Long‐range corrected density functionals combined with local response dispersion: A promising method for weak interactions

Density functional theory, in general, is considered to underestimate the weak van der Waals type of intermolecular interactions. We optimized parameters of the local response dispersion (LRD) method applied to the long‐range corrected exchange‐correlation functionals (LC‐BOP12+LRD and LCgau‐BOP+LRD) on the interaction energy for the complexes in the recently compiled S66 database and found to be comparable with the high‐level wave function‐based methods reported in ?ezá? et al. (J. Chem. Theory Comput. 2011 , 7, 2427). Our calculations with the S66 intermolecular complexes at equilibrium geometries suggests that the LC‐BOP12+LRD and LCgau‐BOP+LRD are well‐balanced and lower cost alternatives to the methods reported in the database. Further, test on the S66X8 database (with eight nonequilibrium points) and the HBC6 and NBC10 database shows LC+LRD method with newly optimized parameters is a promising candidate for dealing such weak interactions. Finally, the new parameterized LC+LRD method was tested on X40 benchmark halogenated complexes.Copyright © 2013 Wiley Periodicals, Inc.     

Evaluation of the performance of post-Hartree-Fock methods in terms of intermolecular distance in noncovalent complexes

Dissociation curves calculated using multiple correlated QM methods for 66 noncovalent complexes (?ezá? et al., J Chem Theory Comput 2011, 7, 2427) have allowed us to interpolate equilibrium intermolecular distances for each studied method. Comparison of these data with CCSD(T)/complete basis set reference geometries provides information on how these methods perform in geometry optimizations. The large set of systems considered here is necessary for reliable statistical evaluation of the results and assessment of the robustness of the studied methods. Our results show that advanced methods such as MP3 and CCSD provide significant improvement over MP2 only when empirical scaling is used. The best results can be achieved with spin component scaled CCSD optimized for noncovalent interactions, with a root mean square error of 0.4% of the equilibrium distance. Scaled MP3, the MP2.5 method, yields comparably good results (error 0.5%) while being substantially cheaper. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2012     

Comparison of intermolecular interaction energies from SAPT and DFT including empirical dispersion contributions

The dispersion correction based on damped atom-atom long-range interaction contributions has been tested for an extended S22 database of intermolecular complexes using density functional theory (DFT) and symmetry adapted perturbation theory (SAPT) to account for the remaining interaction energy contributions. In the case of DFT, the dispersion correction of Grimme (J. Comput. Chem. 2006, 27, 1787) was used, while for SAPT, another damping function has been developed that has been optimized particularly for the database. It is found that both approaches yield about the same accuracy for the mixed-type complexes, while the DFT plus dispersion method performs better for the hydrogen-bridged systems and the SAPT plus dispersion approach is better for the dispersion-dominated complexes if compared with coupled cluster singles-doubles with perturbative triples interaction energies as a reference.     

Adaptive steered molecular dynamics: validation of the selection criterion and benchmarking energetics in vacuum

The potential of mean force (PMF) for stretching decaalanine in vacuum was determined earlier by Park and Schulten [J. Chem. Phys. 120, 5946 (2004)] in a landmark article demonstrating the efficacy of combining steered molecular dynamics and Jarzynski's nonequilibrium relation. In this study, the recently developed adaptive steered molecular dynamics (ASMD) algorithm [G. Ozer, E. Valeev, S. Quirk, and R. Hernandez, J. Chem. Theory Comput. 6, 3026 (2010)] is used to reproduce the PMF of the unraveling of decaalanine in vacuum by averaging over fewer nonequilibrium trajectories. The efficiency and accuracy of the method are demonstrated through the agreement with the earlier work by Park and Schulten, a series of convergence checks compared to alternate SMD pulling strategies, and an analytical proof. The nonequilibrium trajectories obtained through ASMD have also been used to analyze the intrapeptide hydrogen bonds along the stretching coordinate. As the decaalanine helix is stretched, the initially stabilized i → i + 4 contacts (α-helix) is replaced by i → i + 3 contacts (3(10)-helix). No significant formation of i → i + 5 hydrogen bonds (π-helix) is observed.     

The use of atomic intrinsic polarizabilities in the evaluation of the dispersion energy

The recent approach presented by Becke and Johnson [J. Chem. Phys. 122, 154104 (2005); 123, 024101 (2005); 123, 154101 (2005); 124, 174104 (2006); 124, 014104 (2006)] for the evaluation of dispersion interactions based on the properties of the exchange-hole dipole moment is combined with a Hirshfeld-type partitioning for the molecular polarizabilities into atomic contributions, recently presented by some of the present authors [A. Krishtal et al., J. Chem. Phys. 125, 034312 (2006)]. The results on a series of nine dimers, involving neon, methane, ethene, acetylene, benzene, and CO(2), taken at their equilibrium geometry, indicate that when the C(6), C(8), and C(10) terms are taken into account, the resulting dispersion energies can be obtained deviating 3% or 8% from high level literature data [E. R. Johnson and A. D. Becke, J. Chem. Phys. 124, 174104 (2006)], without the use of a damping function, the only outlier being the parallel face-to-face benzene dimer.     

Comparison of aromatic NH···π, OH···π, and CH···π interactions of alanine using MP2, CCSD, and DFT methods

A comparison of the performance of various density functional methods including long‐range corrected and dispersion corrected methods [MPW1PW91, B3LYP, B3PW91, B97‐D, B1B95, MPWB1K, M06‐2X, SVWN5, ωB97XD, long‐range correction (LC)‐ωPBE, and CAM‐B3LYP using 6‐31+G(d,p) basis set] in the study of CH···π, OH···π, and NH···π interactions were done using weak complexes of neutral (A) and cationic (A⁺) forms of alanine with benzene by taking the Møller–Plesset (MP2)/6‐31+G(d,p) results as the reference. Further, the binding energies of the neutral alanine–benzene complexes were assessed at coupled cluster (CCSD)/6‐31G(d,p) method. Analysis of the molecular geometries and interaction energies at density functional theory (DFT), MP2, CCSD methods and CCSD(T) single point level reveal that MP2 is the best overall performer for noncovalent interactions giving accuracy close to CCSD method. MPWB1K fared better in interaction energy calculations than other DFT methods. In the case of M06‐2X, SVWN5, and the dispersion corrected B97‐D, the interaction energies are significantly overrated for neutral systems compared to other methods. However, for cationic systems, B97‐D yields structures and interaction energies similar to MP2 and MPWB1K methods. Among the long‐range corrected methods, LC‐ωPBE and CAM‐B3LYP methods show close agreement with MP2 values while ωB97XD energies are notably higher than MP2 values. © 2010 Wiley Periodicals, Inc. J Comput Chem 2010     

DFT-D4 counterparts of leading meta-generalized-gradient approximation and hybrid density functionals for energetics and geometries

Previously, we introduced DFT-D3(BJ)ω B97X-V and ω B97M-V functionals and assessed them for the GMTKN55 database [Najibi and Goerigk, J Chem. Theory Comput. 2018, 14, 5725]. In this study, we present DFT-D4 damping parameters to build the DFT-D4 counterparts of these functionals and assess these in comparison. We extend our analysis beyond GMTKN55 and especially turn our attention to enzymatically catalyzed and metal–organic reactions. We find that B97M-D4 is now the second-best performing meta-generalized-gradient approximation functional for the GMTKN55 database and it can provide noticeably better organometallic reaction energies compared to B97M-D3(BJ). Moreover, the aforementioned DFT-D3(BJ)-based functionals have not been thoroughly assessed for geometries and herein we close this gap by analyzing geometries of noncovalently bound dimers and trimers, peptide conformers, water hexamers and transition-metal complexes. We find that several of the B97(M)-based methods—particularly the DFT-D4 versions—surpass the accuracy of previously studied methods for peptide conformer, water hexamer, and transition-metal complex geometries, making them safe-to-use, cost-efficient alternatives to the original methods. The DFT-D4 variants can be easily used with ORCA4.1 and above.     

The structure and binding energies of the van der Waals complexes of Ar and N2 with phenol and its cation, studied by high level ab initio and density functional theory calculations

We have investigated, using both ab initio and density functional theory methods, the minimum energy structures and corresponding binding energies of the van der Waals complexes between phenol and argon or the nitrogen molecule, and the corresponding complexes involving the phenol cation. Structures were obtained at the MP2 level using a large basis, and the corresponding energies were corrected for basis set superposition error (BSSE), higher order electron correlation effects, and for basis set size. The structures of the global minima were further refined for the effects of BSSE and the corresponding binding energies were evaluated. For each neutral species, we find only a single true minimum, pi bonded for argon and OH bonded for nitrogen. For both cationic species, we find that the OH-bonded complex is preferred over other minima which we have identified as having Ar or N(2) between exogeneous atoms. The ab initio calculations are generally in excellent agreement with experimental binding energies and rotational constants. We find that the B3LYP functional is particularly poor at describing these complexes, while a density functional theory (DFT) method with an empirical correction for dispersive interactions (DFT-D) is very successful, as are some of the new functionals proposed by Zhao and Truhlar [J. Phys. Chem. A 109, 5656 (2005); J. Chem. Theory Comput. 2, 1009 (2006); Phys. Chem. Chem. Phys. 7, 2701 (2005); J. Phys. Chem. A 108, 6908 (2004)]. Both the ab initio and DFT-D methods accurately predict the intermolecular vibrational modes.     

Global minimum geometries of acetylene clusters (HCCH)n with n ≤ 55 obtained by a heuristic method combined with geometrical perturbations

Geometry optimization of acetylene clusters (HCCH)_n in the range of n ≤ 55 was carried out with a recently proposed intermolecular potential consisting of Morse potentials, damped dispersion terms, and damped Coulomb terms. The heuristic method developed by the present author was used in the optimization: optimal geometries were searched by using geometry perturbations and subsequent local optimizations. The calculations were repeated until the global minimum was found for each cluster at least three times. The obtained results were analyzed to examine structural evolution of the clusters. The geometries of the clusters with n ≥ 25 were similar to the geometry in the cubic crystal of acetylene whereas smaller clusters take icosahedron‐based geometries. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Conductor‐like screening model for relaxed excited states: Implementation in the semiempirical method MSINDO

Two approaches to treat solvent polarization and reorientation effects for excited states of molecules and surfaces have been implemented in the recently developed MSINDO‐sCIS method (Gadaczek, Krause, Hintze, Bredow, J. Chem. Theory Comput. 2011, 7, 3675). They allow for an efficient calculation of analytical energy gradients and hence open the opportunity to investigate fluorescence effects or photochemical reactions in solution for large molecules that are difficult to treat with high‐level methods. Both approaches are based on the conductor‐like screening model (COSMO) (Klamt and Schüürmann, J. Chem. Soc., Perkin Trans. 1993, 2, 799) in combination with the configuration interaction singles (CIS) method (Foresman, Head‐Gordon, Pople, and Frisch, J. Phys. Chem. 1992, 96, 135). The paper gives a brief outline of the theoretical background. As a first application, solvent shifts of three well‐studied, environment‐sensitive fluorescent dyes (Kucherak, Didier, Mély, and Klymchenko, J. Phys. Chem. Lett. 2010, 1, 616) have been calculated and compared with experimental results and standard time‐dependent density functional theory. A statistical evaluation of MSINDO‐COSMO‐sCIS is provided for a set of 39 molecules suggested recently by Jacquemin et al. (Jacquemin, Planchat, Adamo, and Mennucci, J. Chem. Theory Comput. 2012, 8, 2359). Calculated vertical and adiabatic excitation energies and fluorescence energies are compared to experimental data. © 2014 Wiley Periodicals, Inc.     

Implementation of empirical dispersion corrections to density functional theory for periodic systems

A recently developed empirical dispersion correction (Grimme et al., J. Chem. Phys. 2010, 132, 154104) to standard density functional theory (DFT‐D3) is implemented in the plane‐wave program package VASP. The DFT‐D3 implementation is compared with an implementation of the earlier DFT‐D2 version (Grimme, J. Comput. Chem. 2004, 25, 1463; Grimme, J. Comput. Chem. 2006, 27, 1787). Summation of empirical pair potential terms is performed over all atom pairs in the reference cell and over atoms in shells of neighboring cells until convergence of the dispersion energy is obtained. For DFT‐D3, the definition of coordination numbers has to be modified with respect to the molecular version to ensure convergence. The effect of three‐center terms as implemented in the original molecular DFT‐D3 version is investigated. The empirical parameters are taken from the original DFT‐D3 version where they had been optimized for a reference set of small molecules. As the coordination numbers of atoms in bulk and surfaces are much larger than in the reference compounds, this effect has to be discussed. The results of test calculations for bulk properties of metals, metal oxides, benzene, and graphite indicate that the original parameters are also suitable for solid‐state systems. In particular, the interlayer distance in bulk graphite and lattice constants of molecular crystals is considerably improved over standard functionals. With the molecular standard parameters (Grimme et al., J. Chem. Phys. 2010, 132, 154104; Grimme, J. Comput. Chem. 2006, 27, 1787) a slight overbinding is observed for ionic oxides where dispersion should not contribute to the bond. For simple adsorbate systems, such as Xe atoms and benzene on Ag(111), the DFT‐D implementations reproduce experimental results with a similar accuracy as more sophisticated approaches based on perturbation theory (Rohlfing and Bredow, Phys. Rev. Lett. 2008, 101, 266106). © 2012 Wiley Periodicals, Inc.     

Intermolecular interactions of formic acid with benzene: Energy decomposition analyses with ab initio MP2 and double‐hybrid density functional computations

The intermolecular interactions of formic acid (HCOOH) with benzene (C₆H₆) have been investigated using localized molecular orbital energy decomposition analyses (LMO‐EDA) with ab initio MP2 and several double‐hybrid density functionals. The molecular geometries of five HCOOH…C₆H₆ complexes and corresponding benchmark total interaction energies at the CCSD(T)/CBS level are taken from literature (Zhao et al., J. Chem. Theory Comput. 2009, 5, 2726). According to the results of LMO‐EDA with the MP2 method, the dispersion energies are found to be as important as the electrostatic energies for the total interaction energies of the five HCOOH…C₆H₆ complexes. Based on LMO‐EDA with the double‐hybrid density functionals of B2PLYP, B2K‐PLYP, B2T‐PLYP, and B2GP‐PLYP computations, two new parameters for the framework of B2PLYP are extrapolated. These two new parameters are tested with other 10 complexes involving C₆H₆ (Crittenden, J. Phys. Chem. A 2009, 113, 1663), and they perform well on predicting the corresponding total interaction energies. Interestingly, these two new parameters for the framework of B2PLYP also perform well on the noncovalent complexation energies database (NCCE31/05) developed by Truhlar's group (Zhao and Truhlar, J. Phys. Chem. A 2005, 109, 5656). Therefore, these two new parameters appear to be suitable for investigating the noncovalent interactions, and they are denoted as B2N‐PLYP, where N stands for the noncovalent interaction. This study is expected to provide new insight into the derivation of double‐hybrid density functionals for studying the noncovalent interactions. © 2013 Wiley Periodicals, Inc.     

On the effect of the BSSE on intermolecular potential energy surfaces. Comparison of a priori and a posteriori BSSE correction schemes

A comparative study of geometrical parameters is performed on the complexes HF–HF, H₂O–H₂O, and HF–H₂O using 12 different basis sets at the RHF, MP2, and DFT (BLYP and B3LYP) levels of theory. The equilibrium geometries were obtained from uncorrected, a posteriori (counterpoise, CP) and a priori (Chemical Hamiltonian Approach, CHA) BSSE‐corrected potential energy surfaces. The calculation of equilibrium geometries using the CP and CHA schemes is described in details. The effect of the BSSE on various intermolecular parameters is discussed and the performance of the applied theoretical models is critically evaluated from the BSSE point of view. © 2001 John Wiley & Sons, Inc. J Comput Chem 22: 765–786, 2001     

Density functional theory augmented with an empirical dispersion term. Interaction energies and geometries of 80 noncovalent complexes compared with ab initio quantum mechanics calculations

Standard density functional theory (DFT) is augmented with a damped empirical dispersion term. The damping function is optimized on a small, well balanced set of 22 van der Waals (vdW) complexes and verified on a validation set of 58 vdW complexes. Both sets contain biologically relevant molecules such as nucleic acid bases. Results are in remarkable agreement with reference high-level wave function data based on the CCSD(T) method. The geometries obtained by full gradient optimization are in very good agreement with the best available theoretical reference. In terms of the standard deviation and average errors, results including the empirical dispersion term are clearly superior to all pure density functionals investigated-B-LYP, B3-LYP, PBE, TPSS, TPSSh, and BH-LYP-and even surpass the MP2/cc-pVTZ method. The combination of empirical dispersion with the TPSS functional performs remarkably well. The most critical part of the empirical dispersion approach is the damping function. The damping parameters should be optimized for each density functional/basis set combination separately. To keep the method simple, we optimized mainly a single factor, s(R), scaling globally the vdW radii. For good results, a basis set of at least triple-zeta quality is required and diffuse functions are recommended, since the basis set superposition error seriously deteriorates the results. On average, the dispersion contribution to the interaction energy missing in the DFT functionals examined here is about 15 and 100% for the hydrogen-bonded and stacked complexes considered, respectively.     

Extension of the LAP functional to include parallel spin correlation

The kinetic energy density-dependent correlation functional LAP1 is extended to include parallel-spin correlation beyond the exchange level. Two exchange–correlation schemes are considered, combining the new correlation functional (LAP3) with the GGA exchange of Becke and the GGA exchange of Perdew. Extensive tests on molecules and hydrogen-bonded systems are presented and discussed elucidating the role of parallel–spin correlation in different cases. Its inclusion in the LAP functional leads, on average, to a slight improvement of the calculated binding energies and equilibrium geometries of molecules. Particularly high sensitivity of the energy results on the relative share of parallel-spin correlation is observed for aromatic molecules and for systems involving weak hydrogen bonds. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 64 : 427–446, 1997     

Improved third-order Møller-Plesset perturbation theory

Based on a partitioning of the total correlation energy into contributions from parallel‐ and antiparallel‐spin pairs of electrons, a modified third‐order Møller–Plesset (MP) perturbation theory is developed. The method, termed SCS–MP3 (SCS for spin‐component‐scaled) continues previous work on an improved version of MP2 (S. Grimme, J Chem Phys 2003, 118, 9095). A benchmark set of 32 isogyric reaction energies, 11 atomization energies, and 11 stretched geometries is used to assess to performance of the model in comparison to the standard quantum chemical approaches MP2, MP3, and QCISD(T). It is found, that the new method performs significantly better than usual MP2/MP3 and even outperforms the more costly QCISD method. Opposite to the usual MP series, the SCS third‐order correction uniformly improves the results. Dramatic enhancements are especially observed for the more difficult atomization energies, some of the stretched geometries, and reaction and ionization energies involving transition metal compounds where the method seems to be competitive or even superior to the widely used density functional approaches. Further tests performed for other complex systems (biradicals, C₂₀ isomers, transition states) demonstrate that the SCS–MP3 model yields often results of QCISD(T) accuracy. The uniformity with which the new approach improves for very different correlation problems indicates significant robustness, and suggests it as a valuable quantum chemical method of general use. © 2003 Wiley Periodicals, Inc. J Comput Chem 24: 1529–1537, 2003     

Vibrational corrections to geometries of transition metal complexes from density functional theory

Zero-point vibrational corrections are computed at the BP86/AE1 level for the set of 50 transition-metal/ligand bonds that have recently been proposed as testing ground for DFT methods, because of the availability of precise experimental gas-phase geometries (Bühl and Kabrede, J Chem Theory Comput 2006, 2, 1282). These corrections are indicated to be transferable to a large extent between various density-functional/basis-set combinations, so that they can be used to estimate zero-point averaged r0g distances from re values optimized at other theoretical levels. Applying this approach to a number of popular DFT levels does not, in general, improve their overall accuracy in terms of mean and standard deviations from experiment. The hybrid variant of the meta-functional TPSS is confirmed as promising choice for computing structures of transition-metal complexes.     

NUPACK: Analysis and design of nucleic acid systems

The Nucleic Acid Package (NUPACK) is a growing software suite for the analysis and design of nucleic acid systems. The NUPACK web server ( http://www.nupack.org ) currently enables:

• Analysis: thermodynamic analysis of dilute solutions of interacting nucleic acid strands.
• Design: sequence design for complexes of nucleic acid strands intended to adopt a target secondary structure at equilibrium.
• Utilities: evaluation, display, and annotation of equilibrium properties of a complex of nucleic acid strands.

NUPACK algorithms are formulated in terms of nucleic acid secondary structure. In most cases, pseudoknots are excluded from the structural ensemble. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Resolutions of the Coulomb operator. VI. Computation of auxiliary integrals

We discuss the efficient computation of the auxiliary integrals that arise when resolutions of two-electron operators (specifically, the Coulomb operator [T. Limpanuparb, A. T. B. Gilbert, and P. M. W. Gill, J. Chem. Theory Comput. 7, 830 (2011)] and the long-range Ewald operator [T. Limpanuparb and P. M. W. Gill, J. Chem. Theory Comput. 7, 2353 (2011)]) are employed in quantum chemical calculations. We derive a recurrence relation that facilitates the generation of auxiliary integrals for Gaussian basis functions of arbitrary angular momentum and propose a near-optimal algorithm for its use.     

Quantum mechanical study of regioselectivity of radical additions to substituted olefins

Regiochemical trends in the addition of free radicals to substituted olefins are investigated by different quantum chemical approaches with special reference to oxygen centered radicals. From a methodological point of view, density functional methods provide correct general trends but they do not reach quantitative accuracy, especially for intermediate complexes. More reliable results are obtained by single point post‐Hartree–Fock computations at density functional geometries. A number of test computations show that reoptimization of the geometry and computation of vibrational frequencies by correlated methods can be safely avoided. As a consequence, the overall computational approach has very reasonable computer costs. From a more chemical point of view, a careful analysis of computational results points out the significant role of anomeric and polar effects in tuning the common filicity of carbon centered radicals. © 2000 John Wiley & Sons, Inc. J Comput Chem 21: 675–691, 2000