Third-order Contributions Research Articles

Multipole Expansion of Atomic Electron Density Fluctuation Interactions in the Density-Functional Tight-Binding Method.

The accuracy of the density-functional tight-binding (DFTB) method in describing noncovalent interactions is limited due to its reliance on monopole-based spherical charge densities. In this study, we present a multipole-extended second-order DFTB (mDFTB2) method that takes into account atomic dipole and quadrupole interactions. Furthermore, we combine the multipole expansion with the monopole-based third-order contribution, resulting in the mDFTB3 method. To assess the accuracy of mDFTB2 and mDFTB3, we evaluate their performance in describing noncovalent interactions, proton transfer barriers, and dipole moments. Our benchmark results show promising improvements even when using the existing electronic parameters optimized for the original DFTB3 model. Both mDFTB2 and mDFTB3 outperform their monopole-based counterparts, DFTB2 and DFTB3, in terms of accuracy. While mDFTB2 and mDFTB3 perform comparably for neutral and positively charged systems, mDFTB3 exhibits superior performance over mDFTB2 when dealing with negatively charged systems and proton transfers. Overall, the incorporation of the multipole expansion significantly enhances the accuracy of the DFTB method in describing noncovalent interactions and proton transfers.

Asymptotic theory of Mack-mode receptivity in hypersonic boundary layers due to interaction of a heating/cooling source and a freestream sound wave

In this paper, we study the local receptivity of the inviscid Mack modes in hypersonic boundary layers induced by the interaction between a surface heating or cooling source (HCS) and a freestream acoustic wave. The asymptotic analysis reveals that among the three distinguished layers, i.e. the main, wall and Stokes layers, the leading-order receptivity is attributed to the interaction of the HCS-induced mean-flow distortion and the acoustic signature in the wall layer; the second-order contribution appears in the Stokes layer; the third-order contribution appears in both the main and wall layers. Interestingly, at a moderate Reynolds number, the third-order contribution to the receptivity efficiency may be quantitatively greater than the second-order one, but this does not lead to breakdown of this asymptotic theory. Assuming the HCS intensity to be sufficiently weak, the asymptotic predictions are made for four representative cases involving different Mach numbers and wall temperatures, which are compared with the results obtained by the finite-Reynolds-number theory based on either the extended compressible Orr–Sommerfeld equations or the harmonic linearised Navier–Stokes (HLNS) calculations. Taking into account the first three orders of the receptivity efficiency, the asymptotic predictions are confirmed to be sufficiently accurate even when the Reynolds number is a few thousands, and the agreement with the finite-Reynolds-number calculations is better when the wall temperature of the base flow approaches the adiabatic wall temperature. The HLNS calculations are also conducted for moderate HCS intensities. It is found that the nonlinearity does not affect the receptivity coefficient much even when the temperature distortion of the HCS reaches $80\,\%$ of the temperature at the wall.

Ab initio study of the nonlinear optical properties and dc photocurrent of the Weyl semimetal TaIrTe4

The type-II Weyl semimetal TaIrTe${}_{4}$ exhibits one of the largest nonlinear photoresponses measured to date [Ma $e\phantom{\rule{0}{0ex}}t$ $a\phantom{\rule{0}{0ex}}l$., Nature Materials 18, 476 (2019)]. The present work provides theoretical calculations on several competing candidate effects that can account for it. The third-order contribution known as the jerk current matches the measured magnitude of the photocurrent as well as its angular distribution. The origin of the large photoresponse is traced back to a constructive interference between the effective mass and the dipole matrix element of the Weyl bands crossing the Fermi level.

Indirect nuclear spin-spin couplings with third-order contributions added to the SOPPA method.

In this article, a modification of the second-order polarization propagator approximation (SOPPA) method is introduced and illustrated for the calculation of the indirect nuclear spin-spin couplings. The standard SOPPA method, although cheaper in terms of computational cost, offers less accurate results than the ones obtained with coupled cluster methods. A new method, named SOPPA+A3-3, was therefore developed by adding the terms of the third-order A matrix that rely on the second-order double amplitudes. The performance of this third-order contribution was studied using the coupled cluster singles and doubles method as a reference, calculating the spin-spin couplings of molecules of diverse sizes and compositions, and comparing them to the SOPPA method. The results show that inclusion of this third-order contribution gives more accurate results than the standard SOPPA method with a level of accuracy close to that of the coupled cluster method with only a small increase in the computational cost of the response calculation that dominates the computational cost for small- to medium-sized molecules. The implementation of the first contributions to the third-order polarization propagator approximation in the Dalton program, thus, already shows a significant change in these molecular properties over those obtained with the standard SOPPA method.

Exploring the accuracy and usefulness of semi-empirically scaled ADC schemes by blending second and third order terms.

Different approaches to mixed-order algebraic-diagrammatic construction (ADC) schemes are investigated. The performance of two different strategies for scaling third-order contributions to the ADC secular matrix is evaluated. Both considered schemes employ a single tuning parameter and conserve general properties inherent to all ADC methods, such as hermiticity and size-consistency. The first approach, scaled-matrix ADC[(2) + x(3)], scales all contributions first occurring in ADC(3) equally and leads to an improvement of the accuracy of excitation energies compared to ADC(3) for x = 0.4-0.5. However, with respect to excited state dipole moments, this method provides lower accuracy than ADC(3). The second scaling approach, MP[(1) + x(2)] - ISR(3), scales the second order contributions of the ground-state wavefunction and derives a rigorous ADC scheme via the intermediate state representation formalism. Although the error in excitation energies is not improved, this method provides insight into the relevance of the individual terms of the ADC(3) matrix and indicates that the MP(2) wavefunction is, indeed, the optimal reference wavefunction for deriving a third-order single-reference ADC scheme.

Probing chirality across the electromagnetic spectrum with the full semi-classical light-matter interaction.

We present a formulation and implementation of anisotropic and isotropic electronic circular dichroism (ECD) using the full semi-classical light-matter interaction operator within a four-component relativistic framework. Our treatment uniquely accounts for both beyond-first-order light-matter interactions and relativistic effects, enabling us to investigate the ECD response across the electromagnetic spectrum from optical to x-ray wavelengths where relativistic selection rules and spatial field variations gain increasing importance. We consider the isotropic and oriented ECD across the valence transition and sulfur L- and K-edge transitions in the simplest disulfides, H2S2 and (CH3S)2, and evaluate the influence of the full interaction by comparing to a traditional truncated formulation in the Coulomb gauge (velocity representation). Additionally, we demonstrate that in the relativistic formalism, it is possible to work in the velocity representation, hence keeping order-by-order gauge-origin invariance, contrary to the multipolar gauge, yet being able to distinguish electric and magnetic multipole contributions. Going beyond a first-order treatment in the wave vector is mandatory in the higher-energy end of the soft x-ray region and beyond where the consequent intensity redistribution becomes significant. While the sulfur K-edge absorption spectrum is essentially unaffected by this redistribution, the signed differential counterpart is not: At least third-order contributions are required to describe the differential absorption profile that is otherwise overestimated by a factor of about two. The first-order description deteriorates at higher transition energies (beyond ∼1000eV) where it may even fail to predict the signof individual differential oscillator strengths.

Shear waves and sound attenuation in underwater waveguides.

In addition to dissipation of acoustic energy in the seabed, bottom-interacting normal modes are attenuated by radiation of shear waves into soft sediments, where shear speed is small compared to the sound speed in water. The shear-wave contribution and the dissipation have distinct frequency dependencies, and their relative magnitude affects the observed frequency dependence of mode attenuation. Previous studies suggested that the shear-wave contribution to the attenuation is proportional to the cube of the small ratio of the shear and sound speeds. Here, coupling of compressional and shear waves in layered soft sediments is analyzed. Besides the well-known, third-order contribution to the attenuation due to shear-wave generation at the water-sediment interface, a stronger, first-order, contribution is found to occur due to compressional-to-shear wave conversion at interfaces within the sediment. First-order effects of weak shear on mode travel times are also identified. Stratification of the sediment density and interference of shear waves reflected within the seabed control the frequency dependence of the shear-wave contribution to sound attenuation. With the shear-wave contribution being larger than previously estimated, its effect on the experimentally measured frequency dependence of the sound dissipation may need to be re-assessed.

Self-exciting jumps in the oil market: Bayesian estimation and dynamic hedging

In this paper we propose a novel self-exciting jump-diffusion model for oil price dynamics based on a Hawkes-type process. In particular, the jump intensity is stochastic and path dependent, implying that the occurrence of a jump will increase the probability of observing a new jump and this feature of the model aims at explaining the jumps clustering effect. Moreover, volatility is described by a stochastic process, which can jump simultaneously with prices. The model specification is completed by a stochastic convenience yield. In order to estimate the model we apply the two-stage Sequential Monte Carlo (SMC) sampler (Fulop and Li, 2019) to both spot and futures quotations. From the estimation results we find evidence of self-excitation in the oil market, which leads to an improved fit and a better out of sample futures forecasting performance with respect to jump-diffusion models with constant intensity. Furthermore, we compute and discuss two optimal hedging strategies based on futures trading. The optimality of the first hedging strategy proposed is based on the variance minimization, while the second strategy takes into account also the third-order moment contribution in considering the investors attitudes. A comparison between the two strategies in terms of hedging effectiveness is provided.

Spin-locality of \u03b72 and {overline{eta}}^2 quartic higher-spin vertices

Higher-spin theory contains a complex coupling parameter η. Different higher-spin vertices are associated with different powers of η and its complex conjugate overline{eta} . Using Z-dominance Lemma of [1], that controls spin-locality of the higher-spin equations, we show that the third-order contribution to the zero-form B(Z; Y; K) admits a Z-dominated form that leads to spin-local vertices in the η2 and {overline{eta}}^2 sectors of the higher-spin equations. These vertices include, in particular, the η2 and {overline{eta}}^2 parts of the ϕ4 scalar field vertex.

Generation of highly nonlinear irregular waves in a wave flume experiment: Spurious harmonics and their effect on the wave spectrum

Spurious harmonics generation can occur in wave basins when strongly nonlinear waves are generated with linear wave maker theory. For regular long crested waves, the phenomena is known to cause a beating pattern for the amplitude of the higher harmonics, while for irregular waves the effect is often thought to average out. We here report on recent experimental results for spurious harmonic generation for highly nonlinear irregular waves. Long crested waves at two depths are considered. We find a systematic distortion of the spectral shape which is non-uniform in space. The effect occurs in the linear and super harmonic region of the spectrum and is strongest at the smaller depth. We analyze the phenomenon through re-computations in a fully nonlinear wave model and the further application of a four-phase harmonic separation. We next extend the analysis to allow separation of the third- and first-order content of the first-harmonic wave field, by computation at two different amplitudes. The numerical results resembles the measured spectrum closely and the harmonic separation shows that the spectral distortion in the super-harmonic region is to large extent a property of the second-harmonic wave field. Further, the match between the first-harmonic spectrum and the original target spectrum is shown to improve strongly by the removal of the extracted third-order contribution. The observed spectral distortion in the linear frequency region can thus be explained by this third-order field.Returning to the second-order wave field, we demonstrate that the extracted second-order wave field with associated spectral distortion can be reproduced by application of the Sharma and Dean theory with inclusion of the reconstructed spurious wave field from the linear wave generation. Finally, the effect of the spurious second-order harmonic generation on the measured forces is discussed and quantified. We find that the force peak exceedance probability is less influenced by the spurious waves than the spectral shape. We link this to the smaller phase speed of the spurious waves, that makes them arrive later than the main wave packet and thus leads to a limited enhancement of the local peaks.

A parabolic relaxation model for the Navier-Stokes-Korteweg equations

The isothermal Navier-Stokes-Korteweg system is a classical diffuse interface model for compressible two-phase flow which grounds in Van Der Waals' theory of capillarity. However, the numerical solution faces two major challenges: due to a third-order dispersion contribution in the momentum equations, extended numerical stencils are required for the flux calculation. Furthermore, the equation of state given by a Van-der-Waals law, exhibits non-monotone behaviour in the two-phase state space leading to imaginary eigenvalues of the Jacobian of the first-order fluxes.In this work a lower-order parabolic relaxation model is used to approximate solutions of the classical NSK equations. Whereas an additional parabolic evolution equation for the relaxation variable has to be solved, the system involves no differential operator of higher as second order. The use of a modified pressure function guarantees that the first-order fluxes remain hyperbolic. Altogether, the relaxation system is directly accessible for standard compressible flow solvers.We use the higher-order Discontinuous Galerkin spectral element method as realized in the open source code FLEXI. The relaxation model is validated against solutions of the original NSK model for a variety of 1D and 2D test cases. Three-dimensional simulations of head-on droplet collisions for a range of different collision Weber numbers underline the capability of the approach.

Wavelength-scanning second-harmonic generation for determining absolute charge density at aqueous interfaces.

We develop a new, to the best of our knowledge, optical scheme based on second-harmonic generation (SHG) at multiple wavelengths for unequivocal separation of the second-order and the electric-field-induced third-order nonlinear optical contributions from aqueous interfaces. The third-order SHG originating from the field-induced reorientation order of water molecules in the electrical double layer offers an optical label-free and inherent probe to the surface charge density and surface potential in the absolute scales. We verify this wavelength-scanning SHG scheme both theoretically and experimentally, and show that the approach is applicable to water interfaces with bulk ionic strength below 500 µM and can achieve a detection sensitivity for a surface charge density of ∼10-4C/m2.

Emergent nonlinear phenomena in discrete-time quantum walks

Quantum walks are important tools for the development of quantum algorithms and carrying out quantum simulations. Recent interest in nonlinear discrete-time quantum walks aims to use it as a shortcut through dynamical regimes hard to obtain using current methods. We introduce a model featuring a modified conditional shift operator to carry dependence on the local occupation probability with a given strength we are able to control. It accounts for a third-order nonlinear contribution which is found in many physical contexts. We find a rich set of dynamical profiles, including solitonlike propagation, self-trapping, and chaos, all these arising from rather simple rules. Our tool set goes beyond unitary transformations, thus broadening the possibilities for controlling quantum dynamics.

Self-Exciting Jumps in the Oil Market: Bayesian Estimation and Dynamic Hedging

In this paper we propose a novel self-exciting jump diffusion model for oil price dynamics based on a Hawkes-type jump-diffusion process. In particular, the jump intensity is stochastic and path dependent, implying that the occurrence of a jump will increase the probability of observing a new jump and this feature of the model aims at explaining the jumps clustering effect. Moreover, volatility is described by a stochastic process, which can jump simultaneously with prices. The model specification is completed by a stochastic convenience yield. In order to estimate the model we apply the two-stage Sequential Monte Carlo (SMC) sampler (Fulop and Li, 2019) to both spot and futures quotations. From the estimation results we find evidence of self-excitation in the oil market, which leads to an improved fit and a better out of sample futures forecasting performance with respect to jump-diffusion models with constant intensity. Furthermore, we compute and discuss two optimal hedging strategies based on futures trading. The optimality of the first hedging strategy proposed is based on the variance minimization, while the second strategy takes into account also the third-order moment contribution in considering the investors attitudes. A comparison between the two strategies in terms of hedging effectiveness is provided.

Probing Charged Aqueous Interfaces Near Critical Angles: Effect of Varying Coherence Length

Angle-resolved vibrational sum frequency generation experiments have been used to study the silica–water interface as a function of ionic strength. Well below the critical angle, the sum frequency intensity increases up to 10–4 M NaCl and then drops. However, near the critical angle, a plateau may be observed up to 10–4 M. We first demonstrate that this is a result of the interaction of a long Debye length at low ionic strength with a long coherence length near the critical angles. In order to account for the behavior at the lowest concentrations where surface potentials are typically large, it is necessary to consider an electrostatic potential that extends into the bulk aqueous phase beyond the Debye–Huckel approximation. Because the extent of second- and third-order contributions to the nonlinear polarization can vary with ionic strength, but not with the angle of incidence, we perform a global fit to the experimental data using our proposed model to extract the relative magnitude of the two susceptibi...

RI-MP3 calculations of biomolecules based on the fragment molecular orbital method.

In this study, the third-order Møller-Plesset perturbation (MP3) theory using the resolution of the identity (RI) approximation was combined with the fragment molecular orbital (FMO) method to efficiently calculate a high-order electron correlation energy of biomolecular systems. We developed a new algorithm for the RI-MP3 calculation, which can be used with the FMO scheme. After test calculations using a small molecule, the FMO-RI-MP3 calculations were performed for two biomolecular systems comprising a protein and a ligand. The computational cost of these calculations was only around 5 and 4 times higher than those of the FMO-RHF calculations. The error associated with the RI approximation was around 2.0% of the third-order correlation contribution to the total energy. However, the RI approximation error in the interaction energy between the protein and ligand molecule was insignificantly small, which reflected the negligible error in the inter fragment interaction energy. © 2018 Wiley Periodicals, Inc.

Bulk Contributions Modulate the Sum-Frequency Generation Spectra of Water on Model Sea-Spray Aerosols

Summary Vibrational sum-frequency generation (vSFG) spectroscopy is used to determine the molecular structure of water at a model sea-spray aerosol surface. Both measured and calculated spectra display specific features as a result of third-order contributions to the vSFG response, and these are associated with finite interfacial electric potentials. We demonstrate that theoretical modeling enables separation of the third-order contributions, thus allowing for a systematic analysis of the strictly surface-sensitive, second-order component of the vSFG response. This study provides fundamental insights into the interfacial molecular organization and hydrogen-bonding structure of water, which mediate heterogeneous processes on sea-spray aerosols. Our results emphasize the key role that computer simulations can play in interpreting vSFG spectra and revealing microscopic details at complex aqueous interfaces, which can be difficult to extract from experiments because of the mixing of second-order, surface-sensitive, and third-order bulk-dependent contributions to the vSFG response.

Interelectronic-interaction contribution to the nonlinear Zeeman effect in boronlike ions

Relativistic calculations of the second- and third-order contributions in magnetic field to the Zeeman splitting in boronlike ions are presented for the wide range of nuclear charge numbers $Z=6$--$92$. The interelectronic-interaction correction of the first order in $1/Z$ is evaluated to all orders in $\alpha Z$. The higher-order corrections in $1/Z$ are taken into account approximately by means of effective screening potentials. The obtained results are important for interpretation of experimental data on the Zeeman splitting in boronlike ions, in particular, for the ARTEMIS experiment presently implemented at GSI.

Semiclassical gravitational effects on the gravitational lensing in the spacetime of topological defects

The theory of gravitational lensing has revealed many generic and fundamental properties of compact objects like black holes and wormholes. In this article, we utilize a recent formulation to compute the quantum effects on the deflection angle of a light ray, namely, the Gauss–Bonnet theorem (GBT) to explore the semiclassical gravitational effects in the spacetime of a point-like global monopole and a cosmic string. Previously, the Gauss–Bonnet theorem (Gibbons and Werner, 2008) was proposed as an alternative way to compute the deflection angle of light in a static, spherically symmetric and asymptotically flat spacetime. In the present article we have used the celebrated GBT that applied to the optical metric as well as the geodesic method in computing the deflection angle. Interestingly one can observe that we have found an exact result between GBT and the standard approach up to the third-order contributions terms by modifying the domain of integration for cosmic string and global monopole deflection angles. Finally we have considered the time delay in the cosmic string/global monopole spacetime and found that the delay in time is proportional to the linear mass density of the cosmic string and global monopole parameter, respectively.

High-order optical nonlinearities in nanocomposite films dispersed with semiconductor quantum dots at high concentrations

We describe the nonlinear optical properties of inorganic-organic nanocomposite films in which semiconductor CdSe quantum dots as high as 6.8 vol.% are dispersed. Open/closed Z-scan measurements, degenerate multi-wave mixing and femtosecond pump-probe/transient grating measurements are conducted. It is shown that the observed fifth-order optical nonlinearity has the cascaded third-order contribution that becomes prominent at high concentrations of CdSe QDs. It is also shown that there are picosecond-scale intensity-dependent and nanosecond-scale intensity-independent decay components in absorptive and refractive nonlinearities. The former is caused by the Auger process, while the latter comes from the electron-hole recombination process.