Session B39: Electronic Structure: Calculations I

Acceleration of Hartree-Fock Exchange Computations using Recursive Subspace Bisection

We use the recursive subspace bisection algorithm [1] to accelerate the computation of the Hartree-Fock exchange operator in electronic structure computations involving hybrid density functionals. This approach leads to a reduction of the computational cost of the exchange operator from $O(N^3 \log N)$ to $O(N^2 \log N)$ and allows for controlled accuracy through a threshold parameter. The subspace bisection method is extended to invariant subspaces including excited states. Applications to molecular dynamics simulations and computations of energy band gaps in large systems using the PBE0 hybrid functional will be presented. \\[4pt] [1] F. Gygi, Phys. Rev. Lett. 102, 166406 (2009).   

A Discontinuous Galerkin Framework for Electronic Structure Calculations

It is generally accepted that a good basis set for any calculation should possess a number of salient features, including systematic improvability, adaptive resolution of multiscale features, and fidelity in capturing the pertinent physics. Considering the progenitors of most modern electronic structure basis sets to be Gaussian-type orbitals or planewaves, descendants of these methods have inherited features that address either systematic improvability (planewaves) or adaptive resolution (Gaussians) separately, and use a variety of tricks to differentiate the core and valence physics. Discontinuous Galerkin methods provide a framework for defining adaptive local basis sets, that may be based on these canonical basis sets, that can be mixed and matched to simultaneously achieve all of these goals. Our group is presently developing a new electronic structure code to enable Density Functional and Hartree-Fock calculations within this framework, particularly in the context of all-electron formulations wherein the accurate resolution of both core and valence states is necessary. Numerous implementation details will be addressed, including the incorporation of hardware- and software-based acceleration, such as GPU-based parallelism, and fast electrostatics solvers.   

New method of optimizing the Jastrow factor for solids with the transcorrelated method

Transcorrelated (TC) method[1-5] is one of the promising theories for \textit{ab initio} electronic structure calculation of solids. It is one of the wave-function-based approaches which are considered to be advantageous for high-accuracy calculation. In the TC method, the total wave function is approximated as the Jastrow-Slater-type wave function, and the many-body Hamiltonian is similarity-transformed by the Jastrow factor. Then we solve an SCF equation and optimize one-body orbitals in the Slater determinant with relatively low computational cost. On the other hand, optimization of the Jastrow factor has been computationally much more demanding although it is indispensable to high-accuracy calculation. In this study, a new method of optimizing the Jastrow factor is developed by use of variance minimization of the total energy. It is demonstrated that, by truncating the basis-set expansion of the variance, the optimization is realized with low computational cost. [1] S. F. Boys and N. C. Handy, Proc. R. Soc. London Ser. A 309, 209 (1969). [2] S. Ten-no, Chem. Phys. Lett. 330, 169 (2000). [3] N. Umezawa and S. Tsuneyuki, J. Chem. Phys. 119, 10015 (2003). [4] R. Sakuma and S. Tsuneyuki, J. Phys. Soc. Jpn. 75, 103705 (2006). [5] H. Luo, J. Chem. Phys. 133, 154109 (2010).   

Adaptive local basis set for Kohn-Sham density functional theory in a discontinuous Galerkin framework

Uniform discretization of the Kohn-Sham Hamiltonian generally results in a large number of basis functions per atom in order to resolve the rapid oscillations of the Kohn-Sham orbitals around the nuclei even in the pseudopotential framework. Atomic orbitals and similar objects significantly reduces the number of basis functions, but these basis sets generally require fine tuning of the parameters in order to reach high accuracy. We present a novel discretization scheme that adaptively and systematically builds the rapid oscillations of the Kohn-Sham orbitals around the nuclei as well as environmental effects into the basis functions. The resulting basis functions are localized in the real space, and are discontinuous in the global domain. The continuous Kohn-Sham orbitals and the electron density are evaluated from the discontinuous basis functions using the discontinuous Galerkin (DG) framework. Our method is implemented in parallel and the current implementation is able to handle systems with at least thousands of atoms. Numerical examples indicate that our method can reach very high accuracy (less than $1$meV) with a very small number ($4\sim 40$) of basis functions per atom.   

Retrofit of the HSE density functional

The original parameterization of the Heyd-Scuseria-Ernzerhof (HSE) density functional was dependent on the choice of a hybrid fraction and a range-separation length for separating out a portion of the exchange energy to compute exactly. For backward compatibility with the PBE0 functional, the hybrid fraction was fixed to 0.25. Here, we examine the full hybrid fraction / separation length phase space. With respect to multiple error metrics, the phase space does not have a well-developed point of minimum error. Instead, there is a ``valley'' of functionals with increasing hybrid fraction and decreasing separation length of similarly good quality. This enables a reduction of the separation length without degrading the accuracy of the HSE06 parameterization, which in turn reduces the computational cost of evaluating the exchange energy.   

Projector Augmented Wave formulation of orbital-dependent exchange-correlation functionals

The use of orbital-dependent exchange-correlation functionals within electronic structure calculations has recently received renewed attention for improving the accuracy of the calculations, especially correcting self-interaction errors. Since the Projector Augmented Wave (PAW) method\footnote{ P. Bl\"{o}chl, {\em{Phys. Rev. B}} {\bf{50}}, 17953 (1994).} is an efficient pseudopotential-like scheme which ensures accurate evaluation of all multipole moments of direct and exchange Coulomb integrals, it is a natural choice for implementing orbital-dependent formalisms. Using Fock exchange as an example of an orbital-dependent functional, we developed the formulation and numerical implementation of the approximate optimized effective potential formalism of Kreiger, Li, and Iafrate (KLI)\footnote{ J. B. Krieger, Y. Li, and G. J. Iafrate {\em{Phys. Rev. A}} {\bf{45}}, 101 (1992).} within the PAW method, comparing results with the analogous Hartree-Fock treatment.\footnote{ Xiao Xu and N. A. W. Holzwarth, {\em{Phys. Rev. B}} {\bf{81}}, 245105 (2010); {\bf{84}}, 155113 (2011).} Test results are presented for ground state properties of two well-known materials -- diamond and LiF. This formalism can be extended to treat orbital-dependent functionals more generally.   

Solution of the Bethe-Salpeter equation without empty electronic states: Applications to solids, nanostructures and molecules

A method to solve the Bethe-Salpeter equation that avoids the explicit calculation of empty electronic states and the storage and inversion of dielectric matrices has been recently introduced [1-3]. This approach is suitable to compute the absorption spectra of large systems in a wide energy range and without relying on the Tamm-Dancoff approximation. We show the accuracy and scalability of this method by presenting calculations of absorption spectra of solids, molecules and nanostructures, including Si quantum dots and nanowires. In the case of nanowires, we discuss the influence of size and surface reconstruction on the optical properties.\\[4pt] [1] D. Rocca, D. Lu, and G. Galli, J. Chem. Phys. 133, 164109 (2010)\\[0pt] [2] D. Rocca, Y. Ping, R. Gebauer, and G. Galli, submitted to PRB \\[0pt] [3] Y. Ping, D. Rocca, D. Lu, and G. Galli, submitted to PRB   

Iterative diagonalization of non-Hermitian eigenproblems in time-dependent density-functional and many-body perturbation theory

We present a technique for the iterative diagonalization of random-phase approximation (RPA) matrices, which are encountered in the framework of time-dependent density-functional theory (TDDFT) and in the solution of the Bethe-Salpeter equation (BSE) [1]. The non-Hermitian character of these matrices does not permit a straightforward application of standard iterative techniques used, i.e., for the diagonalization of ground state Hamiltonians. We first introduce a new block variational principle for RPA matrices. We then develop an algorithm for the simultaneous calculation of multiple eigenvalues and eigenvectors, with convergence and stability properties similar to techniques used to iteratively diagonalize Hermitian matrices. The algorithm is validated by computing multiple low-lying excitation energies of molecules at both the TDDFT and BSE level.\\[4pt] [1] D. Rocca, Z. Bai, R.-C. Li, and G. Galli, submitted to J. Chem. Phys.   

New \emph{ab initio} approaches for calculating the microscopic electron-density response matrix

The electron-density response matrix is a key quantity for excitation calculations such as GW-BSE. Typically, the response matrix is obtained at the level of the random phase approximation (RPA), with the wavefunction from local density approximation (LDA) density functional theory. The expense of this approach grows quickly with the number of atoms, and its accuracy depends on both the accuracy of the LDA and the RPA. We investigate a series of approaches to overcome these difficulties, based on an eigenvalue decomposition of the molecular response matrix.   

Cluster expansion of the electron-density response function: GW+BSE with molecular environments

Molecular excitations in dielectric environments have drawn great interest because environmental manipulation provides the possibility to engineer photo-excitation processes. In exciton calculation the environments often are replaced by dielectric continuum media. These approaches have been successful for solvated molecules, but they lack molecular detail, and hence miss microscopic features. We present a new method to represent environments that allows a more accurate treatment of a wide range of systems by employing cluster expansions of environmental response functions. This initial work, at the GW+BSE level, presents results for shifts in excitation energies due to environments consisting of either individual molecules or conjugated polymers.   

Unified description of ground and excited states of finite systems: the self-consistent \textit{GW} approach

Fully self-consistent $GW$ calculations -- based on the iterative solution of the Dyson equation -- provide an approach for consistently describing ground and excited states on the same quantum mechanical level. Based on our implementation in the all-electron localized basis code FHI-aims [1], we show that for finite systems self-consistent $GW$ reaches the same final Green function regardless of the starting point. The results show that self-consistency systematically improves ionization energies and total energies of closed shell systems compared to perturbative $GW$ calculations ($G_0W_0$) based on Hartree-Fock or (semi)local density-functional theory. These improvements also translate to the electron density as demonstrated by a better description of the dipole moments of hetero-atomic dimers and the similarity with the coupled cluster singles doubles (CCSD) density. The starting-point independence of the self-consistent Green function facilitates a systematic and unbiased assessment of the performance of the $GW$ approximation for finite systems. It therefore constitutes an unambiguous reference for the future development of vertex corrections and beyond $GW$ schemes. [1] V. Blum \textit{et al.}, Comp. Phys. Comm. {\bf 180}, 2175 (2009).   

Stress formulation in the all-electron full-potential linearized augmented plane wave method

Stress formulation in the linearlized augmented plane wave (LAPW) method has been proposed in 2002 [1] as an extension of the force formulation in the LAPW method [2]. However, pressure calculations only for Al and Si were reported in Ref.[1] and even now stress calculations have not yet been fully established in the LAPW method. In order to make it possible to efficiently relax lattice shape and atomic positions simultaneously and to precisely evaluate the elastic constants in the LAPW method, we reformulate stress formula in the LAPW method with the Soler-Williams representation [3]. Validity of the formulation is tested by comparing the pressure obtained as the trace of stress tensor with that estimated from total energies for a wide variety of material systems. Results show that pressure is estimated within the accuracy of less than 0.1 GPa. Calculations of the shear elastic constant show that the shear components of the stress tensor are also precisely computed with the present formulation [4].\\[4pt] [1] T. Thonhauser {\it et al.}, Solid State Commun. {\bf 124}, 275 (2002).\\[0pt] [2] R. Yu {\it et al.}, Phys. Rev. B {\bf 43}, 6411 (1991).\\[0pt] [3] J. M. Soler and A. R. Williams, Phys. Rev. B {\bf 40}, 1560 (1989).\\[0pt] [4] N. Nagasako and T. Oguchi, J. Phys. Soc. Jpn. {\bf 80}, 024701 (2011).   

Real Space DFT by Locally Optimal Block Preconditioned Conjugate Gradient Method

Real space approaches solve the Kohn-Sham (KS) DFT problem as a system of partial differential equations (PDE) in real space numerical grids. In such techniques, the Hamiltonian matrix is typically much larger but sparser than the matrix arising in state-of-the-art DFT codes which are often based on directly minimizing the total energy functional. Evidence of good performance of real space methods - by Chebyshev filtered subspace iteration (CFSI) - was reported by Zhou, Saad, Tiago and Chelikowsky [1]. We found that the performance of the locally optimal block preconditioned conjugate gradient method (LOGPCG) introduced by Knyazev [2], when used in conjunction with CFSI, generally exceeds that of CFSI for solving the KS equations. We will present our implementation of the LOGPCG based real space electronic structure calculator. \\[4pt] [1] Y. Zhou, Y. Saad, M. L. Tiago, and J. R. Chelikowsky, ``Self-consistent-field calculations using Chebyshev-filtered subspace iteration,'' J. Comput. Phys., vol. 219,pp. 172-184, November 2006. \\[0pt] [2] A. V. Knyazev, ``Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method,'' SIAM J. Sci. Comput, vol. 23, pp. 517-541, 2001.   

Bridging density-functional and many-body perturbation theory: orbital-density dependence in electronic-structure functionals

Energy functionals which depend explicitly on the orbital densities (ODD), instead of the total charge density, appear when applying self-interaction corrections to density-functional theory. In these cases (e.g. the Perdew-Zunger [1] and the non-Koopmans [2] approaches) the total energy loses invariance under unitary rotations of the orbitals, and the minimization of the functionals leads to orbital-dependent Hamiltonians. We show that it is possible to identify the orbital-dependency of densities and potentials with an effective and discretized frequency-dependency, in close analogy to the quasi-particle approximation of frequency-dependent self-energies and naturally oriented to interpret electronic spectroscopies [3]. Some of the existing ODD functionals are analyzed from this new perspective. Numerical results for the electronic structure of gas-phase molecules (within the Koopmans-corrected class of functionals) are computed and found in excellent agreement with photoemission (UPS) data. [1] J.-P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981). [2] I. Dabo, A. Ferretti, N. Poilvert, Y. Li, N. Marzari, M. Cococcioni, Phys. Rev. B 82, 115121 (2010). [3] M. Gatti, V. Olevano, L. Reining, I.-V. Tokatly, Phys. Rev. Lett. 99, 057401 (2007).   

A new type of pseudopotentials: effective atomic pseudopotentials

We derive a new type of pseudopotentials from conventional norm-conserving pseudopotentials for the treatment of a large number of atoms. The pseudopotentials are not aimed at the calculation of the total enegy, but of band edge states relevant for optical processes. We describe the pseudopotential construction and benchmark its quality and transferability by comparison to standard DFT calculations.