Hellmann-Feynman Theorem Research Articles

Ground-state Bethe root densities and quantum phase transitions

Exactly solvable models provide a unique method, via qualitative changes in the distribution of the ground-state roots of the Bethe ansatz equations, to identify quantum phase transitions. Here we expand on this approach, in a quantitative manner, for two models of Bose–Einstein condensates. The first model deals with the interconversion of bosonic atoms and molecules. The second is the two-site Bose–Hubbard model, widely used to describe tunneling phenomena in Bose–Einstein condensates. For these systems we calculate the ground-state root density. This facilitates the determination of analytic forms for the ground-state energy, and associated correlation functions through the Hellmann–Feynman theorem. These calculations provide a clear identification of the quantum phase transition in each model. For the first model we obtain an expression for the molecular fraction expectation value. For the two-site Bose–Hubbard model we find that there is a simple characterization of condensate fragmentation.

The Hellmann–Feynman theorem, the comparison theorem, and the envelope theory

The envelope theory is a convenient method to compute approximate solutions for bound state equations in quantum mechanics. It is shown that these approximate solutions obey a kind of Hellmann–Feynman theorem, and that the comparison theorem can be applied to these approximate solutions for two ordered Hamiltonians.

Effect of Tensor Correlations on the Density Dependence of the Nuclear Symmetry Energy

We analyze the effect of the tensor force and other components of the nucleon-nucleon interaction on the nuclear symmetry energy and its density dependence by using the Hellmann–Feynman theorem. The analysis is performed within the microscopic Brueckner–Hartree–Fock approach using the Argonne V18 potential plus a Urbana IX three-nucleon force. Our results show that the potential part of the nuclear Hamiltonian, and in particular its tensor component, gives the largest contribution to the symmetry energy. The decomposition of the symmetry energy into a kinetic part and a potential energy part provides physical insight on the correlated nature of the system, indicating that pure neutron matter is less correlated than symmetric nuclear matter.

Correct electrostatic treatment of noncovalent interactions: the importance of polarization

The Hellmann–Feynman theorem assures us that the forces felt by the nuclei in a molecule or complex are purely classically electrostatic. Nevertheless, it is often claimed (incorrectly) that electrostatic considerations are not sufficient to explain noncovalent interactions. Such assertions arise largely from neglecting the polarization that is inherently part of the electrostatic interaction, and must be taken into account. Accordingly, we now outline the requirements for a correct electrostatic treatment and discuss the difference between physical observables and quantities that arise from mathematical models. Polarization and donor–acceptor charge transfer are shown to be equivalent for weak interactions. However, polarization is a physical observable while charge transfer, in this context, is mathematical modelling. We also discuss some popular schemes for analyzing noncovalent interactions.WIREs Comput Mol Sci2015, 5:169–177. doi: 10.1002/wcms.1210This article is categorized under:Electronic Structure Theory > Ab Initio Electronic Structure MethodsElectronic Structure Theory > Density Functional TheoryMolecular and Statistical Mechanics > Molecular Interactions

Molecular Binding in Post-Kohn–Sham Orbital-Free DFT

Molecular binding in post-Kohn-Sham orbital-free DFT is investigated, using noninteracting kinetic energy functionals that satisfy the uniform electron gas condition and which are inhomogeneous under density scaling. A parameter is introduced that quantifies binding, and a series of functionals are determined from fits to near-exact effective homogeneities and/or Kohn-Sham noninteracting kinetic energies. These are then used to investigate the relationship between binding and the accuracy of the effective homogeneity and noninteracting kinetic energy at the equilibrium geometry. For a series of 11 molecules, the binding broadly improves as the effective homogeneity improves, although the extent to which it improves is dependent on the accuracy of the noninteracting kinetic energy; optimal binding appears to require both to be accurate simultaneously. The use of a Thomas-Fermi-von Weizsäcker form, augmented with a second gradient correction, goes some way toward achieving this, exhibiting molecular binding on average. The findings are discussed in terms of the noninteracting kinetic potential and the Hellmann-Feynman theorem. The extent to which the functionals can reproduce the system-dependence of the near-exact effective homogeneity is quantified, and potential energy curves are presented for selected molecules. The study provides impetus for including density scaling homogeneity considerations in the design of noninteracting kinetic energy functionals.

Non-Hermitian-transport effects in coupled-resonator optical waveguides

Coupled-resonator optical waveguides (CROWs) are known to have interesting and useful dispersion properties. Here, we study the transport in these waveguides in the general case where each resonator is open and asymmetric, i.e., is leaky and possesses no mirror-reflection symmetry. Each individual resonator then exhibits asymmetric backscattering between clockwise and counterclockwise propagating waves, which in combination with the losses induces non-orthogonal eigenmodes. In a chain of such resonators, the coupling between the resonators induces an additional source of non-hermiticity, and a complex band structure arises. We show that in this situation the group velocity of wave packets differs from the velocity associated with the probability density flux, with the difference arising from a non-hermitian correction to the Hellmann-Feynman theorem. Exploring these features numerically in a realistic scenario, we find that the complex band structure comprises almost-real branches and complex branches, which are joined by exceptional points, i.e., nonhermitian degeneracies at which not only the frequencies and decay rates coalesce but also the eigenmodes themselves. The non-hermitian corrections to the group velocity are largest in the regions around the exceptional points.

Eigensolution techniques, their applications and Fisherʼs information entropy of the Tietz–Wei diatomic molecular model

In this study, the approximate analytical solutions of Schrödinger, Klein–Gordon and Dirac equations under the Tietz–Wei (TW) diatomic molecular potential are represented by using an approximation for the centrifugal term. We have applied three types of eigensolution techniques: the functional analysis approach, supersymmetry quantum mechanics and the asymptotic iteration method to solve the Klein–Gordon, Dirac and Schrödinger equations, respectively. The energy eigenvalues and the corresponding eigenfunctions for these three wave equations are obtained, and some numerical results and figures are reported. It has been shown that these techniques yielded exactly the same results. some expectation values of the TW diatomic molecular potential within the framework of the Hellmann–Feynman theorem have been presented. The probability distributions that characterize the quantum mechanical states of TW diatomic molecular potential are analyzed by means of complementary information measures of a probability distribution called Fisherʼs information entropy. This distribution has been described in terms of Jacobi polynomials, whose characteristics are controlled by quantum numbers.

A quantum informed continuum model for ferroelectric materials

Quantum mechanical stress relations are studied to guide the development of a continuum scale electromechanical constitutive model for ferroelectric materials. Stresses at the quantum scale are determined through the use of the Hellmann–Feynman theorem to obtain an electrostatic stress that depends on the electric quadrupole density as opposed to polarization dependent electrostriction. The result is integrated into a continuum model using a generalized set of electronic coordinate vector order parameters contained within a Lagrangian density formulation. The new constitutive model is shown to be consistent with both quantum based stress and classical phenomenological electrostriction. This conclusion is verified through a numerical study of lead titanate where calculations of energy, stress and polarization from density functional theory (DFT) are fit to continuum stored energy and electrostatic stresses. The numerical analysis includes uncertainty quantification using Bayesian statistics to gain further insight into material parameter uncertainty when approximating DFT calculations as a reduced-order continuum model.

Calculating Derivative Couplings between Time-Dependent Hartree-Fock Excited States with Pseudo-Wavefunctions.

A pseudo-wavefunction description of time-dependent Hartree-Fock (TDHF) states is proposed and used to develop an analytic expression for derivative couplings between TDHF excited states based on the Hellmann-Feynman theorem. The resulting expression includes Pulay terms associated with using an atom-centered basis as well as a correction to ensure translational invariance. We demonstrate that our formalism recovers the well-known Chernyak-Mukamel expression near a crossing and in the limit of a complete basis, and thus our approach is consistent with time-dependent response theory. In a companion paper (DOI 10.1021/jp5057682 ), we investigate these derivative couplings near conical intersections and show that they behave correctly.

The Hellmann–Feynman theorem for statistical averages

We discuss the Hellmann–Feynman theorem for degenerate states and its application to the calculation of the derivatives of statistical averages with respect to external parameters.

Critical nuclear charge for two-electron atoms.

The critical nuclear charge Z(c) required to bind a nucleus plus two electrons in a heliumlike atom has recently been an area of active study, resulting in a disagreement with earlier calculations and with the value obtained from the radius of convergence 1/Z* of a 1/Z expansion of the energy. In order to resolve the disagreement, have performed high-precision variational calculations in Hylleraas coordinates. With the double basis set method, we have been able to obtain good convergence for Z very close to Z(c), which together with the Hellmann-Feynman theorem yields the value Z(c) = 0.911,028,224,077,255,73(4), corresponding to 1/Z(c) = 1.097,660,833,738,559,80(5). This value is in agreement with the value obtained by Baker et al. [Phys. Rev. A 41, 1247 (1990)]. A significant feature of the results is that the outer electron remains localized near the nucleus, even at Z = Z(c), and the bound state evidently changes smoothly into a shape resonance for Z<Z(c). A qualitative polarization potential is proposed to account for the resonance, and the radial distribution function for the electron density is calculated.

Tensor force effects and high-momentum components in the nuclear symmetry energy

We analyze microscopic many-body calculations of the nuclear symmetry energy and its density dependence. The calculations are performed in the framework of the Brueckner-Hartree-Fock and the Self-Consistent Green's Functions methods. Within Brueckner-Hartree-Fock, the Hellmann-Feynman theorem gives access to the kinetic energy contribution as well as the contributions of the different components of the nucleon-nucleon interaction. The tensor component gives the largest contribution to the symmetry energy. The decomposition of the symmetry energy in a kinetic part and a potential energy part provides physical insight on the correlated nature of the system, indicating that neutron matter is less correlated than symmetric nuclear matter. Within the Self-Consistent Green's Function approach, we compute the momentum distributions and we identify the effects of the high momentum components in the symmetry energy. The results are obtained for the realistic interaction Argonne V18 potential, supplemented by the Urbana IX three-body force in the Brueckner-Hartree-Fock calculations.

σ-Hole bonding: a physical interpretation.

The anisotropic electronic densities of covalently-bonded Group IV-VII atoms frequently give rise to regions of positive electrostatic potential on the extensions of covalent bonds to these atoms. Through such positive "σ-holes," the atoms can interact attractively and highly directionally with negative sites such as the lone pairs of Lewis bases, anions, π electrons, etc. In the case of Group VII this is called "halogen bonding." Hydrogen bonding can be viewed as a less directional subset of σ-hole interactions. Since positive σ-holes often exist in conjunction with regions of negative potential, the atoms can also interact favorably with positive sites. In accordance with the Hellmann-Feynman theorem, all of these interactions are purely Coulombic in nature (which encompasses polarization and dispersion). The strength of σ-hole bonding increases with the magnitudes of the potentials of the positive σ-hole and the negative site; their polarizabilities must sometimes also be taken explicitly into account.

Non-perturbative treatment of molecules in linear magnetic fields: Calculation of anapole susceptibilities

In the present study a non-perturbative approach to ab initio calculations of molecules in strong, linearly varying, magnetic fields is developed. The use of London atomic orbitals (LAOs) for non-uniform magnetic fields is discussed and the standard rationale of gauge-origin invariance is generalized to invariance under arbitrary constant shifts of the magnetic vector potential. Our approach is applied to study magnetically induced anapole moments (or toroidal moments) and the related anapole susceptibilities for a test set of chiral and nonchiral molecules. For the first time numerical anapole moments are accessible on an ab initio level of theory. Our results show that the use of London atomic orbitals dramatically improves the basis set convergence also for magnetic properties related to non-uniform magnetic fields, at the cost that the Hellmann-Feynman theorem does not apply for a finite LAO basis set. It is shown that the mixed anapole susceptibility can be related to chirality, since its trace vanishes for an achiral molecule.

Analytical derivatives, forces, force constants, molecular geometries, and related response properties in electronic structure theory

Analytical calculation of energy derivatives with respect to nuclear coordinates revolutionized applied molecular quantum mechanics by allowing the routine calculation of molecular structures and related properties. The cost of calculating first derivatives (gradients, forces) is comparable to the calculation of the energy for most electronic structure methods. Thus analytical differentiation, compared to numerical one, increases efficiency by a factor proportional to the number of nuclei and greatly improves numerical accuracy. Coordinate derivatives, together with their generalizations to electric and magnetic perturbations, are crucial for the determination of transition states, vibrational frequencies, infrared and Raman intensities, non‐Born–Oppenheimer couplings, and magnetic properties: NMR spectra, magnetizability, vibrational circular dichroism, etc. Derivative theory, unlike perturbation theory, generally requires perturbation‐dependent basis sets. The inclusion of contributions originating from this dependence is a better alternative than using the Hellmann–Feynman theorem with an extended basis set. Analytical second derivatives, compared to the numerical differentiation of first derivatives, do not yield savings similar to first derivatives versus energy, in accordance with Wigner's 2n + 1 rule, but still improve greatly efficiency and numerical accuracy. Third and fourth derivatives have also been implemented for simpler wavefunctions. Analytical gradients were initially formulated for variational wavefunctions. It was realized only later that the penalty for nonvariational wavefunctions is modest. The Lagrangian formulation provides a simple, elegant framework for general derivative theory. Disadvantages of analytical derivatives are increased code complexity, and, particularly for higher derivatives, the requirement of large blocks of computer time and memory, both of which may interfere with code parallelization.This article is categorized under: Electronic Structure Theory &gt; Ab Initio Electronic Structure Methods

Thermal vacuum state for the two-coupled-oscillator model at finite temperature: Derivation and application

Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state (TVS) for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann—Feynman theorem and the entroy variation in the context of the TVS.

Tensor force and the nuclear symmetry energy

We analyze the contribution of the different terms of the nucleon-nucleon interaction to the nuclear symmetry energy So and its slope parameter L by using the Hellmann-Feynman theorem. The analysis is done using the Argonne V18 potential plus the Urbana IX three-body force within the framework of the microscopic Brueckner-Hartree-Fock approach. Our results show that the potential part of the nuclear Hamiltonian, and in particular the tensor component of the nuclear force, gives the main contribution to both S0 and L. The kinetic contribution to S0 is very small and negative, in agreement with recent results.

Adaptive molecular decomposition: Large-scale quantum chemistry for liquids

We present a linear-scaling method based on self-consistent charge non-orthogonal tight-binding. Linear scaling is achieved using a many-body expansion, which is adjusted dynamically to the instantaneous molecular configuration of a liquid. The method is capable of simulating liquids over large length and time scales, and also handles reactions correctly. Benchmarking on typical carbonate electrolytes used in Li-ion batteries displays excellent agreement with results from full tight-binding calculations. The decomposition slightly breaks the Hellmann-Feynman theorem, which is demonstrated by application to water. However, an additional correction also enables dynamical simulation in this case.

The nonrelativistic molecular Tietz potential

We solve the 3-dimensional Schrödinger equation under the Tietz potential by an approximate analytical scheme. A Pekeris-type approximation and the Nikiforov--Uvarov technique are used in the calculations to report the arbitrary-state eigenfunctions and eigenvalues and we calculate some useful expectation values of the Tietz potential using the Hellmann-Feynman theorem and obtain the oscillator strength.

Comparison Theorems for the Position-Dependent Mass Schrödinger Equation

The following comparison rules for the discrete spectrum of the position-dependent mass (PDM) Schrödinger equation are established. (i) If a constant mass m0 and a PDM m(x) are ordered everywhere, that is either, m0≤m(x) or m0≥m(x), then the corresponding eigenvalues of the constant-mass Hamiltonian and of the PDM Hamiltonian with the same potential and the BenDaniel-Duke ambiguity parameters are ordered. (ii) The corresponding eigenvalues of PDM Hamiltonians with the different sets of ambiguity parameters are ordered if ∇2(1/m(x)) has a definite sign. We prove these statements by using the Hellmann-Feynman theorem and offer examples of their application.