Volume 136, Issue 18, 14 May 2012

Ions with likecharges repel each other with a magnitude given by the Coulomb law. The repulsion is also known to persist in aqueous solutions albeit factored by the medium's dielectric constant. In this paper, we report results from molecular dynamics simulations of alkali halides salt solutions indicating an effective attraction between some of the likecharged monovalent ions. The attraction is observed between anions, as well as between cations, leading to the formation of dimers with lifetimes on the order of few picoseconds. Two mechanisms have been identified to drive this counterintuitive attraction. The first is exhibited by highcharge density ions, such as fluoride, at low salt concentrations, yielding effective attractions with magnitude up to the order of 1–2 kT. In this case, the stronger local electric field generated when the two ions are in contact augments the alignment of neighboring waters toward the ions. This results in a gain of substantial favorable ionwater interactionenergy. For fluorides, this interaction constitutes the major change among the different energy components compensating for the anionanion repulsion, and therefore, rendering likecharge association possible. The second mechanism involves mediation by counterions, the attractions increase with salt concentration and are characterized by small magnitudes. In particular, clusters of ion triplets, in which a counterion is either bridging the two likecharged ions or is paired to only one of them, are formed. Although these two mechanisms may not yield net attractions in many cases, they might still be operational and significant, explaining effective repulsions between likecharged ions with magnitudes much smaller than expected based on continuum electrostatics.
 ARTICLES

 Theoretical Methods and Algorithms

An adaptive stepsize method for the chemical Langevin equation
View Description Hide DescriptionMathematical and computational modeling are key tools in analyzing important biological processes in cells and living organisms. In particular, stochastic models are essential to accurately describe the cellular dynamics, when the assumption of the thermodynamic limit can no longer be applied. However, stochastic models are computationally much more challenging than the traditional deterministic models. Moreover, many biochemical systems arising in applications have multiple timescales, which lead to mathematical stiffness. In this paper we investigate the numerical solution of a stochastic continuous model of wellstirred biochemical systems, the chemical Langevin equation. The chemical Langevin equation is a stochastic differential equation with multiplicative, noncommutative noise. We propose an adaptive stepsize algorithm for approximating the solution of models of biochemical systems in the Langevin regime, with small noise, based on estimates of the local error. The underlying numerical method is the Milstein scheme. The proposed adaptive method is tested on several examples arising in applications and it is shown to have improved efficiency and accuracy compared to the existing fixed stepsize schemes.

Hypergeneralizedgradient functionals constructed from the LiebOxford bound: Implementation via local hybrids and thermochemical assessment
View Description Hide DescriptionIn 2009 Odashima and Capelle (OC) showed a way to design a correlationonlydensity functional that satisfies a LiebOxford bound on the correlation energy, without empirical parameters and even without additional theoretical parameters. However, they were only able to test a sizeinconsistent version of it that employs total energies. Here, we show that their alternative sizeconsistent form that employs energy densities, when combined with exact or semilocal exchange, is a local hybrid (lh) functional. We test several variants of this nonempirical OClh functional on standard molecular test sets. Although no variant yields enthalpies of formation with the accuracy of the semilocal TaoPerdewStaroverovScuseria (TPSS) exchangecorrelation, OClh correlation with exact exchange yields rather accurate energy barriers for chemical reactions. Our purpose here is not to advocate for a new density functional, but to explore a previously published idea. We also discuss the importance of nearselfconsistency for fully nonlocal functionals.

A quantum generalization of intrinsic reaction coordinate using path integral centroid coordinates
View Description Hide DescriptionWe propose a generalization of the intrinsic reaction coordinate (IRC) for quantum manybody systems described in terms of the massweighted ring polymer centroids in the imaginarytime path integral theory. This novel kind of reaction coordinate, which may be called the “centroid IRC,” corresponds to the minimum free energy path connecting reactant and product states with a least amount of reversible work applied to the center of masses of the quantum nuclei, i.e., the centroids. We provide a numerical procedure to obtain the centroid IRC based on first principles by combining ab initio path integral simulation with the string method. This approach is applied to NH_{3} molecule and N_{2} ion as well as their deuterated isotopomers to study the importance of nuclear quantum effects in the intramolecular and intermolecular proton transferreactions. We find that, in the intramolecular proton transfer (inversion) of NH_{3}, the free energy barrier for the centroid variables decreases with an amount of about 20% compared to the classical one at the room temperature. In the intermolecular proton transfer of N_{2} , the centroid IRC is largely deviated from the “classical” IRC, and the free energy barrier is reduced by the quantum effects even more drastically.

A temperature behavior of the frustrated translational mode of adsorbate and the nature of the “adsorbate–substrate” interaction
View Description Hide DescriptionA temperature behavior of the frustrated translational mode (Tmode) of a light particle, coupled by different regimes of ohmicity to the surface, is studied within a formalism of the generalized diffusion coefficients. The memory effects of the adsorbate motion are considered to be the main reason of the Tmode origin. Numerical calculations yield a thermally induced shift and broadening of the Tmode, which is found to be linear in temperature for Ohmic and superOhmic systems and nonlinear for strongly subOhmic ones. We obtain analytical expressions for the Tmode shift and width at weak coupling for the systems with integer “ohmicity” indexes n = 0÷2 in zero temperature and high temperature limits. We provide an explanation of the experimentally observed blue or redshifts of the Tmode on the basis of a comparative analysis of two typical times of the system evolution: a time of decay of the “velocity–velocity” autocorrelation function, and a correlation time of the thermal bath random forces. A relation of the Tmode to the multiple jumps of the adsorbate is discussed, and generalization of conditions of the multiple hopping to the case of quantum surface diffusion is performed.

Excitation energies from rangeseparated timedependent density and density matrix functional theory
View Description Hide DescriptionTimedependent density functional theory (TDDFT) in the adiabatic formulation exhibits known failures when applied to predicting excitation energies. One of them is the lack of the doubly excited configurations. On the other hand, the timedependent theory based on a oneelectron reduced density matrix functional (timedependent density matrix functional theory, TDDMFT) has proven accurate in determining single and double excitations of H_{2} molecule if the exact functional is employed in the adiabatic approximation. We propose a new approach for computing excited stateenergies that relies on functionals of electron density and oneelectron reduced density matrix, where the latter is applied in the longrange region of electronelectron interactions. A similar approach has been recently successfully employed in predicting ground state potential energy curves of diatomic molecules even in the dissociation limit, where static correlation effects are dominating. In the paper, a timedependent functional theory based on the rangeseparation of electronic interaction operator is rigorously formulated. To turn the approach into a practical scheme the adiabatic approximation is proposed for the short and longrange components of the coupling matrix present in the linear response equations. In the end, the problem of finding excitation energies is turned into an eigenproblem for a symmetric matrix. Assignment of obtained excitations is discussed and it is shown how to identify double excitations from the analysis of approximate transition density matrix elements. The proposed method used with the shortrange local density approximation (srLDA) and the longrange BuijseBaerends density matrix functional (lrBB) is applied to H_{2} molecule (at equilibrium geometry and in the dissociation limit) and to Be atom. The method accounts for double excitations in the investigated systems but, unfortunately, the accuracy of some of them is poor. The quality of the other excitations is in general much better than that offered by TDDFTLDA or TDDMFTBB approximations if the rangeseparation parameter is properly chosen. The latter remains an open problem.

Equivalence between the mechanical model and energytransfer theory for the classical decay rates of molecules near a spherical particle
View Description Hide DescriptionIn the classical modeling of decay rates for molecules interacting with a nontrivial environment, it is well known that two alternate approaches exist which include: (1) a mechanical model treating the system as a damped harmonic oscillator driven by the reflected fields from the environment; and (2) a model based on the radiative and nonradiative energy transfers from the excited molecular system to the environment. While the exact equivalence of the two methods is not trivial and has been explicitly demonstrated only for planar geometry, it has been widely taken for granted and applied to other geometries such as in the interaction of the molecule with a spherical particle. Here we provide a rigorous proof of such equivalence for the moleculesphere problem via a direct calculation of the decay rates adopting each of the two different approaches.

Evaluation of the grandcanonical partition function using expanded WangLandau simulations. I. Thermodynamic properties in the bulk and at the liquidvapor phase boundary
View Description Hide DescriptionThe WangLandau sampling is a powerful method that allows for a direct determination of the density of states. However, applications to the calculation of the thermodynamic properties of realistic fluids have been limited so far. By combining the WangLandau method with expanded grandcanonical simulations, we obtain a highaccuracy estimate for the grandcanonical partition function for atomic and molecular fluids. Then, using the formalism of statistical thermodynamics, we are able to calculate the thermodynamic properties of these systems, for a wide range of conditions spanning the singlephase regions as well as the vaporliquid phase boundary. Excellent agreement with prior simulation work and with the available experimental data is obtained for argon and CO_{2}, thereby establishing the accuracy of the method for the calculation of thermodynamic properties such as free energies and entropies.

Evaluation of the grandcanonical partition function using expanded WangLandau simulations. II. Adsorption of atomic and molecular fluids in a porous material
View Description Hide DescriptionWe propose to apply expanded WangLandau simulations to study the adsorption of atomic and molecular fluids in porous materials. This approach relies on a uniform sampling of the number of atoms and molecules adsorbed. The method consists in determining a highaccuracy estimate of the grandcanonical partition function for the adsorbed fluids. Then, using the formalism of statistical mechanics, we calculate absolute and excess thermodynamic properties relevant to adsorption processes. In this paper, we examine the adsorption of argon and carbon dioxide in the isoreticular metalorganic framework (IRMOF1). We assess the reliability of the method by showing that the predicted adsorption isotherms and isosteric heats are in excellent agreement with simulation results obtained from grandcanonical Monte Carlo simulations. We also show that the proposed method is very efficient since a single expanded WangLandau simulation run at a given temperature provides the whole adsorption isotherm. Moreover, this approach provides a direct access to a wide range of thermodynamic properties, such as, e.g., the excess Gibbs free energy and the excess entropy of adsorption.

A masterequation approach to simulate kinetic traps during directed selfassembly
View Description Hide DescriptionRobust directed selfassembly of nonperiodic nanoscale structures is a key process that would enable various technological breakthroughs. The dynamic evolution of directed selfassemblies towards structures with desired geometries is governed by the rugged potential energy surface of nanoscale systems, potentially leading the system to kinetic traps. To study such phenomena and to set the framework for the directed selfassembly of nanoparticles towards structures with desired geometries, the development of a dynamic model involving a master equation to simulate the directed selfassembly process is presented. The model describes the probability of each possible configuration of a fixed number of nanoparticles on a domain, including parametric sensitivities that can be used for optimization, as a function of time during selfassembly. An algorithm is presented that solves largescale instances of the model with linear computational complexity. Case studies illustrate the influence of several degrees of freedom on directed selfassembly. A design approach that systematically decomposes the ergodicity of the system to direct selfassembly of a targeted configuration with high probability is illustrated. The prospects for extending such an approach to larger systems using coarse graining techniques are also discussed.

Comparisons of classical chemical dynamics simulations of the unimolecular decomposition of classical and quantum microcanonical ensembles
View Description Hide DescriptionPrevious studies have shown that classical trajectory simulations often give accurate results for shorttime intramolecular and unimolecular dynamics, particularly for initial nonrandom energy distributions. To obtain such agreement between experiment and simulation, the appropriate distributions must be sampled to choose initial coordinates and momenta for the ensemble of trajectories. If a molecule's classical phase space is sampled randomly, its initial decomposition will give the classical anharmonic microcanonical (RRKM) unimolecular rate constant for its decomposition. For the work presented here, classical trajectory simulations of the unimolecular decomposition of quantum and classical microcanonical ensembles, at the same fixed total energy, are compared. In contrast to the classical microcanonical ensemble, the quantum microcanonical ensemble does not sample the phase space randomly. The simulations were performed for CH_{4}, C_{2}H_{5}, and Cl^{−}‐‐‐CH_{3}Br using both analytic potential energy surfaces and direct dynamics methods. Previous studies identified intrinsic RRKM dynamics for CH_{4} and C_{2}H_{5}, but intrinsic nonRRKM dynamics for Cl^{−}‐‐‐CH_{3}Br. Rate constants calculated from trajectories obtained by the time propagation of the classical and quantum microcanonical ensembles are compared with the corresponding harmonic RRKM estimates to obtain anharmonic corrections to the RRKM rate constants. The relevance and accuracy of the classical trajectory simulation of the quantum microcanonical ensemble, for obtaining the quantum anharmonic RRKM rate constant, is discussed.
 Gas Phase Dynamics and Structure: Spectroscopy, Molecular Interactions, Scattering, and Photochemistry

Electron attachment to molecules in a cluster environment
View Description Hide DescriptionLowenergy dissociative electron attachment (DEA) to the CF_{2}Cl_{2} and CF_{3}Cl molecules in a watercluster environment is investigated theoretically. Calculations are performed for the water trimer and water hexamer. It is shown that the DEA cross section is strongly enhanced when the attaching molecule is embedded in a watercluster, and that this cross section grows as the number of water molecules in the cluster increases. This growth is explained by a trapping effect that is due to multiple scattering by water molecules while the electron is trapped in the cluster environment. The trapping increases the resonance lifetime and the negative ion survival probability. This confirms qualitatively existing experiments on electron attachment to the CF_{2}Cl_{2} molecule placed on the surface of H_{2}O ice. The DEA cross sections are shown to be very sensitive to the position of the attaching molecule within the cluster and the orientation of the electron beam relative to the cluster.

Quantum and classical approaches for rotational relaxation and nonresonant laser alignment of linear molecules: A comparison for CO_{2} gas in the nonadiabatic regime
View Description Hide DescriptionA quantum approach and classical molecular dynamics simulations (CMDS) are proposed for the modeling of rotational relaxation and of the nonadiabatic alignment of gaseous linear molecules by a nonresonant laser field under dissipative conditions. They are applied to pure CO_{2} and compared by looking at statetostate collisional rates and at the value of ⟨cos^{2}[θ _{z}(t)]⟩ induced by a 100 fs laser pulse linearly polarized along . The main results are: (i) When properly requantized, the classical model leads to very satisfactory predictions of the permanent and transient alignments under nondissipative conditions. (ii) The CMDS calculations of collisionalbroadening coefficients and rotational statetostate rates are in very good agreement with those of a quantum model based on the energy corrected sudden (ECS) approximation. (iii) Both approaches show a strong propensity of collisions, while they change the rotational energy (i.e., J), to conserve the angular momentum orientation (i.e., M/J). (iv) Under dissipative conditions, CMDS and quantumECS calculations lead to very consistent decays with time of the “permanent” and transient components of the laserinduced alignment. This result, expected from (i) and (ii), is obtained only if a properly J and Mdependent ECS model is used. Indeed, rotational statetostate rates and the decay of the “permanent” alignment demonstrate, for pure CO_{2}, the limits of a Mindependent collisional model proposed previously. Furthermore, computations show that collisions induce a decay of the “permanent” alignment about twice slower than that of the transient revivals amplitudes, a direct consequence of (iii). (v) The analysis of the effects of reorienting and dephasingelastic collisions shows that the latter have a very small influence but that the former play a nonnegligible role in the alignment dynamics. (vi) Rotationtranslation collisionally induced transfers have also been studied, demonstrating that they only slightly change the alignment dissipation for the considered laser energy conditions.

Calibrationquality adiabatic potential energy surfaces for and its isotopologues
View Description Hide DescriptionCalibrationquality ab initio adiabatic potential energy surfaces (PES) have been determined for all isotopologues of the molecular ion . The underlying Born–Oppenheimer electronic structure computations used optimized explicitly correlated shifted Gaussian functions. The surfaces include diagonal Born–Oppenheimer corrections computed from the accurate electronic wave functions. A fit to the 41 655 ab initio points is presented which gives a standard deviation better than 0.1 cm^{−1} when restricted to the points up to 6000 cm^{−1} above the first dissociation asymptote. Nuclear motion calculations utilizing this PES, called GLH3P, and an exact kinetic energy operator given in orthogonal internal coordinates are presented. The rovibrational transition frequencies for , H_{2}D^{+}, and are compared with high resolution measurements. The most sophisticated and complete procedure employed to compute rovibrational energy levels, which makes explicit allowance for the inclusion of nonadiabaticeffects, reproduces all the known rovibrational levels of the isotopologues considered to better than 0.2 cm^{−1}. This represents a significant (orderofmagnitude) improvement compared to previous studies of transitions in the visible. Careful treatment of linear geometries is important for high frequency transitions and leads to new assignments for some of the previously observed lines. Prospects for further investigations of nonadiabaticeffects in the isotopologues are discussed. In short, the paper presents (a) an extremely accurate global potential energy surface of resulting from high accuracy ab initio computations and global fit, (b) very accurate nuclear motion calculations of all available experimental line data up to 16 000 cm^{−1}, and (c) results suggest that we can predict accurately the lines of towards dissociation and thus facilitate their experimental observation.

Efficient quantumclassical method for computing thermal rate constant of recombination: Application to ozone formation
View Description Hide DescriptionEfficient method is proposed for computing thermal rate constant of recombination reaction that proceeds according to the energy transfer mechanism, when an energized molecule is formed from reactants first, and is stabilized later by collision with quencher. The mixed quantumclassical theory for the collisional energy transfer and the rovibrational energy flow [M. Ivanov and D. Babikov, J. Chem. Phys.134, 144107 (2011)]10.1063/1.3576103 is employed to treat the dynamics of molecule + quencher collision. Efficiency is achieved by sampling simultaneously (i) the thermal collision energy, (ii) the impact parameter, and (iii) the incident direction of quencher, as well as (iv) the rotational state of energized molecule. This approach is applied to calculate thirdorder rate constant of the recombination reaction that forms the ^{16}O^{18}O^{16}O isotopomer of ozone. Comparison of the predicted rate vs. experimental result is presented.

Identification of the dimethylaminetrimethylamine complex in the gas phase
View Description Hide DescriptionWe have identified the dimethylaminetrimethylamine complex (DMATMA) at room temperature in the gas phase. The Fourier transform infrared (FTIR)spectrum of DMATMA in the NHstretching fundamental region was obtained by spectral subtraction of spectra of each monomer. Explicitly correlated coupled cluster calculations were used to determine the minimum energy structure and interaction energy of DMATMA. Frequencies and intensities of NHstretching transitions were also calculated at this level of theory with an anharmonic oscillator local mode model. The fundamental NHstretching intensity in DMATMA is calculated to be approximately 700 times larger than that of the DMA monomer. The measured and calculated intensity is used to determine a room temperature equilibrium constant of DMATMA of 1.7 × 10^{−3} atm^{−1} at 298 K.

Experimental and computational investigation of the group 11–group 2 diatomic molecules: First determination of the AuSr and AuBa bond energies and thermodynamic stability of the copper and silveralkaline earth species
View Description Hide DescriptionThe dissociation energies of the intermetallic molecules AuSr and AuBa were for the first time determined by the Knudsen effusion mass spectrometry method. The two species were produced in the vapor phase equilibrated with apt mixtures of the constituent elements, and the dissociation equilibria were monitored massspectrometrically in the temperature range 1406–1971 K (AuSr) and 1505–1971 K (AuBa). The thirdlaw analysis of the equilibrium data gives the following dissociation energies (, in kJ/mol): 244.4 ± 4.8 (AuSr) and 273.3 ± 6.3 (AuBa), so completing the series of s for the AuAE (AE = group 2 element) diatomics. The AuAE species were also studied computationally at the coupled cluster including single, double and perturbative triple excitation [CCSD(T)] level with basis sets of increasing zeta quality, and various complete basis set limit extrapolations were performed to calculate the dissociation energies. Furthermore, the entire series of the heteronuclear diatomic species formed from one group 11 (Cu, Ag) and one group 2 (Be, Mg, Ca, Sr, Ba) metal was studied by DFT with the hybrid metaGGA TPSSh functional and the def2QZVPP basis set, selected after screening a number of functionalbasis set combinations using the AuAE species as benchmark. Dissociation energies, internuclear distances, vibrational frequencies, and anharmonic constants were determined for the CuAE and AgAE species and their thermal functions evaluated therefrom. On this basis, a thermodynamic evaluation of the formation of these species was carried out under various conditions.

Structural and spectroscopic study of the van der Waals complex of CO with HCO^{+} and the isoelectronic complex of CS with HCS^{+}
View Description Hide DescriptionThis work reports the results of high level ab initio calculations of the OCHCO^{+} complex and the SCHCS^{+} complex and their hydrogen migration transition states. Geometry optimizations are performed at the CCSD(T)/augccpV5Z level of theory. Subsequent frequency calculations are carried out at the CCSD(T)/augccpVQZ level of theory. Additional geometry optimizations and harmonic frequency calculations for all the species involved in this study have been done with the explicitly correlated CCSD(T)F12 method with the augccpVTZ and VTZF12 basis set. The geometries, rotational constants, harmonic vibrational frequencies, and energetics of the species involved in the complex are reported. These methods result in accurate computational predictions that have mean deviations for bond lengths, rotational constants, and vibrational frequencies of 0.001 Å, 163 MHz, and 46 cm^{−1}, respectively. These results provide essential spectroscopic properties for the complexes that can facilitate both laboratory and interstellar observations, and they also provide a comparison between oxygen and sulfur complex observability based on thermodynamic stability.

Electron impact total cross section for acetylene over an extensive range of impact energies (1 eV–5000 eV)
View Description Hide DescriptionComprehensive study on electron impact for acetylene molecule is performed in terms of eigenphase diagram, electronic excitation cross sections as well as total cross section calculations from 1 eV to 5000 eV in this article. Computation of cross section over such a wide range of energy is reported for the first time. We have employed two distinct formalisms to derive cross sections in these impact energies. From 1 eV to ionization threshold of the target we have used the ab initio Rmatrix method and then spherical complex optical potential method beyond that. At the crossing point of energy, both theories matched quite well and hence prove that they are consistent with each other. The results presented here expectedly give excellent agreement with other experimental values and theories available. The techniques employed here are well established and can be used to predict cross sections for other targets where data are scarce or not available. Also, this methodology may be integrated to online database such as Virtual Atomic and Molecular Data Centre to provide cross section data required by any user.

The absorption spectrum of D_{2}: Ultrasensitive cavity ring down spectroscopy of the (2–0) band near 1.7 μm and accurate ab initio line list up to 24 000 cm^{−1}
View Description Hide DescriptionEleven very weak electric quadrupole transitions Q(2), Q(1), S(0)S(8) of the first overtone band of D_{2} have been measured by very high sensitivity CWcavity ring down spectroscopy (CRDS) between 5850 and 6720 cm^{−1}. The noise equivalent absorption of the recordings is on the order of α _{ min } ≈ 3 × 10^{−11} cm^{−1}. By averaging a high number of spectra, the noise level was lowered to α _{ min } ≈ 4 × 10^{−12} cm^{−1} in order to detect the S(8) transition which is among the weakest transitions ever detected in laboratory experiments (line intensity on the order of 1.8 × 10^{−31} cm/molecule at 296 K). A Galatry profile was used to reproduce the measured line shape and derive the line strengths. The pressure shift and position at zero pressure limit were determined from recordings with pressures ranging between 10 and 750 Torr. A highly accurate theoretical line list was constructed for pure D_{2} at 296 K. The intensity threshold was fixed to a value of 1 × 10^{−34} cm/molecule at 296 K. The obtained line list is provided as supplementary material. It extends up to 24 000 cm^{−1} and includes 201 transitions belonging to ten v0 cold bands (v = 0–9) and three v1 hot bands (v = 1–3). The energy levels include the relativistic and quantum electrodynamic corrections as well as the effects of the finite nuclear mass. The quadrupole transition moments are calculated using highly accurate adiabatic wave functions. The CRDS line positions and intensities of the first overtone band are compared to the corresponding calculated values and to previous measurements of the S(0)S(3) lines. The agreement between the CRDS and theoretical results is found within the claimed experimental uncertainties (on the order of 1 × 10^{−3} cm^{−1} and 2% for the positions and intensities, respectively) while the previous S(0)S(3) measurements showed important deviations for the line intensities.

A product branching ratio controlled by vibrational adiabaticity and variational effects: Kinetics of the H + transN_{2}H_{2} reactions
View Description Hide DescriptionThe abstraction and addition reactions of H with transN_{2}H_{2} are studied by highlevel ab initio methods and density functional theory.Rate constants were calculated for these two reactions by multistructural variational transition state theory with multidimensional tunneling and including torsional anharmonicity by the multistructural torsion method. Rate constants of the abstraction reaction show large variational effects, that is, the variational transition state yields a smaller rate constant than the conventional transition state; this results from the fact that the variational transition state has a higher zeropoint vibrational energy than the conventional transition state. The addition reaction has a classical barrier height that is about 1 kcal/mol lower than that of the abstraction reaction, but the addition rates are lower than the abstraction rates due to vibrational adiabaticity. The calculated branching ratio of abstraction to addition is 3.5 at 200 K and decreases to 1.2 at 1000 K and 1.06 at 1500 K.