Volume 140, Issue 18, 14 May 2014

Standard density functional approximations often give questionable results for oddelectron radical complexes, with the error typically attributed to selfinteraction. In density corrected density functional theory (DCDFT), certain classes of density functional theory calculations are significantly improved by using densities more accurate than the selfconsistent densities. We discuss how to identify such cases, and how DCDFT applies more generally. To illustrate, we calculate potential energy surfaces of HO·Cl^{−} and HO·H2O complexes using various common approximate functionals, with and without this density correction. Commonly used approximations yield wrongly shaped surfaces and/or incorrect minima when calculated self consistently, while yielding almost identical shapes and minima when density corrected. This improvement is retained even in the presence of implicit solvent.
 SPECIAL TOPIC: ADVANCES IN DENSITY FUNCTIONAL THEORY


Preface: Special Topic on Advances in Density Functional Theory
View Description Hide DescriptionThis Special Topic Issue on the Advances in Density Functional Theory, published as a celebration of the fifty years of density functional theory, contains a retrospective article, a perspective article, and a collection of original research articles that showcase recent theoretical advances in the field. It provides a timely discussion reflecting a cross section of our understanding, and the theoretical and computational developments, which have significant implications in broad areas of sciences and engineering.
 Editorial

Editorial: Reflections on fifty years of density functional theory
View Description Hide Description  Perspective

Perspective: Fifty years of densityfunctional theory in chemical physics
View Description Hide DescriptionSince its formal inception in 1964–1965, KohnSham densityfunctional theory (KSDFT) has become the most popular electronic structure method in computational physics and chemistry. Its popularity stems from its beautifully simple conceptual framework and computational elegance. The rise of KSDFT in chemical physics began in earnest in the mid 1980s, when crucial developments in its exchangecorrelation term gave the theory predictive power competitive with welldeveloped wavefunction methods. Today KSDFT finds itself under increasing pressure to deliver higher and higher accuracy and to adapt to ever more challenging problems. If we are not mindful, however, these pressures may submerge the theory in the wavefunction sea. KSDFT might be lost. I am hopeful the KohnSham philosophical, theoretical, and computational framework can be preserved. This Perspective outlines the history, basic concepts, and present status of KSDFT in chemical physics, and offers suggestions for its future development.
 Articles

Randomphase approximation correlation energies from Lanczos chains and an optimal basis set: Theory and applications to the benzene dimer
View Description Hide DescriptionA new ab initio approach is introduced to compute the correlation energy within the adiabatic connection fluctuation dissipation theorem in the random phase approximation. First, an optimally small basis set to represent the response functions is obtained by diagonalizing an approximate dielectric matrix containing the kinetic energy contribution only. Then, the Lanczos algorithm is used to compute the full dynamical dielectric matrix and the correlation energy. The convergence issues with respect to the number of empty states or the dimension of the basis set are avoided and the dynamical effects are easily kept into account. To demonstrate the accuracy and efficiency of this approach the binding curves for three different configurations of the benzene dimer are computed: Tshaped, sandwich, and slipped parallel.

Introducing constricted variational density functional theory in its relaxed selfconsistent formulation (RSCFCVDFT) as an alternative to adiabatic time dependent density functional theory for studies of charge transfer transitions
View Description Hide DescriptionWe have applied the relaxed and selfconsistent extension of constricted variational density functional theory (RSCFCVDFT) for the calculation of the lowest charge transfer transitions in the molecular complex XTCNE between X = benzene and TCNE = tetracyanoethylene. Use was made of functionals with a fixed fraction (α) of HartreeFock exchange ranging from α = 0 to α = 0.5 as well as functionals with a long range correction (LC) that introduces HartreeFock exchange for longer interelectronic distances. A detailed comparison and analysis is given for each functional between the performance of RSCFCVDFT and adiabatic timedependent density functional theory (TDDFT) within the TammDancoff approximation. It is shown that in this particular case, all functionals afford the same reasonable agreement with experiment for RSCFCVDFT whereas only the LCfunctionals afford a fair agreement with experiment using TDDFT. We have in addition calculated the CT transition energy for XTCNE with X = toluene, oxylene, and naphthalene employing the same functionals as for X = benzene. It is shown that the calculated charge transfer excitation energies are in as good agreement with experiment as those obtained from highly optimized LCfunctionals using adiabatic TDDFT. We finally discuss the relation between the optimization of length separation parameters and orbital relaxation in the RSCFCVDFT scheme.

Analytic energy gradients for constrained DFTconfiguration interaction
View Description Hide DescriptionThe constrained density functional theoryconfiguration interaction (CDFTCI) method has previously been used to calculate groundstate energies and barrier heights, and to describe electronic excited states, in particular conical intersections. However, the method has been limited to evaluating the electronic energy at just a single nuclear configuration, with the gradient of the energy being available only via finite difference. In this paper, we present analytic gradients of the CDFTCI energy with respect to nuclear coordinates, which gives the potential for accurate geometry optimization and molecular dynamics on both the ground and excited electronic states, a realm which is currently quite challenging for electronic structure theory. We report the performance of CDFTCI geometry optimization for representative reaction transition states as well as molecules in an excited state. The overall accuracy of CDFTCI for computing barrier heights is essentially unchanged whether the energies are evaluated at geometries obtained from quadratic configurationinteraction singles and doubles (QCISD) or CDFTCI, indicating that CDFTCI produces very good reaction transition states. These results open up tantalizing possibilities for future work on excited states.

Metallophilic interactions from dispersioncorrected densityfunctional theory
View Description Hide DescriptionIn this article, we present the first comprehensive study of metallophilic (aurophilic) interactions using dispersioncorrected densityfunctional theory. Dispersion interactions (an essential component of metallophilicity) are treated using the exchangehole dipole moment (XDM) model. By comparing against coupledcluster benchmark calculations on simple dimers, we show that LCωPBEXDM is a viable functional to study interactions between closedshell transition metals and that it performs uniformly better than secondorder MøllerPlesset theory, the basic computational technique used in previous works. We apply LCωPBEXDM to address several open questions regarding metallophilicity, such as the interplay between dispersion and relativistic effects, the interaction strength along group 11, the additivity of homo and heterometallophilic effects, the stability of [E(AuPH3)4]^{+} cations (E = N, P, As, Sb), and the role of metallophilic effects in crystal packing. We find that relativistic effects explain the prevalence of aurophilicity not by stabilizing metalmetal contacts, but by preventing gold from forming ionic structures involving bridge anions (which are otherwise common for Ag and Cu) as a result of the increased electron affinity of the metal. Dispersion effects are less important than previously assumed and their stabilization contribution is relatively independent of the metal.

Exact exchange planewavepseudopotential calculations for slabs
View Description Hide DescriptionThe exact exchange of density functional theory is applied to both freestanding graphene and a Si(111) slab, using the planewave pseudopotential (PWPP) approach and a periodic repetition of the supercell containing the slab. It is shown that (i) PWPP calculations with exact exchange for slabs in supercell geometry are basically feasible, (ii) the width of the vacuum required for a decoupling of the slabs is only moderately larger than in the case of the localdensity approximation, and (iii) the resulting exchange potential v x shows an extended region, both far outside the surface of the slab and far from the middle of the vacuum region between the slabs, in which v x behaves as −e ^{2}/z, provided the width of the vacuum is chosen sufficiently large. This last result is corroborated by an analytical analysis of periodically repeated jellium slabs. The intermediate −e ^{2}/z behavior of v x can be used for an absolute normalization of v x and the total KohnSham potential, which, in turn, allows the determination of the work function.

Molecular properties of excited electronic state: Formalism, implementation, and applications of analytical second energy derivatives within the framework of the timedependent density functional theory/molecular mechanics
View Description Hide DescriptionThis work extends our previous works [J. Liu and W. Z. Liang, J. Chem. Phys.135, 014113 (2011); J. Liu and W. Z. Liang, J. Chem. Phys.135, 184111 (2011)] on analytical excitedstate energy Hessian within the framework of timedependent density functional theory (TDDFT) to couple with molecular mechanics (MM). The formalism, implementation, and applications of analytical first and second energy derivatives of TDDFT/MM excited state with respect to the nuclear and electric perturbations are presented. Their performances are demonstrated by the calculations of adiabatic excitation energies, and excitedstate geometries, harmonic vibrational frequencies, and infrared intensities for a number of benchmark systems. The consistent results with the full quantum mechanical method and other hybrid theoretical methods indicate the reliability of the current numerical implementation of developed algorithms. The computational accuracy and efficiency of the current analytical approach are also checked and the computational efficient strategies are suggested to speed up the calculations of complex systems with many MM degrees of freedom. Finally, we apply the current analytical approach in TDDFT/MM to a realistic system, a red fluorescent protein chromophore together with part of its nearby protein matrix. The calculated results indicate that the rearrangement of the hydrogen bond interactions between the chromophore and the protein matrix is responsible for the large Stokes shift.

Accurate and systematically improvable density functional theory embedding for correlated wavefunctions
View Description Hide DescriptionWe analyze the sources of error in quantum embedding calculations in which an active subsystem is treated using wavefunction methods, and the remainder using density functional theory. We show that the embedding potential felt by the electrons in the active subsystem makes only a small contribution to the error of the method, whereas the error in the nonadditive exchangecorrelation energy dominates. We test an MP2 correction for this term and demonstrate that the corrected embedding scheme accurately reproduces wavefunction calculations for a series of chemical reactions. Our projectorbased embedding method uses localized occupied orbitals to partition the system; as with other local correlation methods, abrupt changes in the character of the localized orbitals along a reaction coordinate can lead to discontinuities in the embedded energy, but we show that these discontinuities are small and can be systematically reduced by increasing the size of the active region. Convergence of reaction energies with respect to the size of the active subsystem is shown to be rapid for all cases where the density functional treatment is able to capture the polarization of the environment, even in conjugated systems, and even when the partition cuts across a double bond.

Longrange correlation energy calculated from coupled atomic response functions
View Description Hide DescriptionAn accurate determination of the electron correlation energy is an essential prerequisite for describing the structure, stability, and function in a wide variety of systems. Therefore, the development of efficient approaches for the calculation of the correlation energy (and hence the dispersion energy as well) is essential and such methods can be coupled with many densityfunctional approximations, local methods for the electron correlation energy, and even interatomic force fields. In this work, we build upon the previously developed manybody dispersion (MBD) framework, which is intimately linked to the randomphase approximation for the correlation energy. We separate the correlation energy into shortrange contributions that are modeled by semilocal functionals and longrange contributions that are calculated by mapping the complex allelectron problem onto a set of atomic response functions coupled in the dipole approximation. We propose an effective rangeseparation of the coupling between the atomic response functions that extends the already broad applicability of the MBD method to nonmetallic materials with highly anisotropic responses, such as layered nanostructures. Application to a variety of highquality benchmark datasets illustrates the accuracy and applicability of the improved MBD approach, which offers the prospect of firstprinciples modeling of large structurally complex systems with an accurate description of the longrange correlation energy.

Towards a systematic way to correct density functional approximations
View Description Hide DescriptionIn order to simulate the exact universal density functional, approximations are nowadays constructed by permitting more flexibility in its ansatz. In view of the difficulty of defining a systematically improvable form for it, this paper argues that an alternative way could be considered. It falls within the class of hybrid functionals with multideterminant wave functions. The parameter controlling the hybridization is considered as variable. The invariance of the exact result with respect to changes in this variable is used to introduce information about the system under consideration, and to correct the density functional result. The construction considered in this paper accelerates convergence from the model system to the physical one, in the vicinity of the latter. The method, at the present level of implementation, should be seen as a starting point for further development, and not necessarily as a computationally advantageous tool.

A selfinteractionfree local hybrid functional: Accurate binding energies visàvis accurate ionization potentials from KohnSham eigenvalues
View Description Hide DescriptionWe present and test a new approximation for the exchangecorrelation (xc) energy of KohnSham density functional theory. It combines exact exchange with a compatible nonlocal correlation functional. The functional is by construction free of oneelectron selfinteraction, respects constraints derived from uniform coordinate scaling, and has the correct asymptotic behavior of the xc energy density. It contains one parameter that is not determined ab initio. We investigate whether it is possible to construct a functional that yields accurate binding energies and affords other advantages, specifically KohnSham eigenvalues that reliably reflect ionization potentials. Tests for a set of atoms and small molecules show that within our localhybrid form accurate binding energies can be achieved by proper optimization of the free parameter in our functional, along with an improvement in dissociation energy curves and in KohnSham eigenvalues. However, the correspondence of the latter to experimental ionization potentials is not yet satisfactory, and if we choose to optimize their prediction, a rather different value of the functional's parameter is obtained. We put this finding in a larger context by discussing similar observations for other functionals and possible directions for further functional development that our findings suggest.

Exchangecorrelation energy from pairing matrix fluctuation and the particleparticle random phase approximation
View Description Hide DescriptionDespite their unmatched success for many applications, commonly used local, semilocal, and hybrid density functionals still face challenges when it comes to describing longrange interactions, static correlation, and electron delocalization. Density functionals of both the occupied and virtual orbitals are able to address these problems. The particlehole (ph) Random Phase Approximation (RPA), a functional of occupied and virtual orbitals, has recently known a revival within the density functional theory community. Following up on an idea introduced in our recent communication [H. van Aggelen, Y. Yang, and W. Yang, Phys. Rev. A88, 030501 (2013)], we formulate more general adiabatic connections for the correlation energy in terms of pairing matrix fluctuations described by the particleparticle (pp) propagator. With numerical examples of the ppRPA, the lowestorder approximation to the pppropagator, we illustrate the potential of density functional approximations based on pairing matrix fluctuations. The ppRPA is sizeextensive, selfinteraction free, fully antisymmetric, describes the strong static correlation limit in H2, and eliminates delocalization errors in and other singlebond systems. It gives surprisingly good nonbonded interaction energies – competitive with the phRPA – with the correct R ^{−6} asymptotic decay as a function of the separation R, which we argue is mainly attributable to its correct secondorder energy term. While the ppRPA tends to underestimate absolute correlation energies, it gives good relative energies: much better atomization energies than the phRPA, as it has no tendency to underbind, and reaction energies of similar quality. The adiabatic connection in terms of pairing matrix fluctuation paves the way for promising new density functional approximations.

Construction of a parameterfree doubly hybrid density functional from adiabatic connection
View Description Hide DescriptionIn this work, the adiabatic connection (AC) formalism, coordinate scaling relations, and the second order GörlingLevy perturbation theory (GL2) are first reviewed. Emphasis is laid upon the construction of the AC integrand ( ), with suitable input data and the proper asymptotic behavior on λ^{−1/2} as λ → ∞. This leads to a nonempirical DH functional, namely, PBEACDH. The PBEACDH functional is unique in that it explicitly considers contributions from density scaling and singles, and it utilizes density and orbital information from the PBE functional, which has a local multiplicative potential, being most compatible with the GL2 theory. Systematical tests on heats of formation, bond dissociation enthalpies, reaction barrier heights, and nonbonded interactions, using some wellestablished benchmarking sets, suggest that PBEACDH is a significant improvement over its parent functional PBE, as well as PBE0, Becke's HalfandHalf (PBEHH), and GL2. The physical insight gained in the present work should prove useful for the further development of new functionals.

Selfinteraction corrections in density functional theory
View Description Hide DescriptionSelfinteraction corrections for KohnSham density functional theory are reviewed for their physical meanings, formulations, and applications. The selfinteraction corrections get rid of the selfinteraction error, which is the sum of the Coulomb and exchange selfinteractions that remains because of the use of an approximate exchange functional. The most frequently used selfinteraction correction is the PerdewZunger correction. However, this correction leads to instabilities in the electronic state calculations of molecules. To avoid these instabilities, several selfinteraction corrections have been developed on the basis of the characteristic behaviors of selfinteracting electrons, which have no twoelectron interactions. These include the von Weizsäcker kinetic energy and longrange (farfromnucleus) asymptotic correction. Applications of selfinteraction corrections have shown that the selfinteraction error has a serious effect on the states of core electrons, but it has a smaller than expected effect on valence electrons. This finding is supported by the fact that the distribution of selfinteracting electrons indicates that they are near atomic nuclei rather than in chemical bonds.

Ensemble density variational methods with self and ghostinteractioncorrected functionals
View Description Hide DescriptionEnsemble density functional theory (DFT) offers a way of predicting excitedstates energies of atomic and molecular systems without referring to a density response function. Despite a significant theoretical work, practical applications of the proposed approximations have been scarce and they do not allow for a fair judgement of the potential usefulness of ensemble DFT with available functionals. In the paper, we investigate two forms of ensemble density functionals formulated within ensemble DFT framework: the Gross, Oliveira, and Kohn (GOK) functional proposed by Gross et al. [Phys. Rev. A37, 2809 (1988)] alongside the orbitaldependent eDFT form of the functional introduced by Nagy [J. Phys. B34, 2363 (2001)] (the acronym eDFT proposed in analogy to eHF – ensemble HartreeFock method). Local and semilocal groundstate density functionals are employed in both approaches. Approximate ensemble density functionals contain not only spurious selfinteraction but also the socalled ghostinteraction which has no counterpart in the groundstate DFT. We propose how to correct the GOK functional for both kinds of interactions in approximations that go beyond the exactexchange functional. Numerical applications lead to a conclusion that functionals free of the ghostinteraction by construction, i.e., eDFT, yield much more reliable results than approximate self and ghostinteractioncorrected GOK functional. Additionally, local density functional corrected for selfinteraction employed in the eDFT framework yields excitations energies of the accuracy comparable to that of the uncorrected semilocal eDFT functional.

Kinetic and interaction components of the exact timedependent correlation potential
View Description Hide DescriptionThe exact exchangecorrelation (xc) potential of timedependent density functional theory has been shown to have striking features. For example, step and peak features are generically found when the system is far from its groundstate, and these depend nonlocally on the density in space and time. We analyze the xc potential by decomposing it into kinetic and interaction components and comparing each with their exactadiabatic counterparts, for a range of dynamical situations in model onedimensional twoelectron systems. We find that often, but not always, the kinetic contribution is largely responsible for these features that are missed by the adiabatic approximation. The adiabatic approximation often makes a smaller error for the interaction component, which we write in two parts, one being the Coulomb potential due to the timedependent xc hole. Nonadiabatic features of the kinetic component were also larger than those of the interaction component in cases that we studied when there is negligible step structure. In groundstate situations, step and peak structures arise in cases of static correlation, when more than one determinant is essential to describe the interacting state. We investigate the timedependent natural orbital occupation numbers and find the corresponding relation between these and the dynamical step is more complex than for the groundstate case.

How important is selfconsistency for the dDsC density dependent dispersion correction?
View Description Hide DescriptionThe treatment of dispersion interactions is ubiquitous but computationally demanding for seamless ab initio approaches. A highly popular and simple remedy consists in correcting for the missing interactions a posteriori by adding an attractive energy term summed over all atom pairs to standard density functional approximations. These corrections were originally based on atom pairwise parameters and, hence, had a strong touch of empiricism. To overcome such limitations, we recently proposed a robust systemdependent dispersion correction, dDsC, that is computed from the electron density and that provides a balanced description of both weak inter and intramolecular interactions. From the theoretical point of view and for the sake of increasing reliability, we here verify if the selfconsistent implementation of dDsC impacts groundstate properties such as interaction energies, electron density, dipole moments, geometries, and harmonic frequencies. In addition, we investigate the suitability of the a posteriori scheme for molecular dynamics simulations, for which the analysis of the energy conservation constitutes a challenging tests. Our study demonstrates that the postSCF approach in an excellent approximation.

Response calculations based on an independent particle system with the exact oneparticle density matrix: Polarizabilities
View Description Hide DescriptionRecently, we have demonstrated that the problems finding a suitable adiabatic approximation in timedependent onebody reduced density matrix functional theory can be remedied by introducing an additional degree of freedom to describe the system: the phase of the natural orbitals [K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, Phys. Rev. Lett.105, 013002 (2010); K. J. H. Giesbertz, O. V. Gritsenko, and E. J. Baerends, J. Chem. Phys.133, 174119 (2010)]. In this article we will show in detail how the frequencydependent response equations give the proper static limit (ω → 0), including the perturbation in the chemical potential, which is required in static response theory to ensure the correct number of particles. Additionally we show results for the polarizability for H2 and compare the performance of two different twoelectron functionals: the phaseincluding Löwdin–Shull functional and the density matrix form of the Löwdin–Shull functional.
