Application of a simple quantum chemical approach to ligand fragment scoring for<Emphasis Type="Italic"> Trypanosoma brucei</Emphasis> pteridine reductase 1 inhibition

There is a need for improved and generally applicable scoring functions for fragment-based approaches to ligand design. Here, we evaluate the performance of a computationally efficient model for inhibitory activity estimation, which is composed only of multipole electrostatic energy and dispersion energy terms that approximate long-range ab initio quantum mechanical interaction energies. We find that computed energies correlate well with inhibitory activity for a compound series with varying substituents targeting two subpockets of the binding site of Trypanosoma brucei pteridine reductase 1. For one subpocket, we find that the model is more predictive for inhibitory activity than the ab initio interaction energy calculated at the MP2 level. Furthermore, the model is found to outperform a commonly used empirical scoring method. Finally, we show that the results for the two subpockets can be combined, which suggests that this simple nonempirical scoring function could be applied in fragment–based drug design.     

Accurate modeling of the intramolecular electrostatic energy of proteins

The (?, ψ) energy surface of blocked alanine (N-acetyl–N′-methyl alanineamide) was calculated at the Hartree-Fock (HF)/6-31G* level using ab initio molecular orbital theory. A collection of six electrostatic models was constructed, and the term electrostatic model was used to refer to (1) a set of atomic charge densities, each unable to deform with conformation; and (2) a rule for estimating the electrostatic interaction energy between a pair of atomic charge densities. In addition to two partial charge and three multipole electrostatic models, this collection includes one extremely detailed model, which we refer to as nonspherical CPK. For each of these six electrostatic models, parameters—in the form of partial charges, atomic multipoles, or generalized atomic densities—were calculated from the HF/6-31G* wave functions whose energies define the ab initio energy surface. This calculation of parameters was complicated by a problem that was found to originate from the locking in of a set of atomic charge densities, each of which contains a small polarization-induced deformation from its idealized unpolarized state. It was observed that the collective contribution of these small polarization-induced deformations to electrostatic energy differences between conformations can become large relative to ab initio energy differences between conformations. For each of the six electrostatic models, this contribution was reduced by an averaging of atomic charge densities (or electrostatic energy surfaces) over a large collection of conformations. The ab initio energy surface was used as a target with respect to which relative accuracies were determined for the six electrostatic models. A collection of 42 more complete molecular mechanics models was created by combining each of our six electrostatic models with a collection of seven models of repulsion + dispersion + intrinsic torsional energy, chosen to provide a representative sample of functional forms and parameter sets. A measure of distance was defined between model and ab initio energy surfaces; and distances were calculated for each of our 42 molecular mechanics models. For most of our 12 standard molecular mechanics models, the average error between model and ab initio energy surfaces is greater than 1.5 kcal/mol. This error is decreased by (1) careful treatment of the nonspherical nature of atomic charge densities, and (2) accurate representation of electrostatic interaction energies of types 1—2 and 1—3. This result suggests an electrostatic origin for at least part of the error between standard model and ab initio energy surfaces. Given the range of functional forms that is used by the current generation of protein potential functions, these errors cannot be corrected by compensating for errors in other energy components. © 1995 by John Wiley & Sons, Inc.     

Evaluation of the charge penetration energy between non-orthogonal molecular orbitals using the Spherical Gaussian Overlap approximation

An overlap dependent formula for evaluating the charge penetration energy between non-orthogonal molecular orbitals is derived using the Spherical Gaussian Overlap approximation. When combined with an accurate multipole representation of the electrostatic energy, such as in the effective fragment potential method, ab initio electrostatic energies are generally reproduced to within 0.2 kcal/mol for a variety of molecular dimers and basis sets. The only larger error is for the DMSO dimer, where the electrostatic energy is overestimated by 0.7 kcal/mol.     

Limitations of the molecular multipole expansion treatment of electrostatic interactions for C-H...O and O-H...O hydrogen bonds and application of a general charge density approach

A molecular multipole expansion treatment (up to hexadecapole) is examined for its accuracy in describing hydrogen-bond electrostatic interactions, with particular reference to explaining the differences between blue-shifted C-H...O and red-shifted O-H...O bonds. In interactions of H2O and CH4 with point charges at hydrogen-bonding distances, we find that the molecular multipole treatment not only fails to reproduce ab initio energies but also forces on OH or CH bonds, and therefore cannot properly account for the electrostatic component of the interaction. A treatment based on a molecule's permanent charge density and its derivatives and the charge density and its derivatives induced by an external multipole distribution is in full accord with ab initio results, as shown by application to models of the H2O-H2O and CH4-FH systems. Such a charge density approach provides a fundamental basis for understanding the importance of interaction forces in initiating structural change and thereby altering molecular properties.     

Accuracy of distributed multipoles and polarizabilities: comparison between the LoProp and MpProp models

Localized multipole moments up to the fifth moment as well as localized dipole polarizabilities are calculated with the MpProp and the newly developed LoProp methods for a total of 20 molecules, predominantly derived from amino acids. A comparison of electrostatic potentials calculated from the multipole expansion obtained by the two methods with ab initio results shows that both methods reproduce the electrostatic interaction with an elementary charge with a mean absolute error of approximately 1.5 kJ/mol at contact distance and less than 0.1 kJ/mol at distances 2 A further out when terms up to the octupole moments are included. The polarizabilities are tested with homogenous electric fields and are found to have similar accuracy. The MpProp method gives better multipole moments unless diffuse basis sets are used, whereas LoProp gives better polarizabilities.     

Atomic forces for geometry‐dependent point multipole and Gaussian multipole models

In standard treatments of atomic multipole models, interaction energies, total molecular forces, and total molecular torques are given for multipolar interactions between rigid molecules. However, if the molecules are assumed to be flexible, two additional multipolar atomic forces arise because of (1) the transfer of torque between neighboring atoms and (2) the dependence of multipole moment on internal geometry (bond lengths, bond angles, etc.) for geometry‐dependent multipole models. In this study, atomic force expressions for geometry‐dependent multipoles are presented for use in simulations of flexible molecules. The atomic forces are derived by first proposing a new general expression for Wigner function derivatives . The force equations can be applied to electrostatic models based on atomic point multipoles or Gaussian multipole charge density. Hydrogen‐bonded dimers are used to test the intermolecular electrostatic energies and atomic forces calculated by geometry‐dependent multipoles fit to the ab initio electrostatic potential. The electrostatic energies and forces are compared with their reference ab initio values. It is shown that both static and geometry‐dependent multipole models are able to reproduce total molecular forces and torques with respect to ab initio, whereas geometry‐dependent multipoles are needed to reproduce ab initio atomic forces. The expressions for atomic force can be used in simulations of flexible molecules with atomic multipoles. In addition, the results presented in this work should lead to further development of next generation force fields composed of geometry‐dependent multipole models. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Cumulative coordinates for approximations of high‐order atomic multipole moments in aluminosilicate and aluminophosphate sieves*

A scheme to obtain approximate analytical functions for the atomic distributed multipole moments of the crystallographically different atoms within aluminosilicate and aluminophosphate sieves is discussed. Respective atomic multipole moments are derived within the CRYSTAL95 ab initio periodic Hartree–Fock code with different basis sets, from minimal STO‐3G to 6‐21G*. In order to illustrate the possible applications, distributed analyses are carried out for various structural models from all‐siliceous zeolites and aluminophosphates with ratio Al/P=1 to hydrogen forms of aluminosilicates. Simple approximate forms based on charge and geometry coordinates are proposed for the high‐order moments of each atom, which are further required for the calculation of the electrostatic field within the structures. The possibility to use this analytical approach to evaluate the electrostatic field within embedded cluster models is also shortly discussed. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 70–85, 2001     

Polarizable Atomic Multipole-based Molecular Mechanics for Organic Molecules

An empirical potential based on permanent atomic multipoles and atomic induced dipoles is reported for alkanes, alcohols, amines, sulfides, aldehydes, carboxylic acids, amides, aromatics and other small organic molecules. Permanent atomic multipole moments through quadrupole moments have been derived from gas phase ab initio molecular orbital calculations. The van der Waals parameters are obtained by fitting to gas phase homodimer QM energies and structures, as well as experimental densities and heats of vaporization of neat liquids. As a validation, the hydrogen bonding energies and structures of gas phase heterodimers with water are evaluated using the resulting potential. For 32 homo- and heterodimers, the association energy agrees with ab initio results to within 0.4 kcal/mol. The RMS deviation of hydrogen bond distance from QM optimized geometry is less than 0.06 ?. In addition, liquid self-diffusion and static dielectric constants computed from molecular dynamics simulation are consistent with experimental values. The force field is also used to compute the solvation free energy of 27 compounds not included in the parameterization process, with a RMS error of 0.69 kcal/mol. The results obtained in this study suggest the AMOEBA force field performs well across different environments and phases. The key algorithms involved in the electrostatic model and a protocol for developing parameters are detailed to facilitate extension to additional molecular systems.     

Transferable atom equivalent multicentered multipole expansion method

The transferability of atomic and functional group properties is an implicit concept in chemistry. The work presented here describes the use of Transferable Atom Equivalents (TAE) to represent molecular electrostatic potential fields through the use of integrated atomic multipole moments that are associated with each TAE atom type used in the reconstruction. TAE molecular surface distributions of electrostatic potentials are compared with analytical ab initio and empirical (Gasteiger) partial charge reference models for several conformations of test peptides. Surface electrostatic potential distributions computed using TAE multipole representations were found to converge at the octopole level, with incremental improvement observed when hexadecapoles were included. Molecular electrostatic potential fields that were produced using the TAE method were observed to be responsive to conformational changes and to compare well with ab initio reference distributions. Generation of TAE atom types and their associated multipoles does not involve fitting to sample electrostatic potential fields, but rather utilizes integrated AIM atomic electron density distributions within representative chemical environments. The RECON program was used for TAE reconstruction. RECON is capable of processing 5,000 drug-sized molecules or 25 proteins per minute per 1.7 GHz P4 Linux processor.     

Comparison of charge models for fixed-charge force fields: small-molecule hydration free energies in explicit solvent

In molecular simulations with fixed-charge force fields, the choice of partial atomic charges influences numerous computed physical properties, including binding free energies. Many molecular mechanics force fields specify how nonbonded parameters should be determined, but various choices are often available for how these charges are to be determined for arbitrary small molecules. Here, we compute hydration free energies for a set of 44 small, neutral molecules in two different explicit water models (TIP3P and TIP4P-Ew) to examine the influence of charge model on agreement with experiment. Using the AMBER GAFF force field for nonbonded parameters, we test several different methods for obtaining partial atomic charges, including two fast methods exploiting semiempirical quantum calculations and methods deriving charges from the electrostatic potentials computed with several different levels of ab initio quantum calculations with and without a continuum reaction field treatment of solvent. We find that the best charge sets give a root-mean-square error from experiment of roughly 1 kcal/mol. Surprisingly, agreement with experimental hydration free energies does not increase substantially with increasing level of quantum theory, even when the quantum calculations are performed with a reaction field treatment to better model the aqueous phase. We also find that the semiempirical AM1-BCC method for computing charges works almost as well as any of the more computationally expensive ab initio methods and that the root-mean-square error reported here is similar to that for implicit solvent models reported in the literature. Further, we find that the discrepancy with experimental hydration free energies grows substantially with the polarity of the compound, as does its variation across theory levels.     

Interaction energy of a water molecule with a single-layer graphitic surface modeled by hydrogen- and fluorine-terminated clusters

In ab initio calculations a finite graphitic cluster model is often used to approximate the interaction energy of a water molecule with an infinite single-layer graphitic surface (graphene). In previous studies, the graphitic cluster model is a collection of fused benzene rings terminated by hydrogen atoms. In this study, the effect of using fluorine instead of hydrogen atoms for terminating the cluster model is examined to clarify the role of the boundary. The interaction energy of a water molecule with the graphitic cluster was computed using ab initio methods at the MP2 level of theory and with the 6-31G(d = 0.25) basis set. The interaction energy of a water molecule with graphene is estimated by extrapolation of two series of increasing size graphitic cluster models (C(6n2)H(6n) and C(6n2)F(6n), n = 1-3). Two fixed orientations of water molecule are considered: (a) both hydrogen atoms of water pointing toward the cluster (mode A) and (b) both hydrogen atoms of water pointing away from the cluster (mode B). The interaction energies for water mode A are found to be -2.39 and -2.49 kcal/mol for C(6n2)H(6n) and C(6n2)F(6n) cluster models, respectively. For water mode B, the interaction energies are -2.32 and -2.44 kcal/mol for C(6n2)H(6n) and C(6n2)F(6n) cluster models, respectively.     

Kinetics and Thermodynamics of Reactions Involving Criegee Intermediates: An Assessment of Density Functional Theory and Ab Initio Methods Through Comparison with CCSDT(Q)/CBS Data

Reactions involving Criegee intermediates (CIs, R₁R₂COO) are important in atmospheric ozonolysis models. In recent years, density functional theory (DFT) and CCSD(T)-based ab initio methods are increasingly being used for modeling reaction profiles involving CIs. We obtain highly accurate CCSDT(Q)/CBS reaction energies and barrier heights for ring-closing reactions involving atmospherically important CIs (R₁/R₂ = H, Me, OH, OMe, F, CN, cyclopropene, ethylene, acetaldehyde, and acrolein). We use this benchmark data to evaluate the performance of DFT, double-hybrid DFT (DHDFT), and ab initio methods for the kinetics and thermodynamics of these reactions. We find that reaction energies are more challenging for approximate theoretical procedures than barrier heights. Overall, taking both reaction energies and barrier heights into account, only one of the 58 considered DFT methods (the meta-GGA MN12-L) attains near chemical accuracy, with root-mean-square deviations (RMSDs) of 3.5 (barrier heights) and 4.7 (reaction energies) kJ mol⁻¹. Therefore, MN12-L is recommended for investigations where CCSD(T)-based methods are not computationally feasible. For reaction barrier heights performance does not strictly follow Jacob's Ladder, for example, DHDFT methods do not perform better than conventional DFT methods. Of the ab initio methods, the cost-effective CCSD(T)/CBS(MP2) approach gives the best performance for both reaction energies and barrier heights, with RMSDs of 1.7 and 1.4 kJ mol⁻¹, respectively. All the considered Gaussian-n methods show good performance with RMSDs below the threshold of chemical accuracy for both reaction energies and barrier heights, where G4(MP2) shows the best overall performance with RMSDs of 2.9 and 1.5 kJ mol⁻¹, respectively. © 2019 Wiley Periodicals, Inc.     

First-principles calculation of local atomic polarizabilities

Common methods of determining atomic polarizabilities suffer from the inclusion of nonlocal effects such as charge polarization. A new method is described for determining fully ab initio atomic polarizabilities based on calculating the response of atomic multipoles to the local electrostatic potential. The localized atomic polarizabilities are then used to calculate induction energies that are compared to ab initio induction energies to test their usefulness in practical applications. These polarizabilities are shown to be an improvement over the corresponding molecular polarizabilities, in terms of both absolute accuracy and the convergence of the multipolar induction series. The transferability of localized polarizabilities for the alkane series is also discussed.     

Theory and range of modern semiempirical molecular orbital methods

Semiempirical molecular orbital methods have a long history. They serve to tackle large systems and complicated processes beyond the reach of ab initio or density functional methods. Although their setup is derived from Hartree–Fock theory, the design of approximate energy expressions and the empirical parameters are used to achieve higher accuracy than the underlying ab initio theory. In this way the effect of larger basis sets or correlation can be partially simulated. All widely used semiempirical methods establish their accuracy by error statistics for molecular properties with experimental and high-level ab initio or density functional theory calculations as a reference. Their computational efficiency makes them suitable for the study of biochemical systems and solid materials. The present review presents a variety of applications which demonstrate the need for and success of semiempirical methods.     

A finite field method for calculating molecular polarizability tensors for arbitrary multipole rank

A finite field method for calculating spherical tensor molecular polarizability tensors α_lm;l′m′ = ?Δ_lm/??_l′m′* by numerical derivatives of induced molecular multipole Δ_lm with respect to gradients of electrostatic potential ?_l′m′* is described for arbitrary multipole ranks l and l′. Interconversion formulae for transforming multipole moments and polarizability tensors between spherical and traceless Cartesian tensor conventions are derived. As an example, molecular polarizability tensors up to the hexadecapole–hexadecapole level are calculated for water using the following ab initio methods: Hartree–Fock (HF), Becke three‐parameter Lee‐Yang‐Parr exchange‐correlation functional (B3LYP), Møller–Plesset perturbation theory up to second order (MP2), and Coupled Cluster theory with single and double excitations (CCSD). In addition, intermolecular electrostatic and polarization energies calculated by molecular multipoles and polarizability tensors are compared with ab initio reference values calculated by the Reduced Variation Space method for several randomly oriented small molecule dimers separated by a large distance. It is discussed how higher order molecular polarizability tensors can be used as a tool for testing and developing new polarization models for future force fields. © 2011 Wiley Periodicals, Inc. J Comput Chem, 2011     

Challenges of crystal structure prediction of diastereomeric salt pairs

A methodology for the computational prediction of the crystal structures and resolution efficiency for diastereomeric salt pairs is developed by considering the polymorphic system of the diastereomeric salt pair (R)-1-phenylethylammonium (R/S)-2-phenylpropanoate. To alleviate the mathematical complexity of the search for minima in the lattice energy due to the presence of two flexible entities in the asymmetric unit, the range of rigid-body lattice energy global optimizations was guided by a statistical analysis of the Cambridge Structural Database for common ion-pair geometries and ion conformations. A distributed multipole model for the dominant electrostatic interactions and high-level ab initio calculations for the intramolecular energy penalty for conformational distortions are used to quantify the relative stabilities of the p- and n-salt forms. While the ab initio prediction of the known structure of the p-salt as the most stable structure was insensitive to minor changes in the rigid-ion conformations considered, the relative stabilities of the known polymorphs and hypothetical structures of the n-salt were very sensitive. Although this paper provides a significant advance over traditional search algorithms and empirical force fields in determining the structures and relative stabilities of diastereomeric salt pairs, the sensitivity of the computed lattice energies to the fine details of the ion conformations overtaxes current computational models and renders the design of diastereomeric resolution processes by computational chemistry a challenging problem.     

Ab initio melting curve of copper by the phase coexistence approach

Ab initio calculations of the melting properties of copper in the pressure range 0-100 GPa are reported. The ab initio total energies and ionic forces of systems representing solid and liquid copper are calculated using the projector augmented wave implementation of density functional theory with the generalized gradient approximation for exchange-correlation energy. An initial approximation to the melting curve is obtained using an empirical reference system based on the embedded-atom model, points on the curve being determined by simulations in which solid and liquid coexist. The approximate melting curve so obtained is corrected using calculated free energy differences between the reference and ab initio system. It is shown that for system-size errors to be rendered negligible in this scheme, careful tuning of the reference system to reproduce ab initio energies is essential. The final melting curve is in satisfactory agreement with extrapolated experimental data available up to 20 GPa, and supports the validity of previous calculations of the melting curve up to 100 GPa.     

Gaussian multipoles in practice: Electrostatic energies for intermolecular potentials

A method is presented for calculating the total electrostatic interaction energies between molecules from ab initio monomer wave functions. This approach differs from existing methods, such as Stone's distributed multipole analysis (DMA), in including the short-range penetration energy as well as the long-range multipolar energy. The monomer charge densities are expressed as distributed series of atom-centered functions which we call Gaussian multipoles; these are analogous to the distributed point multipoles used in DMA. Our procedure has been encoded in the GMUL program. Calculations have been performed on the formamide/formaldehyde complex, a model system for N? H …? O hydrogen bonding in biological molecules, and also on guanidinium/benzene, modeling amino/aromatic interactions in proteins. We find that the penetration energy can be significant, especially in its contribution to the variation of the electrostatic energy with interaction geometry. A hybrid method, which uses Gaussian multipoles for short-range atom pair interactions and point multipoles for long-range ones, allows the electrostatic energies, including penetration, to be calculated at a much reduced cost. We also note that the penetration energy may provide the best route to an atom–atom anisotropic model for the exchange-repulsion energy in intermolecular potentials. © 1994 by John Wiley & Sons, Inc.     

Vibrational energies and thermal expansion of layered compounds: MgCl2

Full dynamic free energy minimisation is used to study the stability of different polymorphs of MgCl₂, with a new set of interionic potentials derived from ab initio calculations using density functional theory. The calculated difference in free energies between the - and β-phases is extremely small: the vibrational contribution reverses the relative ordering of these two structures obtained in the static limit. We predict the thermal expansion of both forms is highly anisotropic and that, unusually, above ≈60 K the expansion is larger parallel than perpendicular to the layers.