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Towards Linear-Scaling Hybrid DFT

Kimihiko Hirao

Department of Applied Chemistry, School of Engineering, University of Tokyo, Tokyo Japan 113-8656

DFT is one of the most widely used methods in computational chemistry due to the extremely favorable balance of efficiency versus accuracy in estimating electron correlation. Recently, there has been considerable interest toward the development and assessment of new hybrid functionals based on the long-range correction (LC) scheme [1-6]. In the LC scheme, the exchange functional is partitioned with respect to the interelectronic separation into long-range and short-range parts using a standard error function. Only the short-range part is retained, while the long-range part is replaced with an exact orbital expression using Hartree-Fock (HF) exchange integrals. The LC approach was shown to clearly solve many of the problems which conventional DFT confronted: for example, the underestimation of Rydberg excitation energies and corresponding oscillator strengths, the poor reproduction of charge-transfer excitations in time dependent DFT calculations, and the overestimation of linear and nonlinear polarizabilities of long-chain molecules in coupled-perturbed DFT calculations. Moreover, the LC approach successfully provided a good description of van der Waals interactions [7,8] as well as accurate reaction enthalpies and barrier heights. The practical extension of electronic structure theory to the domain of large-scale molecular systems and nanomaterials using linear-scaling techniques is a promising and active area of research. Over the past decade, approaches have been developed which remove all of the major bottlenecks in a SCF calculation of the ground-state energy using HF or DFT. Recently we have developed a new implementation of a novel approach to treating the Coulomb problem, which is rigorously linear scaling. The Gaussian and finite element Coulomb (GFC) method [9] evaluates the Coulomb potential by direct solution of the Poisson equation. Unlike traditional real-space methods which discretize the Poisson equation on a grid using finite-difference, finite-element, or wavelet basis functions, in our approach we use a mixture of finite-element and Gaussian functions to describe the Coulomb potential. This is the key to the efficiency of the GFC since the Gaussian functions allow for an extremely compact representation of the potential near the nuclei. A drawback of the GFC method is the cost of evaluating the boundary condition for the Poisson equation, which does not scale linearly and becomes a significant cost for systems with several thousands of basis functions. Very recently we have developed a new, efficient O(N) implementation which uses the fast multipole method to evaluate the boundary potential [10]. Although hybrid DFT improves the accuracy, it makes the calculation more expensive since the fast algorithms for Coulomb integrals cannot be employed for HF exchange integrals. The hybrid DFT calculation becomes more time-consuming than the pure DFT for the large molecular system. Recently we have proposed the dual-level approach to DFT [11]. The scheme is based on the low sensitivity of the electron density to the choice of the functional and the basis set. The total electron density in the ground state can be well represented in terms of the density evaluated using the low-quality basis set and the low-cost exchange-correlation functional. The error is remedied by the second-order perturbation theory. The dual-level DFT works quite well and the large reduction of the computer resources can be achieved. Hybrid functional including LC can now be applied to very large systems.

References

[1] H. Iikura, T. Tsuneda, T. Yanai, and K. Hirao, J. Chem. Phys. 115, 3540 (2001).

[2] Y. Tawada, T. Tsuneda, S. Yanagisawa, T. Yanai, and K. Hirao, J. Chem. Phys. 120, 8425 (2004).

[3] M. Kamiya, H. Sekino, T. Tsuneda, and K. Hirao, J. Chem. Phys. 122, 234111 (2005).

[4] J.-W. Song, T. Hirosawa, T. Tsuneda, and K. Hirao, J. Chem. Phys. 126, 154105 (2007).

[5] J-W Song, S. Tokura, and T. Sato, M. A. Watson and K. Hirao, J.Chem.Phys., 127, 154109 (2007).

[6] J-W Song, M. A. Watson, H. Sekino, and K. Hirao, J.Chem.Phys., in press.

[7] T.Sato, T.Tsuneda and K.Hirao, J.Chem.Phys., 123, 104307 (2005).

[8] T. Sato, T. Tsuneda, and K. Hirao, J. Chem. Phys. 126, 234114 (2007).

[9] Y.Kurashige, T. Nakajima and K.Hirao, J.Chem.Phys., 126, 144106 (2007).

[10] M.Watson, Y.Kurashige, T. Nakajima and K.Hirao, J.Chem.Phys.,128, 054105 (2008).

[11] T.Nakajima and K.Hirao, J.Chem.Phys., 124, 184108 (2006).

A density functional theory for symmetric radical cations from bonding to dissociation

Roi Baer

Institute of Chemistry and the Fritz Haber Center for Molecular Dynamics, the Hebrew University of Jerusalem, Jerusalem 91904 Israel

It is known for quite some time that approximate density functional (ADF) theories fail disastrously when describing the dissociative symmetric radical cations R2+. Considering this dissociation limit, previous work has shown that Hartree-Fock (HF) theory favors the R+1---R0 charge distribution while DF approximations favor the R+0.5- R+0.5. Yet, general quantum mechanical principles indicate that both these (as well as all intermediate) average charge distributions are asymptotically energy degenerate. Thus HF and ADF theories mistakenly break the symmetry but in a contradicting way. In this letter we show how to construct system-dependent long-range corrected (LC) density functionals that can successfully treat this class of molecules, avoiding the spurious symmetry breaking. Examples and comparisons to experimental data is given for R=H, He and Ne and it is shown that the new LC theory improves considerably the theoretical description of the R2+ bond properties, the long range form of the asymptotic potential curve as well as the atomic polarizability. The broader impact of this finding is discussed as well and it is argued that the widespread semi-empirical approach which advocates treating the LC parameter as system-independent is in fact inappropriate under general circumstances. Finally, we discuss how to choose the LC parameter for any given system, including bulk systems where the LC parameter is related to the dielectric constant of the solid, enabling improved determination of the band gap.

References

Ester Livshits and Roi Baer, Submitted (2008)

Helen Eisenberg and Roi Baer, in preparation (2008)

SIC with RSH?

Andreas Savin

Laboratoire de Chimie Theorique, CNRS and UPMC, Paris.

Treating long-range exactly is known to improve errors due to self-interaction. Analyzing this for the hydrogen atom shows that this is in part due to a physically correct description of the long range, in the tail of the density, but still relies on error compensation in the inner region of the atom.

Local range-separated hybrids

Gustavo E. Scuseria

Rice University, USA.

I will discuss the work done by my student Aliaksandr Krukau in collaboration with John Perdew and Andreas Savin. Tom Henderson and Ben Janesko have also made valuable contributions to this project. We are developing range-separated hybrids (for exchange) with a screening parameter w that is a function of space w=w(r), ie, the value of w can be different at each and every point in space. Think of this as an approximation to the microscopic dielectric function that gives the "right answer for the right reason "within a Green's Function approach. Our local-range-separation (LRS) or locally-range-separated hybrid is a frequency independent version of this microscopic dielectric function. The implementation of such a beast is non-trivial and require some key approximations. While the idea of a LRS hybrid is something that Andreas and I have talked about for a long time, we needed John to come up with some crucial ingredient to make this thing realizable. Let me say that simple forms of w(r) based on dimensional analysis yield very promising results. I hope to have lots of data to show you in a few weeks by the time we all meet in Paris. For the details of how one goes around implementing this LRS, you will need to come to my talk!

Local admixtures of short- and long-range exact exchange

Benjamin G. Janesko

Scuseria research group, Rice University

An important limitation of range-separated hybrid density functionals is the need to choose whether or not to use long-range exact exchange. Long-range exact exchange (and wavefunction-based correlation) is important for atomic and molecular properties, but is computationally intractable in solids. This will be particularly problematic for calculations on inhomogeneous systems such as reactions at metal surfaces. I present work towards a possible solution of this problem: range-separated hybrids with fixed range-separation parameters w, and a position-dependent fraction of exact exchange in each range. Such functionals complement the position-dependent w discussed by Prof. Scuseria. Three main obstacles to such functionals are evaluating the range-separated exact exchange energy density, self-consistent implementation, and the choice of local hybrid mixing function in each range. The first obstacle can be met by the resolution-of-the-identity treatment of Della Sala and Gorling. I'll discuss progress on the other two obstacles.

A multi-range-separated hybrid functional

Tom Henderson

Rice University

Late last year, Gustavo Scuseria, Artur Izmaylov, Andreas Savin and I proposed a three-range hybrid functional which uses exact exchange only for intermediate electronic separations, the idea being to combine the benefits of long-range and short-range hybrids in one functional. In our paper, we discussed results for a few simple properties. I will review this functional and these results before moving on to discuss new applications, focusing on those for which long-range exact exchange is thought to be essential.

How local range-separation can satisfy many exact constraints

John P. Perdew and Adrienn Ruzsinszky

Tulane University, New Orleans, USA.

Conical Intersections in TDDFT

Mark E. Casida

Laboratoire de Chimie Théorique, Département de Chimie Molécularie (DCM, UMR CNRS/UJF 5250), Institut de Chimie Moléculaire de Grenoble (ICMG, FR-2607), Université Joseph Fourier (Grenoble I), 301 rue de la Chimie, BP 53, F-38041 Grenoble Cedex 9, FRANCE, mark.casida@ujf-grenoble.fr, http://dcm.ujf-grenoble.fr/PERSONNEL/CT/casida/

Over 20 years ago, Runge and Gross proved that the external potential is determined up to an additive spatially-constant function of time by the time-dependent charge density and the initial wave function (which is itself a functional of thedensity if the system starts in its ground stationary state). [RG84] About a decade ago, the linear response form of time-dependent density-functional theory (LR-TDDFT) was introduced into quantum chemistry as a formally rigorous way to obtain information about excited states [C95] and, with few exceptions, has now become the dominant single-determinant method for treating excited states in medium- and large-sized molecules. This has led to obvious hopes that LR-TDDFT could also be used to model photochemical reactions. [C01] However it became clear in the 1990s that many (if not most)photochemical reactions pass through conical intersections (CXs) and many of these have biradicloid character which is normally not easily described by DFT. In fact, Levine et al. have argued that (S0,S1) CXs cannot occur in TDDFT in the adiabatic approximation. Their reasoning will be reviewed and the results of our investigation of the ability of TDDFT to describe a critical (S0,S1) CX in oxirane will be presented. [CDI+07,TTR+08] Important issues will be raised concerning holes below the Fermi level, how to treat two-electron excitations explicitly, and where a multiconfigurational TDDFT could be useful.

References

[RG84] E. Runge and E.K.U. Gross, Phys. Rev. Lett. 52, 997 (1984)."Density functional theory for time-dependent systems"

[C95] M.E. Casida in Recent Advances in Density Functional Methods, Part I, D.P. Chong, editor (World Scientific:Singapore, 1995), p. 155."Time-dependent density-functional response theory for molecules"

[C01] M.E. Casida, in Accurate Description of Low-Lying Molecular States and Potential Energy Surfaces, ACS Symposium Series 828, edited by Mark R. Hoffmann and Kenneth G. Dyall (ACS Press: Washington, D.C., 2002), ISBN 0-8412-3792-1, (Proceedings of ACS Symposium, San Diego, Calif., 2001), pp. 199-220."Jacob's ladder for time-dependent density-functional theory: Some rungs on the way to photchemical heaven"

[LKQM06] B.G. Levine, C. Ko, J. Quenneville, and T.J. Martinez, Molec. Phys. 104, 1039 (2006)."Conical intersections and double excitations in time-dependent density functional theory"

[CDI+07] F. Cordova, L.J. Doriol, A. Ipatov, M.E. Casida, C. Filippi, and A. Vela, J. Chem. Phys. 127, 164111 (2007)."Troubleshooting time-dependent density-functional theory for photochemical applications"

[TTR+08] E. Tapavicza, I. Tavernelli, U. Röthlisberger, C. Filippi, and M.E. Casida, in preparation."Mixed time-dependent density-functional theory/classical trajectory surface hopping study of oxirane photochemistry"

Further documents

M. E. Casida, TDDFT for excited states, in Computational Methods in Catalysis and Materials Science, part of the IDECAT Course book series, Philippe Sautet and Rutger A. van Santen, editors (Wiley-VCH: 2008)

Range-separation in exponential orbital translation: exchange integrals

Philip E. Hoggan

LASMEA UMR 6602 CNRS, Clermont University, 63177 AUBIERE, FRANCE

Range separation in the sense of Ewald's expression of the Coulomb operator may be applied to exchange integrals over exponential type orbitals (ETOs)[1]. This talk concentrates on integrands which are not analytic closed forms. This is generally the case for exchange integrals involving at least one displaced center. The ETOs must then be translated using expressions derived from the Gegenbauer addition theorem [2]. These lead to the definition of so-called Barnett-Coulson-Loewdin Functions (BCLFs) [3]. Naturally, two ranges are involved (in the expansion), according to whether the electron position variable is greater than or less than the translation parameter 'a' (usually a bond distance, fixed by the molecular geometry). Two problems will be discussed: first, how the BCLFs can be used with Ewald range separation (extension of [1]) and second, how are BCLFs related to single range translation procedures [4].

References

[1] J. Angyan et al, J. Phys. A. 39 (2006) 8613.

[2] G. N. Watson, A treatise on the Theory of Bessel Functions, CUP, Cambridge, UK, 1944 (see p366).

[3] M.P. Barnett and Coulson. The evaluation of integrals occuring in the theory of molecular structures. Part I and II. Philos. Trans; R. Soc. London, Ser. A, 243 (1951) 221.

[4] BCLFs. P. E. Hoggan and D. Pinchon, Working note. 10 may 2007.

Response property evaluation with the Range Separation Hybrid DFT scheme

Hideo Sekino

Toyohashi University of Technology, Japan.

Short- and long-range exchange-correlation functionals: asymptotic properties in confined and extended systems

Paola Gori-Giorgi

Laboratoire Chimie Theorique, CNRS and UPMC, Paris.

I will review the properties of short- and long-range exchange-correlation functionals in two limiting cases: when the long-range interaction is approaching the full Coulomb interaction and in the opposite limit, when the short-range interaction becomes the full Coulomb repulsion. The differences between confined and extended systems will be also highlighted. Finally, a short-range local-spin-density functional for correlation based on extensive Quantum Monte Carlo results will be also briefly reviewed. The presentation is based on the work reported in the following papers: PRB 73, 155111 (2006) PRA 73, 032506 (2006) See also review on LR-SR PRA 70, 062505 (2004)

The exact corrrelation energy per particle and Xalpha

Andreas Savin

Laboratoire de Chimie Theorique, CNRS and UPMC, Paris.

An accurate correlation energy per particle can be obtained by performing configuration interaction calculations along adiabatic connections. It is shown for the Be atom, that the values obtained show a similitude with those used in the Xalpha method. However, as fitting showed for Xalpha that the optimal alpha factor in Xalpha is atom-dependent, it is noticed that the alpha factor also feels the influence of the shell.

References

[1] F. Colonna, D. Maynau, A. Savin, Phys. Rev. A. 68, 12505 (2003).

[2] J.Toulouse, F.Colonna, A. Savin, Mol.Phys.103, 2725 (2005).

Explicit long-range correlation in density-functional calculations

Hermann Stoll

Institut fuer Theoretische Chemie, Universitaet Stuttgart

Our current implementation of the coupling between short-range LDA, PBE, and TPSS type functionals with wavefunction-based long-range MP2, (MR)CISD, and CCSD(T) methods [1,2,3] (and local variants thereof [4]) is described in detail. Applications obtained so far for a variety of bonding situations in closed- and open-shell systems (vdW interaction, H-bonds, strong chemical bonds) is discussed.

References

[1] E. Goll, H.-J. Werner, and H. Stoll: A Short-Range Gradient-Corrected Density Functional in Long-Range Coupled-Cluster Calculations for Rare-Gas Dimers, Phys. Chem. Chem. Phys. 7 (2005) 3917

[2] E. Goll, H.-J. Werner, H. Stoll, T. Leininger, P. Gori-Giorgi and A. Savin: A Short-Range Gradient-Corrected Spin Density Functional in Combination with Long-Range Coupled-Cluster Methods: Application to Alkali-Metal Rare-Gas Dimers, Chem. Phys. 329 (2006) 276

[3] E. Goll, H. Stoll, C. Thierfelder and P. Schwerdtfeger: Improved Dipole Moments by Combining Short-Range Gradient-Corrected Density-Functional Theory with Long-Range Wavefunction Methods, Phys. Rev. A 76 (2007) 032507

[4] E. Goll, H.-J. Werner and H. Stoll: Short-Range Density Functionals in Combination with Local Long-Range Ab-Initio Methods: Application to Non-Bonded Complexes, Chem. Phys. 346 (2008) 257

On efficient treatment of both static and dynamic correlation effects with MC-srDFT

Hans Jørgen Aa. Jensen

Department of Physics and Chemistry, University of Southern Denmark, DK-5230 Odense M, Denmark

MCSCF is fairly efficient and accurate at describing static (long-range) correlation effects and Kohn-Sham DFT is fairly efficient and accurate at describing dynamic (short-range) correlation effects. We are investigating how efficient and accurate we can handle both static and dynamic correlation effects with MC-srDFT.

Latest developments in multiconfigurational range-separated density-functional theory: on many-body perturbation theory and actinide chemistry

Emmanuel Fromager

Center for Theoretical and Computational Chemistry, Department of Chemistry, University of Tromsø, 9037 Tromsø, NORWAY, http://www.ctcc.uit.no/people/manu/

A progress report on the development of multiconfigurational range-separated DFT, simply referred to as short-range DFT (srDFT) in the following, is presented. Two different topics are addressed.

(i) The description of dispersion interaction effects in srDFT by means of self-consistent many-body perturbation theory techniques. A general formalism (i.e.valid for any zeroth-order hamiltonian), which is based on the one-electron reduced density matrix, is proposed for the derivation of computable expressions through any order of perturbation [1]. Two applications are then presented: the derivation of a second-order Moeller-Plesset-type one-electron reduced density matrix and a range-separated optimized effective potential method based on second-order Goerling-Levy-type perturbation theory.

(ii) The application of a Multi-Configurational Self-Consistent Field-srDFT (MCSCF-srDFT) approach to actinide chemistry. We focuse on the neptunyl (VII) NpO2+++ ion which has, according to wave-function theory-based calculations, a linear equilibrium geometry. However, due to significant static correlation effects, standard functionals such as LDA, PBE or B3LYP give an equilibrium geometry that is bent. We address this bending problem within the MCSCF-srDFT approach [2].

References

[1] E. Fromager and H. J. Aa. Jensen, submitted to Phys. Rev. A (2008). "Self-consistent many-body perturbation theory in range-separated density-functional theory: a one-electron reduced density matrix-based formulation"

[2] E. Fromager, F. Real, P. Waahlin, U. Wahlgren and H. J. Aa. Jensen, to be submitted to J. Chem. Phys. (2008). "On the universality of the long/short-range separation in multiconfigurational density-functional theory. II. Investigating f0 actinide species"

Short-ranged LDA + long-ranged HF+MP2 applied to solids

Martijn Marsman and Georg Kresse

Faculty of Physics, University of Vienna, Sensengasse 8/12, A-1090 Vienna, Austria.

Together with Iann Gerber and Janos Angyan of the University of Nancy we have implemented a range separated HF/DFT hybrid functional that combines short-range LDA exchange-correlation with a Hartree-Fock + MP2 description of long-ranged exchange-correlation interactions, within the framework of the full-potential Projector-Augmented-Wave (PAW) method, using periodic boundary conditions and a plane wave basis set. We are in the process of characterizing the aforementioned scheme by calculating the lattice constants, bulk moduli, band gaps, and atomization energies of a set of semiconducting and insulating solid state systems. We will report preliminary results, mainly lattice constants and bulk moduli, and compare them to common DFT, Hartree-Fock+MP2, and experiment.

Beyond local and static Kohn-Sham potentials: the point of view of Green's functions theory

Matteo Gatti

European Theoretical Spectroscopy Facility (ETSF) - LSI Ecole Polytechnique, Palaiseau (France)

DFT is an in principle exact theory for the calculation of ground-state properties. But Kohn-Sham eigenvalues cannot be used to interpret photoemission spectra, which involve electronic excitations. Today a successful approach for the calculation of band structures in solids is based on Green's functions theory. In particular, in the standard GW method the band structure is obtained as first-order perturbative correction of the LDA eigenvalues with respect to the GW self-energy. This perturbative scheme fails when the LDA starting point is not adequate. In these cases a solution is to make use of a static but nonlocal potential instead of the LDA one. In this way one obtains different wavefunctions and energies for the subsequent GW calculation. In this context a possible choice is the COHSEX approximation, which is a static approximation to the GW self-energy where the local Coulomb hole (COH) term is added to the nonlocal screened-exchange (SEX) term. This COHSEX+GW scheme is in general successful, but computationally expensive. Therefore one would like to devise cheaper - and still reliable - alternative schemes for the calculation of photoemission spectra. For these purposes, I will show how a generalization of the Sham-Schlueter equation provides a very powerful tool in the design of new efficient computational schemes and discuss the relation to hybrid functionals.

References

For a recent application of the COHSEX+GW scheme see: M. Gatti, F. Bruneval, V. Olevano, and L. Reining, Phys. Rev. Lett. 99, 266402 (2007).

For the generalization of the Sham-Schlueter equation see: M. Gatti, V. Olevano, L. Reining, and I.V. Tokatly, Phys. Rev. Lett. 99, 057401 (2007).

Fractional Spins in Density Functional Theory

Weitao Yang

Duke University, Durham, USA.

Generalized hybrid theory

Mel Levy

Tulane University, New Orleans, USA