Equilibrium Structure Research Articles

A Lagrangian construction of rotating star models

ABSTRACTWe present a new formulation for numerically obtaining axisymmetric equilibrium structures of rotating stars in two spatial dimensions. With a view to apply it to the secular evolution of rotating stars, we base it on the Lagrangian description, i.e. we solve the force-balance equations to find the spatial positions of fluid elements endowed individually with a mass, specific entropy and angular momentum. The system of non-linear equations obtained by finite-differencing the basic equations is solved with the W4 method, which is a new multidimensional root-finding scheme of our own devising. We augment it with a remapping scheme to avoid distortions of the Lagrangian coordinates. In this first one of a series of papers, we will give a detailed description of these methods initially. We then present the results of some test calculations, which include the construction of both rapidly rotating barotropic and baroclinic equilibrium states. We gauge their accuracies quantitatively with some diagnostic quantities as well as via comparisons with the counterparts obtained with an Eulerian code. For a demonstrative purpose, we apply the code to a toy-model cooling calculation of a rotating white dwarf.

Planarization of negatively curved [7]circulene on a graphene monolayer

[7]Circulene (C28H14) is the smallest saddle-shaped [n]circulene with negative Gaussian curvature. The electronic properties of highly curved π-systems change upon planarization, and therefore the induced planarization of these systems has attracted considerable attention in recent years. Here we use dispersion-corrected density functional theory (DFT) calculations (at the PW6B95-D4/def2-QZVP level) to show that the highly nonplanar [7]circulene adopts a completely planar equilibrium structure upon adsorption on a nanographene (modeled as C96H24). This planarization is rooted in strong dispersion and electrostatic interactions between the [7]circulene and C96H24 (est. as △Ee,int = 166.9 kJ mol−1), which are significantly stronger than the planarization energy of the free [7]circulene (est. as △Ee,planar = 40.0 kJ mol−1). Upon planarization, the magnetic nature of [7]circulene changes from nonaromatic to antiaromatic. Thus, this graphene-induced planarization provides a novel approach for a reversible nonaromatic/antiaromatic molecular switch.

Protonation of apolar species: From Cl2H+ to (E)-NCCHCHCNH+ through computational investigations

Radioastronomy is a powerful tool for the discovery of molecules in space but it requires molecular species to be polar. The observation of apolar species can be however enabled by protonation, which occurs from reaction with the abundant H3+ ion whenever the proton affinity of the species under consideration is greater than that of H2. This property can be easily investigated by computational chemistry and, in this work, it has been used to asses the potential protonation of simple homo diatomics, such as Cl2, P2, and Si2, as well as apolar species containing two equivalent CN moieties, such as diisocyanogen (CNNC) and (E)-1,2-dicyanoethene. Quantum chemistry has also been exploited to investigate the mechanisms of three protonation reactions of H3+ with Cl2, P2, and CNNC. To support laboratory measurements and astronomical observations of the resulting transient species, their rotational spectroscopic parameters were accurately computed together with fundamental vibrational frequencies. For this purpose, we have employed CCSD(T)-based computational methodologies, which provide equilibrium structures with errors smaller than 0.001 Å and 0.1° for bond distances and angles, respectively. Such an accuracy is expected to lead to rotational constants predicted, in relative terms, with uncertainties better than 0.2%. Instead, the expected accuracy on vibrational frequencies is ∼10 cm−1, thus being well suited to guide band assignments.

Reparameterization of the chemical-potential equalization model with DFTB3: A practical balance between accuracy and transferability.

To improve the performance of the third-order density-functional tight-binding method (DFTB3) for non-covalent interactions involving organic and biological molecules, a chemical-potential equalization (CPE) approach was introduced [J. Phys. Chem. A, 116, 9131 (2012)] and parameterized for the H, C, N, O, and S chemical elements [J. Chem. Phys., 143, 084123 (2015)]. Based largely on equilibrium structures, the parameterized DFTB3/CPE models were shown to exhibit improvements in molecular polarizabilities and intermolecular interactions. With more extensive analyses, however, we observe here that the available DFTB3/CPE models have two critical limitations: (1) they lead to sharply varying potential energy surfaces, thus causing numerical instability in molecular dynamics (MD) simulations, and (2) they lead to spurious interactions at short distances for some dimer complexes. These shortcomings are attributed to the employed screening functions and the overfitting of CPE parameters. In this work, we introduce a new strategy to simplify the parameterization procedure and significantly reduce free parameters down to four global (i.e., independent of element type) ones. With this strategy, two new models, DFTB3/CPE(r) and DFTB3/CPE(r†) are parameterized. The new models lead to smooth potential energy surfaces, stable MD simulations, and alleviate the spurious interactions at short distances, thus representing consistent improvements for both neutral and ionic hydrogen bonds.

Recombinant Expression and Chemical Amidation of Isotopically Labeled Native Melittin.

Post-translational modifications are ubiquitous in the eukaryotic proteome. However, these modifications are rarely incorporated in NMR studies of eukaryotic proteins, which are typically produced through recombinant expression in E. coli. Melittin is the primary peptide in honey bee venom. Its native C-terminal amide significantly affects its equilibrium structure and dynamics in solution and is thus a prerequisite for studying its native structure and function. Here, we present a method for producing triply isotopically labeled (2H, 13C, and 15N) native melittin through recombinant expression followed by chemical amidation. We then show that structural models produced with AlphaFold-Multimer are in even better agreement with experimental residual dipolar couplings than the 2.0 Å resolution X-ray crystal structure for residues G3-K23.

Kekulé’s Oscillating D3h Cyclohexatriene Structure of Benzene

AbstractKekulé postulated that neighbouring carbon atoms in benzene undergo incessant collision with each other, thereby leading to the interchange of double and single bonds, which amounts to an oscillation between two cyclohexatriene structures in dynamic equilibrium. It has been claimed that Kekulé arrived at a fully symmetric D6h structure of benzene and that the oscillation hypothesis should not be attributed to him. However, Clausius’ collision theory, which was known at the time, implies that, when the absolute temperature approaches zero, the collision frequency will tend toward zero too, i.e. collisions will stop, and a static, D3h cyclohexatriene obtains. The classical collision theory did not allow Kekulé to construct the desired D6h structure as the energy minimum. The theory of harmonic oscillators would have allowed it, but that was not attempted at Kekulé’s time.

Monte Carlo simulations of chiral and achiral nematic droplets: thermal quenches and role of the elastic constants

ABSTRACTWe used Monte Carlo simulations with an elastically anisotropic pairwise potential to simulate the equilibrium structure of chiral and achiral nematic liquid crystals in spherical confinement under perpendicular and tangential strong anchoring. The stable structures are obtained by a quenching process, where the temperature is a parameter of the system. We calculate and map the distribution of elastic energy corresponding to splay, twist, and bend. Furthermore, we study the effect of changing the ratio of the elastic constants for achiral and low chirality droplets and found a diverse scenario of possible structures. The method described here allows the introduction of temperature and elastic anisotropy naturally within its framework. It is simpler to implement and faster to execute than the Landau-de Gennes (LdG) method used for simulating liquid crystal droplets.

Constrained Form-Finding of Tension–Compression Structures using Automatic Differentiation

This paper proposes a computational approach to form-find pin-jointed bar structures subjected to combinations of tension and compression forces. The generated equilibrium states can meet additional structural and geometrical constraints via gradient-based optimization. We achieve this by extending the combinatorial equilibrium modeling (CEM) framework in three important ways. First, we introduce a new topological object, the auxiliary trail, to expand the range of structures that can be form-found with the framework. Then, we leverage automatic differentiation (AD) to obtain an exact value of the gradient of the sequential and iterative calculations of the CEM form-finding algorithm, instead of a numerical approximation. Finally, we encapsulate our research developments into an open-source design tool written in Python that is usable across different CAD platforms and operating systems. After studying four different structures –a self-stressed tensegrity, a tree canopy, a curved bridge, and a spiral staircase– we demonstrate that our approach enables the solution of constrained form-finding problems on a diverse range of structures more efficiently than in previous work. • A computational form-finding method is extended to enhance the form generation of spatial structures in static equilibrium subjected to tension and compression forces. The output forms can satisfy combinations of force and geometric constraints via gradient-based optimization. • A new topological modeling object, the auxiliary trail, is presented which simplifies the representation of a structure as a valid input to the method herein extended. • Automatic differentiation (AD) is leveraged to obtain an exact value of the gradient of a sequential and iterative form-finding algorithm with marginal human intervention. The application of AD leads to significant computational performance gains, especially when solving large constrained form-finding problems. • A standalone, open-source design tool is developed which enables the formulation and the solution of constrained form-finding problems in plain and simple Python code. The design tool is readily usable on three different operating systems and on three distinct 3D modeling software environments.

A Dynamic Proton Bond: MH+·H2O ⇌ M·H3O+ Interconversion in Loosely Coordinated Environments.

The interaction of organic molecules with oxonium cations within their solvation shell may lead to the emergence of dynamic supramolecular structures with recurrently changing host-guest chemical identity. We illustrate this phenomenon in benchmark proton-bonded complexes of water with polyether macrocyles. Despite the smaller proton affinity of water versus the ether group, water in fact retains the proton in the form of H3O+, with increasing stability as the coordination number increases. Hindrance in many-fold coordination induces dynamic reversible (ether)·H3O+ ⇌ (etherH+)·H2O interconversion. We perform infrared action ion spectroscopy over a broad spectral range to expose the vibrational signatures of the loose proton bonding in these systems. Remarkably, characteristic bands for the two limiting proton bonding configurations are observed in the experimental vibrational spectra, superimposed onto diffuse bands associated with proton delocalization. These features cannot be described by static equilibrium structures but are accurately modeled within the framework of ab initio molecular dynamics.

Macrocyclic chemosensors with anthraquinone signaling unit built into ionophore. Experimental and computational studies (part I) - synthesis and effect of proton binding on spectrophotometric and electrochemical properties

Two macrocyclic chemosensors with anthraquinone signaling unit incorporated into ionophore system (via positions 1 and 8) have been synthesized and subsequently their physicochemical properties became the subject of our extensive research. First ligand, labeled in the paper as AQ-Ncrown is characterized by a cyclic structure of a crown ether, while second one AQ-Ncrypt includes an additional ethoxy bridge, which ensures the bicyclic character of a cryptand. The studied macrocycles possess both oxygen and nitrogen heteroatoms in the ionophore cavity. Dualistic (chromophore and electrophore) signaling nature of described compounds, makes them potentially attractive molecular recognition systems. The aim of our research was to synthesize and analyze the spectroscopic, acid-base and redox properties of aforesaid macrocycles. Furthermore, we have combined experimental approach together with theoretical investigations. The equilibrium structures of AQ-Ncrown and AQ-Ncrypt were determined with the use of DFT calculations. The sensitivity of studied macrocycles towards interactions with protons was scrutinized. The complete pH-spectrophotometric characteristic of studied ligands together with their protolytic forms and corresponding pKa values were determined. The influence of medium (aprotic and protic solvent) on spectral effects was described. Furthermore, the molecular electrostatic potential maps for ligands and differential electron densities for their mono and dianions were calculated. The redox reactions was investigated at different pHs by cyclic voltammetry. Electrochemical results have presented intriguing phenomenon: the specific stabilization of the reduced form of the protonated molecules. The calculations have revealed that this is a consequence of barrierless intramolecular proton transfer (from the macrocycle cavity onto the anthraquinone moiety) that might occur during the reduction process in acidic medium.

Hydrogen physisorption on the (BeO)n, B2H4(Be,Ti), and B6Ti3 metal clusters: a computational study of energies and atomic charges.

The equilibrium structures of BeO clusters and Be,Ti-decorated boranes were computed with the ωB97X-D method and the 6-31G + (2d,2p) and aug-cc-pVTZ basis sets to study their intermolecular interactions with hydrogen molecules. Thermochemical and molecular properties such as the harmonic vibrational frequency, dipole and quadrupole moments, and atomic charges are employed to understand the attractive interactions that control the adsorption process. Comparison of molecular properties and atomic charges of the studied compounds before and after H2 molecule adsorption shows that most of the interactions among the BeO clusters and boranes with H2 molecules constitute a combination of dispersion, electrostatic, and weak charge transfer interactions. Calculated values of Hirschfeld atomic charges and ΔEe (in parenthesis) (BeO)4.8H2 (0.028 e and -2.0kcal.mol-1), (BeO)2.12H2 (0.030 e and -2.8kcal.mol-1), B6Ti3.10H2 (0.045 e and -15.4kcal.mol-1), and B6Ti3+.10H2 (0.058 e and -15.3kcal.mol-1) show qualitative correlation between hydrogen atomic charges and electronic energy of hydrogen interaction. The ωB97X-D/6-31 + G(2d,2p) values of Gibbs free energy at 298.15K for (BeO)4.8H2 B2H4Ti.4H2 and B6Ti3.10H2 clusters are equal to -5.0, -4.9, and -5.1kcal.mol-1, respectively, which are within the range of energy parameters of materials that could be employed in hydrogen storage tanks for light vehicles.

Processive Pathways to Metastability in Block Copolymer Thin Films.

Block copolymers (BCPs) self-assemble into intricate nanostructures that enhance a multitude of advanced applications in semiconductor processing, membrane science, nanopatterned coatings, nanocomposites, and battery research. Kinetics and thermodynamics of self-assembly are crucial considerations in controlling the nanostructure of BCP thin films. The equilibrium structure is governed by a molecular architecture and the chemistry of its repeat units. An enormous library of materials has been synthesized and they naturally produce a rich equilibrium phase diagram. Non-equilibrium phases could potentially broaden the structural diversity of BCPs and relax the synthetic burden of creating new molecules. Furthermore, the reliance on synthesis could be complicated by the scalability and the materials compatibility. Non-equilibrium phases in BCPs, however, are less explored, likely due to the challenges in stabilizing the metastable structures. Over the past few decades, a variety of processing techniques were introduced that influence the phase transformation of BCPs to achieve a wide range of morphologies. Nonetheless, there is a knowledge gap on how different processive pathways can induce and control the non-equilibrium phases in BCP thin films. In this review, we focus on different solvent-induced and thermally induced processive pathways, and their potential to control the non-equilibrium phases with regards to their unique aspects and advantages. Furthermore, we elucidate the limitations of these pathways and discuss the potential avenues for future investigations.

To age or not to age: Anatomy of a supercooled liquid's response to a high alternating electric field.

Physical aging and structural recovery are the processes with which the structure of a system approaches equilibrium after some perturbation. Various methods exist, that initiate structural recovery, such as changing the temperature or applying a strong, external static field. This work is concerned with high alternating electric fields and their suitability to study structural recovery and aging. The present work demonstrates that rationalizing the nonlinear dielectric response of a supercooled liquid to high-amplitude ac-fields requires multiple fictive temperatures. This feature is in stark contrast to structural recovery after a temperature down-jump or a considerable increase in the static electric field, for which a single parameter, the fictive temperature or material time, describes the structural change. In other words, the structural recovery from a high ac-field does not adhere to time aging-time superposition, which is so characteristic of genuine aging processes.

Computational Studies on the Interaction of Organophosphorus Pesticides with Acetylcholinesterase and Butyrylcholinesterase: Quantum Chemical Cluster Model and HSAB Approaches

In the present study, the interaction between organophosphorus pesticides and cholinesterase enzymes was investigated by quantum chemical cluster model and hard-soft acid-base (HSAB) approaches. The computational results of the equilibrium structure and reaction enthalpy were used to decipher the mechanism of organophosphorus pesticides coumaphos, dicrotophos, phorate, and terbufos, which interacted with the molecular cluster models of acetylcholinesterase (AChE) and butyrylcholinesterase (BChE) enzymes. In addition, the HOMO-LUMO energy gap and the HSAB descriptors prove that AChE has outstanding electron acceptability, which is suitable as a biosensing material. In terms of the calculated electronic spectrum, because the energy level of the ground state and the excited state are changed after adding pesticides with enzymes, a significant red shift phenomenon will occur.

Strain data augmentation enables machine learning of inorganic crystal geometry optimization

Machine-learning (ML) models offer the potential to rapidly evaluate the vast inorganic crystalline materials space to efficiently find materials with properties that meet the challenges of our time. Current ML models require optimized equilibrium structures to attain accurate predictions of formation energies. However, equilibrium structures are generally not known for new materials and must be obtained through computationally expensive optimization, bottlenecking ML-based material screening. A computationally efficient structure optimizer is therefore highly desirable. In this work, we present an ML model capable of predicting the crystal energy response to global strain by using available elasticity data to augment the dataset. The addition of global strains improves our model's understanding of local strains too, significantly improving the accuracy of energy predictions on distorted structures. This allows us to construct an ML-based geometry optimizer, which we used for improving the predictions of formation energy for structures with perturbed atomic positions.

Retail outsourcing strategy in Cournot & Bertrand retail competitions with economies of scale

<p style='text-indent:20px;'>This paper investigates a manufacturer's retail outsourcing strategies under different competition modes with economies of scale. We focus on the effects of market competition modes, economies of scale and competitor's behavior on manufacturer's retail outsourcing decisions, and then we develop four game models under three competition modes. Firstly, we find the channel structure where both manufacturers choose retail outsourcing cannot be an equilibrium structure under the Cournot competition. The Cournot competition mode is less profitable to the firm than the Bertrand competition when the products are complements. Secondly, under the hybrid Cournot-Bertrand competition mode, there is only one equilibrium supply chain structure where neither manufacturer chooses retail outsourcing, when the substitutability and complementarity levels are not sufficiently high. In addition, setting price (quantity) contracts as the strategic variables is the dominant strategy for the direct-sale manufacturer who provides complementary (substitutable) products. Thirdly, both competitive firms will benefit from the situation where they choose the same competition mode. When the products are substitutes (complements), both of them choose the Cournot (Bertrand) competition mode. Finally, we show that the economies of scale have little impact on the equilibrium of the outsourcing structure but a great impact on the competition mode equilibrium.</p>

An ab initio study on the electronic excited states and photodissociation mechanism of bromocarbene molecule.

We used the internally contracted explicitly correlated multireference configuration interaction (icMRCI-F12) method combined with Davidson correction to conduct a high-precision ab initio study of CHBr. The spin-orbit coupling (SOC) is incorporated into the calculation. The 21 spin-free states split into 53 spin-coupled states of CHBr. The vertical transition energies and oscillator strengths are obtained of these states. The SOC effect on the equilibrium structures and the harmonic vibrational frequencies of the ground state X1A', the lowest triplet state a3A'' and the first excited singlet state A1A'' is investigated. The results reveal a significant effect of the SOC on the bond angle and the frequency of the bending mode of a3A''. The potential energy curves of electronic states of CHBr as functions of the H-C-Br bond angle, C-H bond length, and C-Br bond length, respectively, are also investigated. Based on the calculated results, the interactions between electronic states and photodissociation mechanism involved in CHBr in the ultraviolet region are explored. Our theoretical studies will shed light on the complicated interactions and dynamics of the electronic states of bromocarbenes.

Customers' joining behavior in an unobservable <i>GI/Geo/m</i> queue

<p style='text-indent:20px;'>This paper studies the equilibrium balking strategies of impatient customers in a discrete-time multi-server renewal input queue with identical servers. Arriving customers are unaware of the number of customers in the queue before making a decision whether to join or balk the queue. We model the decision-making process as a non-cooperative symmetric game and derive the Nash equilibrium mixed strategy and optimal social strategies. The stationary system-length distributions at different observation epochs under the equilibrium structure are obtained using the roots method. Finally, some numerical examples are presented to show the effect of the information level together with system parameters on the equilibrium and social behavior of impatient customers.</p>

Rationing and market: Structure and stability of equilibria

In recent years, state control of the economy has increased in many countries. A number of states try to influence prices in key areas of economy, in particular by selling resources at fixed prices within given quotas. However, in real economies the governments cannot prevent economic agents from reselling rationed goods at the free market. The study of impact of rationing on the market prices is a difficult and challenging problem. In the present paper we consider an equilibrium model in which part of the goods within the limits of quotas is sold at fixed prices while the remaining goods are sold at market prices; the goods bought at fixed prices can also be resold at market prices. Economy depends on parameters, viz. total resources, incomes of the participants, quotas, and fixed prices. For special values of parameters, this model reduces to pure exchange and fixed income models and, in a sense, is a combination of these models. Basing on known properties of these special cases and using techniques of elementary differential topology, we study the existence of equilibria and their properties. Depending on the values of parameters, a (sufficiently general) economy may have a finite (even or odd) number of equilibria, and in an important special case when total resources are subject to rationing and the total cost of allocated quotas coincides with the total income of the participants the equilibria form a one-dimensional manifold. We consider a generalized tâtonnement process and study its convergence under certain assumptions. It is shown that in our setup convergence of tâtonnement to an equilibrium may involve endogenous inflation.

"Freedom of design" in chemical compound space: towards rational in silico design of molecules with targeted quantum-mechanical properties.

The rational design of molecules with targeted quantum-mechanical (QM) properties requires an advanced understanding of the structure-property/property-property relationships (SPR/PPR) that exist across chemical compound space (CCS). In this work, we analyze these fundamental relationships in the sector of CCS spanned by small (primarily organic) molecules using the recently developed QM7-X dataset, a systematic, extensive, and tightly converged collection of 42 QM properties corresponding to ≈4.2M equilibrium and non-equilibrium molecular structures containing up to seven heavy/non-hydrogen atoms (including C, N, O, S, and Cl). By characterizing and enumerating progressively more complex manifolds of molecular property space-the corresponding high-dimensional space defined by the properties of each molecule in this sector of CCS-our analysis reveals that one has a substantial degree of flexibility or "freedom of design" when searching for a single molecule with a desired pair of properties or a set of distinct molecules sharing an array of properties. To explore how this intrinsic flexibility manifests in the molecular design process, we used multi-objective optimization to search for molecules with simultaneously large polarizabilities and HOMO-LUMO gaps; analysis of the resulting Pareto fronts identified non-trivial paths through CCS consisting of sequential structural and/or compositional changes that yield molecules with optimal combinations of these properties.