Efficient determination of accurate atomic polarizabilities for polarizeable embedding calculations

We evaluate embedding potentials, obtained via various methods, used for polarizable embedding computations of excitation energies of para‐nitroaniline in water and organic solvents as well as of the green fluorescent protein. We found that isotropic polarizabilities derived from DFTD3 dispersion coefficients correlate well with those obtained via the LoProp method. We show that these polarizabilities in conjunction with appropriately derived point charges are in good agreement with calculations employing static multipole moments up to quadrupoles and anisotropic polarizabilities for both computed systems. The (partial) use of these easily‐accessible parameters drastically reduces the computational effort to obtain accurate embedding potentials especially for proteins. © 2016 The Authors. Journal of Computational Chemistry Published by Wiley Periodicals, Inc.     

Fundamental relationships in the theory of electric multipole moments and multipole polarizabilities in static fields

New derivations are given of equations relating molecular electric multipole moments and polarizabilities of general order to the electrostatic energy. The unabridged moment convention is shown to yield relatively simple relations between derivatives of the energy with respect to field gradients and the multipole moments and polarizabilities. Care is taken to distinguish various forms of these derivatives, and one form leads to a proof of a general symmetry of polarizability tensors with respect to permutations of rank indices. The condition of internal equilibrium is shown to be fundamental to the existence of this symmetry. The transformation of multipole moment and polarizability tensors under translation of the coordinate origin is expressed in relatively simple general form. The traceless multipole and polarizability tensors of Buckingham and McLean and Yoshimine are obtained as linear combinations of the unabridged tensors and their traces.     

Multipole moments for embedding potentials: Exploring different atomic allocation algorithms

Polarizable quantum mechanical (QM)/molecular mechanics (MM)‐embedding methods are currently among the most promising methods for computationally feasible, yet reliable, production calculations of localized excitations and molecular response properties of large molecular complexes, such as proteins and RNA/DNA, and of molecules in solution. Our aim is to develop a computational methodology for distributed multipole moments and their associated multipole polarizabilities which is accurate, computationally efficient, and with smooth convergence with respect to multipole order. As the first step toward this goal, we herein investigate different ways of obtaining distributed atom‐centered multipole moments that are used in the construction of the electrostatic part of the embedding potential. Our objective is methods that not only are accurate and computationally efficient, but which can be consistently extended with site polarizabilities including internal charge transfer terms. We present a new way of dealing with well‐known problems in relation to the use of basis sets with diffuse functions in conventional atomic allocation algorithms, avoiding numerical integration schemes. Using this approach, we show that the classical embedding potential can be systematically improved, also when using basis sets with diffuse functions, and that very accurate embedding potentials suitable for QM/MM embedding calculations can be acquired. © 2016 Wiley Periodicals, Inc.     

Gaussian induced dipole polarization model

A new induced dipole polarization model based on interacting Gaussian charge densities is presented. In contrast to the original induced point dipole model, the Gaussian polarization model is capable of finite interactions at short distances. Aspects of convergence related to the Gaussian model will be explored. The Gaussian polarization model is compared with the damped Thole-induced dipole model and the point dipole model. It will be shown that the Gaussian polarization model performs slightly better than the Thole model in terms of fitting to molecular polarizability tensors. An advantage of the model based on Gaussian charge distribution is that it can be easily generalized to other multipole moments and provide effective damping for both permanent electrostatic and polarization models. Finally, a method of parameterizing polarizabilities is presented. This method is based on probing a molecule with point charges and fitting polarizabilities to electrostatic potential. In contrast to the generic atom type polarizabilities fit to molecular polarizability tensors, probed polarizabilities are significantly more accurate in terms of reproducing molecular polarizability tensors and electrostatic potential, while retaining conformational transferability.     

The effects of atomic multipole moments obtained by the potential-derived method on hydrogen bonding

A potential-derived atomic multipole method called the cumulative potential-derived atomic multipole method is developed, with which electrostatic atomic multipole moments are derived by fitting the molecular electric potential in a cumulative way. It is applied to the hydrides of N , O , F , S , Cl , and methanol and the hydrogen-bonded dimers formed between them. The relationship between atomic multipole moments and molecular charge distributions is found. The structures calculated with Buckingham's electrostatic model are in good agreement with experiments. The phenomena of nonlinear structures of most H -bonded complexes—the deviations of symmetry axes of electron donors from H bonds—and correct distinguishing between two alternative structures are attributed to atomic dipole and quadrupole moments. Compared with other methods, this method has a quantitative and qualitative advantage and simple algorithm. The main conclusion is that the atomic multipole moments play a substantial role, although a potential-derived charge model was deemed sufficient previously. © 1993 John Wiley & Sons, Inc.     

Multipolar electrostatics for proteins: Atom–atom electrostatic energies in crambin

Accurate electrostatics necessitates the use of multipole moments centered on nuclei or extra point charges centered away from the nuclei. Here, we follow the former alternative and investigate the convergence behavior of atom‐atom electrostatic interactions in the pilot protein crambin. Amino acids are cut out from a Protein Data Bank structure of crambin, as single amino acids, di, or tripeptides, and are then capped with a peptide bond at each side. The atoms in the amino acids are defined through Quantum Chemical Topology (QCT) as finite volume electron density fragments. Atom‐atom electrostatic energies are computed by means of a multipole expansion with regular spherical harmonics, up to a total interaction rank of L = ?_A+ ?_B + 1 = 10. The minimum internuclear distance in the convergent region of all the 15 possible types of atom‐atom interactions in crambin that were calculated based on single amino acids are close to the values calculated from di and tripeptides. Values obtained at B3LYP/aug‐cc‐pVTZ and MP2/aug‐cc‐pVTZ levels are only slightly larger than those calculated at HF/6‐31G(d,p) level. This convergence behavior is transferable to the well‐known amyloid beta polypeptide Aβ_1–42. Moreover, for a selected central atom, the influence of its neighbors on its multipole moments is investigated, and how far away this influence can be ignored is also determined. Finally, the convergence behavior of AMBER becomes closer to that of QCT with increasing internuclear distance. © 2013 Wiley Periodicals, Inc.     

A finite expansion method for the calculation and interpretation of molecular electrostatic potentials

Because it is useful to have the molecular electrostatic potential as an element in a complex scheme to assess the toxicity of large molecules, efficient and reliable methods are needed for the calculation and characterization of these potentials. A multicenter multipole expansion of the molecular electron charge density calculated with a limited Gaussian basis set is shown here to have only a finite number of nonzero terms from which the molecular electrostatic potential can be calculated. The discrete contributions to the electrostatic potentials from the terms of this expansion provide a physically meaningful decomposition of the potential and a means for its characterization. With pyrrole as an example, the electrostatic potential calculated from this finite expansion of the electron density is compared to that obtained from exact calculations from the same wave function. Good agreement is obtained at distances greater than 1.5 A from any atom in the molecule. In contrast, rearrangement of the terms into an expansion corresponding only to Mulliken atomic charges and dipoles yields a decomposition that produces electrostatic potentials which agree less well with the exact potential. This discrepancy is attributable to the neglect of terms due to higher moments.     

Progress in the understanding of drug-receptor interactions, part 2: experimental and theoretical electrostatic moments and interaction energies of an angiotensin II receptor antagonist (C30H30N6(O)3S)

A combined experimental and theoretical charge density study of an angiotensin II receptor antagonist (1) is presented focusing on electrostatic properties such as atomic charges, molecular electric moments up to the fourth rank and energies of the intermolecular interactions, to gain an insight into the physical nature of the drug-receptor interaction. Electrostatic properties were derived from both the experimental electron density (multipole refinement of X-ray data collected at T=17 K) and the ab initio wavefunction (single molecule and fully periodic calculations at the DFT level). The relevance of SO and SN intramolecular interactions on the activity of 1 is highlighted by using both the crystal and gas-phase geometries and their electrostatic nature is documented by means of QTAIM atomic charges. The derived electrostatic properties are consistent with a nearly spherical electron density distribution, characterised by an intermingling of electropositive and -negative zones rather than by a unique electrophilic region opposed to a nucleophilic area. This makes the first molecular moment scarcely significant and ill-determined, whereas the second moment is large, significant and highly reliable. A comparison between experimental and theoretical components of the third electric moment shows a few discrepancies, whereas the agreement for the fourth electric moment is excellent. The most favourable intermolecular bond is show to be an NHN hydrogen bond with an energy of about 50 kJ mol(-1). Key pharmacophoric features responsible for attractive electrostatic interactions include CHX hydrogen bonds. It is shown that methyl and methylene groups, known to be essential for the biological activity of the drug, provide a significant energetic contribution to the total binding energy. Dispersive interactions are important at the thiophene and at both the phenyl fragments. The experimental estimates of the electrostatic contribution to the intermolecular interaction energies of six molecular pairs, obtained by a new model proposed by Spackman, predict the correct relative electrostatic energies with no exceptions.     

Counterion and composition effects on discotic nematic lyotropic liquid crystals II. Ion exchange and molecular dynamics

The static fluorescence quenching of pyrene by bromide, at the interface of mixed TTAC/TTAB discotic nematic lyotropic liquid crystals, allowed an estimation of the equilibrium constant for the exchange of chloride by bromide. The affinity of the interface for bromide is much higher than for chloride (K(Br-/Cl-) = 13.2). For a molecular level understanding of the experimental results of this and the preceding paper, 20 ns molecular dynamics (MD) simulations were calculated for samples with TTAB/TTAC molar percent ratios 100/0 (A), 50/50 (B) and 0/100 (C). The increment in the concentration of chloride induces a wider distribution of ammonium headgroups along the axis normal to the bilayer surface, increasing the width of the interface. The charge density profile of simulation B shows that the concentration of bromide is higher than the concentration of chloride in the vicinity the ammonium headgroups. The short range contribution to the electrostatic energy from the ammonium-ammonium repulsion is 291.7 kJ/mol for TTAC and 195.6 kJ/mol for TTAB, and the short range ammonium-halide interaction is -6166 kJ/mol for TTAC and -6607 kJ/mol for TTAB, from simulations A and C, respectively. These results are in agreement with a more neutralized TTAB interface. Consistently, the electric dipole moments of water are significantly more aligned with the larger electric field of the TTAB interface.     

Tryptophan side chain electrostatic interactions determine edge-to-face vs parallel-displaced tryptophan side chain geometries in the designed beta-hairpin "trpzip2"

The interaction geometries of the four tryptophan (Trp) side chains in the 12-residue designed beta-hairpin trpzip2 are investigated using all-atom explicit-solvent molecular dynamics simulations. The experimentally observed edge-to-face (EtF) pairwise interaction geometries are stable on a time scale of 10 ns. However, removing the electrostatic multipoles of the Trp side chains while retaining the dipoles of the side chains' NH moieties induces a conformational change to a geometry in which three of the four side chains interact in a parallel-displaced (PD) manner. Free energy simulations of the Etf to PD conformational change reveal that, with the side chain multipole moments intact (+MP), the EtF conformation is preferred by 5.79 kcal/mol. Conversely, with only the dipole moments of the side chain NH moieties intact (-MP), the PD conformation's free energy is more favorable by 1.71 kcal/mol. In contrast to energetic similarities for Trp side chain-water electrostatic and Trp side chain-Trp side chain and Trp side chain-water van der Waals, +MP Trp side chain-Trp side chain electrostatic interactions are more favorable by 4.21 kcal/mol in the EtF conformation, while in the -MP case the EtF and PD conformations' Trp side chain-Trp side chain electrostatic energies are nearly identical. The results highlight the importance of electrostatic multipole moments in determining aromatic-aromatic interaction geometries in aqueous biomolecular systems and argue for the inclusion of this physics in simplified models used for protein-ligand docking and protein structure prediction, possibly through a truncated Coulomb term between aromatic moieties.     

Atomic multipoles: Electrostatic potential fit,local reference axis systems,and conformational dependence

Currently, all standard force fields for biomolecular simulations use point charges to model intermolecular electrostatic interactions. This is a fast and simple approach but has deficiencies when the electrostatic potential (ESP) is compared to that from ab initio methods. Here, we show how atomic multipoles can be rigorously implemented into common biomolecular force fields. For this, a comprehensive set of local reference axis systems is introduced, which represents a universal solution for treating atom‐centered multipoles for all small organic molecules and proteins. Furthermore, we introduce a new method for fitting atomic multipole moments to the quantum mechanically derived ESP. This methods yields a 50–90% error reduction compared to both point charges fit to the ESP and multipoles directly calculated from the ab initio electron density. It is shown that it is necessary to directly fit the multipole moments of conformational ensembles to the ESP. Ignoring the conformational dependence or averaging over parameters from different conformations dramatically deteriorates the results obtained with atomic multipole moments, rendering multipoles worse than partial charges. © 2012 Wiley Periodicals, Inc.     

Cumulative potential-derived atomic multipoles

A method to derive the atomic multipole moments cumulatively up to quadrupole moments was developed. The multipole moments are obtained by least-square simulating the molecular electrostatic potentials. Only the components of the term of highest order in the atomic multipole expansion are optimized while the lower terms remain fixed. The calculations on HF, H₂O and NH₃ show that the cumulative method can give reasonable qualitative and fairly good quantitative results.     

Long-Range interaction coefficients between H,He+, He,and Li+

Using the new expression for the dynamic multipole polarizabilities in terms of effective oscillator strengths and effective transition energies derived from the variational procedure based on the total energy expansion in terms of induced multipole moments, we have calculated the long-range dispersion force coefficients and the leading relativistic correction to the long-range potential between H, He⁺, He, and Li⁺. We have in addition calculated the oscillator strength sum rules for the above systems. Our results are comparable with those obtained using both a hydrodynamic model to quantum mechanics and double perturbation theory.     

Partitioning of higher multipole polarizabilities: numerical evaluation of transferability

In this work, the partitioning of higher multipole polarizabilities, such as dipole-quadrupole, quadrupole-dipole, and quadrupole-quadrupole polarizabilities, into atomic contributions is studied. Partitioning of higher multipole polarizabilities is necessary in the study of accurate interaction energies where dispersion interactions are of importance. The fractional occupation Hirhsfeld-I (FOHI) method is used to calculate the atomic polarizabilities and is briefly explained together with the methodology for partitioning of the polarizabilities. The atomic multipole polarizabilities are calculated for different sets of molecules, linear alkanes, water clusters, and small organic molecules with different functional groups. It is found that the atomic and group contributions of the dipole and quadrupole polarizabilities are transferable as a function of the functional groups.