Hartree Energy Research Articles

Uniqueness of ground states for nonlinear Hartree equations

We consider ground states of nonlinear Hartree equations with trapping potentials V(x), which can be described equivalently by positive minimizers of L2-critical Hartree energy functional. It is known that ground states exist if and only if the parameter a satisfies a<a*≔‖Q‖22, where Q is the unique ground state of ΔQ−Q+∫R4Q2(y)|x−y|2dyQ=0 in R4. In this paper, we prove the uniqueness of ground states as a ↗ a*, where the potential V(x) = p(x)h(x), 0<C≤p(x)≤1C, h(x) is homogeneous of degree q ≥ 2, and however H(y)=∫R4h(x+y)Q2(x)dx admits a unique and non-degenerate critical point.

Minimizers of mass critical Hartree energy functionals in bounded domains

We consider L2-constraint minimizers of the mass critical Hartree energy functional with a trapping potential V(x) in a bounded domain Ω of R4. We prove that minimizers exist if and only if the parameter a>0 satisfies a<a⁎=‖Q‖22, where Q>0 is the unique positive solution of −Δu+u−(∫R4u2(y)|x−y|2dy)u=0 in R4. By investigating new analytic methods, the refined limit behavior of minimizers as a↗a⁎ is analyzed for both cases where all the mass concentrates either at an inner point x0 of Ω or near the boundary of Ω, depending on whether V(x) attains its flattest global minimum at an inner point x0 of Ω or not. As a byproduct, we also establish two Gagliardo–Nirenberg type inequalities which are of independent interest.

Limit behavior of mass critical Hartree minimization problems with steep potential wells

We consider minimizers of the following mass critical Hartree minimization problem: eλ(N)≔inf{u∈H1(Rd), ‖u‖22=N}Eλ(u), where d ≥ 3, λ &amp;gt; 0, and the Hartree energy functional Eλ(u) is defined by Eλ(u)≔∫Rd|∇u(x)|2dx+λ∫Rdg(x)u2(x)dx−12∫Rd∫Rdu2(x)u2(y)|x−y|2dxdy. Here the steep potential g(x) satisfies 0=g(0)=infRdg(x)≤g(x)≤1 and 1−g(x)∈Ld2(Rd). We prove that there exists a constant N* &amp;gt; 0, independent of λg(x), such that if N ≥ N*, then eλ(N) does not admit minimizers for any λ &amp;gt; 0; if 0 &amp;lt; N &amp;lt; N*, then there exists a constant λ*(N) &amp;gt; 0 such that eλ(N) admits minimizers for any λ &amp;gt; λ*(N) and eλ(N) does not admit minimizers for 0 &amp;lt; λ &amp;lt; λ*(N). For any given 0 &amp;lt; N &amp;lt; N*, the limit behavior of positive minimizers for eλ(N) is also studied as λ → ∞, where the mass concentrates at the bottom of g(x).

A Universal 3D Voxel Descriptor for Solid-State Material Informatics with Deep Convolutional Neural Networks

Material informatics (MI) is a promising approach to liberate us from the time-consuming Edisonian (trial and error) process for material discoveries, driven by machine-learning algorithms. Several descriptors, which are encoded material features to feed computers, were proposed in the last few decades. Especially to solid systems, however, their insufficient representations of three dimensionality of field quantities such as electron distributions and local potentials have critically hindered broad and practical successes of the solid-state MI. We develop a simple, generic 3D voxel descriptor that compacts any field quantities, in such a suitable way to implement convolutional neural networks (CNNs). We examine the 3D voxel descriptor encoded from the electron distribution by a regression test with 680 oxides data. The present scheme outperforms other existing descriptors in the prediction of Hartree energies that are significantly relevant to the long-wavelength distribution of the valence electrons. The results indicate that this scheme can forecast any functionals of field quantities just by learning sufficient amount of data, if there is an explicit correlation between the target properties and field quantities. This 3D descriptor opens a way to import prominent CNNs-based algorithms of supervised, semi-supervised and reinforcement learnings into the solid-state MI.

On Resolution-of-the-Identity Electron Repulsion Integral Approximations and Variational Stability.

The definiteness of the Mulliken and Dirac electron repulsion integral (ERI) matrices is examined for different classes of resolution-of-the-identity (RI) ERI approximations with particular focus on local fitting techniques. For global RI, robust local RI, and nonrobust local RI we discuss the definiteness of the approximated ERI matrices as well as the resulting bounds of Hartree, exchange, and total energies. Lower bounds of Hartree and exchange energy contributions are crucial as their absence may lead to variational instabilities, causing severe convergence problems or even convergence to a spurious state in self-consistent-field optimizations. While the global RI approximation guarantees lower bounds of Hartree and exchange energies, local RI approximations are generally unbounded. The robust local RI approximation guarantees a lower bound of the exchange energy but not of the Hartree energy. The nonrobust local RI approximation guarantees a lower bound of the Hartree energy but not of the exchange energy. These issues are demonstrated by sample calculations on carbon dioxide and benzene using the pair atomic RI approximation.

The effects of gap parameter and spin polarization on electronic Hartree and correlation energies of doped graphene nanoribbon

We study the behaviors of both Hartree and correlation energies of undoped gapped armchair graphene nanoribbon using random phase approximation in the context of Hubbard model Hamiltonian. Specially, the effects of spin polarization and gap parameter on electron density dependence of Hartree and correlation energies of armchair graphene nanoribbon has been addressed. Our results show the variation of gap parameter leads to considerable effect on correlation and Hartree energy behavior of spin unpolarized gapped graphene in the middle electron density region. However local Hubbard interaction parameter affects the behaviors of Hartree and correlation energy on the whole range of electron density in zero magnetization case. We also show that a considerable reduction has been observed for density dependence of Hartree and correlation energies of spin polarized gapped graphene nanoribbon.

Optimizing kernel methods for Poisson integrals on a uniform grid

We analyze the error and error propagation in the calculation of the Poisson integral on a uniform grid within Density Functional Theory (DFT) real-space calculations. We suggest and examine several schemes for near neighbors’ interaction correction for the Green’s function kernel to improve the accuracy. Finally, we demonstrate the effect of the different kernels on DFT eigenvalues and Hartree energy accuracy in systems such as C60 and C40H82.

Long-range interactions of hydrogen atoms in excited states. I.2S−1Sinteractions and Dirac-δperturbations

The theory of the long-range interaction of metastable excited atomic states with ground-state atoms is analyzed. We show that the long-range interaction is essentially modified when quasi-degenerate states are available for virtual transitions. A discrepancy in the literature regarding the van der Waals coefficient C_6(2S;1S) describing the interaction of metastable atomic hydrogen (2S state) with a ground-state hydrogen atom is resolved. In the the van der Waals range a_0 << R << a_0/alpha, where a_0= hbar/(alpha m c) is the Bohr radius and alpha is the fine structure constant, one finds the symmetry-dependent result E_{2S;1S}(R) ~ (-176.75 +/- 27.98) E_h (a_0/R)^6 (E_h denotes the Hartree energy). In the Casimir-Polder range a_0/alpha << R << hbar c/L, where L = E(2S_{1/2})-E(2P_{1/2}) is the Lamb shift energy, one finds E_{2S;1S}(R) ~ (-121.50 +/- 46.61) E_h (a_0/R)^6. In the the Lamb shift range R >> hbar c/L, we find an oscillatory tail with a negligible interaction energy below 10^(-36) Hz. Dirac-delta perturbations to the interaction are also evaluated and results are given for all asymptotic distance ranges; these effects describe the hyperfine modification of the interaction, or, expressed differently, the shift of the hydrogen 2S hyperfine frequency due to interactions with neighboring 1S atoms. The 2S hyperfine frequency has recently been measured very accurately in atomic beam experiments.

Second-order response theory of radio-frequency spectroscopy for cold atoms

We present a theoretical description of the radio-frequency (rf) spectroscopy of fermionic atomic gases, based on the second-order response theory at finite temperature. This approach takes into account the energy resolution due to the envelope of the rf pulse. For a noninteracting final state, the momentum- and energy-resolved rf intensity depends on the fermion spectral function and pulse envelope. The contributions due to interactions in the final state can be classified by means of diagrams. Using this formalism, as well as the local density approximation in two and three dimensions, we study the interplay of inhomogeneities and Hartree energy in forming the line shape of the rf signal. We show that the effects of inhomogeneities can be minimized by taking advantage of interactions in the final state, and we discuss the most relevant final-state effects at low temperature and density, in particular the effect of a finite lifetime.

Constraint minimizers of mass critical Hartree energy functionals: Existence and mass concentration

We consider L2-constraint minimizers of mass critical Hartree energy functionals in ℝN with N ≥ 3. We prove that minimizers exist if and only if the parameter a &amp;gt; 0 satisfies a&amp;lt;a∗=Q22, where Q is a positive radially symmetric ground state of Δu−u+∫RNu(y)2x−y2dyu=0 in ℝN. The blow-up behavior of minimizers as a approaches a∗ is also analyzed, for which all the mass concentrates at a global minimum point x0 of the external potential V(x).

Structural and electronic properties of sodium clusters under confinement

Using real-space density functional theory, electronic structure and equilibrium geometries of sodium clusters in the size range of 2--20 atoms have been calculated as a function of confinement. We have examined the evolution of the five lowest isomers as a function of volume for six different compressions. The minimum volume considered is about $1/15$ to $1/10$ of the free-space box volume. We observe a strong tendency for isomeric transitions in many cases, with the higher isomers evolving into the ground state under confinement. In general the clusters tend to become more spherical. The changes in the total energies and the geometries are not significant until the volume gets reduced beyond the 1/3 of the original volume. In this sense, the clusters are not easy to compress. Once the critical volume is reached, the changes in the total energies and geometries are rapid. It turns out that the increase in the total energy is mainly due to the ion-ion and Hartree energies of electrons. We also address how anisotropic confinement affects the geometry of clusters. We further show that geometries obtained with anisotropic confinement are strongly supported by the simulation of clusters inside a carbon nanotube using a hybrid quantum-mechanical and molecular-mechanics approach.

A SIMPLIFIED ROUTE TO OBTAIN THE HARTREE AND HARTREE-FOCK EQUATIONS

Alternative considerably simpler ways of obtaining the Hartree and Hartree-Fock equations are presented. These alternatives do not replace the formal demonstrations, which should be introduced in undergraduate or graduate courses according to the required level of student training. However, the use of the present approaches allows a student-friendlier introduction of the basic principles of electronic structure calculations as a prior teaching resource to the formal demonstrations. General implications and comparisons between the Hartree and Hartree-Fock energies are discussed.

Excitations and benchmark ensemble density functional theory for two electrons.

A new method for extracting ensemble Kohn-Sham potentials from accurate excited state densities is applied to a variety of two-electron systems, exploring the behavior of exact ensemble density functional theory. The issue of separating the Hartree energy and the choice of degenerate eigenstates is explored. A new approximation, spin eigenstate Hartree-exchange, is derived. Exact conditions that are proven include the signs of the correlation energy components and the asymptotic behavior of the potential for small weights of the excited states. Many energy components are given as a function of the weights for two electrons in a one-dimensional flat box, in a box with a large barrier to create charge transfer excitations, in a three-dimensional harmonic well (Hooke's atom), and for the He atom singlet-triplet ensemble, singlet-triplet-singlet ensemble, and triplet bi-ensemble.

Efficient method for localised functions using domain transformation and Fourier sine series

An efficient approach to handle localised states by using spectral methods (SM) in one and three dimensions is presented. The method consists of transformation of the infinite domain to the bounded domain in (0, π) and using the Fourier sine series as a set of basis functions for the SM. It is shown that with an appropriate choice of transformation functions, this method manages to preserve the good properties of original SMs; more precisely, superb computational efficiency when high level of accuracy is necessary. This is made possible by analytically exploiting the properties of the transformation function and the Fourier sine series. An especially important property of this approach is the possibility of calculating the Hartree energy very efficiently. This is done by exploiting the positive properties of the sine series as a basis set and conducting an extinctive part of the calculations analytically. We illustrate the efficiency of this method and implement it to solve the Poisson’s and Helmholtz equations in both one and three dimensions. The efficiency of the method is verified through a comparison to recently published results for both one- and three-dimensional problems.

Exchange and correlation effects in the strongly interacting He-Ps system

The density functional theory has been used to calculate the total energy of the system comprising a He and a positronium atom. The single particle orbitals have been used to calculate the non-interacting kinetic energy, while the electron and positron densities were used for the Hartree energy, the electron exchange-correlation energy and the external potential energies. The electron exchange-correlation energy has been calculated within the Local Density Approximation and the General Gradient Approximation (PBE). For the electron-positron correlation energy the formula by Boronski and Nieminen has been used. The results have been compared to many-body wavefunction calculations employing the exact diagonalization of an explicitly correlated gaussians basis. While the many-body value result predicts that HePs is not bound, the density functional approach predicts a bound state with mean nucleus-positron distance of ~9 a.u. To discuss the origin of this discrepancy the exchange-correlation energy has been deduced from the many-body result and it has been compared to the density functional scheme.

Large two-dimensional electronic systems: Self-consistent energies and densities at low cost

We derive a self-consistent local variant of the Thomas-Fermi approximation for (quasi-)two-dimensional (2D) systems by localizing the Hartree term. The scheme results in an explicit orbital-free representation of the electron density and energy in terms of the external potential, the number of electrons, and the chemical potential determined upon normalization. We test the method over a variety 2D nanostructures by comparing to the Kohn-Sham 2D-LDA calculations up to 600 electrons. Accurate results are obtained in view of the negligible computational cost. We also assess a local upper bound for the Hartree energy.

Many-body and quantum effects in some thermodynamic properties and infinite shear modulus of HFD-like fluid using the radial distribution function

We have calculated pressure, internal energy, and infinite-frequency shear modulus of two-body Hartree–Fock dispersion (HFD)-like potential using the statistical mechanical formula for neon. Molecular dynamics simulations have been also applied to generate the required radial distribution function (RDF) for HFD-like fluid in the range 1.1≤T⁎≤4.4 and 0.43≤ρ⁎≤0.82. Quantum corrections have been considered using the Feynman–Hibbs (FH) approach. To take many-body forces into account for the HFD-like system, the simple three-body potentials of Wang and Sadus (2006) and Hauschild and Prausnitz (1993) were used with the two-body potential without requiring an expensive three-body calculation. The results show that the quantum and three-body effects improve the prediction of the pressure and internal energy of the HFD-like fluid.

Constraint on the second functional derivative of the exchange-correlation energy

Using the density functional adiabatic connection approach for an N-electron system it is shown that γ is the coupling constant that scales the electron–electron interaction strength. For the non-interacting Kohn–Sham Hamiltonian γ = 0 and for the fully interacting system γ = 1. is the Hartree plus exchange-correlation energy while f 0(r) and fγ(r) are the Fukui functions of the non-interacting and interacting systems, respectively. This identity can serve to test the internal self-consistency or quality of approximate functionals. The quality of some popular approximate exchange and correlation functionals are tested for a simple model system.

Hartree shift in unitary Fermi gases

The Hartree energy shift is calculated for a unitary Fermi gas. By including the momentum dependence of the scattering amplitude explicitly, the Hartree energy shift remains finite even at unitarity. Extending the theory also for spin-imbalanced systems allows calculation of polaron properties. The results are in good agreement with more involved theories and experiments.

Spin projected and improved energy states from unrestricted Hartree–Fock

A simple approach for improving the energy and quality of the unrestricted Hartree–Fock (UHF) approximation is presented. Starting from the UHF Slater determinant (SD), we extract eigenstates of S 2 by means of which the energies are calculated. Some of these energies are lower than that of UHF, thus giving a correlation correction. Because of the simplifications made, the present methodology is applicable for large molecules, where other methods are costly or unreliable. To test our procedure we made applications to small atoms and molecules, where it is easy to control errors and compare results.