The importance of middle-range Hartree-Fock-type exchange for hybrid density functionals

Hybrid functionals are responsible for much of the utility of modern Kohn-Sham density functional theory. When rigorously applied to solid-state metallic and small band gap systems, however, the slow decay of their nonlocal Hartree-Fock-type exchange makes hybrids computationally challenging and introduces unphysical effects. This can be remedied by using a range-separated hybrid which only keeps short-range nonlocal exchange, as in the functional of Heyd et al. [J. Chem. Phys. 118, 8207 (2003)]. On the other hand, many molecular properties require full long-range nonlocal exchange, which can also be included by means of a range-separated hybrid such as the recently introduced LC-omegaPBE functional [O. A. Vydrov and G. E. Scuseria, J. Chem. Phys. 125, 234109 (2006)]. In this paper, we show that a three-range hybrid which mainly includes middle-range Hartree-Fock-type exchange and neglects long- and short-range Hartree-Fock-type exchange yields excellent accuracy for thermochemistry, barrier heights, and band gaps, emphasizing that the middle-range part of the 1/r potential seems crucial to accurately model these properties.     

Comparing modern density functionals for conjugated polymer band structures: screened hybrid, Minnesota, and Rung 3.5 approximations

Semiconducting polymers with π-conjugated backbones show promise in fields such as photovoltaics. Practical applications of conjugated polymers require precise control over the polymer's electronic band structure. Several new classes of density functional approximation, including screened hybrids, semilocal Minnesota functionals, and Rung 3.5 functionals, show potential for improved predictions of conjugated polymer band structures. This work compares these methods to standard global hybrid density functionals for bandgaps and band structures of representative conjugated polymers. The new methods exhibit particular promise for modeling three-dimensionally periodic bulk polymers, which can be problematic for global hybrids.     

The performance of semilocal and hybrid density functionals in 3d transition-metal chemistry

We investigate the performance of contemporary semilocal and hybrid density functionals for bond energetics, structures, dipole moments, and harmonic frequencies of 3d transition-metal (TM) compounds by comparison with gas-phase experiments. Special attention is given to the nonempirical metageneralized gradient approximation (meta-GGA) of Tao, Perdew, Staroverov, and Scuseria (TPSS) [Phys. Rev. Lett. 91, 146401 (2003)], which has been implemented in TURBOMOLE for the present work. Trends and error patterns for classes of homologous compounds are analyzed, including dimers, monohydrides, mononitrides, monoxides, monofluorides, polyatomic oxides and halogenides, carbonyls, and complexes with organic pi ligands such as benzene and cyclopentadienyl. Weakly bound systems such as Ca(2), Mn(2), and Zn(2) are discussed. We propose a reference set of reaction energies for benchmark purposes. Our all-electron results with quadruple zeta valence basis sets validate semilocal density-functional theory as the workhorse of computational TM chemistry. Typical errors in bond energies are substantially larger than in (organic) main group chemistry, however. The Becke-Perdew'86 [Phys. Rev. A 38, 3098 (1988); Phys. Rev. B 33, 8822 (1986)] GGA and the TPSS meta-GGA have the best price/performance ratio, while the TPSS hybrid functional achieves a slightly lower mean absolute error in bond energies. The popular Becke three-parameter hybrid B3LYP underbinds significantly and tends to overestimate bond distances; we give a possible explanation for this. We further show that hybrid mixing does not reduce the width of the error distribution on our reference set. The error of a functional for the s-d transfer energy of a TM atom does not predict its error for TM bond energies and bond lengths. For semilocal functionals, self-interaction error in one- and three-electron bonds appears to be a major source of error in TM reaction energies. Nevertheless, TPSS predicts the correct ground-state symmetry in the vast majority of cases and rarely fails qualitatively. This further confirms TPSS as a general purpose functional that works throughout the periodic table. We also give workstation timing comparisons for the 645-atom protein crambin.     

On the test of different atomic exchange functionals

The yet unknown exchange-correlation energy functional and the lack of a systematic way to find such a functional has led to propose different expressions for exchange, on the one hand, and correlation on the other, by several authors. A common factor in the design of exchange functionals is to reproduce different kinds of energies rather than the important atomic shell structure. We present a strategy for testing the behavior of exchange functionals based on the reproduction of atomic shell structure. A variety of well-known exchange functionals are tested on He (closed shell), Li (open shell), and Be (closed subshell). Important differences are found among them. Finally, we propose an exchange energy functional that includes a Becke-88-type expression and, in addition, a nongradient correction term that leads to improved shell structure behavior. © 1995 John Wiley & Sons, Inc.     

Local hybrid functionals based on density matrix products

We present a novel similarity metric comparing exact and semilocal density functional theory (DFT) exchange holes in real space. This metric is obtained from the product of the one-particle density matrix and the uniform electron gas model density matrix. The metric is bound between 0 and 1, 1 in the uniform electron gas, 0 in regions asymptotically far from finite systems, and can detect delocalization of the exact exchange hole and effective fractional occupations. We also present a parameter-free local hybrid functional that uses this similarity metric to locally mix exact and semilocal DFT exchange energy densities. The resulting functional gives better thermochemistry and reaction barrier heights than our original local hybrids [Jaramillo et al., J. Chem. Phys. 118, 1068 (2003)], while retaining moderate accuracy for symmetric radical cation dimers.     

Comparison of one‐parameter and linearly scaled one‐parameter double‐hybrid density functionals for noncovalent interactions

We have compared the performances of the one‐parameter and linearly scaled one‐parameter double‐hybrid density functionals (1DH‐DFs and LS1DH‐DFs) for noncovalent interactions. The only one parameter related to the Hartree–Fock (HF) exchange for each of the tested 1DH‐DFs and LS1DH‐DFs has been fitted with the well‐designed S66 database. The obtained DHDFs are dubbed as 1DH‐PBE‐NC, LS1DH‐PBE‐NC, 1DH‐TPSS‐NC, LS1DH‐TPSS‐NC, 1DH‐PWB95‐NC, and LS1DH‐PWB95‐NC, where “NC” denotes noncovalent interactions. With a specific combination of exchange and correlation functionals, the dependent parameters related to the nonlocal second‐order perturbative energies are nearly identical for the 1DH and LS1DH models. According to our benchmark computations against the S66, S22B, NCCE31, ADIM6, and L7 databases, we suggest that the 1DH‐PWB95‐NC and LS1DH‐PWB95‐NC functionals are much more suitable for evaluating noncovalent interaction energies. Unlike the versatile DHDFs with dispersion corrections for general purpose, our optimized 1DH‐DFs and LS1DH‐DFs only aim at noncovalent interactions. © 2016 Wiley Periodicals, Inc.     

Comparative performance of exchange and correlation density functionals in determining intermolecular interaction potentials of the methane dimer

We have calculated the interaction potentials of the methane dimer for the minimum-energy D(3d) conformation using the density functional theory (DFT) with 90 density functionals chosen from the combinations of nine exchange and 10 correlation functionals. Several hybrid functionals are also considered. While the performance of an exchange functional is related to the large reduced density gradient of the exchange enhancement factor, the correlation energy is determined by the low-density behavior of a correlation enhancement factor. Our calculations demonstrate that the correlation counterpart plays an equally important role as the exchange functional in determining the van der Waals interactions of the methane dimer. These observations can be utilized to better understand the seemingly unsystematic DFT interaction potentials for weakly bound systems.     

Rapidly converging threshold value calculations in screened coulomb potential systems: Critical values of the screening parameter for the Yukawa case

In this work, the threshold values of the screening parameter for Yukawa potential systems are obtained by means of a basis set constructed from Laguerre type functions. The Laguerre basis set is modified by an appropriately chosen extra function in order to imitate the true behaviour of the solutions at the boundary points. The method used is a variational scheme and the numerical results are accurate to thirty decimal points. A scaling parameter is also inserted into the structure of the basis functions, the optimized values of which accelerate the convergence. The main goal of this paper is to develop a method which enables us to calculate the threshold values of the screening parameter for low-lying states. The method is quite general and can be extended to all systems whose potentials decay exponentially when the radial variable goes to infinity.     

General performance of density functionals

The density functional theory (DFT) foundations date from the 1920s with the work of Thomas and Fermi, but it was after the work of Hohenberg, Kohn, and Sham in the 1960s, and particularly with the appearance of the B3LYP functional in the early 1990s, that the widespread application of DFT has become a reality. DFT is less computationally demanding than other computational methods with a similar accuracy, being able to include electron correlation in the calculations at a fraction of time of post-Hartree-Fock methodologies. In this review we provide a brief outline of the density functional theory and of the historic development of the field, focusing later on the several types of density functionals currently available, and finishing with a detailed analysis of the performance of DFT across a wide range of chemical properties and system types, reviewed from the most recent benchmarking studies, which encompass several well-established density functionals together with the most recent efforts in the field. Globally, an overall picture of the level of performance of the plethora of currently available density functionals for each chemical property is drawn, with particular attention being dedicated to the relative performance of the popular B3LYP density functional.     

Seminumerical calculation of the Hartree-Fock exchange matrix: application to two-component procedures and efficient evaluation of local hybrid density functionals

A two-component extension of the seminumerical procedure for the calculation of the Hartree-Fock (HF) exchange matrix recently presented by Neese et al. (Chem Phys 2009, 356, 98) was implemented into the program system TURBOMOLE. It is demonstrated that this allows for efficient self-consistent treatment of spin-orbit coupling at HF and hybrid density functional theory level. One-component HF calculations were performed to study the accuracy of integration grids and the exploitation of the molecular point group symmetry. The efficiency was tested, and for one-component calculations compared to the implementation realized by Neese. It was further demonstrated that local hybrid density functionals can be evaluated with this technique. The "prototype" of this class of functionals, Lh-BLYP, was applied to an organic molecule with more than 150 atoms.     

Systematic optimization of long-range corrected hybrid density functionals

A general scheme for systematically modeling long-range corrected (LC) hybrid density functionals is proposed. Our resulting two LC hybrid functionals are shown to be accurate in thermochemistry, kinetics, and noncovalent interactions, when compared with common hybrid density functionals. The qualitative failures of the commonly used hybrid density functionals in some "difficult problems," such as dissociation of symmetric radical cations and long-range charge-transfer excitations, are significantly reduced by the present LC hybrid density functionals.     

Efficient hybrid density functional calculations in solids: assessment of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional

The present work introduces an efficient screening technique to take advantage of the fast spatial decay of the short range Hartree-Fock (HF) exchange used in the Heyd-Scuseria-Ernzerhof (HSE) screened Coulomb hybrid density functional. The screened HF exchange decay properties and screening efficiency are compared with traditional hybrid functional calculations on solids. The HSE functional is then assessed using 21 metallic, semiconducting, and insulating solids. The examined properties include lattice constants, bulk moduli, and band gaps. The results obtained with HSE exhibit significantly smaller errors than pure density functional theory (DFT) calculations. For structural properties, the errors produced by HSE are up to 50% smaller than the errors of the local density approximation, PBE, and TPSS functionals used for comparison. When predicting band gaps of semiconductors, we found smaller errors with HSE, resulting in a mean absolute error of 0.2 eV (1.3 eV error for all pure DFT functionals). In addition, we present timing results which show the computational time requirements of HSE to be only a factor of 2-4 higher than pure DFT functionals. These results make HSE an attractive choice for calculations of all types of solids.     

Implementation of screened hybrid density functional for periodic systems with numerical atomic orbitals: basis function fitting and integral screening

We present an efficient O(N) implementation of screened hybrid density functional for periodic systems with numerical atomic orbitals (NAOs). NAOs of valence electrons are fitted with gaussian-type orbitals, which is convenient for the calculation of electron repulsion integrals and the construction of Hartree-Fock exchange matrix elements. All other parts of Hamiltonian matrix elements are constructed directly with NAOs. The strict locality of NAOs is adopted as an efficient two-electron integral screening technique to speed up calculations.     

Assessment and validation of a screened Coulomb hybrid density functional

This paper presents a revised and improved version of the Heyd-Scuseria-Ernzerhof screened Coulomb hybrid functional. The performance of this functional is assessed on a variety of molecules for the prediction of enthalpies of formation, geometries, and vibrational frequencies, yielding results as good as or better than the successful PBE0 hybrid functional. Results for ionization potentials and electron affinities are of slightly lower quality but are still acceptable. The comprehensive test results presented here validate our assumption that the screened, short-range Hartree-Fock (HF) exchange exhibits all physically relevant properties of the full HF exchange. Thus, hybrids can be constructed which neglect the computationally demanding long-range part of HF exchange while still retaining the superior accuracy of hybrid functionals, compared to pure density functionals.     

Performance of range‐separated hybrid exchange–correlation functionals for the calculation of magnetic exchange coupling constants of organic diradicals

The prediction of magnetic behavior is important for the design of new magnetic materials. Kohn–Sham density functional theory is popular for this purpose, although one should be careful about choosing the right exchange–correlation functional. Here, we perform a statistical analysis to test different range‐separated hybrid density functionals for the calculation of magnetic exchange coupling constants J of fourteen organic diradicals. Our analysis suggests that in absolute terms the MN12SX functional performs best among the series of twelve functionals studied here (including the popular B3LYP), followed by N12SX functionals along with Scuseria's HSE series of functionals. LC‐ PBE was found to be the least accurate, which is in contrast with its good performance for calculating J for transition metal complexes. The HSE family of functionals and B3LYP are the only functionals to reproduce the qualitative trends of the coupling constants correctly for the ferromagnetically coupled diradicals under study. © 2017 Wiley Periodicals, Inc.     

Reparameterization of hybrid functionals based on energy differences of states of different multiplicity

 Low-spin/high-spin energy splittings for Fe(II) transition-metal complexes – particularly in weak ligand fields – cannot be well described by density functional methods. Different density functionals yield results which differ by up to 1 eV in transition-metal complexes with sulfur-rich first coordination spheres. We attribute this failure to the fact that the high-spin state is systematically favoured in Hartree–Fock-type theories, because Fermi correlation is included in the exact exchange, while Coulomb correlation is not. We thus expect that the admixture of exact exchange to a given density functional will heavily influence the energy splitting between states of different multiplicity. We demonstrate that the energy splitting depends linearly on the coefficient of exact exchange admixture. This remarkable result is found for all the Fe(II)–S complexes studied. From this observation we conclude in connection with experimental results that Becke's 20% admixture should be reduced to about 15% if meaningful energetics are sought for transition-metal compounds. We rationalize that this reduction by 5% will not affect the quality of the hybrid functional since we arrive at a slightly modified functional, which lies between the pure density functional and the hybrid density functional, which both give good results for “standard” systems. Received: 13 July 2001 / Accepted: 31 August 2001 /  Published online: 16 November 2001     

Effect of the Perdew-Zunger self-interaction correction on the thermochemical performance of approximate density functionals

The Perdew-Zunger self-interaction-corrected density functional theory (SIC-DFT) was implemented self-consistently using a quasi-Newton direct minimization method. We calculated SIC-DFT energies for a number of atoms and molecules using various approximate density functionals, including hybrids. Self-interaction errors (SIE) of these functionals were compared and analyzed in terms of contributions from valence and core orbitals. We also calculated enthalpies of formation of the standard G2-1 set of 55 molecules and found that self-interaction-correction (SIC) improves agreement with experiment only for the LSDA functional, while all other functionals show worse performance upon introducing SIC. This is the first systematic study of the effect of SIC on thermochemical properties. We found no direct connection between the magnitude of the SIE contained in a functional and its performance for thermochemistry. Approximate functionals with large self-interaction errors can accurately reproduce enthalpies of formation. Our results do not support the popular belief that a smaller SIE of hybrid functionals is the main reason for their higher accuracy.     

Short-range exchange and correlation energy density functionals: beyond the local-density approximation

We propose approximations which go beyond the local-density approximation for the short-range exchange and correlation density functionals appearing in a multideterminantal extension of the Kohn-Sham scheme. A first approximation consists of defining locally the range of the interaction in the correlation functional. Another approximation, more conventional, is based on a gradient expansion of the short-range exchange-correlation functional. Finally, we also test a short-range generalized-gradient approximation by extending the Perdew-Burke-Ernzerhof exchange-correlation functional to short-range interactions.