Density Functionals

The thermodynamic properties of molecules are of fundamental interest in chemistry and engineering. This chapter deals with developments made in the last few years in the search for accurate density functional theory-based quantum chemical electronic structure methods for this purpose. The typical target accuracy for reaction energies of larger systems in the condensed phase is realistically about 2 kcal/mol. This level is within reach of modern density functional approximations when combined with appropriate continuum solvation models and slightly modified thermostatistical corrections. Nine higher-level functionals of dispersion corrected hybrid, range-separated hybrid, and double-hybrid type were first tested on four common, mostly small molecule, thermochemical benchmark sets. These results are complemented by four large molecule reaction examples. In these systems with 70–200 atoms, long-range electron correlation is responsible for important parts of the interactions and dispersion-uncorrected functionals fail badly. When used together with properly polarized triple- or quadruple-zeta type AO basis sets, most of the investigated functionals provide accurate gas phase reaction energies close to the values estimated from experiment. The use of theoretical back-correction schemes for solvation and thermal effects, the impact of the self-interaction error for unsaturated systems, and the prospect of local coupled-cluster based reference energies as benchmarks are discussed.

Density functional theory with generalized gradient approximations (GGAs) for the exchange-correlation density functional is widely used to model adsorption and reaction of molecules on surfaces. In other areas of computational chemistry, GGAs have largely been replaced by more accurate meta-GGA and hybrid approximations. Meta-GGAs and hybrids can ameliorate GGAs’ systematic over-delocalization of electrons and systematic underestimate of reaction barriers. This chapter discusses extensions of meta-GGAs and screened hybrids to surface chemistry. It reviews evidence that GGAs’ systematic underestimate of gas-phase reaction barriers carries over to reactions on surfaces, and that meta-GGAs and screened hybrids can improve results. It closes with recent applications and new work towards more accurate functionals for surfaces. These promising results motivate further exploration of meta-GGAs and screened hybrids for surface chemistry.

van der Waals interactions are important in typical van der Waals-bound systems such as noble gas, hydrocarbon, and alkaline earth dimers. The summed-up van der Waals series of Perdew et al. 2012 works well and is asymptotically correct at large separation between two atoms. However, as with the Hamaker 1937 expression, it has a strong singularity at short non-zero separation, where the two atoms touch. In this work we remove that singularity (and most of the short-range contribution) by evaluating the summed-up series at an effective distance between the atom centers. Only one fitting parameter is introduced for this short-range cut-off. The parameter in our model is optimized for each system, and a system-averaged value is used to make the final binding energy curves. This method is applied to different noble gas dimers such as Ar–Ar, Kr–Kr, Ar–Kr, Ar–Xe, Kr–Xe, Xe–Xe, Ne–Ne, He–He, and also to the Be₂ dimer. When this correction is added to the binding energy curve from the semilocal density functional meta-GGA-MS2, we get a vdW-corrected binding energy curve. These curves are compared with the results of other vdW-corrected methods such as PBE-D2 and vdW-DF2, and found to be typically better. Binding energy curves are in reasonable agreement with those from experiment.

Density-functional theory (DFT) methods have achieved widespread popularity for thermochemical predictions, which has lead to extensive benchmarking of functionals. While the use of statistics to judge the quality of various density-functional approximations is valuable and even seems unavoidable, the present chapter suggests some pitfalls of statistical analyses. Several illustrative examples, focusing on analysis of thermochemistry and intermolecular interactions, are presented to show that conclusions can be heavily influenced by both the data-set size and the choice of the criterion used to assess an approximation’s quality. Even with reliable approximations, the risk of publishing inaccurate results naturally increases with the number of calculations reported.

The ring or random-phase approximation (RPA) method combined with Kohn–Sham reference states has become established as an alternative method to common ab initio wave function methods for the description of the electronic structure of molecules and solids. The reason for this lies in the fact that the RPA possesses, in contrast to, for example, configuration interaction or coupled-cluster methods, a favourable scaling behaviour of N ⁴ ⋅ log (N) with the system size and describes a number of thermodynamic and electronic properties with a higher accuracy than standard density-functional theory methods. Moreover, the RPA method is able to describe not only dynamic but also strong static electron correlation effects, in contrast to conventional single-reference methods. The latter also include large systems with a small or vanishing band gap. In this work, the performance of the RPA and some extensions to the RPA, including exchange correlations, are tested for the description of thermochemical properties.

We examine the integer discontinuity (or derivative discontinuity) of the exact energy functionals of Kohn–Sham density-functional theory for the spin-polarized case. The integer discontinuity and its cause, the piecewise linearity of the energy in the grand canonical ensemble, are required to improve the predictive power of density-functional approximations to the exchange-correlation energy. We show how any spin-polarized DFA can be adapted to display the proper integer discontinuity. The formalism we present here can be used to improve functionals further within spin density-functional theory and fragment-based formulations of DFT.

It is not always possible in Kohn–Sham density-functional theory for the non-interacting reference state to have integer-only occupancies. Cases of “strong” correlation, with very small HOMO-LUMO gaps, involve fractional occupancies. At the transition states of symmetric avoided-crossing reactions, for example, representation of the correct density requires a 50/50 mixing of degenerate HOMOs. In a recent paper (Becke, J Chem Phys 139:021104, 2013) the “B13” strong-correlation density functional of Becke (J Chem Phys 138:074109, 2013 and 138:161101, 2013) was shown to give excellent barrier heights in symmetric avoided-crossing reactions. However, the calculations were performed only at reactant and transition-state geometries, where the fractional HOMO-LUMO occupancies in the latter are 50/50 by symmetry. In the present chapter, we compute full reaction curves for avoided crossings in H₂ + H₂, ethylene (twisting around the double bond), and cyclobutadiene (double-bond automerization) by determining fractional occupancies variationally. We adopt a practical strategy for doing so which does not involve self-consistent B13 computations (not yet possible) and involves minimal cost. Single-bond dissociation curves for H₂ and LiH are also presented.