Fourth-order Corrections Research Articles

Shape Energy of Fluid Membranes: Analytic Expressions for the Fourth-Order Variation of the Bending Energy

The shape energy of a fluid membrane was formulated previously in terms of the socalled Helfrich Hamiltonian [W. Helfrich, Z. Naturforsch. C 28 (1973), 693; H. J. Deuling and W. Helfrich, J. de Phys. 37 (1976), 1335]. Although the shape deformation might be minor, it should play an important role in the case of a membrane in the proximity of instabilities. Here, we extend the work of Ou-Yang and Helfrich [Ou-Yang Zhong-Can and Wolfgang Helfrich, Phys. Rev. A 39 (1989), 5280; Ou-Yang Zhong-Can, Liu Ji-Xing and Xie Yu-Zhang, Geometric Methods in the Elastic Theory of Membranes in Liquid Crystal Phases (World Scientific, Singapore, 1999)], expanding the bending energy up to fourth order in the deformation function with the help of theoretical tools from differential geometry. After obtaining the fourth-order correction for the bending energy, we investigate its role in the bending deformation by considering two special cases - those of spherical and cylindrical surfaces. In the case of a sphere, we restrict our consideration to two deformed shapes, prolate and oblate. The results of our analysis indicate that, taking account of fourth-order corrections, the critical value determining the oblate-to-prolate transition behaves in a more complicated way than in the case that only corrections up to third order are considered. We also compare the present results will those obtained previously. In the case of a cylinder, two special surface deformations are considered, and conditions for the stability of the surface under such deformations are found. The formation of possible surface shapes deviating slightly from a cylinder with fourth-order corrections is found, even though no such stable state is found in the calculation up to third order. We also apply the variation of the energy up to third order to the case of a deformed torus.

The Laplace Transform Perturbative Triples Correction Ansatz

We describe the implementation of the spin-unrestricted Laplace transform fourth-order perturbative triples correction. A reduction in the computational scaling with respect to canonical implementations is attained without relying on the large molecule asymptote. The intrinsic scaling difficulties that the Laplace equations exhibit upon increasing the size of the basis sets are properly addressed. The method is suited for medium-size molecules.

QCD CORRECTIONS IN Γsl(B)

Short-distance expansion of the total semileptonic B widths is reviewed for the OPE-conformable scheme employing low-scale running quark masses. The third- and fourth-order BLM corrections are given and the complete resummation of the BLM series presented. The effect of higher perturbative orders with running quark masses is found to be very small. Numerical consequences for |Vcb| are addressed.

On the Ripening of Vesicle Dispersions

We analyze the ripening process where, in a dispersion of vesicles, amphiphile monomers diffuse between vesicles of different sizes, leading to a change in the size distribution. Since, typically, vesicles are not under tension, the driving force for the process comes from the curvature energy. With the conventional expansion of the curvature energy density, gc, to harmonic order we find that the free energy change on adding a monomer is independent of vesicle radius. The monomer exchange is then a purely random process. By introducing corrections to the harmonic curvature energy, either by expanding gc to fourth order in the curvatures, or by a term logarithmic in the vesicle size, one obtains a driving force for ripening. Depending on the correction, different scenarios are predicted. The fourth-order correction and a negative logarithmic correction term result in a ripening toward unimodal vesicle distribution around the mean vesicle size of the initial distribution. For a positive logarithmic correction term, on the other hand, the ripening by itself should result in the formation of many small vesicles and a few very big ones. The rate of the monomer redistribution is primarily determined by the magnitude of the thermodynamic driving force and by the monomer solubility. For vesicles formed by single chain surfactants we estimate that the size redistribution should occur at a measurable rate, providing an opportunity to experimentally study which correction to the leading curvature energy is most significant.

Fourth-order quantum master equation and its Markovian bath limit

Fourth-order quantum master equations (FQMEs) are derived in both time nonlocal and local forms for a general system Hamiltonian, with new detailed expressions for the fourth-order kernel, where the bath correlation functions are explicitly decoupled from the system superoperators. Further simplifications can be made for the model of linearly coupled harmonic oscillator bath. Consideration of the high temperature Ohmic bath limit leads to a general Markovian FQME with compact forms of time independent superoperators. Two examples of this equation are then considered. For the system of a quantum particle in a continuous potential field, the equation reduces to a known form of the quantum Fokker–Planck equation, except for a fourth-order potential renormalization term that can be neglected only in the weak system-bath interaction regime. For a two-level system with off-diagonal coupling to the bath, fourth-order corrections do not alter the relaxation characteristics of the second-order equation and introduce additional coherence terms in the equations for the off-diagonal elements.

Hyperfine structure in hydrogen and helium ion

QED theory of the hyperfine splitting of the $1s$ and $2s$ state in hydrogen isotopes and helium-3 ion is considered. We develop an accurate theory of a specific difference 8E_{HFS}(2s)-E_{HFS}(1s). We take into account fourth order corrections and nuclear structure effects. The theoretical prediction is now of a higher accuracy than the experiment is. The study of the difference provides the most accurate test (on a level of a part in 10^8) of the QED theory of $1s$ HFS up to date. The theory agrees with most of the experimental data.

Aspects of near-threshold neutral pion photoproduction off protons

We investigate near threshold neutral pion photoproduction off protons to fourth order in heavy baryon chiral perturbation theory in the light of the new data from MAMI. We show that the unitarity cusp at the secondary \pi^+ n threshold is in agreement with expectations from the final state theorem. We also analyze the fourth order corrections to the P-wave low-energy theorems and show that potentially large Delta-isobar contributions are cancelled by sizeable pion loop effects. This solidifies the parameter free third order predictions, which are in excellent agreement with the data.

Fourth-Order Rotational Corrections to the Effective Dipole Moments of Nonrigid Asymmetric Rotors

An expression for the fourth-order rotational correction terms of the effective dipole moments of nonrigid asymmetric rotors has been derived using the method of contact transformation. The treatment takes into account the large-amplitude bending motion. The correction terms have been calculated for the different bending states of the H2O molecule. A poor convergence of the rotational series for this molecule has been obtained and some different nonpolynomial forms for these series, obtained by different summation methods, have been proposed and tested.

On the importance of third- and fourth-order corrections in multi-reference Møller–Plesset theory

An implementation and applications of an approximate fourth-order multi-reference Møller–Plesset theory restricted to single and double excitations (MR-MP4(SD)) are presented. The results for small full CI benchmark systems indicate a substantial improvement compared to MR-MP2 and MR-MP3 treatments. For larger molecules we propose a configuration selection procedure to keep the size of the expansion spaces within the limits of personal computer capabilities. As typical applications we consider vertical excitation energies of unsaturated organic molecules (up to 66 correlated electrons) and reaction energies involving biradicals. It is found that the systematic under(over)estimation of relative energies of open- versus closed-shell states at MR-MP2(MR-MP3) levels is in most cases corrected at fourth-order.

Algorithm for obtaining the gradient expansion of the local density of states and the free energy of a superconductor

We present an efficient algorithm for obtaining the gauge-invariant gradient expansion of the local density of states and the free energy of a clean superconductor. Our method is based on a new mapping of the semiclassical linearized Gorkov equations onto a pseudo-Schr\"odinger equation for a three-component wave function $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\psi}}(x),$ where one component is directly related to the local density of states. Because $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\psi}}(x)$ satisfies a linear equation of motion, successive terms in the gradient expansion can be obtained by simple linear iteration. Our method works equally well for real and complex order parameter, and in the presence of arbitrary external fields. We confirm a recent calculation of the fourth order correction to the free energy by Kosztin, Kos, Stone, and Leggett [Phys. Rev. B 58, 9365 (1998)], who obtained a discrepancy with an earlier result by Tewordt [Z. Phys. 180, 385 (1964)]. We also give the fourth order correction to the local density of states, which has not been published before.

Unusual properties of the molecular nanomagnet Mn12ac

The magnetic properties of the Mn12ac molecular cluster were studied by complex susceptibility and specific heat measurements. Extensive ac susceptibility measurements done under a magnetic field on good quality single crystals and zero field powder sample specific heat results indicate that the usual uniaxial Hamiltonian, as well as the recently proposed fourth order corrections, are not enough to account for all the observations. Very low temperature specific heat data indicate the presence of zero field splitting of the ground state which is much larger than expected. Finally, specific heat measurements done under a magnetic field permitted us to observe and measure the lattice spin relaxation by calorimetric methods with time constants that are in agreement with the magnetic relaxation measurements.

Static first hyperpolarizability of small all-trans polymethincimine oligomers. Basis set and electron correlation effects

Ab initio calculations, which include the electronic correlation effects, of the static longitudinal first hyperpolarizability, β L , of small all-trans polymethineimine (PMI) chains (from monomer to trimer) have been performed with a large panel of basis sets. The electron correlation corrections were included via the Møller-Plesset partitioning and the Coupled-Cluster method. We show that the basis set effects are smaller when using correlated techniques than at the Hartree-Fock level. Moreover, the Hartree-Fock approximation does not reproduce the correct sign of β L even for the dimer and the trimer. Addition of d functions on carbon and nitrogen atoms is important in order to obtain a quantitative evaluation of β L , while the use of supplementary p functions on hydrogen atoms does not lead to significant variations. When polarization functions are added, the amplitude of β L decreases by going from a double-ζ to a triple-ζ basis set. Within the Møller-Plesset scheme, the second-order correction is by far the largest and is responsible for the sign change. However, contrary to the third-order correction, the fourth-order correction considerably changes the β L amplitude due to important single and triple contributions. In the Coupled-Cluster approaches, the inclusion of the singles strongly damps the correction due to the doubles, the inclusion of the triples within the CCSD(T) approach modulating it. Comparisons with other works demonstrate that the correlation effects are large in PMI oligomers. We conclude that the 6–31G basis set with d polarization functions on the carbon and nitrogen atoms, used in the Møller-Plesset partitioning consistent with electron-electron interactions through the fourth order, is suitable for the investigation of small PMI chains and would deserve to be considered in the study of longer chains.

Expansion in the width and collective dynamics of a domain wall

We show that collective dynamics of a curved domain wall in a (3+1)-dimensional relativistic scalar field model is represented by a Nambu-Goto membrane and (2+1)-dimensional scalar fields defined on the world-sheet of the membrane. Our argument is based on a recently by us proposed version of the expansion in the width. The derivation of the expansion is significantly reformulated for the present purpose. Third- and fourth-order corrections to the domain wall solution are considered. We also derive an equation of motion for the core of the domain wall. Without the (2+1)-dimensional scalar fields this equation would be non-local.

Analytic Calculation of Anomalous Scaling in Random Shell Models for a Passive Scalar

An exact nonperturbative calculation of the fourth-order anomalous correction to the scaling behavior of a random shell model for passive scalars is presented. Importance of ultraviolet (UV) and infrared (IR) boundary conditions on the inertial scaling properties are determined. We find that anomalous behavior is given by the null space of the inertial operator and we prove strong UV and IR independence of the anomalous exponent. A limiting case where diffusive behavior can influence inertial properties is also presented. [S0031-9007(97)03479-0] PACS numbers: 47.27.Gs, 47.27.Jv

A finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented.

A METHOD SOLVING KEPLER'S EQUATION FOR HYPERBOLIC CASE

We developed a method to solve Kepler’s equation for the hyperboliccase. The solution interval is separated into three regions; F ≪ 1, F≈ 1, F ≫ 1. For the region F is large, we transformed the variablefrom F to exp F and applied the Newton method to the transformed equation.For the region F is small, we expanded the equation with respect to F andapplied the Newton method to the approximated equations. For the middleregion, we adopted a discretization of the Newton method and used the Taylorseries expansion for the evaluation of transcendental functions (Fukushima,1997). Numerical measurements showed that, in the case of Intel Pentiumprocessor, the new method is more than 3.7 times as fast as the combinationof a fourth order correction formula (Fukushima, 1997) and Roy’sstarting procedure (Roy, 1988) and others.

Potential Energy Surfaces for Proton Abstractions from Acetic Acid

The abstractions of hydrogen from both carbon and oxygen in acetic acid by hydride, fluoride, and hydroxide anions have been studied using ab initio electronic structure calculations. Molecular structures were optimized at the Hartree-Fock level of theory using the 6-31++G(d,p) basis set. For energetics, the 6-311++G(d,p) basis set was used, with second- and fourth-order perturbation theory corrections, for both minima and transition states. For the hydride and fluoride ion abstractions of hydrogen from carbon, a small activation energy exists at the Hartree-Fock level, but vanishes when correlation energy corrections are introduced. No other barriers are found for the abstraction reactions, but intermediate minima are found on the F{sup -} + CH{sub 3}COOH {yields} FH + {sup -}CH{sub 2}COOH surface and on the analogous OH{sup -} + CH{sub 3}COOH surface. The calculated heats of formation for both acetic acid anions are in good agreement with the experimental value. The fourth-order perturbation theory calculation of the activation energy for the isomerization of acetate to enolate ion is 50.4 kcal/mol. The G2 values for the gas phase acidities of acetic acid at the OH and CH ends of the molecule are 339.3 and 365.8 kcal/mol, respectively. The former result is in good agreementmore » with the experiment. 26 refs., 5 figs., 5 tabs.« less

Self-consistent intermediate Hamiltonians: A coupled cluster type formulation of the singles and doubles configuration interaction matrix dressing

This paper presents a new self-consistent dressing of a singles and doubles configuration interaction matrix which insures size-consistency, separability into closed-shell subsystems if localized molecular orbitals (MOs) are used, and which includes all fourth order corrections. This method yields, among several schemes, a reformulation of the coupled cluster method, including fully the cluster operators of single and double excitations, and partially those of the triples (Bartlett’s algorithm named CCSDT-1a). Further improvement can be easily included by adding exclusion principle violating corrections. Since it leads to a matrix diagonalization, the method behaves correctly in case of near degeneracies between the reference determinant and some doubles. Due to its flexibility this formulation offers the possibility of consistent combination with less expensive treatments for the study of very large systems.

Choosing between alternative MP2 algorithms in the self-consistent reaction field theory of solvent effects

The MP2 energy and electronic density can be obtained in the framework of the self-consistent reaction field theory of solvent effects by the simple application of the usual MP2 formulae using SCF orbitals which have been optimized in the self-consistent reaction field of the SCF density. Iterative schemes which apply an MP2-corrected density to achieve self-consistency of the reaction field are not only computationally much longer, but inconsistent, since they involve higher (fourth) order corrections in the electron correlation.

Ab initio study of the nonlinear optical properties of urea: Electron correlation and dispersion effects

We present the results of ab initio calculations on the first-, second-, and third-order molecular polarizabilities of urea. An efficacious general finite field perturbation approach, previously applied in the case of paranitroaniline, is extended to the evaluation of axial and nonaxial components of the nonlinear responses. The validity of our numerical procedure is examined at the Hartree–Fock level of theory by comparison with analytical derivative results. The impact of electron correlation is analyzed, by calculating the optical nonlinearities at the Moller-Plesset perturbation theory level. The second-order Moller-Plesset electron correlation correction is shown: (i) to enhance the third-order polarizability y by almost a factor of 2 and (ii) to include the major correlation effects as consideration of the fourth-order correction further improves the y components by less than 20%. We also discuss the frequency-dependence of the nonlinear optical properties of urea by presenting calculations of the dynamic components at the noncorrelated level of theory for different optical processes. © 1995 John Wiley & Sons, Inc.