Session Q35: Focus Session: DFT VI: New Functional Developments

Some thoughs about old and new density functionals

The only theoretical approximation that lies at the heart of DFT appears is the (in)famous exchange-correlation functional. It is therefore not surprising that this quantity has been extensively studied, and that more than 150 different approximations have been put forward in the past 50 years. In this talk I will show some results concerning different families of functionals. The first concerns hybrid functionals, and in particular the role of the mixing parameter. This is commonly assumed to be fixed at a value around 0.2-0.3. However, by noting the similarities between the hybrid functionals and screened Hartree-Fock and ultimately GW theories, we can relate this parameter to the screening properties of the system. In this way we can build a recipe that allows for a considerable improvement on the results obtained by traditional hybrid functionals for solids. As a second topic, I will discuss if it is possible to completely get rid of the Slater integrals present in both Hartree-Fock, hybrid functionals or OEP approaches, and anyway get a proper description of the exchange in terms of reduced densities. I will pay particular attention to the new meta-GGA functionals for the exchange potentials, like, e.g., the Becke-Johnson potential and its more recent variations, like the Tran and Blaha or the Rasanen, Pittalis, and Proetto functionals.   

Self-interaction corrected Kohn--Sham potentials

Exchange-correlation potentials derived from conventional density-functional approximations fail to exhibit the slow Coulombic decay---a problem that is related to the self-interaction error in the potential. We show how the self-interaction error of standard semilocal approximations can be effectively reduced by employing modified electron densities to construct the corresponding Kohn--Sham potentials. Using this correction scheme in the framework of adiabatic time-dependent density-functional theory we obtain significantly improved electronic excitation energies, especially for Rydberg states.   

Density-on-wave-function mapping beyond the Hohenberg-Kohn theorem

Density-functional theory is based on the Hohenberg-Kohn theorem, establishing a one-on-one mapping between ground-state densities and wave functions. That theorem does not, however, make a direct statement on whether two wave functions that are in some sense close are mapped on two densities that are also close, and vice versa. In this work, a metric is defined that allows to quantify the meaning of ``close'' in the preceding sentence. This metric stratifies Hilbert space into concentric spheres on which maximum and minimum distances between states can be defined and geometrically interpreted. Numerical calculations for the Helium atom, Hooke's atom and a lattice Hamiltonian show that the mapping between densities and ground states, which is highly complex and nonlocal in the coordinate description, in metric space becomes a monotonic and nearly linear mapping of vicinities onto vicinities. In this sense, the density-on-wave-function mapping is not only simpler than expected; it is as simple as it could be. \\[4pt] I. D'Amico, J. P. Coe, V. V. Fran\c{c}a, and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011) and Phys. Rev. Lett. 107, 188902 (2011). See also E. Artacho, Phys. Rev. Lett. 107, 188901 (2011).   

Finding density functionals with machine learning

Using standard methods from machine learning, we introduce a novel technique for density functional approximation. We use kernel ridge regression with a Gaussian kernel to approximate the non-interacting kinetic energy of 1-dimensional multi-electron systems. With fewer than 100 training densities, we can achieve mean absolute errors of less than 1 kcal/mol on new densities. We determine densities for which our new functional will fail or perform well. Finally, we use principle component analysis to extract accurate functional derivatives from our functional, enabling an orbital-free minimization of the total energy to find a self-consistent density. This empirical method has two parameters, set via cross-validation, and requires no human intuition. In principle, this general technique can be extended to multi-dimensional systems, and can be used to approximate exchange-correlation density functionals.   

New Density Functionals with Broad Applicability in Chemistry (SOGGA11, SOGGA11-X, M11, M11-L) and Approaches to Open-Shell DFT

The accuracy of density functional theory for practical applications is determined by the quality of the necessarily approximate exchange-correlation functional (``density functional'') being used, and the goal of functional development in chemical physics is to obtain a functional that is accurate for a broad range of chemistry and physics. In our work we consider molecular structures and solid-state lattice constants and band gaps, but we emphasize energetics for main-group and transition-metal chemistry, including thermochemistry and barrier heights, noncovalent interaction energies, and excitation energies. This lecture will discuss four new density functionals, each optimized to give the best across-the-board performance for a broad range of chemistry in their class of functional: SOGGA11, a generalized gradient approximation (GGA); SOGGA11-X, a global hybrid GGA; M11: a range-separated hybrid meta-GGA, and M11-L, a meta-GGA. SOGGA11 and M11-L are local functionals, and SOGGA11-X and M11 include some nonlocal Hartree--Fock exchange. To the extent that time permits, I may also discuss recent progress in the treatment of open-shell systems by density functional theory, including time-dependent DFT, open-shell SCF, and noncollinear DFT. This invited lecture is based on collaborative research carried out with Roberto Peverati, Sijie Luo, Ke Yang, Boris Averkiev, Yan Zhao, and Rosendo Valero.   

Self-Interaction Free and Analytic Treatment of the Coulomb Energy in Kohn-Sham Density Functional Theory

We have developed a new treatment of the LDA functional in Kohn-Sham density functional theory which is expressed in terms of the pair density of a non-interacting system of particles, thus avoiding from the outset self-interaction effects. The pair density is expressed explicitly in terms of the density using a orthonormal and complete basis expressed as a functional of the density. This allows its functional differentiation with respect to the density and therefore the determination of the self-interaction free Coulomb potential by analytic means. The method is illustrated with numerical results for the atom series.   

Electronic structure via potential functional approximations

The universal functional of Hohenberg and Kohn is given as a coupling-constant integral over the density as a functional of the potential [1]. Conditions are derived under which potential-functional approximations are variational. Construction via this method and imposition of these conditions are shown to greatly improve the accuracy of the non-interacting kinetic energy needed for orbital-free Kohn-Sham calculations. This result provides a direct route to a self-consistent, orbital-free theory for the electronic structure of matter within the Kohn-Sham framework. It solely requires an approximation to the non-interacting density as a functional of the potential, which, so far, has been derived for simple systems [2,3]. \\[4pt] [1] A. Cangi, D. Lee, P. Elliott, K. Burke, E. K. U. Gross, Phys. Rev. Lett. 106, 236404, (2011).\\[0pt] [2] A. Cangi, D. Lee, P. Elliott, K. Burke, Phys. Rev. B 81, 235128, (2010).\\[0pt] [3] P. Elliott, D. Lee, A. Cangi, K. Burke, Phys. Rev. Lett. 100, 256406, (2008).   

Self-Consistent Random Phase Approximation

We report self-consistent Random Phase Approximation (RPA) calculations within the Density Functional Theory. The calculations are performed by the direct minimization scheme for the optimized effective potential method developed by Yang et al. [1]. We show results for the dissociation curve of H$_{2}^{+}$, H$_{2}$ and LiH with the RPA, where the exchange correlation kernel has been set to zero. For H$_{2}^{+}$ and H$_{2}$ we also show results for RPAX, where the exact exchange kernel has been included. The RPA, in general, over-correlates. At intermediate distances a maximum is obtained that lies above the exact energy. This is known from non-self-consistent calculations and is still present in the self-consistent results. The RPAX energies are higher than the RPA energies. At equilibrium distance they accurately reproduce the exact total energy. In the dissociation limit they improve upon RPA, but are still too low. For H$_{2}^{+}$ the RPAX correlation energy is zero. Consequently, RPAX gives the exact dissociation curve. We also present the local potentials. They indicate that a peak at the bond midpoint builds up with increasing bond distance. This is expected for the exact KS potential.\\[4pt] [1] W. Yang, and Q. Wu, \emph{Phys. Rev. Lett.}, {\bf 89}, 143002 (2002)   

On the Hohenberg-Kohn and Levy-Lieb Constrained Search Proofs of Density Functional Theory

In HK, a 1-1 relationship between the density $\rho ({\bf{r}})$ and the potential $v({\bf{r}})$ is established. (The relationship between $v({\bf{r}})$ and the ground state $\Psi$ is 1-1.) The proof, valid for $v$-representable densities, shows $\rho({\bf{r}})$ to be a basic variable. The LL proof is independent of $v({\bf{r}})$, and is valid for $N$-representable densities. In,\footnote{Pan and Sahni, IJQC 110, 2833 (2010)} we have proved that in an external magnetic field ${\bf{B}}({\bf{r}})=\mathbf{\nabla} \times {\bf{A}}({\bf{r}})$, there is a 1-1 relationship between $\{\rho({\bf{r}}), {\bf{j}} ({\bf{r}})\}$, with ${\bf{j}}({\bf{r}})$ the physical current density, and the potentials $\{v({\bf{r}}), {\bf{A}}({\bf{r}})\}$. (The relationship between $\{v({\bf{r}}), {\bf{A}}({\bf{r}})\}$ and $\Psi$ is \emph{many-to-one}.) This proves that $\{\rho({\bf{r}}), {\bf{j}}({\bf{r}})\}$ are the basic variables. The LL proof independent of $\{v({\bf{r}}), {\bf{A}}({\bf{r}})\}$ follows readily. However, such a proof also follows if $\{\rho({\bf{r}}), {\bf{j}}_{p}({\bf{r}})\}$, with ${\bf{j}}_{p}({\bf{r}})$ the paramagnetic current density, are considered the basic variables. As such knowledge of the basic variables as determined via HK is a pre-requisite to any LL type proof.   

Efficient van der Waals energy calculations via a continuum mechanics approach

Recent developments in continuum mechanics (CM) [Tao \emph{et al}, PRL{\bf 103},086401] enable the calculation of density response functions from groundstate properties only. Using the direct Random Phase Approximation (dRPA) we develop this CM approach into a third-rung van der Waals energy functional, which we dub the CM-dRPA. The functional requires as input the groundstate Kohn-Sham potential $V^{\rm{KS}}(\vec{r})$, density $n^0(\vec{r})$ and a kinetic stress tensor ${\rm{T}}^0(\vec{r})$ defined via $T^0_{\mu\nu}=Re\sum_{i\rm{ occ}} \psi_{i,\mu}^*\psi_{i,\nu} - n^0_{,\mu,\nu}/4$ where $\psi_i$ is an orbital. We present efficient algorithmic schemes for its evaluation in bulk and molecular systems using the full eigen-solutions of the bare CM equation and a second, simpler evaluation to find the interacting eigenvalues. These eigen-solutions are then used to calculate the correlation energy via a simple summation. The CM-dRPA is \emph{significantly} faster than a full dRPA calculation in systems with many electrons. We then apply the CM-dRPA functional to metallic, slab-like 2D-homogeneous jellium systems and periodic solids, with good results for vdW dispersion. In the metallic case most efficient vdW functionals would fail qualitatively.