Calculation of open p-shell atoms in the algebraic approach of the Hartree–Fock method

Highly precise calculations of analytical Hartree–Fock orbitals and energies have been performed within the limits of the Roothaan–Hartree–Fock atomic theory (Roothaan–Bagus method) for all open p-shell atoms of the Periodic Table. They were calculated in an algebraic approach using Slater-type atomic orbitals (AOs) as basis functions. Nonlinear parameters (orbital exponents) of AOs were optimized with exceptional accuracy by second-order methods. As a result, it was possible to satisfy exactly the virial relation (10^–14–10^–17) with calculated atomic term energies being close to the Hartree–Fock limit.     

Semiempirical fine-tuning for Hartree–Fock ionization potentials of atomic ions with non-integral atomic number

Amovilli and March (2006) [8] used diffusion quantum Monte Carlo techniques to calculate the non-relativistic ionization potential I(Z)$I(Z)$ in He-like atomic ions for the range of (fractional) nuclear charges Z   lying between the known critical value _Zc=0.911$Z_{\mathrm{c}}=0.911$ at which I(Z)$I(Z)$ tends to zero and Z=2$Z=2$. They showed that it is possible to fit I(Z)$I(Z)$ to a simple quadratic expression. Following that idea, we present here a semiempirical fine-tuning of Hartree–Fock ionization potentials for the isoelectronic series of He, Be, Ne, Mg and Ar-like atomic ions that leads to excellent estimations of _Zc$Z_{\mathrm{c}}$ for these series. The empirical information involved is experimental ionization and electron affinity data. It is clearly demonstrated that Hartree–Fock theory provides an excellent starting point for determining I(Z)$I(Z)$ for these series.     

Hartree–Fock self-consistent field method for atoms with two open shells of identical symmetry

A generalization of the Roothaan–Bagus method (Roothaan–Hartree–Fock atomic theory) on atoms with open shells of identical symmetry is given. Using orbital exponents of Slater-type atomic orbitals optimized with high accuracy by methods for the minimization of the first and second orders, energy values for atoms with two open s-type shells are calculated within the limits of the Roothaan–Hartree–Fock atomic theory.     

Fission barrier of actinide nuclei with double-Λ particles within the Skyrme–Hartree–Fock method

Fission barrier of actinide nuclei including two Λ hyperons is studied in the framework of the Skyrme–Hartree–Fock approach at zero temperature. We adopt the zero-range Skyrme-type interaction between Λ–N and Λ–Λ. Fission barrier is investigated using several parameter sets for ΛΛ interactions. We obtain the result that the barrier height becomes higher as the number of Λ particle increases and the barrier width has a dependence on the ΛΛ interaction. A relation between Λ binding energy and density distribution of Λ particle inside a core nucleus is also discussed. We found that the both Λ particles are attracted into heavier fission fragment.     

Differentiable programming and density matrix based Hartree–Fock method

Differentiable programming is an emerging programming paradigm that allows people to take derivative of an output of arbitrary code snippet with respect to its input. It is the workhorse behind several well known deep learning frameworks,and has attracted significant attention in scientific machine learning community. In this paper, we introduce and implement a density matrix based Hartree–Fock method that naturally fits into the demands of this paradigm, and demonstrate it by performing fully variational ground state calculation on several representative chemical molecules.     

Study of Octupole Deformation of Radium Isotopes in the Hartree–Fock–Bogoliubov Approximation with Skyrme Forces

Physics of Atomic Nuclei - The change in the octupole deformation of nuclei in the chain of even–even radium isotopes is studied on the basis of the Hartree–Fock–Bogoliubov method...     

Spinodal instabilities of asymmetric nuclear matter within the Brueckner–Hartree–Fock approach

We study the spinodal instabilities of asymmetric nuclear matter at finite temperature within the microscopic Brueckner–Hartree–Fock (BHF) approximation using the realistic Argonne V18 nucleon–nucleon potential plus a three-body force of Urbana type. Our results are compared with those obtained with the Skyrme force SLy230a and the relativistic mean field models NL3 and TW. We find that BHF predicts a larger spinodal region. This result is a direct consequence of the fact that our Brueckner calculation predicts a larger critical temperature and saturation density of symmetric nuclear matter than the Skyrme and relativistic mean field ones. We find that the instability is always dominated by total density fluctuations, in agreement with previous results of other authors. We study also the restoration of the isospin symmetry in the liquid phase, i.e., the so-called isospin distillation or fragmentation effect, finding that its efficiency increases with increasing proton fraction and decreases as temperature and density increase. In general, we find that the Brueckner results are comparable to those obtained with the Skyrme and the relativistic mean field models, although the restoration of isospin symmetry is not so efficient in this case.     

Nonstationary Theory of Perturbations for Open-Shell Atoms in the Hartree–Fock Approximation

For a multielectron open-shell system exposed to an external time-dependent perturbation, the Hartree–Fock nonstationary equations are obtained in terms of density operators. Using them as a basis, equations of nonstationary coupled perturbation theory are suggested in orbital representation within the framework of the two-operator variant of the Roothaan method for an open shell. The perturbation theory corrections to the orbitals have been found in the form of expansions in unperturbed orbitals which are assumed to be calculated in the LCAO approximation in the basis of Slater-type atomic orbitals. The dynamic polarizability of open-shell atoms of substances from Li to F and Sc has been calculated as an even-power series of the frequency of incident radiation.     

The Hartree–Fock Approximation Calculation of Electric Polarizability of Atoms with an Open Shell

Within the framework of the restricted Hartree–Fock method, equations of stationary coupled perturbation theory have been obtained for atomic-molecular systems with an open shell in orbital representation. Corrections to the Hartree–Fock orbitals are sought in the form of expansions in unperturbed orbitals which are assumed calculated in the LCAO approximation. The resulting inhomogeneous algebraic system for the expansion coefficients admits an exact solution. The static polarizability of atoms with an open shell from Li to Sc has been calculated with the use of an optimized basis set of Slater-type atomic orbitals.     

Relativistic Schrödinger Theory and the Hartree–Fock Approach

Within the framework of Relativistic Schrödinger Theory (RST), the scalar two-particle systems with electromagnetic interactions are treated on the basis of a non-Abelian gauge group U(2) which is broken down to the Abelian subgroup U(1)×U(1). In order that the RST dynamics be consistent with the (non-Abelian) Maxwell equations, there arises a compatibility condition which yields cross relationships for the links between the field strengths and currents of both particles such that self-interactions are eliminated. In the non-relativistic limit, the RST dynamics becomes identical to the well-known Hartree–Fock equations (for spinless particles). Consequently the original RST field equations may be considered as the relativistic generalization of the Hartree–Fock equations, and the exchange interactions of the conventional theory (induced by the anti-symmetrization postulate) do reappear here as ordinary gauge interactions due to a broken symmetry.     

Pseudospectral method with symplectic algorithm for the solution of time-dependent SchrSdinger equations

A pseudospectral method with symplectic algorithm for the solution of time-dependent Schrodinger equations （TDSE） is introduced. The spatial part of the wavefunction is discretized into sparse grid by pseudospectral method and the time evolution is given in symplectic scheme. This method allows us to obtain a highly accurate and stable solution of TDSE. The effectiveness and efficiency of this method is demonstrated by the high-order harmonic spectra of one-dimensional atom in strong laser field as compared with previously published work. The influence of the additional static electric field is also investigated.     

Towards a Skyrme–Hartree–Fock–Bogolyubov Mass Formula

In this work, we explain our astrophysical motivations for deriving a mass formula based on HFB calculations with a Skyrme interaction. We give an overview of existing mass formulae and present briefly the last HF+BCS mass formula [1]. The Skyrme force MSk7 [1] is considered in the study of shell effects at N=82, in the neutron-rich region far from stability, within the HFB and HF+BCS theories, and compared with results obtained using the forces SkP^δ and SkP^δρ [2]. This revised version was published online in July 2006 with corrections to the Cover Date.     

Existence of Hartree–Fock excited states for atoms and molecules

For neutral and positively charged atoms and molecules, we prove the existence of infinitely many Hartree–Fock critical points below the first energy threshold (that is, the lowest energy of the same system with one electron removed). This is the equivalent, in Hartree–Fock theory, of the famous Zhislin–Sigalov theorem which states the existence of infinitely many eigenvalues below the bottom of the essential spectrum of the N-particle linear Schrödinger operator. Our result improves a theorem of Lions in 1987 who already constructed infinitely many Hartree–Fock critical points, but with much higher energy. Our main contribution is the proof that the Hartree–Fock functional satisfies the Palais–Smale property below the first energy threshold. We then use minimax methods in the N-particle space, instead of working in the one-particle space.     

Electronic structure from equivalent differential equations of Hartree–Fock equations

A strict universal method of calculating the electronic structure of condensed matter from the Hartree–Fock equation is proposed. It is based on a partial differential equation(PDE) strictly equivalent to the Hartree–Fock equation, which is an integral–differential equation of fermion single-body wavefunctions. Although the maximum order of the differential operator in the Hartree–Fock equation is 2, the mathematical property of its integral kernel function can warrant the equation to be strictly equivalent to a 4 th-order nonlinear partial differential equation of fermion single-body wavefunctions. This allows the electronic structure calculation to eliminate empirical and random choices of the starting trial wavefunction(which is inevitable for achieving rapid convergence with respect to iterative times, in the iterative method of studying integral–differential equations), and strictly relates the electronic structure to the space boundary conditions of the singlebody wavefunction.     

Similarity solutions of the Fokker–Planck equation with time-dependent coefficients

In this work, we consider the solvability of the Fokker–Planck equation with both time-dependent drift and diffusion coefficients by means of the similarity method. By the introduction of the similarity variable, the Fokker–Planck equation is reduced to an ordinary differential equation. Adopting the natural requirement that the probability current density vanishes at the boundary, the resulting ordinary differential equation turns out to be integrable, and the probability density function can be given in closed form. New examples of exactly solvable Fokker–Planck equations are presented, and their properties analyzed.     

Bogolubov–Hartree–Fock Theory for Strongly Interacting Fermions in the Low Density Limit

We consider the Bogolubov–Hartree–Fock functional for a fermionic many-body system with two-body interactions. For suitable interaction potentials that have a strong enough attractive tail in order to allow for two-body bound states, but are otherwise sufficiently repulsive to guarantee stability of the system, we show that in the low-density limit the ground state of this model consists of a Bose–Einstein condensate of fermion pairs. The latter can be described by means of the Gross–Pitaevskii energy functional.     

Uniform in N global well-posedness of the time-dependent Hartree–Fock–Bogoliubov equations in $$\mathbb {R}^{1+1}$$

We prove the global well-posedness of the time-dependent Hartree–Fock–Bogoliubov (TDHFB) equations in \(\mathbb {R}^{1+1}\) with two-body interaction potential of the form \(N^{-1}v_N(x) = N^{\beta -1} v(N^\beta x)\) where \(v\ge 0\) is a sufficiently regular radial function, i.e., \(v \in L^1(\mathbb {R})\cap C^\infty (\mathbb {R})\). In particular, using methods of dispersive PDEs similar to the ones used in Grillakis and Machedon (Commun Partial Differ Equ 42:24–67, 2017), we are able to show for any scaling parameter \(\beta >0\) the TDHFB equations are globally well-posed in some Strichartz-type spaces independent of N, cf. (Bach et al. in The time-dependent Hartree–Fock–Bogoliubov equations for Bosons, 2016. arXiv:1602.05171).     

Do Dyson Orbitals resemble canonical Hartree–Fock orbitals?

ABSTRACT

Dyson orbitals are overlaps between states with N and N±1 electrons and provide conceptual links between transition probabilities of electron detachment or attachment, density matrices, total energies and general principles of chemical bonding. Canonical, Hartree–Fock orbitals are compared with Dyson orbitals obtained with electron–propagator calculations that retain all elements of the self–energy matrix, wherein all orbital–relaxation and electron–correlation corrections to Koopmans results reside. For valence ionization energies and electron affinities of representative closed–shell molecules, canonical, Hartree–Fock orbitals usually are excellent approximations to Dyson orbitals, although there are some notable cases where the resemblance is not as strong. Numerical relationships between pole strengths and the Koopmans contributions to Dyson orbitals also are inferred from the data.     

Reconciling phase diffusion and Hartree–Fock approximation in condensate systems

Despite the weakly interacting regime, the physics of Bose–Einstein condensates is widely affected by particle–particle interactions. They determine quantum phase diffusion, which is known to be the main cause of loss of coherence. Studying a simple model of two interacting Bose systems, we show how to predict the appearance of phase diffusion beyond the Bogoliubov approximation, providing a self-consistent treatment in the framework of a generalized Hartree–Fock–Bogoliubov perturbation theory.