPU model is not always a good solution when execute on other GPU models, especially when some external conditions were changed., Comment: Accepted to ISISE2009 (Second International Symposium on Information Science and Engineering, 26 - 28,Dec. 2009, Shanghai, China)

There is a pressing need for a description of complex systems that includes considerations of the underlying network of interactions, for a diverse range of biological, technological and other networks. In this work relationships between second-order phase transitions and the power laws associated with scale-free networks are directly quantified. A unique unbiased partitioning of complex networks (exemplified in this work by software architectures) into high- and low-connectivity regions can be made. Other applications to finance and aerogels are outlined., Comment: 12 pages, 5 figures