Exponential Repulsive Potential Research Articles

A comparison of interatomic potentials for noble gases

Prediction of transport properties of noble gases requires the calculation of collision integrals, which depend on interatomic potentials as the input. However the accuracy of transport properties depends largely on the accuracy of interaction potentials. So different interatomic potentials of noble gases are compared in order to get the accurate transport properties. The forms and characteristics of Lennard-Jones, exponential repulsive, Hartree-Fock-Dispersion-B (HFD-B), and phenomenological model potentials that are used to describe the atomic interactions between noble gases are analyzed in this paper. Then the calculation method of transport properties is presented. Viscosities and thermal conductivities of noble gases based on these four potentials are obtained using Chapman-Enskog method in the temperature range for computation from 300 to 5000 K. It can be seen from the results that the interaction potentials have a great influence on the calculated results of transport properties. There are great differences between the results obtained using different interaction potentials. These differences of the calculated results can be explained according to the performance of interaction potentials. Results calculated with Lennard-Jones potential are always much lower in the high temperature range due to its overestimated repulsive part, and the exponential repulsive potential gives unreasonable results at low temperatures because there is no attractive well in this potential. Therefore, the accurate interatomic potentials for noble gases can be obtained only by comparing the calculated results with published experimental and theoretical data of other researchers. It can be found that the results obtained by HFD-B potential agree well with previously experimental and theoretical data. So it is apparent that the HFD-B potential in light of Hartree-Fock repulsion and dispersion theory can provide a realistic description of the trends and features of interatomic potentials, allowing accurate theoretical calculations to be made for transport properties of noble gases.

An algebraic description of anharmonic diatom–diatom inelastic collisions in the semiclassical approximation

An algebraic model to describe inelastic collisions between two anharmonic diatomic molecules in the semiclassical approximation is presented. The interactions for the diatomic systems are modelled in terms of Morse potentials, while an exponential repulsive potential is taken for the interactions between the nearest atoms of the diatomic systems. This problem is treated in the interaction potential framework, where an approximation in terms of the generators of three SU(2) groups is proposed, two corresponding to the Morse oscillators and the other to the interaction. The transition probabilities are given in terms of a sum of the products of three Wigner's d(β) functions corresponding to the three SU(2) groups. As an example the systems N2 + N2 and H2 + H2 are described and compared with exact quantum mechanical calculations.

The Transport Properties of Sodium Atoms and the Heat Capacity of Sodium Dimers at High Temperatures

Including the contribution of excited state atoms can improve calculations of dilute gaseous transport properties at high temperatures. For sodium, experimental and/or theoretical information is available about the potential energy curves associated with each of ten low-lying states of the sodium dimer. These include the \({{\rm X}^{1}\Sigma_{\rm g}{}^{+}}\) and \({^{3}\Sigma_{\rm u}{}{}^{+}}\) states that dissociate to two ground state 2S sodium atoms and the four \({^{3}\Sigma_{\rm g,u}{}^{+},\,^{1}\Sigma _{\rm g,u}{}^{+},\,^{1}\Pi _{\rm g,u},\,^{3}\Pi _{\rm g,u}}\) gerade/ungerade pairs of states that dissociate to a ground state 2S atom and an excited state 2P atom. Nine of these are bound states and have been fitted with the Hulburt–Hirschfelder potential, a very good general purpose atom–atom potential. The 3Πg state is not bound and has been fitted with the exponential repulsive potential. We have used these potentials to calculate viscosity collision integrals as a function of temperature, and employed degeneracy-weighted averaging to determine the viscosity and translational contribution to the thermal conductivity of the sodium atoms. These same potentials have been used to calculate the heat capacity, \({C_{p}^{\rm o}}\), of the sodium dimer using an approach that depends on the second virial coefficient and its first two temperature derivatives. Again, the inclusion of molecular states that dissociate to an excited state atom allows \({C_{p}^{\rm o}}\) to be determined with improved accuracy at higher temperatures. Thus, thermophysical property calculations for sodium have been extended to 25,000 K. These results are compared with previous results, including heat capacities given in the NIST-JANAF Thermochemical Tables.

Linear collision of a classical harmonic oscillator with a mass point in the high-frequency region

Collinear collisions, with exponential repulsive potential, between a mass point and a harmonic oscillator are studied in the limit of high frequency. Multiple encounters during single collisions and an arbitrary value of the mass parameter $m$ are permitted. Methods are analytical and numerical. The main results are the following: For an initially resting oscillator the energy $E$ after the collision decays exponentially with frequency. The limiting decay constant for high frequency is independent of $m$ and identical to the golden rule prediction $2\ensuremath{\pi}∕\ensuremath{\lambda}$ where ${\ensuremath{\lambda}}^{\ensuremath{-}1}$ is a dimensionless frequency. In the high-frequency domain the energy of the oscillator has the form $E=2{\ensuremath{\pi}}^{2}\ensuremath{\phi}{(m)}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}{\ensuremath{-}2\ensuremath{\pi}∕\ensuremath{\lambda}}$ where $\ensuremath{\phi}(m)=2m\ensuremath{-}\frac{2}{3}(1+\mathrm{ln}\phantom{\rule{0.2em}{0ex}}4){m}^{2}+\ensuremath{\cdots}$. The first term is the golden rule result, and the second term is new. A series of polynomials ${\ensuremath{\phi}}_{n}$ is constructed that converges to $\ensuremath{\phi}(m)$. It is suggested that $\ensuremath{\phi}(m)$ is an entire function of the complex variable $m$ which has the form of an infinite product $\ensuremath{\phi}(m)=2m\ensuremath{\prod}(1\ensuremath{-}m∕{m}_{n})$ where the ${m}_{n}$ are the positive roots of $\ensuremath{\phi}(m)$. The first 15 roots have been determined numerically which suggests an asymptotic behavior $\sqrt{{m}_{n}}\ensuremath{\sim}an+b$. Agreement with the numerical data is excellent.

The Possibility of Using Repulsive Interaction Potentials for the Calculation of High-Temperature Integrals of Collisions between Atoms and Molecules

The possibility is demonstrated of using exponential repulsive interaction potentials for the calculation of high-temperature integrals of collisions between atoms and molecules. An analytical formula is suggested, and the values of employed parameters are determined. The difference between the calculation results for complex interaction potentials with due regard for repulsion, potential well, and long-range attraction and the calculation results for the repulsive exponential interaction potential in the entire range of impact parameters is investigated. The dependence of the calculation accuracy on the choice of the region of approximation of the model potential by the exponential one is noted.

A re-evaluation of the means used to calculate transport properties of reacting flows

Simulations of laminar combustion and other reactive flow processes (like chemical vapor deposition, plasma etching, etc.) are presently carried out in most cases using the transport code TRANFIT attached to the CHEMKIN package. The approach used is based on experimental data from 1975 and is now outdated, especially in view of recent work presented in the literature. The new approach described here seeks to remove the deficiencies of former transport models by using the following features: (1) representation of transport data of light species at high temperature by switching to an exponential repulsive potential, (2) use of effective potential parameters to handle the intermolecular forces in an easy and elegant way, if polar molecules are considered, and (3) use of a simplified formula for binary thermal diffusion factors, based on an expansion for large values of the mass ratio of the species included. This paper presents the new transport model in terms of a complete set of equations. The molecular parameters provided allow a complete treatment of the oxidation of H2 and H2/CO mixtures (data for species taking place in the oxidation of hydrocarbons and in other reaction systems are not yet available). To demonstrate the consequences of the new transport model for combustion processes, results have been generated by implementing the model in a code for the simulation of premixed laminar flames.

Critical analysis of viscosity data of thermal argon plasmas at atmospheric pressure

Reliable values of the viscosity in thermal argon plasmas are most important for our understanding of the momentum transfer and for realistic modeling of various plasma applications. Despite numerous attempts to determine reliable viscosity values over the last three decades, discrepancies still exist among the data reported by different authors. In this paper, a critical analysis is undertaken of calculated and experimental data of the argon viscosity based on recent publications. Our recalculation of viscosities in thermal argon plasmas are performed by using Lennard-Jones, Morse, Aziz, and exponential repulsive potentials for Ar-Ar atom interactions in different temperature ranges from 300 to 20,000 K. The contributions of elastic collisions of e-Ar, e-Ar+, and Ar+-Ar, as well as charge exchange of Ar+-Ar, to the viscosity become important with increasing temperature and degree of ionization in argon plasmas. Based on a critical analysis and recalculations, improved values of the argon viscosity are recommended, covering temperatures from 300 to 20,000 K. Polynomial expressions have been developed for calculating argon viscosities, which will be useful for numerical work and other applications of thermal argon plasmas at atmospheric pressure.

Formulae for the first and second derivatives of anisotropie potentials with respect to geometrical parameters

We have derived formulae for the forces and torques arising from the electrostatic energy of a system of rigid molecules, using a rigorous unified approach. We have also obtained systematically the second derivatives with respect to transla-tional and rotational coordinates. Some of the properties of first and second derivatives are discussed for the general case of a system with arbitrary symmetry. Derivatives of a general anisotropic exponential repulsive potential have also been evaluated. All of these have been implemented as Fortran subroutines in the Orient program.

The continuum solid and compliance functions in gas-surface low energy collisions

The paper presents a model and calculations for the scattering of atomic and molecular beams from solid surfaces at low energies. The formulation permits the study of momentum and energy exchange phenomena including multiple collisions and the capture, i.e. adsorption of the projectile. The model uses a hybrid continuum-discrete representation for the system. An elastic continuum is used for the representation of the generation of phonons in the solid through collisions and the resulting momentum and energy exchange processes. On the other hand, the particulate nature of the lattice, as manifested in the corrugation is retained. This hybrid continuum-discrete representation of the solid is a suitable description for relaxation dynamics of adsorbates on surfaces and low energy collisions of gas particles with surfaces. For high energy collisions, the collision time is too short for the phonons created at the target to reach other lattice points. Therefore, the representation of the target solid by a relatively small cluster is an adequate model and there is no need to include the collaborative response of the lattice as a whole. For the low energy collisions on the other hand, the collision time is long enough such that the phonons have sufficient time to propagate over a substantial region in the lattice. Consequently, the collaborative response of the solid as a collection of a large number of lattice particles is essential. The present model uses a continuum model for the response of the solid as an accurate and convenient representation. The continuum hypothesis is validated by the predominance of long or equivalently, low frequency waves among the phonons generated during the collision. The formulation is presented for an atomic particle as projectile. Possible extensions to cover projectiles like molecules or clusters are briefly indicated. The phonons generated by the projectile are described in terms of compliance functions. These are basically Green's functions for the response of a semi-infinite solid to forces acting on its boundary. In physical terms, the compliance functions can be viewed as a frequency dependent effective spring with damping coefficients for the motion of the target point embedded in the solid. This makes the handling of the solid extremely practical by reducing the representation of its collaborative response to that of one point. The full set of compliance coefficients for all possible surface forces and moments are available. Within this picture, the discreteness of the lattice enters through the corrugated gas particle-surface interaction potential. As applications, both exact numerical integration of the equations as well as perturbation calculations are presented within a classical framework, although the formulation and calculations presented here are all within a classical context. Specific results are given for the projectile trajectories as well as momentum and energy exchange. The numerical, i.e. exact, trajectory calculations show the capability of the model in allowing realistic simulations for phenomena that involve rather complicated dynamics, including multiple collisions, skidding along the surface and eventual capture of the projectile. The perturbation solutions on the other hand, while limited to single collisions, provide attractive analytical expressions that capture basic features of the physics in the energy exchange through collision. Particularly, the availability of thermal averages through simple quadratures is a valuable asset considering the gigantic computational tasks involved in exact simulations. The quadratures involved can be evaluated in closed form for both the exponential repulsive and Morse potentials. The specific results presented use the parameters of the He+LiF system as well as models with deeper wells and softer solids.

Sum rule methods for the approximation of continuous electronic absorption spectra

The sum rules that apply to electronic transitions in molecules are used to find expressions for the first four moments of an absorption spectrum. While no simple general form for the fourth moment is found, an equation that can be evaluated for a number of useful special cases is derived. Expressions for the spectral moments for molecules in particular vibrational states and expressions for the moments of a temperature-dependent spectrum are found for the case of a transition from a harmonic lower state potential to a linear repulsive, quadratic, general, and exponential repulsive upper state potential

The hardness factor in rotational inelastic scattering

Inelastic cross-sections for He + HF collisions have been calculated using an exponential repulsive potential in which the exponent can be increased to approach the hard shape limit. Convergence to the hard shape results is slow, particularly for a transition 0 → 3 which is near the threshold. We attribute slow convergence to the different boundary conditions that are applied in the soft and hard calculations.

Discrete-continuum hybrid model for gas-surface collisions: II. Closed form perturbation solutions for single collisions

A perturbation scheme for the scattering of particle beams from single crystal surfaces is presented. The procedure is based on the assumption that displacements of the lattice points are small compared to the lattice spacing and that surface corrugation is weak. The phonon interaction is accounted for by representing the crystal as an elastic continuum following the hybrid model introduced in the preceding paper. The projectile-surface interaction is described in terms of compliance coefficients which are effective spring and damping constants of the solid, This makes the handling of the solid extremely practical. The perturbation theory and the linear solid model permit the study of collision-induced waves and the thermal waves separately. The total energy exchange is obtained as the sum of the energy exchanges between the projectile and the two types of waves. The perturbation solution yields closed form expressions for the energy exchange with both the exponential repulsive and Morse potentials. These results compare favorably with the exact (i.e., numerical) calculations of the proceeding paper when multiple collisions, resonances and capture are absent. The availability of explicit expressions for the energy transfer permits the calculation of the necessary thermal averages in closed form as well. Quantitative results indicate in addition that for low energies the Morse potential results are markedly different from those with the purely repulsive cases that is commonly used for its simplicity. At high energies, the results of the two potentials converge, as expected.

Discrete—continuum hybrid model for gas—surface collisions II. Closed form perturbation solutions for single collisions

Abstract A perturbation scheme for the scattering of particle beams from single crystal surfaces is presented. The procedure is based on the assumption that displacements of the lattice points are small compared to the lattice spacing and that surface corrugation is weak. The phonon interaction is accounted for by representing the crystal as an elastic continuum following the hybrid model introduced in the preceding paper. The projectile-surface interaction is described in terms of compliance coefficients which are effective spring and damping constants of the solid, This makes the handling of the solid extremely practical. The perturbation theory and the linear solid model permit the study of collision-induced waves and the thermal waves separately. The total energy exchange is obtained as the sum of the energy exchanges between the projectile and the two types of waves. The perturbation solution yields closed form expressions for the energy exchange with both the exponential repulsive and Morse potentials. These results compare favorably with the exact (i.e., numerical) calculations of the proceeding paper when multiple collisions, resonances and capture are absent. The availability of explicit expressions for the energy transfer permits the calculation of the necessary thermal averages in closed form as well. Quantitative results indicate in addition that for low energies the Morse potential results are markedly different from those with the purely repulsive cases that is commonly used for its simplicity. At high energies, the results of the two potentials converge, as expected.

Compression of KCl in the B2 structure to 56 GPa

The room temperature static compression of the high pressure B2 phase of KCl has been measured in a diamond anvil cell to 56 GPa. Birch-Murnaghan finite strain analysis extrapolated to zero pressure gives V 02 V 01 = 0.8483 ± 0.0057 and K 02 = 28.7 ± 0.6 GPa, with K 02' constrained to a value of four. Universal equation of state parameters are virtually identical. Although these results generally concur with previous work at lower pressures, significant differences exist in the zero pressure equation of state parameters extrapolated from the two sets of data. Recent ab initio structure calculations predict compressions in good agreement with our experimental results. Born models utilizing exponential repulsive potentials account well for the data, even over the large (50%) range of compression observed.

Resonance Raman spectra of photodissociating CH 3I and CD 3I in solution

Raman spectra of CH 3I and CD 3I in hexane, including absolute cross sections for CH 3I, have been obtained using 266 nm excitation, on resonance with the directly dissociative 3Q 0 state. Both isotopic species exhibit long vibrational progressions in the C-I stretch and its combination bands with the methyl umbrella mode. The relative intensities of the C-I overtones in CH 3I are similar in vapor and solution phases, indicating that the early photodissociation dynamics are not greatly altered by solvation. The higher overtone transitions are significantly broadened and shifted to lower frequencies in solution. The absorption spectra and resonance Raman intensities of CH 3I have been simulated by parameterizing an uncoupled two-oscillator model in which the C-I stretch is taken as a Morse oscillator in the ground state with an exponential repulsive potential in the excited state.

COLLISION-INDUCED DISSOCIATION (CID) (Ⅰ)——TIME-DEPENDENT QUANTUM THEORY

Collision-induced dissociation dynamic processes have been studied in this paper by using Time-dependnet quantum mechanical perturbation theory. The molecular systems are of the type of the Morse oscillator and the intermolecular exponential repulsive interaction potential. The analytical formula established can be conveniently applied to analyse a variety of physical and chemical problems relevent to CID processes. The numerical calculation for the (I2, He) system shows that vibrational energy is more effective than translational energy in enhancing the rate of the CID processes, as the total energy of the system is fixed. In addition, except the energy barrier, there are no extra dynamical restrictions for CID processes.

Transport properties of ground state oxygen atoms

The transport properties of dilute monatomic gases depend on the two body interactions between like atoms. When two ground state oxygen atoms interact, they can follow any of 18 potential energy curves corresponding to O2, all of which contribute to the transport properties of the ground state atoms. Transport collision integrals have been calculated for those interactions with an attractive minimum in the potential by accurately representing ab initio quantum mechanical potential energy curves with the Hulburt–Hirschfelder potential. Repulsive ab initio potential energy curves have been accurately represented either with the exponential repulsive potential or with an exponential repulsive potential with an additional Gaussian term to model a shoulder-like feature on the repulsive wall. Results are given for viscosity, thermal conductivity, and diffusion and they are compared with previous theoretical calculations.

Collisional perturbation of quantum regular and irregular intramolecular states: State-to-state study of a model

Abstract State-to-state analysis is presented of collisional energy transfer in quantum-mechanical collinear collisions between an atom and a triatomic molecule. The molecular potential was that of Henon and Heiles, while the interaction was modelled by both a soft, exponential repulsive potential and a Morse potential. This analysis emphasizes how different categories of molecular states (normal mode, local mode, or mode mixed) are influenced by the collision perturbation. The discussion centers around plots showing the energy dependence of P ??? ( X ), the probability that initial state i ends up in category X , and contour maps of state-to-state transition probabilities, P ??? ( E ). For the exponential repulsive model, there is a resistance to alteration of the molecular energy; “horizontal” transitions occur within bands of quasi-resonant states. By way of contrast, the Morse interaction model exhibits many vertical transitions, with many low-intensity transitions to a large number of final states. The special role played by extreme motion states, emphasized in recent studies by Hose and Taylor, is discussed in relation to their survivability. Finally, these results are related to trends that we found earlier in the energy dependence of the average energy transfer.

Elastic properties of the lower mantle inferred from rigid ion lattice models

Simple theoretical rigid ion lattice models, utilizing two-body central force exponential type repulsive potentials, for first and second nearest neighbor interactions, have been applied to estimate the higher order elastic constants of several alkali halides and NaCl-structure oxides. The results provided by the theoretical models for the first and second pressure derivatives of the bulk modulus ( K′ 0and K″ 0) for the alkali halides are generally consistent with available experimental data. Model calculations of K″ 0 for the oxides MgO, CaO, SrO, and BaO, are about an order of magnitude less than those for the alkali halides. Depending on the details of the lattice model, the present results suggest a value of K″ 0 for MgO in the range −1.7–−3.1 Mbar −1. In addition, the theoretical lattice models indicate values for the shear-related elastic constants C″ s, C″ 44, and μ″ 0 of the NaCl-structure oxides which are markedly lower than those of the alkali halides. The range of values for K″ 0 and the second pressure derivative of the rigidity modulus μ″ 0, which appear from the present results to characterize close-packed oxides, are consistent with estimates of these parameters for the Earth's lower mantle, assuming an adiabatic temperature gradient and compositional homogeneity. Therefore, it is not necessary to invoke a vertical compositional gradient in the deep mantle to account for the observed variation in elastic properties. However, the range of the theoretical values for K′ 0and μ″ 00 which appear to characterize lower mantle oxide materials, make it necessary to use higher order elastic properties in the equation of state in order to obtain accurate and meaningful evaluations of the composition and state of the Earth's deep interior.

Collisional perturbation of regular and irregular intramolecular dynamics: A model quantum-mechanical study

A quantum-mechanical study is presented of energy transfer in collinear atom-molecule inelastic collisions. The molecular potential is the well studied two-mode, Henon-Heiles potential. Two atom-molecule interaction potentials are considered: a weak exponential repulsive potential (previously considered in a classical study by Schatz), and a Morse attractive interaction. For these two potentials, the mean energy transfer and transition probabilities, classified according to the “category of chaos” (normal-mode, local-mode, or mixed-mode states), are compared, as functions of the total energy. The classical critical energy for the molecular potential plays no role in the energy dependence of the mean collisional energy transfer. In addition, preservation of memory (hysteresis) about the initial category of chaos is larger for collisions involving the weak repulsive potential.