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At least 217 records · Page 12

Nitromethane Decomposition via Automated Reaction Discovery and an Ab Initio Corrected Kinetic Model

In the explore the systematic construction of kinetic models from in silico reaction data for the decomposition of nitromethane. Our models are constructed in a computationally affordable manner by using reactions discovered through accelerated molecular dynamics simulations using the ReaxFF reactive force field. The reaction paths are then optimized to determine reaction rate parameters. We introduce a reaction barrier correction scheme that combines accurate thermochemical data from density functional theory with ReaxFF minimal energy paths. We validate our models across different thermodynamic regimes, showing predictions of gas phase CO and NO concentrations and high-pressure induction times that are similar to experimental data. The kinetic models are analyzed to find fundamental decomposition reactions in different thermodynamic regimes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Spatial Mapping of Valence Excited-State Landscapes Using Time-Resolved Shake-Down Spectroscopy

Time-resolved X-ray photoelectron spectroscopy (XPS) is used to track the photodissociation dynamics of 2-iodothiophene following 262 nm excitation. The transient XPS features include both direct ionization of the initially populated excited states and pronounced satellite peaks arising from shake-down processes. While the direct ionization signals exhibit only minimal energy shifts during C−I bond cleavage, the shake-down transitions undergo a substantial, 5 eV, shift over the reaction coordinate. By correlating these shifts with simulated C−I bond lengths, a direct structural mapping is established that reveals the exceptional sensitivity of shake-down channels to molecular geometry. These results demonstrate that shake-down transitions provide a new and powerful probe of ultrafast structural dynamics.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Mechanistic Insights into Processive Polyethylene Hydrogenolysis through In Situ NMR

Chemical polymer upcycling by processive catalysts is a promising plastic waste remediation strategy, with the capability of producing selective, high-value products from waste plastics with minimal energy input. We previously designed a novel processive catalyst with a mesoporous SiO 2 shell/Pt nanoparticle/SiO 2 core architecture (mSiO 2 /Pt/SiO 2 ) that deconstructs polyolefins within narrow pores. Here, in this paper, we elucidate the mechanism of processive polyolefin hydrogenolysis using in situ magic-angle spinning (MAS) nuclear magnetic resonance (NMR) spectroscopy and coarse-grained molecular dynamics simulations. We observe that most polyethylene–Pt interactions do not lead to C–C bond cleavage but rather to the release of the polymer via a dehydrogenation–rehydrogenation cycle. The porous architecture increases the likelihood that a released polymer is later cleaved and enables the catalyst to perform multiple successive cleavages to the same polymer chain. Both experiment and simulation show that the extent of processivity is strongly correlated with the length of the pores, with longer pores leading to a higher processivity.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Oxidation Dynamics of Supported Catalytic Cu Clusters: Coupling to Fluxionality

Copper oxide nanoclusters have a wide range of catalytic applications, such as the selective oxidation of hydrocarbons. O 2 binding to the catalyst, activation, and release upon reagent oxidation are key events in these catalytic chemistries. Furthermore, these events are expected to be accompanied by significant structural changes of the Cu clusters, because O atoms integrate into the cluster, rather than bind to its surface. Topping the complexity of the problem, partially oxidized Cu clusters are known to exhibit strong fluxionality and feature diverse and interconverting structures and oxygen contents in conditions of oxidative dehydrogenation (ODH). Hence, a significant dynamic coupling between the “hot” O 2 molecule impacting the cluster at reaction temperatures and the cluster fluxionality can be expected. In this work, we focus on the dynamics of dioxygen integration into a partially oxidized Cu cluster supported on hydroxylated amorphous alumina–a system recently reported to be an exceptionally selective catalyst for cyclohexane ODH with very little CO produced, whose mechanistic underpinnings are of utmost interest. The statistics over a swarm of adsorption and scattering trajectories where O 2 hits various sites on the cluster at reaction temperature shows that the O 2 binding does not only follow the minimal energy paths. O 2 also rarely integrates into the cluster in a single step and instead first binds to a single Cu atom via either an η 1 -O 2 or an η 2 -O 2 mode. Surprisingly, this step often has a higher barrier than the subsequent O 2 integration and dissociation, which in turn take multiple steps and complete the oxidation process. Dynamic trajectories starting from the key transition state of integration of the adsorbed O 2 can also lead to different intermediate structures during or right after the dissociation, due to the energy released from the transition state and the thermal intracluster effects. From these activated O 2 chemisorbed structures, O 2 dissociation occurs with moderate barriers (~0.5 eV), producing multiple final oxidized Cu 4 O 4 states. Hence, a diversity of reaction profiles for the attack of supported Cu cluster by O 2 emerges due to the dynamic effects, with implications for mechanisms, kinetic models, and catalyst design principles.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NH 3 -Mediated Reactive Capture and Conversion: Integrating CO 2 Absorption from Flue Gas with CO Production via NH 4 HCO 3 Electrolysis

Efficient carbon capture and utilization require strategies that minimize energy penalties of CO 2 regeneration and compression. Reactive capture and conversion (RCC) address this challenge by integrating capture with direct electrochemical conversion. Here, we show an NH3-mediated tandem RCC system that couples capture of CO 2 from simulated flue gas (10% v/v CO 2 in N 2 ) with electroreduction of NH 4 HCO 3 to CO over a Ni single-atom catalyst (Ni-SAC). Speciation modeling and capture experiments revealed that a deep CO 2 capture with C/N ratio of 0.65 was achieved using 2.5 M NH 3 from simulated flue gas. Electrolysis of the resulting NH 4 HCO 3 on the Ni- SAC delivered an 85% CO Faradaic efficiency at 100 mA/cm 2 with excellent tolerance to NH 3 /NH 4 + as confirmed by DFT calculations and ab initio molecular dynamics (AIMD) simulations. Further, the technoeconomic analysis established a levelized total cost of CO manufacturing of $25.43/kmol, gauging the practical viability. Overall, this study holds great potential to decarbonize the chemical manufacturing industry while reducing synthetic production costs.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

A Nitrogen Out-of-Plane (NOOP) Mechanism for Imine-Based Light-Driven Molecular Motors

Light-driven molecular motors have generated considerable interest due to their potential applications in material and biological systems. Recently, Greb and Lehn reported a new class of molecular motors, chiral N-alkyl imines, which undergo unidirectional rotation induced by light and heat. The mechanism of unidirectional motion in molecular motors containing a C═N group has been assumed to consist of photoinduced torsion about the double bond. In this work, we present a computational study of the photoisomerization dynamics of a chiral N-alkyl imine motor. We find that the location and energetics of minimal energy conical intersections (MECIs) alone are insufficient to understand the mechanism of the motor. Furthermore, a key part of the mechanism consists of out-of-plane distortions of the N atom (followed by isomerization about the double bond). Dynamic effects and out-of-plane distortions are critical to understand the observed (rather low) quantum yield for photoisomerization. Finally, our results provide hints as to how the photoisomerization quantum yield might be increased, improving the efficiency of this class of molecular motors.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Thermodynamic and Kinetic Activity Descriptors for the Catalytic Hydrogenation of Ketones

Activity descriptors are a powerful tool for the design of catalysts than can efficiently utilize H 2 with minimal energy losses. In this study, we develop the use of hydricity and H - self-exchange rates as thermodynamic and kinetic descriptors for the hydrogenation of ketones by molecular catalysts. Two complexes with known hydricity, HRh(dmpe) 2 and HCo(dmpe) 2 , were investigated for the catalytic hydrogenation of ketones under mild conditions (1.5 atm, 25 °C). The rhodium catalyst proved to be an efficient catalyst for a wide range of ketones, whereas the cobalt catalyst could only hydrogenate electron-deficient ketones. Using a combination of experiment and electronic structure theory, thermodynamic hydricity values were established for 46 alkoxide/ketone pairs in both MeCN and THF solvent. Through comparison of the hydricities of the catalysts and substrates, it was determined that catalysis was only observed for catalyst/ketone pairs with an exergonic H - transfer step. Mechanistic studies revealed that H - transfer was rate-limiting step for catalysis, allowing for the experimental and computation construction of linear free-energy relationships (LFERs) for H - transfer. Further analysis revealed the LFERs could be reproduced using Marcus theory, in which the H - self-exchange rates for the HRh/Rh + and ketone/alkoxide pairs were used to predict the experimentally measured catalytic barriers within 2 kcal mol -1 . Finally, these studies significantly expand the scope of catalytic reactions that can be analyzed with a thermodynamic hydricity descriptor and firmly establish Marcus theory as a valid approach to develop kinetic descriptors for designing catalysts for H - transfer reactions.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Solvent Polarity Independent Symmetry-Breaking Charge Separation in a Slip-Stacked Covalent Terrylene Monoimide Dimer

The design of organic materials capable of efficient photoinduced charge separation in low-polarity environments is a critical challenge for advancing organic photovoltaics. Symmetry-breaking charge separation (SB-CS) offers a promising route to efficient charge separation with minimal energy loss; however, in conventional organic materials this process typically relies on polar solvents to stabilize the charge-separated state. Here, we investigate a slip-stacked terrylene monoimide dimer (TMI 2 ) engineered with intrinsic electronic asymmetry to circumvent this limitation. We demonstrate through a comprehensive suite of ultrafast spectroscopic techniques that TMI 2 undergoes remarkably efficient SB-CS regardless of solvent polarity enabled by the permanent dipole moment of TMI and the accessible intramolecular charge-transfer (ICT) character of its monomer units. Furthermore, analysis of vibronic coherences reveals that the initial state mixing is actively driven by a 193 cm –1 intermolecular mode, while the final SB-CS state is marked by a distinct vibrational fingerprint of the radical ion pair product. Our findings reveal how the intrinsic electronic asymmetry of the TMI monomers that constitute TMI 2 creates an efficient charge separation pathway that precludes the need for external stabilization from a polar solvent. Here, this work establishes a powerful molecular design principle for leveraging intrinsic monomeric asymmetry to achieve efficient charge separation in low-dielectric environments, with significant implications for the development of next-generation organic optoelectronic materials.

Aromatic compounds↗

Advanced Newton methods for geodynamical models of Stokes flow with viscoplastic rheologies

Strain localization and resulting plasticity and failure play an important role in the evolution of the lithosphere. These phenomena are commonly modeled by Stokes flows with viscoplastic rheologies. The nonlinearities of these rheologies make the numerical solution of the resulting systems challenging, and iterative methods often converge slowly or not at all. Yet accurate solutions are critical for representing the physics. Moreover, for some rheology laws, aspects of solvability are still unknown. We study a basic but representative viscoplastic rheology law. The law involves a yield stress that is independent of the dynamic pressure, referred to as von Mises yield criterion. Two commonly used variants, perfect/ideal and composite viscoplasticity, are compared. We derive both variants from energy minimization principles, and we use this perspective to argue when solutions are unique. We propose a new stress-velocity Newton solution algorithm that treats the stress as an independent variable during the Newton linearization but requires solution only of Stokes systems that are of the usual velocity-pressure form. To study different solution algorithms, we implement 2-D and 3-D finite element discretizations, and we generate Stokes problems with up to 7 orders of magnitude viscosity contrasts, in which compression or tension results in significant nonlinear localization effects. Comparing the performance of the proposed Newton method with the standard Newton method and the Picard fixed-point method, we observe a significant reduction in the number of iterations and improved stability with respect to problem nonlinearity, mesh refinement, and the polynomial order of the discretization.

Rudi, Johann↗

Nontrivial nanostructure, stress relaxation mechanisms, and crystallography for pressure-induced Si-I → Si-II phase transformation

Crystallographic theory based on energy minimization suggests austenite-twinned martensite interfaces with specific orientation, which are confirmed experimentally for various materials. Pressure-induced phase transformation (PT) from semiconducting Si-I to metallic Si-II, due to very large and anisotropic transformation strain, may challenge this theory. Here, unexpected nanostructure evolution during Si-I → Si-II PT is revealed by combining molecular dynamics (MD), crystallographic theory, generalized for strained crystals, and in situ real-time Laue X-ray diffraction (XRD). Twinned Si-II, consisting of two martensitic variants, and unexpected nanobands, consisting of alternating strongly deformed and rotated residual Si-I and third variant of Si-II, form {111} interface with Si-I and produce almost self-accommodated nanostructure despite the large transformation volumetric strain of -0.237. The interfacial bands arrest the {111} interfaces, leading to repeating nucleation-growth-arrest process and to growth by propagating {110} interface, which (as well as {111} interface) do not appear in traditional crystallographic theory.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

On the nonadiabatic collisional quenching of OH(A) by H 2 : a four coupled quasi-diabatic state description

A four-state diabatic potential energy matrix (DPEM), H d , for the description of the nonadiabatic quenching of OH(A 2 Σ + ) by collisions with H 2 is reported. The DPEM is constructed as a fit to adiabatic energies, energy gradients, and derivative couplings obtained exclusively from multireference configuration interaction wave functions. A four-adiabatic-electronic-state representation is used in order to describe all energetically accessible regions of the nuclear coordinate space. Partial permutation-inversion symmetry is incorporated into the representation. Here, the fit is based on electronic structure data at 42 882 points, described by over 1.6 million least squares equations with a root mean square (mean unsigned) error of 178(83) cm –1 . Comparison of ab initio and H d determined minima, saddle points, and energy minimized points on C 2v , C s , C ∞v , and C 1 (noncoplanar) portions of two conical intersection seams are used to establish the accuracy of the H d .

74 ATOMIC AND MOLECULAR PHYSICS↗

Experimentally informed structure optimization of amorphous TiO 2 films grown by atomic layer deposition

Amorphous titanium dioxide TiO 2 (a-TiO 2 ) has been widely studied, particularly as a protective coating layer on semiconductors to prevent corrosion and promote electron–hole conduction in photoelectrochemical reactions. The stability and longevity of a-TiO 2 is strongly affected by the thickness and structural heterogeneity, implying that understanding the structure properties of a-TiO 2 is crucial for improving the performance. This study characterized the structural and electronic properties of a-TiO 2 thin films (~17 nm) grown on Si by atomic layer deposition (ALD). Fluctuation spectra V(k) and angular correlation functions were determined with 4-dimensional scanning transmission electron microscopy (4D-STEM), which revealed the distinctive medium-range ordering in the a-TiO 2 film. A realistic atomic model of a-TiO 2 was established guided by the medium-range ordering and the previously reported short-range ordering of a-TiO 2 film, as well as the interatomic potential. The structure was optimized by the StructOpt code using a genetic algorithm that simultaneously minimizes energy and maximizes the match to experimental short- and medium-range ordering. The StructOpt a-TiO 2 model presents improved agreements with the medium-range ordering and the k-space location of the dominant 2-fold angular correlations compared with a traditional melt-quenched model. The electronic structure of the StructOpt a-TiO 2 model was studied by ab initio calculations and compared to the crystalline phases and experimental results. Finally, this work uncovered the medium-range ordering in a-TiO 2 thin films and provided a realistic a-TiO 2 structure model for further investigation of structure–property relationships and materials design. In addition, the improved multi-objective optimization package StructOpt was provided for structure determination of complex materials guided by experiments and simulations.

36 MATERIALS SCIENCE↗

Deciphering the morphology of transition metal carbonate cathode precursors

The performance and life of Li-ion battery cathode materials are determined by both the composition (crystal structure and transition metal ratio) and the morphology (particle size, size distribution, and surface area). Careful control of these two aspects is the key to long lasting, high-energy batteries that can undergo fast charge. Developing such cathodes requires manipulation of the synthesis conditions, namely the coprecipitation process to develop the precursor and a calcination step to lithiate and convert it to a transition metal oxide. In this paper, we utilize a combination of controlled synthesis, microscopic and spectroscopic characterization, and multi-scale mathematical modeling to shed light on the synthesis of cathode precursors. The complex interplay between the various chemical reactions in the co-precipitation process is studied to provide experimentalists with guidance on achieving composition control during synthesis. Further, the formation of a variety of morphologies of the primary particles and the driving force for agglomeration is mathematically described, for the first time, based on an energy minimization approach. Results suggest that the presence of Ni and/or Co significantly lowers the reaction rate constant compared to Mn, resulting in agglomerated growth in the former and single crystal growth in the latter. Modeling studies are used to provide a phase map describing the synthesis conditions needed to control the secondary particle size and corresponding size distribution. Finally, this paper represents an important step in developing a computationally guided approach to the synthesis of battery cathode materials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Photoinitiated thermoset polymerization through controlled release of metathesis catalysts encapsulated in poly(phthalaldehyde)

Photoinitiated polymerization enables spatiotemporal control of reaction conditions and can thereby generate materials with high complexity while consuming minimal energy. Where ring opening metathesis polymerization (ROMP) is concerned, photo-activated processes are typically enabled by chemical inhibition of ruthenium carbenes via the careful design of complexed ligands such that photoactivation can proceed through an isomerization or ligand dissociation event. In this contribution, we have explored a new approach to photoinitiation of ROMP based on physical inhibition through microencapsulation and controlled release of metathesis catalysts. Micron-sized particles of poly(phthalaldehyde) (PPA), catalyst, and photoacid generator were fabricated by spray drying. The particles were dispersed in dicyclopentadiene monomer, after which polymerization was initiated through temperature or UV exposure, both inducing depolymerization of the PPA particles and in situ catalyst release. The monomer/particle dispersions were found to be stable and reproducibly polymerizable with 3 weeks of storage at room temperature. Furthermore, the dispersions can be used for both photo- and thermal-initiated frontal ROMP, yielding a polymerized thermoset of equivalent properties to conventional bulk- and frontally-polymerized analogues. In conclusion, this work will ultimately enable new manufacturing techniques for ROMP-based materials, due to the modular, easily tunable nature of the underlying initiating system and its unparalleled stability.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

TurboRVB: A many-body toolkit for ab initio electronic simulations by quantum Monte Carlo

TurboRVB is a computational package for ab initio Quantum Monte Carlo (QMC) simulations of both molecular and bulk electronic systems. The code implements two types of well established QMC algorithms: Variational Monte Carlo (VMC) and diffusion Monte Carlo in its robust and efficient lattice regularized variant. A key feature of the code is the possibility of using strongly correlated many-body wave functions (WFs), capable of describing several materials with very high accuracy, even when standard mean-field approaches [e.g., density functional theory (DFT)] fail. The electronic WF is obtained by applying a Jastrow factor, which takes into account dynamical correlations, to the most general mean-field ground state, written either as an antisymmetrized geminal power with spin-singlet pairing or as a Pfaffian, including both singlet and triplet correlations. This WF can be viewed as an efficient implementation of the so-called resonating valence bond (RVB) Ansatz, first proposed by Pauling and Anderson in quantum chemistry [L. Pauling, The Nature of the Chemical Bond (Cornell University Press, 1960)] and condensed matter physics [P.W. Anderson, Mat. Res. Bull 8, 153 (1973)], respectively. The RVB Ansatz implemented in TurboRVB has a large variational freedom, including the Jastrow correlated Slater determinant as its simplest, but nontrivial case. Moreover, it has the remarkable advantage of remaining with an affordable computational cost, proportional to the one spent for the evaluation of a single Slater determinant. Therefore, its application to large systems is computationally feasible. The WF is expanded in a localized basis set. Several basis set functions are implemented, such as Gaussian, Slater, and mixed types, with no restriction on the choice of their contraction. The code implements the adjoint algorithmic differentiation that enables a very efficient evaluation of energy derivatives, comprising the ionic forces. Thus, one can perform structural optimizations and molecular dynamics in the canonical NVT ensemble at the VMC level. For the electronic part, a full WF optimization (Jastrow and antisymmetric parts together) is made possible, thanks to state-of-the-art stochastic algorithms for energy minimization. In the optimization procedure, the first guess can be obtained at the mean-field level by a built-in DFT driver. The code was efficiently parallelized by using a hybrid MPI-OpenMP protocol, which is also an ideal environment for exploiting the computational power of modern Graphics Processing Unit accelerators.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Effects of electron beam irradiation on CrMnV and CrMnTiV high entropy alloys: Nano-mechanical, structural, and thermodynamic perspectives

Beam exit windows are crucial components of any particle accelerator as they provide an interface between the beamline vacuum and target material at atmospheric media. For high beam power machines, special materials and designs are required to withstand high radiation and mechanical loads, while minimizing energy loss during transition and maximizing window lifetime. This research investigates the impact of electron beam exposure to bulk CrMnV and CrMnTiV high entropy alloys (HEAs) with the primary goal of identifying suitable candidate materials for the design of robust and durable exit window settings. The selection criteria include intrinsic characteristics, power dissipation, and mechanical responses. According to the thermodynamic calculations, both equiatomic CrMnV and the addition of 7% of Ti with equiatomic CrMnV yield solid-solutions phases. The structural and mechanical properties of CrMnV and CrMnTiV samples were tested using field emission scanning electron microscopy, atomic force microscopy, scanning electron microcopy with energy dispersive x-ray spectroscopy, x-ray diffraction, and nanoindentation before and after exposure to a dose of ~66 kGy from a 10 MeV e-beam accelerator. Despite exhibiting beam transmission characteristics comparable to Cr and V, the indentation hardness of HEAs exceeded that of the Cr and V samples by five to six times. The examination of the CrMnTiV irradiated samples revealed organized deformation patterns depicting new features, which we suspect twinning and twin boundaries due to the addition of Ti to CrMnV. Ti, a hexagonal-close-packed crystal structure, is commonly known for deformation twinning behavior.

36 MATERIALS SCIENCE↗

Injection locking and coupling dynamics in superconducting nanowire-based cryogenic oscillators

Oscillators designed to function at cryogenic temperatures play a critical role in superconducting electronics and quantum computing by providing stable, low-noise signals with minimal energy loss. Here, in this work, we present a comprehensive numerical study of injection locking and mutual coupling dynamics in superconducting nanowire (ScNW)-based cryogenic oscillators. Using the design space of a standalone ScNW-based oscillator, we investigate two critical mechanisms that govern frequency synchronization and signal coordination in cryogenic computing architectures: (1) injection locking induced by an external AC signal with a frequency near the oscillator's natural frequency, and (2) the mutual coupling dynamics between two ScNW oscillators under varying coupling strengths. We identify key design parameters—such as shunt resistance, nanowire inductance, and coupling strength—that govern the locking range. Additionally, we examine how the amplitude of the injected signal affects the amplitude of the locked oscillation, offering valuable insights for power-aware oscillator synchronization. Furthermore, we analyze mutual synchronization between coupled ScNW oscillators using capacitive and resistive coupling elements. Our results reveal that the phase difference between oscillators can be controlled by tuning the coupling strength, enabling programmable phase-encoded information processing. These findings could enable building ScNW-based oscillatory neural networks, synchronized cryogenic logic blocks, and on-chip cryogenic resonator arrays.

Artificial neural networks↗

Inherent-state melting and the onset of glassy dynamics in two-dimensional supercooled liquids

Below the onset temperature T o , the equilibrium relaxation time of most glass-forming liquids exhibits glassy dynamics characterized by a super-Arrhenius temperature dependence. In this supercooled regime, the relaxation dynamics also proceeds through localized elastic excitations corresponding to hopping events between inherent states, i.e., potential-energy-minimizing configurations of the liquid. Despite its importance in distinguishing the supercooled regime from the high-temperature regime, the microscopic origin of T o is not yet known. Here, we construct a theory for the onset temperature in two dimensions and find that an inherent-state melting transition, described by the binding–unbinding transition of dipolar elastic excitations, delineates the supercooled regime from the high-temperature regime. The corresponding melting transition temperature is in good agreement with the onset temperature found in various two-dimensional (2D) atomistic models of glass formers and an experimental binary colloidal system confined to a water–air interface. Additionally, we find the predictions for the renormalized elastic moduli to agree with the experimentally observed values for the latter 2D colloidal system. We further discuss the predictions of our theory on the displacement and density correlations at supercooled conditions, which are consistent with observations of the Mermin–Wagner fluctuations in experiments and molecular simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗