Session Y24: Density Functional Theory II

Exact Exchange calculations for periodic systems: a real space approach

We present a real-space method for exact-exchange Kohn-Sham calculations of periodic systems. The method is based on self-consistent solutions of the optimized effective potential (OEP) equation on a three-dimensional non-orthogonal grid, using norm conserving pseudopotentials. These solutions can be either exact, using the S-iteration approach, or approximate, using the Krieger, Li, and Iafrate (KLI) approach. We demonstrate, using a variety of systems, the importance of singularity corrections and use of appropriate pseudopotentials.   

Dissociation of diatomic molecules and the exact-exchange Kohn-Sham potential: the case of LiF

The incorrect fractional-charge dissociation of stretched diatomic molecules, predicted by semi-local exchange-correlation functionals, is revisited. This difficulty can be overcome with asymptotically correct non-local potential operators, but should also be absent in exact Kohn-Sham theory, where the potential is local. Here, we show, for the illustrative case of the LiF dimer, that the exact-exchange local Kohn-Sham potential, constructed within the Krieger, Li, and Iafrate (KLI) approximation, can lead to binding energy and charge dissociation curves that are qualitatively correct. This correct behavior is traced back to a characteristic ``step'' structure in the local exchange potential and its relation to the Kohn-Sham eigenvalues is analyzed.   

A Projector Augmented Wave Treatment of Fock Exchange in Hartree-Fock and Optimized Effective Potential Calculations

The use of orbital-dependent exchange-correlation functionals within electronic structure calculations has recently received renewed attention as a means of improving accuracy. In particular, the Fock exchange functional exactly cancels the electron self-interaction error which can be particularly significant in transition metals and other materials with localized orbitals. Since the Projector Augmented Wave (PAW) formalism\footnote{P. E. Bl\"{o}chl, {\em{Phys. Rev. B}} {\bf{50}}, 17953 (1994).} accurately evaluates the interaction integrals including all multiple moments, it is a natural choice for representing the Fock exchange functional within an efficient pseudopotential-like scheme. We have adapted the PAW formalism for use both in Hartree-Fock (HF) theory and in the KLI approximation to Optimized Effective Potential theory.\footnote{J. B. Krieger, Y. Li and G. J. Iafrate {\em{Phys. Rev. A}} {\bf{45}} 101 (1992).} We show that the effects of core electrons are significant and can be accurately treated within a frozen core orbital approximation.\footnote{Xiao Xu and N. A. W. Holzwarth, {\em{Phys. Rev. B}} {\bf{81}} 245105 (2010).} PAW-HF and PAW-KLI basis, projector, and pseudopotential functions are presented for several elements throughout the periodic table.   

Localized resolution of identity for efficient Hartree-Fock exchange, based on numeric atom-centered orbitals

Methods based on an exact exchange operator (EX) are increasingly popular, but are still restricted to analytical basis functions (e.\,g. Gaussians) for medium system sizes. We here introduce a localized resolution-of-identity approach for the two-electron Coulomb operator, based on expanding single-particle basis function products separately into auxiliary atom-centered basis sets that are restricted to two centers. Our approach produces accurate results for all-electron EX, can be applied both to analytical and numeric basis functions, requires only ${\mathcal{O}}(N^2)$ intermediate storage and retains a path towards ${\mathcal{O}}(N)$ EX for large systems. We demonstrate a total-energy accuracy of $<1$\,meV/atom for systems including Alanine chains and the S22 benchmark molecule set [1], using the numeric atom-centered orbital based all-electron electronic structure code FHI-aims [2].\\[4pt] [1] P. Jure\v{c}ka \emph{et al.}, Phys. Chem. Chem. Phys.~\textbf{8}, 1985 (2006).\\[0pt] [2] V. Blum \emph{et al.}, Comput. Phys. Comm.~\textbf{180}, 2175 (2009).   

Analytic Treatment of the Pair Density 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 by analytic means. The method is illustrated with numerical results for the potential in the case of one and three dimensional systems and is compared to the potentials obtained from the Hartree term.   

Wave Function Functionals for the Density

In recent work we have developed\footnote{Pan, Slamet, and Sahni, Phys. Rev. A \textbf{81}, 042524 (2010).} a constrained-search variational method for the construction of wave functions that are functionals of a function $\chi: \Psi = \Psi [\chi]$. These wave function functionals are \emph{simultaneously} normalized, reproduce the \emph{exact} expectation of either single- or two-particle operators, and lead to rigorous upper bounds to the energy. In this paper we extend this method to the construction of wave function functionals $\Psi[\chi]$ that are simultaneously normalized, reproduce the density \emph{exactly}, and lead to rigorous upper bounds to the energy. We apply the method to the ground state of the He atom to obtain wave function functionals that reproduce the density of an accurate correlated wave function. The wave function functionals as expected give rise to the exact expectation of non-differential single particle operators, and lead to accurate two-particle expectations and highly accurate energies.   

Koopmans' condition for density-functional theory

In approximate Kohn-Sham density-functional theory, self-interaction manifests itself as the dependence of the energy of an orbital on its fractional occupation. Here, we first examine self-interaction in terms of the discrepancy between total and partial electron removal energies, and then highlight the importance of imposing the generalized Koopmans' condition to resolve this discrepancy. In the process, we derive a correction to approximate functionals that, in the frozen-orbital approximation, eliminates the unphysical occupation dependence of orbital energies up to the third order in the single-particle densities. This non-Koopmans correction brings physical meaning to single-particle energies; when applied to common local or semilocal density functionals it provides results that are in excellent agreement with experimental data while providing an explicit total energy functional that preserves or improves on the description of established structural properties.   

Density Functional Theory for Open Systems

We provide a formal proof of the convexity relation, $2E[N] \leq E[N - 1] + E[N + 1]$, characterizing the total energy of interacting $N$-electron systems under the action of a given external potential, hitherto assumed to be true only on experimental grounds. This is used to prove the inequality, $I(N) - A(N) \geq 0$, where $I(N)$ and $A(N)$ are, respectively, the ionization potential and electron affinity of a $N$-electron system.   

Volume effects in band gap predictions for solids

The \textit{ab initio} prediction of band gaps for solids is important for fundamental and practical reasons. Many approaches exist to remedy the ``band gap problem'' in Density Functional Theory (DFT) in which band gaps are severely underestimated. We recently proposed the $\Delta$-sol method [1], an adaptation of the $\Delta$SCF method towards solids, in which the fundamental gap is evaluated using total energies from DFT. Using $\Delta$-sol, we obtained band gaps for over 100 crystalline semiconductors at accuracies similar to those of hybrid functionals such as HSE, but at significantly smaller computational costs. However, the accuracy of band gap predictions from first principles remains dependent on accurate determination of lattice parameters and cell volumes. In this talk, we discuss the effects of the accuracy in lattice parameters on predicted band gaps. We present results on the accuracy of cell volumes determined using several exchange-correlation functionals: LDA, PBE, HSE and AM05, and compare the dependence of Kohn-Sham gaps and band gaps predicted using $\Delta$-sol on cell volumes. Finally we discuss optimal approaches for predicting band gaps for compounds with unknown lattice parameters. \\[4pt] [1] M. K. Y. Chan and G. Ceder, Phys. Rev. Lett. 105, 196403 (2010)   

Fundamental gaps in finite systems from the eigenvalues of generalized kohn-sham method

We present a broadly-applicable, physically-motivated first-principles approach to determining the fundamental gap of finite systems. The approach is based on using a range-separated hybrid functional within the generalized Kohn-Sham approach to density functional theory. Its key element is the choice of a range-separation parameter such that Koopmans' theorem for both neutral and anion is obeyed as closely as possible. We demonstrated the validity, accuracy, and advantages of this approach on first, second and third row atoms, the oligoacene family of molecules, and a set of hydrogen-passivated silicon nanocrystals. This extends the quantitative usage of density functional theory to an area long believed to be outside its reach.   

Self-optimizing Kohn-Sham hybrid functional

Recent work using range-separated hybrid functionals has confirmed the importance of including long-range exchange in treatments of phenomena such as charge transfer reactions. Using a self-optimizing [1,2] form of the BNL [3] functional, we present results for structural, electronic, and thermochemical properties of a large set of molecules (including the G2 and G3 test sets). The success of this approach, as well as its ability to describe reaction barriers, will be discussed. \\[4pt] [1] T. Stein, L. Kronik, and R. Baer, JACS, 131 (8), 2818, 2009 \newline [2] T. Stein, H. Eisenberg, L. Kronik, and R. Baer, ``Fundamental gaps of finite systems from the eigenvalues of a generalized Kohn-Sham method'', Phys. Rev. Lett., in press. \newline [3] E. Livshits and R. Baer, PCCP, 9, 2932 , 2007   

TDDFT and qualitative properties of excited states: three illustrative applications using DMol$^3$

Three applications of DMol$^3$ TDDFT [1] are presented to show possible new frontiers in each case. First, excitations involving multiplet structure for the example of the Ti$^{4+}$ ion are discussed, showing that atomic multiplet splitting is fully exhibited within TDDFT. This approach to multiplets exhibits notable similarities and also notable differences with a first principles based Condon-Shortley-Cowan multiplet theory. Second, UV-VIS spectra of benzene and derivative molecules are discussed by comparing experimental log plots of molar extinction with a TDDFT results completed by the Gaussian envelope model for the vibrational progression. The envelope model provides a natural scale for comparing TDDFT excitations with measured absorption spectra. In the third example, excited states of (Fe(CN)$_5$NO)$^{-2}$ are studied along the reaction coordinate connecting the long lived metastable states that can be produced by optical excitation.\\[4pt] [1] B. Delley, J. Phys. Cond. Mat. 22, 384208, 2010.   

Exact Time-Dependent Kohn-Sham Potential for an Interacting Few-Body System

Time-dependent density functional theory enables practical simulations of the dynamic many-electron systems, but one of the biggest obstacles to reliable application is the quality of the approximate potential. It is often difficult to determine whether ever-more sophisticated approximations properly include new physics, as there exist few benchmark exact potentials. Towards this ends, we have developed and tested a scheme to extract the exact (non-adiabatic) time-dependent Kohn-Sham potential for few body systems. We will present some examples on 1D model systems. The approach is general and can be used to back engineer high-level quantum mechanical simulations to gain insight into TDDFT on a broad scale. Sandia National Laboratories is a multi-program laboratory operated by Sandia Corporation, a wholly owned subsidiary of the Lockheed Martin company, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

TDDFT approach to study nonlinear excitonic effects in Four-Wave Mixing processes

We develop a time-dependent density-functional theory (TDDFT) formalism to study nonlinear effects in the processes in Four-Wave Mixing experiments. Namely, we generalize our recently proposed approach to calculate excitonic and biexcitonic eigen-energies within TDDFT on the dynamical case, which includes nonlinear effects, the like exciton-exciton interaction. In particular, we derive the TDDFT version of the nonlinear time-dependent equation for excitonic polarization with terms which correspond to exciton-exciton correlations through the memory functions. We have obtained the formula which relates the memory functions with the TDDFT exchange- correlation (XC) kernel. To test the formalism, we calculate the 2D Fourier spectra of a GaAs multi-quantum well system and compare them with experimenatal results in the case of several XC kernels. In addition, we compare the results with the ones obtained within a many-body method for non-linear effects in semiconductors. It is shown that the results obtained within the TDDFT approach may reproduce semi-quantively the 2D Fourier spectra, including the nonlinear effects, in the case of several phenomenological potentials.   

Long-range corrected time-dependent density functional theory with spin-orbit couplings

Relativistic time-dependent density functional theory (TDDFT) is a powerful tool to include both of relativistic and correlation effects with low computational cost. However, TDDFT with conventional exchange functionals have severe problems in e.g. the reproducibility of charge transfer (CT) and Rydberg excitation energies and oscillator strengths. These problems are due to the lack of long-range exchange interactions in conventional exchange functionals. We have proposed long-range corrected (LC) DFT and have overcome these problems. Especially, LC-TDDFT succeeds in describing CT excitations with remarkable accuracy. CT excitations often play a major role in spin-forbidden transitions, because the spin-orbit couplings are significant for excitations inducing the changes in electron distributions. In this study, LC-DFT has been applied to a spin-orbit TDDFT to describe spin-forbidden transitions appropriately by TDDFT. Our results have demonstrated that LC-DFT accurately reproduces the splitting of ionization energies of heavy atoms and spin-forbidden excitation energies for which electrons are moved to widely-distributed orbitals.