Range-separation Parameter Research Articles

Range-Separation Parameter in Tuned Exchange-Correlation Functionals: Successive Ionizations and the Fukui Function.

The range-separation parameter in tuned, range-separated exchange-correlation functionals is investigated in two contexts. First, the system dependence of the parameter is investigated for a series of systems obtained by successively ionizing a single species, paying particular attention to the degree of linearity in the energy versus electron number curve. The parameter exhibits significant system dependence and, therefore, achieving near-linearity in one segment of the curve leads to strong nonlinearity in other segments. This provides a challenging test case for the development of new functionals designed to overcome the known problems of this class of functional. Next, the study considers whether a range-separation parameter tuned to a Koopmans energy condition is also applicable for the analogous density condition. This is tested by comparing two formulations of the Fukui function of conceptual density functional theory, for three representative systems. Both formulations yield the same general features and are not highly sensitive to the range-separation parameter. However, the agreement between the two is near-optimal when the energy-tuned parameter is used, indicating that this parameter is applicable for the analogous density condition.

Employing Range Separation on the meta-GGA Rung: New Functional Suitable for Both Covalent and Noncovalent Interactions.

We devise a scheme for converting an existing exchange functional into its range-separated hybrid variant. The underlying exchange hole of the Becke-Roussel type has the exact second-order expansion in the interelectron distance. The short-range part of the resulting range-separated exchange energy depends on the kinetic energy density and the Laplacian even if the base functional lacks the dependence on these variables. The most successful practical realization of the scheme, named LC-PBETPSS, combines the range-separated Perdew-Burke-Ernzerhof (PBE) exchange lifted to the hybrid meta-generalized gradient approximation rung and the Tao-Perdew-Staroverov-Scuseria (TPSS) correlation. The value of the range-separation parameter is estimated theoretically and confirmed by empirical optimization. The D3 dispersion correction is recommended for all energy computations employing the presented functional. Numerical tests show remarkably robust performance of the method for noncovalent interaction energies, barrier heights, main-group thermochemistry, and excitation energies.

Ghost-interaction correction in ensemble density-functional theory for excited states with and without range separation

Ensemble density-functional theory (eDFT) suffers from the so-called "ghost interaction" error when approximate exchange-correlation functionals are used. In this work, we present a rigorous ghost interaction correction (GIC) scheme in the context of range-separated eDFT. The method relies on an exact decomposition of the ensemble short-range exchange-correlation energy into a multideterminantal exact exchange term, which involves the long-range interacting ensemble density matrix instead of the Kohn--Sham (KS) one, and a complementary density-functional correlation energy. A generalized adiabatic connection formula is derived for the latter. In order to perform practical calculations, the complementary correlation functional has been simply modeled by its ground-state local density approximation (LDA) while long-range interacting ground- and excited-state wavefunctions have been obtained self-consistently by combining a long-range configuration interaction calculation with a short-range LDA potential. We show that GIC reduces the curvature of approximate range-separated ensemble energies drastically while providing considerably more accurate excitation energies, even for charge-transfer and double excitations. Interestingly, the method performs well also in the context of standard KS-eDFT, which is recovered when the range-separation parameter is set to zero.

Description of the Charge Transfer States at the Pentacene/C60 Interface: Combining Range-Separated Hybrid Functionals with the Polarizable Continuum Model

Density functional theory (DFT) approaches based on range-separated hybrid functionals are currently methods of choice for the description of the charge-transfer (CT) states in organic donor/acceptor solar cells. However, these calculations are usually performed on small-size donor/acceptor complexes and as result do not account for electronic polarization effects. Here, using a pentacene/C60 complex as a model system, we discuss the ability of long-range corrected (LCR) hybrid functionals in combination with the polarizable continuum model (PCM) to determine the impact of the solid-state environment on the CT states. The CT energies are found to be insensitive to the interactions with the dielectric medium when a conventional time-dependent DFT/PCM (TDDFT/PCM) approach is used. However, a decrease in the energy of the CT state in the framework of LRC functionals can be obtained by using a smaller range-separated parameter when going from an isolated donor/acceptor complex to the solid-state case.

Diffusion Monte Carlo Calculations for Rare-earths: Applying the Long-range Corrected Scheme to Minnesota M06 Functional

By employing the diffusion Monte Carlo (DMC) method, we test the success of the unpublished long-range corrected M06 (LC-M06) functional, compared to LC-BLYP functional, in improving the calculations of the ground and excited states energies for some 4f- and 5f-rare-earths at different values of the range separation parameter μ. It is found that LC-M06 functional performs well for both the ground and the excited states of 4f-lanthanides. In addition, our results reveal that the optimal value of μ for the ground state lies between 0.40-0.50 a.u. while a smaller optimal value μ =0.35 a.u. is found for the excited states. However, LC-BLYP ground and excited states results at the optimal value of μ seem to be the best for 5f-actinides.

Improving the Description of the Optical Properties of Carotenoids by Tuning the Long-Range Corrected Functionals.

In this work we use gap-fitting procedure to tune the long-range corrected functionals and accurately investigate the electronic and optical properties of the five main molecules composing Buriti oil (extracted from Mauritia flexuosa L.) in the framework of density functional theory (DFT) and time-dependent (TD) DFT. The characteristic length (1/ω) was observed to be entirely system dependent, though we concluded that its determination is of fundamental importance to rescue geometrical, electronic, and optical properties with accuracy. We demonstrate that our approach of tuning characteristic length for each system resulted in an absorbance spectra in better experimental agreement when compared to the traditional methodology. Therefore, this study indicates that the tuning of the range-separation parameter is crucial to improve the description of the optical properties of conjugated molecules when TDDFT is used. For example, the wavelength of maximum absorption, λmax, for the phytofluene, obtained using B3LYP, is 381 nm, while using the gap-fitting procedure for the tuned-LC-BLYP the estimated λmax changed to 358 nm. The latter estimate is in better agreement with the experimental value of 350 nm.

Diffusion Monte Carlo Calculations for Rare-earths: Hartree-Fock, Hybrid B3LYP, and Long-range Corrected LC-BLYP Functional

Our study aim is to make highlights on the failure or success of both popular B3LYP functional and long-range corrected LC-BLYP functional at different values of the range separation parameter in improving the ground state and excited states energies calculations for 4f-lanthanides and 5f-actinides. The most popular B3LYP functional fails to provide a balanced description of excitation energies for 4f-lanthanides. However, it gives reasonable results for the actinides with exception of Pu and Am atoms. The ground state energies of 5f-actinides are improved significantly at the critical value of the range separation parameter equals 0.35 a.u. while the optimal value for the excited states lies in the critical range of 0.35-0.4 a.u. However, our results show that LC-BLYP functional is not recommended for 4f-lanthanides which have strongly localized nature.

First principles optimally tuned range-separated density functional theory for prediction of phosphorus-hydrogen spin-spin coupling constants.

Optimally tuned range-separated (OT-RS) density functional theory (DFT) is a recent endeavor toward the systematic and non-empirical routes for designing the exchange-correlation functionals. Herein, a detailed analysis of the development and benchmarking of the OT-RS functionals for predicting the experimental nuclear magnetic resonance (NMR) spin-spin coupling constants (SSCCs) in diverse sets of compounds containing phosphorus-hydrogen (P-H) bonds has been done. More specifically, besides analyzing the performances of standard long-range corrected (LC) functionals, two new non-empirical OT-RS functionals are proposed for this purpose. Furthermore, we dissect the importance of both short- and long-range exchange contributions and range separation parameters in LC density functional calculations of P-H SSCCs. It is shown that the proposed functionals not only give an improved description of SSCCs with respect to conventional LC approximations but also in many cases perform better than other functionals from various rungs. The accountability of the new models for predicting the SSCCs and their components in continuum solvents has also been examined and validated. Overall, we hope that this contribution stimulates the development of novel OT-RS DFT approximations based on theoretical arguments as a methodology with both high accuracy and computational efficiency for modeling the NMR parameters.

On describing the optoelectronic characteristics of poly(benzodithiophene-co-quinoxaline)-fullerene complexes: the influence of optimally tuned density functionals.

Here, we investigate the effects of both tuning the range-separation parameter of long-range corrected (LRC) density functionals and including dispersion corrections on describing the local optoelectronic properties of polymer-fullerene interfaces that are critical to the performance of polymer solar cells (PSCs). Focusing on recently studied (Chen, et al., Chem. Mater., 2012, 24, 4766-4772) PSC active layers derived from poly(benzodithiophene-co-quinoxaline) and substituted fullerene PC71BM, we compare the performance of global hybrid functionals (B3LYP and B3LYP-D) alongside two LRC functionals (ωB97X and ωB97X-D) and their optimally tuned (OT) analogs (OT-ωB97X and OT-ωB97X-D). Our results confirm that OT-LRC functionals generally improve the description of the optical properties of the individual materials with respect to experiment. For electron-donor (eD)-electron-acceptor (eA) complexes used to describe the local optoelectronic properties of the material interface, PC71BM is found to preferentially settle near the quinoxaline acceptor units on the copolymer backbone, regardless of the functional, though dispersion corrections have a strong influence on the intermolecular distances and, in turn, the nature of the excited states. All functionals yield very similar descriptions of the transition maxima for the complexes, i.e. predominant local excitations on the copolymer. Importantly, tuning the range-separation parameter of the LRC functional is shown to have a profound effect, as the OT functionals allow for the description of the charge transfer states of the eD-eA complexes, while the non-tuned LRC functionals predict only local intramolecular excitations. These results hold considerable importance for deriving the appropriate physical understanding of the interfacial structure-property-function relationships of PSCs.

Combining linear interpolation with extrapolation methods in range-separated ensemble density functional theory

ABSTRACTThe combination of a recently proposed linear interpolation method (LIM), which enables the calculation of weight-independent excitation energies in range-separated ensemble density functional approximations, with the extrapolation scheme of Savin is presented in this work. It is shown that LIM excitation energies vary quadratically with the inverse of the range-separation parameter μ when the latter is large. As a result, the extrapolation scheme, which is usually applied to long-range interacting energies, can be adapted straightforwardly to LIM. This extrapolated LIM (ELIM) has been tested on a small test set consisting of He, Be, H2 and HeH+. Relatively accurate results have been obtained for the first singlet excitation energies with the typical μ = 0.4 value. The improvement of LIM after extrapolation is remarkable, in particular for the doubly excited 21Σ+g state in the stretched H2 molecule. Three-state ensemble calculations in H2 also show that ELIM does not necessarily improves relative excitation energies, even though individual excitation energies are more accurate after extrapolation. Finally, an alternative decomposition of the short-range ensemble exchange–correlation energy is proposed in order to correct for ghost-interaction errors in multi-determinant range-separated ensemble density functional theory calculations. The implementation and calibration of such a scheme arecurrently in progress.

Tuning range-separated DFT functionals for accurate orbital energy modeling of conjugated molecules

Density functional theory (DFT) calculations with range-separated (RS) functionals are important for orbital energy modeling of conjugated molecules that involve charge transfer excitation. However, the accuracy of the computed results depends on the range-separation parameter (w), and the optimal values of w for a wide range of conjugated systems has hitherto been missing. Herein, orbital energy modeling of twelve representative conjugated molecules, that are promising electron-donors in bulk heterojunction-based organic photovoltaic devices, are benchmarked using DFT and time-dependent DFT (TD-DFT) with three RS functionals (LC-BLYP, wB97XD and CAM-B3LYP). The results using the RS functionals under consideration with default values of w deviate largely from the experimental values (mean signed error (MSE) on HOMO are 2.1eV, 0.93eV and 1.5eV, and MSE on vertical excitation energies are 0.47eV, 0.55eV and 0.82eV, respectively). Computation of orbital energies using tuned range-separation parameter for these RS functionals in the range of 0.05⩽w⩾0.5Bohr−1 indicates that w plays a significant role on the accuracy of the ground and the excited state energies. We found that the accurate orbital energies of conjugated systems can be predicted using values of w between 0.1 and 0.15Bohr−1, much smaller than the default values of 0.47, 0.33 and 0.20Bohr−1 used in LC-BLYP, CAM-B3LYP and wB97XD, respectively.

Linear interpolation method in ensemble Kohn-Sham and range-separated density-functional approximations for excited states

Gross-Oliveira-Kohn density-functional theory (GOK-DFT) for ensembles is, in principle, very attractive but has been hard to use in practice. A practical model based on GOK-DFT for the calculation of electronic excitation energies is discussed. The model relies on two modifications of GOK-DFT: use of range separation and use of the slope of the linearly interpolated ensemble energy, rather than orbital energies. The range-separated approach is appealing, as it enables the rigorous formulation of a multideterminant state-averaged DFT method. In the exact theory, the short-range density functional, which complements the long-range wave-function-based ensemble energy contribution, should vary with the ensemble weights even when the density is held fixed. This weight dependence ensures that the range-separated ensemble energy varies linearly with the ensemble weights. When the (weight-independent) ground-state short-range exchange-correlation functional is used in this context, curvature appears, thus leading to an approximate weight-dependent excitation energy. In order to obtain unambiguous approximate excitation energies, we propose to interpolate linearly the ensemble energy between equiensembles. It is shown that such a linear interpolation method (LIM) can be rationalized and that it effectively introduces weight dependence effects. As proof of principle, the LIM has been applied to He, Be, and ${\text{H}}_{2}$ in both equilibrium and stretched geometries as well as the stretched ${\text{HeH}}^{+}$ molecule. Very promising results have been obtained for both single (including charge transfer) and double excitations with spin-independent short-range local and semilocal functionals. Even at the Kohn-Sham ensemble DFT level, which is recovered when the range-separation parameter is set to 0, LIM performs better than standard time-dependent DFT.

Tuned range separated hybrid functionals for solvated low bandgap oligomers

The description of charge transfer excitations has long been a challenge to time dependent density functional theory. The recently developed concept of "optimally tuned range separated hybrid (OT-RSH) functionals" has proven to describe charge transfer excitations accurately in many cases. However, describing solvated or embedded systems is yet a challenge. This challenge is not only computational but also conceptual, because the tuning requires identifying a specific orbital, typically the highest occupied one of the molecule under study. For solvated molecules, this orbital may be delocalized over the solvent. We here demonstrate that one way of overcoming this problem is to use a locally projected self-consistent field diagonalization on an absolutely localized molecular orbital expansion. We employ this approach to determine ionization energies and the optical gap of solvated oligothiophenes, i.e., paradigm low gap systems that are of relevance in organic electronics. Dioxane solvent molecules are explicitly represented in our calculations, and the ambiguities of straightforward parameter tuning in solution are elucidated. We show that a consistent estimate of the optimal range separated parameter (ω) at the limit of bulk solvation can be obtained by gradually extending the solvated system. In particular, ω is influenced by the solvent beyond the first coordination sphere. For determining ionization energies, a considerable number of solvent molecules on the first solvation shell must be taken into account. We demonstrate that accurately calculating optical gaps of solvated systems using OT-RSH can be done in three steps: (i) including the chemical environment when determining the range-separation parameter, (ii) taking into account the screening due to the solvent, and (iii) using realistic molecular geometries.

Molecules relevant for organic photovoltaics: a range-separated density functional study

Accurate determination of both fundamental and optical gap is necessary for designing molecules relevant for organic photovoltaics. Here, we study how range-separated density functionals reproduce frontier orbital energies, HOMO (highest occupied molecular orbital)–LUMO (lowest unoccupied molecular orbital) gaps, and optical gaps for molecules relevant for organic photovoltaics. In this study, we consider 12 different range-separated density functional for computing HOMO energy, HOMO–LUMO gap, and optical gap which are compared with available experimental and reported GW values. We found that the reproduction of desired photovoltaic properties primarily depend on range separation parameter. Moreover, the tested functionals are comparable with OT-BNL functional.

Range-separated double-hybrid density-functional theory applied to periodic systems.

Quantum chemistry methods exploiting density-functional approximations for short-range electron-electron interactions and second-order Møller-Plesset (MP2) perturbation theory for long-range electron-electron interactions have been implemented for periodic systems using Gaussian-type basis functions and the local correlation framework. The performance of these range-separated double hybrids has been benchmarked on a significant set of systems including rare-gas, molecular, ionic, and covalent crystals. The use of spin-component-scaled MP2 for the long-range part has been tested as well. The results show that the value of μ = 0.5 bohr(-1) for the range-separation parameter usually used for molecular systems is also a reasonable choice for solids. Overall, these range-separated double hybrids provide a good accuracy for binding energies using basis sets of moderate sizes such as cc-pVDZ and aug-cc-pVDZ.

On the Performance of Optimally Tuned Range-Separated Hybrid Functionals for X-ray Absorption Modeling

We investigate the performance of optimally tuned range-separated hybrid functionals (OT-RSH) for modeling X-ray absorption spectra (XAS) of a benchmark set of simple molecules (water, ammonia, methane, hydrogen peroxide, hydrazine, and ethane), using time-dependent density functional theory (TDDFT). Spectra were simulated within the Path Integral based Reflection Principle methodology. Relative intensities, peak positions, and widths were compared with available experimental data. We show that the OT-RSH approach outperforms empirically parametrized functionals in terms of relative peak positions and intensities. Furthermore, we investigate the effect of geometry specific tuning where the range separation parameter is optimized for each geometry. Finally, we propose a simple correction scheme allowing for calculations of XAS on the absolute energy scale using the OT-RSH approach combined with ΔSCF/TDDFT-based calculations of core ionization energies.

Quantum chemical study of the electronic properties of an Iridium-based photosensitizer bound to medium-sized silver clusters

The equilibrium structures and electronic excitation spectra of the Ir(III) photosensitizer [Ir(ppy)2(bpy)]+ bound to medium-sized silver clusters Agn(n=19,20) are investigated using time-dependent density functional theory. The long-range corrected LC-BLYP approach is used with a system-specific range separation parameter. The weak physisorption of the hybrid complexes yields only small changes in the broadened absorption spectra of the hybrid system as compared with its constituents. However, the density of states as well as the fine structure of the spectra is strongly modified upon complexation. It is shown that the standard range separation parameter (0.47bohr−1) cannot predict these properties correctly and the optimized value of 0.16bohr−1 should be used instead.

Calculating excitation energies by extrapolation along adiabatic connections

In this paper, an alternative method to range-separated linear-response time-dependent density-functional theory and perturbation theory is proposed to improve the estimation of the energies of a physical system from the energies of a partially interacting system. Starting from the analysis of the Taylor expansion of the energies of the partially interacting system around the physical system, we use an extrapolation scheme to improve the estimation of the energies of the physical system at an intermediate point of the range-separated or linear adiabatic connection where either the electron--electron interaction is scaled or only the long-range part of the Coulomb interaction is included. The extrapolation scheme is first applied to the range-separated energies of the helium and beryllium atoms and of the hydrogen molecule at its equilibrium and stretched geometries. It improves significantly the convergence rate of the energies toward their exact limit with respect to the range-separation parameter. The range-separated extrapolation scheme is compared with a similar approach for the linear adiabatic connection, highlighting the relative strengths and weaknesses of each approach.

Tuning Range-Separated Density Functional Theory for Photocatalytic Water Splitting Systems.

We discuss the system-specific optimization of long-range-separated density functional theory (DFT) for the prediction of electronic properties relevant for a photocatalytic cycle based on an Ir(III) photosensitizer (IrPS). Special attention is paid to the charge-transfer properties, which are of key importance for the photoexcitation dynamics but cannot be correctly described by means of conventional DFT. The optimization of the range-separation parameter using the ΔSCF method is discussed for IrPS including its derivatives and complexes with electron donors and acceptors used in photocatalytic hydrogen production. Particular attention is paid to the problems arising for a description of medium effects by means of a polarizable continuum model.

Calculating Electron-Transfer Coupling with Density Functional Theory: The Long-Range-Corrected Density Functionals

The density functional theory (DFT) with commonly used functionals is known to be incorrect for charge-transfer problems. With long-range-corrected (LC) density functionals, the asymptotic exchange potential is gradually switched to the Hartree-Fock exchange at a long range, and the prediction for charge-transfer states is greatly improved. In this work, we test LC-DFT's performance on charge-transfer couplings. The range-separation parameter can be tuned nonempirically for properties of a generalized DFT. We propose to minimize the difference of highest-occupied Kohn-Sham orbital energy and the ionization potential (for hole transfer) or the lowest-unoccupied orbital energy and the electron affinity (for electron transfer). For photoinduced charge transfer, the minimum in the sum of such differences for the donor and the acceptor is proposed. With the range-separation parameters optimized, we found that ET couplings derived from the LC-DFT are close to those derived from coupled cluster with singles and doubles. When compared with experimentally derived Mulliken-Hush couplings, LC-DFT couplings are greatly improved as well. We also found that the couplings from BNL and LC-BLYP functionals are generally better than those from LC-ωPBE and LC-ωPBE0. LC-DFT is suitable for calculating ET coupling, especially with this nonempirical approach for the range-separation parameter.