Showing papers in "Molecular Physics in 2014"

Investigating excited electronic states using the algebraic diagrammatic construction (ADC) approach of the polarisation propagator

Abstract: The development of reliable theoretical methods and the provision of efficient computer programs for the investigation of optical spectra and photochemistry of large molecules in general is one of the most important tasks of contemporary theoretical chemistry. Here, we present an overview of the current features of our implementation of the algebraic diagrammatic construction scheme of the polarisation propagator, which is a versatile and robust approach for the theoretical investigation of excited states and their properties.

Hamiltonian replica exchange in GROMACS: a flexible implementation

Abstract: A simple and general implementation of Hamiltonian replica exchange for the popular molecular dynamics software GROMACS is presented. In this implementation, arbitrarily different Hamiltonians can be used for the different replicas without incurring in any significant performance penalty. The implementation was validated on a simple toy model – alanine dipeptide in water – and applied to study the rearrangement of an RNA tetraloop, where it was used to compare recently proposed force-field corrections.

Linear-scaling and parallelisable algorithms for stochastic quantum chemistry

Abstract: For many decades, quantum chemical method development has been dominated by algorithms which involve increasingly complex series of tensor contractions over one-electron orbital spaces. Procedures for their derivation and implementation have evolved to require the minimum amount of logic and rely heavily on computationally efficient library-based matrix algebra and optimised paging schemes. In this regard, the recent development of exact stochastic quantum chemical algorithms to reduce computational scaling and memory overhead requires a contrasting algorithmic philosophy, but one which when implemented efficiently can achieve higher accuracy/cost ratios with small random errors. Additionally, they can exploit the continuing trend for massive parallelisation which hinders the progress of deterministic high-level quantum chemical algorithms. In the Quantum Monte Carlo community, stochastic algorithms are ubiquitous but the discrete Fock space of quantum chemical methods is often unfamiliar, and the methods...

Thermodynamic properties and the approximate solutions of the Schrodinger equation with the shifted Deng–Fan potential model

Abstract: With the introduction of a new improved approximation scheme (Pekeris-type approximation) to deal with the centrifugal term, the energy eigenvalues and the wave functions of the Schrodinger equation of the shifted Deng–Fan molecular potential are obtained, via the asymptotic iteration method. Rotational–vibrational energy eigenvalues of some diatomic molecules are presented, these results are in good agreement with other results in the literature. For these selected diatomic molecules, energy eigenvalues obtained are in much better agreement with the results obtained from the rotating Morse potential model for moderate values of rotational and vibrational quantum numbers. Furthermore, thermodynamic properties such as the vibrational mean U, specific heat C, free energy F and entropy S for the pure vibrational state in the classical limit for these energy eigenvalues are studied.

Multireference electron correlation methods with density matrix renormalisation group reference functions

Abstract: Recent advances in quantum chemical density matrix renormalisation group (DMRG) theory are presented. The DMRG, originally devised as an alternative to the exact diagonalisation in condensed matter physics, has become a powerful quantum chemical method for molecular systems that exhibit a multireference character, e.g., excited states, π-conjugated systems, transition metal complexes, and in particular for large systems by combining it with conventional multireference electron correlation methods. The capability of the current quantum chemical DMRG is demonstrated for an application involving the potential energy curve of the chromium dimer, which is one of the most demanding multireference systems and thus requires the best electronic structure treatment for non-dynamical and dynamical correlation as well as large basis sets.

Comparative assessment of the ELBA coarse-grained model for water

Abstract: The ELBA force field for water consists of a single spherical site embedded with a point dipole. This coarse-grained model is assessed here through the calculation of fundamental properties of bulk liquid water and the water–vapour interface. Accuracy and efficiency are evaluated and compared against simulations of standard three- and four-site atomistic models. For bulk liquid systems, ELBA reproduces accurately most of the investigated properties. However, the radial distribution function deviates from atomistic and experimental data, indicating a loss of local structure. The water–vapour interface, simulated over a range of temperatures from 300 to 600 K, is captured realistically in terms of its density distribution, and the accuracy in reproducing the experimental surface tension is as high as that of the best atomistic model. The critical temperature of ELBA is also found to be in excellent agreement with experiment. However, the interfacial electric field and surface potential are missing. The comp...

The influence of orbital rotation on the energy of closed-shell wavefunctions

Abstract: The orbital dependence of closed-shell wavefunction energies is investigated by performing doubly-occupied configuration interaction (DOCI) calculations, representing the most general class of these wavefunctions. Different local minima are examined for planar hydrogen clusters containing two, four, and six electrons applying (spin) symmetry-broken restricted, unrestricted, and generalised orbitals with real and complex coefficients. Contrary to Hartree–Fock (HF), restricted DOCI is found to properly break bonds and thus unrestricted orbitals, while providing a quantitative improvement of the energy, are not needed to enforce a qualitatively correct bond dissociation. For the beryllium atom and the BH diatomic, the lowest possible HF energy requests symmetry-broken generalised orbitals, whereas accurate results for DOCI can be obtained within a restricted formalism. Complex orbital coefficients are shown to increase the accuracy of HF and DOCI results in certain cases. The computationally inexpensive AP1r...

Ionic liquid confined in silica nanopores: Molecular Dynamics in the isobaric-isothermal ensemble

Abstract: Molecular dynamics simulations in the isobaric–isothermal ensemble are used to investigate the structure and dynamics of an ionic liquid confined at ambient temperature and pressure in hydroxylated amorphous silica nanopores. The use of the isobaric–isothermal ensemble allows estimating the effect of confinement and surface chemistry on the density of the confined ionic liquid. The structure of the confined ionic liquid is investigated using density profiles and structural order parameters while its dynamics is assessed by determining the mobility and ionic conductivity of the confined phase. Despite the important screening of the electrostatic interactions (owing to the small Debye length in ionic liquids), the local structure of the confined ionic liquid is found to be mostly driven by electrostatic interactions. We show that both the structure and dynamics of the confined ionic liquid can be described as the sum of a surface contribution arising from the ions in contact with the surface and a bulk-like...

Modelling of the thermodynamic and solvation properties of electrolyte solutions with the statistical associating fluid theory for potentials of variable range

Abstract: An improved formulation of the extension of the statistical associating fluid theory for potentials of variable range to electrolytes (SAFT-VRE) is presented, incorporating a representation for the dielectric constant of the solution that takes into account the temperature, density and composition of the solvent. The proposed approach provides an excellent correlation of the dielectric-constant data available for a number of solvents including water, representative alcohols and carbon dioxide, and it is shown that the methodology can be used to treat mixed-solvent electrolyte solutions. Models for strong electrolytes of the metal-halide family are considered here. The salts are treated as fully dissociated and ion-specific interaction parameters are presented. Vapour pressure, density, and mean ionic activity coefficient data are used to determine the ion–ion and solvent–ion parameters, and mixed-salt electrolyte solutions (brines) are then treated predictively. We find that the resulting intermolecular p...

VPT2+K spectroscopic constants and matrix elements of the transformed vibrational Hamiltonian of a polyatomic molecule with resonances using Van Vleck perturbation theory

Abstract: Vibrational levels of polyatomic molecules are analysed with Van Vleck perturbation theory to connect experimental energy levels to computed molecular potential energy surfaces. Vibrational matrix elements are calculated from a quartic potential function via second-order Van Vleck perturbation theory, a procedure that treats both weak and strong interactions among vibrational states by approximately block-diagonalising the vibrational Hamiltonian. A clear and complete derivation of anharmonic and resonance constants as well as general expressions for both on- and off-diagonal matrix elements of the transformed Hamiltonian is presented. The equations are written in partial fraction form and as a constant multiplied by a harmonic oscillator matrix element to facilitate removing the effect of strongly interacting resonant states both in analytical formulae and in computer code. The derived equations are validated numerically, and results for the isotopomers of formaldehyde (H2CO, HDCO, D2CO) are included. Th...

Atomic-scale modelling of elastic and failure properties of clays

Abstract: The elastic and failure properties of a typical clay, illite, are investigated using molecular simulation We employ a reactive (ReaxFF) and a non-reactive (ClayFF) force field to assess the elastic properties of the clay As far as failure is concerned, ReaxFF was used throughout the study; however, some calculations were also performed with ClayFF A crack parallel to the clay layers is found to have low fracture resistance when submitted to a tensile loading perpendicular to the crack The mechanism of both yield and fracture failures is decohesion in the interlayer space In contrast, under shear loading, the nanoscale failure mechanism is a stick-slip between clay layers No fracture propagation is observed as the clay layers slide on top of each other The low fracture resistance in mode I and the stick-slip failure in mode II are both the consequence of the lack of chemical bonds between clay layers where the cohesion is provided by non-covalent interactions This work, which provides a description

The (100), (111) and (110) surfaces of diamond: an ab initio B3LYP study

Abstract: We present an accurate ab initio study of the structure and surface energy of the low-index (100), (111) and (110) diamond faces, by using the hybrid Hartree–Fock/density functional B3LYP Hamiltonian and a localised all-electron Gaussian-type basis set. A two-dimensional periodic slab model has been adopted, for which convergence on both structural and energetic parameters has been thoroughly investigated. For all the three surfaces, possible relaxations and reconstructions have been considered; a detailed geometrical characterisation is provided for the most stable structure of each orientation. Surface energy is discussed for all the investigated faces.

Dynamic nuclear polarisation via the integrated solid effect I: theory

Abstract: In the hyperpolarisation method known as dynamic nuclear polarisation (DNP), a small amount of unpaired electron spins is added to the sample containing the nuclear spins and the polarisation of these unpaired electron spins is transferred to the nuclear spins by means of a microwave field. Traditional DNP uses weak continuous wave (CW) microwave fields, so perturbation methods can be used to calculate the polarisation transfer. A much faster transfer of the electron spin polarisation is obtained with the integrated solid effect (ISE) which uses strong pulsed microwave fields. As in nuclear orientation via electron spin locking, the polarisation transfer is coherent, similar to the coherence transfer between nuclear spins. This paper presents a theoretical approach to calculate this polarisation transfer.ISE is successfully used for a fast polarisation transfer from short-lived photo-excited triplet states to the surrounding nuclear spins in molecular crystals. These triplet states are strongly aligned in...

Why colloidal systems can be described by statistical mechanics: some not very original comments on the Gibbs paradox

Abstract: Colloidal particles are distinguishable. Moreover, their thermodynamic properties are extensive. Statistical mechanics predicts such behaviour if one accepts that the configurational integral of a system of N colloids must be divided by N!. In many textbooks, it is argued that the factor N! corrects for the fact that identical particles (in the quantum mechanical sense) are indistinguishable. Clearly, this argument does not apply to colloids. This article explains why, nevertheless, all is well. The point has been made before, but has not yet sunk in. I also discuss the effect of polydispersity.

Generalised adiabatic connection in ensemble density-functional theory for excited states: example of the H2 molecule

Abstract: A generalised adiabatic connection for ensembles (GACE) is presented. In contrast to the traditional adiabatic connection formulation, both ensemble weights and interaction strength can vary along a GACE path while the ensemble density is held fixed. The theory is presented for non-degenerate two-state ensembles but it can in principle be extended to any ensemble of fractionally occupied excited states. Within such a formalism an exact expression for the ensemble exchange–correlation density-functional energy, in terms of the conventional ground-state exchange–correlation energy, is obtained by integration over the ensemble weight. Stringent constraints on the functional are thus obtained when expanding the ensemble exchange–correlation energy through second order in the ensemble weight. For illustration purposes, the analytical derivation of the GACE is presented for the H2 model system in a minimal basis, leading thus to a simple density-functional approximation to the ensemble exchange–correlation ener...

Novel (Li2X)+(LiX2)− supersalts (X = F, Cl) with aromaticity: a journey towards the design of a new class of salts

Abstract: LiF2 and Li2F can be considered as representatives of main group superhalogens and superalkali species, respectively. For the first time, we present a study on the interactions between LiX2 and Li2X, for X = F, Cl. Our findings show that this interaction leads to the formation of ring-shaped Li3X3 supersalts. The quantum theory of atoms in molecule (QTAIM) approach is used to discuss superatomic bonding in these novel species. The aromatic character of Li3X3 rings is established by QTAIM in addition to various chemical-reactivity-based measures. Thus, the present work opens up an avenue to further investigate these new classes of aromatic species, theoretically as well as to synthesise them, experimentally.

Density-fitted singles and doubles coupled cluster on graphics processing units

Abstract: We adapt an algorithm for singles and doubles coupled cluster (CCSD) that uses density fitting or Cholesky decomposition (CD) in the construction and contraction of all electron repulsion integrals (ERIs) for use on heterogeneous compute nodes consisting of a multicore central processing unit (CPU) and at least one graphics processing unit (GPU). The use of approximate three-index ERIs ameliorates two of the major difficulties in designing scientific algorithms for GPUs: (1) the extremely limited global memory on the devices and (2) the overhead associated with data motion across the bus. For the benzene trimer described by an aug-cc-pVDZ basis set, the use of a single NVIDIA Tesla C2070 (Fermi) GPU accelerates a CD-CCSD computation by a factor of 2.1, relative to the multicore CPU-only algorithm that uses six highly efficient Intel Core i7-3930K CPU cores. The use of two Fermi GPUs provides an acceleration of 2.89, which is comparable to that observed when using a single NVIDIA Kepler K20c GPU (2.73).

Adsorption of H2S at Stone–Wales defects of graphene-like BC3: a computational study

Abstract: We employed density functional theory to characterise H2S adsorption, and dissociation on the pristine and Stone–Wales (SW) defected BC3 graphenes. H2S is predicted to be weakly adsorbed on the pristine graphene with the adsorption energy of about 7.11 kcal/mol. Two types of SW defects were generated by rotating a C–C bond (SW-CC) or a B–C bond (SW-BC) by about 90°. We predict that, in contrast to SW-BC, dehydrogenation of H2S is energetically more favourable on the SW-CC compared to the associative adsorption. It is also found that SW-CC formation is more favourable than the formation of SW-BC. Molecular adsorption of H2S on both of the SW defected sheets is more favourable than that on the pristine sheet. The preferable adsorption process on the SW-BC and SW-CC defected graphene sheets is via associative and dissociative mechanisms, respectively. Furthermore, the highest occupied molecular orbital and lowest unoccupied molecular orbital energy gap of the SW-BC defected sheet is highly sensitive to the a...

Hydration of the chloride ion in concentrated aqueous solutions using neutron scattering and molecular dynamics

Abstract: Neutron scattering experiments were performed on 6 m LiCl solutions in order to obtain the solvation structure around the chloride ion. Molecular dynamics simulations on systems mirroring the concentrated electrolyte conditions of the experiment were carried out with a variety of chloride force-fields. In each case the simulations were run with both full ionic charges and employing the electronic continuum correction (implemented through charge scaling) to account effectively for electronic polarisation. The experimental data were then used to assess the successes and shortcomings of the investigated force-fields. We found that due to the very good signal-to-noise ratio in the experimental data, they provide a very narrow window for the position of the first hydration shell of the chloride ion. This allowed us to establish the importance of effectively accounting for electronic polarisation, as well as adjusting the ionic size, for obtaining a force-field which compares quantitatively to the experimental ...

A theoretical study of substitution effects on halogen–π interactions

Abstract: Ab initio calculations are performed to analyse the existence of intermolecular halogen···π interactions in NCX complexes with YC≡CY, where X = Cl, Br and Y = H, CN, F, Cl, OH, NH2, and CH3. Molecular geometries and interaction energies of the complexes are investigated at the MP2/aug-cc-pVTZ level of theory. Our results indicate that the interaction energies for the NCX···YC≡CY complexes lie in the range between −0.5 and −5.9 kcal/mol. The physical nature of the interactions is studied using symmetry-adapted perturbation theory (SAPT). The stability of the X···π interactions is predicted to be attributable mainly to electrostatic and dispersion effects.

Dipolar response and hydrogen-bond kinetics in liquid water in square-wave time-varying electric fields

Abstract: Non-equilibrium molecular dynamics simulations of liquid water have been performed at 298 K in the presence of external time-varying electric fields, approximating a square wave, of varying peak intensity (0.005–0.1 V/A) in the microwave to far-infrared frequency range (20–500 GHz). Significant non-thermal field effects were noted in terms of dipolar response and acceleration of hydrogen-bond kinetics. The coupling between the total dipole moment and the external field has been investigated and autocorrelation functions (ACFs) of both the total dipole moment and the average of the individual molecular dipole moment along the laboratory axis of the applied fields exhibited coupling, with the former showing a stronger coupling and the latter showing coupling to lower magnitude fields. The maximum alignment achieved has been computed as a function of field intensities and frequencies: the lower frequencies show a greater maximum alignment as the system had more time within each field cycle to respond. The no...

New insights into the extraction of uranium(VI) by an N,N-dialkylamide

Abstract: This paper presents the methodology developed in order to thoroughly characterise a solvent extraction system containing high solute concentrations. The chemical system selected is N,N-(2-ethylhexyl)isobutyramide (DEHiBA) diluted in one alkane with increasing concentration of uranium(VI). Combining experiments with theoretical calculations allowed a deeper understanding of the extraction mechanism. A thermodynamic study was performed by the classical van't Hoff method and also by direct calorimetry to provide the enthalpies of extraction and specific heats. Dedicated methods like vapour pressure osmometry and electrospray ionisation mass spectrometry analysis provide information about the stoichiometry of the extracted species. Spectroscopic investigations with ultraviolet–visible and Fourier transform infrared probed the uranium coordination. Finally, a combination of molecular dynamics simulations, and small and wide-angle X-ray scattering experiments investigated the organisation in the organic phase b...

Prediction of the thermophysical properties of molten salt fast reactor fuel from first-principles

Abstract: Molten fluorides are known to show favourable thermophysical properties which make them good candidate coolants for nuclear fission reactors. Here we investigate the special case of mixtures of lithium fluoride and thorium fluoride, which act both as coolant and as fuel in the molten salt fast reactor concept. By using ab initio parameterised polarisable force fields, we show that it is possible to calculate the whole set of properties (density, thermal expansion, heat capacity, viscosity and thermal conductivity) which are necessary for assessing the heat transfer performance of the melt over the whole range of compositions and temperatures. We then deduce from our calculations several figures of merit which are important in helping the optimisation of the design of molten salt fast reactors.

Solid-state NMR for bacterial biofilms

Abstract: Bacteria associate with surfaces and one another by elaborating an extracellular matrix to encapsulate cells, creating communities termed biofilms. Biofilms are beneficial in some ecological niches, but also contribute to the pathogenesis of serious and chronic infectious diseases. New approaches and quantitative measurements are needed to define the composition and architecture of bacterial biofilms to help drive the development of strategies to interfere with biofilm assembly. Solid-state NMR is uniquely suited to the examination of insoluble and complex macromolecular and whole-cell systems. This article highlights three examples that implement solid-state NMR to deliver insights into bacterial biofilm composition and changes in cell-wall composition as cells transition to the biofilm lifestyle. Most recently, solid-state NMR measurements provided a total accounting of the protein and polysaccharide components in the extracellular matrix of an E. coli biofilm and transform our qualitative descriptions of matrix composition into chemical parameters that permit quantitative comparisons among samples. We present additional data for whole biofilm samples (cells plus the extracellular matrix) that complement matrix-only analyses. The study of bacterial biofilms by solid-state NMR is an exciting avenue ripe with many opportunities and we close the article by articulating some outstanding questions and future directions in this area.

Similarity transformed equation of motion coupled cluster theory revisited: a benchmark study of valence excited states

Abstract: The similarity transformed equation of motion coupled cluster (STEOM-CC) method is benchmarked against CC3 and EOM-CCSDT-3 for a large test set of valence excited states of organic molecules studied by Schreiber et al. [M. Schreiber, M.R. Silva-Junior, S.P. Sauer, and W. Thiel, J. Chem. Phys. 128, 134110 (2008)]. STEOM-CC is found to behave quite satisfactorily and provides significant improvement over EOM-CCSD, CASPT2 and NEVPT2 for singlet excited states, lowering standard deviations of these methods by almost a factor of 2. Triplet excited states are found to be described less accurately, however. Besides the parent version of STEOM-CC, additional variations are considered. STEOM-D includes a perturbative correction from doubly excited determinants. The novel STEOM-H (ω) approach presents a sophisticated technique to render the STEOM-CC transformed Hamiltonian hermitian. In STEOM-PT, the expensive CCSD step is replaced by many-body second-order perturbation theory (MBPT(2)), while extended STEOM (EXT-S...

Cooperativity effects betweenσ-hole interactions: a theoretical evidence for mutual influence between chalcogen bond and halogen bond interactions in F2S···NCX···NCY complexes (X = F, Cl, Br, I; Y = H, F, OH)

Abstract: Quantum chemical calculations are performed to study the cooperativity effects between chalcogen bond and halogen bond interactions in F2S···NCX···NCY complexes, where X = F, Cl, Br, I and Y = H, F, OH. These effects are investigated in terms of geometric and energetic features of the complexes, which are computed by second-order Moller–Plesset perturbation theory (MP2). For each F2S···NCX···NCY complex studied, the effect of cooperativity on the chalcogen bond is dependent on the strength of halogen bond. The results indicate that the interaction energies of chalcogen and halogen bonds in the triads are more negative relative to the respective dyads. The interaction energy of chalcogen bond is increased by 31%–49%, whereas that of halogen bond by 28%–62%. The energy decomposition analysis reveals that electrostatic force plays a main role in the cooperativity effects between the chalcogen bond and halogen bond interactions. The topological analysis, based on the quantum theory of atoms in molecules, is u...

Computation of high-order virial coefficients in high-dimensional hard-sphere fluids by Mayer sampling

Abstract: The Mayer sampling method was used to compute the virial coefficients of high-dimensional hard-sphere fluids. The first 64 virial coefficients for dimensions 12 < D ⩽ 100 were obtained to high precision, and several lower dimensional virial coefficients were computed. The radii of convergence of the virial series in 13, 15, 17 and 19 dimensions agreed well with the analytical results from the Percus–Yevick closure.

Hydrogen–vacancy interaction in bcc iron: ab initio calculations and thermodynamics

Abstract: The paper presents results of ab initio modelling of formation energies of vacancy–hydrogen complexes VHn and an extended variant of thermodynamic theory describing equilibrium concentrations of such complexes. A single H atom is shifted from vacancy to a neighbouring O-site by 1.19 A. Two H atoms in a vacancy form a dumbbell with H–H distance of 2.38 A being much greater than in H2 molecule. Configurations of three, four and five H atoms in a vacancy are more complex, and H–H distances gradually increase showing repulsion between hydrogen atoms. Binding energy of a VHn−1 complex with the next hydrogen atom to form VHn is 0.60, 0.61, 0.39, 0.37 and 0.31 for n = 1–5, which is close to other researchers’ data. These results were used to construct an improved variant of thermodynamic description of vacancy–hydrogen interaction in a bcc solid solution taking into account both binding energies and hydrogen atom configurations in different VHn complexes. Calculations show that at low temperatures most vacancies...

Does the G·G*syn DNA mismatch containing canonical and rare tautomers of the guanine tautomerise through the DPT? A QM/QTAIM microstructural study

Abstract: We have established that the asynchronous concerted double proton transfer (DPT), moving with a time gap and without stable intermediates, is the underlying mechanism for the tautomerisation of the G·G*syn DNA base mispair (C1 symmetry), formed by the keto and enol tautomers of the guanine in the anti- and syn-configurations, into the G*·G*syn base mispair (C1), formed by the enol and imino tautomers of the G base, using quantum-mechanical calculations and Bader's quantum theory of atoms in molecules. By constructing the sweeps of the geometric, electron-topological, energetic, polar and natural bond orbital properties along the intrinsic reaction coordinate of the G·G*syn↔G*·G*syn DPT tautomerisation, the nine key points, that are critical for the atomistic understanding of the tautomerisation reaction, were set and comprehensively analysed. It was found that the G·G*syn mismatch possesses pairing scheme with the formation of the O6···HO6 (7.01) and N1H···N7 (6.77) H-bonds, whereas the G*·G*syn mismatch ...

Molecular dynamics simulations of interlayer structure and mobility in hydrated Li-, Na- and K-montmorillonite clays

Abstract: Molecular dynamics (MD) simulations were carried out to study the hydrate structure and diffusion of water and cations in the clay interlayer region The goal of this work was to examine the mechanisms by which the clay counterions influence the swelling and dispersion of shale Three series of MD simulations were performed to study Wyoming montmorillonites containing either Na+, K+ or Li+ counterions and 100, 200 or 300 mg of adsorbed water per gram of clay Here, the TIP4P interaction model for liquid water is used in the MD simulations The TIP4P results are compared with simulations using the MCY interaction model, particularly for the K-clay hydrates For the one- and two-layer K-hydrates, the counterion speciation and water structure observed are in agreement with the MCY model However, we observe that the two-layer hydrate persists up to high water contents and that – contrary to the MCY model – the hypothetical three-layer hydrate is unstable, in agreement with experimental observations If one n