Volume 138, Issue 1, 07 January 2013
Index of content:

Efficient estimates for the preselection of twoelectron integrals in atomicorbital based MøllerPlesset perturbation theory (AOMP2) theory are presented, which allow for evaluating the AOMP2 energy with computational effort that scales linear with molecular size for systems with a significant HOMOLUMO gap. The estimates are based on our recently introduced QQR approach [S. A. Maurer, D. S. Lambrecht, D. Flaig, and C. Ochsenfeld, J. Chem. Phys.136, 144107 (Year: 2012)10.1063/1.3693908], which exploits the asympotic decay of the integral values with increasing braket separation as deduced from the multipole expansion and combines this decay behavior with the common Schwarz bound to a tight and simple estimate. We demonstrate on a diverse selection of benchmark systems that our AOMP2 method in combination with the QQRtype estimates produces reliable results for systems with both localized and delocalized electronic structure, while in the latter case the screening essentially reverts to the common Schwarz screening. For systems with localized electronic structure, our AOMP2 method shows an early onset of linear scaling as demonstrated on DNA systems. The favorable scaling behavior allows to compute systems with more than 1000 atoms and 10 000 basis functions on a single core that are clearly not accessible with conventional MP2 methods. Furthermore, our AOMP2 method is particularly suited for parallelization and we present benchmark calculations on a proteinDNA repair complex comprising 2025 atoms and 20 371 basis functions.
 ARTICLES


Theoretical Methods and Algorithms

Efficient distanceincluding integral screening in linearscaling MøllerPlesset perturbation theory
View Description Hide DescriptionEfficient estimates for the preselection of twoelectron integrals in atomicorbital based MøllerPlesset perturbation theory (AOMP2) theory are presented, which allow for evaluating the AOMP2 energy with computational effort that scales linear with molecular size for systems with a significant HOMOLUMO gap. The estimates are based on our recently introduced QQR approach [S. A. Maurer, D. S. Lambrecht, D. Flaig, and C. Ochsenfeld, J. Chem. Phys.136, 144107 (Year: 2012)10.1063/1.3693908], which exploits the asympotic decay of the integral values with increasing braket separation as deduced from the multipole expansion and combines this decay behavior with the common Schwarz bound to a tight and simple estimate. We demonstrate on a diverse selection of benchmark systems that our AOMP2 method in combination with the QQRtype estimates produces reliable results for systems with both localized and delocalized electronic structure, while in the latter case the screening essentially reverts to the common Schwarz screening. For systems with localized electronic structure, our AOMP2 method shows an early onset of linear scaling as demonstrated on DNA systems. The favorable scaling behavior allows to compute systems with more than 1000 atoms and 10 000 basis functions on a single core that are clearly not accessible with conventional MP2 methods. Furthermore, our AOMP2 method is particularly suited for parallelization and we present benchmark calculations on a proteinDNA repair complex comprising 2025 atoms and 20 371 basis functions.

Interplay of nonMarkov and internal friction effects in the barrier crossing kinetics of biopolymers: Insights from an analytically solvable model
View Description Hide DescriptionConformational rearrangements in biomolecules (such as protein folding or enzymeligand binding) are often interpreted in terms of lowdimensional models of barrier crossing such as Kramers’ theory. Dimensionality reduction, however, entails memory effects; as a result, the effective frictional drag force along the reaction coordinate nontrivially depends on the time scale of the transition. Moreover, when both solvent and “internal” friction effects are important, their interplay results in a highly nonlinear dependence of the effective friction on solvent viscosity that is not captured by common phenomenological models of barrier crossing. Here, these effects are illustrated using an analytically solvable toy model of an unstructured polymer chain involved in an inter or intramolecular transition. The transition rate is calculated using the GroteHynes and Langer theories, which—unlike Kramers’ theory—account for memory. The resulting effective frictional force exerted by the polymer along the reaction coordinate can be rationalized in terms of the effective number of monomers engaged in the transition. Faster transitions (relative to the polymer reconfiguration time scale) involve fewer monomers and, correspondingly, lower friction forces, because the polymer chain does not have enough time to reconfigure in response to the transition.

The flexible nature of exchange, correlation, and Hartree physics: Resolving “delocalization” errors in a “correlation free” density functional
View Description Hide DescriptionBy exploiting freedoms in the definitions of “correlation,” “exchange,” and “Hartree” physics in ensemble systems, we better generalise the notion of “exact exchange” (EXX) to systems with fractional occupations of the frontier orbitals, arising in the dissociation limit of some molecules. We introduce the linear EXX (“LEXX”) theory whose pair distribution and energy are explicitly piecewise linear in the occupations . We provide explicit expressions for these functions for frontier s and p shells. Used in an optimised effective potential (OEP) approach the LEXX yields energies bounded by the piecewise linear “ensemble EXX” (EEXX) energy and standard fractional optimised EXX energy: E ^{EEXX} ⩽ E ^{LEXX} ⩽ E ^{EXX}. Analysis of the LEXX explains the success of standard OEP methods for diatoms at large spacing, and why they can fail when both spins are allowed to be noninteger so that “ghost” Hartree interactions appear between opposite spin electrons in the usual formula. The energy E ^{LEXX} contains a cancellation term for the spin ghost case. It is evaluated for H, Li, and Na fractional ions with clear derivative discontinuities for all cases. The pshell form reproduces accurate correlationfree energies of BF and AlCl. We further test LEXX plus correlation energy calculations on fractional ions of C and F and again we find both derivative discontinuities and good agreement with exact results.

A wavelet analysis for the Xray absorption spectra of molecules
View Description Hide DescriptionWe present a Wavelet transform analysis for the Xray absorption spectra of molecules. In contrast to the traditionally used Fourier transform approach, this analysis yields a 2D correlation plot in both R and kspace. As a consequence, it is possible to distinguish between different scattering pathways at the same distance from the absorbing atom and between the contributions of single and multiple scattering events, making an unambiguous assignment of the fine structure oscillations for complex systems possible. We apply this to two previously studied transition metal complexes, namely iron hexacyanide in both its ferric and ferrous form, and a rhenium diimine complex, [ReX(CO)_{3}(bpy)], where X = Br, Cl, or ethyl pyridine (Etpy). Our results demonstrate the potential advantages of using this approach and they highlight the importance of multiple scattering, and specifically the focusing phenomenon to the extended Xray absorption fine structure (EXAFS) spectra of these complexes. We also shed light on the low sensitivity of the EXAFS spectrum to the ReX scattering pathway.

Exploiting a semianalytic approach to study first order phase transitions
View Description Hide DescriptionIn a previous contribution [C. E. Fiore and M. G. E. da Luz, Phys. Rev. Lett.107, 230601 (Year: 2011)10.1103/PhysRevLett.107.230601] we have proposed a method to treat first order phase transitions at low temperatures. It describes arbitrary order parameter through an analytical expression W, which depends on few coefficients. Such coefficients can be calculated by simulating relatively small systems, hence, with a low computational cost. The method determines the precise location of coexistence lines and arbitrary response functions (from proper derivatives of W). Here we exploit and extend the approach, discussing a more general condition for its validity. We show that, in fact, it works beyond the low T limit, provided the first order phase transition is strong enough. Thus, W can be used even to study athermal problems, as exemplified for a hardcore lattice gas. We furthermore demonstrate that other relevant thermodynamic quantities, as entropy and energy, are also obtained from W. To clarify some important mathematical features of the method, we analyze in detail an analytically solvable problem. Finally, we discuss different representative models, namely, Potts, BellLavis, and associating gaslattice, illustrating the procedure's broad applicability.

Linear scaling explicitly correlated MP2F12 and ONIOM methods for the longrange interactions of the nanoscale clusters in methanol aqueous solutions
View Description Hide DescriptionA linear scaling quantum chemistry method, generalized energybased fragmentation (GEBF) approach has been extended to the explicitly correlated secondorder MøllerPlesset perturbation theory F12 (MP2F12) method and own Nlayer integrated molecular orbital molecular mechanics (ONIOM) method, in which GEBFMP2F12, GEBFMP2, and conventional density functional tightbinding methods could be used for different layers. Then the longrange interactions in dilute methanol aqueous solutions are studied by computing the binding energies between methanol molecule and water molecules in gasphase and condensed phase methanolwater clusters with various sizes, which were taken from classic molecular dynamics (MD) snapshots. By comparing with the results of force field methods, including SPC, TIP3P, PCFF, and AMOEBA09, the GEBFMP2F12 and GEBFONIOM methods are shown to be powerful and efficient for studying the longrange interactions at a high level. With the GEBFONIOM(MP2F12:MP2) and GEBFONIOM(MP2F12:MP2:cDFTB) methods, the diameters of the largest nanoscale clusters under studies are about 2.4 nm (747 atoms and 10 209 basis functions with augccpVDZ basis set) and 4 nm (3351 atoms), respectively, which are almost impossible to be treated by conventional MP2 or MP2F12 method. Thus, the GEBFF12 and GEBFONIOM methods are expected to be a practical tool for studying the nanoscale clusters in condensed phase, providing an alternative benchmark for ab initio and density functional theory studies, and developing new force fields by combining with classic MD simulations.

Polarization consistent basis sets. VIII. The transition metals ScZn
View Description Hide DescriptionPolarization consistent basis sets, optimized for density functional calculations, are proposed for the transition metals ScZn. The basis set composition in terms of number of primitive functions and the contraction is defined based on energetic analyses of atoms and molecules along the lines used in previous work and on the performance for molecular systems. The performance for atomization energies and dipole moments is compared to other widely used basis sets, and it is shown that the new basis sets allow a systematic reduction of basis set errors and, in general, have basis set errors lower than or at par with existing ones.

The multigrid POTFIT (MGPF) method: Grid representations of potentials for quantum dynamics of large systems
View Description Hide DescriptionIn this article, a new method, multigrid POTFIT (MGPF), is presented. MGPF is a gridbased algorithm which transforms a general potential energy surface into product form, that is, a sum of products of onedimensional functions. This form is necessary to profit from the computationally advantageous multiconfiguration timedependent Hartree method for quantum dynamics. MGPF circumvents the dimensionality related issues present in POTFIT [A. Jäckle and H.D. Meyer, J. Chem. Phys.104, 7974 (Year: 1996)10.1063/1.471513], allowing quantum dynamical studies of systems up to about 12 dimensions. MGPF requires the definition of a fine grid and a coarse grid, the latter being a subset of the former. The MGPF approximation relies on a series of underlying POTFIT calculations on grids which are smaller than the fine one and larger than or equal to the coarse one. This aspect makes MGPF a bit less accurate than POTFIT but orders of magnitude faster and orders of magnitude less memory demanding than POTFIT. Moreover, like POTFIT, MGPF is variational and provides an efficient error control.

Electron affinities and ionisation potentials for atoms via “benchmark” tdDFT calculations with and without exchange kernels
View Description Hide DescriptionOne of the known weaknesses of the adiabatic connection fluctuation dissipation (ACFD) correlation energy functional under the direct randomphase approximation (RPA) is its failure to accurately predict energy differences between dissimilar systems. In this work we evaluate ionisation potentials I and electron affinities A for atoms and ions with one to eighteen electrons using the ACFD functional under the RPA, and with the “PGG (PetersilkaGossmannGross)” and “RXH (radial exchange hole)” model exchange kernels. All calculations are carried out using a realspace, all electron method with an exact exchange groundstate to minimise errors. As expected, the RPA is less accurate even than some regular density functional theory approaches, while the introduction of a dynamical exchange kernel improves results. In contrast to the case of atomic groundstate energies, the PGG kernel outperforms the RXH kernel for I and A. Mean absolute errors for I/A are found to be 3.27/2.38 kcal/mol, 4.38/5.43 kcal/mol, and 9.24/ 8.94 kcal/mol for the PGG, RXH, and RPA, respectively. We thus show that the inclusion of even the simple “RXH” kernel improves both quantities when compared to the RPA.

Efficient methods for including quantum effects in Monte Carlo calculations of large systems: Extension of the displaced points path integral method and other effective potential methods to calculate properties and distributions
View Description Hide DescriptionWe present a procedure to calculate ensemble averages, thermodynamic derivatives, and coordinate distributions by effective classical potential methods. In particular, we consider the displacedpoints path integral (DPPI) method, which yields exact quantal partition functions and ensemble averages for a harmonic potential and approximate quantal ones for general potentials, and we discuss the implementation of the new procedure in two Monte Carlo simulation codes, one that uses uncorrelated samples to calculate absolute free energies, and another that employs Metropolis sampling to calculate relative free energies. The results of the new DPPI method are compared to those from accurate path integral calculations as well as to results of two other effective classical potential schemes for the case of an isolated water molecule. In addition to the partition function, we consider the heat capacity and expectation values of the energy, the potential energy, the bond angle, and the OH distance. We also consider coordinate distributions. The DPPI scheme performs best among the three effective potential schemes considered and achieves very good accuracy for all of the properties considered. A key advantage of the effective potential schemes is that they display much lower statistical sampling variances than those for accurate path integral calculations. The method presented here shows great promise for including quantum effects in calculations on large systems.

NonMarkovian stochastic Schrödinger equation at finite temperatures for charge carrier dynamics in organic crystals
View Description Hide DescriptionA new nonMarkovian stochastic Schrödinger equation at finite temperatures is presented to correctly describe charge carrier dynamics in organic molecular crystals. The electronphonon interactions in both site energies and electronic couplings are incorporated by the timedependent complexvalued random fluctuations which are generated from corresponding spectral density functions. The approach is thus easily extended to investigate coherenttohopping charge transfer in systems with thousands of molecular sites. The capability of present approach is demonstrated by numerical simulations of carrier dynamics in the spinboson model and a realistic FennaMatthewsOlson complex. The results manifest that the nonMarkovian effect and complexvalued random forces are essential to guarantee the detailed balance. In an application to a longchain donoracceptor system, it is also interesting to find a property of coherenttohopping charge transfer from temperature dependence of diffusion coefficients.

Efficient methods and practical guidelines for simulating isotope effects
View Description Hide DescriptionThe shift in chemical equilibria due to isotope substitution is frequently exploited to obtain insight into a wide variety of chemical and physical processes. It is a purely quantum mechanical effect, which can be computed exactly using simulations based on the path integral formalism. Here we discuss how these techniques can be made dramatically more efficient, and how they ultimately outperform quasiharmonic approximations to treat quantum liquids not only in terms of accuracy, but also in terms of computational cost. To achieve this goal we introduce path integral quantum mechanics estimators based on free energy perturbation, which enable the evaluation of isotope effects using only a single path integral molecular dynamics trajectory of the naturally abundant isotope. We use as an example the calculation of the free energy change associated with H/D and ^{16}O/^{18}O substitutions in liquid water, and of the fractionation of those isotopes between the liquid and the vapor phase. In doing so, we demonstrate and discuss quantitatively the relative benefits of each approach, thereby providing a set of guidelines that should facilitate the choice of the most appropriate method in different, commonly encountered scenarios. The efficiency of the estimators we introduce and the analysis that we perform should in particular facilitate accurate ab initio calculation of isotope effects in condensed phase systems.

Advanced Experimental Techniques

The vibrational spectrum of CaCO_{3} aragonite: A combined experimental and quantummechanical investigation
View Description Hide DescriptionThe vibrational properties of CaCO_{3} aragonite have been investigated both theoretically, by using a quantum mechanical approach (all electron Gaussian type basis set and B3LYP HFDFT hybrid functional, as implemented in the CRYSTAL code) and experimentally, by collecting polarized infrared (IR) reflectance and Raman spectra. The combined use of theory and experiment permits on the one hand to analyze the many subtle features of the measured spectra, on the other hand to evidentiate limits and deficiencies of both approaches. The full set of TO and LO IR active modes, their intensities, the dielectric tensor (in its static and high frequency components), and the optical indices have been determined, as well as the Raman frequencies. Tools such as isotopic substitution and graphical animation of the modes are available, that complement the analysis of the spectrum.

Atoms, Molecules, and Clusters

Chirpedpulse millimeterwave spectroscopy: Spectrum, dynamics, and manipulation of Rydberg–Rydberg transitions
View Description Hide DescriptionWe apply the chirpedpulse millimeterwave (CPmmW) technique to transitions between Rydberg states in calcium atoms. The unique feature of Rydberg–Rydberg transitions is that they have enormous electric dipole transition moments (∼5 kiloDebye at n* ∼ 40, where n* is the effective principal quantum number), so they interact strongly with the mmwave radiation. After polarization by a mmwave pulse in the 70–84 GHz frequency region, the excited transitions reradiate free induction decay (FID) at their resonant frequencies, and the FID is heterodynedetected by the CPmmW spectrometer. Data collection and averaging are performed in the time domain. The spectral resolution is ∼100 kHz. Because of the large transition dipole moments, the available mmwave power is sufficient to polarize the entire bandwidth of the spectrometer (12 GHz) in each pulse, and highresolution survey spectra may be collected. Both absorptive and emissive transitions are observed, and they are distinguished by the phase of their FID relative to that of the excitation pulse. With the combination of the large transition dipole moments and direct monitoring of transitions, we observe dynamics, such as transient nutations from the interference of the excitation pulse with the polarization that it induces in the sample. Since the waveform produced by the mmwave source may be precisely controlled, we can populate states with high angular momentum by a sequence of pulses while recording the results of these manipulations in the time domain. We also probe the superradiant decay of the Rydberg sample using photon echoes. The application of the CPmmW technique to transitions between Rydberg states of molecules is discussed.

Photodissociation of gaseous CH_{3}COSH at 248 nm by timeresolved Fouriertransform infrared emission spectroscopy: Observation of three dissociation channels
View Description Hide DescriptionUpon onephoton excitation at 248 nm, gaseous CH_{3}C(O)SH is dissociated following three pathways with the products of (1) OCS + CH_{4}, (2) CH_{3}SH + CO, and (3) CH_{2}CO + H_{2}S that are detected using timeresolved Fouriertransform infrared emission spectroscopy. The excited state ^{1}(n_{O}, π^{*} _{CO}) has a radiative lifetime of 249 ± 11 ns long enough to allow for Ar collisions that induce internal conversion and enhance the fragment yields. The rate constant of collisioninduced internal conversion is estimated to be 1.1 × 10^{−10} cm^{3} molecule^{−1} s^{−1}. Among the primary dissociation products, a fraction of the CH_{2}CO moiety may undergo further decomposition to CH_{2} + CO, of which CH_{2} is confirmed by reaction with O_{2} producing CO_{2}, CO, OH, and H_{2}CO. Such a secondary decomposition was not observed previously in the Ar matrixisolated experiments. The highresolution spectra of CO are analyzed to determine the rovibrational energy deposition of 8.7 ± 0.7 kcal/mol, while the remaining primary products with smaller rotational constants are recognized but cannot be spectrally resolved. The CO fragment detected is mainly ascribed to the primary production. A prior distribution method is applied to predict the vibrational distribution of CO that is consistent with the experimental findings.

Correlation between the variation in observed melting temperatures and structural motifs of the global minima of gallium clusters: An ab initio study
View Description Hide DescriptionWe have investigated the correlation between the variation in the melting temperature and the growth pattern of small positively charged gallium clusters. Significant shift in the melting temperatures was observed for a change of only few atoms in the size of the cluster. Clusters with size between 31−42 atoms melt between 500–600 K whereas those with 46−48 atoms melt around 800 K. Density functional theory based first principles simulations have been carried out on clusters with n = 31, …, 48. At least 150 geometry optimizations have been performed towards the search for the global minima for each size resulting in about 3000 geometry optimizations. For gallium clusters in this size range, the emergence of spherical structures as the ground state leads to higher melting temperature. The wellseparated core and surface shells in these clusters delay isomerization, which results in the enhanced stability of these clusters at elevated temperatures. The observed variation in the melting temperature of these clusters therefore has a structural origin.

Correlated ab initio investigations on the intermolecular and intramolecular potential energy surfaces in the ground electronic state of the complex
View Description Hide DescriptionThis work reports the first highly correlated ab initio study of the intermolecular and intramolecular potential energy surfaces in the ground electronic state of the complex. Accurate electronic structure calculations were performed using the coupled cluster method including single and double excitations with addition of the perturbative triples correction [CCSD(T)] with the Dunning's correlation consistent basis sets augccpVnZ, n = 2–5. Also, the explicitly correlated CCSD(T)F12a level of theory was employed with the AVnZ basis as well as the Peterson and coworkers VnZF12 basis sets with n = 2 and 3. Results of all levels of calculations predicted two equivalent minimum energy structures of planar geometry and C_{s} symmetry along the A^{″} surface of the complex, whereas the A^{′} surface is repulsive. Values of the geometrical parameters and the counterpoise corrected dissociation energies (CpD_{e}) that were calculated using the CCSD(T)F12a/VnZF12 level of theory are in excellent agreement with those obtained from the CCSD(T)/augccpV5Z calculations. The minimum energy structure is characterized by a very short hydrogen bond of length of 1.328 Å, with elongation of the HF bond distance in the complex by 0.133 Å, and D_{e} value of 32.313 Kcal/mol. Mulliken atomic charges showed that 65% of the negative charge is localized on the hydrogen bonded end of the superoxide radical and the HF unit becomes considerably polarized in the complex. These results suggest that the hydrogen bond is an incipient ionic bond. Exploration of the potential energy surface confirmed the identified minimum and provided support for vibrationally induced intramolecular proton transfer within the complex. The Tshaped geometry that possesses C_{2v} symmetry presents a saddle point on the top of the barrier to the inplane bending of the hydrogen above and below the axis that connects centers of masses of the monomers. The height of this barrier is 7.257 Kcal/mol, which is higher in energy than the hydrogen bending frequency by 909.2 cm^{−1}. The calculated harmonic oscillator vibrational frequencies showed that the H–F stretch vibrational transition in the complex is redshifted by 2564 cm^{−1} and gained significant intensity (by at least a factor of 30) with respect to the transition in the HF monomer. These results make the complex an excellent prototype for infrared spectroscopic investigations on openshell complexes with vibrationally induced proton transfer.

Interatomic decay of innervalence ionized states in ArXe clusters: Relativistic approach
View Description Hide DescriptionIn this work we investigate interatomic electronic decay processes taking place in mixed argonxenon clusters upon the innervalence ionization of an argon center. We demonstrate that both interatomic Coulombic decay and electrontransfer mediated decay (ETMD) are important in larger rare gas clusters as opposed to dimers. Calculated secondary electron spectra are shown to depend strongly on the spinorbit coupling in the final states of the decay as well as the presence of polarizable environment. It follows from our calculations that ETMD is a pure interface process taking place between the argonxenon layers. The interplay of all these effects is investigated in order to arrive at a suitable physical model for the decay of innervalence vacancies taking place in mixed ArXe clusters.

DFT study of vibronic properties of d^{8} (Ni, Pd, and Pt) phthalocyanines
View Description Hide DescriptionBy means of density functional theory, we have studied the electronic structure and vibronic properties of single neutral NiPc, PdPc, and PtPc molecules and their singly and doubly ionized cations and anions. In particular, the vibronic couplings and reorganization energies of all systems are compared. Partitioning of the reorganization energy, corresponding to the photoelectron spectra of the first and second ionizations of studied molecules, into normal mode contributions shows that the major contributions are due to several vibrational modes with a_{1g } symmetry and energies lower than 1600 cm^{−1}. The results reveal that the reorganization energy due to the singly positive ionization in the studied molecules is up to about one order of magnitude less than other reorganization energies. This makes these metal phthalocyanines, from the perspective of intramolecular reorganization energies, attractive as electron donor for intramolecular electron transfer in electron acceptordonor systems.

Spectroscopic and thermochemical properties of the cC_{6}H_{7} radical: A highlevel theoretical study
View Description Hide DescriptionThe electronic ground state ( ) of the cyclohexadienyl radical (cC_{6}H_{7}) has been studied by explicitly correlated coupled cluster theory at the RCCSD(T)F12x (x = a, b) level, partly in combination with the doublehybrid density functional method B2PLYP. An accurate equilibrium structure has been established and the groundstate rotational constants are predicted to be A_{0} = 5347.3 MHz, B_{0} = 5249.7 MHz, and C_{0} = 2692.5 MHz. The calculated vibrational wavenumbers agree well with the recent pH_{2} matrix IR data [M. Bahou, Y.J. Wu, and Y.P. Lee, J. Chem. Phys.136, 154304 (Year: 2012)10.1063/1.3703502] and several predictions have been made. A low value of 6.803 ± 0.005 eV is predicted for the adiabatic ionization energy of cC_{6}H_{7}. Owing to a moderately large change in the equilibrium structure upon ionization, the first band of the photoelectron spectrum is dominated by the adiabatic peak (100%) and only the peaks corresponding to excitation of the two lowest totally symmetric vibrations (ν_{12} and ν_{11}) by one vibrational quantum have relative intensities of more than 15%. The C_{6}H_{6}H dissociation energy is calculated to be D_{0} = 85.7 kJ mol^{−1}, with an estimated error of ∼2 kJ mol^{−1}.
