Second-order Contributions Research Articles

The shape of large surface waves on the open sea and the Draupner New Year wave

Ocean surface waves are both random in time and nonlinear. For a linear random model of the sea surface, the average shape of a large crest is found to be the scaled auto-correlation function, as is shown mathematically by Lindgren [Lindgren G. Some properties of a normal process near a local maximum. Ann Math Stat 1970;41:1870–83] and Boccotti [Boccotti P. Some new results on statistical properties of wind waves. Appl Ocean Res 1983;5:134–40]. Using field data from the North Sea, the applicability of this simple result is discussed. Stokes-type corrections up to fifth order are incorporated in an approximate but robust manner, valid at least locally in space and time. Exact second-order wave theory is used to check the accuracy of the approximate Stokes treatment and to clarify the magnitude and character of the second-order contributions. A giant wave, called the New Year wave, recorded at the Draupner platform in the North Sea at 15:20 on 1st January 1995, is examined. In a sea-state with a significant wave height of approximately 12m, this freak wave, which has a peak elevation of 18.5m above still water level, was unambiguously recorded. The local asymmetries of this wave are removed using the Stokes model, allowing the probability of occurrence of this remarkable event to be estimated, based on the standard Rayleigh distribution for linear crest amplitude. It is concluded that new physics, not incorporated in standard approaches to offshore engineering design, may have played an important role in the generation of this freak wave.

Role of the Solvent in Computing the 1,4-Benzosemiquinone g-Tensor by the Coupled-Perturbed Kohn−Sham Hybrid Density Functional Method

The 1,4-benzosemiquinone (BzSQ) g-tensor components, in the gas phase and in methanol, are computed. Neese's coupled-perturbed Kohn−Sham method is used, in conjunction with the UB1LYP and UPBE0 exchange-correlation hybrid density functionals. The computed g-tensor elements are analyzed and broken down into their individual first- and second-order contributions. The inclusion of electron correlation, via the UB1LYP and UPBE0 hybrid density functionals, improves these calculations over those carried out at the Hartree−Fock level. Further improvement is obtained if the SCF reaction field method (SCRF) of Tomasi, as implemented by Barone, is employed to mimic the solvent effects. The best agreement with experiment is achieved using a model in which the BzSQ hydrogen bonds to four methanol solvent molecules to form a supermolecule. This occurs because the main contribution to the g-tensor components arises from the net spin densities on the carbonyl oxygen atoms, which are highly influenced by hydrogen bonding...

Spectroscopy of the OC-HF hydrogen-bonded complex at vHF=3.

The v(HF)=3 levels of the linear OC-HF complex are observed in the range of 10,800-11,500 cm(-1) using intracavity Ti-sapphire laser-induced fluorescence. The vibrational predissociation linewidths of both (30000) and (3001(1)0) states exceed 5 GHz; thus, the measured spectra are not rotationally resolvable. Under the assumption that these levels are not strongly perturbed, the rotational constants of the two levels are determined to be 0.1100(1) cm(-1) for (30000), 0.1081(1), and 0.1065(1) cm(-1) for f and e sublevels of (3001(1)0), respectively, through band contour fitting. The (30000)<--(00000) band origin is at 10,894.46(1) cm(-1), showing a HF wave number redshift of 478.3 cm(-1). The 4.07 redshift ratio of v(HF)=3 to that of v(HF)=1 indicates a significantly nonlinear increase of the intermolecular interaction energy through HF valence excitation. An ab initio interaction potential surface for HF valence coordinates varying from 0.8 to 1.25 A is used to examine vibrational dynamics. The HF valence vibration v(1) is treated perturbatively, showing that the vibrational redshifts are determined essentially in first order with only a very small second-order contribution. The (3001(1)0)<--(00000) combination transition is observed with the band origin at 11,432.66(1) cm(-1), giving the HF intermolecular bending mode to be 538.2 cm(-1). The high frequency of this vibration, compared to that in similar HF complexes, shows the strong angular anisotropy of the intermolecular interaction potential of OC-HF with respect to the HF subunit. The lifetime of the (3001(1)0) level increases to 28 ps from 14 ps for (30000).

Magnetic Nanoparticles as Many‐Spin Systems

AbstractFor Abstract see ChemInform Abstract in Full Text.

Neutrino bremsstrahlung in neutron matter from effective nuclear interactions

We revisit the emissivity from neutrino pair bremsstrahlung in neutron–neutron scattering, nn→nnνν̄, which was calculated from the one-pion exchange potential including correlation effects by Friman and Maxwell. Starting from the free-space low-momentum nucleon–nucleon interaction Vlowk, we include tensor, spin-orbit and second-order medium-induced non-central contributions to the scattering amplitude in neutron matter. We find that the screening of the nucleon–nucleon interaction reduces the emissivity from neutrino bremsstrahlung for densities below nuclear matter density. We discuss the implications of medium modifications for the cooling of neutron stars via neutrino emission, taking into account recent results for the polarization effects on neutron superfluidity.

Relativistic spin-orbit effects on hyperfine coupling tensors by density-functional theory.

A second-order perturbation theory treatment of spin-orbit corrections to hyperfine coupling tensors has been implemented within a density-functional framework. The method uses the all-electron atomic mean-field approximation and/or spin-orbit pseudopotentials in incorporating one- and two-electron spin-orbit interaction within a first-principles framework. Validation of the approach on a set of main-group radicals and transition metal complexes indicates good agreement between all-electron and pseudopotential results for hyperfine coupling constants of the lighter nuclei in the system, except for cases in which scalar relativistic effects become important. The nonrelativistic Fermi contact part of the isotropic hyperfine coupling constants is not always accurately reproduced by the exchange-correlation functionals employed, particularly for the triplet and pi-type doublet radicals in the present work. For this reason, ab initio coupled-cluster singles and doubles with perturbative triples results for the first-order contributions have been combined in the validation calculations with the density-functional results for the second-order spin-orbit contributions. In the cases where spin-orbit corrections are of significant magnitude relative to the nonrelativistic first-order terms, they improve the agreement with experiment. Antisymmetric contributions to the hyperfine tensor arise from the spin-orbit contributions and are discussed for the IO2 radical, whereas rovibrational effects have been evaluated for RhC, NBr, and NI.

Coherent and Dissipative Spin Dynamics inN-Electron Systems

We investigate the ensemble averaged evolution of N-electron systems dynamically coupled to a statistical environment. The electrons are characterized by their spatial and by their spin properties. While the Hilbert space for single electrons is given by the tensor product of the Hilbert spaces associated with both properties, the corresponding Hilbert space for N-electron systems cannot be factorized. Consequently, quantum correlations between spatial and spin properties become extremely important. We assume that the evolution of the spin properties is controlled by spin-orbit interaction and that the spatial properties take the part of a bath held near some equilibrium. This description is appropriate for magnetic systems where the electronic states near the ground state correspond to different spin configurations, whereas electronic states with large excitation energies belong to different spatial-orbital configurations. In order to determine the coarse grained evolution of the spin properties, we have to know the evolution of the N-electron system over time intervals larger than the bath-correlation time. This is obtained from the first- and second-order contributions in the interaction picture. We show that, in spite of the strong quantum correlations between spin properties and spatial properties, the coarse grained statistical evolution of the electronic spin properties may be described by a set of coupled master equations.

Second-order post-newtonian equations of light ray

A series of space astrometric missions have been proposed recently, in some of which second-order contribution to light propagation in multiple systems should be considered, i.e., second-order post-Newtonian ( 2PN ) equations of light ray have to be discussed under the recent extension of the DSX scheme, founded in early 1990's to investigete a complete first-order post-Newtonian (1PN) celestial mechanics for N arbitrarily composed and shaped, rotating deformable bodies. In this paper, the iterative method is taken to establish and calculate the 2PN equations of light ray. The 2PN equations for the solar system have been deduced from the metric tensor and Christoffel symbols. When the higherorder terms are neglected, the 2PN equations reduce to the well-known 1PN equations. By using these equations, the light propagation in the solar system can be calculated.

Disappearance and reappearance of magnetic ordering upon lanthanide substitution in(Er1−xDyx)Al2

Multiple magnetic ordering phenomena, when substituting Dy for Er in $({\mathrm{Er}}_{1\ensuremath{-}x}{\mathrm{Dy}}_{x}){\mathrm{Al}}_{2}$ over a broad range of concentrations $0&lt;~x&lt;~0.85,$ observed in low-temperature heat capacity measurements have been explained theoretically. The Hamiltonian that describes the system includes crystalline electric field effects, exchange interaction, and second-order contributions such as quadrupolar and magnetoelastic effects. We show that the discontinuity in the heat capacity, which arises at certain concentrations of the alloying element Dy, is the result of the competition between the magnetoelastic coupling and quadrupolar effects. The existence of the disappearance and reappearance of the magnetic phases in other lanthanide-lanthanide systems is also discussed.

Transverse nuclear spin relaxation due to director fluctuations in liquid crystals. III. A slow-motional theory for the angular dependence in pulsed experiments

In the previous article, we have proposed a slow-motional theory for second-order effects of director fluctuations on transverse spin relaxation of quadrupolar nuclei in liquid crystals [D. Frezzato, G.J. Moro, and G. Kothe, J. Chem. Phys. 119, 6931 (2003), preceding paper]. This methodology is now generalized to arbitrary orientations of director and magnetic field. The characteristic functions are evaluated for the free induction decay and the echo intensities in Carr–Purcell–Meiboom–Gill (CPMG) multipulse sequences. From the solution of the corresponding integral equations, the relative magnitude of first and second-order contributions can be assessed. This enables a complete characterization of the angular and pulse spacing dependent transverse relaxation rates observed in CPMG multipulse experiments.

Transverse nuclear spin relaxation due to director fluctuations in liquid crystals. II. Second-order contributions of the fluctuating director

Recently, we have introduced a slow-motional theory for transverse nuclear spin relaxation due to director fluctuations [D. Frezzato, G. Kothe, and G. J. Moro, J. Phys. Chem. B 105, 1281 (2001)]. This method is now generalized to second-order contributions of the fluctuating director. We consider the specific case in which the director is aligned orthogonal to the magnetic field. By exploiting the Gaussian character of director fluctuations, the stochastic Liouville equation for the coupled spin and director dynamics is solved in terms of a characteristic function whose time dependence is determined by a nonlinear integral equation. A convenient solution of the integral equation is obtained by decomposing the characteristic function according to the relaxation rates of the director fluctuations. In a first application, we evaluate the free induction decay and the corresponding absorption spectrum for quadrupolar probe nuclei in nematic liquid crystals. It is shown that the transverse magnetization is well represented by a monoexponential decay, i.e., a Lorentzian lineshape in the frequency domain. Explicit relations are derived for the linewidths and frequency shifts under slow-motional conditions where the Redfield theory cannot be applied anymore.

Fine and hyperfine structure in three low-lying 3Σ+ states of molecular hydrogen

The fine structure constant (electron spin-spin coupling) and the hyperfine structure parameters (electron-nuclear spin coupling, including spin-rotation and electron-nuclear quadrupole coupling) in the low-lying triplet states b3Σ+ u, a3Σ+ g and e3Σ+ u of molecular hydrogen and deuterium are calculated using a recently developed technique with full configuration interaction and multiconfiguration self-consistent field wave functions. The second-order spin-orbit coupling contribution to the 3Σ+ states splitting is negligible, and the calculations therefore provide a good estimate of the zero-field splitting based only on the electron spin-spin coupling values. For the bound a3Σ+ g state a negligible zero-field splitting is found, in qualitative agreement with the e-a spectrum. The zero-field splitting parameter is considerable for the repulsive b3Σ+ u state (≃1 cm−1) and of intermediate size for the bound e3Σ+ u state. The isotropic hyperfine coupling constant is very large not only for the valence b3Σ+ u state (1580 MHz) but also for the Rydberg a and e triplet states (≃1400 MHz). The quadrupole coupling constants for the deuterium isotopes are negligible (0.04–0.07 MHz) for all studied triplet states. The electric dipole activity of the spin sublevels in the triplet-singlet transitions to the ground state is estimated by means of the quadratic response technique.

Ab initio study of nonhomogeneous broadening of the zero-field splitting of triplet guest molecules in diluted glasses

Nonhomogeneous broadening of phosphorescence lines and microwave signals in optical detection of magnetic resonance (ODMR) has been calculated using multiconfigurational self-consistent field wave functions and the polarized continuum model. The solvent effects on the zero-field splitting (ZFS) parameters in the low-lying triplet states of aza-aromatic molecules are found to be linearly dependent on the solvent-induced shifts in the phosphorescence frequency in agreement with experimental data. The main contribution to the ZFS originates in the dipolar interaction of the two electron spins, the spin–spin coupling. The spin–orbit coupling (SOC) contribution to the ZFS parameter is much larger for the 3nπ* state of pyrazine compared to the 3ππ* states of quinoline. The second-order SOC contribution to the splitting of the 3nπ* state in the pyrazine molecule does not show any appreciable dependence on the dielectric constant of the solvent. This raises doubts about earlier theories for explaining the inhomogeneous broadening in triplet-state spectra based exclusively on the SOC-induced mixing of the singlet and triplet states. We complete the interpretation of the ODMR spectrum of pyrazine by calculating the hyperfine coupling (HFC) tensors in the lowest triplet state using the UB3LYP hybrid functional. An appreciable solvent-induced rotation of the anisotropic HFC tensor axes has been obtained for the 3nπ* state of pyrazine, in particular for 13C and 14N nuclei. This produces additional nonhomogeneous broadening not only in electron–nuclear double resonance spectra, but also in electron paramagnetic resonance signals because the anisotropic HFC perturbation results in an intensity redistribution among the magnetic transitions between the spin sublevels. A small in-plane rotation of the ZFS tensor axes upon solvation has been predicted for quinoline. Rotation of the magnetic axes induced by the interaction with isotropic solvents can provide a new mechanism for spin-lattice relaxation in the triplet state.

Theory and Computer Simulation of the First- and Second-order Perturbative Contributions to the Free Energy of Square-well Fluids

We have performed extensive MC simulations in the NVT ensemble of the first- and second-order perturbative contributions, F 1 and F 2 , to the free energy of square-well (SW) fluids for wide ranges of densities and potential widths. These data are used to test the accuracy of theoretical calculations performed within the framework of the Barker-Henderson perturbation theory. For F 2 the performance of several approximations is analyzed.

Antisymmetric double exchange in trimeric mixed-valence clusters

Abstract The theory of antisymmetric double exchange (AS DE) interaction is developed for the mixed-valence (MV) trimeric [dn–dn–dn±1] clusters of orbitally non-degenerate ions. Strong isotropic double exchange (DE) and Heisenberg exchange interactions (t-J model) form isotropic trigonal states 2S+1 Γ i (Γi=A1, A2, E) of the MV trimers. Taking into account the spin–orbit coupling results in the vector-type AS DE interaction in these clusters. For the MV trimers with si=1/2, the Hamiltonian of the AS DE coupling has the form H AS DE Z =2 i ∑ αβ K αβ Z T αβ (S β Z −S α Z ) , where KαβZ is the antisymmetric ( K → ij =− K → ji ) vector parameter of the AS DE interaction and T αβ is the isotropic transfer operator. The operator H AS DE Z describes the vector-type spin-transfer interaction induced by the spin–orbit coupling. The microscopic consideration of the AS DE coupling shows that the vector AS DE parameter K → ij is linearly proportional to the spin–orbit coupling constant λ and electron transfer parameter t. The AS DE coupling results in the new effect: the linear AS DE splittings Δ of the 2S+1 E DE terms, Δ are proportional to the cluster AS DE parameter KZ=(KabZ+KbcZ+KcaZ)/3. The vector of the AS DE interaction is directed along the trigonal Z-axis: K Z ≠0, K X =K Y =0 . For the delocalized [d9–d10–d10] ([Cu(II)Cu2(I)]) cluster, the linear AS DE fine splitting Δ=2K Z 3 of the ground 2 E DE term determines strong anisotropy of the Zeeman splitting, axial anisotropy of g-factors, and magnetic properties. The AS DE coupling mixes the 2S+1 A 1 and 2S ′ +1 A 2 , 2S+1 E and 2S ′ +1 E ′ DE terms, ΔS=0,±1. The mixing of the DE levels 2S+1 Γ i by the AS DE coupling and Dzialoshinsky–Moriya AS exchange (H DM =∑ αβ G → αβ [ S → α × S → β ]) determines the second-order AS contributions {∼[(n1KZ+n2GZ)2/(n3t+n4J)], K Z ≫G Z , G Z =(G ab Z +G bc Z +G ca Z )/3} to the cluster zero-field splitting (ZFS) parameters D S ( 2S+1 Γ i ) of the axial anisotropy ( H AN =D S ( 2S+1 Γ i )[S z 2 −S(S+1)/3] , DS≪KZ). The second-order AS DE contributions to the ZFS parameters D S ( 2S+1 Γ i ) are different for the 2S+1 A 1 , 2S+1 A 2 and 2S+1 E DE terms.

Complete basis set limit studies of conventional and R12 correlation methods: The silicon dicarbide (SiC2) barrier to linearity

The problematic SiC2 barrier to linearity is investigated in a benchmark study of one-electron basis set convergence properties of both the conventional and linear R12/A formulations of second-order Møller–Plesset (MP2) perturbation theory. A procedure for computational molecular partial-wave expansions is constructed and applied to the T-shaped and linear forms of SiC2. The largest basis set used [Si(22s17p14d6f5g2h2i1k)/C(19s14p14d6f5g2h2i1k)] included functions of orbital angular momentum as large as l=7 (k), and systematic saturation was performed through l=6 (i). With respect to angular momentum l, correlation energy increments are found to decay in accord with analytical models that suggest (l+1/2)−6 and (l+1/2)−4 functional forms for the R12/A and conventional methods, respectively. A benchmark complete basis set (CBS) limit for the second-order correlation contribution to the SiC2 barrier to linearity, 5.66 kcal mol−1, was determined via MP2-R12/A partial-wave expansions. Conventional MP2 calculations, using both the standard cc-pV6Z and the [Si(22s17p14d6f5g2h2i1k)/C(19s14p14d6f5g2h2i1k)] basis sets, underestimate MP2 correlation energies by at least 3 kcal mol−1, while the barrier is underestimated by at least 0.1 kcal mol−1. Both X−3 cc-pVXZ extrapolations and partial-wave extrapolations greatly improve conventional correlation energies, with the cc-pVXZ extrapolated barrier in error by only 0.07 kcal mol−1. While the absolute accuracy of the conventional partial-wave extrapolations is substantially better than the cc-pVXZ extrapolated values, unbalanced errors result in an overestimation of the barrier by nearly 0.2 kcal mol−1. The CBS-limit MP2 contribution is combined via a focal-point analysis with conventional coupled cluster computations through triple excitations (CCSDT), resulting in an inferred CBS CCSDT barrier of 5.45 kcal mol−1 after accounting for core correlation and relativistic effects. The critical question of post-CCSDT corrections is approached through explicit coupled cluster computations perturbatively accounting for connected quadruple excitations [BD(TQ) and CCSD(2)], as well as shifted [2,1] Padé approximants of MPn series and continued fraction and quadratic Padé approximants of coupled-cluster series. The best available post-CCSDT correction, extracted from BD(TQ)/cc-pVTZ theory, of 0.87 kcal mol−1, results in a final prediction near 6.3 kcal mol−1 for the SiC2 barrier to linearity.

Theoretical investigation of EPR parameters for two trigonal Yb 3+ centers in LiNbO 3 and LiNbO 3:MgO crystals

The perturbation formulas of EPR parameters ( g factors g ∥, g ⊥ and hyperfine structure constants A ∥, A ⊥) for the lowest Kramers doublet Γ 6 or Γ 7 of 2F 7/2 of 4f 13 ion in trigonal symmetry are established. In these formulae, the contributions of the covalency effects, the admixture between J=7/2 and 5/2 states as well as the second-order perturbation (which is not considered in the previous works) are all included. From these formulae, the EPR parameters for two trigonal Yb 3+ centers in LiNbO 3 and MgO codoping LiNbO 3 crystals are reasonably explained by considering suitable defect structures of these centers. The results (including the relative importance of second-order perturbation contribution) are discussed.

Extending the First-Order Post-Newtonian Scheme in MultipleSystems to the Second-Order Contributions to Light Propagation

We extend the first-order post-Newtonian scheme in multiple systemspresented by Damour-Soffel-Xu to the second-order contribution to lightpropagation without changing the virtue of the scheme on the linearpartial differential equations of the potential and vector potential.The spatial components of the metric are extended to the second-orderlevel both in a global coordinates (qij/c4) and a localcoordinates (Qab/c4). The equations of qij (or Qab)are obtained from the field equations. The relationship between qijand Qab are also presented. In the special case of the solar system(isotropic condition is applied (qij = δijq)), we obtainthe solution of q. Finally, a further extension of the second-ordercontributions in the parametrized post-Newtonian formalism is discussed.

Green's Function Expansions in Dyadic Root Functions for Shielded Layered Waveguides — Abstract

Dyadic Green's functions for inhomogeneous parallel-plate waveguides are considered. The usual residue series form of the Green's function is examined in the case of modal degeneracies, where secondorder poles are encountered. The corresponding second-order residue contributions are properly interpreted as representing "associated functions" of the structure by constructing a new dyadic root function representation of the Hertzian potential Green's dyadic. The dyadic root functions include both eigenfunctions (corresponding to first-order residues) and associated functions, analogous to the idea of Jordan chains in finite-dimensional spaces. Numerical results are presented for the case of a two-layer parallel-plate waveguide.

The helium dimer potential from a combined density functional theory and symmetry-adapted perturbation theory approach using an exact exchange–correlation potential

Starting with an analytic representation of the electron density from a Hylleraas wavefunction we have obtained an analytic representation of the exchange–correlation potential of the helium atom. This, essentially exact, exchange–correlation potential has been employed to test a recently developed approach, named DFT-SAPT, which combines symmetry-adapted perturbation theory of intermolecular interaction energies with a density functional theory description of the interacting monomers. In DFT-SAPT all of the second-order contributions including the exchange corrections are determined from coupled-perturbed density functional theory. Comparison of the results for the helium dimer to previous high-quality supermolecular and intermolecular perturbation theory results demonstrates the success of the new approach.