Form Of Potential Function Research Articles

(Compressed) sensing and sensibility

For decades, researchers have built computer models of molecular interactions to predict properties of new molecules (1). These models take the form of potential functions, equations that can be used predict the molecular energy of interaction. Potential functions have very broad applications. Other than ab initio quantum mechanics-based approaches (2), all simulations studies rely on some sort of model for potential functions. Thus, the ability to improve the accuracy of potential functions would have very wide sweeping impact. In PNAS, AlQuraishi and McAdams (3) propose a new approach to this challenging and important problem and demonstrate considerable gains in accuracy with their new method.

Catastrophe Theory Analysis on Asymmetric Mining Pillar Dynamic Failure

Based on the simplified analytic model of asymmetric mining rock beam-pillar system, the catastrophe model of dynamic failure of the rock beam-pillar system in asymmetric mining has been established by way of the differential form of total potential function deduced by law of conservation of energy. Analysis and illustration show it is consistent in discriminating the system stability by energy importing rate J—presenting the descending or increasing tendency. Outside or suroundings imports energy into rock beam-pillar system when J> 0; and the system exports energy to outside or surroundings when J<0.J=0 is the critical condition of system destabilization,which implicates Cook stiffness criterion and is a dynamic generalization of Cook criterion in the form of energy.

On the Educational Aspects of Potential Functions for the System Analysis and Control

This paper focuses on the education aspect of potential functions and presents the concept as a general approach for formulation of dynamic processes in different areas of engineering, e.g., dynamics of motion, electrical fields, power systems, control, and robotics. In power systems, the potential functions have been used in transient stability analysis in capacities such as 1) transient energy function and 2) the theoretical backbone of the equal-area criterion. The unified representation introduced in this paper is to help students (especially global learners by providing them with the big picture of the subject) discover similarities between different disciplines, which facilitates inter-application of the analytical methods. Recognizing the importance of digital computer based simulation techniques in education, the paper also introduces a software tool for assisting students/researchers in experimenting with different forms of potential functions and their application to a dynamic environment consisting of mobile robots, targets, and obstacles.

Rebalancing IMF Quotas

AbstractIn this paper, we address the issue of IMF quota calculation in several directions. We first discuss the relevance of various specifications regarding the choice of variables, the way to measure them and the potential functional forms. Then, relying on long‐run projections for GDP, trade and foreign direct investment, we compare the ‘old’ system of formulas to the new one adopted in 2008 at different time horizons until 2030. Two major results emerge from our analysis. First, a single chair for the Eurozone can free 2–3 percentage points of quota shares, but this amount fades over time because intra‐Eurozone trade is relatively less dynamic than world trade. Second, introducing population in quota formulas would be the most transparent and efficient way of significantly raising the quota shares for less developed countries.

Variability in fish size/otolith radius relationships among populations of Chinook salmon

Back-calculation of growth trajectories from otolith microstructure is a valuable tool for understanding mechanisms underlying variability in growth among fish populations. We analyzed fish length/otolith radius relationships for Snake River spring/summer Chinook and Snake River fall Chinook salmon, listed as separate “Evolutionarily Significant Units” (ESUs) under the US Endangered Species Act, to determine whether these ESUs shared relationships. In addition, we analyzed otoliths from seven separate populations within the Snake River spring/summer Chinook ESU to assess the variability in relationships among populations, which are much more closely related than ESUs. We also examined several potential functional forms for the equations. We found that the separate ESUs had significantly different fish length/otolith radius relationships, but that variability in otolith growth rate could not explain the difference. Relationships among populations within the spring/summer Chinook ESU did not vary nearly as much as those between ESUs. The quadratic model and the power model fit the data equally well, and constraining these models to pass through a biological intercept (estimated fish length and otolith radius at hatching) resulted in only a slight decrease in model fit. To test the ability of the models to back-calculate fish lengths, we predicted the length at tagging for 17 PIT-tagged fall Chinook that were measured at release and at recapture. The back-calculation demonstrated little bias (<1 mm FL, on average) and relatively small standard deviation (∼3.5 mm) for the best model. When we repeated the back-calculation with data from both ESUs combined, bias increased substantially (to 15 mm FL), demonstrating the importance of determining the proper taxonomic level at which to combine data within a species.

Analytic Morse/long-range potential energy surfaces and predicted infrared spectra for CO2–H2

Five-dimensional ab initio potential energy surfaces (PESs) for CO(2)-H(2) that explicitly incorporate dependence on the Q(3) asymmetric-stretch normal-mode coordinate of the CO(2) monomer and are parametrically dependent on its Q(1) symmetric-stretch coordinate have been calculated. Analytic four-dimensional PESs are obtained by least-squares fitting vibrationally averaged interaction energies for v(3)(CO(2)) = 0, and 1 to the Morse/long-range potential function form. These fits to 23,113 points have root-mean-square (rms) deviations of 0.143 and 0.136 cm(-1), and require only 167 parameters. The resulting vibrationally averaged PESs provide good representations of the experimental infrared data: for infrared transitions of para- and ortho-H(2)-CO(2), the rms discrepancies are only 0.004 and 0.005 cm(-1), respectively. The calculated infrared band origin shifts associated with the nu(3) fundamental of CO(2) are -0.179 and -0.092 cm(-1) for para-H(2)-CO(2) and ortho-H(2)-CO(2), in good agreement with the (extrapolated) experimental values of -0.198 and -0.096 cm(-1).

Theoretical evaluation of crystalline state properties of chalcogenide crystals with sodium chloride structure

Hellmann and Ali-Hasan forms of repulsive potential functions have been applied to compute the values of cohesive energy, atomization energy, force constant, IR absorption frequency, Debye temperature, Gruneisen parameter, Anderson-Gruneisen parameter and Moelwyn-Hughes parameter for 45 chalcogenide crystals of NaCl-structure. The results are reported here. Calculations are also performed for the estimation of first order volume dependence of Gruneisen parameter commonly known as second Gruneisen parameter using expressions of higher order derivatives of interaction potential within the frame-work of Dugdale and MacDonald theory. The high pressure behaviour of these crystals have also been studied.

Four-parameter analytical local model potential for atoms

Analytical local model potential for modeling the interaction in an atom reduces the computational effort in electronic structure calculations significantly. A new four-parameter analytical local model potential is proposed for atoms Li through Lr, and the values of four parameters are shell-independent and obtained by fitting the results of Xa method. At the same time, the energy eigenvalues, the radial wave functions and the total energies of electrons are obtained by solving the radial Schrödinger equation with a new form of potential function by Numerov's numerical method. The results show that our new form of potential function is suitable for high, medium and low Z atoms. A comparison among the new potential function and other analytical potential functions shows the greater flexibility and greater accuracy of the present new potential function.

Small spacecraft formation using potential functions

A group of small spacecraft able to change its orbital formation through using the potential function is discussed. Spacecraft shapes, sizes, and maneuvering capabilities in general are not identical. All objects are assumed to maneuver under discrete thruster effects. A hyperbolic form of attractive potential function is then used to reduce the control intervention by using the natural orbital motion for approaching goal configuration. A superquadric repulsive potential with 3D rigid object representation is then used to have more accurate mutual sensing between objects. As spacecraft start away from their goals, the original parabolic attractive potential becomes inefficient as the continuous control force increases with distance linearly. The hyperbolic attractive potential offers good representation of the control force independent of the distance to goal, ensuring global stability as well.

Molecular Dynamics Simulation on Mechanics of Skutterudite CoSb3 Nanowire

For the binary thermoelectric material CoSb3 with a complex crystal structure, the Morse potential functional form is employed to describe its three-dimensional atomic interactions. The mechanical responses and deformation behavior of a rectangular cross-section CoSb3 nanowire subjected to uniaxial tensile strain are simulated at constant temperature by the molecular dynamics method. The deformation is strain controlled with constant strain rate. When the strain increases, necking gradually becomes distinct near the middle of the model, and complete damage occurs at around 60% strain. The single-crystal CoSb3 nanowire exhibits properties distinct from those of single-crystal CoSb3 bulk previously studied. Comparison of the stress–strain curves and configuration evolutions of the CoSb3 nanowire and bulk during tensile loading indicate that an interesting brittle–ductile transition phenomenon occurs when the single-crystal CoSb3 varies from bulk to nanowire. Future efforts should be devoted to seeking the critical dimension at which this transition happens and the mechanism behind it.

Analytic three-dimensional ‘MLR’ potential energy surface for CO2–He, and its predicted microwave and infrared spectra

A three-dimensional, analytic potential energy surface for CO(2)-He that explicitly incorporates its dependence on the Q(3) asymmetric-stretch normal-mode coordinate of the CO(2) monomer has been obtained by least-squares fitting new ab initio interaction energies to a new three-dimensional Morse/Long-Range (3D-MLR) potential function form. This fit to 2832 points has a root-mean-square (RMS) deviation of 0.032 cm(-1) and requires only 55 parameters. The resulting pure ab initio potential provides a good representation of the experimental microwave and infrared data: for 51 pseudo microwave and 49 infrared transitions the RMS discrepancies are 0.0110 and 0.0445 cm(-1), respectively. Scaling this surface using only two morphing parameters yields an order of magnitude better agreement with experiments, with RMS discrepancies of only 0.0025 and 0.0038 cm(-1), respectively. The calculated infrared band origin shift associated with the nu(3) fundamental of CO(2) is 0.109 cm(-1), in good agreement with the (extrapolated) experimental value of 0.095 cm(-1).

A generalized Petviashvili iteration method for scalar and vector Hamiltonian equations with arbitrary form of nonlinearity

The Petviashvili’s iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power-law nonlinearity: −Mu+up=0, where M is a positive definite self-adjoint operator and p=const. In this paper, we propose a systematic generalization of this method to both scalar and vector Hamiltonian equations with arbitrary form of nonlinearity and potential functions. For scalar equations, our generalized method requires only slightly more computational effort than the original Petviashvili method.

From Atomic-Level Lattice Structure to Estimate the Silicon Mechanical Bulk Behaviour Using the Atomistic-Continuum Mechanics

This paper proposes a novel atomistic-continuum mechanics (ACM) based on the finite element method (FEM) to investigate the mechanical bulk behavior of atomic-level single crystal silicon under uniaxial tensile loading. The ACM could be reduced efficiently the computational time and maintained the simulation accuracy. A general form of Stillinger-Weber potential function was used for interaction between the silicon atoms in the ACM simulations. Simulation results shows that the Young’s modulus of single crystal silicon were 121.8, 153 and 174.6 GPa along the (100), (110) and (111) crystallographic plane, respectively. These results are in reasonable agreement with the experiment and simulation results reported in the literature.

Parametrization of interatomic potential functions using a genetic algorithm accelerated with a neural network

A genetic algorithm (GA) can be used to fit highly nonlinear functional forms, such as empirical interatomic potentials from a large ensemble of data. The performance of a GA for fitting such functional forms is enhanced through an approach that is based on the use of a neural network (NN) to accelerate the computation of the fitness function for the GA. Application of the new approach for fitting an ensemble of potentials computed from ab initio calculations to a specified functional form (here, the Tersoff potential functional form is used as an example) has shown that the computational efficiency achieved through the use of a NN can reduce computational time by over two orders of magnitude. The potentials estimated from functions thus fitted were within 0.1% of the actual potential values. Specifically, the mean squared error (MSE) on molecular potentials was $&lt;{10}^{\ensuremath{-}5}\phantom{\rule{0.3em}{0ex}}{\mathrm{eV}}^{2}$ for fitting Tersoff potentials, and $&lt;0.0025\phantom{\rule{0.3em}{0ex}}{\mathrm{eV}}^{2}$ for fitting ab initio potential energies of isolated, 5-atom silicon clusters. Furthermore, since the potential was fitted to a physically meaningful Tersoff functional form, the resulting potential function appears to have the ability to extrapolate over a reasonable range of the parameter space, and may have a better accuracy in estimating the forces compared to that obtained from neural networks, which are often highly inaccurate when extrapolated. Hence, the method can be useful for rendering various molecular dynamics (MD) simulations more tractable. It is also apparent, based on the present investigation, that a Tersoff potential, albeit with different (GA parametrized) coefficients, is adequate for representing the ab initio potentials of 5-atom Si clusters.

An accurate analytic potential function for ground-state N2 from a direct-potential-fit analysis of spectroscopic data

Two types of combined-isotopologue analysis have been performed on an extensive spectroscopic data set for ground-state N2 involving levels up to v=19, which is bound by half the well depth. Both a conventional Dunham-type analysis and a direct-potential-fit (DPF) analysis represent the data within (on average) the estimated experimental uncertainties. However, the Dunham-type parameters do not yield realistic predictions outside the range of the data used in the analysis, while the potential function obtained from the DPF treatment yields quantum mechanical accuracy over the data region and realistic predictions of the energies and properties of unobserved higher vibrational levels. Our DPF analysis also introduces a compact new analytic potential function form which incorporates the two leading inverse-power terms in the long-range potential.

Development of bond-order potentials that can reproduce the elastic constants and melting point of silicon for classical molecular dynamics simulation

The Tersoff potential is one of the most widely used interatomic potentials for silicon. However, its poor description of the elastic constants and melting point of diamond silicon is well known. In this research, three bond-order type interatomic potentials have been developed: the first one is fitted to the elastic constants by employing the Tersoff potential function form, the second one is fitted to both the elastic constants and melting point by employing the Tersoff potential function form and the third one is fitted to both the elastic constants and melting point by employing the modified Tersoff potential function form in which the angular-dependent term is improved. All of developed potentials well reproduce the elastic constants of diamond silicon as well as the cohesive energies and equilibrium bond lengths of silicon polytypes. The third potential can reproduce the melting point, while the second one cannot reproduce that. The elastic constants and melting point calculated using the third potential turned out to be C 11 = 166.4 GPa, C 12 = 65.3 GPa, C 44 = 77.1 GPa and T m = 1681 K. It was also found that only elastic constants can be reproduced using the original Tersoff potential function, and that our proposed angular-dependent term is a key to reproducing the melting point.

Quantum Mechanical Analysis of 1,2-Ethanediol Conformational Energetics and Hydrogen Bonding

A proper understanding of the conformational energetics of 1,2-ethanediol (ethylene glycol) is important to the construction of molecular mechanics force fields for the treatment of carbohydrates since these biologically important molecules have a prevalence of vicinal hydroxyl groups. In the present study, quantum mechanical analysis of the 10 unique minimum-energy conformations of ethylene glycol is performed by using 10 model chemistries ranging from HF/6-311++G(d,p) up to a hybrid method that approximates CCSD(T)/cc-pVQZ. In addition, natural bond orbital (NBO) analysis of these conformations with deletion of pairings of CO bond/antibonding and lone pair/antibonding orbitals is used to investigate contributions from the "gauche" effect to ethylene glycol conformational energetics. MP2 with the "correlation consistent" basis sets and DFT/6-311++G(d,p) do the best job of matching the approximate CCSD(T)/cc-pVQZ energies while MP2/6-31G(d) and Hartree-Fock both fare poorly. NBO analysis shows the conformational energies to be independent of the deletion of matrix elements associated with (i) CO bonding and antibonding orbital interactions and (ii) lone pair and antibonding orbital interactions, whereas the energetic ordering correlates with geometric parameters consistent with internal hydrogen bonds. Thus, the present results suggest that standard molecular mechanics potential energy functional forms, which lack explicit terms to account for stereoelectronic effects, are appropriate for carbohydrates.

Solving inverse two-point boundary value problems using collage coding

The method of collage coding, with its roots in fractal imaging, is the central tool in a recently established rigorous framework for solving inverse initial value problems for ordinary differential equations (Kunze and Vrscay 1999 Inverse Problems 15 745–70). We extend these ideas to solve the following inverse problem: given a function u(x) on [A, B] (which may be the interpolation of data points), determine a two-point boundary value problem on [A, B] which admits u(x) as a solution as closely as desired. The solution of such inverse problems may be useful in parameter estimation or determination of potential functional forms of the underlying differential equation. We discuss ways to improve results, including the development of a partitioning scheme. Several examples are considered.

A new approach to the pseudo-orthogonal properties of eigenfunction expansion form of the crack-tip complex potential function in anisotropic and piezoelectric fracture mechanics

Abstract Over the past twenty years, the well-known weight function theory based on the Bueckner work conjugate integral has been widely used to calculate crack tip fracture dominant parameter such as the stress intensity factor, the energy release rate (or the J -integral) and the T -stress in various kinds of cracked materials (e.g. isotropic materials, anisotropic materials and piezoelectric materials). Meanwhile, the pseudo-orthogonal property of the eigenfunction expansion form of the crack tip stress complex potential function has been proved to play a very important role in the theory. In this paper, we provide a new approach to establish the pseudo-orthogonal properties for crack problems in anisotropic and/or piezoelectric materials. In the latter case associated with mechanical-electric coupling, the electrical boundary conditions under both impermeable and permeable crack models are considered. The approach developed is much simpler than the classical complex variable separation technique proposed by previous researchers and hence the cumbersome and lengthy manipulations are avoided. Moreover, it is shown that, unlike previous works, the orthogonal properties of the material characteristic matrices A and B induced by the Stroh theory are no longer necessary in establishing the pseudo-orthogonal properties of eigenfunction expansion form in cracked piezoelectric materials. The approach can be easily extended to treat many other different crack problems concerning the Bueckner integral involving several complex arguments.

Constructing a small strain potential for multi-scale modeling

Simulation of bulk materials with some components locally far from equilibrium usually requires a computationally intensive quantum mechanical description to capture the relevant mechanisms (e.g. effects of chemistry). Multi-scale modeling entails a compromise whereby, the most accurate quantum description is used only where needed and the remaining bulk of the material is replaced by a simpler classical system of point particles. The problem of constructing an appropriate potential energy function for this classical system is addressed here. For problems relating to fracture, a consistent embedding of a quantum (QM) domain in its classical (CM) environment requires that the classical system should yield the same structure and elastic properties as the QM domain for states near equilibrium. It is proposed that an appropriate classical potential can be constructed using ab initio data on the equilibrium structure and weakly strained configurations calculated from the quantum description, rather than the more usual approach of fitting to a wide range of empirical data. This scheme is illustrated in detail for a model system, a silica nanorod that has the proper stiochiometric ratio of Si:O as observed in real silica. The potential energy is chosen to be pair wise additive, with the same pair potential functional form as familiar phenomenological TTAM potential. Here, the parameters are determined using a genetic algorithm with force data obtained directly from a quantum calculation. The resulting potential gives excellent agreement with properties of the reference quantum calculations both for structure (bond lengths, bond angles) and elasticity (Young's modulus). The proposed method for constructing the classical potential is carried out for two different choices for the quantum mechanical description: A transfer Hamiltonian method (NDDO with coupled-cluster parameterization) and density functional theory (with plane wave basis set and PBE exchange correlation functional). The quality of the potentials obtained in both cases is quite good, although the two quantum rods have significant differences.