Third-order Contributions Research Articles

Third-order particle-hole ring diagrams with contact-interactions and one-pion exchange

The third-order particle-hole ring diagrams are evaluated for a NN-contact interaction of the Skyrme type. The pertinent four-loop coefficients in the energy per particle $\bar E(k_f) \sim k_f^{5+2n}$ are reduced to double-integrals over cubic expressions in euclidean polarization functions. Dimensional regularization of divergent integrals is performed by subtracting power-divergences and the validity of this method is checked against the known analytical results at second-order. The complete ${\cal O}(p^2)$ NN-contact interaction is obtained by adding two tensor terms and their third-order ring contributions are also calculated in detail. The third-order ring energy arising from long-range $1\pi$-exchange is computed and it is found that direct and exchange contributions are all attractive. The very large size of the pion-ring energy, $\bar E(k_{f0})\simeq -92\,$MeV at saturation density, is however in no way representative for that of realistic chiral NN-potentials. Moreover, the third-order (particle-particle and hole-hole) ladder diagrams are evaluated with the full ${\cal O}(p^2)$ contact interaction and the simplest three-ring contributions to the isospin-asymmetry energy $A(k_f)\sim k_f^5$ are studied.

A reference-modified density functional theory: An application to solvation free-energy calculations for a Lennard-Jones solution.

In the conventional classical density functional theory (DFT) for simple fluids, an ideal gas is usually chosen as the reference system because there is a one-to-one correspondence between the external field and the density distribution function, and the exact intrinsic free-energy functional is available for the ideal gas. In this case, the second-order density functional Taylor series expansion of the excess intrinsic free-energy functional provides the hypernetted-chain (HNC) approximation. Recently, it has been shown that the HNC approximation significantly overestimates the solvation free energy (SFE) for an infinitely dilute Lennard-Jones (LJ) solution, especially when the solute particles are several times larger than the solvent particles [T. Miyata and J. Thapa, Chem. Phys. Lett. 604, 122 (2014)]. In the present study, we propose a reference-modified density functional theory as a systematic approach to improve the SFE functional as well as the pair distribution functions. The second-order density functional Taylor series expansion for the excess part of the intrinsic free-energy functional in which a hard-sphere fluid is introduced as the reference system instead of an ideal gas is applied to the LJ pure and infinitely dilute solution systems and is proved to remarkably improve the drawbacks of the HNC approximation. Furthermore, the third-order density functional expansion approximation in which a factorization approximation is applied to the triplet direct correlation function is examined for the LJ systems. We also show that the third-order contribution can yield further refinements for both the pair distribution function and the excess chemical potential for the pure LJ liquids.

Nonlinear Rashba spin splitting in transition metal dichalcogenide monolayers.

Single-layer transition-metal dichalcogenides (TMDs) such as MoS2 and MoSe2 exhibit unique electronic band structures ideal for hosting many exotic spin-orbital orderings. It has been widely accepted that Rashba spin splitting (RSS) is linearly proportional to the external field in heterostructure interfaces or to the potential gradient in polar materials. Surprisingly, an extraordinary nonlinear dependence of RSS is found in semiconducting TMD monolayers under a gate field. In contrast to small and constant RSS in polar materials, the potential gradient in non-polar TMDs gradually increases with the gate bias, resulting in nonlinear RSS with a Rashba coefficient an order-of-magnitude larger than the linear one. Most strikingly, under a large gate field MoSe2 demonstrates the largest anisotropic spin splitting among all known semiconductors to our knowledge. Based on the k·p model via symmetry analysis, we identify that the third-order contributions are responsible for the large nonlinear Rashba splitting. The gate tunable spin splitting found in semiconducting pristine TMD monolayers promises future spintronics applications in that spin polarized electrons can be generated by external gating in an experimentally accessible way.

Time-dependent density-functional tight-binding method with the third-order expansion of electron density.

We develop a formalism for the calculation of excitation energies and excited state gradients for the self-consistent-charge density-functional tight-binding method with the third-order contributions of a Taylor series of the density functional theory energy with respect to the fluctuation of electron density (time-dependent density-functional tight-binding (TD-DFTB3)). The formulation of the excitation energy is based on the existing time-dependent density functional theory and the older TD-DFTB2 formulae. The analytical gradient is computed by solving Z-vector equations, and it requires one to calculate the third-order derivative of the total energy with respect to density matrix elements due to the inclusion of the third-order contributions. The comparison of adiabatic excitation energies for selected small and medium-size molecules using the TD-DFTB2 and TD-DFTB3 methods shows that the inclusion of the third-order contributions does not affect excitation energies significantly. A different set of parameters, which are optimized for DFTB3, slightly improves the prediction of adiabatic excitation energies statistically. The application of TD-DFTB for the prediction of absorption and fluorescence energies of cresyl violet demonstrates that TD-DFTB3 reproduced the experimental fluorescence energy quite well.

Electron-related optical properties in T-shaped AlxGa1−xAs/GaAs quantum wires and dots

The electronic structure and the intersubband optical absorption and relative refractive index change coefficients in T-shaped two-dimensional quantum dot and one-dimensional quantum wire are studied. The T-shaped quantum dot is embedded in AlxGa1−xAs, with x = 0.35, the arm region has x = 0 whereas different values of the Al molar fraction are present for the T-stem region (x = 0, 0.7, 0.14, and 0.21). The model calculation is useful for studying both a 1D quantum wire of T-shaped cross-section and a 2D T-shaped quantum dot. The conduction and valence band states are described within the effective mass and parabolic band approximations. The agreement between calculated photoluminescence peak energy transitions and previously reported experimental values in such T-shaped quantum well wires is discussed. The electron-related optical coefficients are calculated using a density-matrix expansion with the inclusion of the linear and third-order nonlinear contributions to the dielectric susceptibility. The results for this optical response are presented as functions of the Al molar fraction, as well as of the polarization, and intensity of the incident light.

Third-order effect in magnetic small-angle neutron scattering by a spatially inhomogeneous medium

Magnetic small-angle neutron scattering (SANS) is a powerful tool for investigating nonuniform magnetization structures inside magnetic materials. Here, we consider a ferromagnetic medium with weakly inhomogeneous uniaxial magnetic anisotropy, saturation magnetization, and exchange stiffness, and derive, to second order in the amplitudes of the inhomogeneities, the micromagnetic solutions for the equilibrium magnetization textures. Further, we compute the corresponding magnetic SANS cross section up to the third order. For the special case of scattering geometry where the incident neutron beam is perpendicular to the applied magnetic field, twice the cross section along the direction orthogonal to both the field and the neutron beam cancels the cross section along the field direction in the second order. This cancellation does not depend on the defect shape and amplitudes of the exchange inhomogeneities. Hence, such a cross-section difference has only a third-order contribution in the amplitudes of the inhomogeneities. It provides a separate gateway for a deeper analysis of the sample's magnetic structure. We derive and analyze analytical expressions for the dependence of this difference on the scattering-vector magnitude for the case of spherical Gaussian inhomogeneities.

Time Correlation Function Modeling of Third-Order Sum Frequency Vibrational Spectroscopy of a Charged Surface/Water Interface.

Sum frequency vibrational spectroscopy (SFVS), a second-order optical process, is interface-specific in the dipole approximation [Perry, A.; Neipert, C.; Moore, P.; Space, B. Chem. Rev. 2006, 106, 1234-1258; Richmond, G. L. Chem. Rev. 2002, 102, 2693-2724; Byrnes, S. J.; Geissler, P. L.; Shen, Y. R. Chem. Phys. Lett. 2011, 516, 115-124]. At charged interfaces, the experimentally detected signal is a combination of enhanced second-order and static-field-induced third-order contributions due to the existence of a static field. Evidence of the importance/relative magnitude of this third-order contribution is seen in the literature [Ong, S.; Zhao, X.; Eisenthal, K. B. Chem. Phys. Lett. 1992, 191, 327-335; Zhao, X.; Ong, S.; Eisenthal, K. B. Chem. Phys. Lett. 1993, 202, 513-520; Shen, Y. R. Appl. Phys. B: Laser Opt. 1999, 68, 295-300], but a molecularly detailed approach to separately calculating the second- and third-order contributions is difficult to construct. Recent work presented a novel molecular dynamics (MD)-based theory that provides a direct means to calculate the third-order contributions to SFVS spectra at charged interfaces [Neipert, C.; Space, B. J. Chem. Phys. 2006, 125, 224706], and a hyperpolarizability model for water was developed as a prerequisite to practical implementation [Neipert, C.; Space, B. Comput. Lett. 2007, 3, 431-440]. Here, these methods are applied to a highly abstracted/idealized silica/water interface, and the results are compared to experimental data for water at a fused quartz surface. The results suggest that such spectra have some quite general spectral features.

Prediction of standard enthalpy of formation in the solid state by a third-order group contribution method

New group contribution model for the prediction of the standard enthalpy of formation in the solid phase of pure compounds is presented. An extensive set of 1222 experimental enthalpies of formation data was collected from different literature sources, evaluated and used to develop this model (85% as the training set and 15% as the test set). The compounds in the data set are composed of hydrogen, carbon, oxygen and nitrogen. The prediction is based exclusively on the molecular structure of the compound and the model results show a good agreement with the experimental data. Compared to the most accurate models for estimating the standard enthalpy of formation in the solid state, this model demonstrates significant improvements in accuracy and applicability.

Dipole moments of CH3F in the ν3 and ν6 vibrational excited states from the Stark effect

Both the parallel and perpendicular rotational Stark components of the methyl fluoride J, k, l: 2, ±1, 0←1, ±1, 0 and 2, ±1, ∓1←1, ±1, ∓1 lines in the ν3 and ν6 excited vibrational states were measured in the microwave region of ca 100GHz at the Stark fields in the range of ca 150–1100V/cm. The analysis of the Stark components was complicated by unexpected hyperfine structure patterns spanning a few MHz, which were observed for MJ components with large Stark shifts (>300MHz), most probably due to an enhancement through the Stark effect. The permanent electric dipole moments for these two vibrational states were derived from selected separated hyperfine components of the Stark components: μ3=1.90345(46)D, μ6=1.86394(32)D (the expanded uncertainties are given with the coverage factor k=2). The significant perturbational second-order Stark contributions were simply eliminated using a Stark component difference approach. Third-order Stark contributions were included in the analysis, and higher-order Stark contributions and polarizability contributions were neglected since they are inconsequential in this study.

Electrostatics-based finite-size corrections for first-principles point defect calculations

Finite-size corrections for charged defect supercell calculations typically consist of image-charge and potential alignment corrections. A wide variety of schemes for both corrections have been proposed for decades. Regarding the image-charge correction, Freysoldt, Neugebauer, and Van de Walle (FNV) recently proposed a novel method that enables us to accurately estimate the correction energy a posteriori through alignment of the defect-induced potential to the model charge potential [Freysoldt et al., Phys. Rev. Lett. 102, 016402 (2009)]. This method, however, still has two issues in practice. Firstly, it uses planar-averaged potential for determining the potential offset, which cannot be readily applied to relaxed system. Secondly, the long-range Coulomb interaction is assumed to be screened by a macroscopic dielectric constant. This is valid only for cubic systems and can bring forth huge errors for defects in anisotropic materials. In this study, we use the atomic site electrostatic potential as a potential marker instead of the planar-averaged potential, and extend the FNV scheme by adopting the point charge model in an anisotropic medium for estimating long-range interactions. We also revisit the conventional potential alignment correction and show that it is fully included in the image-charge correction and therefore unnecessary. In addition, we show that the potential alignment corresponds to a part of first-order and full of third-order image-charge correction; thus the third-order image-charge contribution is absent after the potential alignment. Finally, a systematic assessment of the accuracy of the extended FNV correction scheme is performed for a wide range of material classes. The defect formation energies calculated using around 100-atom supercells are successfully corrected even after atomic relaxation within a few tenths of eV compared to those in the dilute limit.

Energy, decay rate, and effective masses for a moving polaron in a Fermi sea: Explicit results in the weakly attractive limit

We study the properties of an impurity of mass M moving through a spatially homogeneous three-dimensional fully polarized Fermi gas of particles of mass m. In the weakly attractive limit, where the effective coupling constant and perturbation theory can be used, both for a broad and a narrow Feshbach resonance, we obtain an explicit analytical expression for the complex energy of the moving impurity up to order two included in g. This also gives access to its longitudinal and transverse effective masses , , as functions of the impurity wave vector K. Depending on the modulus of K and on the impurity-to-fermion mass ratio M/m we identify four regions separated by singularities in derivatives with respect to K of the second-order term of , and we discuss the physical origin of these regions. Remarkably, the second-order term of presents points of non-differentiability, replaced by a logarithmic divergence for M = m, when K is on the Fermi surface of the fermions. We also discuss the third-order contribution and relevance for cold-atom experiments.

Two-Exciton States and Coherent Third-Order Response from Semiconductor Quantum Wells

We present a microscopic description of the coherent third-order response from a quantum well, treated as a quasi-two-dimensional semiconductor, in the exciton representation. We establish a closed system of dynamic equations including those for coherent spin-polarized photons and excitons and for correlated two-exciton structures, called molecules. We employ such a system of equations as a starting point for calculating all third-order contributions to the semiconductor coherent polarization in the case where the eigenenergies and eigenfunctions of the molecules are available. Considering the molecular problem, we calculate the interaction potential between excitons in molecules for quantum-well samples in the limit of small exciton momentum taking place in the close-to-normal-incidence excitation geometry. The potential computed allows an approximate explicit determination of the molecular eigenenergies and eigenfunctions in the slow scattering limit. The criterion for a coherent pump-probe experiment to support the limit is pointed out.

Nonlinear absorption coefficient and relative refraction index change for an asymmetrical double [formula omitted]-doped quantum well in GaAs with a Schottky barrier potential

In this work we are reporting the energy level spectrum for a quantum system consisting of an n-type double δ-doped quantum well with a Schottky barrier potential in a Gallium Arsenide matrix. The calculated states are taken as the basis for the evaluation of the linear and third-order nonlinear contributions to the optical absorption coefficient and to the relative refractive index change, making particular use of the asymmetry of the potential profile. These optical properties are then reported as a function of the Schottky barrier height (SBH) and the separation distance between the δ-doped quantum wells. Also, the effects of the application of hydrostatic pressure are studied. The results show that the amplitudes of the resonant peaks are of the same order of magnitude of those obtained in the case of single δ-doped field effect transistors; but tailoring the asymmetry of the confining potential profile allows the control the resonant peak positions.

Rovibrational effects on NMR shieldings in a heavy-element system: XeF2

Fully quantum-mechanical treatment of the effects of thermal rovibrational motion in a heavy-element molecule with relativistic effects is carried out for the heavy (129/131)Xe and light (19)F nuclear shieldings in the linear XeF(2) molecule. More importantly, purely quantum-mechanical, intramolecular phenomena, the primary and secondary isotope effect on these shieldings, respectively, are treated with including both the zero-point vibrational and finite-temperature effects. While large solvent effects influence the experimental absolute shielding constants and chemical shifts (thereby making comparison of experiment and theory very difficult), they are not significant for the isotope shifts. We study the role of electron correlation at both nonrelativistic (NR) and relativistic [Breit-Pauli perturbational theory (BPPT) as well as 4-component Dirac theory] level. We obtain quantitative agreement with the nearly solvent-independent experimental (19)F secondary isotope shifts. This implies a promising accuracy for our predictions of the experimentally so far non-existing primary Xe isotope shift and the temperature dependence of Xe and F chemical shifts corresponding to a low pressure gas phase. To achieve this, a combination of high-level ab initio NR shielding surface is found necessary, in the present work supplemented by relativistic corrections by density-functional theory (DFT). Large errors are demonstrated to arise due to DFT in the NR shielding surface, explaining findings in recent computational studies of heavy-element isotope shifts. Besides a high-quality property hypersurface, the inclusion of thermal effects (in addition to zero-point motion) is also necessary to compare with experimental results. The geometry dependence of the different relativistic influences on the wave function, Zeeman interaction, and hyperfine interaction, as well as their role in the temperature dependence of both the Xe and F shielding constants and their isotope shifts, are discussed. The relativistic rovibrational effects arise from the same individual contributions as previously found for the chemical shifts and shielding anisotropies. In general, the spin-orbit interactions are more sensitive to rovibrational motion than the scalar relativistic contributions. A previously suggested third-order BPPT contribution to shielding anisotropy is shown to be important for a better agreement with experiment.

Role of entropy perturbations in third-order nonlinear acoustic tomography problems

The effect of the entropy perturbations on the acoustic pressure, which arises due to third-order nonlinear interaction, has been considered. The contribution to entropy perturbation due to the presence of viscosity and thermal conductivity (by means of the general equation of heat transfer) and also due to the nonlinearity of the considered medium has been estimated. It has been shown that when all possible factors responsible for the change in entropy are taken into account, the third-order contribution to the acoustic pressure due to density perturbations in primary waves strongly exceeds that due to entropy perturbations. It has been concluded that an isoentropic mathematical description of third-order nonlinear effects in nonlinear acoustic tomography problems is appropriate.

Surface second harmonic generation from coumarin 343 dye-attached TiO2 nanoparticles at liquid–liquid interface

The nonlinear optical properties of coumarin 343 (C343) dye-attached TiO2 nanoparticles in the size range 5–8 nm adsorbed at the interface of water/1,2-dichloroethane have been studied by using the surface second harmonic generation technique. No second harmonic (SH) response was observed from the bare TiO2 nanoparticles adsorbed at the interface, however, a strong SH response was measured from the dye molecules attached at the surfaces of the nanoparticles. The increase in the SH intensity with the increase of TiO2 nanoparticle concentration in the aqueous solution of C343 is mainly due to the pre-alignment of the dye molecules at the surfaces of nanoparticles and is partly due to the third-order polarization contribution of the nanoparticles to the observed total SH response.

On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions

We present the third-order contributions to the quark–gluon and gluon–quark timelike splitting functions for the evolution of fragmentation functions in perturbative QCD. These quantities have been derived by studying physical evolution kernels for photon- and Higgs-exchange structure functions in deep-inelastic scattering and their counterparts in semi-inclusive annihilation, together with constraints from the momentum sum rule and the supersymmetric limit. For this purpose we have also calculated the second-order coefficient functions for one-hadron inclusive Higgs decay in the heavy-top limit. A numerically tolerable uncertainty remains for the quark–gluon splitting function, which does not affect the endpoint logarithms for small and large momentum fractions. We briefly discuss these limits and illustrate the numerical impact of the third-order corrections. Compact and accurate parametrizations are provided for all third-order timelike splitting functions.

Higher-order contributions to transport coefficients in two-temperature hydrogen thermal plasma

Within the framework of Chapman-Enskog method, electron transport properties and their higher-order contributions have been studied in temperature range 5000–40 000 K at different pressures for hydrogen thermal plasma in local thermodynamic equilibrium (LTE) and non-local thermodynamic equilibrium (NLTE) regimes. Two cases of thermal plasma have been considered: (i) Ground state (GS) plasma in which all atomic hydrogen has been assumed to be in ground state and (ii) the excited state (ES) plasma in which hydrogen atoms are distributed in various possible electronically excited states (EES). The plasma composition is calculated by modified Saha equation of van de Sanden et al. The influence of non-equilibrium parameter θ (=Te/Th) on these properties has been examined in both the cases. It has been observed that both EES and θ modify the plasma composition and consequently affect the electron transport properties (viz., electron thermal conductivity, electrical conductivity, thermal diffusion and thermal diffusion ratio). It is shown that non-equilibrium parameter θ has meager effect on the higher-order convergence in comparison to EES. The unique behaviour observed for third-order contribution to these transport properties in GS plasma for small values of θ could be explained only when EES are taken into account. It is noted that EES show their influence on higher-orders to a considerable extent even when e-H(n) cross-sections are replaced by the ground state ones. Thus electron transport coefficients and their higher-order contributions are affected significantly due to inclusion of EES in LTE and NLTE plasmas.

Third-order corrections to random-phase approximation correlation energies

Several random-phase approximation (RPA) correlation methods were compared in third order of perturbation theory. While all of the considered approaches are exact in second order of perturbation theory, it is found that their corresponding third-order correlation energy contributions strongly differ from the exact third-order correlation energy contribution due to missing interactions of the particle-particle-hole-hole type. Thus a simple correction method is derived which makes the different RPA methods also exact to third-order of perturbation theory. By studying the reaction energies of 16 chemical reactions for 21 small organic molecules and intermolecular interaction energies of 23 intermolecular complexes comprising weakly bound and hydrogen-bridged systems, it is found that the third-order correlation energy correction considerably improves the accuracy of RPA methods if compared to coupled-cluster singles doubles with perturbative triples as a reference.

Separating surface structure and surface charge with second-harmonic and sum-frequency scattering

We analyze the effect of an electrostatic surface charge on the angular nonlinear light scattering pattern from spherical particles in solution. An electrostatic field near a charged interface leads to a bulk-allowed third-order process, the strength of which is proportional to the electrostatic surface potential. The commonly isotropic nature of the bulk leads to a fixed angular scattering pattern with fixed intensity ratios between polarizations. We show that second- and third-order scattering effects are separable due to their different angular radiation patterns. Furthermore, nonlinear light scattering from a third-order contribution is strongest in the $ppp$ polarization combination.