Hellmann-Feynman Theorem Research Articles

Accurate Calculation of Interatomic Forces with Neural Networks Based on a Generative Transformer Architecture.

Using neural networks to express electronic wave functions represents a new paradigm for solving the Schrödinger equation in quantum chemistry. For practical quantum chemistry simulations, one needs to know not only energies of molecules, but also accurate forces acting on constituent atoms. In this work, we achieve the accurate calculation of interatomic forces on QiankunNet, a platform that combines transformer-based deep neural networks with efficient batched autoregressive sampling. Our approach permits the application of the Hellmann-Feynman theorem to force calculations without introducing corrective Pulay terms. The results show that the calculated interatomic forces are in close agreement with those derived from the full configuration interaction method, irrespective of whether the system is a simple molecule or a strongly correlated electron system like a linear hydrogen chain. Furthermore, the calculated interatomic forces are employed for atomic relaxation in the torsional rotation process of ethylene, and the energy barrier obtained from the scanned potential energy surface is in excellent agreement with the experiment. Our work contributes to the application of artificial intelligence to broader quantum chemistry simulations, such as modeling challenging chemical transformations where electron correlations are difficult to describe.

Determination of reduced density matrices in the doubly occupied configuration interaction space: A Hellmann-Feynman theorem approach.

In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.

Non-relativistic energy spectra of and diatomic molecules confined in a modified Scarf potential via supersymmetric WKB approach

In this research, the energy equation of the Schrodinger equation with the modified Scarf potential has been obtained using the supersymmetric WKB (SWKB) approach. The Pekeris approximation has been applied to enable the analytical solution. The energy equation was applied to determine the rovibrational states of two diatomic molecules such as O 2 + ( X 2 Π g ) and N 2 ( X 1 Σ g + ) . The average kinetic and potential energies were computed via the Hellman-Feynman theorem. The analytical result of the energy equation obtained coincides with the work reported in earlier literature where the authors had applied a different approach. The energy eigenvalues for the studied molecules were found to agree with the ones obtained using the Teitz-Hua molecular potential, the deformed modified Rosen-Morse (DMRM) potential, the Morse potential and the Rydberg-Klein-Rees data points. For the energy spectra, we found average absolute deviations of 0.129515% for O 2 + ( X 2 Π g ) and 1.834993% for N 2 ( X 1 Σ g + ) molecules respectively. The energy spectra increase as the quantum numbers increase up to the dissociation limit before decreasing. For the molecules, the atoms were found to possess an oscillatory-type motion where an increase in the mean kinetic energy resulted in a decrease in mean potential energy and vice-visa.

ENERGY SPECTRA, EXPECTATION VALUES, AND THERMODYNAMIC PROPERTIES OF HCl AND LiH DIATOMIC MOLECULES

The Schrödinger equation is solved by applying the Nikiforov-Uvarov-Functional Analysis method to the Hulthén plus screened Kratzer Potential. The Greene-Aldrich approximation is employed to determine the closed form expressions for the energy equation and the wave function. The Hellmann-Feynman theorem was employed to calculate the energy spectra and expectation values of various quantum states for diatomic moleculesof HCl and LiH. Subsequently, we employed the energy equation that we had previously derived to compute the partition function, which in turn enabled us to determine the thermodynamic properties associated with the diatomic molecules. The partition function for the diatomic molecules of 2Hand LiHwas calculated at different temperatures. The results indicate that the partition function of the two diatomic molecules rose as the temperature increased.The findings we obtained align with the results documented in the literature.

Hellmann-Feynman theorem in non-Hermitian systems

Hellmann-Feynman theorem in non-Hermitian systems

Beyond Cavity Born-Oppenheimer: On Nonadiabatic Coupling and Effective Ground State Hamiltonians in Vibro-Polaritonic Chemistry.

The emerging field of vibro-polaritonic chemistry studies the impact of light-matter hybrid states known as vibrational polaritons on chemical reactivity and molecular properties. Here, we discuss vibro-polaritonic chemistry from a quantum chemical perspective beyond the cavity Born-Oppenheimer (CBO) approximation and examine the role of electron-photon correlation in effective ground state Hamiltonians. We first quantitatively review ab initio vibro-polaritonic chemistry based on the molecular Pauli-Fierz Hamiltonian in dipole approximation and a vibrational strong coupling (VSC) Born-Huang expansion. We then derive nonadiabatic coupling elements arising from both "slow" nuclei and cavity modes compared to "fast" electrons via the generalized Hellmann-Feynman theorem, discuss their properties, and reevaluate the CBO approximation. In the second part, we introduce a crude VSC Born-Huang expansion based on adiabatic electronic states, which provides a foundation for widely employed effective Pauli-Fierz Hamiltonians in ground state vibro-polaritonic chemistry. Those do not strictly respect the CBO approximation but an alternative scheme, which we name crude CBO approximation. We argue that the crude CBO ground state misses electron-photon correlation relative to the CBO ground state due to neglected cavity-induced nonadiabatic transition dipole couplings to excited states. A perturbative connection between both ground state approximations is proposed, which identifies the crude CBO ground state as a first-order approximation to its CBO counterpart. We provide an illustrative numerical analysis of the cavity Shin-Metiu model with a focus on nonadiabatic coupling under VSC and electron-photon correlation effects on classical activation barriers. We finally discuss the potential shortcomings of the electron-polariton Hamiltonian when employed in the VSC regime.

Accurate and efficient calculations of Hellmann-Feynman forces for quantum computation.

First-order derivatives of energies with respect to atomic coordinates are widely computed and used in quantum chemistry simulations. The rapidly emerging technology of quantum computing offers a new paradigm for solving relevant quantum chemistry equations. In this work, we have achieved analytical calculations of atomic forces based on the Hellmann-Feynman theorem within the framework of the variational quantum eigensolver. The accuracy of the approach is demonstrated by calculating the atomic forces of H2, LiH, H2O, and NH3 molecules, which are in excellent agreement with values obtained from full configuration interaction calculations. In particular, for systems with degenerate molecular orbitals, the analytical approach has a significant accuracy advantage over finite-difference-based methods and will not involve additional computational effort on a quantum computer. The calculated forces are further used to optimize the geometries of NH3 and CH4 molecules and to perform abinitio molecular dynamics simulations for the umbrella inversion of NH3, demonstrating the feasibility of the approach in practical quantum chemistry simulations.

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.

Energy spectra and asymptotic laws of the radial screened Coulomb potential

The energy spectra and corresponding asymptotic laws of the radial screened Coulomb potential V(r)=−exp⁡(−C/r)/r are investigated by employing the generalized pseudospectral method. Our numerical calculations manifest full convergence for all investigated bound states with the screening parameter C ranging from zero to infinity, but show significant discrepancies with previous predictions. Based on the Hellmann-Feynman theorem, we analyze the asymptotic behavior of the energy spectra for screening parameters close to zero. We further propose an empirical reciprocal law for energies when the screening parameter approaches infinity. We finally conclude that for the radial screened Coulomb potential there do not exist finite values of critical screening parameters at which the bound states merge into the continuum.

Computation of forces and stresses in solids: Towards accurate structural optimization with auxiliary-field quantum Monte Carlo

The accurate computation of forces and other energy derivatives has been a long-standing challenge for quantum Monte Carlo methods. A number of technical obstacles contribute to this challenge. We discuss how these obstacles can be removed with the auxiliary-field quantum Monte Carlo (AFQMC) approach. AFQMC is a general, high-accuracy, many-body total-energy method for molecules and solids. The implementation of back-propagation for pure estimators allows direct calculation of gradients of the energy via the Hellmann-Feynman theorem. A planewave basis with norm-conserving pseudopotentials is used for the study of periodic bulk materials. Completeness of the planewave basis minimizes the effect of so-called Pulay terms. The ionic pseudopotentials, which can be incorporated in AFQMC in exactly the same manner as in standard independent-electron methods, regulate the force and stress estimators and eliminate any potential divergence of the Monte Carlo variances. The resulting approach allows applications of full geometry optimizations in bulk materials. It also paves the way for many-body computations of the phonon spectrum in solids.

Incommensurate charge density wave in multiband intermetallic systems exhibiting competing orders

The appearance of an incommensurate charge density wave vector $\textbf{Q} = (Q_x,Q_y)$ on multiband intermetallic systems presenting commensurate charge density wave (CDW) and superconductivity (SC) orders is investigated. We consider a two-band model in a square lattice, where the bands have distinct effective masses. The incommensurate CDW (inCDW) and CDW phases arise from an interband Coulomb repulsive interaction, while the SC emerges due to a local intraband attractive interaction. For simplicity, all the interactions, the order parameters and hybridization between bands are considered $\textbf{k}$-independent. The multiband systems that we are interested are intermetallic systems with a $d$-band coexisting with a large $c$-band, for which a mean-field approach has proved suitable. We obtain the eigenvalues and eigenvectors of the Hamiltonian numerically and minimize the free energy density with respect to the diverse parameters of the model by means of the Hellmann-Feynman theorem. We investigate the system in real as well as momentum space and we find an inCDW phase with wave vector $\textbf{Q} = (\pi, Q_y) = (Q_x, \pi)$. Our numerical results show that the arising of an inCDW state depends on parameters, such as the magnitude of the inCDW and CDW interactions, band filling, hybridization and the relative depth of the bands. In general, inCDW tends to emerge at low temperatures, away from half-filling. We also show that, whether the CDW ordering is commensurate or incommensurate, large values of the relative depth between bands may suppress it. We discuss how each parameter of the model affects the emergence of an inCDW phase.

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.

Analytical Forces for the Optimized Effective Potential Calculations.

The optimized effective potential (OEP) equation is an ill-conditioned linear system when using finite basis sets. Without any special treatment, the obtained exchange-correlation (XC) potential may have unphysical oscillations. One way to alleviate this problem is to regularize the solutions; however, a regularized XC potential is not the exact solution to the OEP equation. As a result, the system's energy is no longer variational against the Kohn-Sham (KS) potential, and the analytical forces cannot be derived from the Hellmann-Feynman theorem. In this work, we develop a robust and nearly black-box OEP method to ensure that the system's energy is variational against the KS potential. The basic idea is to add a penalty function that regularizes the XC potential to the energy functional. Analytical forces can then be derived based on the Hellmann-Feynman theorem. Another key result is that the impact of the regularization can be much reduced by regularizing the difference between the XC potential and an approximate XC potential rather than regularizing the XC potential. Numerical tests show that forces and the energy differences between systems are not sensitive to the regularization coefficient, which indicates that in practice accurate structural and electronic properties can be obtained without extrapolating the regularization coefficient to zero. We expect this new method to be found useful for calculations that employ advanced, orbital-based functionals, especially for applications that require efficient force calculations.

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.

Expectation Values and Energy Spectra of the Varshni Potential in Arbitrary Dimensions

Abstract: The Klein-Gordon equation with Varshni potential was solved through the Nikiforov-Uvarov method. The Greene and Aldrich approximation schemes were employed to overcome the centrifugal barrier. The energy eigenvalues were obtained in relativistic and non-relativistic regimes, as well as the corresponding normalized wave functions. Energy spectra and expectation values of the square of inverse of position kinetic energy and the square of the momentum for five selected diatomic molecules: H2, HCl, TiH, I2 and CO, using their separate spectroscopic parameters were computed through Hellmann-Feynman Theorem. Bound-state energy eigenvalues were also computed for Varshni potential and the numerical results agree with the already existing literature. Keywords: Expectation values, Varshni potential, Nikiforov-Uvarov method, Klein-Gordon equation.

On the solutions and applications of the generalised Cornell potential

This work presents the solutions to the non-relativistic Schrödinger equation for the generalised Cornell potential using the Nikiforov–Uvarov functional analysis method and the Greene–Aldrich approximation. Energy eigenvalues and eigen functions are obtained in closed form. Eigenvalue spectra of some physical potentials are also deduced from the generalised results. Further, within the framework of the Kratzer and the Coulomb perturbed potentials, two variants of the generalised potential, energy eigenvalue spectra of diatomic molecules viz CH, ScH, N, I and H and the mass spectra of and mesons are determined, respectively. We also compute the expectation values of position inverse and its square, kinetic energy, and square of momentum for the above diatomic molecules by employing the Hellman–Feynman theorem. A close agreement is observed between the present results and some other past works.

Simple approach to current-induced bond weakening in ballistic conductors

We present a simple, first principles scheme for calculating mechanical properties of nonequilibrium bulk systems assuming an ideal ballistic distribution function for the electronic states described by the external voltage bias. This allows for fast calculations of estimates of the current-induced stresses inside bulk systems carrying a ballistic current. The stress is calculated using the Hellmann-Feynman theorem, and is in agreement with the derivative of the nonequilibrium free energy. We illustrate the theory and present results for one-dimensional (1D) metal chains. We find that the susceptibility of the yield stress to the applied voltage agrees with the ordering of break voltages among the metals found in experiments. In particular, gold is seen to be the most stable under strong current, while aluminum is the least stable.

Investigating Some Diatomic Molecules Bounded by the Two-Dimensional Isotropic Oscillator plus Inverse Quadratic Potential in an External Magnetic Field

We investigate the nonrelativistic magnetic effect on the energy spectra, expectation values of some quantum mechanical observables, and diamagnetic susceptibility for some diatomic molecules bounded by the isotropic oscillator plus inverse quadratic potential. The energy eigenvalues and normalized wave functions are obtained via the parametric Nikiforov-Uvarov method. The expectation values square of the position r 2 , square of the momentum p 2 , kinetic energy T , and potential energy V are obtained by applying the Hellmann-Feynman theorem, and an expression for the diamagnetic susceptibility X is also derived. Using the spectroscopic data, the low rotational and low vibrational energy spectra, expectation values, and diamagnetic susceptibility X for a set of diatomic molecules (I2, H2, CO, and HCl) for arbitrary values, Larmor frequencies are calculated. The computed energy spectra, expectation values, and diamagnetic susceptibility X were found to be more influenced by the external magnetic field strength and inverse quadratic potential strength g than the vibrational frequencies and the masses of the selected molecules.