Three-body Terms Research Articles

Full-dimensional, ab initio potential energy and dipole moment surfaces for water

We report full-dimensional, ab initio potential energy (PES) and dipole moment surfaces (DMS) for water. The PES is a sum of one-, two- and three-body terms. The three-body potential is a fit, reported here, to roughly 30,000 intrinsic three-body energies obtained with second-order Møller-Plesset perturbation theory (MP2) and using the aug-cc-pVTZ basis set (avtz). The one- and two-body potentials are from an ab initio water dimer potential [Shank et al., J. Chem. Phys. 130, 144314 (2009)]. The predictive accuracy of the PES is demonstrated for the water trimer, tetramer, and hexamer by comparing the energies and harmonic frequencies obtained from the PES and new high level ab initio calculations at the respective global minima. The DMS is constructed from one- and two-body dipole moments, based on fits to MP2/avtz dipole moments. It is shown to be very accurate for the hexamer by comparison with direct calculations of the hexamer dipole. To illustrate the anharmonic character of the PES one-mode calculations of the 18 monomer fundamentals of the hexamer are reported in normal coordinates.

Accuracy of fragmentation in ab initio calculations of hydrated sodium cation

We discuss the accuracy of fragmentation in fragment molecular orbital (FMO) calculations for the hydrated sodium ion. Two-body expansion shows a considerable error in total energy even if water molecules in the second hydration shell are included in the same fragment as the sodium ion. Inclusion of the three-body term significantly improves both the total energy and the charge distributions. We also illustrate the dependence of the net charge of sodium ion on solvent size and the interfacial property of water molecules. The present study will thus provide fundamental information about hydrated ion to facilitate further theoretical and experimental studies.

Quantum chemical investigation of the adsorption of methanol on a cluster model of faujasite

The adsorption of methanol in zeolites of the faujasite type with sodium and calcium counter ions is studied with quantum chemical methods. The zeolite is represented with a cluster model allowing calculations at the Møller-Plesset as well as the DFT level of theory. An adsorption energy of −62.4 kJ/mol is calculated at the MP2/TZVP//BP86/TZVP level of theory for one methanol molecule at one site II sodium cation. Insight into the adsorption process is obtained with a three-body decomposition which reveals a strong destabilisation of the adsorption strength by large, positive three-body terms, which is important for force field development.

Development of generalized potential-energy surfaces using many-body expansions, neural networks, and moiety energy approximations

A general method for the development of potential-energy hypersurfaces is presented. The method combines a many-body expansion to represent the potential-energy surface with two-layer neural networks (NN) for each M-body term in the summations. The total number of NNs required is significantly reduced by employing a moiety energy approximation. An algorithm is presented that efficiently adjusts all the coupled NN parameters to the database for the surface. Application of the method to four different systems of increasing complexity shows that the fitting accuracy of the method is good to excellent. For some cases, it exceeds that available by other methods currently in literature. The method is illustrated by fitting large databases of ab initio energies for Si(n) (n=3,4,...,7) clusters obtained from density functional theory calculations and for vinyl bromide (C(2)H(3)Br) and all products for dissociation into six open reaction channels (12 if the reverse reactions are counted as separate open channels) that include C-H and C-Br bond scissions, three-center HBr dissociation, and three-center H(2) dissociation. The vinyl bromide database comprises the ab initio energies of 71 969 configurations computed at MP4(SDQ) level with a 6-31G(d,p) basis set for the carbon and hydrogen atoms and Huzinaga's (4333/433/4) basis set augmented with split outer s and p orbitals (43321/4321/4) and a polarization f orbital with an exponent of 0.5 for the bromine atom. It is found that an expansion truncated after the three-body terms is sufficient to fit the Si(5) system with a mean absolute testing set error of 5.693x10(-4) eV. Expansions truncated after the four-body terms for Si(n) (n=3,4,5) and Si(n) (n=3,4,...,7) provide fits whose mean absolute testing set errors are 0.0056 and 0.0212 eV, respectively. For vinyl bromide, a many-body expansion truncated after the four-body terms provides fitting accuracy with mean absolute testing set errors that range between 0.0782 and 0.0808 eV. These errors correspond to mean percent errors that fall in the range 0.98%-1.01%. Our best result using the present method truncated after the four-body summation with 16 NNs yields a testing set error that is 20.3% higher than that obtained using a 15-dimensional (15-140-1) NN to fit the vinyl bromide database. This appears to be the price of the added simplicity of the many-body expansion procedure.

Approximating quantum many-body intermolecular interactions in molecular clusters using classical polarizable force fields

Many-body intermolecular interaction expansions provide a promising avenue for the efficient quantum mechanical treatment of molecular clusters and condensed-phase systems, but the computationally expensive three-body and higher terms are often nontrivial. When polar molecules are involved, these many-body terms are typically dominated by electrostatic induction effects, which can be approximated relatively easily. We demonstrate an accurate and inexpensive hybrid quantum/classical model in which one- and two-body interactions are computed quantum mechanically, while the many-body induction effects are approximated with a simple classical polarizable force field. Whereas typical hybrid quantum/classical models partition a system spatially into distinct quantum and classical regions, the model demonstrated here partitions based on the order in the many-body interaction series. This enables a spatially homogeneous treatment of the entire system, which could prove advantageous in studying a wide range of condensed-phase molecular systems.

Interfacial potentials for Al/SiC(111)

To study the metal/semiconductor interface by means of atomistic simulation, an effectiveinterfacial potential is an important issue. In this work, ab initio adhesive energies are usedto derive interfacial potentials for the Al/SiC(111) interface. In order to describe thedirectional covalent bonds at the interface, we suggest a potential model comprisingboth two-body and three-body terms. The former is a parameter-free potentialobtained by a lattice inversion method and the latter is assigned in modifiedStillinger–Weber potential form. The obtained potentials are used to study theposition of misfit dislocations in the Al/SiC(111) interface. There is a coherentAl interlayer on the interface plane and the dislocation appears on the Al side.

Interaction potential for indium phosphide: a molecular dynamics and first-principles studyof the elastic constants, generalized stacking fault and surface energies

Indium phosphide is investigated using molecular dynamics (MD) simulations anddensity-functional theory calculations. MD simulations use a proposed effective interactionpotential for InP fitted to a selected experimental dataset of properties. The potentialconsists of two- and three-body terms that represent atomic-size effects, charge–charge,charge–dipole and dipole–dipole interactions as well as covalent bond bending andstretching. Predictions are made for the elastic constants as a function of density andtemperature, the generalized stacking fault energy and the low-index surface energies.

Adiabatic hyperspherical study of triatomic helium systems

The $^{4}\mathrm{He}_{3}$ system is studied using the adiabatic hyperspherical representation. We adopt the current state-of-the-art helium interaction potential including retardation and the nonadditive three-body term to calculate all low-energy properties of the triatomic $^{4}\mathrm{He}$ system. The bound state energies of the $^{4}\mathrm{He}$ trimer are computed as well as the $^{4}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the three-body recombination, and collision induced dissociation rates at finite temperatures. We also treat the system that consists of two $^{4}\mathrm{He}$ and one $^{3}\mathrm{He}$ atoms, and compute the spectrum of the isotopic trimer $^{4}\mathrm{He}_{2}^{3}\mathrm{He}$, the $^{3}\mathrm{He}+^{4}\mathrm{He}_{2}$ elastic scattering cross sections, the rates for three-body recombination, and the collision induced dissociation rate at finite temperatures. The effects of retardation and the nonadditive three-body term are investigated. Retardation is found to be significant in some cases, while the three-body term plays only a minor role for these systems.

Application of Double Ionization State-Specific Equation of Motion Coupled Cluster Method to Organic Diradicals

The state-specific equation of motion coupled cluster method is applied to three systems of diradical character: automerization of cyclobutadiene, singlet-triplet gaps of trimethylmethylene, and Bergman reaction. The aim of the paper is to assess the performance of the method and test numerically the importance of orbital optimization, three-body terms in transformed Hamiltonian, and the choice of cluster equations.

Solids and Supersolids of Three-Body Interacting Polar Molecules on an Optical Lattice

We study the physics of cold polar molecules loaded into an optical lattice in the regime of strong three-body interactions, as put forward recently by Büchler et al. [Nature Phys. 3, 726 (2007)]. To this end, quantum Monte Carlo simulations, exact diagonalization, and a semiclassical approach are used to explore hard-core bosons on the 2D square lattice which interact solely by long-ranged three-body terms. The resulting phase diagram shows a sequence of solid and supersolid phases. Our findings are directly relevant for future experimental implementations and open a new route towards the discovery of a lattice supersolid phase in experiment.

Incremental expansions for SCF interaction energies: A comparison for hydrogen-bonded clusters

We present an analysis of several types of embedding potentials for use within incremental expansions of SCF interaction energies. The test set comprises hydrogen-bonded clusters (water, ammonia, methanol). It is shown that an expansion up to two-body terms is sufficient for high accuracy, in all cases, as long as the environment is described by accurate embedded-monomer Coulomb and exchange potentials. A two-body expansion with atom-centered point charges for embedding leads to larger errors, but the relative differences are kept within bounds; inclusion of three-body terms often overcompensates the deviations at the two-body level, due to basis set superposition effects.

Potential energy surfaces for the 1 A4′, 2 A4′ 1 A4″ and 2 A4″ states of Li3

Global potential energy surfaces for the 1 (4)A('), 2 (4)A('), 1 (4)A("), and 2 (4)A(") spin-aligned states of Li(3) are constructed as sums of a diatomics-in-molecules (DIM) term plus a three-body term. The DIM model, using a large basis set of 15 (4)A(") and 22 (4)A(') states, is used to obtain a "mixed-pairwise additive" contribution to the potential. A global fit of the three-body terms conserves the accuracy of the ab initio points of a full configuration-interaction calculation. The resulting fit accurately describes conical intersections for both the 1 (4)A(') and 2 (4)A(') surfaces with a root-mean-square (rms) deviation of 5.4x10(-5) hartree in D(infinityh) geometries and 1.2x10(-4) hartree in C(infinityv) geometries. The global fit appears to be quantitatively correct with a rms deviation of 1.8x10(-4)hartree for 1 (4)A('), 9.2x10(-4) hartree for 2 (4)A('), 2.5x10(-4) hartree for 1 (4)A("), and 5.1x10(-4) hartree for 2 (4)A("). A possible diabolic conical intersection, also called an accidental degeneracy, in C(2v) geometries, indicating a seam of conical intersections in C(s) geometries, is also found in ab initio calculations for A(2) states. As shown in this example, the DIM procedure can be optimized to describe the geometric phase and nonadiabatic effects in multisurface potentials.

Hyperchaos of two coupled Bose–Einstein condensates with a three-body interaction

We investigate the dynamics of two tunnel-coupled Bose-Einstein condensates (BECs) in a double-well potential. The effects of the three-body recombination loss and the feeding of the condensates from the thermal cloud are studied in the case of attractive interatomic interaction. An imaginary three-body interaction term is considered and a two-mode approximation is used to derive three coupled equations which describe the total atomic numbers of the two condensates, the relative population and relative phase respectively. Theoretical analyses and numerical calculations demonstrate the existence of chaotic and hyperchaotic behaviour by using a periodically time-varying scattering length.

Unstable Cycle of Two Coupled Bose-Einstein Condensates with Three-Body Interaction

We investigate the dynamics of two Bose-Einstein condensates (BECs) tunnel-coupled by a double-well potential. The effects of the three-body recombination losses and the feeding of the condensates from the thermal cloud are studied in the case of attractive interatomic interaction. An imaginary three-body interaction term is considered and a two-mode approximation is used to derive three coupled equations, which describe the total atomic numbers of the two condensates, the relative population and relative phase. Theoretical analysis and numerical calculations demonstrate the existence of chaotic and unstable cycle behavior for a periodically time-varying scattering length.

A parametrization of the baryon octet and decuplet masses

We construct a general parametrization of the baryon octet and decuplet mass operators including the three-body terms using the unit operator and the symmetry-breaking factors Md = diag(0, 1, 0) and Ms = diag(0, 0, 1) in conjunction with the spin operators. Our parametrization has the minimal number of operators needed to describe all the octet and decuplet masses. Investigating the likely size of the three-body terms, we find that contributions of the three-body hypercharge splittings are comparable to those from the one- and two-body isospin splittings and that contributions of the three-body isospin splitting operators are very small. We prove that, in dynamical calculations, one must go to three loops to get the three-body terms. We also find that the suggested hierarchy of sizes for terms in the most general expression for baryon masses that involve multiple factors of Md and/or Ms does not hold strictly for dynamical calculations in heavy-baryon chiral perturbation theory: terms of a given order in a meson loop expansion may appear both with the expected factors of Md and Ms, and with one factor more.

Molecular dynamics simulations on the local order of liquid and amorphous ZnTe

Molecular dynamics studies of structural and dynamical correlations of molten and vitreous states under several conditions of density and temperature were performed. We use an effective recently proposed interatomic potential, consisting of two- and three-body covalent interactions which has successfully described the structural, dynamical, and structural phase transformation induced by pressure in ZnTe [D. S. Borges and J. P. Rino, Phys. Rev. B 72, 014107 (2005)]. The two-body term of the interaction potential consists of Coulomb interaction resulting from charge transfer, steric repulsion due to atomic sizes, charge-dipole interaction to include the effect of electronic polarizability of anions, and dipole-dipole (van der Waals) interactions. The three-body covalent term is a modification of the Stillinger-Weber potential. Molecular dynamics simulations in isobaric-isenthalpic ensemble have been performed for systems amounting to 4096 and 64 000 particles. Starting from a crystalline zinc-blende (ZB) structure, the system is initially heated until a very homogeneous liquid is obtained. The vitreous zinc telluride phase is attained by cooling the liquid at sufficiently fast cooling rates, while slower cooling rates lead to a disordered ZB crystalline structure. Two- and three-body correlations for the liquid and vitreous phases are analyzed through pair distribution functions, static structure factors, and bond angle distributions. In particular, the neutron static structure factor for the liquid phase is in very good agreement with both the reported experimental data and first-principles simulations.

Interaction potentials for alumina and molecular dynamics simulations of amorphous and liquid alumina

Structural and dynamical properties of crystalline alumina α-Al2O3 and amorphous and molten alumina are investigated with molecular dynamics simulation based on an effective interatomic potentials consisting of two- and three-body terms. Structural correlations are examined through pair distribution functions, coordination numbers, static structure factors, bond angle distributions, and shortest-path ring analyses. The calculated results for neutron and x-ray static structure factors are in good agreement with experimental results. Dynamical correlations, such as velocity autocorrelation function, vibrational density of states, current-current correlation function, and frequency-dependent conductivity, are also discussed.

Note on spin–orbit interactions in nuclei and hypernuclei

A detailed comparison is made between the spin–orbit interactions in Λ hypernuclei and ordinary nuclei. We argue that there are three major contributions to the spin–orbit interaction: (1) a short-range component involving scalar and vector mean fields; (2) a “wrong-sign” spin–orbit term generated by the pion exchange tensor force in second order; and (3) a three-body term induced by two-pion exchange with excitation of virtual Δ(1232)-isobars (à la Fujita–Miyazawa). For nucleons in nuclei the long-range pieces related to the pion-exchange dynamics tend to cancel, leaving room dominantly for spin–orbit mechanisms of short-range origin (parametrized, e.g., in terms of relativistic scalar and vector mean fields terms). In contrast, the absence of an analogous 2π-exchange three-body contribution for Λ hyperons in hypernuclei leads to an almost complete cancellation between the short-range (relativistic mean-field) component and the “wrong-sign” spin–orbit interaction generated by second order π-exchange with an intermediate Σ hyperon. These different balancing mechanisms between short- and long-range components are able to explain simultaneously the very strong spin–orbit interaction in ordinary nuclei and the remarkably weak spin–orbit splitting in Λ hypernuclei.

Correction: Signal Propagation in Proteins and Relation to Equilibrium Fluctuations

Correction for: Chennubhotla C, Bahar I (2007) Signal propagation in proteins and relation to equilibrium fluctuations. PLoS Comput Biol 3(9): e172. 10.1371/journal.pcbi.0030172 Four mathematical expressions appeared incorrectly. The correct expressions follow. The hitting time expression Equation 14 involves three different types of contributions: a one-body term that depends on the destination node, ; a two-body term that depends on the initial and final nodes, ; and a series of three-body terms that depend on intermediate nodes, in addition to the two end points, . Derivation of Equation 14. The discussion below borrows from results in [12,13]. Deriving from is a three-step process: …

Coupled-cluster theory for three-body Hamiltonians

We derive coupled-cluster equations for three-body Hamiltonians. The equations for the one- and two-body cluster amplitudes are presented in a factorized form that leads to an efficient numerical implementation. We employ low-momentum two- and three-nucleon interactions and calculate the binding energy of $^{4}\mathrm{He}$. The results show that the main contribution of the three-nucleon interaction stems from its density-dependent zero-, one-, and two-body terms that result from the normal ordering of the Hamiltonian in coupled-cluster theory. The residual three-body terms that remain after normal ordering can be neglected.