Quasiclassical Method Research Articles

Quasi-classical approaches to vibronic spectra revisited.

The framework to approach quasi-classical dynamics in the electronic ground state is well established and is based on the Kubo-transformed time correlation function (TCF), being the most classical-like quantum TCF. Here we discuss whether the choice of the Kubo-transformed TCF as a starting point for simulating vibronic spectra is as unambiguous as it is for vibrational ones. Employing imaginary-time path integral techniques in combination with the interaction representation allowed us to formulate a method for simulating vibronic spectra in the adiabatic regime that takes nuclear quantum effects and dynamics on multiple potential energy surfaces into account. Further, a generalized quantum TCF is proposed that contains many well-established TCFs, including the Kubo one, as particular cases. Importantly, it also provides a framework to construct new quantum TCFs. Applying the developed methodology to the generalized TCF leads to a plethora of simulation protocols, which are based on the well-known TCFs as well as on new ones. Their performance is investigated on 1D anharmonic model systems at finite temperatures. It is shown that the protocols based on the new TCFs may lead to superior results with respect to those based on the common ones. The strategies to find the optimal approach are discussed.

Nonadiabatic Dynamics via the Symmetrical Quasi-Classical Method in the Presence of Anharmonicity.

The symmetrical quasi-classical (SQC) method recently proposed by Miller and Cotton allows one to simulate nonadiabatic dynamics based on an algorithm with classical-like scaling with respect to system size. This is made possible by casting the electronic degrees of freedom in terms of mapping variables that can be propagated in a classical-like manner. While SQC was shown to be rather accurate when applied to benchmark models with harmonic electronic potential energy surfaces, it was also found to become inaccurate and to suffer numerical instabilities when applied to anharmonic systems. In this paper, we propose an extended SQC (E-SQC) methodology for overcoming those discrepancies by describing the anharmonic nuclear modes, which are coupled to the electronic degrees of freedom, in terms of classical-like mapping variables. The accuracy of E-SQC relative to standard SQC is demonstrated on benchmark models with quartic and Morse potential energy surfaces.

Theoretical insight into the effect of collision energy on the S(3P) + SH(X2∏) → S2(X3∑g−) + H(2S) reaction

In this work, the dynamics of reaction S(3P) + SH(X2∏) → S2(X3∑g−) + H(2S) has been investigated using quasi-classical trajectories method on a newly constructed accurate double many-body expansion potential energy surface of Song et al. [J. Phys. Chem. A 115, 5274 (2011)]. The integral cross section and the rate constant are obtained. Furthermore, the distribution reflecting vector correlation and the polarization-dependent differential cross section are investigated. The results show a decreasing behavior of the integral cross section as the increasing of the collision energy, and dominate complex forming mechanism at lower energies, as well as the stripping mechanism at higher energies. Those results indicate that the product S2 is sensitively affected by the collision energy.

Ultrafast Electron Transfer with Symmetrical Quasi-classical Dynamics based on Mapping Hamiltonian and Quantum Dynamics based on ML-MCTDH

Symmetrical quasi-classical (SQC) method based on mapping Hamiltonian is an efficient approach that is potentially useful to treat the nonadiabatic dynamics of very large systems. We try to evaluate the performance of this method in the ultrafast electron transfer processes involving a few of electronic states and a large number of vibrational modes. The multilayer multiconfigurational time-dependent Hartree (ML-MCTDH) method was used to get the accurate dynamical results for benchmark. Although the population dynamics in the longtime limit show differences in the ML-MCTDH and SQC calculations, the SQC method gives acceptable results.

Quantum limit in a quasi-one-dimensional conductor in a high tilted magnetic field

Recently, we have suggested Fermi-liquid - non-Fermi-liquid angular crossovers which may exist in quasi-one-dimensional (Q1D)conductors in high tilted magnetic fields [see A.G. Lebed, Phys. Rev. Lett. $\textbf{115}$, 157001 (2015).] All calculations in the Letter, were done by using the quasi-classical Peierls substitution method, whose applicability in high magnetic fields was questionable. Here, we solve a fully quantum mechanical problem and show that the main qualitative conclusions of the above mentioned Letter are correct. In particular, we show that in high magnetic fields, applied along one of the two main crystallographic axis, we have 2D electron spectrum, whereas, for directions of high magnetic fields far from the axes, we have 1D electron spectrum. The later is known to promote non-Fermi-liquid properties. As a result, we expect the existence of Fermi-liquid - non-Fermi-liquid angular crossovers or phase transitions. Electronic parameters of Q1D conductor (Per)$_2$Pt(mnt)$_2$ show that such transitions can appear in feasible high magnetic fields of the order of $H \simeq 20-25 \ T$.

Vibrational Excitation of H2 Scattering from Cu(111): Effects of Surface Temperature and of Allowing Energy Exchange with the Surface

In scattering of H2 from Cu(111), vibrational excitation has so far defied an accurate theoretical description. To expose the causes of the large discrepancies with experiment, we investigate how the feature due to vibrational excitation (the “gain peak”) in the simulated time-of-flight spectrum of (v = 1, j = 3) H2 scattering from Cu(111) depends on the surface temperature (Ts) and the possibility of energy exchange with surface phonons and electron–hole pairs (ehp’s). Quasi-classical dynamics calculations are performed on the basis of accurate semiempirical density functionals for the interaction with H2 + Cu(111). The methods used include the quasi-classical trajectory method within the Born–Oppenheimer static surface model, the generalized Langevin oscillator (GLO) method incorporating energy transfer to surface phonons, the GLO + friction (GLO+F) method also incorporating energy exchange with ehp’s, and ab initio molecular dynamics with electronic friction (AIMDEF). Of the quasi-classical methods tested, comparison with AIMDEF suggests that the GLO+F method is accurate enough to describe vibrational excitation as measured in the experiments. The GLO+F calculations also suggest that the promoting effect of raising Ts on the measured vibrational excitation is due to an electronically nonadiabatic mechanism. However, by itself, enabling energy exchange with the surface by modeling surface phonons and ehp’s leads to reduced vibrational excitation, further decreasing the agreement with experiment. The simulated gain peak is quite sensitive to energy shifts in calculated vibrational excitation probabilities and to shifts in a specific experimental parameter (the chopper opening time). While the GLO+F calculations allow important qualitative conclusions, comparison to quantum dynamics results suggests that, with the quasi-classical way of describing nuclear motion and the present box quantization method for assigning the final vibrational state, the gain peak is not yet described with quantitative accuracy. Ways in which this problem might be resolved in the future are discussed.

Erratum to: Proximity Effect on the Density of States and Majorana Bound States in a Thin Superconducting Ferromagnetic Nanowire

This article presents analysis on the identification of Majorana states in a model of a heterogeneous superconducting ferromagnetic nanowire. It is demonstrated the significant effect that the exchange field has on the energy of bound in-gap states and on the density of states (DOS) itself, which appear as a periodic dependence of the DOS on the magnitude of the field as well as the dislocation in energy of the Majorana state candidate, located at zero energy in absence of the exchange field. Furthermore, the proximity effect is demonstrated to have observable consequences in a superconductor even in the clean limit. The model is a unidimensional system of fermions subject to a p-wave order parameter. The quasiclassical method is employed through the Eilenberger equations resulting in explicit expressions for the DOS, readily testable experimentally.

Quantization of conductance minimum and index theorem

We discuss the minimum value of the zero-bias differential conductance $G_{\textrm{min}}$ in a junction consisting of a normal metal and a nodal superconductor preserving time-reversal symmetry. Using the quasiclassical Green function method, we show that $G_{\textrm{min}}$ is quantized at $ (4e^2/h) N_{\mathrm{ZES}}$ in the limit of strong impurity scatterings in the normal metal. The integer $N_{\mathrm{ZES}}$ represents the number of perfect transmission channels through the junction. An analysis of the chiral symmetry of the Hamiltonian indicates that $N_{\mathrm{ZES}}$ corresponds to the Atiyah-Singer index in mathematics.

Robust zero-energy bound states around a pair-density-wave vortex core in locally noncentrosymmetric superconductors

We numerically investigate the electronic structures around a vortex core in a bilayer superconducting system, with s-wave pairing, Rashba spin-orbit coupling and Zeeman magnetic field, with use of the quasiclassical Green's function method. The Bardeen-Cooper-Schrieffer (BCS) phase and the so-called pair-density wave (PDW) phase appear in the temperature-magnetic-field phase diagram in a bulk uniform system [Phys. Rev. B 86, 134514 (2012)]. In the low magnetic field perpendicular to the layers, the zero-energy vortex bound states in the BCS phase are split by the Zeeman magnetic field. On the other hand, the PDW state appears in the high magnetic field, and sign of the order parameter is opposite between the layers. We find that the vortex core suddenly shrinks and the zero-energy bound states appear by increasing the magnetic field through the BCS-PDW transition. We discuss the origin of the change in vortex core structure between the BCS and PDW states by clarifying the relation between the vortex bound states and the bulk energy spectra. In the high magnetic field region, the PDW state and vortex bound states are protected by the spin-orbit coupling. These characteristic behaviors in the PDW state can be observed by scanning tunneling microscopy/spectroscopy.

Comparative analysis of the role of atom and ion spectral lines in radiative heating of four types of space capsules

This paper presents the results of two-dimensional calculation of radiative heating of the Fire-II, Stardust, Orion, and PPTS (Prospected Piloted Transport System, Russia) spacecraft entering the dense atmosphere of the Earth with orbital and superorbital velocities. A specificity of the simulation is the allowance for atom and ion spectral lines with use of the NERAT-ASTEROID computer platform. This computer platform is destined for solving the complete system of equations of radiative gas dynamics of viscous, heatconducting, and physically and chemically nonequilibrium gas and radiative transfer in two- and threedimensional geometries. The spectral and optical properties of high-temperature gases are calculated in the entire flow field by ab-initio quasi-classical and quantum mechanical methods. The selective heat radiation transfer was calculated by the line-by-line method on specially generated computational grids over the radiation wavelength, which make possible appreciable saving of computer resources while providing detailed description of the contours of atomic lines.

Radiative gas dynamics of the Fire-II superorbital space vehicle

The rates of convective and radiative heating of the Fire-II reentry vehicle are calculated, and the results are compared with experimental flight data. The computational model is based on solving a complete set of equations for (i) the radiative gas dynamics of a physically and chemically nonequilibrium viscous heatconducting gas and (ii) radiative transfer in 2D axisymmetric statement. The spectral optical parameters of high-temperature gases are calculated using ab initio quasi-classical and quantum-mechanical methods. The transfer of selective thermal radiation in terms of atomic lines is calculated using the line-by-line method on a specially generated computational grid that is nonuniform in radiation wavelength.

A Quasi-Classical Model of the Hubbard Gap in Lightly Compensated Semiconductors

A quasi-classical method for calculating the narrowing of the Hubbard gap between the A 0 and A + acceptor bands in a hole semiconductor or the D 0 and D – donor bands in an electron semiconductor is suggested. This narrowing gives rise to the phenomenon of a semiconductor transition from the insulator to metal state with an increase in doping level. The major (doping) impurity can be in one of three charge states (–1, 0, or +1), while the compensating impurity can be in states (+1) or (–1). The impurity distribution over the crystal is assumed to be random and the width of Hubbard bands (levels), to be much smaller than the gap between them. It is shown that narrowing of the Hubbard gap is due to the formation of electrically neutral acceptor (donor) states of the quasicontinuous band of allowed energies for holes (electrons) from excited states. This quasicontinuous band merges with the top of the valence band (v band) for acceptors or with the bottom of the conduction band (c band) for donors. In other words, the top of the v band for a p-type semiconductor or the bottom of the c band for an n-type semiconductor is shifted into the band gap. The value of this shift is determined by the maximum radius of the Bohr orbit of the excited state of an electrically neutral major impurity atom, which is no larger than half the average distance between nearest impurity atoms. As a result of the increasing dopant concentration, the both Hubbard energy levels become shallower and the gap between them narrows. Analytical formulas are derived to describe the thermally activated hopping transition of holes (electrons) between Hubbard bands. The calculated gap narrowing with increasing doping level, which manifests itself in a reduction in the activation energy e2 is consistent with available experimental data for lightly compensated p-Si crystals doped with boron and n-Ge crystals doped with antimony.

Radiation aerothermodynamics of the Stardust space vehicle

This paper presents a radiation-gasdynamic model and the results of calculating the radiative-convective heating of the Stardust space vehicle, obtained using the NERAT(2D)-ASTEROID computer code designed for solving the full system of equations of the radiation-gas dynamics of a viscous, conductive, physically and chemically non-equilibrium gas and radiation transfer in a two-dimensional geometry. The system of continuum mechanics equations is solved by the explicit method using a simplified procedure for solving the problem of the decomposition of an arbitrary discontinuity. The system of equations of energy conservation of translational degrees of freedom of gas particles and vibrational degrees of freedom of nitrogen, oxygen, and nitrogen oxide molecules is solved by the implicit finite difference method. A multiblock and multigrid procedure is used to obtain a numerical solution. Calculation of the transfer of selective thermal radiation is done using the line-by-line method on specially generated inhomogeneous calculation grids over the wavelength of radiation, making it possible to achieve an appreciable saving of computing resources. In a single cycle of the computing process, the spectral optical properties of high-temperature gases are calculated using ab initio quasi-classical and quantum mechanics methods.

Product Angular and Rovibrational State Distributions of the Li + HF(v=0, j=0) → LiF(v', j') + H Reaction

A state-to-state dynamics analysis of the title reaction has been investigated via the quasiclassical trajecory method. Results of the product state resolved differen- tial cross sections, polarization parameters, as well as rovibrational state distributions were revealed and discussed, most of which agree well with the recent quantum me- chanics study by Roncero and co-workers. It was found that more than 82.28% of reactive trajectories undergo the direct reaction mechanism. The title reaction occurs predominantly in the head-end collision.

Quantum and quasi-classical calculations for the S⁺ + H₂(v,j) → SH⁺(v',j') + H reactive collisions.

State-to-state cross-sections for the S(+) + H2(v,j) → SH(+)(v',j') + H endothermic reaction are obtained using quantum wave packet (WP) and quasi-classical (QCT) methods for different initial ro-vibrational H2(v,j) over a wide range of translation energies. The final state distribution as a function of the initial quantum number is obtained and discussed. Additionally, the effect of the internal excitation of H2 on the reactivity is carefully studied. It appears that energy transfer among modes is very inefficient that vibrational energy is the most favorable for the reaction, and rotational excitation significantly enhances the reactivity when vibrational energy is sufficient to reach the product. Special attention is also paid to an unusual discrepancy between classical and quantum dynamics for low rotational levels while agreement improves with rotational excitation of H2. An interesting resonant behaviour found in WP calculations is also discussed and associated with the existence of roaming classical trajectories that enhance the reactivity of the title reaction. Finally, a comparison with the experimental results of Stowe et al. for S(+) + HD and S(+) + D2 reactions exhibits a reasonably good agreement with those results.

Josephson effect in spin-singlet superconductor/ferromagnetic insulator/spin-triplet superconductor junctions with helical p-wave states

We study the Josephson effect in spin-singlet superconductor/helical p-wave superconductor junctions with a ferromagnetic barrier using the quasiclassical Green function method. It is found that both sin ϕ-type and cos ϕ-type current-phase relations always exist, irrespective of the gap symmetries in superconductors. The indispensable condition for the sin ϕ-type and cos ϕ-type current is that the magnetization must have a component parallel to the crystallographic a- or b-axis, which is distinct from the case of the p-wave superconductor described by a d-vector with a uniform direction. The relation between the condition and the symmetries of the gap functions is analysed. We investigate in detail the symmetries and the sign reversal of the Josephson current when the magnetization is rotated.

Quasiclassical Approximation in the Non-Relativistic and Relativistic Problems of Tunneling Ionization of a Hydrogen-Like Atom in a Uniform Electric Field

A recurrent scheme for finding the quasiclassical solution of the onedimensional equation obtained after the separation of variables in the Schrodinger equation in parabolic coordinates is derived. The method of quasiclassical localized states is developed for the Dirac equation with an arbitrary axially symmetric potential of barrier type which does not allow complete separation of the variables. By means of the proposed quasiclassical methods the non-relativistic and relativistic wavefunctions for hydrogenlike (H-like) atoms in an external uniform electrostatic field of intensity F are constructed in the classically forbidden and allowed regions. The general analytical expressions of the leading term of the asymptotic behaviour (at small F ) of the ionization rate of an H-like atom in the uniform electrostatic field are obtained for the non-relativistic and relativistic cases.

The role of atomic lines in radiation heating of the experimental space vehicle Fire-II

The results of calculating the convective and radiation heating of the Fire-II experimental space vehicle allowing for atomic lines of atoms and ions using the NERAT-ASTEROID computer platform are presented. This computer platform is intended to solve the complete set of equations of radiation gas dynamics of viscous, heat-conductive, and physically and chemically nonequilibrium gas, as well as radiation transfer. The spectral optical properties of high temperature gases are calculated using ab initio quasi-classical and quantum-mechanical methods. The calculation of the transfer of selective thermal radiation is performed using a line-by-line method using specially generated computational grids over the radiation wavelengths, which make it possible to attain a noticeable economy of computational resources.

Magnetization induced by odd-frequency spin-triplet Cooper pairs in a Josephson junction with metallic trilayers

We theoretically study the magnetization inside a normal metal induced in an s-wave superconductor/ferromagnetic metal/normal metal/ferromagnetic metal/s-wave superconductor (S/F1/N/F2/S) Josephson junction. Using quasiclassical Green's function method, we show that the magnetization becomes finite inside N. The origin of this magnetization is due to odd-frequency spin-triplet Cooper pairs formed by electrons of equal and opposite spins, which are induced by proximity effect in the S/F1/N/F2/S junction. We find that the magnetization M(d,q) in N can be decomposed into two parts, M(d,q)=MI(d)+MII(d,q), where q is the superconducting phase difference between two Ss and d is the thickness of N. MI(d) exists generally in S/F junctions, while MII(d,q) carries all q dependence and represents the fingerprint of phase coherence between two Ss in Josephson junctions. The q dependence thus allows us to control the magnetization in N by tuning q for a fixed d. We show that MI(d) weakly decreases with increasing d, while the q dependent magnetization MII(d,q) rapidly decays with d. Moreover, we find that the time-averaged magnetization <MII(d,q)> exhibits discontinuous peak at each resonance DC voltage Vn=nhw_S/2e(n: integer) when DC voltage V as well as AC voltage v_ac(t) with frequency w_S are both applied to the S/F1/N/F2/S junction. This is because MII(d,q) oscillates generally in time t (AC magnetization) with dq/dt=2e[V+v_ac(t)]/h and thus <MII(d,q)>=0, but can be converted into the time-independent DC magnetization for DC voltage at Vn. We also discuss that the magnetization induced in N can be measurably large in realistic systems. Therefore, the measurement of the induced magnetization serves as an alternative way to detect the phase coherence between two Ss in Josephson junctions. Our results also provide a basic concept for tunable magnetization in superconducting spintronics devices.

Long rangep-wave proximity effect into a disordered metal

We use quasiclassical methods of superconductivity to study the superconducting proximity effect from a topological $p$-wave superconductor into a disordered one-dimensional metallic wire. We demonstrate that the corresponding Eilenberger equations with disorder reduce to a closed non-linear equation for the superconducting component of the matrix Green's function. Remarkably, this equation is formally equivalent to a classical mechanical system (i.e., Newton's equations), with the Green function corresponding to a coordinate of a fictitious particle and the coordinate along the wire corresponding to time. This mapping allows to obtain exact solutions in the disordered nanowire in terms of elliptic functions. A surprising result that comes out of this solution is that the $p$-wave superconductivity proximity-induced into the disordered metal remains long-range, decaying as slowly as the conventional $s$-wave superconductivity. It is also shown that impurity scattering leads to the appearance of a zero-energy peak.