Exact Exchange Method Research Articles

Machine Learning K-Means Clustering of Interpolative Separable Density Fitting Algorithm for Accurate and Efficient Cubic-Scaling Exact Exchange Plus Random Phase Approximation within Plane Waves.

The exact-exchange plus random-phase approximation (EXX+RPA) method has emerged as a crucial tool for precisely characterizing electronic structures in molecular and solid systems. We present an accurate and efficient implementation of EXX+RPA calculations that scale cubically and are conducted within plane waves. Our approach incorporates the interpolative separable density fitting (ISDF) algorithm, effectively mitigating the computational challenges associated with the plane wave basis set. To overcome the constraints of the conventional ISDF algorithm, characterized by the exceptionally high prefactor in QR factorization for interpolation point selection, we introduce an enhanced machine learning K-means method. This method incorporates a novel empirical weight function called "SSM+" for more precise interpolation point selection, capturing physical information more accurately across diverse systems. Our machine learning approach offers a quasiquadratic scaling alternative, effectively replacing the computationally demanding cubic-scaling QRCP algorithm in plane-wave-based EXX+RPA calculations. Furthermore, we enhance the method's capabilities by optimizing GPU acceleration using MATLAB's integrated GPU toolkit. In particular, our approach reduces the computational scaling of χ0 from 3.80 to 2.13 and the overall computational scaling of EXX from 2.74 to 2.10. We achieve a remarkable GPU acceleration speedup of up to 35×. Regarding CPU computation time, the standard quartic-scaling method requires 22 h to compute Si128, while QRCP completes the calculation in only around 1 h, achieving a speedup up to 20×. However, the utilization of the K-means algorithm reduces the time to 800 s, a substantial improvement of 100× compared to the standard algorithm. By employing the K-means algorithm, the computational time for interpolative point calculation using QRCP decreases from 1 h to 1 min, resulting in a 55× speed increase. With this improved algorithm, we successfully computed the dissociation curve of H2 and the equilibrium polyynic geometry of C18 molecules.

Highly Efficient, Linear-Scaling Seminumerical Exact-Exchange Method for Graphic Processing Units.

We present a highly efficient and asymptotically linear-scaling graphic processing unit accelerated seminumerical exact-exchange method (sn-LinK). We go beyond our previous central processing unit-based method (Laqua, H.; Kussmann, J.; Ochsenfeld, C. J. Chem. Theory Comput. 2018, 14, 3451-3458) by employing our recently developed integral bounds (Thompson, T. H.; Ochsenfeld, C. J. Chem. Phys. 2019, 150, 044101) and high-accuracy numerical integration grid (Laqua, H.; Kussmann, J.; Ochsenfeld, C. J. Chem. Phys. 2018, 149, 204111). The accuracy is assessed for several established test sets, providing errors significantly below 1mEh for the smallest grid. Moreover, a comprehensive performance analysis for large molecules between 62 and 1347 atoms is provided, revealing the outstanding performance of our method, in particular, for large basis sets such as the polarized quadruple-zeta level with diffuse functions.

Efficient Semi-Numerical Implementation of Relativistic Exact Exchange within the Infinite-Order Two-Component Method Using a Modified Chain-of-Spheres Method.

We present an efficient implementation of relativistic exact exchange within the infinite-order two-component method (IOTC) by employing a state-of-the-art seminumerical integration technique. For accurate consideration of the picture change, inherent to two-component methods, we propose a new scheme based on a relativistic or picture-change transformation of the density matrix, which provides a simple and efficient formulation of relativistically transformed quantities such as the electron density or exact exchange and thus avoids expensive integral transformations. We show that the new scheme does not introduce additional numerical or theoretical errors beyond the approximations of the IOTC method. For the efficient implementation of exact-exchange integrals, we build upon a modified version of the chain-of-spheres exact-exchange (COSX) method. In addition to the conventional overlap and density matrix screening by S- and P-junctions, respectively, we introduce a new simple screening technique in the sense of the original COSX method by additionally considering the asymptotic decay of the integrals over the Coulomb operator within the new F-junctions. Together with the picture-change transformation of the density matrix, this modified COSX method is shown to provide superior efficiency for the calculation of relativistic exact exchange compared to a conventional analytical direct self-consistent-field implementation of exact exchange.

Assessment of quality and reliability of band structures from exact-exchange-only Kohn–Sham, hybrid, and GW methods

Band gaps for a large number of semiconductors obtained with the exact-exchange(EXX)-only Kohn–Sham method are compared to those from various generalized Kohn–Sham methods based on hybrid functionals and with experimental values. EXX band gaps are shown to be of superior quality than the band gaps from most of the considered hybrid methods, band gaps obtained with the hybrid functional HSE06 have a similar quality as EXX ones. The EXX method is found to yield d -electron binding energies that are clearly superior to those determined within the local density approximation and to be comparable to d -electron binding energies from GW approaches. EXX plane-wave pseudopotential calculations and EXX all-electron calculations are shown to yield very similar KS transition energies demonstrating that the plane-wave pseudopotential approach is well-suited for EXX methods. Previous discrepancies between pseudopotential and all-electron calculations, most likely, have to be attributed to technical problems in early all-electron EXX calculations.

Exact exchange: A pathway for a density functional theory of the integer quantum Hall effect

It is shown here that the exact-exchange (EE) formalism provides a natural and rigorous approach for a density functional theory (DFT) of the integer quantum Hall effect (IQHE). Application of a novel EE method to a quasi–two-dimensional electron gas (q2DEG) subjected to a perpendicular magnetic field leads to the following main findings. i) the microscopic exchange energy functional of the IQHE has been obtained, whose main feature being that it minimizes with a discontinuous derivative at every integer filling factor ν; ii) an analytical solution is found for the magnetic-field dependent EE potential, in the one-subband regime; iii) as a consequence of i), the EE potential display sharp discontinuities at every integer ν; and iv) the widely used local spin density approximation (LSDA) is strongly violated for filling factors close to integer values.

Self-interaction free local exchange potentials applied to metallic systems

We extend the formalism of local exchange methods to calculate and investigate the electronic structure of metals. It is well-known that the Hartree–Fock method when applied to metals shows unphysical behaviour, however the accurate treatment of exchange via DFT’s exact exchange method and using our local Fock exchange method can be used to describe metallic band structures accurately.

Exact exchange with non-orthogonal generalized Wannier functions.

The evaluation of exact exchange (EXX) is an important component of quantum chemical calculations performed with ab initio and hybrid density functional methods. While evaluating exact exchange is routine in molecular quantum chemical calculations performed with localized basis sets, the non-local nature of the exchange operator presents a major impediment to the efficient use of exact exchange in calculations that employ planewave basis sets. Non-orthogonal generalized Wannier functions (NGWFs) corresponding to planewave expansions of localized basis functions are an alternative form of basis set that can be used in quantum chemical calculations. The periodic nature of these functions renders them suitable for calculations of periodic systems, while the contraction of sets of planewaves into individual basis functions reduces the number of variational parameters, permitting the construction and direct diagonalization of the Fock matrix. The present study examines how NGWFs corresponding to Fourier series representations of conventional atom-centered basis sets can be used to evaluate exact exchange in periodic systems. Specifically, an approach for constructing the exchange operator with NGWFs is presented and used to perform Hartree-Fock calculations with a series of molecules in periodically repeated simulation cells. The results demonstrate that the NGWF approach is significantly faster than the EXX method, which is a standard approach for evaluating exact exchange in periodic systems.

Simple screened exact-exchange approach for excitonic properties in solids

We present a screened exact-exchange (SXX) method for the efficient and accurate calculation of the optical properties of solids, where the screening is achieved through the zero-wavevector limit of the inverse dielectric function. The SXX approach can be viewed as a simplification of the Bethe-Salpeter equation (BSE) or, in the context of time-dependent density-functional theory, as a first step towards a new class of hybrid functionals for the optical properties of solids. SXX performs well for bound excitons and continuum spectra in both small-gap semiconductors and large-gap insulators, with a computational cost much lower than that of the BSE.

Stability conditions for exact-exchange Kohn-Sham methods and their relation to correlation energies from the adiabatic-connection fluctuation-dissipation theorem.

The occurrence of instabilities, in particular singlet-triplet and singlet-singlet instabilities, in the exact-exchange (EXX) Kohn-Sham method is investigated. Hessian matrices of the EXX electronic energy with respect to the expansion coefficients of the EXX effective Kohn-Sham potential in an auxiliary basis set are derived. The eigenvalues of these Hessian matrices determine whether or not instabilities are present. Similar as in the corresponding Hartree-Fock case instabilities in the EXX method are related to symmetry breaking of the Hamiltonian operator for the EXX orbitals. In the EXX methods symmetry breaking can easily be visualized by displaying the local multiplicative exchange potential. Examples (N2, O2, and the polyyne C10H2) for instabilities and symmetry breaking are discussed. The relation of the stability conditions for EXX methods to approaches calculating the Kohn-Sham correlation energy via the adiabatic-connection fluctuation-dissipation (ACFD) theorem is discussed. The existence or nonexistence of singlet-singlet instabilities in an EXX calculation is shown to indicate whether or not the frequency-integration in the evaluation of the correlation energy is singular in the EXX-ACFD method. This method calculates the Kohn-Sham correlation energy through the ACFD theorem theorem employing besides the Coulomb kernel also the full frequency-dependent exchange kernel and yields highly accurate electronic energies. For the case of singular frequency-integrands in the EXX-ACFD method a regularization is suggested. Finally, we present examples of molecular systems for which the self-consistent field procedure of the EXX as well as the Hartree-Fock method can converge to more than one local minimum depending on the initial conditions.

Efficient exact-exchange time-dependent density-functional theory methods and their relation to time-dependent Hartree–Fock

A recently introduced time-dependent exact-exchange (TDEXX) method, i.e., a response method based on time-dependent density-functional theory that treats the frequency-dependent exchange kernel exactly, is reformulated. In the reformulated version of the TDEXX method electronic excitation energies can be calculated by solving a linear generalized eigenvalue problem while in the original version of the TDEXX method a laborious frequency iteration is required in the calculation of each excitation energy. The lowest eigenvalues of the new TDEXX eigenvalue equation corresponding to the lowest excitation energies can be efficiently obtained by, e.g., a version of the Davidson algorithm appropriate for generalized eigenvalue problems. Alternatively, with the help of a series expansion of the new TDEXX eigenvalue equation, standard eigensolvers for large regular eigenvalue problems, e.g., the standard Davidson algorithm, can be used to efficiently calculate the lowest excitation energies. With the help of the series expansion as well, the relation between the TDEXX method and time-dependent Hartree-Fock is analyzed. Several ways to take into account correlation in addition to the exact treatment of exchange in the TDEXX method are discussed, e.g., a scaling of the Kohn-Sham eigenvalues, the inclusion of (semi)local approximate correlation potentials, or hybrids of the exact-exchange kernel with kernels within the adiabatic local density approximation. The lowest lying excitations of the molecules ethylene, acetaldehyde, and pyridine are considered as examples.

Extension of exact-exchange density functional theory of solids to finite temperatures

A general finite-temperature exact-exchange (EXX) formalism derived within the framework of finite-temperature density functional theory for grand canonical ensembles is presented. Based on this formalism a finite-temperature EXX method for solids using plane-wave basis sets is presented. The method is generally applicable, i.e., applicable to insulators, semiconductors, or metals and enables the investigation of temperature effects. More important, however, is that the finite-temperature EXX method enables an EXX treatment of metals by introducing a physically motivated Fermi broadening technique. We tested the method by applying it to sodium, magnesium, and aluminum and compare EXX and LDA (local density approximation) band structures as well as the density of states for the three metals. Differences between LDA and EXX band structures are negligible up to the Fermi level. Above the Fermi level, however, differences between LDA and EXX band structures of magnitudes of 1--2 eV start to build up for energetically higher bands. The magnitude of these differences is of the same order as that of the increases in EXX band gaps compared to LDA band gaps as they are reported for semiconductors and insulators.

Density-Functional Theory with Orbital-Dependent Functionals: Exact-exchange Kohn-Sham and Density-Functional Response Methods

Abstract Kohn-Sham methods with orbital-dependent exchange-correlation functionals, in particular exact-exchange methods, are discussed. The numerical stability of optimized-effective-potential methods to construct the exchange-correlation potential in the case of orbital-dependent energy functionals is considered. A numerically stable exact-exchange Kohn-Sham method is briefly presented. A new ansatz for time-dependent density-functional methods in the response regime is considered and an exact-exchange time-dependent density-functional method based on it is discussed and it is demonstrated that this methods is able to treat qualitatively correctly charge-transfer excitations.

Towards an accurate description of the electronic properties of the biphenylthiol/gold interface: The role of exact exchange

We investigate the role of the exact exchange in describing the biphenylthiol/gold interface. The study is performed by simulating the electronic properties of mercaptobiphenylthiol and aminobiphenylthiol molecules adsorbed on a Au(23) cluster, using local, semilocal and hybrid functionals and an effective exact exchange method, namely, the localized Hartree-Fock (LHF). We find that the local/semilocal functionals strongly underestimate the charge transfer and the bond dipole at the interface due to the self-interaction-error (SIE), which alters the correct level alignment. On the other hand the LHF method is SIE free and predicts a larger charge transfer and bond dipole. We also found that LHF results can be reproduced using hybrid functionals and that conventional local/semilocal correlation functionals are unable to improve over the exchange-only description.

Density functional theory investigation of the polarizability and second hyperpolarizability of polydiacetylene and polybutatriene chains: Treatment of exact exchange and role of correlation

The static polarizability and second hyperpolarizability of increasingly large polydiacetylene and polybutatriene (PBT) chains have been evaluated using the optimized effective potential for exact exchange (OEP-EXX) method developed by Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)], where the unknown part of the effective potential is expressed as a linear combination of Gaussian functions. Various conventional atomic orbital basis sets were employed for the exchange potential (X basis) as well as for the Kohn-Sham orbitals [molecular orbital (MO) basis]. Our results were compared to coupled-perturbed Hartree-Fock (CPHF) calculations and to ab initio correlated values obtained at various levels of approximation. It turns out that (a) small conventional basis sets are, in general, unsatisfactory for the X basis; (b) the performance of a given X basis depends on the MO basis and is generally improved when using a larger MO basis; (c) these effects are exaggerated for the second hyperpolarizability compared to the polarizability; (d) except for the second hyperpolarizability of PBT chains, using 6-311++G** for the X basis gives reasonable agreement with the CPHF results for all MO basis sets; (e) our results suggest that in the limit of a complete X basis the OEP-EXX values may approach the CPHF data; and (f) in general, the quality of a given conventional X basis degrades with the length of the oligomer, which correlates with the fact that the number of X basis functions becomes a smaller fraction of the number required to reproduce exactly the finite-basis-set Hartree-Fock energies. Linear and especially nonlinear electric field responses constitute a very stringent test for assessing the quality of functionals and potentials; appropriately tailored basis sets are needed to describe the latter. Finally, this study further highlights the importance of electron correlation effects on linear and nonlinear responses, for which correlated functionals with OEP are required.

Exact-exchange Kohn-Sham formalism applied to one-dimensional periodic electronic systems

The Exact-Exchange (EXX) Kohn-Sham formalism, which treats exchange interactions exactly within density-functional theory, is applied to one-dimensional periodic systems. The underlying implementation does not rely on specific symmetries of the considered system and can be applied to any kind of periodic structure in one to three dimensions. As a test system, $trans$-polyacetylene, both in form of an isolated chain and in the bulk geometry has been investigated. Within the EXX scheme, bandstructures and independent particle response functions are calculated and compared to experimental data as well as to data calculated by several other methods. Compared to results from the local-density approximation, the EXX method leads to an increased value for the band gap, in line with similar observations for three-dimensional semiconductors. An inclusion of correlation potentials within the local density approximation or generalized gradient approximations leads to only negligible effects in the bandstructure. The EXX band gaps are in good agreement with experimental data for bulk $trans$-polyacetylene. Packing effects of the chains in bulk $trans$-polyacetylene are found to lower the band gap by about 0.5 eV.

First-principles investigation of isolated band formation in half-metallicTixGa1−xP(x=0.3125–0.25)

Several alloy semiconductors containing a transition metal atom of the type $M{\mathrm{Ga}}_{4}{X}_{3}$ with $X=\mathrm{As}$ or P and $M=\mathrm{Sc}$ or Ti have been found previously to present an isolated partially filled intermediate band within the usual band gap of the host semiconductors and have been proposed as highly efficient photovoltaic materials. In this paper, we carry out an ab initio investigation of band structures and electronic properties for the more chemically stable ${\mathrm{Ti}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{P}$ compound as a candidate for isolated intermediate band formation. We have calculated the electronic structures using self-consistent density functional theory method in both local density approximation and generalized gradient approximation (GGA) approaches and compared the GGA results with those obtained with the exact exchange method that we have implemented in the code SIESTA. Using spin-polarized localized wave functions to represent the valence electrons states and nonlocal pseudopotentials for the core electrons, we have also studied in detail the ${\mathrm{Ti}}_{x}{\mathrm{Ga}}_{1\ensuremath{-}x}\mathrm{P}$ compounds at different dilution levels of the Ti transition metal atom $(x=3.125%,6.25%,12.5%,25.0%)$ and for two (cubic and tetragonal) different crystal cells. Results at the different dilutions show in all cases a fully spin-polarized structure and, except for the case of immediate Ti neighbors, indicate a rather small spin coupling between Ti atoms and confirm the presence of the isolated narrow partially filled intermediate band for this compound. They also show the higher suitability of isotropic crystal structures for obtaining in these materials the intermediate band with the desired small band width.

All-Electron Exact Exchange Treatment of Semiconductors: Effect of Core-Valence Interaction on Band-Gap andd-Band Position

We present the first all-electron full-potential exact exchange (EXX) Kohn-Sham density functional calculations on a range of semiconductors and insulators (Ge, GaAs, CdS, Si, ZnS, C, BN, Ne, Ar, Kr, and Xe). We remove one of the main computational obstacles of such calculations by the use of a highly efficient basis for inversion of the response function. We find that the band gaps are not as close to experiment as those obtained from previous pseudopotential EXX calculations. The locations of d bands, determined using the full-potential EXX method, are in excellent agreement with experiment, irrespective of whether these are core, semicore, or valence states. We conclude that the inclusion of the core-valence interaction is necessary for accurate determination of EXX Kohn-Sham band structures and that EXX alone is not a complete answer to the band-gap problem in semiconductors.

Advances in the theory of electronic structure of semiconductors

This paper reviews the present status of recently developed ab-initio as well as semiempirical electronic structure methods that are particularly suited for semiconductors and mesoscopic semiconductor structures. In assessing each method, we provide some additional and hitherto unpublished details about its implementation. In the first part of the paper, we discuss two important extensions of the local density functional theory, the screened-exchange and the exact exchange method namely. We proceed with a discussion of the relativistic local density method that is essential for a prediction of spin-related phenomena in semiconductors. Finally, we review a rather complete and extensve implementation of the envelope function approach for arbitrary three-dimensional nanometer device structures. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Exact-exchange density-functional calculations for noble-gas solids

The electronic structure of noble-gas solids is calculated within density functional theory's exact-exchange method (EXX) and compared with the results from the local-density approximation (LDA). It is shown that the EXX method does not reproduce the fundamental energy gaps as well as has been reported for semiconductors. However, the EXX-Kohn-Sham energy gaps for these materials reproduce about 80 % of the experimental optical gaps. The structural properties of noble-gas solids are described by the EXX method as poorly as by the LDA one. This is due to missing Van der Waals interactions in both, LDA and EXX functionals.

Accurate polymer polarizabilities with exact exchange density-functional theory

The long-standing problem of the large overestimation of polymer polarizabilities in density-functional theory is reexamined and largely solved using an exact exchange method. We have built an accurate optimized effective potential as the sum of a fixed potential and a linear combination of basis sets based on our direct optimization method. This effective potential properly develops a linear counteracting depolarization field, and it significantly improves recent results from approximate optimized potentials. The controversial case of hydrogen chains is now correctly described and the failure of the local density approach is attributed to the large self-interaction error in systems with a non-integer number of electrons.