Density functional approximations for the exchange‐correlation energy EDFAxc of an electronic system are often improved by admixing some exact exchange Ex: Exc≊EDFAxc+(1/n)(Ex−EDFAx). This procedure is justified when the error in EDFAxc arises from the λ=0 or exchange end of the coupling‐constant integral ∫10 dλ EDFAxc,λ. We argue that the optimum integer n is approximately the lowest order of Gorling–Levy perturbation theory which provides a realistic description of the coupling‐constant dependence Exc,λ in the range 0≤λ≤1, whence n≊4 for atomization energies of typical molecules. We also propose a continuous generalization of n as an index of correlation strength, and a possible mixing of second‐order perturbation theory with the generalized gradient approximation.

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Rationale for mixing exact exchange with density functional approximations

Electronic excitations lie at the origin of most of the commonly measured spectra. However, the first-principles computation of excited states requires a larger effort than ground-state calculations, which can be very efficiently carried out within density-functional theory. On the other hand, two theoretical and computational tools have come to prominence for the description of electronic excitations. One of them, many-body perturbation theory, is based on a set of Green’s-function equations, starting with a one-electron propagator and considering the electron-hole Green’s function for the response. Key ingredients are the electron’s self-energy S and the electron-hole interaction. A good approximation for S is obtained with Hedin’s GW approach, using density-functional theory as a zero-order solution. First-principles GW calculations for real systems have been successfully carried out since the 1980s. Similarly, the electron-hole interaction is well described by the Bethe-Salpeter equation, via a functional derivative of S. An alternative approach to calculating electronic excitations is the time-dependent density-functional theory (TDDFT), which offers the important practical advantage of a dependence on density rather than on multivariable Green’s functions. This approach leads to a screening equation similar to the Bethe-Salpeter one, but with a two-point, rather than a four-point, interaction kernel. At present, the simple adiabatic local-density approximation has given promising results for finite systems, but has significant deficiencies in the description of absorption spectra in solids, leading to wrong excitation energies, the absence of bound excitonic states, and appreciable distortions of the spectral line shapes. The search for improved TDDFT potentials and kernels is hence a subject of increasing interest. It can be addressed within the framework of many-body perturbation theory: in fact, both the Green’s functions and the TDDFT approaches profit from mutual insight. This review compares the theoretical and practical aspects of the two approaches and their specific numerical implementations, and presents an overview of accomplishments and work in progress.

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Electronic excitations: density-functional versus many-body Green's-function approaches

A new hybrid density functional for general chemistry applications is proposed. It is based on a mixing of standard generalized gradient approximations GGAs for exchange by Becke B and for correlation by Lee, Yang, and Parr LYP with Hartree-Fock HF exchange and a perturbative second-order correlation part PT2 that is obtained from the Kohn-Sham GGA orbitals and eigenvalues. This virtual orbital-dependent functional contains only two global parameters that describe the mixture of HF and GGA exchange ax and of the PT2 and GGA correlation c, respectively. The parameters are obtained in a least-squares-fit procedure to the G2/97 set of heat of formations. Opposed to conventional hybrid functionals, the optimum ax is found to be quite large 53% with c=27% which at least in part explains the success for many problematic molecular systems compared to conventional approaches. The performance of the new functional termed B2-PLYP is assessed by the G2/97 standard benchmark set, a second test suite of atoms, molecules, and reactions that are considered as electronically very difficult including transition-metal compounds, weakly bonded complexes, and reaction barriers and comparisons with other hybrid functionals of GGA and meta-GGA types. According to many realistic tests, B2-PLYP can be regarded as the best general purpose density functional for molecules e.g., a mean absolute deviation for the two test sets of only 1.8 and 3.2 kcal/mol compared to about 3 and 5 kcal/mol, respectively, for the best other density functionals. Very importantly, also the maximum and minium errors outliers are strongly reduced by about 10‐20 kcal/mol. Furthermore, very good results are obtained for transition state barriers but unlike previous attempts at such a good description, this definitely comes not at the expense of equilibrium properties. Preliminary calculations of the equilibrium bond lengths and harmonic vibrational frequencies for diatomic molecules and transition-metal complexes also show very promising results. The uniformity with which B2-PLYP improves for a wide range of chemical systems emphasizes the need of virtual orbital-dependent terms that describe nonlocal electron correlation in accurate exchange-correlation functionals. From a practical point of view, the new functional seems to be very robust and it is thus suggested as an efficient quantum chemical method of general purpose. © 2006 American Institute of Physics. DOI: 10.1063/1.2148954

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Semiempirical hybrid density functional with perturbative second-order correlation

Preface Acknowledgements Notation Part I. Overview and Background Topics: 1. Introduction 2. Overview 3. Theoretical background 4. Periodic solids and electron bands 5. Uniform electron gas and simple metals Part II. Density Functional Theory: 6. Density functional theory: foundations 7. The Kohn-Sham ansatz 8. Functionals for exchange and correlation 9. Solving the Kohn-Sham equations Part III. Important Preliminaries on Atoms: 10. Electronic structure of atoms 11. Pseudopotentials Part IV. Determination of Electronic Structure, The Three Basic Methods: 12. Plane waves and grids: basics 13. Plane waves and grids: full calculations 14. Localized orbitals: tight binding 15. Localized orbitals: full calculations 16. Augmented functions: APW, KKR, MTO 17. Augmented functions: linear methods Part V. Predicting Properties of Matter from Electronic Structure - Recent Developments: 18. Quantum molecular dynamics (QMD) 19. Response functions: photons, magnons ... 20. Excitation spectra and optical properties 21. Wannier functions 22. Polarization, localization and Berry's phases 23. Locality and linear scaling O (N) methods 24. Where to find more Appendixes References Index.

Electronic Structure: Basic Theory and Practical Methods

First-principles computation of material properties: the ABINIT software project

As an alternative to the standard Kohn-Sham procedure, other exact realizations of density-functional theory ~generalized Kohn-Sham methods! are presented. The corresponding generalized Kohn-Sham eigenvalue gaps are shown to incorporate part of the discontinuity D xc of the exchange-correlation potential of standard KohnSham theory. As an example, a generalized Kohn-Sham procedure splitting the exchange contribution to the total energy into a screened, nonlocal and a local density component is considered. This method leads to band gaps far better than those of local-density approximation and to good structural properties for the materials Si, Ge, GaAs, InP, and InSb.

Generalized Kohn-Sham schemes and the band-gap problem

An exact formal Kohn-Sham scheme is derived with the help of perturbation theory. Through the introduction of a basis set this Kohn-Sham scheme can be used to perform, in principle, exact Kohn-Sham calculations. As a demonstration, only zeroth- and first-order terms in the underlying perturbation theory are considered. As a result an exact basis set ``exchange-only'' method is obtained. The presented perturbation theory expansions of the exchange-correlation energy and potential may serve as a starting point for the development of new approximate exchange-correlation functionals based on Kohn-Sham orbitals and eigenvalues and may be used to check conventional exchange-correlation functionals. The formal structures of the ab initio and the introduced density-functional treatment of electronic systems are compared.

Exact Kohn-Sham scheme based on perturbation theory

A Kohn-Sham formalism for the treatment of excited states within the density-functional theory (DFT) is presented. DFT exchange and correlation energy functionals for excited states are defined. Explicit expressions for these functionals are derived by generalizing a recent DFT perturbation theory. A computational scheme for the treatment of excited states within the DFT is suggested. Differences of Kohn-Sham eigenvalues are shown to be well-defined approximations for excitation energies. Correction terms to these approximations are presented. Perturbation theory expansions for band gaps are discussed. \textcopyright{} 1996 The American Physical Society.

Density-functional theory for excited states

We present a Kohn-Sham method that allows one to treat exchange interactions exactly within density-functional theory. The method is used to calculate lattice constants, cohesive energies, Kohn-Sham eigenvalues, dielectric functions, and effective masses of various zinc-blende semiconductors (Si, Ge, C, SiC, GaAs, AlAs, GaN, and AlN). The results are compared with values obtained within the local-density approximation, generalized gradient approximations, the Krieger-Li-Iafrate approximation for the Kohn-Sham exchange potential, and the Hartree-Fock method. We find that the exact exchange formalism, augmented by local density or generalized gradient correlations, yields both structural and optical properties in excellent agreement with experiment. Exact exchange-only calculations are found to lead to densities and energies that are close to Hartree-Fock values but to eigenvalue gaps that agree with experiment within 0.2 eV. The generalized gradient approximations for exchange yield energies that are much improved compared to local-density values. The exact exchange contribution to the discontinuity of the exchange-correlation potential is computed and discussed in the context of the band-gap problem.

Exact exchange Kohn-Sham formalism applied to semiconductors

A new Kohn-Sham method that treats exchange interactions within density functional theory exactly is applied to Si, diamond, GaN, and InN. The exact local exchange potential leads to significantly increased band gaps that are in good agreement with experimental data. Generalized gradient approximations yield exchange energies that are much closer to the exact values than those predicted by the local density approximation. The exchange contribution to the derivative discontinuity of the exchange-correlation potential is found to be very large (of the order of 5--10 eV).

Andreas Görling

Papers

Generalized Kohn-Sham schemes and the band-gap problem

Exact Kohn-Sham scheme based on perturbation theory

Density-functional theory for excited states

Exact exchange Kohn-Sham formalism applied to semiconductors

Exact Kohn-Sham Exchange Potential in Semiconductors