Kinetic Energy Operator Research Articles

One- and two-dimensional solitons in spin–orbit-coupled Bose–Einstein condensates with fractional kinetic energy

We address effects of spin–orbit coupling (SOC), phenomenologically added to a two-component Bose–Einstein condensate composed of particles moving by Lévy flights, in one- and two-dimensional (1D) and (2D) settings. The corresponding system of coupled Gross–Pitaevskii equations includes fractional kinetic-energy operators, characterized by the Lévy index, α < 2 (the normal kinetic energy corresponds to α = 2). The SOC terms, with strength λ, produce strong effects in the 2D case: they create families of stable solitons of the semi-vortex and mixed-mode types in the interval of 1 < α < 2, where the supercritical collapse does not admit the existence of stable solitons in the absence of the SOC. At λ → 0, amplitudes of these solitons vanish ∼λ 1/(α−1).

Quasienergy operators and generalized squeezed states for systems of trapped ions

Collective many-body dynamics for time-dependent quantum Hamiltonians is investigated for a dynamical system that exhibits multiple degrees of freedom, in this case a combined (Paul and Penning) trap. Quantum stability is characterized by a discrete quasienergy spectrum, while the quasienergy states are symplectic coherent states. We introduce the generators of the Lie algebra of the symplectic group SL(2,R), which we use to build the coherent states (CS) associated to the system under investigation. The trapped ion is treated as a harmonic oscillator (HO) to which we associate the quantum Hamilton function. We obtain the kinetic and potential energy operators as functions of the Lie algebra generators and supply the expressions for the classical coordinate, momentum, kinetic and potential energy, along with the total energy. Moreover, we also infer the dispersions for the coordinate and momentum, together with the asymmetry and the flatness parameter for the distribution. The system interaction with laser radiation is also examined for a system of identical two-level atoms. The Hamilton function for the Dicke model is derived. The optical system is modelled as a HO (trapped ion) that undergoes interaction with an external laser field and we use it to engineer a squeezed state of the electromagnetic (EM) field. We consider coherent and squeezed states associated to both ion dynamics and to the EM field. The approach used enables one to build CS in a compact and smart manner by use of the group theory.

Constructing Hermitian Hamiltonians for spin zero neutral and charged particles on a curved surface: physical approach

The Hamiltonians for a spin zero neutral particle and a charged particle in an electromagnetic field pinned to an arbitrary 2D surface embedded in 3D space are constructed. The new approach we follow starts with an expression for the 3D momentum operators whose components along the surface and the normal to the surface are separately Hermitian. The normal part of the kinetic energy operator is a Hermitian operator in this case. When this operator is dropped and the thickness of the layer is set to zero, one automatically gets the Hermitian surface Hamiltonian that contains the geometric potential term as expected. We show that Hermitian surface and normal momenta emerge automatically once one symmetrizes the usual normal and surface momentum operators; the geometrical potential originates from this symmetrization, too. Comparing our approach with the standard thin-layer quantization (TLQ), we show that Hermicity is also at the heart of TLQ.

Reduced-dimensional vibrational models of the water dimer.

A model based on the finite-basis representation of a vibrational Hamiltonian expressed in internal coordinates is developed. The model relies on a many-mode, low-order expansion of both the kinetic energy operator and the potential energy surface (PES). Polyad truncations and energy ceilings are used to control the size of the vibrational basis to facilitate accurate computations of the OH stretch and HOH bend intramolecular transitions of the water dimer (H2 16O)2. Advantages and potential pitfalls of the applied approximations are highlighted. The importance of choices related to the treatment of the kinetic energy operator in reduced-dimensional calculations and the accuracy of different water dimer PESs are discussed. A range of different reduced-dimensional computations are performed to investigate the wavenumber shifts in the intramolecular transitions caused by the coupling between the intra- and intermolecular modes. With the use of symmetry, full 12-dimensional vibrational energy levels of the water dimer are calculated, predicting accurately the experimentally observed intramolecular fundamentals. It is found that one can also predict accurate intramolecular transition wavenumbers for the water dimer by combining a set of computationally inexpensive reduced-dimensional calculations, thereby guiding future effective-Hamiltonian treatments.

Quantum dynamics with curvilinear coordinates: models and kinetic energy operator.

In order to simplify the numerical solution of the time-dependent or time-independent Schrödinger equations associated with atomic and molecular motions, the use of well-adapted coordinates is essential. Usually, this set of curvilinear coordinates leads to a Hamiltonian operator that is as separable as possible. Although their corresponding kinetic energy operator (KEO) expressions can be derived analytically for small systems or special kinds of coordinates, a numerical and exact approach allows one to compute them in terms of sophisticated curvilinear coordinates. Furthermore, the numerical approach enables one to easily define reduced-dimensionality or constrained models. We present here a recent implementation of this numerical approach that allows nested coordinate transformations, therefore leading to great flexibility in the definition of the curvilinear coordinates. Furthermore, this implementation has no limitations in terms of numbers of atoms or coordinate transformations. The quantum dynamics of the cis-trans photoisomerization of part of the retinal chromophore illustrates the construction of the coordinates and KEO part of a three-dimensional model. This article is part of the theme issue 'Chemistry without the Born-Oppenheimer approximation'.

Chemistry without the Born-Oppenheimer approximation.

Chemistry without the Born-Oppenheimer approximation.

Computational simulation and nonlinear vibration motions of isolated waves localized in small part of space

Nonlinear Schrödinger equations are examined to see if any new solution structures can be found for three different kinds of nonlinear equations. The generalized Korteweg–de Vries (gKdV) equation, the (2 + 1)–Ablowitz–Kaup–Newwell–Segur (AKNS) equation, and the Maccari system are all investigated. These models commonly characterize kinetic energy operators, which are affected by mass location and time dependence and the numerous representations of kinetic energy operators. Soliton wave solutions for the three new models are created by using the analytical Khater II approach. Figures depicting the numerical simulations of the resulting solutions may be seen all over the place. Mathematica 12 software is used to verify the approaches’ accuracy and re-incorporate all new results into the original models.

Generalized semiconfined harmonic oscillator model with a position-dependent effective mass

By using a point canonical transformation starting from the constant-mass Schr\"odinger equation for the isotonic potential, it is shown that a semiconfined harmonic oscillator model with a position-dependent mass in the BenDaniel-Duke setting and the same spectrum as the standard harmonic oscillator can be easily constructed and extended to a semiconfined shifted harmonic oscillator, which could result from the presence of a uniform gravitational field. A further generalization is proposed by considering a $m$-dependent position-dependent mass for $0<m<2$ and deriving the associated semiconfined potential. This results in a family of position-dependent mass and potential pairs, to which the original pair belongs as it corresponds to $m=1$. Finally, the potential that would result from a general von Roos kinetic energy operator is presented and the examples of the Zhu-Kroemer and Mustafa-Mazharimousavi settings are briefly discussed.

Kinetic Energy Operator of Linear Symmetric Molecules of the A2B2 Type in Polyspherical Orthogonal Coordinates

Kinetic Energy Operator of Linear Symmetric Molecules of the A2B2 Type in Polyspherical Orthogonal Coordinates

On the position-dependent mass Schrödinger equation for Mie-type potentials

The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.

Mapping Quantum Chemical Dynamics Problems to Spin-Lattice Simulators.

The accurate computational determination of chemical, materials, biological, and atmospheric properties has a critical impact on a wide range of health and environmental problems, but is deeply limited by the computational scaling of quantum mechanical methods. The complexity of quantum chemical studies arises from the steep algebraic scaling of electron correlation methods and the exponential scaling in studying nuclear dynamics and molecular flexibility. To date, efforts to apply quantum hardware to such quantum chemistry problems have focused primarily on electron correlation. Here, we provide a framework that allows for the solution of quantum chemical nuclear dynamics by mapping these to quantum spin-lattice simulators. Using the example case of a short-strong hydrogen-bonded system, we construct the Hamiltonian for the nuclear degrees of freedom on a single Born-Oppenheimer surface and show how it can be transformed to a generalized Ising model Hamiltonian. We then demonstrate a method to determine the local fields and spin-spin couplings needed to identically match the molecular and spin-lattice Hamiltonians. We describe a protocol to determine the on-site and intersite coupling parameters of this Ising Hamiltonian from the Born-Oppenheimer potential and nuclear kinetic energy operator. Our approach represents a paradigm shift in the methods used to study quantum nuclear dynamics, opening the possibility to solve both electronic structure and nuclear dynamics problems using quantum computing systems.

On the inclusion of cusp effects in expectation values with explicitly correlated Gaussians

AbstractThis paper elaborates the integral transformation technique and uses it for the case of the non‐relativistic kinetic and Coulomb potential energy operators, as well as for the relativistic mass‐velocity and Darwin terms. The techniques are tested for the ground electronic state of the helium atom and perturbative relativistic energies are reported for the ground electronic state of the molecular ion near its equilibrium structure.

Quantum Dissipative Dynamics (QDD): A real-time real-space approach to far-off-equilibrium dynamics in finite electron systems

In this paper, we present “QDD” (Quantum Dissipative Dynamics), a code package for simulating the dynamics of electrons and ions in finite electron systems (atoms, molecules, clusters) under the influence of external electromagnetic fields. Electron emission is properly accounted for. The novel feature of the present code is that it also covers the description of dissipative dynamics induced by dynamical correlations generated by electron-electron collisions.The paper reviews the underlying theoretical as well as numerical methods and demonstrates the code's capabilities on a selection of typical examples. Program summaryProgram Title: QDDCPC Library link to program files:https://doi.org/10.17632/jpwzm9knmb.1Licensing provisions: GPLv3Programming language: Fortran 90 (with Fortran 2008 standard)Supplementary material: User Manual, input and output filesNature of problem: The QDD code serves to simulate the dynamics of finite electron systems (atoms, molecules, clusters) excited by strong electromagnetic fields as delivered by lasers or highly charged ions. Basis of the description is Time-Dependent Density Functional Theory (TDDFT) at the level of the Time-Dependent Local-Density Approximation (TDLDA) augmented by an approximate Self-Interaction Correction (SIC), the latter being crucial for proper description of electron emission. The novel feature of the present code is that it allows one to track the dissipative dynamics induced by dynamical correlations from the earliest times of excitation on. This is done here at a fully quantum mechanical level within the Relaxation-Time Approximation (RTA). Electron dynamics is also coupled to ionic motion treated by classical molecular dynamics.Solution method: The numerical representation uses a 3D coordinate-space grid for electronic wave functions and fields. The kinetic energy operator is evaluated in momentum space connected by a Fast Fourier Transform (FFT). Standard schemes for electronic ground state, ionic ground state, and propagation of TDLDA in real time as well as ionic dynamics are used. Electron emission is enabled by absorbing boundary conditions using a mask function near the boundaries. The time evolution of dissipation (by RTA) is evaluated in a large space of occupied and unoccupied single-electron states.Additional comments including restrictions and unusual features: Only the actual computing hardware (RAM, number of nodes on board) limits the system size which can be treated. The code package allows sequential and parallel (OpenMP) computation. The simulation can be continued from a previously saved dynamical configuration.

Encapsulation of a Water Molecule inside C60 Fullerene: The Impact of Confinement on Quantum Features.

We introduce an efficient quantum fully coupled computational scheme within the multiconfiguration time-dependent Hartree (MCTDH) approach to handle the otherwise extremely costly computations of translational–rotational–vibrational states and energies of light-molecule endofullenes. Quantum calculations on energy levels are reported for a water molecule inside C60 fullerene by means of such a systematic approach that includes all nine degrees of freedom of H2O@C60 and does not consider restrictions above them. The potential energy operator is represented as a sum of natural potentials employing the n-mode expansion, along with the exact kinetic energy operator, by introducing a set of Radau internal coordinates for the H2O molecule. On the basis of the present rigorous computations, various aspects of the quantized intermolecular dynamics upon confinement of H2O@C60 are discussed, such as the rotational energy level splitting and the significant frequency shifts of the encapsulated water molecule vibrations. The impact of water encapsulation on quantum features is explored, and insights into the nature of the underlying forces are provided, highlighting the importance of a reliable first-principles description of the guest–host interactions.

Mass gap in strongly coupled infinite derivative non-local Higgs: Dyson–Schwinger approach

We investigate the non-perturbative degrees of freedom (d.o.f.) in the class of non-local Higgs theories that have been proposed as an ultraviolet completion 4D quantum field theory generalizing the kinetic energy operators to an infinite series of higher derivatives inspired by string field theory and ghost-free non-local approaches to quantum gravity. At the perturbative level, the d.o.f. of non-local Higgs are the same of the local theory. We prove that, at the non-perturbative level, the physical spectrum of the strongly-coupled Higgs mass is actually corrected from the ‘infinite number of derivatives’ present in the action. The technique we use permits to derive the set of Dyson–Schwinger equations in differential form. This proves essentially useful when exact solutions to the local equations are known. We show that all the formalism of the local theory involving the Dyson–Schwinger approach extends quite naturally to the non-local case. Using these methods, the spectrum of the strongly-coupled non-local theories become accessible in the non-perturbative regimes. We calculate the N-point correlation functions and predict the mass-gap in the spectrum arising purely from the self-interaction and the non-local scale M. The proper weak perturbation limit is correctly recovered from the strong coupling regime in the limit of the coupling of the theory goes to zero. Our results show the mass gap generated in the non-local theories gets damped in the UV and it reaches conformal limit. We discuss the implications of our result in particle physics and cosmology.

Diluted mass gap in strongly coupled non-local Yang-Mills

We investigate the non-perturbative regimes in the class of non-Abelian theories that have been proposed as an ultraviolet completion of 4-D Quantum Field Theory (QFT) generalizing the kinetic energy operators to an infinite series of higher-order derivatives inspired by string field theory. We prove that, at the non-perturbative level, the physical spectrum of the theory is actually corrected by the “infinite number of derivatives” present in the action. We derive a set of Dyson-Schwinger equations in differential form, for correlation functions till two-points, the solution for which are known in the local theory. We obtain that just like in the local theory, the non-local counterpart displays a mass gap, depending also on the mass scale of non-locality, and show that it is damped in the deep UV asymptotically. We point out some possible implications of our result in particle physics and cosmology and discuss aspects of non-local QCD-like scenarios.

Wave Packet Dynamical Simulation of Quasiparticle Interferences in 2D Materials

Materials consisting of single- or a few atomic layers have extraordinary physical properties, which are influenced by the structural defects. We present two calculation methods based on wave packet (WP) dynamics, where we compute the scattering of quasiparticle WPs on localized defects. The methods are tested on a graphene sheet: (1) We describe the perfect crystal lattice and the electronic structure by a local atomic pseudopotential, then calculate the Bloch eigenstates and build a localized WP from these states. The defect is represented by a local potential, then we compute the scattering by the time development of the WP. (2) We describe the perfect crystal entirely by the kinetic energy operator, then we calculate the scattering on the local defect described by the potential energy operator. The kinetic energy operator is derived from the dispersion relation, which can be obtained from any electronic structure calculation. We also verify the method by calculating Fourier transform images and comparing them with experimental FFT-LDOS images from STM measurements. These calculation methods make it possible to study the quasiparticle interferences, inter- and intra-valley scattering, anisotropic scattering, etc., caused by defect sites for any 2D material.

Vibrational energy levels of the S 0 and S 1 states of formaldehyde using an accurate ab initio based global diabatic potential energy matrix

Low-lying vibrational energy levels of both the ground (S 0) and first excited singlet (S 1) states of formaldehyde have been determined using an exact kinetic energy operator and adiabatic potential energy surfaces derived from a recently developed full-dimensional global diabatic potential energy matrix (DPEM) based on a neural network representation of high level ab initio data. The agreement with available experimental band origins provides strong evidence in support of the accuracy of the DPEM, which can be used in the future for understanding the internal conversion of H2CO(S 1) and the subsequent roaming dynamics on its S 0 state.

Artificial Symmetries for Calculating Vibrational Energies of Linear Molecules

Linear molecules usually represent a special case in rotational-vibrational calculations due to a singularity of the kinetic energy operator that arises from the rotation about the a (the principal axis of least moment of inertia, becoming the molecular axis at the linear equilibrium geometry) being undefined. Assuming the standard ro-vibrational basis functions, in the 3N−6 approach, of the form ∣ν1,ν2,ν3ℓ3;J,k,m⟩, tackling the unique difficulties of linear molecules involves constraining the vibrational and rotational functions with k=ℓ3, which are the projections, in units of ℏ, of the corresponding angular momenta onto the molecular axis. These basis functions are assigned to irreducible representations (irreps) of the C2v(M) molecular symmetry group. This, in turn, necessitates purpose-built codes that specifically deal with linear molecules. In the present work, we describe an alternative scheme and introduce an (artificial) group that ensures that the condition ℓ3=k is automatically applied solely through symmetry group algebra. The advantage of such an approach is that the application of symmetry group algebra in ro-vibrational calculations is ubiquitous, and so this method can be used to enable ro-vibrational calculations of linear molecules in polyatomic codes with fairly minimal modifications. To this end, we construct a—formally infinite—artificial molecular symmetry group D∞h(AEM), which consists of one-dimensional (non-degenerate) irreducible representations and use it to classify vibrational and rotational basis functions according to ℓ and k. This extension to non-rigorous, artificial symmetry groups is based on cyclic groups of prime-order. Opposite to the usual scenario, where the form of symmetry adapted basis sets is dictated by the symmetry group the molecule belongs to, here the symmetry group D∞h(AEM) is built to satisfy properties for the convenience of the basis set construction and matrix elements calculations. We believe that the idea of purpose-built artificial symmetry groups can be useful in other applications.

Using nondirect product Wigner D basis functions and the symmetry-adapted Lanczos algorithm to compute the ro-vibrational spectrum of CH4-H2O.

By doing calculations on the methane-water van der Waals complex, we demonstrate that highly converged energy levels and wavefunctions can be obtained using Wigner D basis functions and the Symmetry-Adapted Lanczos (SAL) method. The Wigner D basis is a nondirect product basis and, therefore, efficient when the kinetic energy operator has accessible singularities. The SAL method makes it possible to exploit symmetry to label energy levels and reduce the cost of the calculation, without explicitly using symmetry-adapted basis functions. Line strengths are computed, and new bands are identified. In particular, we find unusually strong transitions between states associated with the isomers of the global minimum and the secondary minimum.