Further analysis and comparative study of intermolecular interactions using dimers from the S22 database

Correction to DFT interaction energies by an empirical dispersion term valid for a range of intermolecular distances

The computation of intermolecular interaction energies via commonly used density functionals is hindered by their inaccurate inclusion of medium and long range dispersion interactions. Practical computation of inter- and intra-macrobiomolecule interaction energies, in particular, requires a fairly accurate yet not overly expensive methodology. It is also desirable to compute intermolecular energies not only at their equilibrium (lowest energy) configurations but also over a range of biophysically relevant distances. We present a method to compute intermolecular interaction energies by including an empirical correction for dispersion which is valid over a range of intermolecular distances. This is achieved by optimizing parameters that moderate the empirical correction by explicit comparison of density functional (B3LYP) energies with distance-dependent (DD) reference values obtained at the CCSD(T)/CBS limit. The resulting method, hereafter referred to as B3LYP-DD, yields interaction energies with an accuracy generally better than 1 kcal mol(-1) for different types of noncovalent complexes, over a range of intermolecular distances and interaction strengths, relative to the expensive CCSD(T)/CBS standard. For a training set of dispersion interacting complexes, B3LYP-DD interaction energies in combination with diffuse functions display absolute errors equal to or smaller than 0.68 kcal mol(-1). The empirical correction does not significantly increase the computational cost as compared to standard density functional calculations. Applications relevant to biomolecular energy and structure, such as prediction of DNA base-pair interactions, are also presented.

Computational insight into the electronic structure and absorption spectra of lithium complexes of N-confused tetraphenylporphyrin

The present work is a theoretical investigation on lithium complexes of N-confused tetraphenylporphyrins (aka inverted) employing density functional theory (DFT) and time-dependent DFT, using the B3LYP, CAM-B3LYP, and M06-2X functionals in conjunction with the 6-31G(d,p) basis set. The purpose of the present study is to calculate the electronic structure and the bonding of the complexes to explain the unusual coordination environment in which Li is found experimentally and how the Li binding affects the Q and the Soret bands. The calculations show that, unlike a typical tetrahedral Li(+) cation, this Li forms a typical bond with one N and interacts with the remaining two N atoms, and it is located in the right place to form an agostic-like interaction with the internal C atom. The reaction energy, the enthalpy for the formation of the lithium complexes of N-confused porphyrins, and the effect of solvation are also calculated. The insertion of Li into N-confused porphyrin, in the presence of tetrahydrofuran, is exothermic with a reaction energy calculated to be as high as -72.4 kcal/mol using the lithium bis(trimethylsilyl)amide reagent. Finally, there is agreement in the general shape among the vis-UV spectra determined with different functionals and the experimentally available ones. The calculated geometries are in agreement with crystallographic data, where available.

Accounting for non-optimal interactions in molecular recognition: a study of ion-π complexes using a QM/MM model with a dipole-polarisable MM region

For a quantitative understanding of molecular structure, interaction and dynamics, accurate modelling of the energetics of both near-equilibrium and less optimal contacts is important. In this work, we explore the potential energy surfaces of representative ion-π complexes. We examine the performance of a semi-empirical QM/MM approach and the corresponding QM/MMpol model, where inducible point dipoles are additionally employed in the MM region. The predicted potential energy surfaces of cation-benzene complexes are improved by inclusion of explicit MM polarisation of the π-molecule. For cation-formamide complexes, inducible dipoles appreciably improve energetic estimates at geometries forming non-optimal interactions. Energetic component analysis suggests that the implicit MM polarisation of the fixed charge QM/MM model mirrors the behaviour of the QM/MMpol dipole model for the energetics of near-equilibrium conformations. However, for complexes at less optimal orientations, the QM/MM model exhibits higher errors than the QM/MMpol approach, being unable to capture orientation-dependent variations in polarisation energy.

S66: A Well-balanced Database of Benchmark Interaction Energies Relevant to Biomolecular Structures

With numerous new quantum chemistry methods being developed in recent years and the promise of even more new methods to be developed in the near future, it is clearly critical that highly accurate, well-balanced, reference data for many different atomic and molecular properties be available for the parametrization and validation of these methods. One area of research that is of particular importance in many areas of chemistry, biology, and material science is the study of noncovalent interactions. Because these interactions are often strongly influenced by correlation effects, it is necessary to use computationally expensive high-order wave function methods to describe them accurately. Here, we present a large new database of interaction energies calculated using an accurate CCSD(T)/CBS scheme. Data are presented for 66 molecular complexes, at their reference equilibrium geometries and at 8 points systematically exploring their dissociation curves; in total, the database contains 594 points: 66 at equilibrium geometries, and 528 in dissociation curves. The data set is designed to cover the most common types of noncovalent interactions in biomolecules, while keeping a balanced representation of dispersion and electrostatic contributions. The data set is therefore well suited for testing and development of methods applicable to bioorganic systems. In addition to the benchmark CCSD(T) results, we also provide decompositions of the interaction energies by means of DFT-SAPT calculations. The data set was used to test several correlated QM methods, including those parametrized specifically for noncovalent interactions. Among these, the SCS-MI-CCSD method outperforms all other tested methods, with a root-mean-square error of 0.08 kcal/mol for the S66 data set.

The energy computation paradox and ab initio protein folding

The routine prediction of three-dimensional protein structure from sequence remains a challenge in computational biochemistry. It has been intuited that calculated energies from physics-based scoring functions are able to distinguish native from nonnative folds based on previous performance with small proteins and that conformational sampling is the fundamental bottleneck to successful folding. We demonstrate that as protein size increases, errors in the computed energies become a significant problem. We show, by using error probability density functions, that physics-based scores contain significant systematic and random errors relative to accurate reference energies. These errors propagate throughout an entire protein and distort its energy landscape to such an extent that modern scoring functions should have little chance of success in finding the free energy minima of large proteins. Nonetheless, by understanding errors in physics-based score functions, they can be reduced in a post-hoc manner, improving accuracy in energy computation and fold discrimination.

Integration Grid Errors for Meta-GGA-Predicted Reaction Energies: Origin of Grid Errors for the M06 Suite of Functionals

We have assessed integration grid errors arising from the use of popular DFT quadrature schemes for a set of 34 organic reaction energies. The focus is primarily on M05-2X and the M06 suite of functionals (M06-L, M06, M06-2X, and M06-HF). M05-2X, M06, and M06-2X outperform popular older DFT functionals for the reaction energies studied, and offer accuracies comparable to results from perturbative hybrid DFT functionals. However, these new functionals are more sensitive to the choice of quadrature grid than previous generations of DFT functionals. Errors in predicted reaction energies arising from the use of the popular SG-1 grid, which is the default in the Q-Chem package, are significant. In particular, M06-HF reaction energies computed with the SG-1 grid exhibit errors ranging from -6.7 to 3.2 kcal mol(-1) relative to results computed with a very fine integration grid. This grid-sensitivity is not a problem for meta-GGA functionals in general, but is instead due to the specific functional forms used in these functionals. The large grid errors are traced to the kinetic energy density enhancement factor utilized in the exchange component of the M05-2X and M06 functionals. This term contains empirically adjusted parameters that are of large magnitude for all of the M06 functionals and for M06-HF in particular. The product of these large constants with modest integration errors for the kinetic energy density results in very large errors in individual contributions to the exchange energy. This gives rise to the troubling large errors exhibited by these functionals for certain integration grids.

Limits of Free Energy Computation for Protein-Ligand Interactions

A detailed error analysis is presented for the computation of protein-ligand interaction energies. In particular, we show that it is probable that even highly accurate computed binding free energies have errors that represent a large percentage of the target free energies of binding. This is due to the observation that the error for computed energies quasi-linearly increases with the increasing number of interactions present in a protein-ligand complex. This principle is expected to hold true for any system that involves an ever increasing number of inter or intra-molecular interactions (e.g. ab initio protein folding). We introduce the concept of best-case scenario errors (BCS(errors)) that can be routinely applied to docking and scoring exercises and used to provide errors bars for the computed binding free energies. These BCS(errors) form a basis by which one can evaluate the outcome of a docking and scoring exercise. Moreover, the resultant error analysis enables the formation of an hypothesis that defines the best direction to proceed in order to improve scoring functions used in molecular docking studies.

Importance of dispersion and electron correlation in ab initio protein folding

Dispersion is well-known to be important in biological systems, but the effect of electron correlation in such systems remains unclear. In order to assess the relationship between the structure of a protein and its electron correlation energy, we employed both full system Hartree-Fock (HF) and second-order Møller-Plesset perturbation (MP2) calculations in conjunction with the Polarizable Continuum Model (PCM) on the native structures of two proteins and their corresponding computer-generated decoy sets. Because of the expense of the MP2 calculation, we have utilized the fragment molecular orbital method (FMO) in this study. We show that the sum of the Hartree-Fock (HF) energy and force field (LJ6)-derived dispersion energy (HF + LJ6) is well correlated with the energies obtained using second-order Møller-Plesset perturbation (MP2) theory. In one of the two examples studied, the correlation energy as well as the empirical dispersive energy term was able to discriminate between native and decoy structures. On the other hand, for the second protein we studied, neither the correlation energy nor dispersion energy showed discrimination capabilities; however, the ab initio MP2 energy and the HF+LJ6 both ranked the native structure correctly. Furthermore, when we randomly scrambled the Lennard-Jones parameters, the correlation between the MP2 energy and the sum of the HF energy and dispersive energy (HF+LJ6) significantly drops, which indicates that the choice of Lennard-Jones parameters is important.