Effect of self-interaction error in the evaluation of the bond length alternation in trans-polyacetylene using density-functional theory

The calculation of the bond-length alternation (BLA) in trans-polyacetylene has been chosen as benchmark to emphasize the effect of the self-interaction error within density-functional theory (DFT). In particular, the BLA of increasingly long acetylene oligomers has been computed using the M?ller-Plesset wave-function method truncated at the second order and several DFT models. While local-density approximation (LDA) or generalized gradient corrected (GGA) functionals strongly underestimate the BLA, approaches including self-interaction corrections (SIC) provide significant improvements. Indeed, the simple averaged-density SIC scheme (ADSIC), recently proposed by Legrand et al. [J. Phys. B 35, 1115 (2002)], provides better results for the structure of large oligomers than the more complex approach of Krieger et al. [Phys. Rev. A 45, 101 (1992)]. The ADSIC method is particularly promising since both the exchange-correlation energy and potential are improved with respect to standard LDA/GGA using a physically appealing correction, through a different route than the more popular approach through the Hartree-Fock exchange inclusion within the hybrid functionals.     

Testing the TPSS meta-generalized-gradient-approximation exchange-correlation functional in calculations of transition states and reaction barriers

We have studied the performance of local and semilocal exchange-correlation functionals [meta-generalized-gradient-approximation (GGA)-TPSS, GGA-Perdew-Burke-Ernzerhof (PBE), and local density approximation (LDA)] in the calculation of transition states, reaction energies, and barriers for several molecular and one surface reaction, using the plane-wave pseudopotential approach. For molecular reactions, these results have been compared to all-electron Gaussian calculations using the B3LYP hybrid functional, as well as to experiment and high level quantum chemistry calculations, when available. We have found that the transition state structures are accurately identified irrespective of the level of the exchange-correlation functional, with the exception of a qualitatively incorrect LDA prediction for the H-transfer reaction in the hydrogen bonded complex between a water molecule and a OH radical. Both the meta-GGA-TPSS and the GGA-PBE functionals improve significantly the calculated LDA barrier heights. The meta-GGA-TPSS further improves systematically, albeit not always sufficiently, the GGA-PBE barriers. We have also found that, on the Si(001) surface, the meta-GGA-TPSS barriers for hydrogen adsorption agree significantly better than the corresponding GGA-PBE barriers with quantum Monte Carlo cluster results and experimental estimates.     

Density-functional expansion methods: evaluation of LDA, GGA, and meta-GGA functionals and different integral approximations

We extend the Kohn-Sham potential energy expansion (VE) to include variations of the kinetic energy density and use the VE formulation with a 6-31G* basis to perform a "Jacob's ladder" comparison of small molecule properties using density functionals classified as being either LDA, GGA, or meta-GGA. We show that the VE reproduces standard Kohn-Sham DFT results well if all integrals are performed without further approximation, and there is no substantial improvement in using meta-GGA functionals relative to GGA functionals. The advantages of using GGA versus LDA functionals becomes apparent when modeling hydrogen bonds. We furthermore examine the effect of using integral approximations to compute the zeroth-order energy and first-order matrix elements, and the results suggest that the origin of the short-range repulsive potential within self-consistent charge density-functional tight-binding methods mainly arises from the approximations made to the first-order matrix elements.     

Binding energy curves from nonempirical density functionals. I. Covalent bonds in closed-shell and radical molecules

Binding or potential energy curves have been calculated for the ground-state diatomics H(2)(+), He(2)(+), LiH(+), H(2), N(2), and C(2), for the transition state H(3), and for the triplet first excited state of H(2) using the nonempirical density functionals from the first three rungs of a ladder of approximations: the local spin density (LSD) approximation, the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA), and the Tao-Perdew-Staroverov-Scuseria (TPSS) meta GGA. Good binding energy curves in agreement with coupled cluster or configuration interaction calculations are found from the PBE GGA and especially from the TPSS meta GGA. Expected exceptions are the symmetric radicals H(2)(+) and He(2)(+), where the functionals suffer from self-interaction error, and the exotically bonded C(2). Although the energy barrier for the reaction H(2) + H --> H + H(2) is better in PBE than in TPSS, the transition state H(3) is a more properly positioned and curved saddle point of the energy surface in TPSS. The triplet first excited state of H(2) obeys the Aufbau principle and thus is one of the exceptional excited states that are computable in principle from the ground-state functional. The PBE GGA and TPSS meta GGA are useful not only for chemical applications but also for the construction of higher-rung nonempirical functionals that can further improve the binding energy curves.     

Energy landscapes of nucleophilic substitution reactions: a comparison of density functional theory and coupled cluster methods

We have carried out a detailed evaluation of the performance of all classes of density functional theory (DFT) for describing the potential energy surface (PES) of a wide range of nucleophilic substitution (SN2) reactions involving, amongst others, nucleophilic attack at carbon, nitrogen, silicon, and sulfur. In particular, we investigate the ability of the local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA as well as hybrid DFT to reproduce high-level coupled cluster (CCSD(T)) benchmarks that are close to the basis set limit. The most accurate GGA, meta-GGA, and hybrid functionals yield mean absolute deviations of about 2 kcal/mol relative to the coupled cluster data, for reactant complexation, central barriers, overall barriers as well as reaction energies. For the three nonlocal DFT classes, the best functionals are found to be OPBE (GGA), OLAP3 (meta-GGA), and mPBE0KCIS (hybrid DFT). The popular B3LYP functional is not bad but performs significantly worse than the best GGA functionals. Furthermore, we have compared the geometries from several density functionals with the reference CCSD(T) data. The same GGA functionals that perform best for the energies (OPBE, OLYP), also perform best for the geometries with average absolute deviations in bond lengths of 0.06 A and 0.6 degrees, even better than the best meta-GGA and hybrid functionals. In view of the reduced computational effort of GGAs with respect to meta-GGAs and hybrid functionals, let alone coupled cluster, we recommend the use of accurate GGAs such as OPBE or OLYP for the study of SN2 reactions.     

Analytic derivatives for perturbatively corrected "double hybrid" density functionals: theory, implementation, and applications

A recently proposed new family of density functionals [S. Grimme, J. Chem. Phys. 124, 34108 (2006)] adds a fraction of nonlocal correlation as a new ingredient to density functional theory (DFT). This fractional correlation energy is calculated at the level of second-order many-body perturbation theory (PT2) and replaces some of the semilocal DFT correlation of standard hybrid DFT methods. The new "double hybrid" functionals (termed, e.g., B2-PLYP) contain only two empirical parameters that have been adjusted in thermochemical calculations on parts of the G2/3 benchmark set. The methods have provided the lowest errors ever obtained by any DFT method for the full G3 set of molecules. In this work, the applicability of the new functionals is extended to the exploration of potential energy surfaces with analytic gradients. The theory of the analytic gradient largely follows the standard theory of PT2 gradients with some additional subtleties due to the presence of the exchange-correlation terms in the self-consistent field operator. An implementation is reported for closed-shell as well as spin-unrestricted reference determinants. Furthermore, the implementation includes external point charge fields and also accommodates continuum solvation models at the level of the conductor like screening model. The density fitting resolution of the identity (RI) approximation can be applied to the evaluation of the PT2 part with large gains in computational efficiency. For systems with approximately 500-600 basis functions the evaluation of the double hybrid gradient is approximately four times more expensive than the calculation of the standard hybrid DFT gradient. Extensive test calculations are provided for main group elements and transition metal containing species. The results reveal that the B2-PLYP functional provides excellent molecular geometries that are superior compared to those from standard DFT and MP2.     

Optical Spectra of Oligofurans: A Theoretical Approach to the Transition Energies,Reorganization Energies,and the Vibronic Activity

There is experimental evidence of high vibronic activity that accompanies the allowed transition between the ground state and the lowest electronic singlet excited state of oligofurans that contain two, three, and four furan rings. The absorption and emission spectra of the three lowest oligofurans measured at liquid nitrogen temperature show distinct fine structures that are reproduced using the projection-based model of vibronic coupling (with Dushinsky rotation included) parameterized utilizing either Density Functional Theory (DFT, with several different exchange-correlation functionals) or ab initio (CC2) quantum chemistry calculations. Using as a reference the experimental data concerning the electronic absorption and fluorescence for the eight lowest oligofurans, we first analyzed the performance of the exchange-correlation functionals for the electronic transition energies and the reorganization energies. Subsequently, we used the best functionals alongside with the CC2 method to explore how the reorganization energies are distributed among the totally symmetric vibrations, identify the normal modes that dominate in the fine structures present in the absorption and emission bands, and trace their evolution with the increasing number of rings in the oligofuran series. Confrontation of the simulated spectra with the experiment allows for the verification of the performance of the selected DFT functionals and the CC2 method.     

Structure and energetics of Fe2(CO)8 singlet and triplet electronic states

The singlet and triplet state potential energy surfaces (PES) of Fe2(CO)8 are explored by means of density functional theory (DFT) techniques. The two PES have different global mimima: the dibriged C(2v) isomer for the singlet and the unbridged D(2d) isomer for the triplet. The sign of the energy gap between singlet and triplet global minima depends on the type of adopted DFT functional: hybrid functionals predict the triplet is more stable than the singlet, but the opposite applies to generalized gradient approximated (GGA) functionals. The analysis of the computed CO stretching frequencies demonstrates that the experimental data for the unbridged form is compatible also with the unbridged triplet D(2d) isomer. Starting from these two facts, the electronic structure of unbridged D(2d) Fe2(CO)8 is discussed herein. Single-point energy computations at the coupled-cluster single and double (CCSD) level favor the D(2d) triplet state.     

Low-lying electronic states of M(3)O(9)(-) and M(3)O(9)(2-) (M = Mo, W)

Multiple low-lying electronic states of M(3)O(9)(-) and M(3)O(9)(2-) (M = Mo, W) arise from the occupation of the near-degenerate low-lying virtual orbitals in the neutral clusters. We used density functional theory (DFT) and coupled cluster theory (CCSD(T)) with correlation consistent basis sets to study the structures and energetics of the electronic states of these anions. The adiabatic and vertical electron detachment energies (ADEs and VDEs) of the anionic clusters were calculated with 27 exchange-correlation functionals including one local spin density approximation functional, 13 generalized gradient approximation (GGA) functionals, and 13 hybrid GGA functionals, as well as the CCSD(T) method. For M(3)O(9)(-), CCSD(T) and nearly all of the DFT exchange-correlation functionals studied predict the (2)A(1) state arising from the Jahn-Teller distortion due to singly occupying the degenerate e' orbital to be lower in energy than the (2)A(1)' state arising from singly occupying the nondegenerate a(1)' orbital. For W(3)O(9)(-), the (2)A(1) state was predicted to have essentially the same energy as the (2)A(1)' state at the CCSD(T) level with core-valence correlation corrections included and to be higher in energy or essentially isoenergetic with most DFT methods. The calculated VDEs from the CCSD(T) method are in reasonable agreement with the experimental values for both electronic states if estimates for the corrections due to basis set incompleteness are included. For M(3)O(9)(2-), the singlet state arising from doubly occupying the nondegenerate a(1)' orbital was predicted to be the most stable state for both M = Mo and W. However, whereas M(3)O(9)(2-) was predicted to be less stable than M(3)O(9)(-), W(3)O(9)(2-) was predicted to be more stable than W(3)O(9)(-).     

Self-interaction artifacts on structural features of uranyl monohydroxide from Kohn�CSham calculations

Invoking a DFT?+?U approach, we explored self-interaction artifacts in results from Kohn?CSham (KS) density functional calculations on the geometry and the vibrational frequencies of uranyl monohydroxide and the corresponding tetra-aqua complex. Exchange?Ccorrelation functionals based on the local density approximation (LDA) and the generalized-gradient approximation (GGA) predict equilibrium geometries for [UO₂(OH)]⁺ that deviate from the results of hybrid DFT calculations and high-level wavefunction-based methods such as CCSD(T). LDA?+?U and GGA?+?U functionals with corrections for the insufficient localization of the U 5f shell yield better agreement, in particular for the angle U-O_h-H. At the LDA level, a linear coordination of the OH ligand results; with the +U correction, the angle U-O_h-H is reduced by ~35°, in good agreement with CCSD(T) results. At the GGA level, the bending angle is changed by ~20°. This relatively strong self-interaction artifact is traced back to a spurious ?? interaction between U 5f and O(p) orbitals which is less pronounced in the presence of further (aqua) ligands.     

Adiabatic corrections to density functional theory energies and wave functions

The adiabatic finite-nuclear-mass-correction (FNMC) to the electronic energies and wave functions of atoms and molecules is formulated for density-functional theory and implemented in the deMon code. The approach is tested for a series of local and gradient corrected density functionals, using MP2 results and diagonal-Born-Oppenheimer corrections from the literature for comparison. In the evaluation of absolute energy corrections of nonorganic molecules the LDA PZ81 functional works surprisingly better than the others. For organic molecules the GGA BLYP functional has the best performance. FNMC with GGA functionals, mainly BLYP, show a good performance in the evaluation of relative corrections, except for nonorganic molecules containing H atoms. The PW86 functional stands out with the best evaluation of the barrier of linearity of H2O and the isotopic dipole moment of HDO. In general, DFT functionals display an accuracy superior than the common belief and because the corrections are based on a change of the electronic kinetic energy they are here ranked in a new appropriate way. The approach is applied to obtain the adiabatic correction for full atomization of alcanes C(n)H(2n+2), n = 4-10. The barrier of 1 mHartree is approached for adiabatic corrections, justifying its insertion into DFT.     

Modeling continuous changes in substituent electronegativity and chemical hardness using fictitious nuclear potentials

The performances of a family of recently developed generalized gradient approximation (GGA) functionals based on the Tognetti–Cortona–Adamo (TCA) family and making use of the gradient-regulated connection (GRAC) approach are here tested on an uncommon benchmark set for the prediction of transition state (TS) structures and energies of a series of four reactions involving an early transition metal (Zr, d ⁰). This benchmark test thus represents the first step in the organometallic world in which d ⁿ ions allowing complex phenomena such as spin crossover represent the higher level of complexity. The results obtained show that the performances of the GRAC-xxx functionals are comparable to those of global hybrid functionals both in the prediction of reaction barriers and of structural features of TSs. More complex functional forms (such as range-separated hybrids) in average enhance the energetic features, but not necessarily the overall accuracy on calculated structures. On the other hand, and as expected, purposely developed functionals for the prediction of chemical reactivity provide both structural and energetic features in good agreement with post-HF results. The present study, besides proving the good performances of GGA functionals of the GRAC-TCA family for the prediction of TS structural parameters and energetics of metal containing systems, also underlines the importance of the use of diversified benchmark sets to allow a fair evaluation of functionals performances.     

Electronic structure and excitations in oligoacenes from ab initio calculations

Oligoacenes C(4n+2)H(2n+4) (n=2,...,6) are studied using a variety of ab initio methods. Density functional theory (DFT) optimized geometries were in good agreement with experiment. Vertical and adiabatic ionization potentials and electron affinities were computed with DFT and it was found that standard exchange-correlation (xc) functionals underestimate ionization potentials in oligoacenes. Possible reasons for this underestimation are discussed. Low lying electronic excitations were computed using time-dependent density functional theory, configuration interaction singles, and configuration interaction singles with approximate treatment of doubles. In agreement with earlier work, time-dependent DFT in conjunction with standard xc-energy functionals substantially underestimates the lowest (p) singlet-singlet electronic transition.     

Scaled density functional theory correlation functionals

We show that a simple one-parameter scaling of the dynamical correlation energy estimated by the density functional theory (DFT) correlation functionals helps increase the overall accuracy for several local and nonlocal functionals. The approach taken here has been described as the "scaled dynamical correlation" (SDC) method [Ramachandran, J. Phys. Chem. A 2006, 110, 396], and its justification is the same as that of the scaled external correlation (SEC) method of Brown and Truhlar. We examine five local and five nonlocal (hybrid) DFT functionals, the latter group including three functionals developed specifically for kinetics by the Truhlar group. The optimum scale factors are obtained by use of a set of 98 data values consisting of molecules, ions, and transition states. The optimum scale factors, found with a linear regression relationship, are found to differ from unity with a high degree of correlation in nearly every case, indicating that the deviation of calculated results from the experimental values are systematic and proportional to the dynamic correlation energy. As a consequence, the SDC scaling of dynamical correlation decreases the mean errors (signed and unsigned) by significant amounts in an overwhelming majority of cases. These results indicate that there are gains to be realized from further parametrization of several popular exchange-correlation functionals.     

Modelling charge transfer reactions with the frozen density embedding formalism

The frozen density embedding (FDE) subsystem formulation of density-functional theory is a useful tool for studying charge transfer reactions. In this work charge-localized, diabatic states are generated directly with FDE and used to calculate electronic couplings of hole transfer reactions in two π-stacked nucleobase dimers of B-DNA: 5'-GG-3' and 5'-GT-3'. The calculations rely on two assumptions: the two-state model, and a small differential overlap between donor and acceptor subsystem densities. The resulting electronic couplings agree well with benchmark values for those exchange-correlation functionals that contain a high percentage of exact exchange. Instead, when semilocal GGA functionals are used the electronic couplings are grossly overestimated.