Hartree-Fock studies of atoms in strong magnetic fields

Extending conceptual DFT to include external variables: the influence of magnetic fields

An extension of conceptual DFT to include the influence of an external magnetic field is proposed in the context of a program set up to cope with the ever increasing variability of reaction conditions and concomitant reactivity.

Optimizing Molecular Geometries in Strong Magnetic Fields

An efficient implementation of geometrical derivatives at the Hartree-Fock (HF) and current-density functional theory (CDFT) levels is presented for the study of molecular structure in strong magnetic fields. The required integral derivatives are constructed using a hybrid McMurchie-Davidson and Rys quadrature approach, which combines the amenability of the former to the evaluation of derivative integrals with the efficiency of the latter for basis sets with high angular momentum. In addition to its application to evaluating derivatives of four-center integrals, this approach is also applied to gradients using the resolution-of-the-identity approximation, enabling efficient optimization of molecular structure for many-electron systems under a strong magnetic field. The CDFT contributions have been implemented for a wide range of density functionals up to and including the meta-GGA level with current-density dependent contributions and (range-separated) hybrids for the first time. Illustrative applications are presented to the OH and benzene molecules, revealing the rich and complex chemistry induced by the presence of an external magnetic field. Challenges for geometry optimization in strong fields are highlighted, along with the requirement for careful analysis of the resulting electronic structure at each stationary point. The importance of correlation effects is examined by comparison of results at the HF and CDFT levels. The present implementation of molecular gradients at the CDFT level provides a cost-effective approach to the study of molecular structure under strong magnetic fields, opening up many new possibilities for the study of chemistry in this regime.

Accurate and balanced anisotropic Gaussian type orbital basis sets for atoms in strong magnetic fields

In high magnetic field calculations, anisotropic Gaussian type orbital (AGTO) basis functions are capable of reconciling the competing demands of the spherically symmetric Coulombic interaction and cylindrical magnetic (B field) confinement. However, the best available a priori procedure for composing highly accurate AGTO sets for atoms in a strong B field [W. Zhu et al., Phys. Rev. A 90, 022504 (2014)] yields very large basis sets. Their size is problematical for use in any calculation with unfavorable computational cost scaling. Here we provide an alternative constructive procedure. It is based upon analysis of the underlying physics of atoms in B fields that allow identification of several principles for the construction of AGTO basis sets. Aided by numerical optimization and parameter fitting, followed by fine tuning of fitting parameters, we devise formulae for generating accurate AGTO basis sets in an arbitrary B field. For the hydrogen iso-electronic sequence, a set depends on B field strength, nuclear charge, and orbital quantum numbers. For multi-electron systems, the basis set formulae also include adjustment to account for orbital occupations. Tests of the new basis sets for atoms H through C (1 ≤ Z ≤ 6) and ions Li⁺, Be⁺, and B⁺, in a wide B field range (0 ≤ B ≤ 2000 a.u.), show an accuracy better than a few μhartree for single-electron systems and a few hundredths to a few mHs for multi-electron atoms. The relative errors are similar for different atoms and ions in a large B field range, from a few to a couple of tens of millionths, thereby confirming rather uniform accuracy across the nuclear charge Z and B field strength values. Residual basis set errors are two to three orders of magnitude smaller than the electronic correlation energies in multi-electron atoms, a signal of the usefulness of the new AGTO basis sets in correlated wavefunction or density functional calculations for atomic and molecular systems in an external strong B field.

Many-body calculations of low-energy eigenstates in magnetic and periodic systems with self-healing diffusion Monte Carlo: steps beyond the fixed phase

The self-healing diffusion Monte Carlo algorithm (SHDMC) [F. A. Reboredo, R. Q. Hood, and P. R. C. Kent, Phys. Rev. B 79, 195117 (2009); F. A. Reboredo, ibid. 80, 125110 (2009)] is extended to study the ground and excited states of magnetic and periodic systems. The method converges to exact eigenstates as the statistical data collected increase if the wave function is sufficiently flexible. It is shown that the dimensionality of the nodal surface is dependent on whether phase is a scalar function or not. A recursive optimization algorithm is derived from the time evolution of the mixed probability density, which is given by an ensemble of electronic configurations (walkers) with complex weight. This complex weight allows the phase of the fixed-node wave function to move away from the trial wave function phase. This novel approach is both a generalization of SHDMC and the fixed-phase approximation [G. Ortiz, D. M. Ceperley, and R. M. Martin, Phys Rev. Lett. 71, 2777 (1993)]. When used recursively it simultaneously improves the node and the phase. The algorithm is demonstrated to converge to nearly exact solutions of model systems with periodic boundary conditions or applied magnetic fields. The computational cost is proportional to the number of independent degrees of freedom of the phase. The method is applied to obtain low-energy excitations of Hamiltonians with magnetic field. Periodic boundary conditions are also considered optimizing wave functions with twisted boundary conditions which are included in a many-body Bloch phase. The potential applications of this new method to study periodic, magnetic, and complex Hamiltonians are discussed.

Dipole in a magnetic field, work, and quantum spin

The behavior of an atom in a nonuniform magnetic field is analyzed, as well as the motion of a classical magnetic dipole (a spinning charged ball) and a rotating charged ring. For the atom it is shown that, while the magnetic field does no work on the electron-orbital contribution to the magnetic moment (the source of translational kinetic energy being the internal energy of the atom), whether or not it does work on the electron-spin contribution to the magnetic moment depends on whether the electron has an intrinsic rotational kinetic energy associated with its spin. A rotational kinetic energy for the electron is shown to be consistent with the Dirac equation. If the electron does have a rotational kinetic energy, the acceleration of a silver atom in a Stern-Gerlach experiment or the emission of a photon from an electron spin flip can be explained without requiring the magnetic field to do work. For a constant magnetic field gradient along the z axis, it is found that the classical objects oscillate in simple harmonic motion along the z axis, the total kinetic energy -- translational plus rotational -- being a constant of the motion. For the charged ball, the change in rotational kinetic energy is associated only with a change in the precession frequency, the rotation rate about the figure axis remaining constant.

Exact density functionals for two-electron systems in an external magnetic field

In principle, the extension of density functional theory (DFT) to Coulombic systems in a nonvanishing magnetic field is via current DFT (CDFT). Though CDFT is long established formally, relatively little is known with respect to any generally applicable, reliable approximate E(XC) and A(XC) functionals analogous with the workhorse approximate functionals (local density approximation and generalized gradient approximation) of ordinary DFT. Progress can be aided by having benchmark studies on a solvable correlated system. At zero field, the best-known finite system for such purposes is Hooke's atom. Recently we extended the exact ground state solutions for this two-electron system to certain combinations of nonzero external magnetic fields and confinement strengths. From those exact solutions, as well as high-accuracy numerical results for other field and confinement combinations, we construct the correlated electron density and paramagnetic current density, the exact Kohn-Sham orbitals, and the exact DFT and CDFT exchange-correlation energies and potentials. We compare with results from several widely used approximate functionals, all of which exhibit major qualitative failures, whether in CDFT or in naive application of ordinary DFT. We also illustrate how the CDFT vorticity variable nu is a computationally difficult quantity which may not be appropriate in practice to describe the external B field effects on E(XC) and A(XC).