Meta-generalized gradient approximation: explanation of a realistic nonempirical density functional

Tao, Perdew, Staroverov, and Scuseria (TPSS) have constructed a nonempirical meta-generalized gradient approximation (meta-GGA) [Phys. Rev. Lett. 91, 146401 (2003)] for the exchange-correlation energy, imposing exact constraints relevant to the paradigm densities of condensed matter physics and quantum chemistry. Results of their extensive tests on molecules, solids, and solid surfaces are encouraging, suggesting that this density functional achieves uniform accuracy for diverse properties and systems. In the present work, this functional is explained and details of its construction are presented. In particular, the functional is constructed to yield accurate energies under uniform coordinate scaling to the low-density or strong-interaction limit. Its nonlocality is displayed by plotting the factor F(xc) that gives the enhancement relative to the local density approximation for exchange. We also discuss an apparently harmless order-of-limits problem in the meta-GGA. The performance of this functional is investigated for exchange and correlation energies and shell-removal energies of atoms and ions. Non-self-consistent molecular atomization energies and bond lengths of the TPSS meta-GGA, calculated with GGA orbitals and densities, agree well with those calculated self-consistently. We suggest that satisfaction of additional exact constraints on higher rungs of a ladder of density functional approximations can lead to further progress.     

Description of core excitations by time-dependent density functional theory with local density approximation, generalized gradient approximation, meta-generalized gradient approximation, and hybrid functionals

Time-dependent density functional theory (TDDFT) is employed to investigate exchange-correlation-functional dependence of the vertical core-excitation energies of several molecules including H, C, N, O, and F atoms. For the local density approximation (LDA), generalized gradient approximation (GGA), and meta-GGA, the calculated X1s-->pi* excitation energies (X = C, N, O, and F) are severely underestimated by more than 13 eV. On the other hand, time-dependent Hartree-Fock (TDHF) overestimates the excitation energies by more than 6 eV. The hybrid functionals perform better than pure TDDFT because HF exchange remedies the underestimation of pure TDDFT. Among these hybrid functionals, the Becke-Half-and-Half-Lee-Yang-Parr (BHHLYP) functional including 50% HF exchange provides the smallest error for core excitations. We have also discovered the systematic trend that the deviations of TDHF and TDDFT with the LDA, GGA, and meta-GGA functionals show a strong atom-dependence. Namely, their deviations become larger for heavier atoms, while the hybrid functionals are significantly less atom-dependent.     

Reparameterization of a meta-generalized gradient approximation functional by combining TPSS exchange with τ1 correlation

We present a new local meta-GGA exchange-correlation density functional by combining the TPSS meta-GGA exchange and the τ1 meta-GGA correlation functionals. The TPSS meta-GGA exchange-correlation and the τ1 meta-GGA correlation functionals have been implemented in the deMon code. The parameters in the τ1 meta-GGA correlation model are reoptimized in a synchronized way to match the original TPSS meta-GGA exchange counterpart. This reparametrized meta-GGA functional is referred to as “TPSSτ3”. The TPSSτ3 and TPSSτ1 meta-GGAs are validated using a test set that consists of covalent molecules, hydrogen-bonded complexes, and van der Waals interactions. The calculated results from TPSSτ1 and TPSSτ3 are analyzed and compared with reliable experimental data and theoretical data, as well as with those from Bmτ1 and TPSS calculations. The τ1 correlation model describes the aromatic compounds better than TPSS. TPSSτ3 yields satisfactory results for the covalent molecules, the hydrogen-bonded complexes, and the van der Waals complexes in the test set compared with TPSS, Bmτ1 and TPSSτ1. Contribution to the Serafin Fraga Memorial Issue.     

Time-dependent four-component relativistic density functional theory for excitation energies

Time-dependent four-component relativistic density functional theory within the linear response regime is developed for calculating excitation energies of heavy element containing systems. Since spin is no longer a good quantum number in this context, we resort to time-reversal adapted Kramers basis when deriving the coupled Dirac-Kohn-Sham equation. The particular implementation of the formalism into the Beijing density functional program package utilizes the multipolar expansion of the induced density to facilitate the construction of the induced Coulomb potential. As the first application, pilot calculations on the valence excitation energies and fine structures of the rare gas (Ne to Rn) and Group 12 (Zn to Hg) atoms are reported. To the best of our knowledge, it is the first time to be able to account for spin-orbit coupling within time-dependent density functional theory for excitation energies.     

Average excitation energies from time-dependent density functional response theory

The authors present an occupation number averaging scheme for time-dependent density functional response theory (TD-DFRT) in frequency domain. The known problem that TD-DFRT within the local (spin) density approximation (LDA/LSDA) inaccurately predicts Rydberg and charge-transfer excitation energies has been reexamined from the methodology of linear response, without explicit correction of the exchange-correlation potential. The working equations of TD-DFRT are adapted to treat arbitrary difference of orbital occupation numbers, using the nonsymmetric matrix form of Casida's formulation of TD-DFRT [M. E. Casida, in Recent Advances in Density Functional Methods, edited by D. P. Chong (World Scientific, Singapore, 1995), Pt. I, p. 155]. The authors' scheme is applied to typical closed-shell and open-shell molecular systems by examining the dependence of excitation energies on the fraction of excited electron. Good performance of this modified linear response scheme is shown, and is consistent with the authors' previous examination by the real-time propagation approach, suggesting that the calculation of average excitation energies might be one of the ways to better decode excitation energies from LDA/LSDA. Different techniques for treating singlet, triplet, and doublet states are discussed.     

Self-consistent meta-generalized gradient approximation study of adsorption of aromatic molecules on noble metal surfaces

The adsorption of benzene, pyridine, and two nucleobases on the Au(111) surface has been investigated using a fully relaxed, self-consistent meta-generalized gradient approximation (meta-GGA) density functional theory setup with the M06-L functional. The meta-GGA based molecule-surface separations are shortened and the adsorption bond strengths of the molecules are greatly improved over the virtually non-interacting results obtained when using a plain GGA exchange-correlation functional. The nucleobases containing oxygen atoms show higher corrugation with adsorption site and orientation than the other aromatic molecules considered. The adsorption of pentacene is studied on Au, Ag, and Cu surfaces. In agreement with experiment, the adsorption energies are found to increase with decreasing nobleness, but the dependency is underestimated. We point out how the kinetic energy density can discriminate between covalent and non-covalent bonding regions of orbital overlap.     

Performance of the M11 and M11-L density functionals for calculations of electronic excitation energies by adiabatic time-dependent density functional theory

Adiabatic time-dependent density functional theory is a powerful method for calculating electronic excitation energies of complex systems, but the quality of the results depends on the choice of approximate density functional. In this article we test two promising new density functionals, M11 and M11-L, against databases of 214 diverse electronic excitation energies, and we compare the results to those for 16 other density functionals of various kinds and to time-dependent Hartree-Fock. Charge transfer excitations are well known to be the hardest challenge for TDDFT. M11 is a long-range-corrected hybrid meta-GGA, and it shows better performance for charge transfer excitations than any of the other functionals except M06-HF, which is a specialized functional that does not do well for valence excitations. Several other long-range-corrected hybrid functionals also do well, and we especially recommend M11, ωB97X, and M06-2X for general spectroscopic applications because they do exceptionally well on ground-state properties as well as excitation energies. Local functionals are preferred for many applications to extended systems because of their significant cost advantage for large systems. M11-L is a dual-range local functional and-unlike all previous local functionals-it has good performance for Rydberg states as well as for valence states. Thus it is highly recommended for excitation energy calculations on extended systems.     

Performance of time-dependent density functional and Green functions methods for calculations of excitation energies in radicals and for Rydberg electronic states

Time-dependent density functional (TD-DFT) and perturbation theory-based outer valence Green functions (OVGF) methods have been tested for calculations of excitation energies for a set of radicals, molecules, and model clusters simulating points defects in silica. The results show that the TD-DFT approach may give unreliable results not only for diffuse Rydberg states, but also for electronic states involving transitions between MOs localized in two remote from each other spatial regions, for example, for charge-transfer excitations. For the. O-SiX(3) clusters, where X is a single-valence group, TD-DFT predicts reasonable excitation energies but incorrect sequence of electronic transitions. For a number of cases where TD-DFT is shown to be unreliable, the OVGF approach can provide better estimates of excitation energies, but this method also is not expected to perform universally well. The OVGF performance is demonstrated to be satisfactory for excitations with predominantly single-determinant wave functions where the deviations of the calculated energies from experiment should not exceed 0.1-0.3 eV. However, for more complicated transitions involving multiple bonds or for excited states with multireference wave functions the OVGF approach is less reliable and error in the computed energies can reach 0.5-1 eV.     

Total energies of molecules with the local density functional approximation and gaussian basis sets

Total energies of small molecules were calculated with a local density functional (LDF) approximation within the LCAO MO SCF scheme. The local spin density functional (LSD) of Gunnarsson and Lundqvist was used. The basis sets used are of contracted gaussian type which allow comparison of LSD with Hartree-Fock (HF) results. The program for calculation of the LSD term was incorporated into the standard ab initio package. The LSD binding energies were in better agreement with experiment than those from HF.     

Time-dependent density functional theory predictions of the vertical excitation energies of silanones as models for the excitation process in porous silicon

Time-dependent density functional theory calculations with a proper treatment of the asymptotic form of the exchange-correlation potential have been performed on R(R')Si=O to predict vertical excitation energies. The species R(R')Si=O is used as a model for the binding of the -(R)Si=O chromophore to a porous silicon surface. The calculated vertical excitation energies are substantially lower than those determined previously and show that vertical excitation of the lone chromophore is possible for all types of substituents including electronegative ones with KrF laser excitation in contrast to other predictions. If the substituents are electropositive, the chromophore can also be excited by a nitrogen laser. These results, in concert with the effect of the porous silicon surface on the R(R')Si=O excited states, confirm our previous explanation of the photoluminescence of porous silicon as being due to the presence of Si=O chromophores and provide new insights into the photoexcitation process. The results show that the differences in the vertical and adiabatic excitation energies are strongly dependent on whether the substituents are electronegative or electropositive with the former leading to larger differences and the latter leading to smaller differences. The results for the energy differences are explained in terms of the changes in the Si=O bond length on vertical excitation and on the changes in bond angles, which are related to the ability of the Si center in the excited state to undergo an inversion process.     

Assessment of time-dependent density functional theory for predicting excitation energies of bichromophoric peptides: case of tryptophan-phenylalanine

The ability of applied time-dependent density functional theory to predict the near-ultraviolet absorption spectrum of bichomophoric peptides in the gas phase has been tested by calculating the vertical excitation energies of the Tryptophan-Phenylalanine (Trp-Phe) dipeptide. We show that the contamination of the low-frequency part of the spectrum by spurious charge-transfer excitations depends both on the conformation of the peptide chain and the exchange-correlation approximation. For the most stable structure investigated, a hybrid density functional appears to eliminate a large proportion of the spurious states.     

Benchmarking approximate density functional theory for s/d excitation energies in 3d transition metal cations

Holthausen has recently provided a comprehensive study of density functional theory for calculating the s/d excitation energies of the 3d transition metal cations. This study did not include the effects of scalar relativistic effects, and we show here that the inclusion of scalar relativistic effects significantly alters the conclusions of the study. We find, contrary to the previous study, that local functionals are more accurate for the excitation energies of 3d transition method cations than hybrid functionals. The most accurate functionals, of the 38 tested, are SLYP, PBE, BP86, PBELYP, and PW91.     

Density functional calculations of optical excitation energies by a transition-state method

Optical excitation energies of MnO⁻₄, CrO²⁻₄, and RuO₄ are calculated using the density functional methodology. A short outline of some important developments in this theory for the determination of excited-state properties is given. A practical working procedure for the calculation of transition energies including multiplet splitting is described. This method is based on a transition-state approach which is connected, as will be shown, to Slater's transition-state concept. Results obtained by this working procedure are compared to the energy differences between separately converged configurations of ground and excited states and the corresponding multiplet structure, denoted as the ΔSCF calculation in the following. © 1997 John Wiley & Sons, Inc.     

The performance of time-dependent density functional theory based on a noncollinear exchange-correlation potential in the calculations of excitation energies

In the present work we have studied the accuracy of excitation energies calculated from spin-flip transitions with a formulation of time-dependent density functional theory based on a noncollinear exchange-correlation potential proposed in a previous study. We compared the doublet-doublet excitation energies from spin-flip transitions and ordinary transitions, calculated the multiplets splitting of some atoms, the singlet-triplet gaps of some diradicals, the energies of excited quartet states with a doublet ground state. In addition, we attempted to calculate transition energies with excited states as reference. We compared the triplet excitation energies and singlet-triplet separations of the excited state from spin-flip and ordinary transitions. As an application, we show that using excited quartet state as reference can help us fully resolve excited states spin multiplets. In total the obtained excitation energies calculated from spin-flip transitions agree quite well with other theoretical results or experimental data.     

Atomic multiplet energies from density functional calculations

Atomic multiplet term energies for dⁿ configurations have been estimated within density functional theory (DFT) exploiting symmetry to the largest possible extent. The electrostatic two-electron integrals, as well as term energies, are expressed in function of only three non-redundant single determinants (NRSDs), each of them being obtained from density functional calculations. The influence of correlation effects described with a gradient-corrected functional (GGA) is examined and discussed. Comparison with experimental data shows the reliability of this symmetry-based density functional approach.     

Spin-multiplet energies from time-dependent density functional theory

Starting from a formally exact density-functional representation of the frequency-dependent linear density response and exploiting the fact that the latter has poles at the true excitation energies, we develop a density-functional method for the calculation of excitation energies. Simple additive corrections to the Kohn-Sham single-particle transition energies are derived whose actual computation only requires the ordinary static Kohn-Sham orbitals and the corresponding eigenvalues. Numerical results are presented for spin-singlet and triplet energies. © 1996 John Wiley & Sons, Inc.     

Cluster free energies and nucleation: a density functional approach

Density functional methods of statistical mechanics are applied to calculate the average density profiles and free energies of small (50–500 particles) liquid clusters in unstable or metastable equilibrium with respect to the vapor. The results are compared with those obtained from experiments on homogeneous nucleation of liquids from the vapor.