Two-body Terms Research Articles

Dynamic Charge Equilibration-Morse stretch force field: application to energetics of pure silica zeolites.

We present the Dynamic Charge Equilibration (DQEq) method for a self-consistent treatment of charge transfer in force field modeling, where atomic charges are designed to reproduce electrostatic potentials calculated quantum mechanically. Force fields coupled with DQEq allow charges to readjust as geometry changes in classical simulations, using appropriate algorithms for periodic boundary conditions. The full electrostatic energy functional is used to derive the corresponding forces and the second derivatives (hessian) for vibrational calculations. Using DQEq electrostatics, we develop a simple nonbond force field for simulation of silica molecular sieves, where nonelectrostatic interactions are described by two-body Morse stretch terms. Energy minimization calculations with the new force field yield accurate unit cell geometries for siliceous zeolites. Relative enthalpies with respect to quartz and third-law entropies calculated from harmonic vibrational analysis agree very well with available calorimetric data: calculated SiO(2) enthalpies relative to alpha-quartz are within 2 kJ/mol and entropies at 298 K are within 3 J/mol K of the respective experimental values. Contributions from the zero point energy and vibrational degrees of freedom were found to be only about 1 kJ/mol for the free energy of mutual transformations between microporous silica polymorphs. The approach presented here can be applied to interfaces and other oxides as well and it is suitable for development of force fields for accurate modeling of geometry and energetics of microporous and mesoporous materials, while providing a realistic description of electrostatic fields near surfaces and inside pores of adsorbents and catalysts.

Detailed analysis of the three-quark potential in SU(3) lattice QCD

The static three-quark (3Q) potential is studied in detail using SU(3) lattice QCD with $12^3 \times 24$ at $\beta=5.7$ and $16^3 \times 32$ at $\beta=5.8$, 6.0 at the quenched level. For more than 300 different patterns of the 3Q systems, we perform the accurate measurement of the 3Q Wilson loop with the smearing method, which reduces excited-state contaminations, and present the lattice QCD data of the 3Q ground-state potential $V_{\rm 3Q}$. We perform the detailed fit analysis on $V_{\rm 3Q}$ in terms of the Y-ansatz both with the continuum Coulomb potential and with the lattice Coulomb potential, and find that the lattice QCD data of the 3Q potential $V_{\rm 3Q}$ are well reproduced within a few % deviation by the sum of a constant, the two-body Coulomb term and the three-body linear confinement term $\sigma_{\rm 3Q} L_{\rm min}$, with $L_{\rm min}$ the minimal value of the total length of color flux tubes linking the three quarks. From the comparison with the Q-$\bar {\rm Q}$ potential, we find a universality of the string tension as $\sigma_{\rm 3Q} \simeq \sigma_{\rm Q \bar Q}$ and the one-gluon-exchange result for the Coulomb coefficients as $A_{\rm 3Q} \simeq \frac12 A_{\rm Q \bar Q}$. We investigate also the several fit analyses with the various ans\"atze: the Y-ansatz with the Yukawa potential, the $\Delta$-ansatz and a more general ansatz including the Y and the $\Delta$ ans\"atze in some limits. All these fit analyses support the Y-ansatz on the confinement part in the 3Q potential $V_{\rm 3Q}$, although $V_{\rm 3Q}$ seems to be approximated by the $\Delta$-ansatz with $\sigma_\Delta \simeq 0.53 \sigma$.

Weak transitions inA=6and 7 nuclei

The ${}^{6}\mathrm{He}$ $\ensuremath{\beta}$ decay and ${}^{7}\mathrm{Be}$ electron capture processes are studied using variational Monte Carlo wave functions, derived from a realistic Hamiltonian consisting of the Argonne ${v}_{18}$ two-nucleon and Urbana-IX three-nucleon interactions. The model for the nuclear weak axial current includes one- and two-body operators with the strength of the leading two-body term---associated with $\ensuremath{\Delta}$-isobar excitation of the nucleon---adjusted to reproduce the Gamow-Teller matrix element in tritium $\ensuremath{\beta}$ decay. The measured half-life of ${}^{6}\mathrm{He}$ is underpredicted by theory by $\ensuremath{\simeq}8%,$ while that of ${}^{7}\mathrm{Be}$ for decay into the ground and first excited states of ${}^{7}\mathrm{Li}$ is overpredicted by $\ensuremath{\simeq}9%.$ However, the experimentally known branching ratio for these latter processes is in good agreement with the calculated value. Two-body axial current contributions lead to a $\ensuremath{\simeq}1.7%$ (4.4%) increase in the value of the Gamow-Teller matrix element of ${}^{6}\mathrm{He}$ ${(}^{7}\mathrm{Be}),$ obtained with one-body currents only, and slightly worsen (appreciably improve) the agreement between the calculated and measured half-life. Corrections due to retardation effects associated with the finite lepton momentum transfers involved in the decays, as well as contributions of suppressed transitions induced by the weak vector charge and axial current operators, have also been calculated and found to be negligible.

Static three- and four-quark potentials

We present results for the static three- and four- quark potentials in SU(3) and SU(4) respectively. Using a variational approach, combined with multi-hit for the time-like links, we determine the ground state of the baryonic string with sufficient accuracy to test the Y − and Δ − ansätze for the baryonic Wilson area law. Our results favor the Δ ansatz, where the potential is the sum of two-body terms.

Variational quantum Monte Carlo calculations for solid surfaces

Quantum Monte Carlo methods have proven to predict atomic and bulk properties of light and non-light elements with high accuracy. Here we report on the first variational quantum Monte Carlo (VMC) calculations for solid surfaces. Taking the boundary condition for the simulation from a finite layer geometry, the Hamiltonian, including a nonlocal pseudopotential, is cast in a layer resolved form and evaluated with a two-dimensional Ewald summation technique. The exact cancellation of all Jellium contributions to the Hamiltonian is ensured. The many-body trial wave function consists of a Slater determinant with parameterized localized orbitals and a Jastrow factor with a common two-body term plus a new confinement term representing further variational freedom to take into account the existence of the surface. We present results for the ideal (110) surface of Galliumarsenide for different system sizes. With the optimized trial wave function, we determine some properties related to a solid surface to illustrate that VMC techniques provide standard results under full inclusion of many-body effects at solid surfaces.

Density functional theory of charged colloidal systems.

The phase behavior of charged colloidal systems has been studied recently by the density functional theory formalism (DFT) [R. van Roij, M. Dijkstra, and J. P. Hansen, Phys. Rev. E 59, 2010 (1999)]. A key feature of this approach is the appearance of a density and temperature-dependent effective Hamiltonian between the charged colloids. Under certain approximations, the effective Hamiltonian is made up only of a sum of position-independent one-body or volume terms and two-body colloid-separation dependent terms. In the limit of low colloidal densities, the DFT results do not reduce to the familiar Debye-Hückel limiting law nor do the results agree with previous work based on an identical approach but were developed using traditional statistical-mechanical methods [B. Beresford-Smith, D. Y. C. Chan, and D. J. Mitchell J. Colloid Interface Sci. 105, 216 (1985)]. This paper provides a reconciliation of these differences and comments on the significance of the one-body volume terms in the effective Hamiltonian of a system of charged colloids in determining thermodynamics and phase behavior.

Inhomogeneous random-phase approximation and many-electron trial wave functions

The long-range electronic correlations in a uniform electron gas may be deduced from the random-phase approximation (RPA) of Bohm and Pines [Phys. Rev. 92, 609 (1953)]. Here we generalize the RPA to nonuniform systems and use it to derive many-electron Slater-Jastrow trial wave functions for quantum Monte Carlo simulations. The RPA theory fixes the long-range behavior of the inhomogeneous two-body terms in the Jastrow factor and provides an accurate analytic expression for the one-body terms. It also explains the success of Slater-Jastrow trial functions containing determinants of Hartree-Fock or density-functional orbitals, even though these theories do not include Jastrow factors. After adjusting the RPA Jastrow factor to incorporate the known short-range behavior, we test it using variational Monte Carlo simulations. In the small inhomogeneous electron gas system we consider, the analytic RPA-based Jastrow factor slightly outperforms the standard numerically optimized form. The inhomogeneous RPA theory therefore enables us to reduce or even avoid the costly numerical optimization process.

Ab initio studies of π-water tetramer complexes: Evolution of optimal structures, binding energies, and vibrational spectra of π-(H2O)n (n=1–4) complexes

The optimal structures, binding energies, and harmonic vibrational frequencies of clusters containing a substituted benzene molecule microsolvated by four water molecules, termed as π-(water tetramer) clusters (π: p-difluorobenzene, fluorobenzene, benzene, toluene) have been evaluated at the second order perturbation level of theory (MP2) using both the 6-31+G* and aug-cc-pVDZ basis sets. In sharp contrast to the complexes of smaller water clusters with these π systems, wherein the water subcluster is most strongly bound to toluene, the water tetramer is most strongly bound to fluorobenzene. This exceptionally high binding energy results from both a π⋅⋅⋅OH H-bond and a competing σ F⋅⋅⋅OH bond between the water tetramer moiety and the aromatic molecule. The magnitudes of the many-body energy terms and their contribution to the binding energies of these π-(water tetramer) systems indicates that the contributions of three- and higher-order terms are much smaller when compared to the neutral water clusters. The two-body terms associated with the π- and σ-type of interaction indicates that in both the fluorobenzene and p-difluorobenzene complexes, the increase in the size of the water cluster enhances the π-H-bonding interaction and weakens the σ F⋅⋅⋅H interaction. This observation is in consonance with the calculated and experimentally observed redshifts of the OH vibrational frequencies. Thus, with an increase in the size of a water cluster bound to the fluorinated π system, there is a lowering of the redshift induced by the σ F⋅⋅⋅H interaction and an increase in the redshift due to the π-H interaction. The calculated redshift of the π H-bonded OH mode is very much dependent on the basis set, with larger basis sets yielding shifts which are in better agreement with the experimentally determined shifts.

σ to π conformational transition: Interactions of the water trimer with π systems

We investigate the interaction of the water trimer with the aromatic π systems (benzene, toluene, fluorobenzene, and p-difluorobenzene) at the second-order Møller–Plesset level of theory using both the 6-31+G* and aug-cc-pVDZ basis sets. The minimum-energy structures, binding energies, and the harmonic vibrational frequencies were calculated for all these systems. The minimum-energy structures reveal that the water trimer has only a π type of interaction with benzene and toluene, but has both a π and σ type (H-bond with the fluorine atom) of interaction with fluorobenzene and p-difluorobenzene. The binding energies reveal that, contrary to what has been observed in previous theoretical studies, the strength of the interaction of these π systems with the water trimer is greater than with the water dimer. A good agreement is obtained between the calculated and experimentally measured vibrational frequencies of these complexes. An analysis of the calculated OH stretching modes reveals some unusual facets of the π and σ type of interaction observed in the fluorobenzene and p-difluorobenzene complexes. Thus one observes that the OH red-shift associated with the π type of interaction is larger than with the σ type of interaction. This is particularly interesting because in both the water monomer and water dimer complexes, the OH red-shift associated with the interaction of the water cluster with fluorobenzene and p-difluorobenzene is due to the formation of the H-bond with the fluorine atom. Thus the increase in the size of the water cluster seems to favor the π type of interaction over the σ type of interaction. This inference is also supported by the magnitude of the two-body terms associated with the π and σ type of interaction in both the fluorobenzene and p-difluorobenzene complexes.

Three-Quark Potential in SU(3) Lattice QCD.

The static three-quark ( 3Q) potential is studied in SU(3) lattice QCD with 12(3)x24 and beta = 5.7 at the quenched level. From the 3Q Wilson loop, 3Q ground-state potential V(3Q) is extracted using the smearing technique for ground-state enhancement. With accuracy better than a few percent, V(3Q) is well described by a sum of a constant, the two-body Coulomb term, and the three-body linear confinement term sigma(3Q)L(min), with L(min) the minimal value of total length of color flux tubes linking the three quarks. Comparing with the Q-&Qmacr; potential, we find a universal feature of the string tension, sigma(3Q) approximately sigma(Q&Qmacr;), and the OGE result for Coulomb coefficients, A(3Q) approximately 1 / 2A(Q&Qmacr;).

One-body density matrix and momentum distribution ins−pands−dshell nuclei

Analytical expressions of the one- and two-body terms in the cluster expansion of the one-body density matrix and momentum distribution of the $s\ensuremath{-}p$ and $s\ensuremath{-}d$ shell nuclei with $N=Z$ are derived. They depend on the harmonic oscillator parameter b and the parameter $\ensuremath{\beta}$ which originates from the Jastrow correlation function. These parameters have been determined by least squares fit to the experimental charge form factors. The inclusion of short-range correlations increases the high momentum component of the momentum distribution $n(\mathbf{k})$ for all nuclei we have considered while there is an A dependence of $n(\mathbf{k})$ both at small values of k and the high momentum component. The A dependence of the high momentum component of $n(\mathbf{k})$ becomes quite small when the nuclei ${}^{24}\mathrm{Mg},$ ${}^{28}\mathrm{Si},$ and ${}^{32}\mathrm{S}$ are treated as $1d\ensuremath{-}2s$ shell nuclei having the occupation probability of the $2s$ state as an extra free parameter in the fit to the form factors.

Empirical two-body potential for solid silicon nitride, boron nitride, and borosilazane modifications

A short-ranged classical force-field for the modeling of silicon nitrides, boron nitrides, and borosilazanes (Si/B/N) has been derived on the basis of experimental and ab initio solid state and diatomic molecular data. The potential is a sum of pure two-body interaction terms with functional forms apt for fraction simulation. Structural parameters of hypothetical ${\mathrm{Si}}_{3}{\mathrm{B}}_{3}{\mathrm{N}}_{7}$ polymorphs and crystalline boron and silicon nitrides containing threefold or fourfold coordinated boron and nitrogen and fourfold or sixfold coordinated silicon are well reproduced. The quality of second-order properties such as vibrational frequencies, bulk moduli, and elastic constants is more sensitive to the atomic coordination. Very good agreement with experiment is observed for BN and ${\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ modifications with trigonal nitrogen coordination.

Towards a test particle description of transport processes for states with continuous mass spectra

Based on a first order gradient expansion a consistent transport equation is derived for a nonrelativistic system beyond the quasiparticle approximation, i.e. for a regime where the dynamically generated width of the states is allowed to be large. An exactly conserved quantity is identified which is interpreted as an effective particle number obtained by coarse graining. Using a test particle ansatz for this conserved quantity allows to rewrite the transport equation into equations of motion for test particles. The two-body collision terms are formulated in terms of the test particles which gain non-trivial renormalization factors due to the coarse graining process.

The isospin-dependent quadrupole-quadrupole interaction used in shell model calculations – the effects of including ΔN = 2 excitations and a two-body spin-orbit interaction term

The effect of the “B” term in the interaction −χQ(1) · Q(2)[1+Bτ(1) ·τ(2)] was previously considered in the 0p shell (small space). It is now studied in a larger space which additionally includes ΔN = 2 excitations. When B is made sufficiently negative we still obtain for 10Be, even in the larger space, an unphysical collapse of some of the low-lying states so that their energies are less than the energy of the conventional J = 0+ ground state. This effect, however, occurs for values of B considerably more negative than was the case in the smaller space. It is shown that the inclusion of an additional two-body spin-orbit interaction term prevents this unrealistic collapse in both the large and small spaces.

Nonadditive three-body polarizabilities of molecules interacting at long range: Theory and numerical results for the inert gases, H2, N2, CO2, and CH4

Collision-induced light scattering spectra of the inert gases and hydrogen at high densities provide evidence of nonadditive three-body interaction effects, for which a quantitative theory is needed. In this work, we derive and evaluate the three-body polarizability Δα(3) for interacting molecules with negligible electronic overlap. Our results, based on nonlocal response theory, account for dipole-induced-dipole (DID) interactions, quadrupolar induction, dispersion, and concerted induction-dispersion effects. The contribution of leading order comes from a DID term that scales as α3d−6 in the molecular polarizability α and a representative distance d between the molecules in a cluster. Quadrupolar induction effects are also large, however, ranging from ∼35% to 104% of the leading DID terms for equilateral triangular configurations of the species studied in this work, at separations approximately 1 a.u. beyond the van der Waals minima in the isotropic pair potentials. For the same configurations, the dispersion terms range from 2% to 7% of the total Δᾱ(3). The dispersion and induction-dispersion contributions are derived analytically in terms of integrals over imaginary frequency, with integrands containing the polarizability α(iω) and the γ hyperpolarizability. For H, He, and H2, the integrals have been evaluated accurately by 64-point Gauss–Legendre quadrature; for heavier species, we have developed approximations in terms of static polarizabilities, static hyperpolarizabilities, and van der Waals interaction energy coefficients (C6 and C9). In the isotropic interaction-induced polarizability Δᾱ, the three-body terms are comparable in magnitude to the two-body terms, due to a cancellation of the first-order, two-body DID contributions to Δᾱ. For the heavier species in this work (Ar, Kr, Xe, N2, CH4, and CO2) in the configurations studied, the three-body contributions to Δᾱ range from −7 to −9% of the two-body terms for equilateral triangular arrays and from 35% to 47% of the two-body terms for linear, centrosymmetric systems.

Pseudopotential study of electron-hole excitations in colloidal free-standing InAs quantum dots

Excitonic spectra are calculated for free-standing, surface passivated InAs quantum dots using atomic pseudopotentials for the single-particle states and screened Coulomb interactions for the two-body terms. We present an analysis of the single particle states involved in each excitation in terms of their angular momenta and Bloch-wave parentage. We find that (i) in agreement with other pseudopotential studies of CdSe and InP quantum dots, but in contrast to k.p calculations, dot states wavefunction exhibit strong odd-even angular momentum envelope function mixing (e.g. $s$ with $p$) and large valence-conduction coupling. (ii) While the pseudopotential approach produced very good agreement with experiment for free-standing, colloidal CdSe and InP dots, and for self-assembled (GaAs-embedded) InAs dots, here the predicted spectrum does {\em not} agree well with the measured (ensemble average over dot sizes) spectra. (1) Our calculated excitonic gap is larger than the PL measure one, and (2) while the spacing between the lowest excitons is reproduced, the spacings between higher excitons is not fit well. Discrepancy (1) could result from surface states emission. As for (2), agreement is improved when account is taken of the finite size distribution in the experimental data. (iii) We find that the single particle gap scales as $R^{-1.01}$ (not $R^{-2}$), that the screened (unscreened) electron-hole Coulomb interaction scales as $R^{-1.79}$ ($R^{-0.7}$), and that the eccitonic gap sclaes as $R^{-0.9}$. These scaling laws are different from those expected from simple models.

The isospin-dependent quadrupole-quadrupole interaction used in shell model calculations – the effects of including ΔN = 2 excitations and a two-body spin-orbit interaction term

The isospin-dependent quadrupole-quadrupole interaction used in shell model calculations – the effects of including ΔN = 2 excitations and a two-body spin-orbit interaction term

Calculation of rotational states of methane molecules in a square lattice

Methane molecules in a square lattice perform hindered rotation due to the molecular interaction. First, the one-body term in the interaction gives the new pattern of rotational energy level scheme. In the case of strong interaction, this scheme becomes similar to tunneling levels calculated from the pocket state formalism. Secondly, the two-body term produces a potential with lower symmetry. We investigate the case of a ferro-orientational phase. Four levels remain in the lowest part of rotational energy spectrum under the field strength estimated from the sum of atom–atom pair potentials. This small number of levels approaches to more realistic systems.

A QUASI-EXACTLY SOLVABLE N-BODY PROBLEM WITH THE sl(N+1) ALGEBRAIC STRUCTURE

Starting from a one-particle quasi-exactly solvable system, which is characterized by an intrinsic sl (2) algebraic structure and the energy-reflection symmetry, we construct a daughter N-body Hamiltonian presenting a deformation of the Calogero model. The features of this Hamiltonian are (i) it reduces to a quadratic combination of the generators of sl (N+1); (ii) the interaction potential contains two-body terms and interaction with the force center at the origin; (iii) for quantized values of a certain cohomology parameter n it is quasi-exactly solvable, the multiplicity of states in the algebraic sector is (N+n)!/(N!n!); (iv) the energy-reflection symmetry of the parent system is preserved.

Systematic study of the effect of short range correlations on the form factors and densities ofs−pands−dshell nuclei

Analytical expressions of the one- and two-body terms in the cluster expansion of the charge form factors and densities of the s-p and s-d shell nuclei with N=Z are derived. They depend on the harmonic oscillator parameter b and the parameter $\beta$ which originates from the Jastrow correlation function. These expressions are used for the systematic study of the effect of short range correlations on the form factors and densities and of the mass dependence of the parameters b and $\beta$. These parameters have been determined by fit to the experimental charge form factors. The inclusion of the correlations reproduces the experimental charge form factors at the high momentum transfers ($q\geq 2 1/fm$). It is found that while the parameter $\beta$ is almost constant for the closed shell nuclei, $^4$He, $^{16}$O and $^{40}$Ca, its values are larger (less correlated systems) for the open shell nuclei, indicating a shell effect in the closed shell nuclei.