Cherif Farid Matta

Quantum chemical topology (QCT) extracts chemical information from the electron density, ρr, and its derivatives. In this context, however, a direct Helmholtz decomposition of the gradient field of the density, ∇ρr, is uninformative because a gradient field is purely longitudinal. We introduce the density structural torsional vector (DSTV), Fρr=∇lnρrρ0×∇∇2ρr, where ρ0 is a constant positive reference density introduced to make the logarithm dimensionless; its value does not affect the resulting field. The DSTV measures the local directional noncollinearity between the relative density gradient and the spatial variation of the Laplacian of the density and is divergence-free by construction. Its associated scalar descriptors, Sr=sinθr and Sρr=ρrsinθr, where θ(r) is the local angle between the relative density gradient and the gradient of the Laplacian of the density quantify, respectively, the purely angular and density-weighted forms of this noncollinearity, with the latter emphasizing regions of appreciable electron density. Illustrative molecular calculations show that these descriptors reveal anisotropic density–curvature organization, shell structure, and bond-region features that are not immediately evident from the electron density, its gradient, or its Laplacian considered separately. Numerical examples are presented for simple diatomics, including Li2 with a non-nuclear attractor, and the water molecule. The method is proposed as a complementary electron density-derived vectorial descriptor for the real-space analysis of electronic structure.

This article examines, from historical and pedagogical perspectives, Alphonse Berget’s anticipation of Earth–Moon travel in Le Ciel (Larousse, 1923), decades before the beginning of the space age. The discussion is triggered by Le Ciel , a richly illustrated French popular science work, which has a devoted chapter examining lunar and interplanetary travel within Newtonian physics. Although Berget’s treatment was not developed in isolation and reflects a broader early 20th century context that included pioneers such as French aero-engineer Robert Esnault-Pelterie, the book provides a striking pedagogical synthesis of elementary celestial mechanics and scientific popularization. Unlike earlier fictional treatments such as Jules Verne’s De la Terre à la Lune , Berget approached space travel using physical reasoning grounded in Newtonian gravitation. Using qualitative and semi-quantitative arguments based on the inverse-square law, he identified the principal phases of an Earth–Moon trajectory: escape from Earth, inertial translunar motion, transition through competing Earth–Moon gravitational fields, and final lunar capture and deceleration. His estimated Earth–Moon travel time of approximately 49 h is of the same order of magnitude as Apollo missions transit times (~72 h). We compare these early ideas with modern elementary concepts of astrodynamics and with the trajectories of the Apollo and Artemis programs. We also examine Berget’s discussion of interplanetary travel, lunar landscapes, and human factors associated with prolonged voyages, including confinement, food supply, and travel duration. The analysis highlights the pedagogical value of historically grounded scientific reasoning underpinning spaceflight mechanics.

Molecular shape is routinely explained through bonding language: bonds impose angles, lone pairs repel ligands, steric contacts force torsion, orbital interactions favor conformations, and energy-decomposition components stabilize structures. These useful explanations can reverse the order of the underlying theory. In the full molecular Coulomb problem, a fixed geometry is not an intrinsic label of a stationary eigenstate. Standard quantum chemistry uses clamped-nuclei/Born-Oppenheimer approximation whereby a nuclear configuration, number of electrons, and any additional externally applied potentials, define the electronic Hamiltonian. The quantum state then yields energy and bonding descriptors at each geometry. Repeated calculations at different geometries generate the potential-energy landscape whose stationary structures are identified. We examine VSEPR, steric and Pauli repulsion, biphenyl and cis-2-butene, EDA auxiliary states and path/reference dependence, NCI, orbital representation dependence, QTAIM/IQA, localization-delocalization matrices, stable/metastable metals, nuclear transmutation, and the one-electron Bohm quantum potential. We also separate practical Born-Oppenheimer chemistry from the unresolved full-Coulomb structure problem. Stronger causal claims begin with an independent perturbation of the Hamiltonian, followed by responses of geometry, energy, density, and descriptors. Molecular shape is, therefore, a stationary-state or ensemble property, while bonding descriptors provide higher-level explanations rather than independent microscopic agents. This raises a deeper question: why does quantum matter generate robust, transferable chemical regularities, including recurring hydrogen-bonding patterns?

Enzymes are widely understood to accelerate chemical reactions through transition‐state stabilization and optimizing the orientation of reactants within their active sites. A growing body of work highlights the central role of strong electrostatic fields in these processes with magnitudes reaching ∼10 ⁸ –10 ¹⁰ V/m in enzyme active sites. Here, we explore the hypothesis that such fields may, in some cases, facilitate catalysis by inducing field‐assisted electron transfer or partial ionization of bound substrates. To assess the plausibility of this mechanism, a semiclassical tunneling expression in the quasistatic limit was applied to two datasets of enzymatic substrates: a curated set of 54 substrates with experimentally known ionization potentials and an extended set of 3168 enzymatic substrates with machine‐learning‐predicted ionization potentials matched to 5272 enzymes. At field strengths of an order ≳ 10 ¹⁰ V/m, estimated ionization rates span the nanosecond‐to‐picosecond regime for a subset of substrates, suggesting that field‐induced tunneling may render electron‐transfer events statistically accessible within the residence time of enzyme–substrate complexes. These results are recast in probabilistic terms, showing that even modest tunneling rates can lead, under certain circumstances, to non‐negligible ionization probabilities over biologically relevant timescales. Electronic‐structure calculations on the ketosteroid isomerase substrate (5‐androstene‐3,17‐dione) indicate that one‐electron oxidation leads to a redistribution of electron density consistent with the enhancing of the electrophilic character of the molecule's reactive site. While the tunneling expressions employed here are not intended to provide quantitatively predictive rates in condensed enzymatic environments, they offer a physically motivated metric for assessing the sensitivity of ionization barriers to strong electrostatic fields. Taken together, these results suggest that field‐assisted substrate ionization or oxidation may represent a plausible, though not yet established, mechanistic component in certain enzyme‐catalyzed reactions. This hypothesis remains qualitative and is intended to stimulate further theoretical and experimental testing.

Temperature and pH are among the most familiar intensive variables in chemistry and biology and are typically introduced as smooth continuum quantities that can be assigned locally without ambiguity. Yet modern chemistry increasingly probes systems whose dimensions approach those of organelles, vesicles, enzyme active sites, and nanoscopic aqueous compartments, where molecular discreteness and statistical fluctuations become unavoidable. This article examines how continuum thermodynamic concepts emerge from underlying molecular statistics and under what conditions they begin to lose their classical interpretation using intracellular thermometry and proton statistics as pedagogical examples. A central case study is the “hot mitochondrion paradox,” in which fluorescent thermometry experiments suggest mitochondrial temperatures substantially higher (by several degrees) than the surrounding cytoplasm, whereas steady-state heat-diffusion models predict temperature differences of only ∼10–5 K. The pH example is even more striking. At physiological pH, a cell-sized compartment may contain tens of thousands of protons, whereas an organelle-sized volume may contain only tens, rendering pH a fluctuating proton-counting variable rather than a smooth concentration field. Classroom discussions of local equilibrium, thermalization, thermal baths, finite-size fluctuations, transient local heating, operational definitions of temperature and pH, and electronic temperature in quantum systems are suggested. Numerical examples, scaling estimates, and student exercises are also included. The goal is to help students recognize that temperature and pH do not abruptly “fail” at nanoscopic scales; rather, they become statistical, time scale-dependent, probe-dependent, and operationally defined quantities. The accompanying examples, discussion questions, and numerical exercises are intended for upper-level undergraduate physical chemistry and related courses.

It is suggested that, under restrictive ideal circumstances, the interacting electronic energy of a system can be obtained directly from experimentally accessible x‐ray orbitals, defined here as orbitals obtained from direct fitting of the one‐body density matrix to measured x‐ray structure factors, through inverse Kohn–Sham equation functional integration. The argument rests on the identity, within a unitary transformation, between experimentally fitted x‐ray orbitals and theoretically calculated Kohn–Sham orbitals, both of which—in ideal circumstances—yield the same electron density. Using x‐ray orbitals, one may evaluate both the Kohn–Sham kinetic energy and the interacting potential energy by functional integration of the exact Kohn–Sham potential, taken directly from the Kohn–Sham equations, expressed in atomic units (a.u.), . This approach allows the total interacting Kohn–Sham energy: , where E _KS is the interacting electronic energy, is the non‐interacting kinetic‐energy, and is the interacting electronic potential‐energy obtained from the Kohn–Sham potential , to be reconstructed entirely from orbital data measurable through high‐resolution x‐ray diffraction. A theorem is enunciated stating that for a non‐degenerate ground state, absent magnetic fields, in a real finite one‐particle basis, if the experimentally accessible constraints uniquely determine an idempotent rank‐ N x‐ray projector P _XR , and if a Kohn–Sham projector P _KS satisfies the same set of constraints, then the two projectors must be identical ( P _XR = P _KS ). Any remaining freedom corresponds only to unitary rotations within the occupied subspace and does not affect the projector itself. A mathematical development and an illustrative application to the beryllium atom are presented. Since this procedure reconstructs Kohn–Sham energies from diffraction‐constrained electronic structure, it offers an experimentally anchored route for benchmarking and evaluating approximate DFT exchange–correlation functionals via their associated potentials and energy decompositions.

In a 2018 paper and a subsequent article published in 2023, researchers reported that mitochondria maintain temperatures 10°C–15°C higher than the surrounding cytoplasm—a finding that deviates by five to six orders of magnitude from theoretical predictions based on Fourier's law of heat conduction. In 2022, we proposed a solution to this apparent paradox. In the present perspective, we build upon that framework and introduce new ideas to further unravel how a biological membrane—whether of an organelle or a whole cell—can become significantly warmer than its environment. We propose that ion‐translocating proteins embedded in the inner mitochondrial membrane (IMM) can be modeled as ratchet engines, introducing a novel, previously overlooked mode of heat transfer. This mechanism, coupled with localized heat release during the cyclical dehydration‐translocation‐hydration of ions through membrane proteins, may generate transient but substantial temperature spikes. The cumulative thermal occupancy of these microscopic events across the three‐dimensional surface of the IMM can account for the elevated temperatures detected by molecular probes.