Density Fitting Approximation Research Articles

Integral-Direct and Parallel Implementation of the CCSD(T) Method: Algorithmic Developments and Large-Scale Applications.

A completely integral-direct, disk I/O, and network traffic economic coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] implementation has been developed relying on the density-fitting approximation. By fully exploiting the permutational symmetry, the presented algorithm is highly operation count and memory-efficient. Our measurements demonstrate excellent strong scaling achieved via hybrid MPI/OpenMP parallelization and a highly competitive, 60-70% utilization of the theoretical peak performance on up to hundreds of cores. The terms whose evaluation time becomes significant only for small- to medium-sized examples have also been extensively optimized. Consequently, high performance is also expected for systems appearing in extensive data sets used, e.g., for density functional or machine learning parametrizations, and in calculations required for certain reduced-cost or local approximations of CCSD(T), such as in our local natural orbital scheme [LNO-CCSD(T)]. The efficiency of this implementation allowed us to perform some of the largest CCSD(T) calculations ever presented for systems of 31-43 atoms and 1037-1569 orbitals using only four to eight many-core CPUs and 1-3 days of wall time. The resulting 13 correlation energies and the 12 corresponding reaction energies and barrier heights are added to our previous benchmark set collecting reference CCSD(T) results of molecules at the applicability limit of current implementations.

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Stress tensor.

A full implementation of the analytical stress tensor for periodic systems is reported in the TURBOMOLE program package within the framework of Kohn-Sham density functional theory using Gaussian-type orbitals as basis functions. It is the extension of the implementation of analytical energy gradients (Lazarski et al., Journal of Computational Chemistry 2016, 37, 2518-2526) to the stress tensor for the purpose of optimization of lattice vectors. Its key component is the efficient calculation of the Coulomb contribution by combining density fitting approximation and continuous fast multipole method. For the exchange-correlation (XC) part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097-3104) is extended to XC weight derivatives and stress tensor. The computational efficiency and favorable scaling behavior of the stress tensor implementation are demonstrated for various model systems. The overall computational effort for energy gradient and stress tensor for the largest systems investigated is shown to be at most two and a half times the computational effort for the Kohn-Sham matrix formation. © 2019 Wiley Periodicals, Inc.

Efficient Implementation of the Second-Order Quasidegenerate Perturbation Theory with Density-Fitting and Cholesky Decomposition Approximations: Is It Possible To Use Hartree-Fock Orbitals for a Multiconfigurational Perturbation Theory?

The high cost of common multireference second-order perturbation theory (MRPT2) methods compared with the single-reference variant (MP2) arises from the expensive complete active space self-consistent field (CASSCF) orbital optimization step. Furthermore, the use of conventional four-index electron repulsion integrals (ERIs) prevents their application to larger molecular systems due to expensive I/O procedures. To address these bottlenecks of the multiconfigurational second-order quasidegenerate perturbation theory (MC-QDPT2), an efficient implementation of QDPT2 with the density-fitting (DF) and Cholesky decomposition (CD) approximations, denoted by DF-QDPT2 and CD-QDPT2, is reported. For the DF/CD-QDPT2 methods, the Hose-Kaldor approach is used. The DF-QDPT2 method, with the cc-pwCVTZ basis set, dramatically reduces the computational cost compared to conventional multiconfigurational QDPT2 (MC-QDPT2, from the Gamess 2017.R2 package), with a more than 122-fold reduction for the largest member of the diradical test set considered. The DF approximation enables substantially accelerated energies to be obtained for the QDPT2 approach due to the significantly reduced I/O time. The performance of the DF-QDPT2 and CD-QDPT2 methods is compared with that of CASSCF, the multireference second-order perturbation theory (MRMP2), MC-QDPT2, and CASSCF-based second-order perturbation theory (CASPT2) methods for singlet-triplet energy splitting (EST) in O2 and C2 molecules and for the dissociation energy of F2. For the O2 and C2 molecules, the performance of the DF-QDPT2 and CD-QDPT2 methods is significantly better than that of CASSCF, MRMP2, MC-QDPT2, and CASPT2; while for the F2 case, the results of DF-QDPT2, CD-QDPT2, MRMP2, MC-QDPT2, and CASPT2 are similar and remarkably better than that of CASSCF, which fails dramatically. Moreover, the DF-QDPT2, CASSCF, CASPT2, and MRCI+Q methods are applied to potential energy curves (PECs) for N2, CH4, and F2 molecules. Our results demonstrate that the performance of DF-QDPT2 is substantially better than that of CASSCF and is comparable with that of CASPT2 for the molecules considered. Overall, the present application results demonstrate that the DF-QDPT2 and CD-QDPT2 methods are very promising for electronically challenging molecular systems suffering from (quasi)degeneracy problems, where the single-reference methods cannot provide an accurate electronic description, but the DF-QDPT2 and CD-QDPT2 methods can do so at significantly reduced computational costs.

Improving the Efficiency of the Multireference Driven Similarity Renormalization Group via Sequential Transformation, Density Fitting, and the Noninteracting Virtual Orbital Approximation.

This study examines several techniques to improve the efficiency of the linearized multireference driven similarity renormalization group truncated to one- and two-body operators [MR-LDSRG(2)]. We propose a sequential MR-LDSRG(2) [sq-MR-LDSRG(2)] scheme, in which one-body substitutions are folded exactly into the Hamiltonian. This new approach is combined with density fitting (DF) to reduce the storage cost of two-electron integrals. To further avoid storage of large four-index intermediates, we propose a noninteracting virtual orbital (NIVO) approximation of the Baker-Campbell-Hausdorff series that neglects commutators terms with three and four virtual indices. The NIVO approximation reduces the computational prefactor of the MR-LDSRG(2), bringing it closer to that of coupled cluster with singles and doubles (CCSD). We test the effect of the DF and NIVO approximations on the MR-LDSRG(2) and sq-MR-LDSRG(2) methods by computing properties of eight diatomic molecules. The diatomic constants obtained by DF-sq-MR-LDSRG(2)+NIVO are found to be as accurate as those from the original MR-LDSRG(2) and coupled cluster theory with singles, doubles, and perturbative triples. Finally, we demonstrate that the DF-sq-MR-LDSRG(2)+NIVO scheme can be applied to chemical systems with more than 550 basis functions by computing the automerization energy of cyclobutadiene with a quintuple-ζ basis set. The predicted automerization energy is found to be similar to the value computed with Mukherjee's state-specific multireference coupled cluster theory with singles and doubles.

Coupled‐cluster singles, doubles and perturbative triples with density fitting approximation for massively parallel heterogeneous platforms

AbstractA high‐performance implementation of the coupled‐cluster singles, doubles, and perturbative triples [CCSD(T)] is developed in the Massively Parallel Quantum Chemistry program. Novel features include: (1) reduced memory requirements via a density‐fitting (DF) CCSD implementation utilizing distributed lazy evaluation for tensors with more than two unoccupied indices and (2) the ability to utilize efficiently many‐core nodes (Intel Xeon Phi) and heterogeneous nodes with multiple NVIDIA GPUs on each node. All data that are greater than quadratic in the system size are distributed among processes. Excellent strong scaling is observed on distributed‐memory computers equipped with conventional CPUs, Intel Xeon Phi processors, and heterogeneous nodes with multiple NVIDIA GPUs Canonical CCSD(T) energies can be evaluated for systems containing 200 electrons and 1000 basis functions in a few days using a small size commodity cluster, with even larger computations possible on leadership‐class computing resources.

Analytic Energy Gradients for Variational Two-Electron Reduced-Density Matrix Methods within the Density Fitting Approximation

Analytic energy gradients are presented for a variational two-electron reduced-density-matrix-driven complete active space self-consistent field (v2RDM-CASSCF) procedure that employs the density fitting (DF) approximation to the two-electron repulsion integrals. The DF approximation significantly reduces the computational cost of v2RDM-CASSCF gradient evaluation, in terms of both the number of floating-point operations and memory requirements, enabling geometry optimizations on much larger chemical systems than could previously be considered at this level of theory [ Maradzike et al., J. Chem. Theory Comput. , 2017 , 13 , 4113 - 4122 ]. The efficacy of v2RDM-CASSCF for computing equilibrium geometries and harmonic vibrational frequencies is assessed using a set of 25 small closed- and open-shell molecules. Equilibrium bond lengths from v2RDM-CASSCF differ from those obtained from configuration-interaction-driven CASSCF (CI-CASSCF) by 0.62 and 0.05 pm, depending on whether the optimal reduced-density matrices from v2RDM-CASSCF satisfy two-particle N-representability conditions (PQG) or PQG plus partial three-particle conditions (PQG+T2), respectively. Harmonic vibrational frequencies, which are obtained by finite differences of v2RDM-CASSCF analytic energy gradients, similarly demonstrate that quantitative agreement between v2RDM- and CI-CASSCF requires the consideration of partial three-particle N-representability conditions. Lastly, optimized geometries are obtained for the lowest-energy singlet and triplet states of the linear polyacene series up to dodecacene (C50H28), in which case the active space is comprised of 50 electrons in 50 orbitals. The v2RDM-CASSCF singlet-triplet energy gap extrapolated to an infinitely long linear acene molecule is found to be 7.8 kcal mol-1.

A pair natural orbital based implementation of CCSD excitation energies within the framework of linear response theory.

We present a pair natural orbital (PNO)-based implementation of coupled cluster singles and doubles (CCSD) excitation energies that builds upon the previously proposed state-specific PNO approach to the excited state eigenvalue problem. We construct the excited state PNOs for each state separately in a truncated orbital specific virtual basis and use a local density-fitting approximation to achieve an at most quadratic scaling of the computational costs for the PNO construction. The earlier reported excited state PNO construction is generalized such that a smooth convergence of the results for charge transfer states is ensured for general coupled cluster methods. We investigate the accuracy of our implementation by applying it to a large and diverse test set comprising 153 singlet excitations in organic molecules. Already moderate PNO thresholds yield mean absolute errors below 0.01 eV. The performance of the implementation is investigated through the calculations on alkene chains and reveals an at most cubic cost-scaling for the CCSD iterations with the system size.

State-of-the-Art Computations of Vertical Ionization Potentials with the Extended Koopmans' Theorem Integrated with the CCSD(T) Method.

The accurate computation of ionization potentials (IPs), within 0.10 eV, is one of the most challenging problems in modern computational chemistry. The extended Koopmans' theorem (EKT) provides a systematic direct approach to compute IPs from any level of theory. In this study, the EKT approach is integrated with the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method for the first time. For efficiency, the density-fitting (DF) approximation is employed for electron repulsion integrals. Further, the EKT-CCSD(T) method is applied to a set of 23 molecules, denoted as IP23, for comparison with the experimental ionization potentials. For the IP23 set, the EKT-CCSD(T) method, along with the aug-cc-pV5Z basis set, provides a mean absolute error of 0.05 eV. Hence, our results demonstrate that direct computations of IPs at high-accuracy levels can be achieved with the EKT-CCSD(T) method. We believe that the present study may open new avenues in IP computations.

Analytical energy gradients for explicitly correlated wave functions. I. Explicitly correlated second-order Møller-Plesset perturbation theory.

We present the theory and algorithms for computing analytical energy gradients for explicitly correlated second-order Møller-Plesset perturbation theory (MP2-F12). The main difficulty in F12 gradient theory arises from the large number of two-electron integrals for which effective two-body density matrices and integral derivatives need to be calculated. For efficiency, the density fitting approximation is used for evaluating all two-electron integrals and their derivatives. The accuracies of various previously proposed MP2-F12 approximations [3C, 3C(HY1), 3*C(HY1), and 3*A] are demonstrated by computing equilibrium geometries for a set of molecules containing first- and second-row elements, using double-ζ to quintuple-ζ basis sets. Generally, the convergence of the bond lengths and angles with respect to the basis set size is strongly improved by the F12 treatment, and augmented triple-ζ basis sets are sufficient to closely approach the basis set limit. The results obtained with the different approximations differ only very slightly. This paper is the first step towards analytical gradients for coupled-cluster singles and doubles with perturbative treatment of triple excitations, which will be presented in the second part of this series.

Analytic energy gradients for orbital-optimized MP3 and MP2.5 with the density-fitting approximation: An efficient implementation.

Efficient implementations of analytic gradients for the orbital-optimized MP3 and MP2.5 and their standard versions with the density-fitting approximation, which are denoted as DF-MP3, DF-MP2.5, DF-OMP3, and DF-OMP2.5, are presented. The DF-MP3, DF-MP2.5, DF-OMP3, and DF-OMP2.5 methods are applied to a set of alkanes and noncovalent interaction complexes to compare the computational cost with the conventional MP3, MP2.5, OMP3, and OMP2.5. Our results demonstrate that density-fitted perturbation theory (DF-MP) methods considered substantially reduce the computational cost compared to conventional MP methods. The efficiency of our DF-MP methods arise from the reduced input/output (I/O) time and the acceleration of gradient related terms, such as computations of particle density and generalized Fock matrices (PDMs and GFM), solution of the Z-vector equation, back-transformations of PDMs and GFM, and evaluation of analytic gradients in the atomic orbital basis. Further, application results show that errors introduced by the DF approach are negligible. Mean absolute errors for bond lengths of a molecular set, with the cc-pCVQZ basis set, is 0.0001-0.0002 Å. © 2017 Wiley Periodicals, Inc.

Analytic energy gradients for the coupled-cluster singles and doubles with perturbative triples method with the density-fitting approximation.

An efficient implementation of analytic gradients for the coupled-cluster singles and doubles with perturbative triples [CCSD(T)] method with the density-fitting (DF) approximation, denoted as DF-CCSD(T), is reported. For the molecules considered, the DF approach substantially accelerates conventional CCSD(T) analytic gradients due to the reduced input/output time and the acceleration of the so-called "gradient terms": formation of particle density matrices (PDMs), computation of the generalized Fock-matrix (GFM), solution of the Z-vector equation, formation of the effective PDMs and GFM, back-transformation of the PDMs and GFM, from the molecular orbital to the atomic orbital (AO) basis, and computation of gradients in the AO basis. For the largest member of the molecular test set considered (C6H14), the computational times for analytic gradients (with the correlation-consistent polarized valence triple-ζ basis set in serial) are 106.2 [CCSD(T)] and 49.8 [DF-CCSD(T)] h, a speedup of more than 2-fold. In the evaluation of gradient terms, the DF approach completely avoids the use of four-index two-electron integrals. Similar to our previous studies on DF-second-order Møller-Plesset perturbation theory and DF-CCSD gradients, our formalism employs 2- and 3-index two-particle density matrices (TPDMs) instead of 4-index TPDMs. Errors introduced by the DF approximation are negligible for equilibrium geometries and harmonic vibrational frequencies.

Reduced-cost linear-response CC2 method based on natural orbitals and natural auxiliary functions.

A reduced-cost density fitting (DF) linear-response second-order coupled-cluster (CC2) method has been developed for the evaluation of excitation energies. The method is based on the simultaneous truncation of the molecular orbital (MO) basis and the auxiliary basis set used for the DF approximation. For the reduction of the size of the MO basis, state-specific natural orbitals (NOs) are constructed for each excited state using the average of the second-order Møller-Plesset (MP2) and the corresponding configuration interaction singles with perturbative doubles [CIS(D)] density matrices. After removing the NOs of low occupation number, natural auxiliary functions (NAFs) are constructed [M. Kállay, J. Chem. Phys. 141, 244113 (2014)], and the NAF basis is also truncated. Our results show that, for a triple-zeta basis set, about 60% of the virtual MOs can be dropped, while the size of the fitting basis can be reduced by a factor of five. This results in a dramatic reduction of the computational costs of the solution of the CC2 equations, which are in our approach about as expensive as the evaluation of the MP2 and CIS(D) density matrices. All in all, an average speedup of more than an order of magnitude can be achieved at the expense of a mean absolute error of 0.02 eV in the calculated excitation energies compared to the canonical CC2 results. Our benchmark calculations demonstrate that the new approach enables the efficient computation of CC2 excitation energies for excited states of all types of medium-sized molecules composed of up to 100 atoms with triple-zeta quality basis sets.

Nuclei-selected atomic-orbital response-theory formulation for the calculation of NMR shielding tensors using density-fitting

An atomic orbital density matrix based response formulation of the nuclei-selected approach of Beer, Kussmann, and Ochsenfeld [J. Chem. Phys. 134, 074102 (2011)] to calculate nuclear magnetic resonance (NMR) shielding tensors has been developed and implemented into LSDalton allowing for a simultaneous solution of the response equations, which significantly improves the performance. The response formulation to calculate nuclei-selected NMR shielding tensors can be used together with the density-fitting approximation that allows efficient calculation of Coulomb integrals. It is shown that using density-fitting does not lead to a significant loss in accuracy for both the nuclei-selected and the conventional ways to calculate NMR shielding constants and should thus be used for applications with LSDalton.

An Integral-Direct Linear-Scaling Second-Order Møller-Plesset Approach.

An integral-direct, iteration-free, linear-scaling, local second-order Møller-Plesset (MP2) approach is presented, which is also useful for spin-scaled MP2 calculations as well as for the efficient evaluation of the perturbative terms of double-hybrid density functionals. The method is based on a fragmentation approximation: the correlation contributions of the individual electron pairs are evaluated in domains constructed for the corresponding localized orbitals, and the correlation energies of distant electron pairs are computed with multipole expansions. The required electron repulsion integrals are calculated directly invoking the density fitting approximation; the storage of integrals and intermediates is avoided. The approach also utilizes natural auxiliary functions to reduce the size of the auxiliary basis of the domains and thereby the operation count and memory requirement. Our test calculations show that the approach recovers 99.9% of the canonical MP2 correlation energy and reproduces reaction energies with an average (maximum) error below 1 kJ/mol (4 kJ/mol). Our benchmark calculations demonstrate that the new method enables MP2 calculations for molecules with more than 2300 atoms and 26000 basis functions on a single processor.

Spin-Adapted Formulation and Implementation of Density Cumulant Functional Theory with Density-Fitting Approximation: Application to Transition Metal Compounds.

Density cumulant functional theory (DCT) has recently emerged as an attractive ab initio approach for the treatment of electron correlation. In its orbital-optimized formulation (ODC-12) [J. Chem. Phys. 139, 204110 (2013)], DCT has been shown to provide reliable results for a variety of challenging chemical systems. Among the attractive properties of DCT are its size-consistency and size-extensivity, as well as the efficient computation of the molecular properties and analytic gradients. In this work, we present a new formulation and implementation of DCT that takes advantage of spin adaptation and the density-fitting approximation (DF-ODC-12). Our new spin-adapted DF-ODC-12 implementation is more efficient than the previous ODC-12 implementation with up to a ∼12-fold speed-up. We demonstrate the capabilities of DF-ODC-12 with a study of transition metal compounds, which require high levels of electron correlation treatment. For transition metal carbonyl complexes [Fe(CO)5, Cr(CO)6] and the ferrocene molecule [Fe(Cp)2], the DF-ODC-12 equilibrium parameters and bond dissociation energies extrapolated to the complete basis set limit are in very good agreement with reference data derived from experiment.

Density functional theory for molecular and periodic systems using density fitting and continuous fast multipole method: Analytical gradients.

A full implementation of analytical energy gradients for molecular and periodic systems is reported in the TURBOMOLE program package within the framework of Kohn-Sham density functional theory using Gaussian-type orbitals as basis functions. Its key component is a combination of density fitting (DF) approximation and continuous fast multipole method (CFMM) that allows for an efficient calculation of the Coulomb energy gradient. For exchange-correlation part the hierarchical numerical integration scheme (Burow and Sierka, Journal of Chemical Theory and Computation 2011, 7, 3097) is extended to energy gradients. Computational efficiency and asymptotic O(N) scaling behavior of the implementation is demonstrated for various molecular and periodic model systems, with the largest unit cell of hematite containing 640 atoms and 19,072 basis functions. The overall computational effort of energy gradient is comparable to that of the Kohn-Sham matrix formation. © 2016 Wiley Periodicals, Inc.

Analytic energy gradients for the coupled-cluster singles and doubles method with the density-fitting approximation.

An efficient implementation is presented for analytic gradients of the coupled-cluster singles and doubles (CCSD) method with the density-fitting approximation, denoted DF-CCSD. Frozen core terms are also included. When applied to a set of alkanes, the DF-CCSD analytic gradients are significantly accelerated compared to conventional CCSD for larger molecules. The efficiency of our DF-CCSD algorithm arises from the acceleration of several different terms, which are designated as the "gradient terms": computation of particle density matrices (PDMs), generalized Fock-matrix (GFM), solution of the Z-vector equation, formation of the relaxed PDMs and GFM, back-transformation of PDMs and GFM to the atomic orbital (AO) basis, and evaluation of gradients in the AO basis. For the largest member of the alkane set (C10H22), the computational times for the gradient terms (with the cc-pVTZ basis set) are 2582.6 (CCSD) and 310.7 (DF-CCSD) min, respectively, a speed up of more than 8-folds. For gradient related terms, the DF approach avoids the usage of four-index electron repulsion integrals. Based on our previous study [U. Bozkaya, J. Chem. Phys. 141, 124108 (2014)], our formalism completely avoids construction or storage of the 4-index two-particle density matrix (TPDM), using instead 2- and 3-index TPDMs. The DF approach introduces negligible errors for equilibrium bond lengths and harmonic vibrational frequencies.

Large-Scale Variational Two-Electron Reduced-Density-Matrix-Driven Complete Active Space Self-Consistent Field Methods.

A large-scale implementation of the complete active space self-consistent field (CASSCF) method is presented. The active space is described using the variational two-electron reduced-density-matrix (v2RDM) approach, and the algorithm is applicable to much larger active spaces than can be treated using configuration-interaction-driven methods. Density fitting or Cholesky decomposition approximations to the electron repulsion integral tensor allow for the simultaneous optimization of large numbers of external orbitals. We have tested the implementation by evaluating singlet-triplet energy gaps in the linear polyacene series and two dinitrene biradical compounds. For the acene series, we report computations that involve active spaces consisting of as many as 50 electrons in 50 orbitals and the simultaneous optimization of 1892 orbitals. For the dinitrene compounds, we find that the singlet-triplet gaps obtained from v2RDM-driven CASSCF with partial three-electron N-representability conditions agree with those obtained from configuration-interaction-driven approaches to within one-third of 1 kcal mol(-1). When enforcing only the two-electron N-representability conditions, v2RDM-driven CASSCF yields less accurate singlet-triplet energy gaps in these systems, but the quality of the results is still far superior to those obtained from standard single-reference approaches.

Improved Segmented All-Electron Relativistically Contracted Basis Sets for the Lanthanides.

Improved versions of the segmented all-electron relativistically contracted (SARC) basis sets for the lanthanides are presented. The second-generation SARC2 basis sets maintain efficient construction of their predecessors and their individual adaptation to the DKH2 and ZORA Hamiltonians, but feature exponents optimized with a completely new orbital shape fitting procedure and a slightly expanded f space that results in sizable improvement in CASSCF energies and in significantly more accurate prediction of spin-orbit coupling parameters. Additionally, an extended set of polarization/correlation functions is constructed that is appropriate for multireference correlated calculations and new auxiliary basis sets for use in resolution-of-identity (density-fitting) approximations in combination with both DFT and wave function based treatments. Thus, the SARC2 basis sets extend the applicability of the first-generation DFT-oriented basis sets to routine all-electron wave function-based treatments of lanthanide complexes. The new basis sets are benchmarked with respect to excitation energies, radial distribution functions, optimized geometries, orbital eigenvalues, ionization potentials, and spin-orbit coupling parameters of lanthanide systems and are shown to be suitable for the description of magnetic and spectroscopic properties using both DFT and multireference wave function-based methods.

Orbital-Optimized MP3 and MP2.5 with Density-Fitting and Cholesky Decomposition Approximations.

Efficient implementations of the orbital-optimized MP3 and MP2.5 methods with the density-fitting (DF-OMP3 and DF-OMP2.5) and Cholesky decomposition (CD-OMP3 and CD-OMP2.5) approaches are presented. The DF/CD-OMP3 and DF/CD-OMP2.5 methods are applied to a set of alkanes to compare the computational cost with the conventional orbital-optimized MP3 (OMP3) [Bozkaya J. Chem. Phys. 2011, 135, 224103] and the orbital-optimized MP2.5 (OMP2.5) [Bozkaya and Sherrill J. Chem. Phys. 2014, 141, 204105]. Our results demonstrate that the DF-OMP3 and DF-OMP2.5 methods provide considerably lower computational costs than OMP3 and OMP2.5. Further application results show that the orbital-optimized methods are very helpful for the study of open-shell noncovalent interactions, aromatic bond dissociation energies, and hydrogen transfer reactions. We conclude that the DF-OMP3 and DF-OMP2.5 methods are very promising for molecular systems with challenging electronic structures.