An Accurate Description of the Bergman Reaction Using Restricted and Unrestricted DFT:  Stability Test, Spin Density, and On-Top Pair Density

Abstract

DFT calculations provide a reliable description of the Bergman reaction of (Z)-hex-3-ene-1,5-diyne 1 provided the following are considered. (a) Restricted DFT (RDFT) calculations along the reaction path have to be replaced by unrestricted DFT (UDFT) calculations at those locations where the former description becomes unstable. This is the case in the region of the p-didehydrobenzene biradical 2, which possesses significant multireference character. (b) LSD and pure GGA functionals are more stable than hybrid functionals, which can be directly related to the composition of these functionals. With increasing instability, RDFT calculations lead to increasing errors in the S−T splitting and the geometry of 2 as well as in the energetics of the Bergman reaction. (c) LSD and GGA functionals underestimate the energy barrier of the Bergman reaction of 1. This becomes obvious when the correct experimental barrier is considered, which was not done in previous DFT investigations. (d) The best description of the Bergman reaction is provided by a mixed RDFT/UDFT description using the B3LYP functional (average error of 2.7 kcal/mol). Although the B3LYP functional is rather unstable, its semiempirical calibration helps to compensate for the typical underestimation of barriers by GGA functionals, which demonstrates that the performance of a hybrid functional does not necessarily have to do with its stability. (e) Application of the sum formula to the UB3LYP energy of biradical 2 improves the description of the Bergman reaction so that the most reliable data are obtained at RB3LYP-UB3LYP(sum)/6-311+G(3df,3pd). Activation enthalpies at 470 K for forward and backward reaction are 29.9 and 21.4 kcal/mol, respectively (exptl values, 28.23 ± 0.5 and 19.75 ± 0.7 kcal/mol), while the calculated reaction enthalpy at 298 K is 8.5 kcal/mol (exptl value, 8.5 ± 1.0 kcal/mol) in reasonable agreement with experiment. The calculated S−T splitting is 2.6 kcal/mol (after correction, 4.9 kcal/mol; exptl value, 3.8 ± 0.5 kcal/mol at 298 K). It is shown that the UDFT description covers static correlation effects needed for the correct treatment of 2S. Total and on-top pair density reflect this, while Kohn−Sham orbitals and spin density have to be considered as physically not meaningful intermediates in line with the interpretation given by Perdew, Savin, and Burke (Phys. Rev. A 1995, 51, 4531).

Detailed Kinetic Study of the Ring Opening of Cycloalkanes by CBS-QB3 Calculations

“…It is worth noting that the use of one determinantal wavefunction to describe open shell singlet biradicals can be questionable. However, previous studies have shown that the geometries obtained in this way compare well with those obtained at more refined computational levels [34][35][36][37][38]. Second, in CBS-QB3 calculations, a correction for spin-contamination in open-shell species is added to the total energy.…”

Can Unrestricted Density-Functional Theory Describe Open Shell Singlet Biradicals?

“…The electrons in localized orbitals ϕ r and ϕ s (Figure 2a) of the BS-UDFT wave function (6a) avoid each other in a similar way as in the type II reference state (12). As was discussed in detail by Gräfenstein and co-workers [11], the static correlation energy is in this way expressed as a part of the exchange energy. Hence, UDFT can lead to a reasonable description of type II biradicals provided a number of points are considered.…”

“…Type II systems possess long-range non-dynamic electron correlations, which contradicts assumption (2) of standard Kohn-Sham DFT. An RDFT treatment of these systems, although formally a correct approach, will in practice seriously underestimate electron correlation and yield too high an energy for the singlet state [11,30] as is indicated in Figure 1. One approach to treat type II systems with DFT is to start from a two-configuration reference function that covers the static correlation effects and to use DFT for the description of dynamic correlation effects only as is done in the GVB-DFT approach [53,56], or the two-configuration (TC)-DFT methods suggested by Borowski et al Filatov and Shaik [58] developed the restricted-ensemble KS (REKS) method, which represents the total energy of a multi-configuration system as a weighted average over an ensemble of single-determinant states.…”

Density Functional Theory in Transition-Metal Chemistry: Relative Energies of Low-Lying States of Iron Compounds and the Effect of Spatial Symmetry Breaking

“…23 A large literature has developed on this and related broken-symmetry approaches. [24][25][26][27][28][29][30][31][32][33] Broken-symmetry solutions have also been studied in other contexts, and it has often been found that it is important to allow the orbitals to break symmetry in order to get a qualitatively correct description of the system. [33][34][35][36][37][38][39][40][41][42] The energies of the broken-symmetry states are often used without correcting for their spin character.…”