Soft-core Coulomb Research Articles

Grid-Based Ehrenfest Model To Study Electron-Nuclear Processes.

The two-dimensional electron-nuclear Schrödinger equation using soft-core Coulomb potentials has been a cornerstone for modeling and predicting the behavior of one-active-electron diatomic molecules, particularly for processes where both bound and continuum states are important. The model, however, is computationally expensive to extend to more electron or nuclear coordinates. Here we propose use of the Ehrenfest approach to treat the nuclear motion, while the electronic motion is still solved by quantum propagation on a grid. In this work, we present results for a one-dimensional treatment of H2+, where the quantum and semiclassical dynamics can be directly compared, showing remarkably good agreement for a variety of situations. The advantage of the Ehrenfest approach is that it can be easily extended to treat as many nuclear degrees of freedom as needed.

Effects of a soft-core coulomb potential on the dynamics of a hydrogen atom near a metal surface

The goal of this paper is to investigate the effects that the replacement of the Coulomb potential by a soft-core Coulomb potential produces in the classical dynamics of a perturbed Rydberg hydrogen atom. As example, we consider a Rydberg hydrogen atom near a metal surface subjected to a constant electric field in the electron-extraction regime. Thence, the dynamics of the real perturbed Coulomb system, studied by applying the Levi–Civita regularization, is compared with that of the softened one. The results of this study show that the global behavior of the system is significantly altered when the original Coulomb potential part is replaced by a soft-core potential.

Spectra generated by a confined softcore Coulomb potential

Analytic and approximate solutions for the energy eigenvalues generated by a confined softcore Coulomb potentials of the form a/(r + β) in d > 1 dimensions are constructed. The confinement is effected by linear and harmonic-oscillator potential terms, and also through “hard confinement” by means of an impenetrable spherical box. A byproduct of this work is the construction of polynomial solutions for a number of linear differential equations with polynomial coefficients, along with the necessary and sufficient conditions for the existence of such solutions. Very accurate approximate solutions for the general problem with arbitrary potential parameters are found by use of the asymptotic iteration method.

Laser-assisted ultrafast photoassociation in HeH2+

In this work we report control on the photoassociation of a proton and a Helium cation, or an alpha particle and an Hydrogen atom, to form HeH2+ assisted by strong femtosecond laser pulses. The results follow from the numerical solution of the time-dependent Schrödinger equation using soft-core Coulomb potentials in a (1+1)D model (one dimension for the electronic motion plus one dimension for the nuclear motion) and for a nuclear Hamiltonian describing the dynamics in four electronic states. We study the dependence of the photoassociation yield on the initial nuclear wave function for a hot collision (impact kinetic energy of the order of a eV) and on the laser parameters. We predict high sensitivity on the synchronization of the laser with the collision time and Rabi oscillations for the yield as a function of the pulse amplitude and duration, while the ionization and inelastic scattering can be minimized with pulses of 50fs or shorter.

Effect of Ionization on the Wavelength Scaling of High Harmonic Generation Efficiency

The physical origin of the recently reported wavelength scaling law of high harmonics generation (HHG) yield [Tate et al., Phys. Rev. Lett. 98, 013901 (2007); Shiner et al., Phys. Rev. Lett. 103, 073902 (2009)] is not fully understood. We re-investigate theoretically the wavelength (from 800 nm to 2000 nm) scaling of HHG yield in atoms with different soft-core Coulomb potentials. It is found that the power x of the wavelength scaling λ^x surprisingly oscillates in the region -3.63 ∼ -6.32. A similar oscillation is found in the wavelength scaling of ionization ratio upon the ionization potential, indicating that the wavelength scaling of HHG yield is closely relevant to the ionization. The origin of this oscillation in ionization is also identified.

A classical model of H in an intense laser field

Soft-core Coulomb potentials for the pairwise interactions among electrons and protons are introduced so that , H2, and H can take stable geometrical configurations in classical mechanics, and responses of to an ultrashort intense laser pulse are investigated by integrating the classical equations of motion. The Monte Carlo simulation shows that the single and double ionization as well as the dissociation proceed, indicating that classical mechanics can give insight into dynamics of small molecular systems interacting with an intense laser field.

An Ansatz Solution of Dirac Equation under Scalar and Vector Soft-Core Coulomb and Coulomb Tensor Interactions

We study Dirac equation for scalar and vector soft-core Coulomb potentials and a Tensor Coulomb term in spin and pseudospin symmetries via the quasi-analytical ansatz technique.

Quasi-exactly solvable relativistic soft-core Coulomb models

By considering a unified treatment, we present quasi exact polynomial solutions to both the Klein–Gordon and Dirac equations with the family of soft-core Coulomb potentials Vq(r)=−Z/(rq+βq)1/q, Z>0, β>0, q≥1. We consider cases q=1 and q=2 and show that both cases are reducible to the same basic ordinary differential equation. A systematic and closed form solution to the basic equation is obtained using the Bethe ansatz method. For each case, the expressions for the energies and the allowed parameters are obtained analytically and the wavefunctions are derived in terms of the roots of a set of Bethe ansatz equations.

Quantum phase-space analysis of electronic rescattering dynamics in intense few-cycle laser fields

We present a quantum phase-space analysis of the ionization and rescattering dynamics of a one-electron atom in an intense few-cycle laser field. Snapshots of the Wigner function W(x, p) will be analysed for both soft-core Coulomb and short-ranged potentials tuned to yield identical ground state energies and similar energies for the first lowest electronic states. The influence of the long-range Coulomb potential on the rescattering wavepacket in the continuum can be disentangled from that of the induced bound-state polarization. Short- and long-range atomic potentials entail differences in the rescattering dynamics, manifesting themselves in high-order harmonic generation spectra.

Escape behavior of a quantum particle in a loop coupled to a lead

We consider a one-dimensional loop of circumference $L$ crossed by a constant magnetic flux $\ensuremath{\Phi}$ and connected to an infinite lead with coupling parameter $\ensuremath{\varepsilon}$. Assuming that the initial state ${\ensuremath{\psi}}_{0}$ of the particle is confined inside the loop and evolves freely, we analyze the time evolution of the nonescape probability $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)$, which is the probability that the particle will still be inside the loop at some later time $t$. In appropriate units, we found that $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)={P}_{\ensuremath{\infty}}({\ensuremath{\psi}}_{0},\ensuremath{\Phi})+{\ensuremath{\sum}}_{k=1}^{\ensuremath{\infty}}{C}_{k}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})/{t}^{k}$. The constant ${P}_{\ensuremath{\infty}}({\ensuremath{\psi}}_{0},\ensuremath{\Phi})$ is independent of $L$ and $\ensuremath{\varepsilon}$, and vanishes if ${\ensuremath{\psi}}_{0}$ has no bound state components or if $|\mathrm{cos}(\ensuremath{\Phi})|\ensuremath{\ne}1$. The coefficients ${C}_{1}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ and ${C}_{3}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ depend on the initial state ${\ensuremath{\psi}}_{0}$ of the particle, but only the momentum $k=\ensuremath{\Phi}/L$ is involved. There are initial states ${\ensuremath{\psi}}_{0}$ for which $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)\ensuremath{\sim}{C}_{\ensuremath{\delta}}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})/{t}^{\ensuremath{\delta}}$, as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, where $\ensuremath{\delta}=1$ if $\mathrm{cos}(\ensuremath{\Phi})=1$ and $\ensuremath{\delta}=3$ if $\mathrm{cos}(\ensuremath{\Phi})\ensuremath{\ne}1$. Thus, by submitting the loop to an external magnetic flux, one may induce a radical change in the asymptotic decay rate of $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)$. Interestingly, if $\mathrm{cos}(\ensuremath{\Phi})=1$, then ${C}_{1}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ decreases with $\ensuremath{\varepsilon}$ (i.e., the particle escapes faster in the long run) while in the case $\mathrm{cos}(\ensuremath{\Phi})\ensuremath{\ne}1$, the coefficient ${C}_{3}({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon})$ increases with $\ensuremath{\varepsilon}$ (i.e., the particle escapes slower in the long run). Assuming the particle to be initially in a bound state of the loop with $\ensuremath{\Phi}=0$, we compute explicit relations and present some numerical results showing a global picture in time of $P({\ensuremath{\psi}}_{0},L,\ensuremath{\Phi},\ensuremath{\varepsilon},t)$. Finally, by using the pseudospectral method, we consider the interacting case with soft-core Coulomb potentials.

A water-swap reaction coordinate for the calculation of absolute protein–ligand binding free energies

The accurate prediction of absolute protein-ligand binding free energies is one of the grand challenge problems of computational science. Binding free energy measures the strength of binding between a ligand and a protein, and an algorithm that would allow its accurate prediction would be a powerful tool for rational drug design. Here we present the development of a new method that allows for the absolute binding free energy of a protein-ligand complex to be calculated from first principles, using a single simulation. Our method involves the use of a novel reaction coordinate that swaps a ligand bound to a protein with an equivalent volume of bulk water. This water-swap reaction coordinate is built using an identity constraint, which identifies a cluster of water molecules from bulk water that occupies the same volume as the ligand in the protein active site. A dual topology algorithm is then used to swap the ligand from the active site with the identified water cluster from bulk water. The free energy is then calculated using replica exchange thermodynamic integration. This returns the free energy change of simultaneously transferring the ligand to bulk water, as an equivalent volume of bulk water is transferred back to the protein active site. This, directly, is the absolute binding free energy. It should be noted that while this reaction coordinate models the binding process directly, an accurate force field and sufficient sampling are still required to allow for the binding free energy to be predicted correctly. In this paper we present the details and development of this method, and demonstrate how the potential of mean force along the water-swap coordinate can be improved by calibrating the soft-core Coulomb and Lennard-Jones parameters used for the dual topology calculation. The optimal parameters were applied to calculations of protein-ligand binding free energies of a neuraminidase inhibitor (oseltamivir), with these results compared to experiment. These results demonstrate that the water-swap coordinate provides a viable and potentially powerful new route for the prediction of protein-ligand binding free energies.

Soft-core Coulomb potentials and Heun’s differential equation

Schrödinger’s equation with the attractive potential V(r)=−Z/(rq+βq)1/q, Z>0, β>0, q≥1, is shown, for general values of the parameters Z and β, to be reducible to the confluent Heun equation in the case q=1 and to the generalized Heun equation in the case q=2. In a formulation with correct asymptotics, the eigenstates are specified a priori up to an unknown factor. In certain special cases, this factor becomes a polynomial. The asymptotic iteration method is used either to find the polynomial factor and the associated eigenvalue explicitly, or to construct accurate approximations for them. Detailed solutions for both cases are provided.

Energies and wave functions for a soft-core Coulomb potential

For the family of model soft-core Coulomb potentials represented by $V(r)=\ensuremath{-}[Z/{({r}^{q}+{\ensuremath{\beta}}^{q})}^{1/q}]$, with the parameters $Z>0,\text{ }\ensuremath{\beta}>0,\text{ }q\ensuremath{\ge}1$, it is shown analytically that the potentials and eigenvalues, ${E}_{\ensuremath{\nu}\ensuremath{\ell}}$, are monotonic in each parameter. The potential envelope method is applied to obtain approximate analytic estimates in terms of the known exact spectra for pure power potentials. For the case $q=1$, the asymptotic iteration method is used to find exact analytic results for the eigenvalues ${E}_{\ensuremath{\nu}\ensuremath{\ell}}$ and corresponding wave functions, expressed in terms of $Z$ and $\ensuremath{\beta}$. A proof is presented establishing the general concavity of the scaled electron density near the nucleus resulting from the truncated potentials for all $q$. Based on an analysis of extensive numerical calculations, it is conjectured that the crossing between the pair of states $[(\ensuremath{\nu},\ensuremath{\ell}),({\ensuremath{\nu}}^{\ensuremath{'}},{\ensuremath{\ell}}^{\ensuremath{'}})]$ is given by the conditions ${\ensuremath{\nu}}^{\ensuremath{'}}\ensuremath{\ge}(\ensuremath{\nu}+1)$ and ${\ensuremath{\ell}}^{\ensuremath{'}}\ensuremath{\ge}(\ensuremath{\ell}+3)$. The significance of these results for the interaction of intense laser field with an atom is pointed out. Differences in the observed level-crossing effects between the soft-core potentials and the hydrogen atom confined inside an impenetrable sphere are discussed.