Hellmann Feynman Research Articles

Fisher Information for a System Composed of a Combination of Similar Potential Models

The solutions to the radial Schrödinger equation for a pseudoharmonic potential and Kratzer potential have been studied separately in the past. Despite different reports on the Kratzer potential, the fundamental theoretical quantities such as Fisher information have not been reported. In this study, we obtain the solution to the radial Schrödinger equation for the combination of the pseudoharmonic and Kratzer potentials in the presence of a constant-dependent potential, utilizing the concepts and formalism of the supersymmetric and shape invariance approach. The position expectation value and momentum expectation value are calculated employing the Hellmann–Feynman Theory. These expectation values are then used to calculate the Fisher information for both position and momentum spaces in both the absence and presence of the constant-dependent potential. The results obtained revealed that the presence of the constant-dependent potential leads to an increase in the energy eigenvalue, as well as in the position and momentum expectation values. Additionally, the constant-dependent potential increases the Fisher information for both position and momentum spaces. Furthermore, the product of the position expectation value and the momentum expectation value, along with the product of the Fisher information, satisfies both Fisher’s inequality and Cramer–Rao’s inequality.

Exchange–correlation entropy from the generalized thermal adiabatic connection

Warm dense matter is a highly energetic phase characterized by strong correlations, thermal effects, and quantum mechanical electrons. Thermal density functional theory is commonly used in simulations of this challenging phase, driving the development of temperature-dependent approximations to the exchange–correlation free energy. Approaches using the adiabatic connection formula are well known at zero temperature and have been recently leveraged at non-zero temperatures as well. In this work, a generalized thermal adiabatic connection (GTAC) formula is proposed, introducing a fictitious temperature parameter. This allows extraction of the exchange–correlation entropy SXC using simulated interaction strength scaling. This procedure uses a Hellmann–Feynman approach to express the exchange–correlation entropy in terms of a temperature- and interaction strength-dependent exchange–correlation potential energy. In addition, analysis of SXC as a function of interaction strength suggests new forms for approximations, and GTAC itself offers a new framework for exploring both the exact and approximate interplay of temperature, density, and interaction strength across a wide range of conditions.

Ro-vibrational energy spectra and thermal properties of some diatomic molecules

The present work deals with the solutions to the radial Schrodinger equation for harmonic plus screened Kratzer potential (HSKP) within the framework of the Nikiforov–Uvarov functional analysis method. The Greene–Aldrich approximation is used to handle the inverse square terms of the HSKP. Reducing the HSKP into Kratzer and screened Kratzer potentials, the energy spectra of selected diatomic molecules, i.e. LiH, HCl, O 2 and I 2 are computed. Further, the equations for expectation values of several parameters, including inverse of position ( r − 1 ), square of inverse of position ( r − 2 ), kinetic energy (T) and the square of momentum ( p 2 ) are obtained invoking the Hellmann–Feynman theorem. Results of this study are consistent with other similar previous studies. The analytical expressions for partition function and other thermodynamic properties of the diatomic molecules are determined.

Stability of the state of two-electron atoms near to critical nuclear charge

Critical stability of the metastable bound state of two-electron atoms near the critical nuclear charge Z c (smaller than which the bound state ceases to exist) is investigated by employing the explicitly correlated Hylleraas-configuration interaction basis function. Our numerical calculation shows that, when the nuclear charge decreases to Z c , the ionization energy of the system approaches zero linearly and this behavior is fully consistent with the Hellmann–Feynman theorem. Radial and angular physical quantities as well as the inner and outer electron radial density distributions are calculated to reveal the classical geometric structure of the two-electron atom. The inner electron can be well modeled by a hydrogenic model in the 2 p state and its density distribution does not change much as the nuclear charge is changed. The outer electron, which is still localized in a finite region near the nucleus before the system transforms into a well-defined resonance, is shifted notably into a far-distance region. The critical stability of the system near the bound-continuum limit is therefore dominated by the outer electron.

Vortices number and the critical atoms number of a rotating Bose–Einstein condensates in two-dimensional optical lattice

In this paper, the vortices number and the critical atoms number are calculated. Calculations are performed for ultracold trapped Bose gas in two-dimensional deep optical lattice. Hellmann–Feynman theorem is used here to calculate the expectation values of the angular momentum and thermal average of square radius (in situ size). The above-mentioned parameters are calculated from the thermodynamic grand potential [Formula: see text]. The latter is calculated using a conventional semi-classical approximation. This thermodynamic grand potential allows us to investigate the impacts of the finite size and inter-atomic interaction effects. The obtained result for the vortices number has a peak with increase in the rate of rotation. While the critical atoms number has increase in nature with increase in the critical temperature. These parameters show a major dependence on the inter-atomic interaction, depth of optical lattice and number of particles.

Hellmann–Feynman theorem and internal pressure for atoms, molecules and plasmas under pressure

Since the beginning of the twentieth century, confinement of atoms, molecules and plasma inside impenetrable sharp (hard) and smooth (soft) cavities has been studied with utmost interest. Physically, such situations introduce the trapping of a quantum system under high pressure. The internal pressure () of the system under multi-megabar external pressure has not yet been explored. Also, there is a lack of Hellmann–Feynman theorem (HFT) in such a scenario, which is essential to derive a concrete analytical expression of under stressed condition. Here using a scaling concept we provide a general HFT in terms of expectation values of total energy, potential and kinetic energy of a given confined system. A change in boundary condition (from smooth to sharp and vice versa) does not affect this. Applying this proposed HFT, a closed-form expression of and has been obtained for confined and shell-confined quantum systems. Based on this , several virial-like equations are also modeled. This HFT and are demonstrated for one- and many-electron atoms, plasmas and molecules using both analytical and numerical methods. Their applicability in several other confined systems is discussed from simple arguments.

Accurate Hellmann-Feynman forces from density functional calculations with augmented Gaussian basis sets.

The Hellmann-Feynman (HF) theorem provides a way to compute forces directly from the electron density, enabling efficient force calculations for large systems through machine learning (ML) models for the electron density. The main issue holding back the general acceptance of the HF approach for atom-centered basis sets is the well-known Pulay force which, if naively discarded, typically constitutes an error upward of 10 eV/Å in forces. In this work, we demonstrate that if a suitably augmented Gaussian basis set is used for density functional calculations, the Pulay force can be suppressed, and HF forces can be computed as accurately as analytical forces with state-of-the-art basis sets, allowing geometry optimization and molecular dynamics to be reliably performed with HF forces. Our results pave a clear path forward for the accurate and efficient simulation of large systems using ML densities and the HF theorem.

Variationally consistent Hellmann–Feynman forces in the finite element formulation of Kohn–Sham density functional theory

Hellmann–Feynman forces are derived within the numerical framework of the finite element method for density functional theory in the Kohn–Sham formalism. The variational consistency of the force expressions in all-electron and pseudopotential settings are carefully examined, with a particular focus on the implications arising from different representations for interaction terms that are associated with electrostatics. Numerical investigations in nonperiodic systems which range from diatomic molecules to carbon allotropes demonstrate the systematic convergence that is offered by the finite element framework, not only for energy and force but also for geometric configuration and molecular statics parameters. A range of higher-order discretizations employing fixed meshes are invoked within these examples based on classical finite elements as well as on isogeometric analysis. Overall, this work contributes to recent advances which demonstrate the viability of the finite element method for carrying out ab initio molecular dynamics.

Approximate Solutions of the Schrodinger Equation for a Momentum-Dependent potential

The solution of one-dimensional Schrodinger equation for a newly proposed potential called modified shifted Deng-Fan momentum-dependent potential is obtained via supersymmetric approach. The expectation values of momentum and position were calculated using Hellmann Feynman Theorem. The effects of momentum-dependent parameter on the solutions of the system as well as the expectation values were studied. Finally, the special cases of the interacting potential were obtained.

The conceptual power of the Hellmann–Feynman theorem

The conceptual power of the Hellmann–Feynman theorem

Potential parameters, ro-vibrational energy spectra, and expectation values of the improved deformed exponential-type potential

In this contribution, potential parameters were derived for the improved deformed exponential-type potential (IDEP) using Varshni conditions for a diatomic molecule potential. Expression for bound state ro-vibrational energies was obtained by solving the Schrödinger equation in the presence of the IDEP. In modeling the centrifugal term of the Schrödinger equation, a Pekeris-like approximation scheme was employed. Expectation values of potential and kinetic energies were obtained for the IDEP within the framework of Hellmann–Feynman theorem. The IDEP was used to model the spectroscopic data of four diatomic molecules, viz., ICl (X 1Σg+), NaK (c 3Σ+), O2+ (X 2Πg), and NaLi (A 1Σ+). The average absolute deviation from the dissociation energy obtained was 0.1235%, 0.9227%, 1.2275%, and 1.2651%, respectively. The results demonstrated that the IDEP was an ideal model for the diatomic molecules. Data obtained for expectation values of kinetic and potential energies showed that the total energy of the system was conserved and that at upper bound vibration quantum number, the energy of the molecules was entirely kinetic.

Room-Temperature Persistent Luminescence in Metal Halide Perovskite Nanocrystals for Solar-Driven CO 2 Bioreduction

Room-Temperature Persistent Luminescence in Metal Halide Perovskite Nanocrystals for Solar-Driven CO ₂ Bioreduction

Variational formulation of the bond capacity charge polarization model.

We present an alternative energy formulation of the bond capacity charge polarization model to be used in molecular dynamics simulations. The energy expression consists of a Coulombic charge-charge interaction contribution as well as a quadratic Coulomb potential term, which can be seen as the electrostatic energy stored in the system's bond capacities. This formulation is shown to be variational in the potential space, although, it shares the same set of charges with the original non-variational formulation of the model. This variational formulation is compared with the non-variational one in terms of few selected observables showing the underlying distinctiveness of the two approaches. Being variational, this formulation allows for the computation of forces by invoking the classical analog of the Hellmann-Feynman theorem, which makes this approach two times faster than the non-variational one.

Real-space density kernel method for Kohn–Sham density functional theory calculations at high temperature

Kohn-Sham density functional theory calculations using conventional diagonalization based methods become increasingly expensive as temperature increases due to the need to compute increasing numbers of partially occupied states. We present a density matrix based method for Kohn-Sham calculations at high temperatures that eliminates the need for diagonalization entirely, thus reducing the cost of such calculations significantly. Specifically, we develop real-space expressions for the electron density, electronic free energy, Hellmann-Feynman forces, and Hellmann-Feynman stress tensor in terms of an orthonormal auxiliary orbital basis and its density kernel transform, the density kernel being the matrix representation of the density operator in the auxiliary basis. Using Chebyshev filtering to generate the auxiliary basis, we next develop an approach akin to Clenshaw-Curtis spectral quadrature to calculate the individual columns of the density kernel based on the Fermi operator expansion in Chebyshev polynomials and employ a similar approach to evaluate band structure and entropic energy components. We implement the proposed formulation in the SPARC electronic structure code, using which we show systematic convergence of the aforementioned quantities to exact diagonalization results, and obtain significant speedups relative to conventional diagonalization based methods. Finally, we employ the new method to compute the self-diffusion coefficient and viscosity of aluminum at 116 045K from Kohn-Sham quantum molecular dynamics, where we find agreement with previous more approximate orbital-free density functional methods.

Study of dynamic and thermodynamic properties of zinc-blend and wurtzite cadmium sulfide (CdS) using density functional theory

Abstract The dynamic and thermodynamic properties of wurtzite (wz) and zinc-blend (zb) CdS are investigated within the density functional theory using different approximation methods such as LDA, PBE, and DFT+U. Hellmann–Feynman approach is implemented for the relaxation of atomic position for both phases. To guarantee the accuracy of calculation, the convergence test of total energy with respect to energy cutoff and k-point sampling is performed. The dynamic properties such as phonon dispersion, phonon density of state, frequency along with high symmetry points, static and dynamic polarizability, and dielectric constants are calculated. The obtained values are compared with previous theoretical results. DFT + U approximation gives a good result that is consistent with the available theory. Moreover, the vibrational energy, vibrational free energy, entropy, electron chemical potential, and constant-volume specific heat are obtained within LDA, PBE, and DFT + U approximations.

Boosting the Rate Performance of Li–S Batteries via Highly Dispersed Cobalt Nanoparticles Embedded into Nitrogen-Doped Hierarchical Porous Carbon

Boosting the Rate Performance of Li–S Batteries via Highly Dispersed Cobalt Nanoparticles Embedded into Nitrogen-Doped Hierarchical Porous Carbon

Critical screening parameters and critical behaviors of one-electron systems with screened Coulomb potentials

The critical screening parameters for one-electron systems screened by Hulthén, Debye–Hückel, and exponential cosine screened Coulomb potentials are calculated with an accuracy close to the precision of numerical arithmetic. The results for a H atom with an infinitely heavy nucleus are reported from the ground to high-lying excited states, and those for arbitrary two-body charged systems are derived from the Zm-scaling law. A thorough comparison of the critical screening parameters for the ground and the first p-wave excited states with previous predictions is made to demonstrate the accuracy of our calculations. The critical behaviors of system-bound and pseudo-continuum eigenenergies for s- and non-s-wave states are shown to follow the quadratic and linear laws, respectively. The variation of the corresponding wave functions is analyzed in detail. For systems with non-zero orbital angular momenta, the bound states convert into shape-type resonances when the screening parameter exceeds the critical value. The resonance energy shares the same linear law as the pseudo-continuum state, while the resonance width varies by an l-dependent power law. It is further shown that the different asymptotic behaviors of the resonance energy and width are consistent with the complex analog of the Hellmann–Feynman theorem.

Interatomic Repulsion and the Pauli Principle

Atoms may repel or attract each other, but bound systems get virtually all of the attention in the classroom. To address the imbalance, instructors can explore the basis of repulsion in a few simple systems and highlight the important role played by the Pauli principle. A good example is the first triplet excited state of H2 wherein both electrons have the same spin. In light of the Hellmann–Feynman theorem, the net interaction is repulsive due to the low electron density that results in the region between the two nuclei. Analogous destabilization occurs in σ-bonding systems involving 3 electrons because two-thirds of the electrons have the same spin. Symmetrical systems like He2+ are exceptions and actually do form bonds because the minority spin is able to delocalize over both centers. In contrast, the ground state of the neutral He2 system entails 2 α and 2 β electrons and is strictly repulsive. Finally, the lowest energy triplet state of ethylene twists toward a D2d structure to avoid a destabilizing interaction akin to that identified in the triplet state of H2.

Energy spectra and expectation values of selected diatomic molecules through the solutions of Klein–Gordon equation with Eckart–Hellmann potential model

We solved the Klein–Gordon equation using the Nikiforov–Uvarov (NU) method with Eckart–Hellmann potential (EHP) model with approximation to the centrifugal term. The analytical expression of the ro-vibrational energy spectra in relativistic, non-relativistic and the corresponding normalised wavefunction were obtained. The numerical bound states energy for various screening parameters at different quantum states and ro-vibrational energies of EHP for VH, TiH, NiC, TiC and CuLi diatomicmolecules were computed. We have also obtained the expectation values of , and using Hellmann–Feynman theorem for the selected diatomic molecules. Graphically, we have illustrated the variation of ro-vibrational energies and expectation values of , and . Four special cases of EHP where obtained and the results were in excellent agreements with the existing literature.

First-principles calculations on ferroelectricity and lattice dynamics of Type-II multiferroic SmMn2O5

In this work, we report on the structural, electronic, and ferroelectric properties of SmMn2O5 by using first-principles density functional theory plus on-site Coulomb interaction (DFT + U) calculations. A thorough analysis was preformed to reveal the competing characteristics of different high-temperature (T) phases and the polarization mechanism in the low-T multiferroic phase. We show that the structural characteristics of the high-T phases have a strong influence on the low-T multiferroicity. In addition to the spin-induced lattice distortion that reduces substantially the purely electronic ferroelectricity, the dominant polarization mechanism in low-T SmMn2O5 still originates from the electronic polarization. By performing mode decomposition of the Hellmann–Feynman forces and the lattice distortion induced by the q = (0.5, 0, 0) magnetic order, we find that the Raman-active Ag mode characterized by the Mn4+O6 octahedron distortion and synergistic displacement of Mn3+ and Sm ions is of primary importance, while the infrared (IR)-active B2u mode plays a secondary role. These findings provide a theoretical foundation for future studies concerning the enhanced magnetoelectric effects of SmMn2O5 due to its pure exchange–striction mechanism.