Solution Of Hartree Research Articles

Mathematical modeling of the electronic structure of Titanium dioxide \((TiO_2 )_6\) nanoparticles

The calculation of the number of atoms of the given dimensional nanoparticle, composed of different type atoms has been researched in this work. The calculations have been carried out for nanoparticles of titanium dioxide. Theoretical visual models have been configured, and quantum – mechanical calculations have been carried out for \((TiO_2 )_6\) nanoparticle. The calculations for titanium dioxide nanoparticle have been carried out on the basis of Gaussian atomic orbitals. Besides, Gaussian functions have been used as atomic orbitals. The numerical values of unknown coefficients of the linear combination of atomic orbitals of the atoms of the titanium nanoparticle have been found from the solution of Hartree–Fock–Roothaan (HFR) equations.The values of orbital energies, ionization potential, and the total electronic energy of titanium dioxide nanoparticles have been determined. The calculations show that ,titanium dioxide nanoparticle is tough, electrophile, and stable dielectric, material. The effective charge of atoms have been calculated, and the theoretical visual mode of titanium dioxide nanopartical have been constructed.

The Hartree and Vlasov equations at positive density

We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space.

Highly accurate numerical solution of Hartree\u2013Fock equation with pseudospectral method for closed-shell atoms

The Hartree–Fock (HF) equation for atoms with closed (sub)shells is transformed with the pseudospectral (PS) method into a discrete eigenvalue equation for scaled orbitals on a finite radial grid. The Fock exchange operator and the Hartree potential are obtained from the respective Poisson equations also discretized using the PS representation. The numerical solution of the discrete HF equation for closed-(sub)shell atoms from He to No is robust, fast and gives extremely accurate results, with the accuracy superior to that of the previous HF calculations. A very moderate number of 33 to 71 radial grid points is sufficient to obtain total energies with 14 significant digits and occupied orbital energies with 12 to 14 digits in numerical calculations using the double precision (64-bit) of the floating-point format.The electron density at the nucleus is then determined with 13 significant digits and the Kato condition for the density and s orbitals is satisfied with the accuracy of 11 to 13 digits. The node structure of the exact HF orbitals is obtained and their asymptotic dependence, including the common exponential decay, is reproduced very accurately. The accuracy of the investigated quantities is further improved by performing the PS calculations in the quadruple precision (128-bit) floating-point arithmetic which provides the total energies with 25 significant digits while using only 80 to 130 grid points.

Nuclear equation of state and incompressibility in a model with correlations from giant monopole vibrations

The nuclear equation of state and incompressibility are studied in a model with correlations. We first consider a Hartree solution and then add one-particle--one-hole (1p-1h) and two-particle--two-hole (2p-2h) admixtures into the wave function using a separable monopole-monopole interaction $\ensuremath{\lambda}{r}^{2}(i){r}^{2}(j)$ which induces giant monopole vibrations. Stable and unstable points arise in the nuclear equation of state as a function of the oscillator size parameter from pure 1p-1h admixtures. Addition of 2p-2h admixtures leads to one stable point. However, the incompressibility is found to be quite different from the Hartree case with 1p-1h admixtures, even with 2p-2h states included.

Numerical Solution of the Hartree–Fock Equation in Multilevel Tensor-Structured Format

In this paper, we describe a novel method for a robust and accurate iterative solution of the self-consistent Hartree–Fock equation in $\mathbb{R}^3$ based on the idea of tensor-structured computation of the electron density and the nonlinear Hartree and (nonlocal) exchange operators at all steps of the iterative process. We apply the self-consistent field (SCF) iteration to the Galerkin discretization in a set of low separation rank basis functions that are solely specified by the respective values on a three-dimensional Cartesian grid. The approximation error is estimated by $O(h^3)$, where $h=O(n^{-1})$ is the mesh size of an $n\times n\times n$ tensor grid, while the numerical complexity to compute the Galerkin matrices scales linearly in $n\log n$. We propose the tensor-truncated version of the SCF iteration using the traditional direct inversion in the iterative subspace scheme enhanced by the multilevel acceleration with the grid-dependent termination criteria at each discretization level. This implies that the overall computational cost scales almost linearly in the univariate problem size n. Numerical illustrations are presented for the all electron case of H$_2$O and the pseudopotential case of CH$_4$ and CH$_3$OH molecules. The proposed scheme is not restricted to a priori given rank-1 basis sets, allowing analytically integrable convolution transform with the Newton kernel that opens further perspectives for promotion of the tensor-structured methods in computational quantum chemistry.

On Blowup for Time-Dependent Generalized Hartree–Fock Equations

We prove finite-time blowup for spherically symmetric and negative energy solutions of Hartree–Fock and Hartree–Fock–Bogoliubov-type equations, which describe the evolution of attractive fermionic systems (e.g. white dwarfs). Our main results are twofold: first, we extend the recent blowup result of Hainzl and Schlein (Comm. Math. Phys. 287:705–714, 2009) to Hartree–Fock equations with infinite rank solutions and a general class of Newtonian type interactions. Second, we show the existence of finite-time blowup for spherically symmetric solutions of a Hartree–Fock–Bogoliubov model, where an angular momentum cutoff is introduced. We also explain the key difficulties encountered in the full Hartree–Fock–Bogoliubov theory.

Self-consistent solution of Kohn-Sham equations for infinitely extended systems with inhomogeneous electron gas

The density functional approach in the Kohn-Sham approximation is widely used to study properties of many-electron systems. Due to the nonlinearity of the Kohn-Sham equations, the general self-consistence searching method involves iterations with alternate solving of the Poisson and Schr\"{o}dinger equations. One of problems of such an approach is that the charge distribution renewed by means of the Schr\"{o}dinger equation solution does not conform to boundary conditions of Poisson equation for Coulomb potential. The resulting instability or even divergence of iterations manifests itself most appreciably in the case of infinitely extended systems. The published attempts to deal with this problem are reduced in fact to abandoning the original iterative method and replacing it with some approximate calculation scheme, which is usually semi-empirical and does not permit to evaluate the extent of deviation from the exact solution. In this work, we realize the iterative scheme of solving the Kohn-Sham equations for extended systems with inhomogeneous electron gas, which is based on eliminating the long-range character of Coulomb interaction as the cause of tight coupling between charge distribution and boundary conditions. The suggested algorithm is employed to calculate energy spectrum, self-consistent potential, and electrostatic capacitance of the semi-infinite degenerate electron gas bounded by infinitely high barrier, as well as the work function and surface energy of simple metals in the jellium model. The difference between self-consistent Hartree solutions and those taking into account the exchange-correlation interaction is analyzed. The case study of the metal-semiconductor tunnel contact shows this method being applied to an infinitely extended system where the steady-state current can flow.

Tensor decomposition in electronic structure calculations on 3D Cartesian grids

In this paper, we investigate a novel approach based on the combination of Tucker-type and canonical tensor decomposition techniques for the efficient numerical approximation of functions and operators in electronic structure calculations. In particular, we study applicability of tensor approximations for the numerical solution of Hartree–Fock and Kohn–Sham equations on 3D Cartesian grids. We show that the orthogonal Tucker-type tensor approximation of electron density and Hartree potential of simple molecules leads to low tensor rank representations. This enables an efficient tensor-product convolution scheme for the computation of the Hartree potential using a collocation-type approximation via piecewise constant basis functions on a uniform n × n × n grid. Combined with the Richardson extrapolation, our approach exhibits O ( h 3 ) convergence in the grid-size h = O ( n - 1 ) . Moreover, this requires O ( 3 rn + r 3 ) storage, where r denotes the Tucker rank of the electron density with r = O ( log n ) , almost uniformly in n. For example, calculations of the Coulomb matrix and the Hartree–Fock energy for the CH 4 molecule, with a pseudopotential on the C atom, achieved accuracies of the order of 10 - 6 hartree with a grid-size n of several hundreds. Since the tensor-product convolution in 3D is performed via 1D convolution transforms, our scheme markedly outperforms the 3D-FFT in both the computing time and storage requirements.

Construction of Different Kinds of Atomic and Molecular Orbitals Using Complete Orthonormal Sets of Ψα-ETO in Single Exponent Approximation

Using complete orthonormal sets of Ψα-exponential type orbitals in single exponent approximation the new approach has been suggested for construction of different kinds of functions which can be useful in the theory of linear combination of atomic orbitals. These functions can be chosen properly according to the nature of the problems under consideration. This is rather important because the choice of the basis set may be play a crucial role in applications to atomic and molecular problems. As an example of application, different atomic orbitals for the ground states of the neutral and the first ten cationic members of the isoelectronic series of He atom are constructed by the solution of Hartree—Fock—Roothaan equations using Ψ1, Ψ0 and Ψ−1 basis sets. The calculated results are close to the numerical Hartree-Fock values. The total energy, expansion coefficients, orbital exponents and virial ratio for each atom are presented.

Light-induced programming of Si nanocrystal flash memories

We show theoretically that it is possible to achieve much faster programming performances in nanocrystal flash memory devices by means of a light-induced mechanism that inverts the direction of charge transfer between nanocrystals and device substrate in comparison to conventional voltage-induced programming. Our method is based on the self-consistent solutions of Hartree and Poisson equation for both electrons and holes with open boundary conditions.

Reformulation in mathematical programming: An application to quantum chemistry

This paper concerns the application of reformulation techniques in mathematical programming to a specific problem arising in quantum chemistry, namely the solution of Hartree–Fock systems of equations, which describe atomic and molecular electronic wave functions based on the minimization of a functional of the energy. Their traditional solution method does not provide a guarantee of global optimality and its output depends on a provided initial starting point. We formulate this problem as a multi-extremal nonconvex polynomial programming problem, and solve it with a spatial Branch-and-Bound algorithm for global optimization. The lower bounds at each node are provided by reformulating the problem in such a way that its convex relaxation is tight. The validity of the proposed approach was established by successfully computing the ground-state of the helium and beryllium atoms.

Dependence of shake probability on nuclear charge in Li-, Na- and K-like ions

In sudden perturbation approximation, the probability of the shake-up process accompanying inner-shell ionization is calculated for the isoelectronic sequences of Li-, Na- and K-like ions in the ground and excited np and nd states. Numerical solutions of Hartree–Fock equations and hydrogen-like radial orbitals are used. Very large differences between the results of both approximations for all ions and strong dependences on ion charge are obtained at the beginning of the isoelectronic sequences.

Solution of the Skyrme–Hartree–Fock–Bogolyubov equations in the Cartesian deformed harmonic-oscillator basis. (IV) HFODD (v2.08i): a new version of the program

We describe the new version (v2.07f) of the code HFODD which solves the nuclear Skyrme-Hartree-Fock or Skyrme-Hartree-Fock-Bogolyubov problem by using the Cartesian deformed harmonic-oscillator basis. In the new version, all symmetries can be broken, which allows for calculations with angular frequency and angular momentum tilted with respect to the mass distribution. The new version contains an interface to the LAPACK subroutine ZHPEVX.

On Asymmetric Quasiperiodic Solutions of Hartree–Fock Systems

The time-dependent Hartree–Fock system is considered in the presence of external magnetic and electric fields, and with a self-consistent potential including anisotropies. A suitable ansatz reduces a quasiperiodic time-dependent problem to an eigenvalue problem, which is then solved by minimization of an energy functional.

Bosons in a shell: Hartree solution

The Hartree solution for the ground state of N>>1 bosons placed within a spherical impenetrable shell is calculated. The bosons are assumed to interact through Coulomb repulsive forces and therefore occupy a thin volume adjacent to the shell wall. The resulting density profile is compared with its parallel Thomas-Fermi solution for fermions.

Correlated electrons in an external magnetic field

Increasing external magnetic field B gradually forces the electron spins to align in the direction of the applied field. The Hartree solution becomes exact for B⩾ B s( U). A new small parameter Δ B= B s− B enables one to control the transition between weak- and strong-coupling regimes and the metal–insulator transition. Necessity for dynamical vertex corrections at intermediate and strong coupling is demonstrated.

A Semiclassical Approach to Relativistic Nuclear Mean Field Theory

Semiclassical relativistic particle and energy densities for a set of fermions submitted to a scalar field and to the time-like component of a four-vector field are derived in the Wigner-Kirkwood and Thomas-Fermi mean field theories, including gradient corrections up to order ħ 2. As a test of the validity of the semiclassical method, a simple case is studied, where the scalar potential and the zero component of the four-vector potential are identical and correspond to a harmonic oscillator. The semiclassical approach is then applied to the non-linear σ - ω model and the resulting variational equations are solved for finite nuclei and semi-infinite symmetric nuclear matter. In this case Hartree solutions are available and allow for an additional test of the reliability of the ħ 2 corrections.

Analytical properties of molecular harmonics, orbitals and potential energy surfaces

This article addresses an analysis of the solutions of Hartree—Fock equations for molecules, based on examining a bilinear expansion of an integral equation kernel in momentum space. It has been shown that the asymptotic part of molecular orbitals defines a global behaviour of orbital potential surfaces by nuclear coordinates. A notion of molecular harmonics has been introduced which aids the study of both the global and local analytical structure of potential surfaces of molecules, especially those possessing spatial symmetry. In the Appendix some important integrals of the problem are calculated.

Stability of relativistic Hartree states

It is shown that the Hartree solution of a relativistic hamiltonian of the type considered by Walecka and collaborators is unstable with respect to fluctuations of the nucleon effective mass, unless the negative energy states (Dirac sea) are taken into account and the corresponding renormalization counter terms are properly treated. The imaginary solution of the RPA equation for zero momentum excitations arising from the ordinary renormalization is also discussed, in relation to the instability of the vacuum.

Hartree solutions for the self-Yukawian boson sphere.

The ground state of an N-boson assembly interacting through two-body attractive Yukawa forces is analyzed in the Hartree approximation. The solution, which is not universal, can be adequately characterized by a unique parameter \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{} proportional to \ensuremath{\mu}${N}^{\mathrm{\ensuremath{-}}1}$, where ${\ensuremath{\mu}}^{\mathrm{\ensuremath{-}}1}$ is the range of the force. It is found that for \ensuremath{\mu}\ifmmode \tilde{}\else \~{}\fi{} larger than a critical value, there is no bound solution for the system. Our solutions are shown to rigorously fulfill the virial theorem.