Session L35: Focus Session: DFT IV: Ground-State DFT: New Directions

Progress in fractional perspectives of density functional theory

Density functional theory of electronic structure is widely and successfully applied in simulations throughout engineering and sciences. However, there are major failures for many predicted properties. These errors can be characterized and understood through the perspective of fractional charges and fractional spins introduced recently. The fractional perspectives offer a possible pathway forward. I will report progress in excited states, spin state splitting, open-shell singlet states, fukui functions and band gaps.   

Generalized-gradient approximations with non-vanishing exchange-correlation magnetic torque

The description of systems of interacting electrons in the presence of magnetic fields, within spin-density functional theory (SDFT), requires the non-collinear magnetization density vector $\mathbf{m}(\mathbf{r})$ to be used as basic variable, along with the particle density $n(\mathbf{r})$. Futhermore, for a meaningful description of spin-dynamics, the magnetization density and its conjugate exchange-correlation (xc) field $\mathbf{B_{xc}}(\mathbf{r})$ must not be constrained to be locally parallel at every point in space. It is well known that the local density approximation (LDA) cannot, by construction, provide such a non-collinear configuration of the two vector fields. Here we show how popular generalized gradient approximations (GGAs), developed assuming collinear spin-density, can be used to describe non-collinear magnetization states, including the presence of non-vanishing local torque between $\mathbf{m}(\mathbf{r})$ and $\mathbf{B_{xc}}(\mathbf{r})$. Unlike previous attempts to extend the use of collinear GGAs to the domain of non-collinear magnetization densities, the approach we introduce is invariant with respect of spin-rotations, globally satisfies the \emph{zero-torque theorem}, reduces to the proper collinear limit and is numerically stable.   

New gradient functional for SDFT derived from spin spirals in the uniform electron gas

The development of functionals in SDFT depending on gradients of the spin magnetization is a long standing challenge. We present a new functional based on the spin-spiral state of the uniform electron gas. Comparing the principal idea of the new functional to the LSDA and GGAs, we highlight the intrinsic way non-collinearity is built into the proposed approximation. As key feature the functional yields exchange-correlation magnetic fields that are non-collinear w.r.t.~the spin magnetization, while obeying the zero-torque theorem by construction. This means that an adiabatic application of the functional within TD-SDFT accounts for the local torque exerted by the exchange-correlation field and retains the numerical simplicity of explicit density functionals. An implementation of the functional based on the RPA-treatment of spin spirals in the electron gas is shown.   

Tuning the correlation energy in second-order M{\o}ller-Plesset perturbation theory with an effective electron-electron interaction potential

For practical electronic structure calculations on large molecules, wave function theories (WFT) remain less popular than density functional theory (DFT) despite the systematic improvability of WFT. The greater computational cost of WFT is largely to blame and is two-fold in nature: WFT generally presents unfavorable scaling with system size relative to DFT, {\em and} WFT results converge more slowly than those of DFT with respect to basis set size. We propose that each of these issues can be partially mitigated by the introduction of an effective electron-electron interaction potential, in lieu of the true $1/r$ potential, for evaluating the correlation energy in WFT. We discuss the design and optimization of such an effective interaction, in the context of second-order M{\o}ller-Plesset perturbation theory (MP2), for two distinct purposes: to accelerate basis set convergence of MP2 correlation energies, and to re-scale the MP2 correlation to improve agreement with experiment and/or higher-level methods.   

Toward Improved Semilocal and Nonlocal Density Functionals for Atoms, Molecules, and Solids

Semilocal density functionals construct the exchange-correlation energy density at a point from the electron density and orbitals in the neighborhood of that point. They can be constructed nonempirically, and work best for sp-bonded systems near equilibrium. They increase in sophistication from the local spin density approximation to the generalized gradient approximation to the meta-GGA. For a molecule like CO on a transition metal surface, it appears that only a meta-GGA can give a good simultaneous description of the lattice constant and surface energy of the metal, on the one hand, and the adsorption energy of the molecule on the other [1]. I will discuss two remaining deficiencies of the revised TPSS meta-GGA [2]: its artificial order-of-limits problem, and its need for more information about non-bonded interaction. When electrons are shared over stretched bonds, full nonlocality is needed, and typically empirical parameters are also needed. This suggests that we don't yet know enough about the full nonlocality of the density functional for the exchange-correlation energy. \\[4pt] [1] J. Sun, M. Marsman, A. Ruzsinszky, G. Kresse, and J.P. Perdew, Phys. Rev. B 83, 121410 (2011). \\[0pt] [2] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, L.A. Constantin, and J. Sun, Phys. Rev. Lett. 103, 026403 (2009).   

Energy expressions for model exchange potentials: Beyond the Levy--Perdew virial relation

The common way to assign energies to Kohn--Sham exchange potentials is by using the Levy--Perdew virial relation. However, for model potentials that are not functional derivatives, this approach leads to energy expressions that lack translational invariance. We point out that there is a more general procedure for constructing density functionals from model potentials, of which the Levy--Perdew relation is just a special case. Using this generalization we propose a method for converting model potentials into density functionals that ensures translational invariance of the energy. To illustrate our approach we construct a competitively accurate exchange functional from the model potential of van Leeuwen and Baerends.   

Laplacian-based models for the exchange energy

Recent Quantum Monte Carlo data for the exchange-correlation energy density of pseudopotential systems strongly suggest the value of using the Laplacian of the density as a variable for constructing first order corrections to the local density approximation of density functional theory. We report on an exchange functional built upon these observations and extended to the all-electron case. THe model keeps the typical properties of constraint-based generalized gradient approximations (GGA) and also has a finite-valued potential at the nucleus, unlike the GGA. Problems with oscillatory behavior in the potential due to higher order derivatives are controlled by a curvature minimization constraint. The results are tested against exact potentials for the He and Ne atom. A combination of gradient and Laplacian as suggested by a gradient expansion of the exchange hole gives the best overall results.   

A non-empirical improvement of PBE and its hybrid PBE0

We present a non-empirical re-parameterization of the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) exchange-correlation functional and of the related PBE hybrid (PBE0) obtained by imposing the constraint that, for the hydrogen atom, the exchange energy cancels the Coulomb repulsion energy. The new parameterization is validated with well-known test sets. The results for the re-parameterized PBE GGA, called PBEmol, show a substantial improvement over the original PBE in predicted heats of formation, while retaining the quality of the original PBE functional for description of all the other properties considered. The results for the hybrids indicate that, although the PBE0 functional provides a rather good description of those properties, the predictions of the re-parameterized functional are, except in the case of the ionization potentials, modestly better. Also, the results are better than B3LYP, except for the case of the ionization potentials and the harmonic frequencies.   

Effect of the orbital-overlap dependence on Meta Generalized Gradient Approximation

The dimensionless inhomogeneity parameter, $\alpha $, characterizing the extent of orbital overlap, is disentangled from the other dimensionless inhomogeneity parameter, s, the reduced density gradient, in terms of constructing a meta generalized gradient approximation (MGGA) for the exchange functional. We show that the formation of the intershell region inside an atom is associated with increase of$\alpha $, which suggests MGGA should expect a monotonically decreasing $\alpha $ dependence for a wide range of density. This leads to a simple nonempirical MGGA exchange functional, which interpolates between the sigle-orbital regime for confinement systems, where $\alpha $=0, and the slowly varying density regime, where $\alpha \approx \mbox{1}$, and then extrapolates to $\alpha \to \infty $. The new MGGA exchange functional, combined with the variant of the Perdew-Burke-Erzerhof (PBE) GGA correlation as used in the revised Tao-Perdew-Staroverov-Scuseria (revTPSS) MGGA [1], performs equally well for atoms, molecules, surfaces, and solids, with an implication of a tight Lieb-Oxford bound. \\[4pt] [1] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, L.A. Constantin, and J. Sun, Phys. Rev. Lett. 103, 026403 (2009).   

The derivative discontinuity in density functional theory from an electrostatic description of the exchange and correlation potential

We present an approach that we have recently proposed [{\it Phys. Rev. Lett.} {\bf 107}, 183002 (2011)] to approximate the exchange and correlation (XC) term in density functional theory. In our approach the XC potential is considered as an electrostatic potential, generated by a fictitious XC density, which is in turn a functional of the electronic density. In this picture, the exact asymptotic limit for low density regions, wrongly predicted by many XC functionals, can be imposed as a local condition. Based on this XC density representation we develop a correction scheme that fixes the asymptotic behavior of any approximated XC potential for finite systems. The procedure is simple, computationally inexpensive, and does not depend on adjusted parameters. Additionally, from the correction procedure it is possible to extract an approximation to the derivative discontinuity of XC energy. This value can be used to directly obtain the gap of the system as a ground-state property. Results are presented for the application of the method to atoms and small molecules. The correction results in a significant improvement in the value of the ionization energy and the gap, with errors that are comparable to the results given by orbital dependent functionals.   

Multi-Scale Approach to Simulations of Kelvin Probe Force Microscopy

The distance dependence and atomic-scale contrast recently observed in nominal contact potential difference(CPD) signals simultaneously recorded during non-contact atomic force microscopy on surfaces of insulating and semiconducting samples have stimulated theoretical attempts to explain how the applied bias voltage affects electrostatic forces acting on the atomic scale. We attack the problem in two steps. First, the electrostatics of the macroscopic tip-cantilever-sample system is treated by a finite-difference method on an adjustable nonuniform mesh. It has the advantages of getting a systematically increasable accuracy as well as the ability of considering the cantilever which turns out to be important for insulating samples. The resulting electric field near the tip apex is then inserted into a series of wavelet-based density functional theory calculations. Results are obtained for a reactive disordered neutral silicon nano-scale tip interacting with a NaCl(001) sample. Bias-dependent forces and resulting atomic displacements are computed to an accuracy of 1 pN. Theoretical expressions for Kelvin signals and local contact potential difference (LCPD) are obtained combining both contributions to the electrostatic force and evaluated for several tip oscillation amplitudes.