Applications of the Kinetic Energy Partition Method to Model Atoms

We generalize the recently developed kinetic energy partition (KEP) method for systems with two competing interactions to solve the quantum eigenvalue problems of a particle interacting with a number of harmonic-oscillator potentials. First, the original formulation of the KEP method is extended to the case with many interaction potentials. Second, we apply the method to the two and three harmonic-oscillators problems, respectively. Finally, the general N harmonic-oscillators system is considered and solved. We show that the KEP method can yield the exact solutions very accurately and efficiently.

Based on the idea of adiabatic symmetry, we present a novel basis set expansion method—the kinetic energy partition (KEP) method—for solving quantum eigenvalue problems. Broken symmetry is responsible for quantum entanglement in many-body systems via parametric non-adiabatic corrections. Starting from simple one-particle-in-one-dimension problems, we gradually increase the complexity in the number of particles and the interaction patterns. Our goal in the mini-review is to advocate for the utility of the KEP method in front-line research, in particular for research beginners in quantum many-body problems.

Solving many-body Schrödinger equations with kinetic energy partition method

We have used the recently developed kinetic energy partition (KEP) method to solve the quantum eigenvalue problems for helium-like atoms and obtain precise ground state energies and wave-functions. The key to treating properly the electron-electron (repulsive) Coulomb potential energies for the KEP method to be applied is to introduce a "negative mass" term into the partitioned kinetic energy. A Hartree-like product wave-function from the subsystem wave-functions is used to form the initial trial function, and the variational search for the optimized adiabatic parameters leads to a precise ground state energy. This new approach sheds new light on the all-important problem of solving many-electron Schrödinger equations and hopefully opens a new way to predictive quantum chemistry. The results presented here give very promising evidence that an effective one-electron model can be used to represent a many-electron system, in the spirit of density functional theory.

We present a new basis set expansion method for quantum dynamics systems with two competing modes where the interaction potentials are equally dominant. The new idea introduced here is a kinetic energy partition scheme instead of the usual division of the potential energy. The partition results in two kinetic energy terms with their effective masses. By distributing each partial kinetic energy to the respective potential, the full Hamiltonian can be expressed as the sum of the two competing modes. The solution procedure is illustrated by using a system consisting of a particle under the action of two harmonic potentials with different equilibrium distances and force constants. Next we apply this method to obtain the potential energy curves for the prototype hydrogen molecule ion. This new expansion converges very fast to the exact solutions for both eigenvalues and eigenfunctions.

Solving challenging problems in quantum chemistry is one of the most promising applications of quantum computers. Within the quantum algorithms proposed for problems in excited-state quantum chemistry, subspace-based quantum algorithms, including quantum subspace expansion (QSE), quantum equation of motion (qEOM), and quantum self-consistent equation-of-motion (q-sc-EOM), are promising for pre-fault-tolerant quantum devices. The working equation of QSE and qEOM requires solving a generalized eigenvalue equation with associated matrix elements measured on a quantum computer. Our careful analytical and numerical analysis of the standard and generalized eigenvalue problems, especially in the context of excited-state methods, shows that the errors in eigenvalues magnify drastically with an increase in the condition number of the overlap matrix when a generalized eigenvalue equation is solved in the presence of statistical sampling errors. This makes methods such as QSE unstable for errors that are unavoidable when using quantum computers. Further, at very high condition numbers of the overlap matrix, the QSE's working equation could not be solved without any additional steps in the presence of sampling errors, as it becomes ill-conditioned. It was possible to use the thresholding technique in this case to solve the equation, but the solutions achieved had missing excited states, which may be a problem for future chemical studies. We also show that excited-state methods that have an eigenvalue equation as the working equation, such as q-sc-EOM, do not have the problems associated with the condition number and could be generally more stable to errors and therefore more suitable candidates for excited-state quantum chemistry calculations using quantum computers.

We show that the recently proposed kinetic energy partition method for the approximate solution of the Schrodinger equation is a particular case of the well known Rayleigh–Ritz variational method with an extremely restricted implementation. In addition to it, the quantum-mechanical example chosen by the authors to illustrate their approach is unsuitable for the proposed application.

Comment on: “Solving many-body Schrödinger equations with kinetic energy partition method”, Ann. Phys. 388 (2018) 54–68 by Y-H. Chen and S.D. Chao

The kinetic energy partition method applied to a confined quantum harmonic oscillator in a one-dimensional box

The veracity of a low temperature plasma model is limited by the accuracy of the electron transport coefficient, which can be solved by simulating the electron transport process. When simulating the transport properties of electrons, there are a variety of approaches to dealing with the scattering of electrons and energy partition between the primary-electrons and secondary-electrons after electron-neutral particles’ collision. In this paper used is a model based on the Monte Carlo collision method to investigate the influence of scattering method and energy partition method on the electron transport coefficient. The electron energy distribution function, electron mean energy, flux mobility and diffusion coefficients, as well as the Townsend ionization coefficients are calculated in the hydrogen atom gas under a reduced electric field from 10 to 1000 Td. The calculation results show that the influence of the isotropic scattering assumption on the electron transport coefficients increases with reduced electric field increasing. However, even under a relatively low reduced electric field (10 Td), the calculated mean energy, flux mobility, and flux diffusion coefficient of electrons under the assumption of anisotropic scattering are 39.68%, 17.38% and 119.18% higher than those under the assumption of the isotropic scattering. The different energy partition methods have a significant influence on the electron transport coefficient under a medium-to-high reduced electric field (> 200 Td). Under a high electric field, the mean energy, flux mobility and flux diffusion coefficient calculated by the equal-partition method (the primary and secondary electrons equally share the available energy) are all less than the values from the zero-partition method (the energy of secondary-electrons is assigned to zero). While the change of Townsend ionization coefficient with reduced electric fields shows a different trend. The electron transport coefficient obtained by the Opal method lies between the values from the equal-partition method and the zero-partition method. In addition, considering the anisotropic scattering, the influence of energy partition method on the transport coefficient is higher than that under the assumption of isotropic scattering. This study shows the necessity of considering the anisotropic electron scattering for calculating the electron transport coefficient, and special attention should be paid to the choice of energy partition method under a high reduced electric field.

We experimentally and numerically investigate the early-time hydrodynamic response of tin microdroplets driven by a ns-laser-induced plasma. Experimentally, we use stroboscopic microscopy to record the laser-induced dynamics of liquid tin droplets and determine the propulsion speed $(U)$ and initial radial expansion rate $({\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}}_{0})$. The ratio of these two quantities is a key parameter to be optimized for applications in nanolithography, where laser-impacted tin droplets serve as targets for generating extreme ultraviolet light. We explore a large parameter space to investigate the influence of the tin droplet diameter, laser beam diameter, and laser energy on the ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}}_{0}/U$ ratio. We find good agreement when comparing the experimentally obtained $U$ and ${\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{R}}_{0}$ values to those obtained by detailed radiation-hydrodynamic simulations using RALEF-2D. From the validated simulations, we extract the spatial distribution of the plasma-driven pressure impulse at the droplet-plasma interface to quantify its influence on the partitioning of kinetic energy channeled into propulsion or deformation. Our findings demonstrate that the width of the pressure impulse is the sole pertinent parameter for extracting the kinetic energy partitioning, which ultimately determines the late-time target morphology. We find good agreement between our full radiation-hydrodynamic modeling and a generalized analytical fluid-dynamics model [H. Gelderblom et al., J. Fluid Mech. 794, 676 (2016)]. These findings can be used to optimize the kinetic energy partition and tailor the features of tin targets for nanolithography.

Summary form only given. In the process of plasma ablation from wires in a wire array, consisting of deuterium and palladium, plasma in the precursor consists of heavy and light ions. In this multi-component current carrying plasma global MHD instabilities can develop. Dependence of the growth rates of such instabilities from the density ratio of heavy and light ions as well as kinetic energy partition between the two plasma components is discussed. We also present results of 3D hybrid simulations of sausage and kink instabilities in the multi-component plasma of the precursor.

A phase-space approach is proposed for molecular dynamics simulations, which serve as a bridge between detailed descriptions of microscopic world and macroscopic properties of matter. The introduction-aside from the angular momentum of spatial rotations-of other "hyperangular" momenta (the overall grand angular momentum of a cluster of particles and those describing the deformation and rearrangement modes) permits one to analyze different degrees of freedom and to extract, from simulation data, a kinetic energy partition in terms of phase-space invariants. Model calculations illustrate how these provide specific signatures of critical behavior, such as energy thresholds for openings of chaotic pathways in small clusters and for phase transitions in nanoaggregates.

The energies of atomic and molecular systems with a screened Coulomb interaction are expressed in terms of properties of the system with a conventional Coulomb interaction to the third order in the screening constant. The energy levels of an H atom, an H− ion, and nonadiabatic systems, namely, the positronium ion e+e−e− and the positronium molecule e+e+e−e−, are calculated for screened Coulomb interactions of charged particles. It is demonstrated that, as the screening constant of the Coulomb interaction increases, the stability of the hydrogen atom, the H− ion, and the positronium molecule e+e+e−e− decreases, whereas the stability of the positronium ion e+e−e− increases after a slight decrease.

Solutions of the multiconfiguration time-dependent Hartree–Fock equations with Coulomb interactions