N. Helbig

Real-space grids and the Octopus code as tools for the development of new simulation approaches for electronic systems

Real-space grids are a powerful alternative for the simulation of electronic systems. One of the main advantages of the approach is the flexibility and simplicity of working directly in real space where the different fields are discretized on a grid, combined with competitive numerical performance and great potential for parallelization. These properties constitute a great advantage at the time of implementing and testing new physical models. Based on our experience with the Octopus code, in this article we discuss how the real-space approach has allowed for the recent development of new ideas for the simulation of electronic systems. Among these applications are approaches to calculate response properties, modeling of photoemission, optimal control of quantum systems, simulation of plasmonic systems, and the exact solution of the Schrödinger equation for low-dimensionality systems.

Performance of one-body reduced density-matrix functionals for the homogeneous electron gas

The subject of this study is the exchange-correlation-energy functional of reduced density-matrix functional theory. Approximations of this functional are tested by applying them to the homogeneous electron gas. We find that two approximations recently proposed by Gritsenko , [J. Chem. Phys. 122, 204102 (2005)] yield considerably better correlation energies and momentum distributions than previously known functionals. We introduce modifications to these functionals, which, by construction, reproduce the exact correlation energy of the homogeneous electron gas.

Nonlinear phenomena in time-dependent density-functional theory: What Rabi oscillations can teach us

Through the exact solution of a two-electron singlet system interacting with a monochromatic laser we prove that all adiabatic density functionals within time-dependent density-functional theory are not able to discern between resonant and nonresonant (detuned) Rabi oscillations. This is rationalized in terms of a fictitious dynamical exchange-correlation (xc) detuning of the resonance while the laser is acting. The nonlinear dynamics of the Kohn-Sham system shows the characteristic features of detuned Rabi oscillations even if the exact resonant frequency is used. We identify the source of this error in a contribution from the xc functional to the set of nonlinear equations that describes the electron dynamics in an effective two-level system. The constraint of preventing the detuning introduces a new strong condition to be satisfied by approximate xc functionals.

Exact Kohn–Sham potential of strongly correlated finite systems

The dissociation of molecules, even the most simple hydrogen molecule, cannot be described accurately within density functional theory because none of the currently available functionals accounts for strong on-site correlation. This problem led to a discussion of properties that the local Kohn-Sham potential has to satisfy in order to correctly describe strongly correlated systems. We derive an analytic expression for the nontrivial form of the Kohn-Sham potential in between the two fragments for the dissociation of a single bond. We show that the numerical calculations for a one-dimensional two-electron model system indeed approach and reach this limit. It is shown that the functional form of the potential is universal, i.e., independent of the details of the two fragments.

Density functional theory beyond the linear regime: Validating an adiabatic local density approximation

We present a local density approximation (LDA) for one-dimensional (1D) systems interacting via the soft-Coulomb interaction based on quantum Monte Carlo calculations. Results for the ground-state energies and ionization potentials of finite 1D systems show excellent agreement with exact calculations obtained by exploiting the mapping of an N-electron system in d dimensions onto a single electron in Nxd dimensions, properly symmetrized by the Young diagrams. We conclude that 1D LDA is of the same quality as its three-dimensional (3D) counterpart, and we infer conclusions about 3D LDA. The linear and nonlinear time-dependent responses of 1D model systems using LDA, exact exchange, and the exact solution are investigated and show very good agreement in both cases, except for the well-known problem of missing double excitations. Consequently, the 3D LDA is expected to be of good quality beyond the linear response. In addition, the 1D LDA should prove useful in modeling the interaction of atoms with strong laser fields, where this specific 1D model is often used.

Discontinuities of the chemical potential in reduced density matrix functional theory

Abstract Using the discontinuity of the chemical potential as a function of excess charge, the fundamental gaps for finite systems and the band gaps of extended solids are determined within reduced density matrix functional theory. We also present the necessary and sufficient conditions for the one-body reduced density matrix of a system with fractional charge to be ensemble N-representable. The performance of most modern day reduced density matrix functionals is assessed for the gaps and the correlation energy of finite systems. Our results show that for finite systems the PNOF, BBC3, and power functionals yield very accurate correlation energies while for a correct description of the fundamental gap the removal of self-interaction terms is essential. For extended solids we find that the power functional captures the correct band gap behavior for conventional semiconductors as well as strongly correlated Mott insulators, where a gap is obtained in absence of any magnetic ordering.

A functional of the one-body-reduced density matrix derived from the homogeneous electron gas: Performance for finite systems

An approximation for the exchange-correlation energy of reduced-density-matrix-functional theory was recently derived from a study of the homogeneous electron gas [N. N. Lathiotakis, N. Helbig, and E. K. U. Gross, Phys. Rev. B 75, 195120 (2007)]. In the present work, we show how this approximation can be extended appropriately to finite systems, where the Wigner Seitz radius r(s), the parameter characterizing the constant density of the electron gas, needs to be replaced. We apply the functional to a variety of molecules at their equilibrium geometry and also discuss its performance at the dissociation limit. We demonstrate that, although originally derived from the uniform gas, the approximation performs remarkably well for finite systems.

Size consistency of explicit functionals of the natural orbitals in reduced density matrix functional theory

We report a size-inconsistency problem for several functionals within reduced density matrix functional theory. Being explicit functionals of the natural orbitals and occupation numbers, instead of the one-body reduced density matrix, many of the approximate functionals are not invariant under unitary transformations in the subspace of degenerate occupation numbers. One such transformation mixes the degenerate natural orbitals of identical independent subsystems, delocalizing them. Noninvariance under this transformation results in size inconsistency for some of the approximations while others avoid this pathology by favoring orbital localization.

Open shells in reduced-density-matrix-functional theory

Reduced-density-matrix-functional theory is applied to open-shell systems. We introduce a spin-restricted formulation by appropriately expressing approximate correlation-energy functionals in terms of spin-dependent occupation numbers and spin-independent natural orbitals. We demonstrate that the additional constraint of total-spin conservation is indispensable for the proper treatment of open-shell systems. The formalism is applied to the first-row open-shell atoms. The obtained ground-state energies are in very good agreement with the exact values as well as other state of the art quantum chemistry calculations.

Generalized Pauli constraints in reduced density matrix functional theory

Functionals of the one-body reduced density matrix (1-RDM) are routinely minimized under Colemans ensemble N-representability conditions. Recently, the topic of pure-state N-representability conditions, also known as generalized Pauli constraints, received increased attention following the discovery of a systematic way to derive them for any number of electrons and any finite dimensionality of the Hilbert space. The target of this work is to assess the potential impact of the enforcement of the pure-state conditions on the results of reduced density-matrix functional theory calculations. In particular, we examine whether the standard minimization of typical 1-RDM functionals under the ensemble N-representability conditions violates the pure-state conditions for prototype 3-electron systems. We also enforce the pure-state conditions, in addition to the ensemble ones, for the same systems and functionals and compare the correlation energies and optimal occupation numbers with those obtained by the enforcement of the ensemble conditions alone.