A new density functional (DF) of the generalized gradient approximation (GGA) type for general chemistry applications termed B97-D is proposed. It is based on Becke's power-series ansatz from 1997 and is explicitly parameterized by including damped atom-pairwise dispersion corrections of the form C(6) x R(-6). A general computational scheme for the parameters used in this correction has been established and parameters for elements up to xenon and a scaling factor for the dispersion part for several common density functionals (BLYP, PBE, TPSS, B3LYP) are reported. The new functional is tested in comparison with other GGAs and the B3LYP hybrid functional on standard thermochemical benchmark sets, for 40 noncovalently bound complexes, including large stacked aromatic molecules and group II element clusters, and for the computation of molecular geometries. Further cross-validation tests were performed for organometallic reactions and other difficult problems for standard functionals. In summary, it is found that B97-D belongs to one of the most accurate general purpose GGAs, reaching, for example for the G97/2 set of heat of formations, a mean absolute deviation of only 3.8 kcal mol(-1). The performance for noncovalently bound systems including many pure van der Waals complexes is exceptionally good, reaching on the average CCSD(T) accuracy. The basic strategy in the development to restrict the density functional description to shorter electron correlation lengths scales and to describe situations with medium to large interatomic distances by damped C(6) x R(-6) terms seems to be very successful, as demonstrated for some notoriously difficult reactions. As an example, for the isomerization of larger branched to linear alkanes, B97-D is the only DF available that yields the right sign for the energy difference. From a practical point of view, the new functional seems to be quite robust and it is thus suggested as an efficient and accurate quantum chemical method for large systems where dispersion forces are of general importance.

Semiempirical GGA-type density functional constructed with a long-range dispersion correction.

We present the theoretical and technical foundations of the Amsterdam Density Functional (ADF) program with a survey of the characteristics of the code (numerical integration, density fitting for the Coulomb potential, and STO basis functions). Recent developments enhance the efficiency of ADF (e.g., parallelization, near order-N scaling, QM/MM) and its functionality (e.g., NMR chemical shifts, COSMO solvent effects, ZORA relativistic method, excitation energies, frequency-dependent (hyper)polarizabilities, atomic VDD charges). In the Applications section we discuss the physical model of the electronic structure and the chemical bond, i.e., the Kohn–Sham molecular orbital (MO) theory, and illustrate the power of the Kohn–Sham MO model in conjunction with the ADF-typical fragment approach to quantitatively understand and predict chemical phenomena. We review the “Activation-strain TS interaction” (ATS) model of chemical reactivity as a conceptual framework for understanding how activation barriers of various types of (competing) reaction mechanisms arise and how they may be controlled, for example, in organic chemistry or homogeneous catalysis. Finally, we include a brief discussion of exemplary applications in the field of biochemistry (structure and bonding of DNA) and of time-dependent density functional theory (TDDFT) to indicate how this development further reinforces the ADF tools for the analysis of chemical phenomena. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 931–967, 2001

Chemistry with ADF

The successful isolation and characterization of an N-heterocyclic carbene in 1991 opened up a new class of organic compounds for investigation. From these beginnings as academic curiosities, N-heterocyclic carbenes today rank among the most powerful tools in organic chemistry, with numerous applications in commercially important processes. Here we provide a concise overview of N-heterocyclic carbenes in modern chemistry, summarizing their general properties and uses and highlighting how these features are being exploited in a selection of pioneering recent studies.

An overview of N-heterocyclic carbenes

Combined quantum-mechanics/molecular-mechanics (QM/MM) approaches have become the method of choice for modeling reactions in biomolecular systems. Quantum-mechanical (QM) methods are required for describing chemical reactions and other electronic processes, such as charge transfer or electronic excitation. However, QM methods are restricted to systems of up to a few hundred atoms. However, the size and conformational complexity of biopolymers calls for methods capable of treating up to several 100,000 atoms and allowing for simulations over time scales of tens of nanoseconds. This is achieved by highly efficient, force-field-based molecular mechanics (MM) methods. Thus to model large biomolecules the logical approach is to combine the two techniques and to use a QM method for the chemically active region (e.g., substrates and co-factors in an enzymatic reaction) and an MM treatment for the surroundings (e.g., protein and solvent). The resulting schemes are commonly referred to as combined or hybrid QM/MM methods. They enable the modeling of reactive biomolecular systems at a reasonable computational effort while providing the necessary accuracy.

QM/MM Methods for Biomolecular Systems

Seven different types of Slater type basis sets for the elements H (Z = 1) up to E118 (Z = 118), ranging from a double zeta valence quality up to a quadruple zeta valence quality, are tested in their performance in neutral atomic and diatomic oxide calculations. The exponents of the Slater type functions are optimized for the use in (scalar relativistic) zeroth-order regular approximated (ZORA) equations. Atomic tests reveal that, on average, the absolute basis set error of 0.03 kcal/mol in the density functional calculation of the valence spinor energies of the neutral atoms with the largest all electron basis set of quadruple zeta quality is lower than the average absolute difference of 0.16 kcal/mol in these valence spinor energies if one compares the results of ZORA equation with those of the fully relativistic Dirac equation. This average absolute basis set error increases to about 1 kcal/mol for the all electron basis sets of triple zeta valence quality, and to approximately 4 kcal/mol for the all electron basis sets of double zeta quality. The molecular tests reveal that, on average, the calculated atomization energies of 118 neutral diatomic oxides MO, where the nuclear charge Z of M ranges from Z = 1-118, with the all electron basis sets of triple zeta quality with two polarization functions added are within 1-2 kcal/mol of the benchmark results with the much larger all electron basis sets, which are of quadruple zeta valence quality with four polarization functions added. The accuracy is reduced to about 4-5 kcal/mol if only one polarization function is used in the triple zeta basis sets, and further reduced to approximately 20 kcal/mol if the all electron basis sets of double zeta quality are used. The inclusion of g-type STOs to the large benchmark basis sets had an effect of less than 1 kcal/mol in the calculation of the atomization energies of the group 2 and group 14 diatomic oxides. The basis sets that are optimized for calculations using the frozen core approximation (frozen core basis sets) have a restricted basis set in the core region compared to the all electron basis sets. On average, the use of these frozen core basis sets give atomic basis set errors that are approximately twice as large as the corresponding all electron basis set errors and molecular atomization energies that are close to the corresponding all electron results. Only if spin-orbit coupling is included in the frozen core calculations larger errors are found, especially for the heavier elements, due to the additional approximation that is made that the basis functions are orthogonalized on scalar relativistic core orbitals.

Optimized Slater-type basis sets for the elements 1-118.

Analytical expressions are derived for the evaluation of energy gradients in the zeroth order regular approximation (ZORA) to the Dirac equation. The electrostatic shift approximation is used to avoid gauge dependence problems. Comparison is made to the quasirelativistic Pauli method, the limitations of which are highlighted. The structures and first metal-carbonyl bond dissociation energies for the transition metal complexes W(CO)6, Os(CO)5, and Pt(CO)4 are calculated, and basis set effects are investigated.

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Geometry optimizations in the zero order regular approximation for relativistic effects.

In a recent evaluation of density functional theory (DFT) functionals OPBE, which combines Handy's optimized exchange (OPTX) with the PBE correlation, was shown to correctly predict the spin states (singlet through sextet) of seven different iron complexes (2004, J. Phys. Chem. A, 108, 5479). The present study provides a further test of OPBE as well as that of the OPerdew and OLYP functionals, in which OPTX is combined with the Perdew and LYP correlations, respectively. These three are compared to other pure DFT functionals for their performance in calculating the atomization energies for the G2-set of up to 148 molecules, six reaction barriers of SN2 reactions, geometry optimizations of 19 small molecules and four metallocenes, and zero-point vibrational energies for 13 small molecules. OPBE performs exceptional well in all cases.

Performance of the OPBE exchange-correlation functional

Catch it! Geminal phosphorus/aluminum-based frustrated Lewis pairs (FLPs) are easily obtained by hydroalumination of alkynylphosphines. These FLPs can activate terminal acetylenes by two competitive pathways, which were analyzed by DFT calculations, and they can bind carbon dioxide reversibly. Therefore, alongside polyfluorinated boranes, alanes are also ideal Lewis acids for FLP chemistry. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Geminal phosphorus/aluminum-based frustrated Lewis pairs: C-H versus C≡C activation and CO2 fixation

The geometries of the hexacarbonyls and pentacarbonyls of chromium, molybdenum, and tungsten are optimized at the Hartree-Fock and ML2 levels of theory using effective core potentials for the metal atoms. The M-CO bond lengths of Mo(CO) 6 and W(CO) 6 predicted at the ML2 level using moderate valence basis sets are in excellent agreement with experimental values. The Cr-CO bond length in Cr(CO) 6 calculated at ML2 is too short. The total bond energies of the metal hexacarbonyls calculated at the CCSD(T) level of theory are slightly lower than the experimentally derived values

Structures and Bond Energies of the Transition Metal Hexacarbonyls M(CO)6 (M = Cr, Mo, W). A Theoretical Study

Geminal frustrated Lewis pairs (FLPs) are expected to exhibit increased reactivity when the donor and acceptor sites are perfectly aligned This is shown for reactions of the nonfluorinated FLP tBu2PCH2BPh2 with H2, CO2, and isocyanates and supported computationally

Andreas W. Ehlers

Papers

Geometry optimizations in the zero order regular approximation for relativistic effects.

Performance of the OPBE exchange-correlation functional

Geminal phosphorus/aluminum-based frustrated Lewis pairs: C-H versus C≡C activation and CO2 fixation

Structures and Bond Energies of the Transition Metal Hexacarbonyls M(CO)6 (M = Cr, Mo, W). A Theoretical Study

Preorganized frustrated Lewis pairs.