Gradient-corrected Functionals Research Articles

Spin Forbidden Chemical Reactions of Transition Metal Compounds. New Ideas and New Computational Challenges

AbstractFor Abstract see ChemInform Abstract in Full Text.

Oxygen adsorption on graphite and nanotubes

We study the binding of molecular oxygen to a graphene sheet and to a (8,0) single walled carbon nanotube, by means of spin-unrestricted density-functional calculations. We find that triplet oxygen retains its spin-polarized state when interacting with graphene or the nanotube. This leads to the formation of a weak bond with essentially no charge transfer between the molecule and the sheet or tube, as one would expect for a physisorptive bond. This result is independent on the approximation used for the exchange-correlation functional. The binding strength, however, depends strongly on the functional, reflecting the inability of current approximation functionals to deal correctly with dispersion forces. Gradient-corrected functionals yield very weak binding at distances around 4 Å, whereas local density functional results yield substantially stronger binding for both graphene and the nanotube at distances of less than 3 Å. The picture of oxygen physisorption is not substantially altered by the presence of topological defects such as 5–7 Stone–Wales pairs.

Spin forbidden chemical reactions of transition metal compounds. New ideas and new computational challenges

Many reactions of transition metal compounds involve a change in spin. These reactions may proceed faster, slower--or at the same rate as--otherwise equivalent processes in which spin is conserved. For example, ligand substitution in [CpMo(Cl)2(PR3)2] is faster than expected, whereas addition of dinitrogen to [Cp*Mo(Cl)(PMe3)2] is slow. Spin-forbidden oxidative addition of ethylene to [Cp*Ir(PMe3)] occurs competitively with ligand association. To explain these observations, we discuss the shape of the different potential energy surfaces (PESs) involved, and the energy of the minimum energy crossing points (MECPs) between them. This computational approach is of great help in understanding the mechanisms of spin-forbidden reactions, provided that accurate calculations can be used to predict the relevant PESs. Density functional theory, especially using gradient-corrected and hybrid functionals, performs reasonably well for the difficult problem of predicting the energy splitting between different spin states of transition metal complexes, although careful calibration is needed.

Failure of density-functional theory and time-dependent density-functional theory for large extended π systems

Density-functional theory (DFT) is widely used for studying large systems such as metals, semiconductors, and large molecules, with time-dependent density-functional theory becoming a very powerful tool for investigating molecular excited states. As part of a systematic study of both the intrinsic weaknesses of DFT and the weaknesses of present implementations, we consider its application to the one and two-dimensional conjugated π systems: polyacetylene fragments and oligoporphyrins, respectively. Very poor results are obtained for the calculated spectra, and polyacetylene is predicted by all functionals considered, including gradient-corrected functionals, to have a triplet ground state. The cause of this is linked to known problems of existing density functionals concerning nonlocality and asymptotic behavior which result in the highest-occupied molecular-orbital being too high in energy so that semiconductors and low-band-gap insulators are predicted to have metal-like properties. The failure of modern density functionals to predict qualitatively realistic molecular hyperpolarizabilities for extended systems is closely related.

Functionals of the square kinetic energy density

We construct exchange energy functionals that depend on the non-interacting square kinetic energy density τ2(r)=14∑k|Δφk(r)|2 rather than the electron density. Since τ2 is a non-local functional of the electron density, it may describe non-local effects beyond standard generalized gradient approximation models. These effects may be essential in dealing with such a non-local quantity as the exchange energy. The problem of v-representability of τ2 for a slightly perturbed electron gas and Coulomb systems is discussed. A gradient expansion technique that permits construction of gradient-corrected functionals of τ2 is developed, and the performance of the new exchange functionals is analyzed.

Electronic Structure and Bonding in Hexacoordinate Silyl–Palladium Complexes Support from the National Science Foundation (CHE-9876792) is gratefully acknowledged.

Trimerization of [Pd(R2PCH2CH2PR2){1,2-C6H4(SiH2)2}] (1; where R Me or Et) has been demonstrated to produce a trinuclear complex 2where two of the Pd atoms can be readily characterized as PdII centers but the nature of the third, central Pd atom is less clear (Scheme 1, by-products not shown).[1] While this Pd atom (Pd1) is drawn in Scheme 1 as bonding directly to two Si atoms and further interacting with two Si Si bonds, the interatomic distances from the X-ray crystal-structure data could also be interpreted to be consistent with an absence of Si Si bonds and instead six Pd Si bonds,[1] that is, the central metal would formally be PdVI. Both structures are without precedent in palladium coordination chemistry,[2, 3] although compounds of Pd[4±7] and Pt[8, 9] have been reported for inorganic compounds of the form [MFn] (M Pd, Pt; n 2,4,6), where the extreme electronegativity of fluorine is exploited for the generation of higher oxidation states.[10] To better understand the nature of the bonding in 2, we have carried out DFT calculations on 2 and relevant model compounds. With the exception of single-point calculations using the X-ray geometries of 2a and 2b, all the structures were fully optimized and verified as minima by analytic frequency calculations. The functional employed was of the hybrid variety[11] and combined exact Hartree ± Fock exchange with the gradient-corrected exchange and correlation functionals of Becke[12] and Lee, Yang, and Parr,[13] respecRu Ru

Improving “difficult” reaction barriers with self-interaction corrected density functional theory

We examined 11 difficult reactions with self-interaction corrected density (SIC) functional theory. The data set includes dissociation of radicals into symmetric fragments (H2+→H+H+, He2+→He+He+), radical hydrogen abstraction (H+H2→H2+H, H+HCl→H2+Cl, H+N2H2→N2H+H2, CH3+H2→CH4+H), proton transfer [HC(OH)CHC(O)H→HC(O)CHC(OH)H], SN2 halogen exchange (X−+CH3X→CH3X+X−, X=F,Cl,Br), and closed-shell unimolecular dissociation of tetrasine (C2N4H2→N2+2HCN). Calculated self-interaction energies cancel, almost identically, for the reaction energies (ΔER), so that SIC functionals do not lead to a systematic improvement in ΔER. Self-interaction correction increases for reaction transition structures, leading to higher calculated activation barriers (ΔE≠). The average absolute deviation in ΔE≠, from ab initio and experimental barriers, is reduced from 14 kcal/mol for Vosko–Wilk–Nusair (VWN) or 12 kcal/mol for revised Perdew–Burke–Ernzerhof (revPBE) functionals to 5.4 (SIC-VWN) or 3.4 (SIC-revPBE) kcal/mol. Reorganization of the electron density, due to removal of self-interaction, appears to be important. When SIC is included as a perturbation, using self-consistent densities of the parent functional, the average absolute deviations for the barriers increase to 7.5 (VWN+SIC) or 5.3 (revPBE+SIC) kcal/mol. Gradient-corrected functionals (revPBE, BP86) reduce the magnitude of the total self-interaction correction, by improving the description of the core orbitals. For the valence orbitals, both the magnitudes of the self-interaction corrections, and their change between reagents and transition structures, are similar for VWN local density approximation, and generalized gradient approximation functionals. Reducing the magnitude of the self-interaction energy for valence electrons thus appears to be a promising direction for the development of chemically accurate exchange-correlation functionals.

Four-component relativistic Kohn-Sham theory.

A four-component relativistic implementation of Kohn-Sham theory for molecular systems is presented. The implementation is based on a nonredundant exponential parametrization of the Kohn-Sham energy, well suited to studies of molecular static and dynamic properties as well as of total electronic energies. Calculations are presented of the bond lengths and the harmonic and anharmonic vibrational frequencies of Au(2), Hg(2+)(2), HgAu(+), HgPt, and AuH. All calculations are based on the full four-component Dirac-Coulomb Hamiltonian, employing nonrelativistic local, gradient-corrected, and hybrid density functionals. The relevance of the Coulomb and Breit operators for the construction of relativistic functionals is discussed; it is argued that, at the relativistic level of density-functional theory and in the absence of a vector potential, the neglect of current functionals follows from the neglect of the Breit operator.

Comparative study of relativistic density functional methods applied to actinide species AcO(2)(2+) and AcF(6) for Ac = U, Np.

A two-component relativistic density functional method based on the Douglas-Kroll-Hess transformation has been applied to the actinyls and hexafluorides of U and Np. All-electron scalar relativistic calculations as well as calculations including spin-orbit interaction have been compared to results obtained with a pseudopotential approach. In addition, several exchange-correlation potentials have been applied to examine their performance for the bond lengths and vibrational frequencies of the title compounds. The calculations confirm the well-known accuracy of the LDA approach for geometries and frequencies, which is corroborated for the hexafluorides where gas phase experimental data are available. Comparison with results of accurate wave function based methods provides further confirmation of this finding. Gradient-corrected functionals tend to overestimate bond lengths and underestimate frequencies also for actinide compounds. The results obtained with Stoll-Preuss (small core) effective core potentials agree very well with those of all-electron calculations, while calculations with Hay-Martin large core pseudopotentials are somewhat less accurate. For all molecules and properties considered, spin-orbit effects have been found negligible concomitant with the closed-shell electronic structure of the U(VI) compounds and the open-shell situation of the Np(VI) compounds with a single valence f electron.

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.

Time-dependent density functional theory employing optimized effective potentials

Exchange-only ab initio (parameter-free) time-dependent density functional calculations for the vertical excitation energies of atoms and polyatomic molecules are performed by employing optimized effective potentials (OEP’s) and their corresponding adiabatic exchange kernels for the first time. Accurate OEP’s are obtained by a novel linear-combination-of-atomic-orbital (LCAO) algorithm [R. Colle and R. K. Nesbet, J. Phys. B 34, 2475 (2001)] in which a potential is represented as a sum of a seed potential having the correct −1/r asymptotic behavior and a small and rapidly decaying correction, the latter being approximated accurately by a linear combination of Gaussian functions. The time-dependent OEP (TDOEP) methods with and without the Tamm–Dancoff approximation are implemented by using a trial-vector algorithm, which allows us to avoid the storage or manipulation of transformed two-electron integrals or the diagonalization of large matrices. No approximation is made to TDOEP, besides the adiabatic approximation to the exchange kernel, the LCAO expansion of the orbitals and potentials, and occasionally the Tamm–Dancoff approximation. The vertical excitation energies of the beryllium atom and the nitrogen and water molecules calculated by TDOEP are compared with those obtained from time-dependent density functional theory (TDDFT) employing conventional local or gradient-corrected functionals, configuration interaction singles (CIS), time-dependent Hartree–Fock (TDHF) theory, similarity-transformed equation-of-motion coupled-cluster with single and double substitutions, and experiments. TDOEP, which neglects electron correlation while treating the exchange contribution rigorously within the Kohn–Sham DFT framework, performs equally well as, or even appreciably better than, CIS or TDHF. The slightly better performance of TDOEP might be attributed to the local nature of the exchange potentials that allows the bare orbital energy differences to approximate excitation energies well. Nevertheless, TDDFT employing local or gradient-corrected functionals outperforms TDOEP for low-lying valence excited states, implying that the former somehow accounts for electron correlation effectively, whereas for high-lying and Rydberg excited states, the latter performs better than the former. By combining the desirable features of OEP and local or gradient-corrected exchange-correlation potentials, we arrive at a simple asymptotic correction scheme to the latter. TDDFT with the asymptotic correction yields uniformly accurate excitation energies for both valence and Rydberg excited states.

Solvation of formic acid and proton transfer in hydrated clusters

In this paper we report detailed theoretical studies of formic acid–water clusters using a Gaussian implementation of Kohn–Sham density functional theory (DFT). Some MP2 calculations were made when necessary to make comparison. The newly developed Laplacian-dependent (LAP) functionals are extensively used although calculations with other traditional gradient-corrected functionals were also made for comparison. To assess our techniques we studied first the formic acid dimer. Good results for structures, vibrational frequencies and proton transfer barrier heights were achieved by the LAP functionals in contrary to other DFT methods, which usually give extremely low barrier heights. We obtained optimized structures of the formic acid–water clusters with up to 4 waters with many possible minimum energy states. The vibrational frequencies, successive hydration energy as well as the corresponding enthalpy were calculated. The interaction energy between formic acid and water was found to be larger than that of water–water. Ring-type structures are among the lowest in energy. Transition states were located for formic acid–water with various solvation patterns to study the effect of hydration on the proton transfer barrier. The transition state structures are of two fundamental types, i.e., a formic acid anion bound to H3O+- and H5O2+-centered structures, respectively. The proton transfer barrier is reduced by proper solvation of the transition states, notably to full and proper solvation of the hydrated proton units.

A Density Functional Study of EPR Parameters for Vanadyl Complexes Containing Schiff Base Ligands

Deviations of the coordination arrangement of vanadyl complexes from a regular square pyramid are thought to influence, among other things, their biological function. Such structural distortions have been found to be reflected characteristically in EPR spectra (Cornman et al. Inorg. Chem. 1997, 366, 6401). In this work, density functional calculations of electronic g tensors and metal hyperfine coupling tensors have been carried out for a series of four of these vanadyl complexes with structures ranging from nearly trigonal bipyramidal (TBP-5) to nearly square pyramidal (SQP-5). The EPR spectroscopic parameters have been rationalized in terms of electronic and geometrical structures. Using all relevant perturbation operators together with local or gradient-corrected density functionals, Δg-tensor components are underestimated systematically by ca. 40%. Good agreement with experiment is obtained for hyperfine tensor components calculated with hybrid functionals (B3PW91 and BHPW91), which account better for the spin polarization of the core orbitals than GGA functionals such as BP86. The rhombicity of the hyperfine tensor is reproduced well at all levels of theory applied. It is mainly determined by the SOMO composition. The latter explains the increasing rhombicity of the A tensor with increasing distortion of the SQP-5 structures along the series of complexes studied. The orientational dependence of the principal tensor components on the local vanadium coordination is much more pronounced for the g tensor than for the A tensor. The principal axes of the g and A tensors are found to be rotated with respect to each other by as much as 41°.

On the interaction of Mo and Mo2 with NH3, C2H4, and C3H6

The reactivity of Mo and Mo2 with ammonia, ethene, and propene molecules has been investigated by using Density Functional Theory. Different gradient-corrected and hybrid exchange-correlation functionals have been employed. Coordination modes, binding energies, geometrical structures, vibrational frequencies have been computed and compared with the available experimental counterparts. The results obtained show that the molybdenum atom is able to react with C2H4 and C3H6, and binds weakly with NH3. The dimer Mo2 gives a stable complexes with ammonia, ethene, and propene. For the Mo2NH3 complex, all the employed levels of theory give binding energies in good agreement with the experimental value, while in the case of the MoC2H4 system, the use of model core potentials coupled with gradient-corrected exchange-correlation functionals overestimates the binding energies. For MoC3H6, Mo2C2H4, and Mo2C3H6 we predict a binding energy of 14–15, 20–24, and 18–20 kcal/mol, respectively. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 1557–1564, 2001

Can optimized effective potentials be determined uniquely?

Local (multiplicative) effective exchange potentials obtained from the linear-combination- of-atomic-orbital (LCAO) optimized effective potential (OEP) method are frequently unrealistic in that they tend to exhibit wrong asymptotic behavior (although formally they should have the correct asymptotic behavior) and also assume unphysical rapid oscillations around the nuclei. We give an algebraic proof that, with an infinity of orbitals, the kernel of the OEP integral equation has one and only one singularity associated with a constant and hence the OEP method determines a local exchange potential uniquely, provided that we impose some appropriate boundary condition upon the exchange potential. When the number of orbitals is finite, however, the OEP integral equation is ill-posed in that it has an infinite number of solutions. We circumvent this problem by projecting the equation and the exchange potential upon the function space accessible by the kernel and thereby making the exchange potential unique. The observed numerical problems are, therefore, primarily due to the slow convergence of the projected exchange potential with respect to the size of the expansion basis set for orbitals. Nonetheless, by making a judicious choice of the basis sets, we obtain accurate exchange potentials for atoms and molecules from an LCAO OEP procedure, which are significant improvements over local or gradient-corrected exchange functionals or the Slater potential. The Krieger–Li–Iafrate scheme offers better approximations to the OEP method.

Curing difficult cases in magnetic properties prediction with self-interaction corrected density functional theory

The Perdew–Zunger self-interaction correction (SIC) was implemented self-consistently within a molecular density functional theory (DFT) program, using the Krieger–Li–Iafrate approximation to the optimized effective potential, and the Vosko–Wilk–Nusair (VWN) functional. The computationally efficient implementation relies on the fitting of orbital densities for the evaluation of orbital Coulomb potentials, and allows for routine applications to large molecules. Due to the use of the effective potential approach, the evaluation of the energy derivatives can be handled by standard Kohn–Sham DFT techniques in a straightforward way. The SIC-VWN technique is applied to the calculation of nuclear magnetic resonance (NMR) parameters in representative small molecules, containing C, H, N, O, and F. Removal of self-interaction leads to a substantial improvement in the calculated isotropic chemical shifts for N, O, and F, where SIC-VWN holds an advantage over both local (VWN), and gradient-corrected functionals (Becke 88-Perdew 86, BP86). For C and H isotropic chemical shifts, which are well described by the gradient-corrected functionals, SIC-VWN performs as well as BP86. SIC-VWN also improves the description of the absolute chemical shielding, and of the principal components of the NMR shielding tensors. The changes arise mainly from adjustments in the Kohn–Sham orbital energies, leading to a better description of the paramagnetic contribution to the shielding tensor. For spin–spin coupling constants, SIC-VWN improves the description of the paramagnetic contribution. At the same time, the magnitude of the Fermi contact term is underestimated, yielding mixed overall results. Slow convergence of the spin-spin coupling results with the basis set size prevents a conclusive statistical evaluation for this property. The clear physical origin of the SIC-VWN effect in the prediction of magnetic properties opens the tantalizing possibility that this technique may be effective in solving problems often encountered in the calculations of NMR parameters of heavier nuclei.

A DFT Study of Methanol Adsorption in 8T Rings of Chabazite

Hybrid B3LYP and gradient-corrected PW91 functionals were used for studying methanol adsorption on a zeolite cluster consisting of an 8T ring of chabazite. The comparison of the results obtained with PW91 with periodic calculations has shown that the adopted ring is an adequate approximation for the Brønsted sides in chabazite. Both physisorbed and chemisorbed methanol were found to be a minimum on the potential energy surface, with an energy difference up to 10 kJ/mol in favor of the hydrogen-bonded complex. It has been shown that compared to B3LYP, the PW91 functional overestimates the hydroxyl bond distance and underestimates the hydrogen bond distance. In the physisorbed mode, the methanol oxygen atom is strongly bonded to the zeolite proton, whereas the distance between the methanol proton and the framework oxygen atoms is 1.912−2.090 Å. We have calculated for hydrogen bonded methanol hydroxyl stretch frequencies in the intervals 3677−3582 and 2358−2187 cm-1 for the methanol and the zeolite OH bonds, respectively.

Theoretical electron densities in transition metal dihydrides

Analysis of the DFT calculated electron density distributions of the first-row transition metal (TM) dihydrides are reported. Nickel dihydride was used to assess the effects of basis set and exchange-correlation functional on calculated density properties, leading to the conclusions that diffuse basis functions and gradient-corrected functionals accurately describe the essential features of NiH2. These calculations indicate a remarkably high degree of covalent character in the Ni–H bond, as measured by most density properties, although the Laplacian of the density and the electron localization function (ELF) apparently show ‘closed-shell’ interaction. Subsequent B3LYP/6-311++G(f,p) studies of the ground states of the 10 first-row TM dihydrides were performed, and similarities and trends across this series examined. The M–H bond becomes less polar and more covalent across the row, as expected on the basis of electronegativity. We go on to show that several simply calculated electron density properties, notably those at the M–H bond critical point, also recover this trend.

Oxygen vacancies at oxide surfaces: ab initio density functional theory studies on vanadium pentoxide

The local electronic structure at the V2O5 (010) surface is studied by ab initio density functional theory (DFT) methods using gradient-corrected functionals (RPBE) where embedded clusters as large as V20O62H24, representing one or two crystal layers of the substrate, are used as models. Results of local binding and charging of differently coordinated surface-oxygen sites as well as densities of states allow a characterization of the detailed electronic structure of the surface. Electronic and geometric details of surface-oxygen vacancies as well as hydrogen adsorption are studied by appropriate clusters. A comparison of the data, concerning vacancy energies, charging, geometric relaxation, and diffusion, shows sizeable variations between different oxygen sites and can give further insight into possible mechanisms of surface relaxation and reconstruction. Hydrogen is found to stabilize at all surface-oxygen sites forming surface-OH and H2O species. As a result, the binding of surface oxygen with its vanadium neighbors is weakened. Therefore, the presence of hydrogen at the oxide surface facilitates oxygen removal and can contribute to the enhanced yield of oxygenated products near vanadia-based surfaces.

Charge Density Analysis of Triplet and Broken Symmetry States Relevant to Magnetic Coupling in Systems with Localized Spin Moments

Comparison of electron charge density for triplet and broken symmetry solutions obtained from different computational methods together with the atoms-in-molecules (AIM) and electron localization function (ELF) topological analyses support the description of these systems through the Heisenberg Hamiltonian. The reduction of the low-energy spectrum to a purely spin Hamiltonian holds for all studied methods, although local density approximation (LDA) exhibits some noticeable deviation. The analysis of charge difference density plots clearly shows that the failure of LDA to describe magnetic coupling is due to the too-strong delocalization that leads to a qualitatively incorrect electron density in the region near the nuclei. Gradient-corrected and hybrid functionals correct this defect but in an exaggerated way. The role of mixing Fock exchange is also discussed.