The objective of this work is the further systematic improvement of the accuracy of Double-Hybrid Density Functionals (DHDF) that add non-local electron correlation effects to a standard hybrid functional by second-order perturbation theory (S. Grimme, J. Chem. Phys., 2006, 124, 034108). The only known shortcoming of these generally highly accurate functionals is an underestimation of the long-range dispersion (van der Waals) interactions. To correct this deficiency, we add a previously developed empirical dispersion term (DFT-D) to the energy expression but leave the electronic part of the functional untouched. Results are presented for the S22 set of non-covalent interaction energies, the G3/99 set of heat of formations and conformational energies of a phenylalanyl-glycyl-glycine peptide model. We furthermore propose seven hydrocarbon reactions with strong intramolecular dispersion contributions as a benchmark set for newly developed density functionals. In general, the proposed composite approach is for many chemically relevant properties of similar quality as high-level coupled-cluster treatments. A significant increase of the accuracy for non-covalent interactions is obtained and the corrected B2PLYP DHDF provides one of the lowest ever obtained Mean Absolute Deviations (MAD) for the S22 set (0.2-0.3 kcal mol(-1)). Unprecedented high accuracy is also obtained for the relative energies of peptide conformations that turn out to be very difficult. The significant improvements found for the G3/99 set (reduction of the MAD from 2.4 to 1.7 kcal mol(-1)) underline the importance of intramolecular dispersion effects in large molecules. In all tested cases the results from the standard B3LYP approach are also significantly improved, and we recommend the general use of dispersion corrections in DFT treatments.

Kohn-Sham density functional theory (KS-DFT) is nowadays the most widely used quantum chemical method for electronic structure calculations in chemistry and physics. Its further application in e.g. supramolecular chemistry or biochemistry has mainly been hampered by the inability of almost all current density functionals to describe the ubiquitous attractive long-range van der Waals (dispersion) interactions. We review here methods to overcome this defect, and describe in detail a very successful correction that is based on damped -C(6).R(-6) potentials (DFT-D). As examples we consider the non-covalent inter- and intra-molecular interactions in unsaturated organic molecules (so-called pi-pi stacking in benzenes and dyes), in biologically relevant systems (nucleic acid bases/pairs, proteins, and 'folding' models), between fluorinated molecules, between curved aromatics (corannulene and carbon nanotubes) and small molecules, and for the encapsulation of methane in water clusters. In selected cases we partition the interaction energies into the most relevant contributions from exchange-repulsion, electrostatics, and dispersion in order to provide qualitative insight into the binding character.

Two hybrid density functionals that include a second-order perturbation correction for non-local correlation effects are tested for the full G3/05 test set. Very large AO basis sets including core-polarization/correlation functions have been employed that yield for the first time results quite close to the basis set limit for this set. The B2-PLYP functional and the new mPW2-PLYP approach with a modified exchange part give by far the lowest MAD over the whole G3/05 set ever reported for a DFT method (2.5 and 2.1 kcal mol(-1), respectively). The big improvement compared to common density functionals is further demonstrated by the reduction of the maximum and minimum errors (outliers) and by much smaller errors for complicated molecular systems.

An efficient implementation of the orbital-optimized second-order Møller-Plesset perturbation theory (OO-MP2) within the resolution of the identity (RI) approximation is reported. Both conventional MP2 and spin-component scaled (SCS-MP2) variants are considered, and an extensive numerical investigation of the accuracy of these approaches is presented. This work is closely related to earlier work of Lochan, R. C.; Head-Gordon, M. J. Chem. Phys. 2007, 126. Orbital optimization is achieved by making the Hylleraas functional together with the energy of the reference determinant stationary with respect to variations of the double excitation amplitudes and the molecular orbital rotation parameters. A simple iterative scheme is proposed that usually leads to convergence within 5-15 iterations. The applicability of the method to larger molecules (up to ∼1000-2000 basis functions) is demonstrated. The numerical results show that OO-SCS-MP2 is a major improvement in electronically complicated situations, such as represented by radicals or by transition states where spin contamination often greatly deteriorates the quality of the conventional MP2 and SCS-MP2 methods. The OO-(SCS-)MP2 approach reduces the error by a factor of 3-5 relative to the standard (SCS-)MP2. For closed-shell main group elements, no significant improvement in the accuracy relative to the already excellent SCS-MP2 method is observed. In addition, the problems of all MP2 variants with 3d transition-metal complexes are not solved by orbital optimization. The close relationship of the OO-MP2 method to the approximate second-order coupled cluster method (CC2) is pointed out. Both methods have comparable computational requirements. Thus, the OO-MP2 method emerges as a very useful tool for computational quantum chemistry.

For the first time, we combine time-dependent double-hybrid density functional approximations (TD-DHDFAs) for the calculation of electronic excitation energies with the concepts of spin-component and spin-opposite scaling (SCS/SOS) of electron-pair contributions to their nonlocal correlation components. Different flavors of this idea, ranging from standard SCS parameters to fully fitted parameter sets, are presented and tested on six different parent DHDFAs. For cross-validation, we assess those methods on three benchmark sets that cover small- to medium-sized chromophores (up to 78 atoms) and different excitation types. For this purpose, we also introduce new CC3 reference values for the popular Gordon benchmark set that we recommend using in future studies. Our results confirm that already the (unscaled) parent TD-DHDFAs are accurate and outperform some wave function methods. Further introduction of SCS/SOS eliminates extreme outliers, reduces deviation spans from reference values by up to 0.5 eV, aligns the performance of the Tamm-Dancoff approximation (TDA) to that of full TD calculations, and also enables a more balanced description of different excitation types. The best-performing TD-based methods in our cross validation have mean absolute deviations as low as 0.14 eV compared to the time- and resource-intensive CC3 approach. A very important finding is that we also obtained SOS variants with excellent performance, contrary to wave function based methods. This opens a future pathway to highly efficient methods for the optimization of excited-state geometries, particularly when paired with computing strategies such as the Laplace transform. We recommend our SCS- and SOS-based variants for further testing and subsequent applications.

The thermodynamic properties of molecules are of fundamental interest in physics, chemistry, and biology. This Account deals with the developments that we have made in the about last five years to find quantum chemical electronic structure methods that have the prospect of being applicable to larger molecules. The typical target accuracy is about 0.5-1 kcal mol(-1) for chemical reaction and 0.1 kcal mol(-1) for conformational energies. These goals can be achieved when a few physically motivated corrections to first-principles methods are introduced to standard quantum chemical techniques. These do not lead to a significantly increased computational expense, and thus our methods have the computer hardware requirements of the corresponding standard treatments. Together with the use of density-fitting (RI) integral approximations, routine computations on systems with about 100 non-hydrogen atoms (2000-4000 basis functions) can be performed on modern PCs. Our improvements regarding accuracy are basically due to the use of modified second-order perturbation theory to account for many-particle (electron correlation) effects. Such nonlocal correlations are responsible for important parts of the interaction in and between atoms and molecules. A common example is the long-range dispersion interaction that lead to van der Waals complexes, but as shown here also the conventional thermodynamics of large molecules is significantly influenced by intramolecular dispersion effects. We first present the basic theoretical ideas behind our approaches, which are the spin-component-scaled Møller-Plesset perturbation theory (SCS-MP2) and double-hybrid density functionals (DHDF). Furthermore, the effect of the independently developed empirical dispersion correction (DFT-D) is discussed. Together with the use of large atomic orbital basis sets (of at least triple- or quadruple-zeta quality), the accuracy of the new methods is even competitive with computationally very expensive coupled-cluster methods, but they still remain routinely applicable for day-to-day chemical problems. This is demonstrated for the G3/99 benchmark set of heats of formation, 34 organic isomerization energies, and barriers for a number of pericyclic reactions. As an electronically complicated example, the relative energies of three isomeric Au(8) clusters are considered. In general, we recommend the very robust B2PLYP-D density functional approach for heat of formation calculations and for electronically complicated situations like transition metal complexes or open-shell species. With B2PLYP-D, an unprecedented low mean absolute deviation for the G3/99 test set with a DFT approach of 1.7 kcal mol(-1) has been achieved. For closed-shell main-group molecules and many relative energies, SCS-MP2 is the method of choice, because it completely avoids the self-interaction error problem that still plagues current DFT. In critical cases, it is recommended to apply SCS-MP2 and B2PLYP-D simultaneously, where also the comparison with standard MP2 and density functionals like...

A recently proposed new family of density functionals [S. Grimme, J. Chem. Phys. 124, 34108 (2006)] adds a fraction of nonlocal correlation as a new ingredient to density functional theory (DFT). This fractional correlation energy is calculated at the level of second-order many-body perturbation theory (PT2) and replaces some of the semilocal DFT correlation of standard hybrid DFT methods. The new "double hybrid" functionals (termed, e.g., B2-PLYP) contain only two empirical parameters that have been adjusted in thermochemical calculations on parts of the G2/3 benchmark set. The methods have provided the lowest errors ever obtained by any DFT method for the full G3 set of molecules. In this work, the applicability of the new functionals is extended to the exploration of potential energy surfaces with analytic gradients. The theory of the analytic gradient largely follows the standard theory of PT2 gradients with some additional subtleties due to the presence of the exchange-correlation terms in the self-consistent field operator. An implementation is reported for closed-shell as well as spin-unrestricted reference determinants. Furthermore, the implementation includes external point charge fields and also accommodates continuum solvation models at the level of the conductor like screening model. The density fitting resolution of the identity (RI) approximation can be applied to the evaluation of the PT2 part with large gains in computational efficiency. For systems with approximately 500-600 basis functions the evaluation of the double hybrid gradient is approximately four times more expensive than the calculation of the standard hybrid DFT gradient. Extensive test calculations are provided for main group elements and transition metal containing species. The results reveal that the B2-PLYP functional provides excellent molecular geometries that are superior compared to those from standard DFT and MP2.

We formulate a new combined quantum mechanics/molecular mechanics (QM/MM) method based on a self-consistent polarizable embedding (PE) scheme. For the description of the QM region, we apply the popular coupled cluster (CC) method detailing the inclusion of electrostatic and polarization effects into the CC Lagrangian. Also, we consider the transformations required to obtain molecular properties from the linear and quadratic response functions and provide an implementation for the calculation of excitation energies, one- and two-photon absorption properties, polarizabilities and hyperpolarizabilities all coupled to a polarizable MM environment. In the process, we identify CC densitylike intermediates that allow for a very efficient implementation retaining a computational low cost of the QM/MM terms even when the number of MM sites increases. The strengths of the new implementation are illustrated by property calculations on different systems representing the frontier of the capabilities of the CC/MM method. We combine our method with a molecular dynamics sampling scheme such that statistical averages of different excited state solvated properties may be obtained. Especially, we systematically investigate the relative importance of multipoles and polarizabilities in the description of two-photon absorption activity for formamide in aqueous solution. Also, we demonstrate the strengths of the CC hierarchies by incorporating correlation effects both at the CC2, CCSD, and at the triples level in the so-called PE-CCSDR(3) model. Finally, we utilize the presented method in the description of a full protein by investigating the shift of the intense electronic excitation energy of the photoactive yellow protein due to the surrounding amino acids.