Second-order Contributions Research Articles

The weak-correlation limits of few-electron harmonium atoms

The weak-correlation asymptotics of electronic properties of harmonium atoms comprising up to four electrons are investigated. In particular, closed-form expressions are derived for the first- and second-order contributions to the Hartree-Fock and correlation energies of eight electronic states that include three singlets, one doublet, two triplets, one quartet, and one quintet, six of which are singly determinantal and two are multi-determinantal. This diversity of states offers a much richer set of benchmarking tools for calibration of approximate electron-correlation methods than the previously published data. The availability of the computed energy contributions due to individual spinorbitals and their pairs present in the dominant Slater determinants further enhances the utility of these benchmarks.

Predicting Temperature-Dependent Aqueous Henry’s Law Constants Using Group Contribution Methods

A first-order temperature-dependent group contribution method was developed to predict Henry’s law constants of hydrocarbons, alcohols, ketones, and formates in which none of the functional groups are attached directly to a benzene ring. Efforts to expand this method to include ester and ether groups were unsuccessful. Second-order groups were developed at a reference condition of 298.15 K and 100 kPa. A second-order temperature-dependent group contribution method was then developed for hydrocarbons, ketones, esters, ethers, and alcohols. These methods were compared to existing literature prediction methods.

Microscopic optical potential from chiral nuclear forces

The energy- and density-dependent single-particle potential for nucleons is constructed in a medium of infinite isospin-symmetric nuclear matter starting from realistic nuclear interactions derived within the framework of chiral effective field theory. The leading-order terms from both two- and three-nucleon forces give rise to real, energy-independent contributions to the nucleon self-energy. The Hartree-Fock contribution from the two-nucleon force is attractive and strongly momentum dependent, in contrast to the contribution from the three-nucleon force which provides a nearly constant repulsive mean field that grows approximately linearly with the nuclear density. Together, the leading-order perturbative contributions yield an attractive single-particle potential that is however too weak compared to phenomenology. Second-order contributions from two- and three-body forces then provide the additional attraction required to reach the phenomenological depth. The imaginary part of the optical potential, which is positive (negative) for momenta below (above) the Fermi momentum, arises at second order and is nearly inversion-symmetric about the Fermi surface when two-nucleon interactions alone are present. The imaginary part is strongly absorptive and requires the inclusion of an effective mass correction as well as self-consistent single-particle energies to improve agreement with phenomenology.

FROM THE MICROSCOPIC TO THE VAN HOVE REGIME IN THE XY CHAIN OUT OF EQUILIBRIUM

Using the framework of rigorous algebraic quantum statistical mechanics, we construct the unique nonequilibrium steady state in the isotropic XY chain in which a sample of arbitrary finite size is coupled by a bond coupling perturbation of arbitrary strength to two infinitely extended thermal reservoirs, and we prove that this state is thermodynamically nontrivial. Moreover, extracting the leading second-order contribution to its microscopic entropy production and deriving its entropy production in the van Hove weak coupling regime, we prove that, in the mathematically and physically important XY chain, the van Hove regime reproduces the leading order contribution to the microscopic regime.

Analysis of double-hybrid density functionals along the adiabatic connection

We present a graphical analysis of the adiabatic connections underlying double-hybrid density-functional methods that employ second-order perturbation theory. Approximate adiabatic connection formulae relevant to the construction of these functionals are derived and compared directly with those calculated using accurate ab initio methods. The discontinuous nature of the approximate adiabatic integrands is emphasised, the discontinuities occurring at interaction strengths which mark the transitions between regions that are: (i) described predominantly by second-order perturbation theory; (ii) described by a mixture of density-functional and second-order perturbation theory contributions; and (iii) described purely by density-functional theory. Numerical examples are presented for a selection of small molecular systems and van der Waals dimers. The impacts of commonly used approximations in each of the three sections of the adiabatic connection are discussed along with possible routes for the development of improved double-hybrid methodologies.

Delineating effects of tensor force on the density dependence of nuclear symmetry energy

In this talk, we report results of our recent studies to delineate effects of the tensor force on the density dependence of nuclear symmetry energy within phenomenological models. The tensor force active in the isosinglet neutron roton interaction channel leads to appreciable depletion/population of nucleons below/above the Fermi surface in the single-nucleon momentum distribution in cold symmetric nuclear matter (SNM). We found that as a consequence of the high momentum tail in SNM the kinetic part of the symmetry energy Ekinsym(ρ) is significantly below the well-known Fermi gas model prediction of approximately 125(ρ/ρ0)2/3. With about 15% nucleons in the high momentum tail as indicated by the recent experiments at J-Lab by the CLAS Collaboration, the Ekinsym(ρ) is negligibly small. It even becomes negative when more nucleons are in the high momentum tail in SNM. These features have recently been confirmed by three independent studies based on the state-of-the-art microscopic nuclear many-body theories. In addition, we also estimate the second-order tensor force contribution to the potential part of the symmetry energy. Implications of these findings in extracting information about nuclear symmetry energy from nuclear reactions are discussed briefly.

Regular Solutions in Vacuum Brans-Dicke Theory Compared to Vacuum Einstein Theory

We will confront some static spherically symmetric vacuum Brans-Dicke solutions in the Jordan and Einstein Frames with the Robertson parameters. While the regular solution in the vacuum Einstein theory is just the Schwarzschild black hole, the same in the Jordan frame Brans-Dicke theory is shown to represent not a black hole but a traversable wormhole. But, in this case, the valid range ofωbecomes too narrow to yield the observed weak field Robertson parameters at the positive mass mouth. The corresponding solution in the Einstein frame also provides a regular wormhole, and it yields the correct parametric values but only up to “one and half order.” We argue that a second-order contribution can in principle distinguish between the signatures of the regular wormhole and the singular Buchdahl solution in the Einstein frame. Thus, at the level of regular solutions, Brans-Dicke theory in each frame predicts effects very different from those of Einstein's theory. To our knowledge, these theoretical distinctions seem not to have received adequate attention so far.

Prediction of the Flash Point Temperature of Organic Compounds with the Positional Distributive Contribution Method

A new universal method to predict the properties of organic compounds named as the positional distributive contribution method has recently been proposed by our group, which has successfully been used for the prediction of critical properties (Tc, Pc, Vc, Zc), normal boiling point (Tb), normal melting point (Tm), enthalpy of vaporization at the normal boiling point (ΔvapHb), and acentric factor (ω). In the following study, the positional distributive contribution method is further extended for the prediction of the flash-point (FP) temperature of various pure organic compounds. Comparison results between experimental and predicted data provide very satisfactory results. The overall average absolute difference (AAD) for the FP prediction of 287 organic compounds is 3.77 K, and the overall average relative deviation (ARD) is 1.16 %. However, Stefanis, Constantinou, and Panayiotou’s second-order group contribution (SCP-GC) method for the prediction of FP temperature of 287 organic compounds leads to the AAD and ARD of 14.16 K and 4.64 %, respectively. In addition, 342 organics missing experimental flash-point data are predicted via the presented procedure. More importantly, the good prediction capability of the proposed method shown in our previous works and this work suggests that it could be realized to use the same universal framework to predict not only Tc, Pc, Vc, Zc, Tb, Tm, ΔvapHb, and ω, but also the FP of organic compounds containing various functional groups, which further demonstrates the universality and stability of our proposed method.

Density Functional Theory Study on Subtriazaporphyrin Derivatives: Dipolar/Octupolar Contribution to the Second-Order Nonlinear Optical Activity

Density functional theory calculations have been carried out on the subtriazaporphyrin skeletons, an excellent prototype for investigating the dipolar/octupolar contribution to the second-order nonlinear optical (second-order NLO) activity, revealing the size effect and clarifying the nature of the limit when expanding the conjugated system is employed to improve the hyper-Rayleigh scattering response coefficient (β(HRS)). The octupolar and dipolar contributions are theoretically separated, rendering it possible to control the dipolar/octupolar second-order NLO contribution ratio by changing the number and orientation of the peripheral fused benzene moieties. In addition, both the dispersion and solvent effect were also revealed to lead to the enhancement of β(HRS).

Localized energetics of linear gravity: Theoretical development

We recently developed a local description of the energy, momentum and angular momentum carried by the linearized gravitational field, wherein the gravitational energy-momentum tensor displays positive energy density and causal energy flux, and the gravitational spin tensor describes purely spatial spin [1,2]. We now investigate the role these tensors play in a broader theoretical context, demonstrating for the first time that (a) they do indeed constitute Noether currents associated with the symmetry of the linearized gravitational field under translation and rotation and (b) they are themselves a source of gravity, analogous to the energy momentum and spin of matter. To prove (a) we construct a Lagrangian for linearized gravity (a covariantized Fierz-Pauli Lagrangian for a massless spin-2 field) and show that our tensors can be obtained from this Lagrangian using a standard variational technique for calculating Noether currents. This approach generates formulae that uniquely generalize our gravitational energy-momentum tensor and spin tensor beyond harmonic gauge: we show that no other generalization can be obtained from a covariantized Fierz-Pauli Lagrangian without introducing second derivatives in the energy-momentum tensor. We then construct the Belinfante energy-momentum tensor associated with our framework (combining spin and energy momentum into a single object) and as our first demonstration of (b) we establish that this Belinfante tensor appears as the second-order contribution to a perturbative expansion of the Einstein field equations, generating the gravitational field in a manner equivalent to the (Belinfante) energy-momentum tensor of matter. By considering a perturbative expansion of the Einstein-Cartan field equations, we then demonstrate that (b) can be realized without forming the Belinfante tensor: our energy-momentum tensor and spin tensor appear as the quadratic terms in separate field equations, generating gravity as distinct entities. Finally, we examine the role of field redefinitions within these perturbative expansions; in contrast to our tensors, the Landau-Lifshitz tensor is found to require a nonlocal field redefinition in order to be cast as a source of the gravitational field. In an appendix, we also give a brief treatment of the global quantities that our framework defines and verify their equivalence (within the quadratic approximation) to the energy momentum and angular momentum of Arnowitt, Deser, and Misner.

Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model

The field theoretic renormalization group and operator product expansion are applied to the Kazantsev-Kraichnan kinematic model for the magnetohydrodynamic turbulence. The anomalous scaling emerges as a consequence of the existence of certain composite fields ("operators") with negative dimensions. The anomalous exponents for the correlation functions of arbitrary order are calculated in the two-loop approximation (second order of the renormalization-group expansion), including the anisotropic sectors. The anomalous scaling and the hierarchy of anisotropic contributions become stronger due to those second-order contributions.

Exact calculation of three-body contact interaction to second order

For a system of fermions with a three-body contact interaction the second-order contributions to the energy per particle $\bar E(k_f)$ are calculated exactly. The three-particle scattering amplitude in the medium is derived in closed analytical form from the corresponding two-loop rescattering diagram. We compare the (genuine) second-order three-body contribution to $\bar E(k_f)\sim k_f^{10}$ with the second-order term due to the density-dependent effective two-body interaction, and find that the latter term dominates. The results of the present study are of interest for nuclear many-body calculations where chiral three-nucleon forces are treated beyond leading order via a density-dependent effective two-body interaction.

Relation between heat kernel method and scattering spectral method

In the present paper, we establish the connection between the heat kernel method and the scattering spectral method in quantum field theory. First, we provide the relation between the heat kernel and the scattering phase shift. Then as applications, we derive a large-time approximation of the heat kernel through the low-energy approximation of the phase shift, and we also derive a high-energy expansion of the phase shift expressed with heat kernel coefficients. Finally, we compare the renormalization schemes in the heat kernel method with that in the scattering spectral method. Concretely, we calculate the first- and second-order contributions to the vacuum energy by heat-kernel and Feynman-diagram methods, respectively, and show the coincidence of the results. Especially, in the heat-kernel framework, we perform both dimensional and zeta-function regularization to calculate the vacuum energy, and the result shows that dimensional renormalization procedure works well in the heat kernel method.

Monotonically convergent optimization in quantum control using Krotov's method

The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.

Second-Order Group Contribution Method for the Determination of the Dipole Moment

An estimation method for the dipole moment is proposed in this work. Polarity is caused by the effect of different neighboring groups in a molecule. A second-order group contribution approach has been chosen due to its ability to account for this effect. It could be shown that the use of second-order groups can cause a significant increase in prediction quality compared to the conventional first-order approach. The prediction quality of the proposed method is good, even for higher values of the dipole moment. Those higher than about 3.5 D are however underestimated. The model also enables the prediction of the small but finite dipole moments of hydrocarbons. Adequate estimations are not limited to compounds from the training set, but also compounds from a test set show good prediction results. Hence, it can be concluded that this method is able to give quick and reliable predictions for not yet measured dipole moments.

A Technique For Removing Second-Order Light Effects From Hyperspectral Imaging Data

The Hyperspectral Imager for the Coastal Ocean (HICO) instrument currently on board the International Space Station is a new sensor designed specifically for the studies of turbid coastal waters and large inland lakes and rivers. It covers the wavelength range between 0.4 and 0.9 μm with a spectral resolution of 5.7 nm and a spatial resolution of approximately 90 m. The HICO sensor is not equipped with a second-order blocking filter in front of the focal plane array. As a result, the second-order light from the shorter visible spectral region falls onto the detectors covering the near-IR spectral region above 0.8 μm. In order to have accurate radiometric calibration of the near-IR channels, the second-order light contribution needs to be removed. The water-leaving radiances of these near-IR channels over clear ocean waters are close to zero because of strong liquid water absorption above 0.8 μm. Through analysis of HICO imaging data containing features of shallow underwater objects, such as coral reefs, we have developed an empirical technique to correct for the second-order light effects in near-IR channels. HICO data acquired over Midway Island in the Pacific Ocean and the Bahamas Banks in the Atlantic Ocean are used to demonstrate the effectiveness of the new technique.

Disentangling multi-level systems: averaging, correlations and memory

We consider two weakly coupled systems and adopt a perturbative approach based on theRuelle response theory to study their interaction. We propose a systematic way ofparameterizing the effect of the coupling as a function of only the variables of a system ofinterest. Our focus is on describing the impacts of the coupling on the long term statisticsrather than on the finite-time behavior. By direct calculation, we find that, at firstorder, the coupling can be surrogated by adding a deterministic perturbation tothe autonomous dynamics of the system of interest. At second order, there areadditionally two separate and very different contributions. One is a term takinginto account the second-order contributions of the fluctuations in the coupling,which can be parameterized as a stochastic forcing with given spectral properties.The other one is a memory term, coupling the system of interest to its previoushistory, through the correlations of the second system. If these correlations areknown, this effect can be implemented as a perturbation with memory on the singlesystem. In order to treat this case, we present an extension to Ruelle’s responsetheory able to deal with integral operators. We discuss our results in the context ofother methods previously proposed for disentangling the dynamics of two coupledsystems. We emphasize that our results do not rely on assuming a time scaleseparation, and, if such a separation exists, can be used equally well to studythe statistics of the slow variables and that of the fast variables. By recursivelyapplying the technique proposed here, we can treat the general case of multi-levelsystems.

Modified Becke’05 method of nondynamic correlation in density functional theory with self-consistent implementation

Becke’s B05 method for nondynamic correlation is simplified for self-consistent implementation. An alternative form is proposed for the nondynamic correlation factors that do not require solving a complicated nonlinear algebraic equation. The four linear parameters of B05 are re-optimized together with one extra parameter in a modified expression for the second-order same-spin energy contribution. The latter is co-linear with the exact-exchange energy density and does not require higher moments of the relaxed exchange hole. Preliminary tests of this method show that it leads to a slight improvement over the resolution-of-identity B05 results reported previously for atomization energies, and to a definite improvement for reaction barriers of Hydrogen abstraction.

High-precision laser-assisted absolute determination of x-ray diffraction angles

A novel technique for absolute wavelength determination in high-precision crystal x-ray spectroscopy recently introduced has been upgraded reaching unprecedented accuracies. The method combines visible laser beams with the Bond method, where Bragg angles (θ and -θ) are determined without any x-ray reference lines. Using flat crystals this technique makes absolute x-ray wavelength measurements feasible even at low x-ray fluxes. The upgraded spectrometer has been used in combination with first experiments on the 1s2p(1)P(1) → 1s(2)(1)S(0) w-line in He-like argon. By resolving a minute curvature of the x-ray lines the accuracy reaches there the best ever reported value of 1.5 ppm. The result is sensitive to predicted second-order QED contributions at the level of two-electron screening and two-photon radiative diagrams and will allow for the first time to benchmark predicted binding energies for He-like ions at this level of precision.

Second-Order Contribution of the Incompressibility in Asymmetric Nuclear Matter

With the complementarity of the nucleonic three-body force, we present the saturation points of symmetric nuclear matter with different interactions adopted within the Brueckner—Hartree—Fock scheme, and a more accurate empirical parameterization function for the equation of state of symmetric nuclear matter and pure neutron matter. On the basis of this fit formula, the symmetry energy and its derivatives are investigated, and ultimately the higher-order coefficient of the isobaric incompressibility for isospin asymmetric nuclear matter is predicted.