Title: Quantum theory of atoms in molecules in condensed charge density space
By leveraging the fundamental doctrine of the quantum theory of atoms in molecules — the partitioning of the electron charge density (ρ) into regions bounded by surfaces of zero flux — we map the gradient field of ρ onto a two-dimensional space called the gradient bundle condensed charge density ([Formula: see text]). The topology of [Formula: see text] appears to correlate with regions of chemical significance in ρ. The bond wedge is defined as the image in ρ of the basin of attraction in [Formula: see text], analogous to the Bader atom, which is the basin of attraction in ρ. A bond bundle is defined as the union of bond wedges that share interatomic surfaces. We show that maxima in [Formula: see text] typically map to bond paths in ρ, though this is not necessarily always true. This observation addresses many of the concerns regarding the chemical significance of bond critical points and bond paths in the quantum theory of atoms in molecules. more »« less
Lipton, Max
(, Journal of Knot Theory and Its Ramifications)
null
(Ed.)
Consider a knot [Formula: see text] in [Formula: see text] with charge uniformly distributed on it. From the standpoint of both physics and knot theory, it is natural to try to understand the critical points of the potential and their behavior. We show the number of critical points of the potential is at least [Formula: see text], where [Formula: see text] is the tunnel number, defined as the smallest number of arcs one must add to [Formula: see text] such that its complement is a handlebody. The result is proven using Morse theory and stable manifold theory.
Spiegel, Daniel; Moreno, Juan; Qi, Marvin; Hermele, Michael; Beaudry, Agnès; Pflaum, Markus J.
(, Reviews in Mathematical Physics)
We consider how the outputs of the Kadison transitivity theorem and Gelfand–Naimark–Segal (GNS) construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation [Formula: see text] of a [Formula: see text]-algebra [Formula: see text] and [Formula: see text], there exists a continuous function [Formula: see text] such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is the set of pairs of [Formula: see text]-tuples [Formula: see text] such that the components of [Formula: see text] are linearly independent. Versions of this result where [Formula: see text] maps into the self-adjoint or unitary elements of [Formula: see text] are also presented. Regarding the GNS construction, we prove that given a topological [Formula: see text]-algebra fiber bundle [Formula: see text], one may construct a topological fiber bundle [Formula: see text] whose fiber over [Formula: see text] is the space of pure states of [Formula: see text] (with the norm topology), as well as bundles [Formula: see text] and [Formula: see text] whose fibers [Formula: see text] and [Formula: see text] over [Formula: see text] are the GNS Hilbert space and closed left ideal, respectively, corresponding to [Formula: see text]. When [Formula: see text] is a smooth fiber bundle, we show that [Formula: see text] and [Formula: see text] are also smooth fiber bundles; this involves proving that the group of ∗-automorphisms of a [Formula: see text]-algebra is a Banach Lie group. In service of these results, we review the topology and geometry of the pure state space. A simple non-interacting quantum spin system is provided as an example illustrating the physical meaning of some of these results.
Olmschenk, Greg; Wang, Xuan; Tang, Hao; Zhu, Zhigang
(, International Journal of Pattern Recognition and Artificial Intelligence)
Gatherings of thousands to millions of people frequently occur for an enormous variety of educational, social, sporting, and political events, and automated counting of these high-density crowds is useful for safety, management, and measuring significance of an event. In this work, we show that the regularly accepted labeling scheme of crowd density maps for training deep neural networks may not be the most effective one. We propose an alternative inverse k-nearest neighbor (i[Formula: see text]NN) map mechanism that, even when used directly in existing state-of-the-art network structures, shows superior performance. We also provide new network architecture mechanisms that we demonstrate in our own MUD-i[Formula: see text]NN network architecture, which uses multi-scale drop-in replacement upsampling via transposed convolutions to take full advantage of the provided i[Formula: see text]NN labeling. This upsampling combined with the i[Formula: see text]NN maps further improves crowd counting accuracy. We further analyze several variations of the i[Formula: see text]NN labeling mechanism, which apply transformations on the [Formula: see text]NN measure before generating the map, in order to consider the impact of camera perspective views, image resolutions, and the changing rates of the mapping functions. To alleviate the effects of crowd density changes in each image, we also introduce an attenuation mechanism in the i[Formula: see text]NN mapping. Experimentally, we show that inverse square root [Formula: see text]NN map variation (iR[Formula: see text]NN) provides the best performance. Discussions are provided on computational complexity, label resolutions, the gains in mapping and upsampling, and details of critical cases such as various crowd counts, uneven crowd densities, and crowd occlusions.
Eltareb, Ali; Lopez, Gustavo E.; Giovambattista, Nicolas
(, The Journal of Chemical Physics)
We perform path integral molecular dynamics (PIMD) simulations of a monatomic liquid that exhibits a liquid–liquid phase transition and liquid–liquid critical point. PIMD simulations are performed using different values of Planck’s constant h, allowing us to study the behavior of the liquid as nuclear quantum effects (NQE, i.e., atoms delocalization) are introduced, from the classical liquid ( h = 0) to increasingly quantum liquids ( h > 0). By combining the PIMD simulations with the ring-polymer molecular dynamics method, we also explore the dynamics of the classical and quantum liquids. We find that (i) the glass transition temperature of the low-density liquid (LDL) is anomalous, i.e., [Formula: see text] decreases upon compression. Instead, (ii) the glass transition temperature of the high-density liquid (HDL) is normal, i.e., [Formula: see text] increases upon compression. (iii) NQE shift both [Formula: see text] and [Formula: see text] toward lower temperatures, but NQE are more pronounced on HDL. We also study the glass behavior of the ring-polymer systems associated with the quantum liquids studied (via the path-integral formulation of statistical mechanics). There are two glass states in all the systems studied, low-density amorphous ice (LDA) and high-density amorphous ice (HDA), which are the glass counterparts of LDL and HDL. In all cases, the pressure-induced LDA–HDA transformation is sharp, reminiscent of a first-order phase transition. In the low-quantum regime, the LDA–HDA transformation is reversible, with identical LDA forms before compression and after decompression. However, in the high-quantum regime, the atoms become more delocalized in the final LDA than in the initial LDA, raising questions on the reversibility of the LDA–HDA transformation.
Miller, Daniel K.; Chernyshov, Ivan Yu.; Torubaev, Yury V.; Rosokha, Sergiy V.
(, Physical Chemistry Chemical Physics)
The relationship between covalent and supramolecular bonding, and the criteria of the assignments of different interactions were explored via the review of selenium and tellurium containing structures in the Cambridge Structural Database and their computational analysis using Quantum Theory of Atoms in Molecules (QTAIM). This combined study revealed continuums of the interatomic Se⋯Br and Te⋯I distances, d Ch⋯X , in the series of associations from the sums of the van der Waals radii of these atoms ( r Ch + r X ) to their covalent bond lengths. The electron densities, ρ ( r ), at Bond Critical Points (BCPs) along the chalcogen bond paths increased gradually from about 0.01 a.u. common for the non-covalent interactions to about 0.1 a.u. typical for the covalent bonds. The log ρ ( r ) values fell on the same linear trend line when plotted against normalized interatomic distances, R XY = d Ch⋯X /( r Ch + r X ). The transition from the positive to negative values of the energy densities, H ( r ), at the BCPs (related to a changeover of essentially non-covalent into partially covalent interactions) were observed at R XY ≈ 0.80. Synchronous changes of bonding characteristics with R XY (similar to that found earlier in the halogen-bonded systems) designated normalized interatomic separation as a critical factor determining the nature of these bondings. The uninterrupted continuums of Te⋯I and Se⋯Br bond lengths and BCPs’ characteristics signified an intrinsic link between limiting types of bonding involving chalcogen atoms and between covalent and supramolecular bonding in general.
Wilson, Timothy R., and Eberhart, M.E. Quantum theory of atoms in molecules in condensed charge density space. Retrieved from https://par.nsf.gov/biblio/10353572. Canadian Journal of Chemistry 97.11 Web. doi:10.1139/cjc-2019-0086.
Wilson, Timothy R., & Eberhart, M.E. Quantum theory of atoms in molecules in condensed charge density space. Canadian Journal of Chemistry, 97 (11). Retrieved from https://par.nsf.gov/biblio/10353572. https://doi.org/10.1139/cjc-2019-0086
@article{osti_10353572,
place = {Country unknown/Code not available},
title = {Quantum theory of atoms in molecules in condensed charge density space},
url = {https://par.nsf.gov/biblio/10353572},
DOI = {10.1139/cjc-2019-0086},
abstractNote = {By leveraging the fundamental doctrine of the quantum theory of atoms in molecules — the partitioning of the electron charge density (ρ) into regions bounded by surfaces of zero flux — we map the gradient field of ρ onto a two-dimensional space called the gradient bundle condensed charge density ([Formula: see text]). The topology of [Formula: see text] appears to correlate with regions of chemical significance in ρ. The bond wedge is defined as the image in ρ of the basin of attraction in [Formula: see text], analogous to the Bader atom, which is the basin of attraction in ρ. A bond bundle is defined as the union of bond wedges that share interatomic surfaces. We show that maxima in [Formula: see text] typically map to bond paths in ρ, though this is not necessarily always true. This observation addresses many of the concerns regarding the chemical significance of bond critical points and bond paths in the quantum theory of atoms in molecules.},
journal = {Canadian Journal of Chemistry},
volume = {97},
number = {11},
author = {Wilson, Timothy R. and Eberhart, M.E.},
}
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