Exact Exchange Energy Research Articles

Assessment of Functionals for TDDFT Calculations of One- and Two-Photon Absorption Properties of Neutral and Anionic Fluorescent Proteins Chromophores

Performance of DFT functionals with different percentages of exact Hartree-Fock exchange energy (EX) is assessed for recovery of the CC2 reference one- (OPA) and two-photon absorption (TPA) spectra of fluorescent proteins chromophores in vacuo. The investigated DFT functionals, together with their EX contributions are BLYP (0%), B3LYP (20%), B1LYP (25%), BHandHLYP (50%), and CAM-B3LYP (19% at short range and 65% at long range). Our test set consists of anionic and neutral chromophores as naturally occurring in the fluorescent proteins. For the first time, we compare TDDFT and CC2 methods for higher excited states than the S1 state, exhibiting relatively large TPA intensity. Our TDDFT results for neutral chromophores reveal an increase in excitation energies as well as TPA and OPA intensities errors, compared to CC2-derived results, as the DFT functional contains less exact exchange. The long-range-corrected CAM-B3LYP functional performs the best, closely followed by BHandHLYP, while BLYP usually significantly underestimates all investigated spectral properties, hence being the worst in reproducing the reference CC2 results. The hybrid B3LYP and B1LYP functionals can be roughly placed in between. We propose that TDDFT may underestimate the TPA intensities for neutral chromophores of fluorescent proteins due to underestimated oscillator strengths between some excited states. In the case of anionic chromophores, we find that B3LYP and B1LYP functionals overcome others in terms of reproducing CC2 excitation energies. On the other hand, however, TPA intensity is usually significantly underestimated, and in this respect, CAM-B3LYP functional seems to be again superior. In contrast to the case of neutral chromophores, it seems that a large magnitude of excited-state dipole moments or changes in dipole moments upon excitation may be the driving force behind high TPA transition moments.

Embedded Cluster Density Approximation for Exchange-Correlation Energy: A Natural Extension of the Local Density Approximation.

We developed a local correlation method in the framework of Kohn-Sham density functional theory (KS-DFT). The method is termed ''embedded cluster density approximation'' (ECDA) and is a logical extension of the local density approximation. In ECDA, an embedded cluster is defined for each atom based on the finite-temperature density functional embedding theory. The clusters' exchange-correlation (XC) energy densities are calculated using high-level XC functionals. The system's XC energy is then constructed by patching these locally computed high-level XC energy densities over the system in an atom-by-atom manner. We derive the relationship between the embedding potential and the system's KS potential. We show how to efficiently compute the system's XC potential, which is the functional derivative of the patched XC energy with respect to the system's electron density. The accuracy of ECDA is examined by patching the exact exchange (EXX) and the random phase approximation (RPA) correlation energy densities in a one-dimensional hydrogen chain, as well as by patching EXX energy densities in several molecules. The agreement between ECDA and KS-DFT on total energies, dipole moments, XC potentials, and KS eigenvalues is good in general as the clusters are made larger. Based on these encouraging results, we expect ECDA to be a simple yet effective method to scale up high-level KS-DFT simulations in large systems.

Density functionals for nondynamical correlation constructed from an upper bound to the exact exchange energy density

ABSTRACTHyper-generalised-gradient approximations (hGGAs) for the exact exchange-correlation functional are increasingly popular in density functional theory. HGGAs model nondynamical correlation using a flexible local combination of exact (Hartree–Fock, HF) exchange and approximate exchange. We present a simplified ‘Rung 3.5’ upper bound to the HF exchange energy density, the essential ingredient of hGGAs. We also present a nonempirical generalised gradient approximation for this upper bound. Both upper bounds go to zero in the high-density and density tail limits, facilitating the construction of hGGAs that recover HF exchange in these limits. The ‘Rung 3.5’ construction enables facile evaluation of analytic derivatives and calculations in periodic boundary conditions. Extensive numerical tests show that the upper bounds capture the critical difference between HF and approximate exchange, showing these ingredients' promise for building simple hGGAs. The tests also indicate a need for more sophisticated semi-local upper bounds.

Exploiting Coordinate Scaling Relations To Accelerate Exact Exchange Calculations.

Exact exchange is an important constituent in many state-of-the-art approximations to the exchange-correlation (xc) functional of Kohn-Sham DFT. However, its evaluation can be computationally intensive, which can be particularly prohibitive in DFT-based molecular dynamics (MD) simulations, often restricted to semilocal functionals. We derive a scheme based on the formal coordinate scaling properties of the exact xc functional that allows for a substantial reduction of the cost of the evaluation of both the exact exchange energy and potential. We show that within a plane-wave/pseudopotential framework, excellent accuracy is retained, while speed ups from up to ∼6 can be reached. The coordinate scaling thus accelerates hybrid-functional-based first-principles MD simulations by nearly one order of magnitude.

Efficient and Linear-Scaling Seminumerical Method for Local Hybrid Density Functionals.

Local hybrid functionals, that is, functionals with local dependence on the exact exchange energy density, generalize the popular class of global hybrid functionals and extend the applicability of density functional theory to electronic structures that require an accurate description of static correlation. However, the higher computational cost compared to conventional Kohn-Sham density functional theory restrained their widespread application. Here, we present a low-prefactor, linear-scaling method to evaluate the local hybrid exchange-correlation potential as well as the corresponding nuclear forces by employing a seminumerical integration scheme. In the seminumerical scheme, one integration in the electron repulsion integrals is carried out analytically and the other one is carried out numerically, employing an integration grid. A high computational efficiency is achieved by combining the preLinK method [ J. Kussmann and C. Ochsenfeld, J. Chem. Phys. 2013 138, 134114 ] with explicit screening of integrals for batches of grid points to minimize the screening overhead. This new method, denoted as preLinX, provides an 8-fold performance increase for a DNA fragment containing four base pairs as compared to existing S- and P-junction-based methods. In this way, our method allows for the evaluation of local hybrid functionals at a cost similar to that of global hybrid functionals. The linear-scaling behavior, efficiency, accuracy, and multinode parallelization of the presented method is demonstrated for large systems containing more than 1000 atoms.

Study of Li Adsorption on Graphdiyne Using Hybrid DFT Calculations.

Promising applications of graphdiyne have often been initiated by theoretical predictions especially using DFT known as the most powerful first-principles electronic structure calculation method. However, there is no systematic study on the reliability of DFT for the prediction of the electronic properties of the graphdiyne. Here, we performed a study of Li adsorption on the graphdiyne using hybrid DFT with LC-ωPBE and compared the results with those of PBE, because accurate prediction of the Li adsorption is important for performance as a Li storage that was first theoretically suggested and then experimentally realized. Our results show that PBE overestimates the adsorption energy inside a pore and the barrier height at the transition state of in-plane diffusion compared to the those of LC-ωPBE. In particular, LC-ωPBE predicted almost barrier-less in-plane diffusion of Li on the graphdiyne because of the presence of both in-plane and out-of-plane π orbitals. Also, LC-ωPBE favors a high spin state due to the exact exchange energy when several Li atoms are adsorbed on the graphdiyne, whereas PBE favors a low spin state. Thus, the use of the hybrid DFT is critical for reliable predictions on the electronic properties of the graphdiyne.

Asymptotic Behavior of Exact Exchange for Slabs: Beyond the Leading Order

Far outside the surface of slabs, the exact exchange (EXX) potential v x falls off as − 1 / z , if z denotes the direction perpendicular to the surface and the slab is localized around z = 0 . Similarly, the EXX energy density e x behaves as − n / ( 2 z ) , where n is the electron density. Here, an alternative proof of these relations is given, in which the Coulomb singularity in the EXX energy is treated in a particularly careful fashion. This new approach allows the derivation of the next-to-leading order contributions to the asymptotic v x and e x . It turns out that in both cases, the corrections are proportional to 1 / z 2 in general.

Simple Modifications of the SCAN Meta-Generalized Gradient Approximation Functional.

We analyzed various possibilities to improve upon the SCAN meta-generalized gradient approximation density functional obeying all known properties of the exact functional that can be satisfied at this level of approximation. We examined the necessity of locally satisfying a strongly tightened lower bound for the exchange energy density in single-orbital regions, the nature of the error cancellation between the exchange and correlation parts in two-electron regions, and the effect of the fourth-order term in the gradient expansion of the correlation energy density. We have concluded that the functional can be modified to separately reproduce the exchange and correlation energies of the helium atom by locally releasing the strongly tightened lower bound for the exchange energy density in single-orbital regions, but this leads to an unbalanced improvement in the single-orbital electron densities. Therefore, we decided to keep the FX ≤ 1.174 exact condition for any single-orbital density, where FX is the exchange enhancement factor. However, we observed a general improvement in the single-orbital electron densities by revising the correlation functional form to follow the second-order gradient expansion in a wider range. Our new revSCAN functional provides more-accurate atomization energies for the systems with multireference character, compared to the SCAN functional. The nonlocal VV10 dispersion-corrected revSCAN functional yields more-accurate noncovalent interaction energies than the VV10-corrected SCAN functional. Furthermore, its global hybrid version with 25% of exact exchange, called revSCAN0, generally performs better than the similar SCAN0 for reaction barrier heights. Here, we also analyzed the possibility of the construction of a local hybrid from the SCAN exchange and a specific locally bounded nonconventional exact exchange energy density. We predict compatibility problems since this nonconventional exact exchange energy density does not really obey the strongly tightened lower bound for the exchange energy density in single-orbital regions.

Intermolecular dispersion energies from coupled exact-exchange Kohn-Sham excitation energies and vectors

The calculation of intermolecular dispersion energies within a time-dependent density-functional theory framework (TDDFT) is reviewed. While the commonly used route to compute dispersion energies is to employ the Casimir-Polder integral transform and thereby describing the dispersion energy as a functional of the response functions of the monomers, this alternative approach leads to a particularly simple form for the long-range interaction between two subsystems in terms of the TDDFT eigenvectors and excitation energies. We present two different schemes to reduce the high computational cost for calculating the excitation energies and vectors of the monomers to be used to compute the dispersion energies via the TDDFT method. This is achieved by a decoupling of the occupied-virtual orbital product basis functions which are associated with single-particle excitations belonging to large and small oscillator strengths. It will be shown that this approach can lead to large speedups of 90% and more compared to a full diagonalisation of the hessian, while the fully coupled dispersion energies can be reproduced with a reasonable accuracy. We also investigate the role of the dispersion energy for the description of σ- and π-stacking interactions and discuss various dispersion energy approximations, including the D3 model by Grimme.

First-principles stability ranking of molecular crystal polymorphs with the DFT+MBD approach.

The ability to accurately calculate the relative stabilities of numerous polymorphs of a given molecular crystal is crucial for the success of any molecular crystal structure prediction (CSP) approach. We have recently presented a hierarchical CSP procedure based on van-der-Waals-inclusive density functional theory [Hoja et al., 2018, arXiv:1803.07503], which yields excellent stability rankings for molecular crystals involving rigid molecules, salts, co-crystals, and highly polymorphic drug-like molecules. This approach includes many-body dispersion effects, exact exchange, as well as vibrational free energies. Here, we discuss in detail the impact of these effects on the obtained stability rankings. In addition, we assess the impact of the approximations used in our hierarchical procedure. We show that our procedure is generally robust to 1-2 kJ mol-1 for the systems in the latest CSP blind test but vibrational free energies for crystals involving flexible molecules would benefit from directly including many-body dispersion interactions. In addition, we also discuss the effect of temperature on the structure of molecular crystals and a simple but effective method for estimating anharmonic effects.

Efficient Computation of Exchange Energy Density with Gaussian Basis Functions.

Density functional theory (DFT) is widely applied in chemistry and physics. Still it fails to correctly predict quantitatively or even qualitatively for systems with significant nondynamic correlation. Several DFT functionals were proposed in recent years to treat the nondynamic correlation, most of which added the exact exchange energy density as a new variable. This quantity, calculated as Hartree-Fock (HF) exchange energy density, is the computational bottleneck for calculations with these new functionals. We present an implementation of an efficient seminumerical algorithm in this paper as a solution for this computational bottleneck. The method scales quadratically with respect to the molecular size and the basis set size. The scheme, exact for the purpose of computing the HF exchange energy density, is favored for medium-sized basis sets and can be competitive even for large basis sets with efficient grids when compared with our previous approximate resolution-of-identity scheme. It can also be used as a seminumerical integration scheme to compute the HF exchange energy and matrix on a standard atom-centered grid. Calculations on a series of alanine peptides show that for large basis sets the seminumerical scheme becomes competitive to the conventional analytical method and can be about six times faster for aug-cc-pvtz basis. The practicality of the algorithm is demonstrated through a local hybrid self-consistent calculation of the acenes-20 molecule.

Testing Exact Upper Bounds to Exact Exchange.

The exact exchange energy and its energy density are useful but computationally expensive ingredients in density functional approximations for Kohn-Sham density functional theory. We present detailed tests of some exact nonempirical upper bounds to exact exchange. These "Rung 3.5" upper bounds contract the Kohn-Sham one-particle density matrix with model density matrices used to construct semilocal model exchange holes and invoke the Cauchy-Schwarz inequality. The contraction automatically eliminates the computationally expensive long-range component of the exact exchange hole. Numerical tests show that the exchange upper bounds underestimate total exchange energies while predicting other properties with accuracy approaching standard hybrid approximations.

Communication: Near-locality of exchange and correlation density functionals for 1- and 2-electron systems.

The uniform electron gas and the hydrogen atom play fundamental roles in condensed matter physics and quantum chemistry. The former has an infinite number of electrons uniformly distributed over the neutralizing positively charged background, and the latter only one electron bound to the proton. The uniform electron gas was used to derive the local spin density approximation to the exchange-correlation functional that undergirds the development of the Kohn-Sham density functional theory. We show here that the ground-state exchange-correlation energies of the hydrogen atom and many other 1- and 2-electron systems are modeled surprisingly well by a different local spin density approximation (LSDA0). LSDA0 is constructed to satisfy exact constraints but agrees surprisingly well with the exact results for a uniform two-electron density in a finite, curved three-dimensional space. We also apply LSDA0 to excited or noded 1-electron densities, where it works less well. Furthermore, we show that the localization of the exact exchange hole for a 1- or 2-electron ground state can be measured by the ratio of the exact exchange energy to its optimal lower bound.

Gauge effects in local hybrid functionals evaluated for weak interactions and the GMTKN30 test set

ABSTRACTThe so-called ‘gauge problem’, due to the non-uniqueness of exchange-energy densities, is a fundamental challenge for density functionals depending on these energy densities, such as local hybrid functionals. We have recently demonstrated how gauge effects influence the potential-energy curves of the argon dimer, and other quantities depending on ‘non-physical’ Pauli repulsions introduced by incompatible gauges of (semi-)local and exact-exchange energy densities . Introduction of suitable calibration functions depending only on semi-local quantities allowed to correct for these deficiencies and suggested ways to obtain more accurate local hybrid functionals beyond the local spin density approximation (LSDA) exchange-energy density. Here we extend the study of the gauge problem by comparing a number of uncalibrated and calibrated local hybrids for (1) the potential-energy curves of further noble-gas dimers and (2) for the entire GMTKN30 test set and its individual subsets. We find that DFT-D3 dispersion corrections fitted to be compatible with uncalibrated local hybrids have to correct not only for missing London dispersion but also for gauge artefacts that make weak interactions too repulsive. This burden is taken away when using properly calibrated local hybrids, which perform much better for dispersion-sensitive quantities already without D3 corrections, and which require only the physically relevant dispersion to be corrected for. The present results suggest directions for further improvement of calibration functions for local hybrids.

Design of exchange-correlation functionals through the correlation factor approach.

The correlation factor model is developed in which the spherically averaged exchange-correlation hole of Kohn-Sham theory is factorized into an exchange hole model and a correlation factor. The exchange hole model reproduces the exact exchange energy per particle. The correlation factor is constructed in such a manner that the exchange-correlation energy correctly reduces to exact exchange in the high density and rapidly varying limits. Four different correlation factor models are presented which satisfy varying sets of physical constraints. Three models are free from empirical adjustments to experimental data, while one correlation factor model draws on one empirical parameter. The correlation factor models are derived in detail and the resulting exchange-correlation holes are analyzed. Furthermore, the exchange-correlation energies obtained from the correlation factor models are employed to calculate total energies, atomization energies, and barrier heights. It is shown that accurate, non-empirical functionals can be constructed building on exact exchange. Avenues for further improvements are outlined as well.

Communication: A new class of non-empirical explicit density functionals on the third rung of Jacob's ladder.

We construct an orbital-free non-empirical meta-generalized gradient approximation (GGA) functional, which depends explicitly on density through the density overlap regions indicator [P. de Silva and C. Corminboeuf, J. Chem. Theory Comput. 10, 3745 (2014)]. The functional does not depend on either the kinetic energy density or the density Laplacian; therefore, it opens a new class of meta-GGA functionals. By construction, our meta-GGA yields exact exchange and correlation energy for the hydrogen atom and recovers the second order gradient expansion for exchange in the slowly varying limit. We show that for molecular systems, overall performance is better than non-empirical GGAs. For atomization energies, performance is on par with revTPSS, without any dependence on Kohn-Sham orbitals.

Fast, accurate evaluation of exact exchange: The occ-RI-K algorithm.

Construction of the exact exchange matrix, K, is typically the rate-determining step in hybrid density functional theory, and therefore, new approaches with increased efficiency are highly desirable. We present a framework with potential for greatly improved efficiency by computing a compressed exchange matrix that yields the exact exchange energy, gradient, and direct inversion of the iterative subspace (DIIS) error vector. The compressed exchange matrix is constructed with one index in the compact molecular orbital basis and the other index in the full atomic orbital basis. To illustrate the advantages, we present a practical algorithm that uses this framework in conjunction with the resolution of the identity (RI) approximation. We demonstrate that convergence using this method, referred to hereafter as occupied orbital RI-K (occ-RI-K), in combination with the DIIS algorithm is well-behaved, that the accuracy of computed energetics is excellent (identical to conventional RI-K), and that significant speedups can be obtained over existing integral-direct and RI-K methods. For a 4400 basis function C68H22 hydrogen-terminated graphene fragment, our algorithm yields a 14× speedup over the conventional algorithm and a speedup of 3.3× over RI-K.

Towards improved local hybrid functionals by calibration of exchange-energy densities.

A new approach for the calibration of (semi-)local and exact exchange-energy densities in the context of local hybrid functionals is reported. The calibration functions are derived from only the electron density and its spatial derivatives, avoiding spatial derivatives of the exact-exchange energy density or other computationally unfavorable contributions. The calibration functions fulfill the seven more important out of nine known exact constraints. It is shown that calibration improves substantially the definition of a non-dynamical correlation energy term for generalized gradient approximation (GGA)-based local hybrids. Moreover, gauge artifacts in the potential-energy curves of noble-gas dimers may be corrected by calibration. The developed calibration functions are then evaluated for a large range of energy-related properties (atomization energies, reaction barriers, ionization potentials, electron affinities, and total atomic energies) of three sets of local hybrids, using a simple one-parameter local-mixing. The functionals are based on (a) local spin-density approximation (LSDA) or (b) Perdew-Burke-Ernzerhof (PBE) exchange and correlation, and on (c) Becke-88 (B88) exchange and Lee-Yang-Parr (LYP) correlation. While the uncalibrated GGA-based functionals usually provide very poor thermochemical data, calibration allows a dramatic improvement, accompanied by only a small deterioration of reaction barriers. In particular, an optimized BLYP-based local-hybrid functional has been found that is a substantial improvement over the underlying global hybrids, as well as over previously reported LSDA-based local hybrids. It is expected that the present calibration approach will pave the way towards new generations of more accurate hyper-GGA functionals based on a local mixing of exchange-energy densities.

Communication: A non-empirical correlation factor model for the exchange-correlation energy.

A persistent challenge in density functional theory is the construction of a nonempirical correlation functional that is compatible with the exact exchange energy. To solve this problem, we develop a correlation factor approach in which an exchange hole model, yielding the exact exchange energy, is multiplied by a correlation factor that turns the exchange hole into an exchange-correlation hole. This results in an accurate correlation energy functional that is determined solely through physical constraints. Subject to the properties of the employed exchange hole model, the proposed correlation factor model to the exchange-correlation energy becomes exact in the high-density limit. In this limit, the exchange-correlation energy is dominated by exchange.

Computational Modeling of the Dielectric Function of Silicon Slabs with Varying Thickness

The dynamical dielectric function of a silicon slab, in the region from near IR to UV light frequencies, is expected to vary with its thickness, and it is important to know whether its optical properties are similar to those of bulk silicon. Slabs of varying thickness are modeled starting from their atomic structure. Modeled Si(111) surfaces are terminated by hydrogen atoms to compensate the dangling bonds, and optical properties have been obtained for Si slabs with 4, 8, and 12 layers. Real and imaginary parts of the dielectric function are obtained from the polarization of the slab, expressed in terms of a delayed response function constructed from a reduced density matrix (RDM) which includes electronic dissipative effects due to coupling of photoexcited electrons to the substrate lattice and to electronic excitations. The related index of refraction and absorption coefficient have also been calculated from the above treatment. These optical properties are obtained using density functional theory (DFT) and plane wave basis sets to construct the equation of motion of a RDM, solved for steady light absorption. Both GGA (PBE) and hybrid (HSE) DFT exchange-correlation density functionals are employed to calculate the optical properties from the RDM. The imaginary part of the dielectric function is related to the light absorbance, and has been compared to measurements showing that better agreement is obtained with the HSE hybrid functional containing part of the exact short-range electronic exchange energy. We present a procedure by which one can reproduce the HSE results for the dielectric function from the computationally less expensive GGA PBE functional calculations, using a single photon energy shift parameter and results from PBE calculations. Our treatment shows that the onset of light absorption and strong diffraction are similar for thin slabs and bulk silicon, and that they have similar peak structure as functions of photon energy. Both properties increase with slab thickness at most photon energies. This makes silicon slabs reliable structures for photovoltaic applications.