Session E31: Advances in Density Functional Theory III

Orbitals and orbital energies in DFT and TDDFT

The status and meaning of orbitals and orbital energies in the Kohn-Sham one-electron model of DFT has been controversial, in contrast to Hartree-Fock orbitals and orbital energies. We will argue the opposite: the exact Kohn-Sham orbitals of DFT are "better" than HF orbitals and their orbital energies are much closer to ionization energies than HF orbital energies are. This follows from the relation between the KS potential and the wavefunction, which can be cast in the form $v_s=v_{c,kin}+v_H+v_{xc}^{hole}+v^{resp}$ \footnote{M. A. Buijse, E. J. Baerends, J. G. Snijders, \textbf{Phys. Rev. A} 40 (1989) 4190}, where each term depends on the KS orbitals and the wavefunction (the one- or two-particle density matrices). The response potential \begin{equation} v^{resp}(r)=\sum_j^\infty \frac{|d_j(r)|^2}{\rho(r)}I_j - \sum_i^H \frac{|\psi_{s,i}(r)|^2}{\rho(r)}(-\epsilon_i) \end{equation} ($d_j$ is the Dyson orbital corresponding to ion state $\Psi_j^{N-1}$, $\psi_{s,i}$ is a Kohn-Sham orbital) enables the connection between ionization energies $I_i$ and orbital energies $\epsilon_i$ to be made \footnote{D. P. Chong, O. V. Gritsenko, E. J. Baerends, \textbf{J. Chem. Phys.} 116 (2002) 1760; O. V. Gritsenko, B. Bra\"ida, E. J. Baerends, \textbf{J. Chem. Phys.} 119 (2003) 1937}. For virtual orbitals and orbital energies similar statements can be made: the shapes and energies of the (exact) KS orbitals are much more realistic than those of the Hartree-Fock model or hybrid functionals \footnote{R. van Meer, O. V. Gritsenko and E. J. Baerends,\textbf{ J. Chem. Theor. Comp.} 10 (2014) 4432}. The HOMO-LUMO gap in molecules is very close to the optical gap, and very different from the fundamental gap. In solids the situation is very different, which is the well-known "KS gap problem". Again the response potential $v^{resp}$ (a good approximation to it) helps to solve this problem, affording a straigtforward correction method of the KS gap to the fundamental gap \footnote{O. Gritsenko, R. van Leeuwen, E. van Lenthe, E. J. Baerends, \textbf{Phys. Rev. A} 51 (1995) 1944; M. Kuisma, J. Ojanen, J. Enkovaara, T. T. Rantala, \textbf{Phys. Rev. B} 82 (2010) 115106}.   

Revealing Open Quantum Systems with Subsystem DFT

The traditional quantum chemical methods, wave function or density based, are designed to solve for a closed system, where the Hamiltonian contains all relevant interactions. The closed system is, however, not realistic, as in real life the system is embedded in an environment with which it interacts to some degree. Including the description of the environment at the full quantum mechanical level leads to the Open Quantum Systems (OQS) theory: the only theory which can describe non-Markovian dynamics between the system and the environment. By allowing the flow of information in both directions phenomena such as quantum entanglement, relevant for the design of quantum computers, become available. While most OQS theories rely on the density matrix to describe the system-bath interaction, time-dependent subsystem DFT[1,2] allows to approach the problem using the electron density. Through Dyson-like equations connecting the density-density response kernels of the OQS and its environment, the extent to which non-Markovian dynamics is present can be revealed. We illustrate this for the process of excitation energy transfer in coupled chromophores embedded in explicit solvent. [1] M. Pavanello, J. Chem. Phys. 138, 204118 (2013). [2] A. Krishtal et al. J. Chem Phys. 142, 154116 (2015).   

An Open Source Embedding Code for the Condensed Phase

Work from our group [1,2] as well as others [3,4] has shown that for many systems such as molecular aggregates, liquids, and complex layered materials, subsystem Density-Functional Theory (DFT) is capable of immensely reducing the computational cost  while providing a better and more intuitive insight into the underlying physics. We developed a massively parallel implementation of Subsystem DFT for the condensed phase [1] into the open-source Quantum ESPRESSO software package. In this talk, we will discuss how we: (1) implemented such a flexible parallel framework aiming at the optimal load balancing; (2) simplified the solution of the electronic structure problem by allowing a fragment specific sampling of the first Brillouin Zone [2]; (3) achieve enormous speedups by solving the electronic structure of each fragment in a unit cell smaller than the supersystem simulation cell, effectively introducing a fragment specific basis set, with no deterioration of the fully periodic simulation. As of March 14, 2016, the code has been released and is available to the public. [1] A. Genova et al., JCP 2014, 141, 174101 [2] A. Genova et al., JPCM 2015, Accepted [3] S. Luber, JCP 2014, 141, 234110 [4] C. Jacob, et al., WIREs 2014, 4, 325   

Scaling properties of the kinetic energy density of atoms -- towards an orbital-free meta-GGA.

The scaling properties of atoms, combining periodicity with gradual increase in density, make a fruitful probe of relationships in density functional theory, and have driven advances in understanding the exchange and correlation energy. Although focus is normally upon the properties of integrated energies, insights can be generated from studying energy density functions as well. We visualize the behavior of the positive-definite kinetic energy density (KED) in closed-shell atoms, in comparison to invariant quantities based upon the gradient and Laplacian of the density. The latter are potential variables for constructing orbital-free functionals for the KE and can be used for analyzing the electronic structure of atoms and molecules. We notice a striking fit of the KED within the core of any atom to a gradient expansion model using both the gradient and the Laplacian, but one different from that derived from first principles for a slowly-varying electron gas. Correlated with this feature, we notice unexpected structure to the KED near the nucleus that cannot be explained simply by the von Weizsacker model, as is often presumed. These unexpected features provide potential insights for developing better orbital-free meta-GGA models for the kinetic energy.   

On the transferability of a parametrized kinetic functional for orbital-free density functional theory calculations

Because of issues with accuracy and transferability of existing orbital-free (OF) density functionals, OF functional development remains an active research area. Due to numerical difficulties, all-electron self-consistent assessment of OF functionals is limited. Using the projector augmented wave method we compute OFDFT all-electron values \footnote{Lehtomaki \textit{et al}., JCP., 141, 234102 (2014).} and we evaluate the performance of a parametrized OF functional for atoms and molecules. We combine the parametrized Thomas-Fermi-Weizs{\" a}cker (TF-W) kinetic model $\lambda$ and $\gamma$ for the fractions of Weizs{\" a}cker and TF functionals, respectively, with LDA for atoms \footnote{Espinosa \textit{et al}., PCCP., (2015).}. We found that one-to-one relation between $\lambda$ and $\gamma$ values defines a region in parameter space that allows the atomic energies and eigenvalues to be approximated with a small average error with respect to the Kohn-Sham values. The optimum values is however different for every property and for every atom. Recently, these results have been combined to test parameter transferability from atoms to molecules \footnote{Karpenko \textit{et al}., in preparation.} and we expect will help for further systematic improvement of OF density functionals.   

Exact Expressions for Exchange-Correlation Potentials

The Baerends and Staroverov groups have devised various exact expressions for the exchange-correlation potential in terms of wave-function and Kohn--Sham ingredients. We show that all these expressions can be obtained as special cases of one general approach. One particular expression derived by us involves at most the two-electron reduced density matrix and is ideally suited for practical calculations of exchange-correlation potentials from many-electron wave functions, as demonstrated by numerical examples. Interesting identities emerging from our derivation are also presented and discussed.   

Stringent test for non-additive, non-interacting, kinetic energy functionals

Partition Density Functional Theory (PDFT) provides an ideal framework for testing and developing new approximations to the non-additive and non-interacting kinetic energy functional ($T_s^{nadd}[\{n_\alpha\}]$), understood as a functional of the set of fragment ground-state densities. We present our progress on both of these fronts: (1) Systematic comparison of the performance of various existing approximations to $T_s^{nadd}[\{n_\alpha\}]$; and (2) Development of new approximations. We find that a re-parametrization of the GGA enhancement factor employed for the construction of $T_s^{nadd}[\{n_\alpha\}]$ through the conjointness conjecture captures essential features of the functional derivatives of $T_s^{nadd}[\{n_\alpha\}]$. A physically-motivated two-orbital approximation for $T_s^{nadd}[\{n_\alpha\}]$ is shown to outperform most other approximations for the case of He$_2$, and an intriguing one-parameter formula makes this approximation accurate for all noble-gas diatomics.   

Partition Theory for Periodic and Semi-Infinite Systems

Standard approximations to the exchange-correlation (XC) functional of Kohn-Sham Density-Functional Theory are insufficiently accurate to describe charge transfer at metal-atom interfaces and other systems requiring proper treatment of fractional electron charges. The root of the problem is connected to the lack of derivative discontinuities in the approximate XC functionals at integer numbers of electrons. Partition Theory (PT) is a promising, formally exact method to correct this issue. We study the simplest model for an atom adsorbed at a metal surface: A one-dimensional step potential separated a fixed distance from an attractive well that admits only one bound state when isolated. The semi-infinite metal is populated with non-interacting electrons up to the Fermi energy. We derive the PT-equations for this problem and indicate how the associated partition potential can be calculated. PT is also a promising method for improving the computational scaling of other large and/or periodic systems. We study the partition potential for periodic 1-D chains of identical attractive wells and comment on the uniqueness of the partition potential when going from finite to periodic systems.   

Petascale orbital-free density functional theory enabled by small-box techniques

Orbital-free density functional theory (OFDFT) is a quantum-mechanics-based method that utilizes electron density as its sole variable. The main computational cost in OFDFT is use of the ubiquitous fast Fourier transform (FFT), which is mainly used to evaluate the kinetic energy density functional (KEDF) and electron-electron Coulomb interaction terms. We design and implement a small-box FFT (SBFFT) algorithm to overcome the parallelization limitations of traditional FFT algorithms. In addition, a real-space truncation of the non-local Wang-Teter KEDF kernel is proposed. The scalability of SBFFT is demonstrated by efficiently simulating one full optimization step (electron density, forces, and stresses) of 1,024,000 lithium (Li) atoms on up to 131,072 cores. Other tests include calculations of physical properties of different phases of bulk Li, geometry optimizations of nanocrystalline Li, and molecular dynamics simulations of liquid Li samples. All of the tests yield excellent accuracy compared to the original OFDFT calculations, suggesting that the OFDFT-SBFFT algorithm opens the door to first-principles simulations of materials containing millions of atoms.   

Degenerate Open Shell Density Perturbation Theory

The density perturbation theory (DPT) methodology we have developed applies the Hohenberg-Kohn theorem to perturbations in density functional theory. At each order, the energy is directly minimized with respect to the density at all lower orders. The difference between the perturbed and unperturbed densities is expanded in terms of a finite number of basis functions, and a single matrix inversion in this space reduces the complexity of the problem to that of non-interacting perturbation theory. For open-shell systems with symmetry, however, the situation becomes more complex. Typically, the perturbation will break the symmetry leading to a zeroth-order shift in the Kohn-Sham potential. Because the symmetry breaking is independent of the strength of the perturbation, the mapping from the initial to the perturbed KS potential is discontinuous and techniques from perturbation theory for noninteracting particles fail. We describe a rigorous formulation of DPT for use in systems that display an initial degeneracy, such as atoms and Fe$_\mathrm{55}$Cp*$_\mathrm{12}$ clusters and present initial calculations on these systems.