Why the generalized gradient approximation works and how to go beyond it

The local spin density (LSD) approximation, while of only moderate accuracy, has proven extremely reliable over three decades of use. We argue that any gradient-corrected functional should preserve the correct features of LSD even if the system under study contains no regions of small density gradient. The Perdew-Wang 1991 (PW91) functional respects this condition, while, e.g., the Lee-Yang-Parr (LYP) correlation functional violates it. We extend this idea to the next generation of density functionals, those which incorporate exact exchange via the optimized effective potential (OEP), with a model in which the correlation hole is constructed from the exact exchange hole. The resulting exchange-correlation hole is deeper and less diffuse than the exact exchange hole. We denote such a functional as “locally correlated Hartree-Fock” and list a variety of conditions such a functional should satisfy. We demonstrate the promise of this approach with a crude simple model. © 1997 John Wiley & Sons, Inc.     

Diminished gradient dependence of density functionals: constraint satisfaction and self-interaction correction

The Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation for the exchange-correlation energy functional has two nonempirical constructions, based on satisfaction of universal exact constraints on the hole density or on the energy. We show here that, by identifying one possible free parameter in exchange and a second in correlation, we can continue to satisfy these constraints while diminishing the gradient dependence almost to zero (i.e., almost recovering the local spin density approximation or LSDA). This points out the important role played by the Perdew-Wang 1991 nonempirical hole construction in shaping PBE and later constructions. Only the undiminished PBE is good for atoms and molecules, for reasons we present, but a somewhat diminished PBE could be useful for solids; in particular, the surface energies of solids could be improved. Even for atoms and molecules, a strongly diminished PBE works well when combined with a scaled-down self-interaction correction (although perhaps not significantly better than LSDA). This shows that the undiminished gradient dependence of PBE and related functionals works somewhat like a scaled-down self-interaction correction to LSDA.     

B2-PPW91: a promising double-hybrid density functional for the electric response properties

A new double-hybrid density functional, termed B2-PPW91, is presented which includes the Becke88 (B88) exchange in conjunction with Perdew-Wang91 (PW91) gradient-corrected correlation functional. The fitting parameters are obtained by minimization of mean absolute error of the static dipole polarizability of 4d transition metal monohalides against the CCSD(T)∕aug-cc-pVTZ∕SDD results. The performance of proposed functional has been assessed for estimation of other response properties, such as dipole moment and excitation energy, for the same species. We then proceed to explore the validity of B2-PPW91 method for calculation of the dipole polarizability of some 5d transition metal monofluorides. In all cases, the improvement compared to common density functional methods and even previously reported double-hybrid functionals such as B2-PLYP and mPW2-PLYP has been observed. This indicates that the utility of double-hybrid density functional methods can be further extended to study linear and non-linear optical properties of transition metal containing molecules.     

Calculation of electronic g-tensors for transition metal complexes using hybrid density functionals and atomic meanfield spin-orbit operators

We report the first implementation of the calculation of electronic g-tensors by density functional methods with hybrid functionals. Spin-orbit coupling is treated by the atomic meanfield approximation. g-Tensors for a set of small main group radicals and for a series of ten 3d and two 4d transition metal complexes have been compared using the local density approximation (VWN functional), the generalized gradient approximation (BP86 functional), as well as B3-type (B3PW91) and BH-type (BHPW91) hybrid functionals. For main group radicals, the effect of exact-exchange mixing is small. In contrast, significant differences between the various functionals arise for transition metal complexes. As has been shown previously, local and in particular gradient-corrected functionals tend to underestimate the "paramagnetic" contributions to the g-tensors in these cases and thereby recover only about 40-50% of the range of experimental g-tensor components. This is improved to ca. 60% by the B3PW91 functional, which also gives slightly reduced standard deviations. The range increases to almost 100% using the half-and-half functional BHPW91. However, the quality of the correlation with experimental data worsens due to a significant overestimate of some intermediate g-tensor values. The worse performance of the BHPW91 functional in these cases is accompanied by spin contamination. Although none of the functionals tested thus appears to be ideal for the treatment of electronic g-tensors in transition metal complexes, the B3PW91 hybrid functional exhibited the overall most satisfactory performance. Apart from the validation of hybrid functionals, some aspects in the treatment of spin-orbit contributions to the g-tensor are discussed.     

Performance of density functionals for calculation of reductive ring-opening reaction energies of Li<Superscript>+</Superscript>-EC and Li<Superscript>+</Superscript>-VC

Reaction energies were determined for reductive ring-opening reactions of Li⁺-coordinated ethylene carbonate (EC) and vinylene carbonate (VC) by using various density functional theory (DFT) and ab-initio methods applying the basis sets up to Dunnings aug-cc-pVQZ. The methods examined include the local density functional (SVWN), the pure gradient-corrected density functionals (BLYP and BPW91), and the hybrid density functionals (B3LYP, B1LYP, B3PW91, and mPW1PW91). Comparison of the DFT results with ab-initio results indicates that the mPW1PW91 approach introduced by Adamo and Barone, is superior to all the other DFT methods (including B3LYP). The performance of more cost-effective Pople-type basis sets ranging from 6-31G(d,p) to 6-311++G(3df,3pd) was assessed at DFT levels of theory by calibrating them with the aug-cc-pVQZ results     

Prescription for the design and selection of density functional approximations: more constraint satisfaction with fewer fits

We present the case for the nonempirical construction of density functional approximations for the exchange-correlation energy by the traditional method of "constraint satisfaction" without fitting to data sets, and present evidence that this approach has been successful on the first three rungs of "Jacob's ladder" of density functional approximations [local spin-density approximation (LSD), generalized gradient approximation (GGA), and meta-GGA]. We expect that this approach will also prove successful on the fourth and fifth rungs (hyper-GGA or hybrid and generalized random-phase approximation). In particular, we argue for the theoretical and practical importance of recovering the correct uniform density limit, which many semiempirical functionals fail to do. Among the beyond-LSD functionals now available to users, we recommend the nonempirical Perdew-Burke-Ernzerhof (PBE) GGA and the nonempirical Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA, and their one-parameter hybrids with exact exchange. TPSS improvement over PBE is dramatic for atomization energies of molecules and surface energies of solids, and small or moderate for other properties. TPSS is now or soon will be available in standard codes such as GAUSSIAN, TURBOMOLE, NWCHEM, ADF, WIEN, VASP, etc. We also discuss old and new ideas to eliminate the self-interaction error that plagues the functionals on the first three rungs of the ladder, bring up other related issues, and close with a list of "do's and don't's" for software developers and users.     

Ground-state properties and static dipole polarizabilities of the alkali dimers from K2 n to Fr2 n(n=0,+1) from scalar relativistic pseudopotential coupled cluster and density functional studies

The newly adjusted energy-consistent nine-valence-electron pseudopotentials for K to Fr are used to calculate spectroscopic properties for the neutral and positively charged alkali dimers using coupled cluster and density functional theory. For the neutral dimers the static dipole polarizability was calculated. The coupled cluster results are all in excellent agreement with experimental values. The density functionals used can give quite different spectroscopic properties especially for the dipole polarizability, with the Perdew-Wang PW91 functional performing best.     

Does the gradient-regulated connection improve the description of correlated metal bond properties?

Gradient-regulated connection (GRAC) is a generalized gradient approximation exchange density functional designed by combining the revPBE and PW91 exchange functionals to impose their behaviors in the slowly- and fast-varying density regions, respectively. Such a construction allows one single density functional to accurately estimate both covalent and weak interactions occurring in main-group-based molecular systems. For the first time, the assessment of the performance of the GRAC exchange functional is extended to the modeling of various metal bond energy and structure properties. This assessment shows that when GRAC is coupled with the Perdew, Burke, Ernzerhof (PBE) correlation, the resulting exchange-correlation density functional is an excellent alternative to global hybrids to model bond dissociation energy, atomic electronic excitation energy, and bond length structure properties of single-reference metal bonds. It also shows that coupling with the Tognetti, Cortona, Adamo (TCA) correlation constitutes a robust approach to tackle energy bond properties of organometallic complexes with multi-reference character.     

Interaction of Dihydrogen with Transition Metal (Pd,Ni, Ag,Cu) Clusters

Quantum calculations of interaction of the molecular hydrogen with transition-metal clusters have been performed. The aim of the project is to compare the results for different metals and different methods of calculations. The calculations have been mostly based on the gradient-corrected methods of the density functional theory. The list of the exchange-correlation functionals includes: the gradient corrected functional BP86, the hybrid functionals B3P86, B3LYP, B3PW9, and the local SVWN functional. The calculations of the potential energy surface (PES) for the hydrogen molecule positioned over the planar Pd₅ clusters have been performed. It was found that the H–H bond activation is without barrier for most of the functionals used. However, the results obtained for the B3LYP functional suggest very small potential barrier, of the order of 0.003 eV. The calculations of the PES for dihydrogen in contact with metal clusters have been performed for Ni₅, Ag₅, Cu₅ clusters and for mixed clusters Ag₄Pd, AgPd₄, NiCu₄, and NiPd₄. The dissociation paths for all the cases with the exception of Ag₅ and Cu₅ have been found and the dissociation energies and activation barriers have been estimated.     

Performance assessment of density-functional methods for study of charge-transfer complexes

Various density functionals are applied to a number of weakly bound intermolecular pi-pi charge-transfer (CT) complexes. Most functionals, including the recently developed mPWPW91 and mPW1PW91, grossly underestimate experimental excitation energies; good agreement is obtained only with the half-and-half hybrid BH&HLYP functional. PW91PW91 provides the best agreement with intermolecular distances measured in crystal, while the BH&HLYP values are about 0.1 A too long. Various hybrid functionals with nonlocal exchange correction provide binding energies that compare favorably with the experimental heats of formation measured in solution.     

Electronic structures of 4d transition metal monoxides by density functional theory

Bond distances, dissociation energies, ionization potentials and electron affinities of 4d transition metal monoxides from YO to CdO and their positive and negative ions were studied by use of density functional methods B3LYP, BLYP, B3PW91, BPW91, B3P86, BP86, SVWN, MPW1PW91 and PBE1PBE. It was found that calculated properties are highly dependent on the functionals employed, especially for dissociation energy. For most neutral species, pure density functionals BLYP, BPW91 and BP86 have good performance in predicting dissociation energy than hybrid density functionals B3LYP, B3PW91 and B3P86. In addition, BLYP gives the largest bond distance compared with other density functional methods, while SVWN gives shortest bond distance, largest dissociation energy and electron affinity. For the ground state, the spin multiplicity of the charged species can be obtained by ± 1 of their corresponding neutral species.     

Calculation of molecular geometries,relative conformational energies,dipole moments,and molecular electrostatic potential fitted charges of small organic molecules of biochemical interest by density functional theory

Density functional theory is tested on a large ensemble of model compounds containing a wide variety of functional groups to understand better its ability to reproduce experimental molecular geometries, relative conformational energies, and dipole moments. We find that gradient-corrected density functional methods with triple-ζ plus polarization basis sets reproduce geometries well. Most bonds tend to be approximately 0.015 Å longer than the experimental results. Bond angles are very well reproduced and most often fall within a degree of experiment. Torsions are, on average, within 4 degrees of the experimental values. For relative conformational energies, comparisons with Hartree-Fock calculations and correlated conventional ab initio methods indicate that gradient-corrected density functionals easily surpass the Hartree-Fock approximation and give results which are nearly as accurate as MP2 calculations. For the 35 comparisons of conformational energies for which experimental data was available, the root mean square (rms) deviation for gradient-corrected functionals was approximately 0.5 kcal mol^?1. Without gradient corrections, the rms deviation is 0.8 kcal mol^?1, which is even less accurate than the Hartree-Fock calculations. Calculations with extended basis sets and with gradient corrections incorporated into the self-consistent procedure generate dipole moments with an rms deviation of 5%. Dipole moments from local density functional calculations, with more modest basis sets, can be scaled down to achieve roughly the same accuracy. In this study, all density functional geometries were generated by local density functional self-consistent calculations with gradient corrections added in a perturbative fashion. Such an approach generates results that are almost identical to the self-consistent gradient-corrected calculations, which require significantly more computer time. Timings on scalar and vector architectures indicate that, for moderately sized systems, our density functional implementation requires only slightly less computer resources than established Hartree-Fock programs. However, our density functional calculations scale much better and are significantly faster than their MP2 counterparts, whose results they approach. © 1995 John Wiley & Sons, Inc.     

Density functional theory study of CO2 capture with transition metal oxides and hydroxides

We have used density functional theory (DFT) employing several different exchange-correlation functionals (PW91, PBE, PBEsol, TPSS, and revTPSS) coupled with lattice dynamics calculations to compute the thermodynamics of CO(2) absorption/desorption reactions for selected transition metal oxides, (TMO), and hydroxides, TM(OH)(2), where TM = Mn, Ni, Zn, and Cd. The van't Hoff plots, which describe the reaction equilibrium as a function of the partial pressures of CO(2) and H(2)O as well as temperature, were computed from DFT total energies, complemented by the free energy contribution of solids and gases from lattice dynamics and statistical mechanics, respectively. We find that the PBEsol functional calculations are generally in better agreement with experimental phase equilibrium data compared with the other functionals we tested. In contrast, the formation enthalpies of the compounds are better computed with the TPSS and revTPSS functionals. The PBEsol functional gives better equilibrium properties due to a partial cancellation of errors in the enthalpies of formation. We have identified all CO(2) capture reactions that lie on the Gibbs free energy convex hull as a function of temperature and the partial pressures of CO(2) and H(2)O for all TMO and TM(OH)(2) systems studied here.     

DFT studies using supercells and projector-augmented waves for structure, energetics, and dynamics of glycine, alanine, and cysteine

A large variety of gas phase conformations of the amino acids glycine, alanine, and cysteine is studied by numerically efficient semi-local gradient-corrected density functional theory calculations using a projector-augmented wave scheme and periodic boundary conditions. Equilibrium geometries, conformational energies, dipole moments, vibrational modes, and IR optical spectra are calculated from first principles. A comparison of our results with values obtained from quantum-chemistry methods with localized basis sets and nonlocal exchange-correlation functionals as well as with experimental data is made. For conformations containing strong intramolecular hydrogen bonds deviations in their energetic ordering occur, which are traced back to different treatments of spatial nonlocality in the exchange-correlation functional. However, even for these structures, the comparison of calculated and measured vibrational frequencies shows satisfying agreement.     

Systematic studies on the computation of nuclear magnetic resonance shielding constants and chemical shifts: the density functional models

We present a systematic density functional investigation on the prediction of the 13C, 15N, 17O, and 19F NMR properties of 23 molecules with 21 density functionals. Extensive comparisons are made for both 13C magnetic shieldings and chemical shifts with respect to the gas phase experimental data and the best CCSD(T) results. We find that the OPBE and OPW91 exchange-correlation functionals perform significantly better than some popular functionals such as B3LYP and PBE1PBE, even surpassing, in many cases, the standard wavefunction-based method MP2. Further analysis has been performed to explore the individual role played by various exchange and correlation functionals. We find that the B88 and PBE exchange functionals have a too strong tendency of deshielding, leading to too deshielded magnetic shielding constants; whereas the OPTX exchange functional performs remarkably well. We claim that the main source of error arises from the exchange functional, but correlation functional also makes important contribution. We find that the correlation functionals may be grouped into two classes. class A, such as LYP and B98, leads to deshielded NMR values, deteriorating the overall performance; whereas class B, such as PW91 and PBE, generally increases the absolute shieldings, which complements the exchange functionals, leading to improved results in the calculation of NMR data.     

Generalized gradient approximation adjusted to transition metals properties: Key roles of exchange and local spin density

Perdew-Burke-Ernzerhof (PBE) and PBE adapted for solids (PBEsol) are exchange-correlation (xc) functionals widely used in density functional theory simulations. Their differences are the exchange, μ, and correlation, β, coefficients, causing PBEsol to lose the Local Spin Density (LSD) response. Here, the μ/β two-dimensional (2D) accuracy landscape is analyzed between PBE and PBEsol xc functional limits for 27 transition metal (TM) bulks, as well as for 81 TM surfaces. Several properties are analyzed, including the shortest interatomic distances, cohesive energies, and bulk moduli for TM bulks, and surface relaxation degree, surface energies, and work functions for TM surfaces. The exploration, comparing the accuracy degree with respect experimental values, reveals that the found xc minimum, called VV, being a PBE variant, represents an improvement of 5% in mean absolute percentage error terms, whereas this improvement reaches ~11% for VVsol, a xc resulting from the restoration of LSD response in PBEsol, and so regarded as its variant.     

Benchmark calculations on the adiabatic ionization potentials of M-NH(3) (M=Na,Al,Ga,In,Cu,Ag)

The ground states of the M-NH(3) (M=Na,Al,Ga,In,Cu,Ag) complexes and their cations have been studied with density functional theory and coupled cluster [CCSD(T)] methods. The adiabatic ionization potentials (AIPs) of these complexes are calculated, and these are compared to results from high-resolution zero-electron kinetic energy photoelectron spectroscopy. By extrapolating the CCSD(T) energies to the complete basis set (CBS) limit and including the core-valence, scalar relativistic, spin-orbit, and zero-point corrections, the CCSD(T) method is shown to be able to predict the AIPs of these complexes to better than 6 meV or 0.15 kcal/mol. 27 exchange-correlation functionals, including one in the local density approximation, 13 in the generalized gradient approximation (GGA), and 13 with hybrid GGAs, were benchmarked in the calculations of the AIPs. The B1B95, mPW1PW91, B98, B97-1, PBE1PBE, O3LYP, TPSSh, and HCTH93 functionals give an average error of 0.1 eV for all the complexes studied, with the B98 functional alone yielding a maximum error of 0.1 eV. In addition, the calculated metal-ammonia harmonic stretching frequencies with the CCSD(T) method are in excellent agreement with their experimental values, whereas the B3LYP method tends to underestimate these stretching frequencies. The metal-ammonia binding energies were also calculated at the CCSD(T)/CBS level, and are in excellent agreement with the available experimental values considering the error limits, except for Ag-NH(3) and Ag(+)-NH(3), where the calculations predict stronger bond energies than measured by about 4 kcal/mol, just outside the experimental error bars of +/-3 kcal/mol.     

Construction of a generalized gradient approximation by restoring the density-gradient expansion and enforcing a tight Lieb-Oxford bound

Recently, a generalized gradient approximation (GGA) to the density functional, called PBEsol, was optimized (one parameter) against the jellium-surface exchange-correlation energies, and this, in conjunction with changing another parameter to restore the first-principles gradient expansion for exchange, was sufficient to yield accurate lattice constants of solids. Here, we construct a new GGA that has no empirical parameters, that satisfies one more exact constraint than PBEsol, and that performs 20% better for the lattice constants of 18 previously studied solids, although it does not improve on PBEsol for molecular atomization energies (a property that neither functional was designed for). The new GGA is exact through second order, and it is called the second-order generalized gradient approximation (SOGGA). The SOGGA functional also differs from other GGAs in that it enforces a tighter Lieb-Oxford bound. SOGGA and other functionals are compared to a diverse set of lattice constants, bond distances, and energetic quantities for solids and molecules (this includes the first test of the M06-L meta-GGA for solid-state properties). We find that classifying density functionals in terms of the magnitude mu of the second-order coefficient of the density gradient expansion of the exchange functional not only correlates their behavior for predicting lattice constants of solids versus their behavior for predicting small-molecule atomization energies, as pointed out by Perdew and co-workers [Phys. Rev. Lett. 100, 134606 (2008); Perdew ibid. 80, 891 (1998)], but also correlates their behavior for cohesive energies of solids, reaction barriers heights, and nonhydrogenic bond distances in small molecules.     

Spin densities in two-component relativistic density functional calculations: noncollinear versus collinear approach

With present day exchange-correlation functionals, accurate results in nonrelativistic open shell density functional calculations can only be obtained if one uses functionals that do not only depend on the electron density but also on the spin density. We consider the common case where such functionals are applied in relativistic density functional calculations. In scalar-relativistic calculations, the spin density can be defined conventionally, but if spin-orbit coupling is taken into account, spin is no longer a good quantum number and it is not clear what the "spin density" is. In many applications, a fixed quantization axis is used to define the spin density ("collinear approach"), but one can also use the length of the local spin magnetization vector without any reference to an external axis ("noncollinear approach"). These two possibilities are compared in this work both by formal analysis and numerical experiments. It is shown that the (nonrelativistic) exchange-correlation functional should be invariant with respect to rotations in spin space, and this only holds for the noncollinear approach. Total energies of open shell species are higher in the collinear approach because less exchange energy is assigned to a given Kohn-Sham reference function. More importantly, the collinear approach breaks rotational symmetry, that is, in molecular calculations one may find different energies for different orientations of the molecule. Data for the first ionization potentials of Tl, Pb, element 113, and element 114, and for the orientation dependence of the total energy of I+2 and PbF indicate that the error introduced by the collinear approximation is approximately 0.1 eV for valence ionization potentials, but can be much larger if highly ionized open shell states are considered. Rotational invariance is broken by the same amount. This clearly indicates that the collinear approach should not be used, as the full treatment is easily implemented and does not introduce much more computational effort.