Time-independent Theory Research Articles

A Kernel-Based Least-Squares Collocation Method for Surface Diffusion

There are plenty of applications and analysis for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be necessary for stability and convergence of widely used unsymmetric Kansa-type strong-form collocation methods. We consider kernel-based meshfree methods, which is a method of lines with collocation and overtesting spatially, for solving parabolic partial differential equations on surfaces without parametrization. In this paper, we extend the time-independent convergence theories for overtesting techniques to the parabolic equations on smooth and closed surfaces.

Dispersion Energy from the Time-Independent Coupled-Cluster Polarization Propagator.

We present a new method of calculation of the dispersion energy in the second-order symmetry-adapted perturbation theory. Using the Longuet-Higgins integral and time-independent coupled-cluster response theory, one shows that the general expression for the dispersion energy can be written in terms of cluster amplitudes and the excitation operators σ, which can be obtained by solving a linear equation. We introduced an approximate scheme dubbed CCPP2(T) for the dispersion energy accurate to the second order of intramonomer correlation, which includes certain classes to be summed to infinity. Assessment of the accuracy of the CCPP2(T) dispersion energy against the FCI dispersion for He2 demonstrates its high accuracy. For more complex systems, CCPP2(T) matches the accuracy of the best methods introduced for calculations of dispersion so far. The method can be extended to higher-order levels of excitations, providing a systematically improvable theory of dispersion interaction.

An Introduction to PT-Symmetric Quantum Mechanics-Time-Dependent Systems

I will provide a pedagogical introduction to non-Hermitian quantum systems that are PT-symmetric, that is they are left invariant under a simultaneous parity transformation (P) and time-reversal (T). I will explain how generalised versions of this antilinear symmetry can be utilised to explain that these type of systems possess real eigenvalue spectra in parts of their parameter spaces and how to set up a consistent quantum mechanical framework for them that enables a unitary time-evolution. In the second part I will explain how to extend this framework to explicitly time-dependent Hamiltonian systems and report in particular on recent progress made in this context. I will explain how to construct the essential key quantity in this framework, the time-dependent Dyson map and metric and solutions to the time-dependent Schrödinger equation, in an algebraic fashion, using time-dependent Darboux transformations, utilising Lewis-Riesenfeld invariants, point transformations and some approximation methods. I comment on the ambiguities of this metric and demonstrate that this can even lead to infinite series of metric operators. I conclude with some applications to PT-symmetrically coupled oscillators, demonstrate the equivalence of the time-dependent double wells and unstable anharmonic oscillators and show how the unphysical PT-symmetrically broken regions in the parameter space for the time-independent theory becomes physical in the explicitly time-dependent systems. I discuss how this leads to a prolongation of the otherwise rapidly decaying von Neumann entropy. The so-called sudden death of the entropy is stopped at a finite value.1

A constitutive framework for rocks undergoing solid dissolution

We formulate a time-dependent damage theory for rocks subjected to mechanical deformation and solid dissolution. The constitutive description is inspired by the transition state theory, which states that the rate of dissolution is a function of the reactive surface area measured through the crack density in the volume. We use a gradient-enhanced damage framework in which damage depends on the deformation of the material as well as on the amount of solid mass dissolved over time. The gradient-enhanced formulation is characterized by a three-field variational formulation with the solid displacement, nonlocal equivalent strain, and nonlocal rate of solid dissolution as the basic state variables. Traditionally, time-independent damage theories have only allowed damage to increase with increasing external load. In the proposed framework, the degree of damage may increase due to solid dissolution even when the external load is held fixed. In this way, solid dissolution is viewed as a process that is responsible for bringing about rate-dependent effects such as creep and stress-relaxation, which are two common features of geomaterial behavior.

Linear and Nonlinear Optical Properties from TDOMP2 Theory

We present a derivation of real-time (RT) time-dependent orbital-optimized Møller–Plesset (TDOMP2) theory and its biorthogonal companion, time-dependent non-orthogonal OMP2 theory, starting from the time-dependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time-independent molecular electronic structure theory. We apply the TDOMP2 method to extract absorption spectra and frequency-dependent polarizabilities and first hyperpolarizabilities from RT simulations, comparing the results with those obtained from conventional time-dependent coupled-cluster singles and doubles (TDCCSD) simulations and from its second-order approximation, TDCC2. We also compare our results with those from CCSD and CC2 linear and quadratic response theories. Our results indicate that while TDOMP2 absorption spectra are of the same quality as TDCC2 spectra, including core excitations where optimized orbitals might be particularly important, frequency-dependent polarizabilities and hyperpolarizabilities from TDOMP2 simulations are significantly closer to TDCCSD results than those from TDCC2 simulations.

A Time-Independent Quantum Approach to Ro-vibrationally Inelastic Scattering between Atoms and Triatomic Molecules

A full-dimensional time-independent quantum mechanical theory for ro-vibrationally inelastic scattering of triatomic molecules with atoms is formulated. The Jacobi-Radau coordinate system used in the calculation allows not only a near perfect description of the vibrational problem but also the adaptation of the exchange symmetry for A2B type triatoms. The S-matrix elements are obtained by solving the close-coupling equations with contracted basis using the log-derivative method. This method is applied to the inelastic scattering of the water molecule by a chlorine atom, which sheds light on the energy gap law in energy transfer in atom-triatom collisions.

Time-independent quantum theory on vibrational inelastic scattering between atoms and open-shell diatomic molecules: Applications to NO + Ar and NO + H scattering

A full-dimensional rigorous quantum mechanical treatment of non-reactive inelastic scattering of an open-shell diatom [e.g., NO(2Π)] with a structureless and spinless atom is presented within the time-independent close-coupling framework. The inclusion of the diatomic vibrational degree of freedom allows the investigation of transitions between different vibrational manifolds, in addition to those between different rotational, spin-orbit, and Λ-doublet states. This method is applied to the scattering of vibrationally excited NO(2Π) with Ar and H (with its spin ignored). The former has negligible vibrational inelasticity, thanks to the weak interaction between the two collisional partners. This conclusion justifies the commonly used two-dimensional approximation in treating NO scattering with rare gas atoms. The latter, on the other hand, is shown to undergo significant vibrational relaxation, even in the ultra-cold regime, owing to a chemically bonded (HNO) complex on the lowest-lying singlet potential energy surfaces.

Theoretical Investigation on Reaction Pathway, Biological Activity, Toxicity and NLO Properties of Diclofenac Drug and Its Ionic Carriers

The present study included the use of the approximate semi-experimental method, the time-independent density function theory (unrestricted), the time-dependent density function theory, and Hartree-Fock method to calculate the reaction pathway of the anti-inflammatory drug diclofenac with its common ionic carriers (sodium and potassium). The basis sets used were STO-3G, 3-21G, 6-31G, and 6-311G. The drug was studied with two new proposed carrier ions (lithium and calcium) which were compared with common carriers. The calculations included the optimized geometrical structure and some physical properties such as standard heat of formation, dipole moment, total energies, and analytical spectra of IR, UV-VIS and 1H NMR. The biological and toxicological activities and the nonlinear optical (NLO) properties were also studied theoretically for the drug and for its proposed and common carriers. All calculations were performed using Gaussian-09 program. The results of the proposed carriers were compared with the common carriers in terms of activation energies, transition states, and products. This study is considered as a step to develop diclofenac prodrugs and find new carriers for diclofenac. The proposed lithium showed a good result and a potential for use as a drug carrier. The results also showed the convergence of the values of the common carriers (Na, K) and those of the proposed carrier (Ca), with their preference over it.

Account of non-Condon effects in time-independent Raman wavefunction theory: Calculation of the S1← S0 vibronic absorption spectrum of formaldehyde.

The time-independent eigenstate-free Raman wavefunction approach for calculating anharmonic vibronic spectra has been extended for the calculation of Herzberg-Teller contributions on the basis of an n-mode expansion of the transition electric dipole moment surface. This allows for the efficient simulation of Franck-Condon dark vibronic spectra. In addition, vibrational angular momentum terms have been implemented into this formalism, as they are important for an accurate description of vibrational wavefunctions spanning double-well potentials. This approach has been used to compute the FC-forbidden vibronic spectrum of the n → π* transition of formaldehyde based on a potential energy and transition dipole moment surfaces obtained from explicitly correlated multi-reference configuration interaction calculations. An extensive analysis of the resulting vibronic structure is provided, which allows for a detailed assignment and interpretation of the experimental spectrum.

Study of delayed creep fracture initiation and propagation based on semi-analytical fractional model

• Corresponding J and C * integrals have the same structure in Laplace space . • A semi-analytical inverse Laplace transform method was developed for Heaviside function . • Delayed initiation and propagation is very important for improving the well production. • Incremental fracture length created the creep is highly controlled by the net pressure in the operation. Fractional calculus has been widely used in the study of constitutive equation of geomaterials. However, a simple and convenient method for dealing with the viscoelastic problem based on fractional model efficiently and accurately is still absent. In this paper, constant stress assumption was employed to obtain the corresponding J and C * integrals in Laplace space and a semi-analytical inverse Laplace transform method based on Stehfest inverse Laplace transform was developed to apply inverse Laplace transform to Heaviside function. In conjunction with the semi-analytical superposition model for calculating the stress and displacement fields around the crack (or fracture), a model for simulating the delayed initiation and propagation of creep fracture was developed. The simulation results show that: the incremental fracture length enlarged by the delayed initiation and propagation is very important for connecting the reservoir rocks and improving the well production. J integral of hydraulic fracture tip will decrease first and then become to be periodic in the later shut-in period; its amplitude and frequency are governed by time-independent elastic theory and time-dependent creep rate. The incremental creep fracture length is not determined by the fractional order. However, the propagation rate of the creep fracture will be highly risen by the increasing of the fractional order.

Variational estimates of the parameters of a thermal explosion of a stationary medium in an arbitrary domain

The variational formulation of the nonlinear problem of steady heat conduction in an arbitrary configuration domain is applied to investigate the conditions for the existence of a steady-state temperature distribution in a stationary medium with the intensity of volumetric energy release rising with increasing temperature. Based on the relations of the time-independent theory of thermal explosion, a variational model of this phenomenon is constructed, which makes it possible to obtain estimates of the critical values of the parameters that determine the temperature state of the medium preceding the thermal explosion. The examples of a comparative analysis of such values for a solid and a hollow cylinder of finite length are given.

Measuring order in disordered systems and disorder in ordered systems: Random matrix theory for isotropic and nematic liquid crystals and its perspective on pseudo-nematic domains.

Surprisingly long-ranged intermolecular correlations begin to appear in isotropic (orientationally disordered) phases of liquid crystal forming molecules when the temperature or density starts to close in on the boundary with the nematic (ordered) phase. Indeed, the presence of slowly relaxing, strongly orientationally correlated, sets of molecules under putatively disordered conditions ("pseudo-nematic domains") has been apparent for some time from light-scattering and optical-Kerr experiments. Still, a fully microscopic characterization of these domains has been lacking. We illustrate in this paper how pseudo-nematic domains can be studied in even relatively small computer simulations by looking for order-parameter tensor fluctuations much larger than one would expect from random matrix theory. To develop this idea, we show that random matrix theory offers an exact description of how the probability distribution for liquid-crystal order parameter tensors converges to its macroscopic-system limit. We then illustrate how domain properties can be inferred from finite-size-induced deviations from these random matrix predictions. A straightforward generalization of time-independent random matrix theory also allows us to prove that the analogous random matrix predictions for the time dependence of the order-parameter tensor are similarly exact in the macroscopic limit, and that relaxation behavior of the domains can be seen in the breakdown of the finite-size scaling required by that random-matrix theory.

Hamiltonian term for a uniform dc electric field under the adiabatic approximation

In this work, we show that the disorder-free Kubo formula for the non-equilibrium value of an observable due to a DC electric field, represented by $E_x\hat{x}$ in the Hamiltonian, can be interpreted as the standard time-independent theory response of the observable due to a time- and position-\textit{independent} perturbation $H_{MF}$. We derive the explicit expression for $H_{MF}$ and show that it originates from the adiabatic approximation to $\langle k|E_x\hat{x}$ in which transitions between the different eigenspinor states of a system are forbidden. The expression for $H_{MF}$ is generalized beyond the real spin degree of freedom to include other spin-like discrete degrees of freedom (e.g. valley and pseudospin). By direct comparison between Kubo formula and the time-independent perturbation theory, as well as the Sundaram-Niu wavepacket formalism, we show that $H_{MF}$ reproduces the effect of the E-field, i.e. $E_x\hat{x}$, up to the first order. This replacement suggests the emergence of a new spin current term that is not captured by the standard Kubo formula spin current calculation. We illustrate this via the exemplary spin current for the heavy hole spin 3/2 Luttinger system. Finally, we apply the formalism and derive an analogous $H_{MF}$ for the effects of a weakly position-dependent coupling to the spin-like internal degrees of freedom. This gives rise to an anomalous velocity as well as spin accumulation terms in spin$\otimes$pseudospin space in addition to those contained explicitly in the unperturbed Hamiltonian.

Simple Closed-Form Expression for Penning Reaction Rate Coefficients for Cold Molecular Collisions by Non-Hermitian Time-Independent Adiabatic Scattering Theory.

We present a simple expression and its derivation for reaction rate coefficients for cold anisotropic collision experiments based on adiabatic variational theory and time-independent non-Hermitian scattering theory. We demonstrate that only the eigenenergies of the resulting one-dimensional Schrödinger equation for different complex adiabats are required. The expression is applied to calculate the Penning ionization rate coefficients of an excited metastable helium atom with molecular hydrogen in an energy range spanning from hundreds of kelvins down to the millikelvin regime. Except for trivial quantities like the masses of the nuclei and the bond length of the diatomic molecule participating in the collision, one needs as input data only the complex potential energy surface (CPES). In calculations, we used recently obtained ab initio CPES by D. Bhattacharya et al. ( J. Chem. Theory Comput. 2017 , 13 , 1682 - 1690 ) without fitting parameters. The results show good accord with current measurements ( Nat. Phys. 2017 , 13 , 35 - 38 ).

Residual Stresses with Time-Independent Cyclic Plasticity in Finite Element Analysis of Welded Joints

Due to the intense concentration of heat in a reduced area when Gas Metal Arc Welding (GMAW) is used to join mechanical components, the regions near the weld bead are subjected to severe thermal cycles. Firstly, the region close to the weld bead that is heated tends to be in compression and, when it cools, tends to be in tension. According to Pilipenko, the material is exposed to elastic compression and, then, reaching the yield limit, undergoes plastic deformation with the appearance of residual stresses followed by elastic-plastic unloading. This could be considered as a strain-stress cycle. This paper applies plastic-strain-range memorization based on time-independent cyclic plasticity theory for butt joints with single V-groove Finite Element (FE) models that were manufactured by GMAW. The theory combines both the isotropic hardening and the nonlinear kinematic hardening rule (Chaboche model) to reproduce the behavior of cyclic plasticity and thus to obtain the residual stresses in welded joint FE models. As a practical example, the proposed theory is validated by three welded joint specimens that were manufactured with different input parameters of speed, voltage, and current. An agreement between the residual stresses obtained by the FE model proposed and those obtained experimentally by the hole drilling method at different depths demonstrates that the proposed theory could be valid for modelling the residual stresses in welded joints when cyclic plasticity is considered over the range of speed, voltage, and current studied.

Unconventional fermionic pairing states in a monochromatically tilted optical lattice

We study the one-dimensional attractive fermionic Hubbard model under the influence of periodic driving with the time-dependent density matrix renormalization group method. We show that the system can be driven into an unconventional pairing state characterized by a condensate made of Cooper pairs with a finite center-of-mass momentum similar to a Fulde-Ferrell state. We obtain results both in the laboratory and the rotating reference frames demonstrating that the momentum of the condensate can be finely tuned by changing the ratio between the amplitude and the frequency of the driving. In particular, by quenching this ratio to the value corresponding to suppression of the tunneling and the Coulomb interaction strength to zero, we are able to ``freeze'' the condensate. We finally study the effects of different initial conditions and compare our numerical results to those obtained from a time-independent Floquet theory in the large frequency regime. Our work offers the possibility of engineering and controlling unconventional pairing states in fermionic condensates.

Insights into the Photoelectron Spectroscopy of Chlorofluoroethenes Studied by Density-Functional and Coupled-Cluster Theories.

The first two ionic states of chlorofluoroethenes were studied by using both time-independent and time-dependent density-functional theories. We calculated the equilibrium geometries and harmonic vibrational frequencies of 1,1-, cis-, and trans-C2H2FCl and their cations by using the B3LYP and B3PW91 functionals together with the cc-pVTZ and aug-cc-pVTZ basis sets. Franck-Condon factors were computed by the method developed in our group, in which the Duschinsky effect was treated explicitly. A new technique, named alignment transformation, followed by Euler transformations was developed to achieve the Eckart conditions. The adiabatic ionization energies were calculated by the CCSD(T) method extrapolated to the complete basis set limit. Insights into the simulated photoelectron spectra of C2H2FCl indicate that the resolutions of recent threshold photoelectron experiments are not high enough to detect individual transitions. The high-resolution photoelectron spectra of C2H2FCl are predicted for future reference. The computed adiabatic ionization energies of the three isomers of C2H2FCl are in accord with the experiments with the absolute deviations ranging from 0.004 to 0.021 eV. We suggest that the agreement between experimental and theoretical spectra should be a key criterion to judge whether a spectral assignment is reasonable.

Scattering theory, multiparticle detection, and time

We consider the theory of multiple-particle fragmentation processes in the light of modern multihit position-sensitive detection. First, we give a formulation of time-independent many-body scattering theory as a direct generalization of standard textbook two-body potential scattering but in such a way as to emphasize position rather than momentum detection. Noteworthy is that classical asymptotic motion of fragments is shown to emerge from this quantum-mechanical time-independent theory and enables the definition of a classical time parameter. This in turn allows a transition to be made to a time-dependent scattering theory, even in the case where all Hamiltonians are time independent. Such a time-dependent description is the basis of the imaging theorem, which connects position detection to momentum detection.

Classical-trajectory Monte Carlo model calculations for the antiproton-induced ionization of atomic hydrogen at low impact energy

The three-body dynamics of the ionization of the atomic hydrogen by 30 keV antiproton impact has been investigated by calculation of fully differential cross sections (FDCS) using the classical trajectory Monte Carlo (CTMC) method. The results of the calculations are compared with the predictions of quantum mechanical descriptions: The semi-classical time-dependent close-coupling theory, the fully quantal, time-independent close-coupling theory, and the continuum-distorted-wave-eikonal-initial-state model. In the analysis particular emphasis was put on the role of the nucleus-nucleus (NN) interaction played in the ionization process. For low-energy electron ejection CTMC predicts a large NN interaction effect on FDCS, in agreement with the quantum mechanical descriptions. By examining individual particle trajectories it was found that the relative motion between the electron and the nuclei is coupled very weakly with that between the nuclei, consequently the two motions can be treated independently. A simple procedure is presented by which the NN interaction effect can be included into the calculations carried out without it.

On the possible control of tunneling in symmetric triple-well systems: the role of symmetry and driving

The possibility of controlling tunneling of a quantum particle localized in the central well (M) of a symmetric triple well potential by carefully choosing the driving parameters of a mono or bichromatic field is considered through careful numerical experiments. Conditions for enhancement and suppression of tunneling rate from the middle (M) to the side wells (L, R) are established. The observed behaviour is sought to be analysed by invoking the time-independent Floquet theory to follow the movement of quasi-energy levels as the driving parameters are changed. We demonstrate how the particle can be tunnel-transported from the central to one particular side well by using a mixed symmetry bichromatic field with delay between the applications of the two frequencies.