Short-range Functionals Research Articles

Asymptotic Theory for a General Class of Short-Range Interaction Functionals

Asymptotic Theory for a General Class of Short-Range Interaction Functionals

Taming the fixed-node error in diffusion Monte Carlo via range separation.

By combining density-functional theory (DFT) and wave function theory via the range separation (RS) of the interelectronic Coulomb operator, we obtain accurate fixed-node diffusion Monte Carlo (FN-DMC) energies with compact multi-determinant trial wave functions. In particular, we combine here short-range exchange-correlation functionals with a flavor of selected configuration interaction known as configuration interaction using a perturbative selection made iteratively (CIPSI), a scheme that we label RS-DFT-CIPSI. One of the take-home messages of the present study is that RS-DFT-CIPSI trial wave functions yield lower fixed-node energies with more compact multi-determinant expansions than CIPSI, especially for small basis sets. Indeed, as the CIPSI component of RS-DFT-CIPSI is relieved from describing the short-range part of the correlation hole around the electron-electron coalescence points, the number of determinants in the trial wave function required to reach a given accuracy is significantly reduced as compared to a conventional CIPSI calculation. Importantly, by performing various numerical experiments, we evidence that the RS-DFT scheme essentially plays the role of a simple Jastrow factor by mimicking short-range correlation effects, hence avoiding the burden of performing a stochastic optimization. Considering the 55 atomization energies of the Gaussian-1 benchmark set of molecules, we show that using a fixed value of μ = 0.5 bohr-1 provides effective error cancellations as well as compact trial wave functions, making the present method a good candidate for the accurate description of large chemical systems.

Density Functional Theory Based Methods for the Calculation of X-ray Spectroscopy.

The availability of new light sources combined with the realization of the unique capabilities of spectroscopy in the X-ray region has driven tremendous advances in the field of X-ray spectroscopy. Currently, these techniques are emerging as powerful analytical tools for the study of a wide range of problems encompassing liquids, materials, and biological systems. Time-resolved measurements add a further dimension to X-ray spectroscopy, opening up the potential to resolve ultrafast chemical processes at an atomic level. X-ray spectroscopy encompasses a range of techniques which provide complementary information, and these include X-ray photoelectron spectroscopy (XPS), X-ray absorption spectroscopy (XAS), X-ray emission spectroscopy (XES), and resonant inelastic X-ray scattering (RIXS). In many studies, the interpretation of the experimental data relies upon calculations to enable the nature of the underlying molecular structure, electronic structure, and bonding to be revealed. Density functional theory (DFT) based methods are some of the most widely used methods for the simulation of X-ray spectra. In this Account, we focus on our recent contributions to the simulation of a range of X-ray spectroscopic techniques using DFT and linear-response time-dependent density functional theory (TDDFT) and show how these methods can provide a computational toolkit for the simulation of X-ray spectroscopy. The importance of the exchange-correlation functional for the calculation of XAS is discussed, and the introduction of short-range corrected functionals is described. The application of these calculations to study large systems through the use of efficient implementations of TDDFT will be highlighted, with the use of these methods illustrated through studies of ionic liquids and transition metal complexes. The extension of TDDFT to calculate XES through the use of a reference determinant for the core-ionized state will be described, and the factors that affect the accuracy of the computed spectra discussed. The application of these approaches will be illustrated through the study of a range of organic molecules and transition metal complexes, which also show how going beyond the dipole approximation in determining the transition intensities can be critical. The application of these approaches to the simulation of the RIXS spectrum of water will also be described, highlighting how ultrafast dynamics on the femtoscale time scale are evident in the measured spectra. In these calculations, the description of the core-ionized and core-excited states becomes increasingly important, and the role of the basis set in accurately describing these states will be explored.

Density-Based Basis-Set Incompleteness Correction for GW Methods.

Similar to other electron correlation methods, many-body perturbation theory methods based on Green's functions, such as the so-called GW approximation, suffer from the usual slow convergence of energetic properties with respect to the size of the one-electron basis set. This displeasing feature is due to the lack of explicit electron-electron terms modeling the infamous Kato electron-electron cusp and the correlation Coulomb hole around it. Here, we propose a computationally efficient density-based basis-set correction based on short-range correlation density functionals which significantly speeds up the convergence of energetics toward the complete basis set limit. The performance of this density-based correction is illustrated by computing the ionization potentials of the 20 smallest atoms and molecules of the GW100 test set at the perturbative GW (or G0W0) level using increasingly large basis sets. We also compute the ionization potentials of the five canonical nucleobases (adenine, cytosine, thymine, guanine, and uracil) and show that, here again, a significant improvement is obtained.

Band Gap of 3D Metal Oxides and Quasi-2D Materials from Hybrid Density Functional Theory: Are Dielectric-Dependent Functionals Superior?

Reproduction of the band gaps of semiconductors and insulators represents a well-known problem for standard DFT approaches based on semilocal functionals. The problem can be partly solved using hybrid functionals, in which a given portion of exact exchange is mixed with the DFT exchange. Recently, a new class of dielectric-dependent functionals has been introduced in which the amount of exact exchange is derived from the static dielectric function of a given compound. In this study we considered in a systematic way on an equal footing a set of 24 nonmagnetic three-dimensional (3D) bulk metal oxides and 24 quasi-two-dimensional (quasi-2D) semiconductors (oxides, hydroxides, chlorides, oxyhalides, nitrides, and transition metal dichalcogenides) and computed the corresponding Kohn-Sham band gaps with three global hybrid functionals and four range-separated hybrid functionals. These in turn were divided into standard (PBE0, B3LYP, HSE06, SC-BLYP) and dielectric-dependent (DD-B3LYP, DD-SC-BLYP, DD-CAM-B3LYP) functionals. We also performed a statistical analysis of the DFT data set along with structural parameters of these 2D and 3D materials. The surprising result is that overall there is no real improvement with the use of dielectric-dependent functionals compared to PBE0, HSE06, and B3LYP. Short-range DD-SC-BLYP gives a minor improvement in the band gaps for bulk metal oxides compared with standard SC-BLYP, but the mean absolute error is still 0.12 eV higher than with B3LYP. The use of dielectric-dependent standard or short-range functionals such as DD-B3LYP or DD-HSE06 worsens the situation. However, the dielectric-dependent version of the long-range-separated functional implemented with the Coulomb attenuating method (CAM), DD-CAM-B3LYP, leads to a clear improvement for band gaps of quasi-2D materials. On the basis of this analysis, the conclusion is that the use of a standard hybrid functional such as B3LYP or HSE06 is recommended for nonmagnetic bulk 3D metal oxides. On the other hand, the treatment of layered materials such as MoO3 or V2O5 benefits from the use of dielectric-dependent range-separated functionals.

Triplet excitation energies from multiconfigurational short-range density-functional theory response calculations.

Linear response theory for the multiconfigurational short-range density functional theory (MC-srDFT) model is extended to triplet response with a singlet reference wave function. The triplet linear response equations for MC-srDFT are derived for a general hybrid srGGA functional and implemented in the Dalton program. Triplet excitation energies are benchmarked against the CC3 model of coupled cluster theory and the complete-active-space second-order perturbation theory using three different short-range functionals (srLDA, srPBE, and srPBE0), both with full linear response and employing the generalized Tamm-Dancoff approximation (gTDA). We find that using gTDA is required for obtaining reliable triplet excitations; for the CAS-srPBE model, the mean absolute deviation decreases from 0.40 eV to 0.26 eV, and for the CAS-srLDA model, it decreases from 0.29 eV to 0.21 eV. As expected, the CAS-srDFT model is found to be superior to the HF-srDFT model when analyzing the calculated triplet excitations for molecules in the benchmark set where increased static correlation is expected.

A Density-Based Basis-Set Correction for Wave Function Theory.

We report a universal density-based basis-set incompleteness correction that can be applied to any wave function method. This correction, which appropriately vanishes in the complete basis-set (CBS) limit, relies on short-range correlation density functionals (with multideterminant reference) from range-separated density-functional theory (RS-DFT) to estimate the basis-set incompleteness error. Contrary to conventional RS-DFT schemes that require an ad hoc range-separation parameter μ, the key ingredient here is a range-separation function μ(r) that automatically adapts to the spatial nonhomogeneity of the basis-set incompleteness error. As illustrative examples, we show how this density-based correction allows us to obtain CCSD(T) atomization and correlation energies near the CBS limit for the G2 set of molecules with compact Gaussian basis sets.

Short-range density functional correlation within the restricted active space CI method.

In the present work, I introduce a hybrid wave function-density functional theory electronic structure method based on the range separation of the electron-electron Coulomb operator in order to recover dynamic electron correlations missed in the restricted active space configuration interaction (RASCI) methodology. The working equations and the computational algorithm for the implementation of the new approach, i.e., RAS-srDFT, are presented, and the method is tested in the calculation of excitation energies of organic molecules. The good performance of the RASCI wave function in combination with different short-range exchange-correlation functionals in the computation of relative energies represents a quantitative improvement with respect to the RASCI results and paves the path for the development of RAS-srDFT as a promising scheme in the computation of the ground and excited states where nondynamic and dynamic electron correlations are important.

Density functional theory calculations of the non-resonant and resonant X-ray emission spectroscopy of carbon fullerenes and nanotubes

The non-resonant X-ray emission spectroscopy of fullerenes and carbon nanotubes is studied with density functional theory in conjunction with short-range corrected functionals. For C60 and C70 the X-ray emission spectra are insensitive to modest structural changes, and absorption onto the fullerene cage has the greatest effect with a broader less structured band observed with the high energy π band reduced in intensity. For carbon nanotubes the X-ray emission spectra are shown to be weakly dependent on the length and chirality of the nanotube. However, some variation with the diameter of the tube is observed in both resonant and non-resonant spectra.

Distance-including rigorous upper bounds and tight estimates for two-electron integrals over long- and short-range operators.

We introduce both rigorous and non-rigorous distance-dependent integral estimates for four-center two-electron integrals derived from a distance-including Schwarz-type inequality. The estimates are even easier to implement than our so far most efficient distance-dependent estimates [S. A. Maurer et al., J. Chem. Phys. 136, 144107 (2012)] and, in addition, do not require well-separated charge-distributions. They are also applicable to a wide range of two-electron operators such as those found in explicitly correlated theories and in short-range hybrid density functionals. For two such operators with exponential distance decay [e-r12 and erfc(0.11⋅r12)/r12], the rigorous bound is shown to be much tighter than the standard Schwarz estimate with virtually no error penalty. The non-rigorous estimate gives results very close to an exact screening for these operators and for the long-range 1/r12 operator, with errors that are completely controllable through the integral screening threshold. In addition, we present an alternative form of our non-rigorous bound that is particularly well-suited for improving the PreLinK method [J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 138, 134114 (2013)] in the context of short-range exchange calculations.

Screening methods for linear-scaling short-range hybrid calculations on CPU and GPU architectures.

We present screening schemes that allow for efficient, linear-scaling short-range exchange calculations employing Gaussian basis sets for both CPU and GPU architectures. They are based on the LinK [C. Ochsenfeld et al., J. Chem. Phys. 109, 1663 (1998)] and PreLinK [J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 138, 134114 (2013)] methods, but account for the decay introduced by the attenuated Coulomb operator in short-range hybrid density functionals. Furthermore, we discuss the implementation of short-range electron repulsion integrals on GPUs. The introduction of our screening methods allows for speedups of up to a factor 7.8 as compared to the underlying linear-scaling algorithm, while retaining full numerical control over the accuracy. With the increasing number of short-range hybrid functionals, our new schemes will allow for significant computational savings on CPU and GPU architectures.

Performance of dispersion-corrected density functional theory for thermochemistry and non-covalent interactions

The accuracy of non-local van der Waals density functional [1] is tested for the thermochemical properties of 1200+ atoms and molecules in the GMTKN30 database. Five (hybrid)GGA functionals are augmented by the non-local (NL) part of the VV10 functional. The widely used atom-pair wise dispersion correction DFT-D3 [2] is considered for comparison. The addition of the NL dispersion energy definitely improves the results of all tested short-range functionals. Based on little empiricism and basic physical insight, DFT-NL can be recommended as robust electronic structure method. For more detailed insight into non-covalent bonding, potential energy curves [3] for five complexes with weak to medium strong hydrogen bonds have been computed with dispersion corrected DFT methods VV10, DFT-D3 and vdW-DF2 [4]. All dispersion corrected methods perform reasonably well for these hydrogen bonds. For the fluorinated complexes, the VV10 method gives remarkably good results. The vdW-DF2 method yields good interaction energies similar to the other methods, but fails to provide accurate equilibrium separations. For large-scale applications we can recommend DFT-D3 based structure optimizations with subsequent checking of interaction energies by single-point VV10 computations.

Electronic level alignment at a metal-molecule interface from a short-range hybrid functional

Hybrid functionals often exhibit a marked improvement over semi-local functionals in the description of the electronic structure of organic materials. Because short-range hybrid functionals, notably the Heyd-Scuseria-Ernzerhof (HSE) functional, can also describe the electronic structure of metals reasonably well, it is interesting to examine to which extent they can correctly describe the electronic structure at metal-organic interfaces. Here, we address this question by comparing HSE calculations with many-body perturbation theory calculations in the GW approximation, or with experimental photoemission data, for two prototypical systems: benzene on graphite and benzene diamine on gold. For both cases, we find that while HSE yields results that are somewhat closer to experiment than those of semi-local functionals, the HSE prediction is still lacking quantitatively by ∼1 eV. We show that this quantitative failure arises because HSE does not correctly capture the fundamental gap of the organic or its renormalization by the metal. These discrepancies are traced back to missing long-range exchange and correlation components, an explanation which applies to any conventional or short-range hybrid functional.

Coupling of Short-range Density-functional with Long-range Post-Hartree-Fock Methods

Abstract A review is given on recent work with range-separated hybrid density-functional approaches, using accurate wavefunction-based ab initio methods for the treatment of the long-range inter-electronic interaction. It is shown that in many cases results can be improved with respect to the borderline methods, since the hybrid approach is less basis-set dependent than in the pure ab initio case, and deficiencies of pure DFT for weak interactions are efficiently removed. However, there is still need for improvement of currently available short-range density functionals.

Robust Periodic Hartree-Fock Exchange for Large-Scale Simulations Using Gaussian Basis Sets.

Hartree-Fock exchange with a truncated Coulomb operator has recently been discussed in the context of periodic plane-waves calculations [Spencer, J.; Alavi, A. Phys. Rev. B: Solid State, 2008, 77, 193110]. In this work, this approach is extended to Gaussian basis sets, leading to a stable and accurate procedure for evaluating Hartree-Fock exchange at the Γ-point. Furthermore, it has been found that standard hybrid functionals can be transformed into short-range functionals without loss of accuracy. The well-defined short-range nature of the truncated exchange operator can naturally be exploited in integral screening procedures and makes this approach interesting for both condensed phase and gas phase systems. The presented Hartree-Fock implementation is massively parallel and scales up to ten thousands of cores. This makes it feasible to perform highly accurate calculations on systems containing thousands of atoms or ten thousands of basis functions. The applicability of this scheme is demonstrated by calculating the cohesive energy of a LiH crystal close to the Hartree-Fock basis set limit and by performing an electronic structure calculation of a complete protein (rubredoxin) in solution with a large and flexible basis set.

On the universality of the long-/short-range separation in multiconfigurational density-functional theory. II. Investigating f actinide species

In a previous paper [Fromager et al., J. Chem. Phys. 126, 074111 (2007)], some of the authors proposed a recipe for choosing the optimal value of the mu parameter that controls the long-range/short-range separation of the two-electron interaction in hybrid multiconfigurational self-consistent field short-range density-functional theory (MC-srDFT) methods. For general modeling with MC-srDFT methods, it is clearly desirable that the same universal value of mu can be used for any molecule. Their calculations on neutral light element compounds all yielded mu(opt)=0.4 a.u. In this work the authors investigate the universality of this value by considering "extreme" study cases, namely, neutral and charged isoelectronic f(0) actinide compounds (ThO(2), PaO(2)(+), UO(2)(2+), UN(2), CUO, and NpO(2)(3+)). We find for these compounds that mu(opt)=0.3 a.u. but show that 0.4 a.u. is still acceptable. This is a promising result in the investigation of a universal range separation. The accuracy of the currently best MC-srDFT (mu=0.3 a.u.) approach has also been tested for equilibrium geometries. Though it performs as well as wave function theory and DFT for static-correlation-free systems, it fails in describing the neptunyl (VII) ion NpO(2)(3+) where static correlation is significant; bending is preferred at the MC-srDFT (mu=0.3 a.u.) level, whereas the molecule is known to be linear. This clearly shows the need for better short-range functionals, especially for the description of the short-range exchange. It also suggests that the bending tendencies observed in DFT for NpO(2)(3+) cannot be fully explained by the bad description of static correlation effects by standard functionals. A better description of the exchange seems to be essential too.

Development and assessment of a short-range meta-GGA functional

Short-range DFT/long-range ab initio methods allow for a combination of the weak basis-set dependency of DFT with an accurate ab initio treatment of long-range effects like van der Waals interaction. In order to improve existing short-range LDA and GGA density functionals, we developed a TPSS-like short-range meta-GGA exchange-correlation functional and checked its performance in long-range CCSD(T) calculations for thermodynamical properties of the G2 set of molecules.

Time-dependent density functional theory calculations of near-edge X-ray absorption fine structure with short-range corrected functionals

We report calculations of core excitation energies and near-edge X-ray absorption fine structure (NEXAFS) spectra computed with time-dependent density functional theory (TDDFT). TDDFT with generalized gradient approximation and standard hybrid exchange-correlation functionals is known to underestimate core excitation energies. This failure is shown to be associated with the self-interaction error at short interelectronic distances. Short-range corrected hybrid functionals are shown to reduce the error in the computed core excitation energies for first and second row nuclei in a range of molecules to a level approaching that observed in more traditional excited states calculations in the ultraviolet region. NEXAFS spectra computed with the new functionals agree well with experiment and the pre-edge features in the NEXAFS spectra of plastocyanin are correctly predicted.

Short-range density functionals in combination with local long-range ab initio methods: Application to non-bonded complexes

Within this work, we introduce local-correlation approximations into a hybrid approach which couples short-range density functional-based with long-range wave-function-based methods. The local treatment of long-range ab initio correlation is expected to greatly enhance the efficiency of the mixed approach for large systems. We start from our previous implementation, where a short-range gradient-corrected density functional of the Perdew–Burke–Ernzerhof (PBE) type is coupled with long-range Møller–Plesset second order perturbation theory and coupled-cluster methods, and extend the scheme to include the corresponding local long-range ab initio methods. The accuracy of the resulting approach is tested using the benchmark database for non-bonded (or, more precisely, weakly bonded) interactions of Zhao and Truhlar. The effect of the local-correlation approximation as well as the influence of counterpoise corrections is discussed for the different types of methods. It is shown that the mixed short-range/long-range approach is less sensitive to the ab initio correlation level and the one-particle basis-set used than pure ab initio methods, while the overall performance is significantly improved with respect to pure PBE.

Local and density fitting approximations within the short-range/long-range hybrid scheme: application to large non-bonded complexes

We introduce local and density fitting approximations into a hybrid approach which couples short-range density functionals with long-range wave-functions. The use of density fitting methods within the local-correlation approximation makes it possible to tackle bigger systems than without these methods, so that we are now able to treat systems of biological interest within the DFT/ab initio scheme. In first benchmark calculations, we apply it to the S22 database of Hobza and co-workers for binding energies of weakly bound molecular clusters.