Optimization and Basis-Set Dependence of a Restricted-Open-Shell Form of B2-PLYP Double-Hybrid Density Functional Theory David C. Graham, † Ambili S. Menon, † Lars Goerigk, †,§,‡ Stefan Grimme, ‡ and Leo Radom* ,† School of Chemistry and ARC Center of Excellence for Free Radical Chemistry and Biotechnology, UniVersity of Sydney, Sydney, NSW 2006, Australia, NRW Graduate School of Chemistry, Wilhelm-Klemm-Strasse 10, D-48149 Mu ¨nster, Germany, and Theoretische Organische Chemie, Organisch-Chemisches Institut der UniVersita ¨t Mu ¨nster, Correnstrasse 40, D-48149 Mu ¨nster, Germany ReceiVed: May 8, 2009 The performance of the restricted-open-shell form of the double-hybrid density functional theory (DHDFT) B2- PLYP procedure has been compared with that of its unrestricted counterpart using the G3/05 test set. Additionally, the influence of basis set on the parametrization and performance of ROB2-PLYP, and the further improvement of ROB2-PLYP through augmentation with a long-range dispersion function, have been investigated. We find that, after optimization of the two empirical DHDFT parameters, the ROB2-PLYP method (HF exchange ) 59% and MP2 correlation ) 28%) performs slightly better than the corresponding UB2-PLYP method (HF exchange ) 62% and MP2 correlation ) 35%), with mean absolute deviations (MADs) from the experimental energies in the G3/05 test set of 9.1 and 9.9 kJ mol -1 , respectively, when the cc-pVQZ basis set is employed. Separate optimizations of the parameters for the RO and U procedures are crucial for a fair comparison. For example, for the G2/97 test set, ROB2-PLYP(53,27) and ROB2-PLYP(62,35) show MADs of 12.2 and 13.5 kJ mol -1 , respectively, compared with the 6.6 kJ mol -1 for (the optimized) ROB2-PLYP(59,28). The performance of ROB2- PLYP deteriorates significantly as the basis-set size is decreased, reflecting the enhanced basis-set dependence of the MP2 contribution compared with standard DFT. We find that this deficiency can be partly overcome through reparametrization. However, when the basis set drops below triple-, the improvements made on reoptimizing the ROB2-PLYP parameters are not sufficient to warrant their general use. We find that the dispersion- and BSSE- corrected ROB2-PLYP(59,28)-D HCP procedure performs significantly better than ROB2-PLYP(59,28) for the S22 test set of interaction energies in which dispersion interactions are particularly important, with the MAD falling from 6.1 to 1.6 kJ mol -1 . However, when the same D correction is applied to the G3/05 test set, the performance of ROB2-PLYP(59,28)-D deteriorates slightly compared with ROB2-PLYP(59,28), with the MAD increasing from 9.1 to 9.5 kJ mol -1 . 1. Introduction The development of highly accurate quantum chemical methods that can reliably predict thermochemical data when experimental data are unavailable or uncertain is highly desirable. High-level composite procedures, such as the Gn(X) methods of Curtiss, Raghavachari, and Pople et al., 1 the complete-basis-set (CBS) methods of Petersson et al., 2 and the Wn methods of Martin et al., 3 have made the task of achieving chemical accuracysfor small systems at leastsvery feasible. However, as a consequence of their cost, calculations on systems larger than those containing a few non-hydrogen atoms are typically out of reach with these methods. 4 Thus it may be necessary to sacrifice both accuracy and the capacity for systematic improvement to work on systems that are more chemically relevant. 5 Density functional theory (DFT) provides an alternative method for the calculation of accurate energies, at a cost comparable to that of Hartree-Fock calculations. 6 Although DFT methods do not offer a systematic way to improve the Hamiltonian, 5 Perdew et al. 7 have put forward a “Jacob’s Ladder” analogy for improving the critical exchange-correlation component of DFT functionals. In this strategy, each higher rung on the ladder is expected to offer an additional improve- ment in the accuracy over the basic (bottom rung) local density approximation (LDA) approach. Grimme has recently proposed a new family of methods, 4,8,9 termed “double-hybrid” density functional theory (DHDFT) procedures, that are consistent with a fifth rung approach in the “Jacob’s Ladder” of DFT methods. These procedures not only incorporate Hartree-Fock exchange admixture in the same way that hybrid DFT procedures such as the immensely popular B3- LYP 10 do but also combine contributions from the semilocal correlation density functional with a proportion of MP2 cor- relation derived using the Kohn-Sham (KS) reference orbitals (KS-PT2). 11 The DHDFT energy is obtained by first solving the KS equations self-consistently using a hybrid density functional containing a semilocal generalized gradient ap- proximation (GGA) for exchange (X) and correlation (C). Second, the MP2 energy (E2) is calculated in the space of the converged KS orbitals. The total exchange-correlation energy for the DHDFT procedure is then obtained by summation of the various parts of the functional: 4 where E X is the exchange energy and E C is the correlation energy. The Møller -Plesset perturbation correction term (E2) is given by † University of Sydney. ‡ Universita ¨t Mu ¨nster. § NRW Graduate School of Chemistry. E XC DHDFT ) (1 - a X )E X DFT + a X E X HF + (1 - a C )E C DFT + a C E2 (1) J. Phys. Chem. A 2009, 113, 9861–9873 9861 10.1021/jp9042864 CCC: $40.75  2009 American Chemical Society Published on Web 07/31/2009