Density functionals from many-body perturbation theory: the band gap for semiconductors and insulators

Theoretically the Kohn-Sham band gap differs from the exact quasiparticle energy gap by the derivative discontinuity of the exchange-correlation functional. In practice for semiconductors and insulators the band gap calculated within any local or semilocal density approximations underestimates severely the experimental energy gap. On the other hand, calculations with an "exact" exchange potential derived from many-body perturbation theory via the optimized effective potential suggest that improving the exchange-correlation potential approximation can yield a reasonable agreement between the Kohn-Sham band gap and the experimental gap. The results in this work show that this is not the case. In fact, we add to the exact exchange the correlation that corresponds to the dynamical (random phase approximation) screening in the GW approximation. This accurate exchange-correlation potential provides band structures similar to the local density approximation with the corresponding derivative discontinuity that contributes 30%-50% to the energy gap. Our self-consistent results confirm substantially the results for Si and other semiconductors obtained perturbatively [R. W. Godby et al., Phys. Rev. B 36, 6497 (1987)] and extend the conclusion to LiF and Ar, a wide-gap insulator and a noble-gas solid.     

Improved supermolecular second order Møller-Plesset intermolecular interaction energies using time-dependent density functional response theory

The supermolecular second order Moller-Plesset (MP2) intermolecular interaction energy is corrected by employing time-dependent density functional (TDDFT) response theory. This is done by replacing the uncoupled second order dispersion contribution contained in the supermolecular MP2 energy with the coupled dispersion energy obtained from the TDDFT approach. Preliminary results for the rare gas dimers He2, Ne2, and Ar2 and a few structures of the (HF)2 and (H2O)2 dimers show that the conventional MP2 interaction energies are considerably improved by this procedure if compared to coupled cluster singles doubles with perturbative triples [CCSD(T)] interaction energies. However, the quality of the interaction energies obtained in this way strongly depends on the exchange-correlation potential employed in the monomer calculations: It is shown that an exact exchange-only potential surprisingly often performs better than an asymptotically corrected hybrid exchange-correlation potential. Therefore the method proposed in this work is similar to the method by Cybulski and Lytle [J. Chem. Phys., 127, 141102 (2007)] which corrects the supermolecular MP2 energies with a scaled dispersion energy from time-dependent Hartree-Fock. The results in this work are also compared to the combination of density functional theory and intermolecular perturbation theory.     

Chemical hardness and the discontinuity of the Kohn-Sham exchange-correlation potential

Chemical hardness, identified as the difference between the vertical first ionization potential I and the vertical electron affinity A, is analyzed in the context of the ionization theorems to derive expressions for its evaluation at different levels of approximation that arise as a direct consequence of the derivative discontinuity of the exchange-correlation potential. The quantities involved in these expressions incorporate indirectly the effects of the discontinuity, but their values may be calculated with any functional of the local density approximation, generalized gradient approximation, or optimized effective potential type, with or without derivative discontinuity, and with or without the correct asymptotic behavior. By comparison with the vertical energy difference values of I and A, which requires the calculation of the N-, (N-1)-, and (N+1)-electron systems, it is found, for a set of 14 closed shell molecules, that the difference between the eigenvalues of the highest occupied molecular orbitals of the N- and (N+1)-electron systems leads to rather accurate values, when the correct asymptotic behavior is incorporated, and that a second-order one-body perturbation approach that only requires information from the N-electron system leads to reasonable values.     

Ab initio studies of properties of small potassium clusters

We have studied the properties of various isomers of potassium clusters containing even number of atoms ranging from 2 to 20 at the ab initio level. The geometry optimization calculations of the isomers of each cluster are performed by using all-electron density functional theory with gradient corrected exchange-correlation functional. Using the optimized geometries of different isomers we investigate the evolution of binding energy, ionization potential, and static polarizability with the increasing size of the clusters. The polarizabilities are calculated by employing M?ller-Plesset perturbation theory and time-dependent density functional theory. The polarizabilities of dimer and tetramer are also calculated by employing large basis set coupled cluster theory with single and double excitations and perturbative triple excitations. The time-dependent density functional theory calculations of polarizabilities are carried out with two different exchange-correlation potentials: (i) an asymptotically correct model potential and (ii) within the local density approximation. A systematic comparison with the other available theoretical and experimental data for various properties of small potassium clusters mentioned above has been performed. These comparisons reveal that both the binding energy and the ionization potential obtained with gradient-corrected potential match quite well with the already published data. Similarly, the polarizabilities obtained with M?ller-Plesset perturbation theory and with model potential are quite close to each other and also close to experimental data.     

元素电负性和硬度的密度泛函理论研究

应用密度泛函理论的DFT LDA、DFT LDA/NL和改进的Slater过渡态方法,把元素的电离能和电子亲合能的计算扩展到周期表的103种元素.并用有限差分方法计算了这103种元素的电负性和硬度.计算中考虑了相对论效应.计算结果比以前Robles等用密度泛函理论的XGL和Xα近似的交换相关泛函的计算结果有所改进,更接近实验值.     

Orbital-dependent correlation energy in density-functional theory based on a second-order perturbation approach: success and failure

We have developed a second-order perturbation theory (PT) energy functional within density-functional theory (DFT). Based on PT with the Kohn-Sham (KS) determinant as a reference, this new ab initio exchange-correlation functional includes an exact exchange (EXX) energy in the first order and a correlation energy including all single and double excitations from the KS reference in the second order. The explicit dependence of the exchange and correlation energy on the KS orbitals in the functional fits well into our direct minimization approach for the optimized effective potential, which is a very efficient method to perform fully self-consistent calculations for any orbital-dependent functionals. To investigate the quality of the correlation functional, we have applied the method to selected atoms and molecules. For two-electron atoms and small molecules described with small basis sets, this new method provides excellent results, improving both second-order Moller-Plesset expression and any conventional DFT results significantly. For larger systems, however, it performs poorly, converging to very low unphysical total energies. The failure of PT based energy functionals is analyzed, and its origin is traced back to near degeneracy problems due to the orbital- and eigenvalue-dependent algebraic structure of the correlation functional. The failure emerges in the self-consistent approach but not in perturbative post-EXX calculations, emphasizing the crucial importance of self-consistency in testing new orbital-dependent energy functionals.     

How accurate is the density functional theory combined with symmetry-adapted perturbation theory approach for CH-pi and pi-pi interactions? A comparison to supermolecular calculations for the acetylene-benzene dimer

Five different orientations of the acetylene-benzene dimer including the T-shaped global minimum structure are used to assess the accuracy of the density functional theory combined with symmetry adapted perturbation theory (DFT-SAPT) approach in its density-fitting implementation (DF-DFT-SAPT) for the study of CH-pi and pi-pi interactions. The results are compared with the outcome of counterpoise corrected supermolecular calculations employing second-order M?ller-Plesset (MP2), spin-component scaled MP2 (SCS-MP2) and single and double excitation coupled cluster theory including perturbative triple excitations (CCSD(T)). For all considered orientations MP2 predicts much deeper potential energy curves with considerably shifted minima compared to CCSD(T) and DFT-SAPT. In spite of being an improvement over the results of MP2, SCS-MP2 tends to underestimate the well depth while DFT-SAPT, employing an asymptotically corrected hybrid exchange-correlation potential in conjunction with the adiabatic local density approximation for the exchange-correlation kernel, is found to be in excellent agreement with CCSD(T). Furthermore, DFT-SAPT provides a detailed understanding of the importance of the electrostatic, induction and dispersion contributions to the total interaction energy and their repulsive exchange corrections.     

First principles calculations of electronic and optical properties of Ag2S

A theoretical study of structural, electronic and optical properties of Ag₂S is presented using the full potential linearized augmented plane wave (FP-LAPW) method within density functional theory (DFT). In this approach, the modified Becke Johnson (MBJ) potential coupled with Local Density Approximation (LDA) was used for the exchange-correlation potential calculation. Ground state properties are determined for the bulk material in monoclinic phase. Band structure reveals that this compound is a direct energy band gap semiconductor. MBJLDA results for the band gap of this compound are much better than those obtained using LDA, Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA) and Engel–Vosko's GGA (EV-GGA). A very good agreement is observed between MBJLDA band gap and corresponding experimental values as compared to other calculations. Optical constants including the dielectric function, refractive index, extinction coefficient, electron energy loss function, reflectivity and absorption coefficient are obtained and discussed.     

Derivative discontinuity, bandgap and lowest unoccupied molecular orbital in density functional theory

The conventional analysis of Perdew and Levy, and Sham and Schlu?ter shows that the functional derivative discontinuity of the exchange-correlation density functional plays a critical role in the correct prediction of bandgaps, or the chemical hardness. In a recent work by the present authors, explicit expressions for bandgap prediction with all common types of exchange-correlation functionals have been derived without invoking the concept of exchange-correlation energy functional derivative discontinuity at all. We here analyze the two approaches and establish their connection and difference. The present analysis further leads to several important results: (1) The lowest unoccupied molecular orbital (LUMO) in DFT has as much meaning in describing electron addition as the highest occupied molecular orbital (HOMO) in describing electron removal. (2) Every term in the total energy functional contributes to the energy gap because of the discontinuity of the derivative of the density (or density matrix) with respect to the number of electrons, ((?ρ(s)(r('),r))/?N)(v(s) ), at integers. (3) Consistent with the Perdew-Levy-Sham-Schlu?ter conclusion that the exact Kohn-Sham energy gap differs from the fundamental bandgap by a finite correction due to the functional derivative discontinuity of the exchange-correlation energy, we show that the exchange-correlation functional cannot be an explicit and differentiable functional of the electron density, either local or nonlocal. The last result is further strengthened when we consider Mott insulators. There, the exact exchange-correlation functional needs to have an explicitly discontinuous (nondifferentiable) dependence on the density or the density matrix. (4) We obtain exact conditions on the derivatives of total energy with respect to the spin-up and spin-down number of electrons.     

An extended PCILO method

The perturbative configuration interaction using localized orbitals (the PCILO method) was extended in the way that current limitations to the two-centre bond approach were overcome. The localized molecular orbitals contain an arbitrary number of the basis set components; this follows from the a priori stated localized bonding model of a molecule. The extended PCILO method was formulated for the CNDO, INDO and NDDO Hamiltonian approximations. The configuration interaction was performed using the Rayleigh-Schrödinger many-body perturbation theory with the Møller-Plesset type of Hamiltonian partitioning, similar to that used in the so-called modified PCILO method. Applications to molecules with semi-localized and/or semi-delocalized bonds, as benzene or diborane, are presented.     

Comparative theoretical investigation of the vertical excitation energies and the electronic structure of [MoVOCl4]-: influence of basis set and geometry

The electronic structures, geometries, and vibration frequencies of the open-shell molybdenum(V) ion, [MoOCl(4)](-), have been calculated at the extended Hückel, semiempirical ZINDO/1, ZINDO/S, and PM3(tm), as well as ab initio and DFT theoretical levels. Electronic structure calculations suggest that the expected metal-fold orbital order can be satisfied only at the DFT level. The time-dependent density functional theory (TDDFT) approach has been used for the calculation of the vertical excitation energies in the UV-vis region with different basis sets, starting geometries, and exchange-correlation functionals. A good agreement between the predicted and the experimental electronic absorption and MCD spectra of the complex, [MoOCl(4)](-), was observed when the B3LYP and B3P86 exchange-correlation functionals were used with a full electron valence double-zeta with polarization basis set for the molybdenum and 6-311G(d) for all other atoms. Similar results were obtained when the LANL2DZ effective core potential for molybdenum atom and 6-31G(d) for all other atoms were used. The best absolute deviation of 0.13 and mean deviation of 0.01 eV were calculated for the bands in the UV-vis region by B3P86, while the results for the B3LYP exchange-correlation functional were less satisfactory. Compared to polarization functions, the inclusion of diffuse functions resulted in little improvement. The calculated excitations energies and charge-transfer band intensities are found to be sensitive to the Mo=O distance and O-Mo-Cl angle.     

Local-orbital-based correlated ab initio band structure calculations in insulating solids: LiF

 A local-orbital-based ab initio approach to calculate correlation effects on quasi-particle energies in insulating solids is presented. The use of localized Wannier-type Hartree–Fock orbitals allows correlation effects to be efficiently assessed. First a Green's function approach based on exact diagonalization is introduced and this is combined with an incremental scheme, while subsequently different levels of perturbative approximations are derived from the general procedure. With these methods the band structure of LiF is calculated and good agreement with experiment is found. By comparing the different approximations proposed, including the exact diagonalization procedure, their relative quality is established. Received: 25 June 2001 / Accepted: 31 August 2001 / Published online: 19 December 2001     

Numerically stable optimized effective potential method with balanced Gaussian basis sets

A solution to the long-standing problem of developing numerically stable optimized effective potential (OEP) methods based on Gaussian basis sets is presented by introducing an approach consisting of an exact exchange OEP method with an accompanying construction and balancing scheme for the involved auxiliary and orbital Gaussian basis sets that is numerically stable and that properly represents an exact exchange Kohn-Sham method. The method is a purely analytical method that does not require any numerical grid, scales like Hartree-Fock or B3LYP procedures, is straightforward to implement, and is easily generalized to take into account orbital-dependent density functionals other than the exact exchange considered in this work. Thus, the presented OEP approach opens the way to the development and application of novel orbital-dependent exchange-correlation functionals. It is shown that adequately taking into account the continuum part of the Kohn-Sham orbital spectrum is crucial for numerically stable Gaussian basis set OEP methods. Moreover, it is mandatory to employ orbital basis sets that are converged with respect to the used auxiliary basis representing the exchange potential. OEP calculations in the past often did not meet the latter requirement and therefore may have led to erroneously low total energies.     

The exchange-correlation potential in ab initio density functional theory

From coupled-cluster theory and many-body perturbation theory we derive the local exchange-correlation potential of density functional theory in an orbital dependent form. We show the relationship between the coupled-cluster approach and density functional theory, and connections and comparisons with our previous second-order correlation potential [OEP-MBPT(2) (OEP-optimized effective potential)] [I. Grabowski, S. Hirata, S. Ivanov, and R. J. Bartlett, J. Chem. Phys. 116, 4415 (2002)]. Starting from a general theoretical framework based on the density condition in Kohn-Sham theory, we define a rigorous exchange-correlation functional, potential and orbitals. Specifying initially to second-order terms, we show that our ab initio correlation potential provides the correct shape compared to those from reference quantum Monte Carlo calculations, and we demonstrate the superiority of using Fock matrix elements or more general infinite-order semicanonical transformations. This enables us to introduce a method that is guaranteed to converge to the right answer in the correlation and basis set limit, just as does ab initio wave function theory. We also demonstrate that the energies obtained from this generalized second-order method [OEP-MBPT2-f] and [OEP-MBPT2-sc] are often of coupled-cluster accuracy and substantially better than ordinary Hartree-Fock based second-order MBPT=MP2.     

Structural Stability and Electronic Properties of the I4_1/amd Vanadium Dioxide

By using LDA+U approach based on the density functional theory, the structural stability of I41/amd VO_2 is investigated. According to the phonon dispersion and stability criteria, the I41/amd is suggested to be another possible and stable structure for the VO_2. Lattice parameters of the I41/amd VO_2 are determined by geometry optimization. The energy band structure shows that the I41/amd VO_2 should be a metal. Furthermore, the upper valence band has dominant 2p-orbital characters, but the lower conduction band shows distinctive 3d-orbital characters. Obvious hybridization between the O-2p and V-3d orbitals is observed.     

Unambiguous optimization of effective potentials in finite basis sets

The optimization of effective potentials is of interest in density-functional theory (DFT) in two closely related contexts. First, the evaluation of the functional derivative of orbital-dependent exchange-correlation functionals requires the application of optimized effective potential methods. Second, the optimization of the effective local potential that yields a given electron density is important both for the development of improved approximate functionals and for the practical application of embedding schemes based on DFT. However, in all cases this optimization turns into an ill-posed problem if a finite basis set is introduced for the Kohn-Sham orbitals. So far, this problem has not been solved satisfactorily. Here, a new approach to overcome the ill-posed nature of such finite-basis set methods is presented for the optimization of the effective local potential that yields a given electron density. This new scheme can be applied with orbital basis sets of reasonable size and makes it possible to vary the basis sets for the orbitals and for the potential independently, while providing an unambiguous potential that systematically approaches the numerical reference.     

Embedding wave function theory in density functional theory

We present a framework for embedding a highly accurate coupled-cluster calculation within a larger density functional calculation. We use a perturbative buffer to help insulate the coupled-cluster region from the rest of the system. Regions are defined, not in real space, but in Hilbert space, though connection between the two can be made by spatial localization of single-particle orbitals. Relations between our embedding approach and some similar techniques are discussed. We present results for small sample systems for which we can extract essentially exact results, demonstrating that our approach seems to work quite well and is generally more reliable than some of the related approaches due to the introduction of additional interaction terms.