Generalized Hartree-Fock Theory Research Articles

Time-dependent nuclear-electronic orbital Hartree-Fock theory in a strong uniform magnetic field.

In an ultrastrong magnetic field, with field strength B ≈ B0 = 2.35 × 105 T, molecular structure and dynamics differ strongly from that observed on the Earth. Within the Born-Oppenheimer (BO) approximation, for example, frequent (near) crossings of electronic energy surfaces are induced by the field, suggesting that nonadiabatic phenomena and processes may play a more important role in this mixed-field regime than in the weak-field regime on Earth. To understand the chemistry in the mixed regime, it therefore becomes important to explore non-BO methods. In this work, the nuclear-electronic orbital (NEO) method is employed to study protonic vibrational excitation energies in the presence of a strong magnetic field. The NEO generalized Hartree-Fock theory and time-dependent Hartree-Fock (TDHF) theory are derived and implemented, accounting for all terms that result as a consequence of the nonperturbative treatment of molecular systems in a magnetic field. The NEO results for HCN and FHF- with clamped heavy nuclei are compared against the quadratic eigenvalue problem. Each molecule has three semi-classical modes owing to the hydrogen-two precession modes that are degenerate in the absence of a field and one stretching mode. The NEO-TDHF model is found to perform well; in particular, it automatically captures the screening effects of the electrons on the nuclei, which are quantified through the difference in energy of the precession modes.

Spin-orbit coupling from a two-component self-consistent approach. I. Generalized Hartree-Fock theory.

Formal and computational aspects are discussed for a self-consistent treatment of spin-orbit coupling within the two-component generalization of the Hartree-Fock theory. A molecular implementation into the CRYSTAL program is illustrated, where the standard one-component code (typical of Hartree-Fock and Kohn-Sham spin-unrestricted methodologies) is extended to work in terms of two-component spinors. When passing from a one- to a two-component description, the Fock and density matrices become complex. Furthermore, apart from the αα and ββ diagonal spin blocks, one has also to deal with the αβ and βα off-diagonal spin blocks. These latter blocks require special care as, for open-shell electronic configurations, certain constraints of the one-component code have to be relaxed. This formalism intrinsically allows us to treat local magnetic torque as well as noncollinear magnetization and orbital current-density. An original scheme to impose a specified noncollinear magnetization on each atomic center as a starting guess to the self-consistent procedure is presented. This approach turns out to be essential to surpass local minima in the rugged energy landscape and allows possible convergence to the ground-state solution in all of the discussed test cases.

Multiple Helical Spin Density Waves and Magnetic Skyrmions in Itinerant Electron System

Magnetic structures and their stability of the multiple helical spin density waves (MHSDW) and related magnetic skyrmions in itinerant electron system have been investigated on the basis of both the phenomenological and microscopic theories. It is shown by using the Ginzburg-Landau (GL) theory and the Application Visualization System (AVS) that the magnetic skyrmion structures emerge as a MHSDW and can be stabilized in itinerant electron system on the fcc lattice even if there is no Dzyaloshinskii-Moriya interaction. Moreover, by using the generalized Hartree-Fock theory for the Hubbard model, the antiferromagnetic skyrmion structure found in the GL theory is shown to be stabilized on the fcc lattice in the vicinity of the half-filled electron number.

Spatial and Spin Symmetry Breaking in Semidefinite-Programming-Based Hartree–Fock Theory

The Hartree-Fock problem was recently recast as a semidefinite optimization over the space of rank-constrained two-body reduced-density matrices (RDMs) [ Phys. Rev. A 2014 , 89 , 010502(R) ]. This formulation of the problem transfers the nonconvexity of the Hartree-Fock energy functional to the rank constraint on the two-body RDM. We consider an equivalent optimization over the space of positive semidefinite one-electron RDMs (1-RDMs) that retains the nonconvexity of the Hartree-Fock energy expression. The optimized 1-RDM satisfies ensemble N-representability conditions, and ensemble spin-state conditions may be imposed as well. The spin-state conditions place additional linear and nonlinear constraints on the 1-RDM. We apply this RDM-based approach to several molecular systems and explore its spatial (point group) and spin ( Ŝ2 and Ŝ3) symmetry breaking properties. When imposing Ŝ2 and Ŝ3 symmetry but relaxing point group symmetry, the procedure often locates spatial-symmetry-broken solutions that are difficult to identify using standard algorithms. For example, the RDM-based approach yields a smooth, spatial-symmetry-broken potential energy curve for the well-known Be-H2 insertion pathway. We also demonstrate numerically that, upon relaxation of Ŝ2 and Ŝ3 symmetry constraints, the RDM-based approach is equivalent to real-valued generalized Hartree-Fock theory.

Generalized Hartree-Fock theory for the dispersion relations of interacting fermionic lattice systems

We study the variational solution of generic interacting fermionic lattice systems using fermionic Gaussian states and show that the process of "gaussification", leading to a nonlinear closed equation of motion for the covariance matrix, is locally optimal in time by relating it to the time-dependent variational principle. By linearising our nonlinear equation of motion around the ground-state fixed point we describe a method to study low-lying excited states leading to a variational method to determine the dispersion relations of generic interacting fermionic lattice systems. This procedure is applied to study the attractive and repulsive Hubbard model on a two-dimensional lattice.

Ground states of fermionic lattice Hamiltonians with permutation symmetry

We study the ground states of lattice Hamiltonians that are invariant under permutations, in the limit where the number of lattice sites, N -> \infty. For spin systems, these are product states, a fact that follows directly from the quantum de Finetti theorem. For fermionic systems, however, the problem is very different, since mode operators acting on different sites do not commute, but anti-commute. We construct a family of fermionic states, \cal{F}, from which such ground states can be easily computed. They are characterized by few parameters whose number only depends on M, the number of modes per lattice site. We also give an explicit construction for M=1,2. In the first case, \cal{F} is contained in the set of Gaussian states, whereas in the second it is not. Inspired by that constructions, we build a set of fermionic variational wave functions, and apply it to the Fermi-Hubbard model in two spatial dimensions, obtaining results that go beyond the generalized Hartree-Fock theory.

Symmetry and broken symmetries in molecular orbital descriptions of unstable molecules II. Alignment, flustration and tunneling of spins in mesoscopic molecular magnets

As an extention of previous work [Yamaguchi K (1983) J Mol Structure (Theochem) 103:101], our theoretical efforts toward molecular materials are briefly reviewed. Axial, helical and general (cone, tetrahedron, etc) spin structures of carbon clusters, manganese oxide clusters and iron-sulfur clusters were investigated using the classical Heisenberg model and generalized Hartree-Fock theory. The spin flustrations in the clusters were also studied in relation to theoretical descriptions of the magnetism and the chemical bonds in active sites of several enzymes. The macroscopic quantum tunneling and coherence of spins in the manganese oxide clusters were analyzed using the instanton model, and the tunneling rate of spins was calculated by the coherent state path integral method. The spin transitions induced by an external magnetic field were studied by the path integral Monte Carlo method. The theoretical results were explained in terms of the symmetry and broken symmetry of the wavefunctions in the mesoscopic molecular systems, which are intermediate in the scale factor. The results were also applied to molecular design of mesoscopic clusters of clusters in the intermediate and strong correlation regime. The active control of spins is finally discussed from the viewpoint of functionalities in molecular and biological materials, and technological applications of mesoscopic molecular magnets are discussed with regard to quantum computing.

A preliminary analysis of the superconductivity of Ba1-xKxBiO by the generalized Hartree-Fock theory

The gap and the renormalization functions of Ba1−x K x BiO have been numerically analyzed by means of the simplified equations of the generalized Hartree–Fock (GHF) theory. Measured functions and parameters have been used as inputs to the GHF integral equations, and all the relevant functions have been iterated self-consistently over the large energy range of 9 eV. The results show a reasonably large gap function which does not reverse its sign below 2 eV, despite static Coulomb repulsion and despite the low density of states at the Fermi level. It is also shown that these results are inconsistent with the conventional Eliashberg equations.

The Coulomb interaction in the generalized Hartree-Fock theory of superconductivity

The matrix self-energy equations of the generalized Hartree-Fock (GHF) theory of HTSC are reevaluated and rederived, while avoiding some shortcomings of the former derivation. The density of states is given for an interacted and renormalized electronic system. The self-energy equations are essentially different in several respects from the conventional Eliashberg equations. The matrix self-energy is analyzed with respect to its dependence on ɛ(p). It is found to depend one very weakly, which results in self-energy equations which depend on a single variable,ω, as in the conventional theory. However, it is found that the bare Coulomb interaction contributes extra terms to the pairing self-energy and to the renormalization function. In the simplified version of the equations, the effect of these two extra terms is incorporated as a single extra term of the renormalization function.

Generalized Hartree-Fock theory and the Hubbard model

The familiar unrestricted Hartree-Fock variational principle is generalized to include quasi-free states. As we show, these are in one-to-one correspondence with the one-particle density matrices and these, in turn provide a convenient formulation of a generalized Hartree-Fock variational principle, which includes the BCS theory as a special case. While this generalization is not new, it is not well known and we begin by elucidating it. The Hubbard model, with its particle-hole symmetry, is well suited to exploring this theory because BCS states for the attractive model turn into usual HF states for the repulsive model. We rigorously determine the true, unrestricted minimizers for zero and for nonzero temperature in several cases, notably the half-filled band. For the cases treated here, we can exactly determine all broken and unbroken spatial and gauge symmetries of the Hamiltonian.

The generalized Hartree-Fock theory of nonmagnetic high-temperature superconductors

The Eliashberg equations are generalized to apply to high-temperature superconductors without spin correlations. The generalization assumes any general electronic density of states. Consequently, it treats the Coulomb interaction dynamically, and takes into account the averaged momentum dependence of the self-energy and of the interactions. Unlike in the conventional Eliashberg equation, the bare Coulomb interaction yields a frequency-independent term in the renormalization function. This term breaks the symmetry of the self-energy, and changes the renormalization.

Variational Principle and Slater's Generalized Hartree-Fock Theory for Nuclei

Variational Principle and Slater's Generalized Hartree-Fock Theory for Nuclei