Long-range Hartree-Fock Exchange Research Articles

Long-range Corrected Density Functional Theory Including a Two-Gaussian Hartree-Fock Operator for High Accuracy Core-excitation Energy Calculations of Both the Second- and Third-Row Atoms (LC2gau-core-BOP).

In the previous work, LCgau-core-BOP, which includes the short-range interelectronic Gaussian attenuating Hartree-Fock (HF) exchange to the long-range HF exchange, showed high accuracy core-excitation energies from 1s orbitals of the second-row atoms (1s → π*, 1s → σ*, 1s → n*, and 1s → Rydberg), but underestimates the core-excitation energies from 1s orbitals of the third-row atoms. To improve this, we added one more Gaussian attenuating HF exchange to LCgau-core-BOP. We named it LC2gau-core-BOP, which achieves a mean absolute error (MAE) of 0.6 and 0.3 eV for core excitation energies of the second- and third-row atoms of the tested small molecules, respectively. We found that the inclusion of the short-range interelectronic HF exchange at a distance ranging from 0.2 to 0.6 a.u. contributes to the increase of performances on 1s orbital energy calculations of the second-row atoms, while the inclusion of more short-range interelectronic HF exchange at a distance ranging from 0 to 0.2 a.u. does to the increase of performance on 1s orbital energy calculations of the third-row atoms. It is notable that all of these improvements were accomplished using flexible Gaussian attenuating HF exchange inclusion. LC2gau-core-BOP shows deviations of less than 0.8 eV from experimental values for all of the core-excitation energies of the tested medium-size molecules consisting of thymine, oxazole, glycine, and dibenzothiophene sulfone. Moreover, by optimizing one parameter of the OP correlation functional, LC2gau-core-BOP provides atomization energies over the G3 test set with an accuracy comparable to that of B3LYP.

Band Gap Energy of Periodic Anatase TiO2 System Evaluated with the B2PLYP Double Hybrid Functional

The electronic properties of anatase titanium dioxide (TiO2) materials are of paramount importance for photocatalytic application. Ab initio calculation is performed on anatase TiO2 with various cluster sizes and shape, using Gaussian 09 program employing the standard 6-311G(d) and 3-21G basis set. Hartree-Fock (HF) theory, exchange-functional of density functional theory including hybrid (B3LYP, B3PW91, PBE1PBE and PBEh1PBE) and double-hybrid (B2PLYP), together with 2nd order Møller-Plesset perturbation theory are used to predict the band gap energy of anatase TiO2. With the inclusion of long-range HF exchange, double hybrid B2PLYP functional is able to predict band gap energy value (3.06 eV) which as compared to the experimental value (3.20 eV). Besides, this double hybrid exchange-correlation functional shows good compromise by obtaining an accurate description for cohesive energy of anatase TiO2. Thus, double-hybrid B2PLYP functional is suggested to be a practical choice for predicting the electronic properties for anatase TiO2.

Stacked Ensemble Machine Learning for Range-Separation Parameters

Density functional theory-based high-throughput materials and drug discovery has achieved tremendous success in recent decades, but its power on organic semiconducting molecules suffered catastrophically from the self-interaction error until the nonempirical but expensive optimally tuned range-separated hybrid (OT-RSH) functionals were developed. An OT-RSH transitions from a short-range (semi)local functional to a long-range Hartree-Fock exchange at a distance characterized by a molecule-specific range-separation parameter (ω). Herein, we propose a stacked ensemble machine learning model that provides an accelerated alternative of OT-RSH based on system-dependent structural and electronic configurations. We trained ML-ωPBE, the first functional in our series, using a database of 1970 molecules with sufficient structural and functional diversity, and assessed its accuracy and efficiency using another 1956 molecules. Compared with nonempirical OT-ωPBE, ML-ωPBE reaches a mean absolute error of 0.00504a0-1 for optimal ω's, reduces the computational cost by 2.66 orders of magnitude, and achieves comparable predictive power in optical properties.

Role of exchange and correlation in high-harmonic generation spectra of H2, N2, and CO2: Real-time time-dependent electronic-structure approaches.

This study arises from the attempt to answer the following question: how different descriptions of electronic exchange and correlation affect the high-harmonic generation (HHG) spectroscopy of H2, N2, and CO2 molecules? We compare HHG spectra for H2, N2, and CO2 with different ab initio electronic structure methods: real-time time-dependent configuration interaction and real-time time-dependent density functional theory (RT-TDDFT) using truncated basis sets composed of correlated wave functions expanded on Gaussian basis sets. In the framework of RT-TDDFT, we employ Perdew-Burke-Ernzerhof (PBE) and long-range corrected Perdew-Burke-Ernzerhof (LC-ωPBE) functionals. We study HHG spectroscopy by disentangling the effect of electronic exchange and correlation. We first analyze the electronic exchange alone, and in the case of RT-TDDFT with LC-ωPBE, we use ω = 0.3 and ω = 0.4 to tune the percentage of long-range Hartree-Fock exchange and short-range exchange PBE. Then, we added the correlation as described by the PBE functional. All the methods give very similar HHG spectra, and they seem not to be particularly sensitive to the different description of exchange and correlation or to the correct asymptotic behavior of the Coulomb potential. Despite this general trend, some differences are found in the region connecting the cutoff and the background. Here, the harmonics can be resolved with different accuracy depending on the theoretical schemes used. We believe that the investigation of the molecular continuum and its coupling with strong fields merits further theoretical investigations in the near future.

UV/Vis absorption spectrum calculations of benzo-1,2-dipyrene isomer using long-range corrected density functional theory

Anti- and syn-isomers of benzo-1,2-dipyrene were synthesized, and nonidentical experimental absorption spectra were observed. The B3LYP, B3LYP-D3, ωB97XD, and optimally tuned long-range corrected (LC) BOP functionals were assessed. B3LYP and B3LYP-D3 underestimated and ωB97XD overestimated the fundamental gaps and their following optical gaps in both isomers. The LC-BOP functional with the range separation parameter (µ=0.18) using the geometries optimized with LCgau-BOP + LRD reproduced the experimental absorption spectra of both isomers well. Therefore, in the case of the absorption spectrum calculation of the excimer-like isomers, it is significant to include both the weak interaction term and long-range Hartree-Fock exchange interaction appropriately.

Application of accelerated long-range corrected exchange functional [LC-DFT(2Gau)] to periodic boundary condition systems: CO adsorption on Cu(111) surface.

Several different types of density functional theory (DFT) exchange correlation functionals were applied to a periodic boundary condition (PBC) system [carbon monoxide (CO) adsorbed on Cu(111): CO/Cu(111)] and the differences in the results calculated using these functionals were compared. The exchange correlation functionals compared were those of Perdew-Burke-Ernzerhof (PBE) and those of long-range corrected density functional theory (LC-DFT), such as LC-ωPBE(2Gau) and LC-BLYP(2Gau). Solid state properties such as the partial density of states were calculated in order to elucidate the detailed adsorption mechanisms and back-bonding peculiar to the CO/Cu(111) system. In addition, our benchmark analysis of the correlations among the orbitals of CO and Cu metal using LC-DFT reasonably was in line with the experimentally observed adsorption site. The computation time was reasonable, and other numerical results were found to agree well with the experimental results and also with the theoretical results of other researchers. This suggests that the long-range Hartree-Fock exchange integral should be included to correctly predict the electronic nature of PBC systems.

Benchmarking the Performance of Time-Dependent Density Functional Theory Methods on Biochromophores.

Quantum chemical calculations are important for elucidating light-capturing mechanisms in photobiological systems. The time-dependent density functional theory (TDDFT) has become a popular methodology because of its balance between accuracy and computational scaling, despite its problems in describing, for example, charge transfer states. As a step toward systematically understanding the performance of TDDFT calculations on biomolecular systems, we study here 17 commonly used density functionals, including seven long-range separated functionals, and compare the obtained results with excitation energies calculated at the approximate second order coupled-cluster theory level (CC2). The benchmarking set includes the first five singlet excited states of 11 chemical analogues of biochromophores from the green fluorescent protein, rhodopsin/bacteriorhodopsin (Rh/bR), and the photoactive yellow protein. We find that commonly used pure density functionals such as BP86, PBE, M11-L, and hybrid functionals with 20-25% of Hartree-Fock (HF) exchange (B3LYP, PBE0) have a tendency to consistently underestimate vertical excitation energies (VEEs) relative to the CC2 values, whereas hybrid density functionals with around 50% HF exchange such as BHLYP, PBE50, and M06-2X and long-range corrected functionals such as CAM-B3LYP, ωPBE, ωPBEh, ωB97X, ωB97XD, BNL, and M11 overestimate the VEEs. We observe that calculations using the CAM-B3LYP and ωPBEh functionals with 65% and 100% long-range HF exchange, respectively, lead to an overestimation of the VEEs by 0.2-0.3 eV for the benchmarking set. To reduce the systematic error, we introduce here two new empirical functionals, CAMh-B3LYP and ωhPBE0, for which we adjusted the long-range HF exchange to 50%. The introduced parameterization reduces the mean signed average (MSA) deviation to 0.07 eV and the root mean square (rms) deviation to 0.17 eV as compared to the CC2 values. In the present study, TDDFT calculations using the aug-def2-TZVP basis sets, the best performing functionals relative to CC2 are ωhPBE0 (rms = 0.17, MSA = 0.06 eV); CAMh-B3LYP (rms = 0.16, MSA = 0.07 eV); and PBE0 (rms = 0.23, MSA = -0.14 eV). For the popular range-separated CAM-B3LYP functional, we obtain an rms value of 0.31 eV and an MSA value of 0.25 eV, which can be compared with the rms and MSA values of 0.37 and -0.31 eV, respectively, as obtained at the B3LYP level.

Linear-response range-separated density-functional theory for atomic photoexcitation and photoionization spectra.

We investigate the performance of the range-separated hybrid (RSH) scheme, which combines long-range Hartree-Fock (HF) and a short-range density-functional approximation (DFA), for calculating the photoexcitation/photoionization spectra of the H and He atoms, using a B-spline basis set in order to correctly describe the continuum part of the spectra. The study of these simple systems allows us to quantify the influence on the spectra of the errors coming from the short-range exchange-correlation DFA and from the missing long-range correlation in the RSH scheme. We study the differences using the long-range HF exchange (nonlocal) potential and the long-range exact exchange (local) potential. Contrary to the former, the latter supports a series of Rydberg states and gives reasonable photoexcitation/photoionization spectra, even without applying linear-response theory. The most accurate spectra are obtained with the linear-response time-dependent RSH (TDRSH) scheme. In particular, for the He atom at the optimal value of the range-separation parameter, TDRSH gives slightly more accurate photoexcitation and photoionization spectra than the standard linear-response time-dependent HF. More generally, this work shows the potential of range-separated density-functional theory for calculating linear and nonlinear optical properties involving continuum states.

Long-range corrected density functional theory with accelerated Hartree-Fock exchange integration using a two-Gaussian operator [LC-ωPBE(2Gau)

Since the advent of hybrid functional in 1993, it has become a main quantum chemical tool for the calculation of energies and properties of molecular systems. Following the introduction of long-range corrected hybrid scheme for density functional theory a decade later, the applicability of the hybrid functional has been further amplified due to the resulting increased performance on orbital energy, excitation energy, non-linear optical property, barrier height, and so on. Nevertheless, the high cost associated with the evaluation of Hartree-Fock (HF) exchange integrals remains a bottleneck for the broader and more active applications of hybrid functionals to large molecular and periodic systems. Here, we propose a very simple yet efficient method for the computation of long-range corrected hybrid scheme. It uses a modified two-Gaussian attenuating operator instead of the error function for the long-range HF exchange integral. As a result, the two-Gaussian HF operator, which mimics the shape of the error function operator, reduces computational time dramatically (e.g., about 14 times acceleration in C diamond calculation using periodic boundary condition) and enables lower scaling with system size, while maintaining the improved features of the long-range corrected density functional theory.

Description of plasmon-like band in silver clusters: the importance of the long-range Hartree-Fock exchange in time-dependent density-functional theory simulations.

Absorption spectra of Ag20 and Ag55(q) (q = +1, -3) nanoclusters are investigated in the framework of the time-dependent density functional theory in order to analyse the role of the d electrons in plasmon-like band of silver clusters. The description of the plasmon-like band from calculations using density functionals containing an amount of Hartree-Fock exchange at long range, namely, hybrid and range-separated hybrid (RSH) density functionals, is in good agreement with the classical interpretation of the plasmon-like structure as a collective excitation of valence s-electrons. In contrast, using local or semi-local exchange functionals (generalized gradient approximations (GGAs) or meta-GGAs) leads to a strong overestimation of the role of d electrons in the plasmon-like band. The semi-local asymptotically corrected model potentials also describe the plasmon as mainly associated to d electrons, though calculated spectra are in fairly good agreement with those calculated using the RSH scheme. Our analysis shows that a portion of non-local exchange modifies the description of the plasmon-like band.

Optimizing calculations of electronic excitations and relative hyperpolarizabilities of electrooptic chromophores.

CONSPECTUS: Organic glasses containing chromophores with large first hyperpolarizabilities (β) are promising for compact, high-bandwidth, and energy-efficient electro-optic devices. Systematic optimization of device performance requires development of materials with high acentric order and enhanced hyperpolarizability at operating wavelengths. One essential component of the design process is the accurate calculation of optical transition frequencies and hyperpolarizability. These properties can be computed with a wide range of electronic structure methods implemented within commercial and open-source software packages. A wide variety of methods, especially hybrid density-functional theory (DFT) variants have been used for this purpose. However, in order to provide predictions useful to chromophore designers, a method must be able to consistently predict the relative ordering of standard and novel materials. Moreover, it is important to distinguish between the resonant and nonresonant contribution to the hyperpolarizabiliy and be able to estimate the trade-off between improved β and unwanted absorbance (optical loss) at the target device's operating wavelength. Therefore, we have surveyed a large variety of common methods for computing the properties of modern high-performance chromophores and compared these results with prior experimental hyper-Rayleigh scattering (HRS) and absorbance data. We focused on hybrid DFT methods, supplemented by more computationally intensive Møller-Plesset (MP2) calculations, to determine the relative accuracy of these methods. Our work compares computed hyperpolarizabilities in chloroform relative to standard chromophore EZ-FTC against HRS data versus the same reference. We categorized DFT methods used by the amount of Hartree-Fock (HF) exchange energy incorporated into each functional. Our results suggest that the relationship between percentage of long-range HF exchange and both βHRS and λmax is nearly linear, decreasing as the fraction of long-range HF exchange increases. Mild hybrid DFT methods are satisfactory for prediction of λmax. However, mild hybrid methods provided qualitatively incorrect predictions of the relative hyperpolarizabilities of three high-performance chromophores. DFT methods with approximately 50% HF exchange, and especially the Truhlar M062X functional, provide superior predictions of relative βHRS values but poorer predictions of λmax. The observed trends for these functionals, as well as range-separated hybrids, are similar to MP2, though predicting smaller absolute magnitudes for βHRS. Frequency dependence for βHRS can be calculated using time-dependent DFT and HF methods. However, calculation quality is sensitive not only to a method's ability to predict static hyperpolarizability but also to its prediction of optical resonances. Due to the apparent trade-off in accuracy of prediction of these two properties and the need to use static finite-field methods for MP2 and higher-level hyperpolarizability calculations in most codes, we suggest that composite methods could greatly improve the accuracy of calculations of β and λmax.

Assessment of range-separated time-dependent density-functional theory for calculating C6 dispersion coefficients

We assess a variant of linear-response range-separated time-dependent density-functional theory (TDDFT), combining a long-range Hartree-Fock (HF) exchange kernel with a short-range adiabatic exchange-correlation kernel in the local-density approximation (LDA) for calculating isotropic C6 dispersion coefficients of homodimers of a number of closed-shell atoms and small molecules. This range-separated TDDFT tends to give underestimated C6 coefficients of small molecules with a mean absolute percentage error of about 5%, a slight improvement over standard TDDFT in the adiabatic LDA which tends to overestimate them with a mean absolute percentage error of 8%, but close to time-dependent Hartree-Fock which has a mean absolute percentage error of about 6%. These results thus show that introduction of long-range HF exchange in TDDFT has a small but beneficial impact on the values of C6 coefficients. It also confirms that the present variant of range-separated TDDFT is a reasonably accurate method even using only a LDA-type density functional and without adding an explicit treatment of long-range correlation.

Computational Study of Molecules with High Intrinsic Hyperpolarizabilities

In the current manuscript we present the results of a computational study on a series of chromophores with enhanced intrinsic hyperpolarizability. The high hyperpolarizability values of these molecules were previously reported and were achieved by making use of aromatic moieties in order to modulate the aromatic stabilization energy along the conjugated bridge between the donor and the acceptor. Calculations were performed using semiempirical, DFT, and TDDFT methods, and the results reproduce the trend determined experimentally for the first hyperpolarizability values. Several calculation schemes were used, and the best agreement was achieved when long-range Hartree-Fock exchange corrections and solvent effects are included in the DFT calculations. The long-range corrections proved to be especially important for the azobenzene derivatives, which otherwise have their hyperpolarizability overestimated considerably in the DFT calculations. The results are also analyzed within the framework of a two-level model, which correctly reproduces the trend in the hyperpolarizabilities of the molecules under study.

Range-separated density-functional theory with the random-phase approximation: Detailed formalism and illustrative applications

Using Green's-function many-body theory, we present the details of a formally exact adiabatic-connection fluctuation-dissipation density-functional theory based on range separation, which was sketched in Toulouse et al. [Phys. Rev. Lett. 102, 096404 (2009)]. Range-separated density-functional theory approaches combining short-range density-functional approximations with long-range random-phase approximations (RPAs) are then obtained as well-identified approximations on the long-range Green's-function self-energy. Range-separated RPA-type schemes with or without long-range Hartree-Fock exchange response kernel are assessed on rare-gas and alkaline-earth-metal dimers and compared to range-separated second-order perturbation theory and range-separated coupled-cluster theory.

Range separated hybrid density functional with long-range Hartree-Fock exchange applied to solids

We report a plane wave-projector augmented wave implementation of the recently proposed exchange-only range separated hybrid (RSHX) density functional [Gerber and Angyan, Chem. Phys. Lett. 415, 100 (2005)] and characterize its performance in the local density approximation (RSHXLDA) for a set of archetypical solid state systems, as well as for some transition metal oxides. Lattice parameters, bulk moduli, band gaps, and magnetic moments of the transition metal oxides have been calculated at different values of the range separation parameter and compared with results obtained with standard local density approximation (LDA), gradient corrected (PBE), and hybrid (HSE) functionals. The RSHX functional, which has the main feature of providing a correct asymptotic behavior of the exchange potential, has a tendency to improve the description of structural parameters with respect to local and generalized gradient approximations. The band gaps are too strongly opened by the presence of the long-range Hartree-Fock exchange in all but wide-gap systems. In the difficult case of transition metal oxides, the gap is overestimated, while magnetic moments and lattice constants are slightly underestimated. The optimal range separation parameter has been found around 0.4 a.u., slightly lower than the value of 0.5 a.u., recommended earlier for molecular systems.

Importance of short-range versus long-range Hartree-Fock exchange for the performance of hybrid density functionals

We consider a general class of hybrid density functionals with decomposition of the exchange component into short-range and long-range parts. The admixture of Hartree-Fock (HF) exchange is controlled by three parameters: short-range mixing, long-range mixing, and range separation. We study how the variation of these parameters affects the accuracy of hybrid functionals for thermochemistry and kinetics. For the density functional component of the hybrids, we test three nonempirical approximations: local spin-density approximation, generalized gradient approximation (GGA), and meta-GGA. We find a great degree of flexibility in choosing the mixing parameters in range-separated hybrids. For the studied properties, short-range and long-range HF exchange seem to have a similar effect on the errors. One may choose to treat the long-range portion of the exchange by HF to recover the correct asymptotic behavior of the exchange potential and improve the description of density tail regions. If this asymptote is not important, as in solids, one may use screened hybrids, where long-range HF exchange is excluded. Screened hybrids retain most of the benefits of global hybrids but significantly reduce the computational cost in extended systems.