Auxiliary functions for molecular integrals with Slater‐type orbitals. II. Gauss transform methods

The Gauss transform of Slater‐type orbitals is used to express several types of molecular integrals involving these functions in terms of simple auxiliary functions. After reviewing this transform and the way it can be combined with the shift operator technique, a master formula for overlap integrals is derived and used to obtain multipolar moments associated to fragments of two‐center distributions and overlaps of derivatives of Slater functions. Moreover, it is proved that integrals involving two‐center distributions and irregular harmonics placed at arbitrary points (which determine the electrostatic potential, field and field gradient, as well as higher order derivatives of the potential) can be expressed in terms of auxiliary functions of the same type as those appearing in the overlap. The recurrence relations and series expansions of these functions are thoroughly studied, and algorithms for their calculation are presented. The usefulness and efficiency of this procedure are tested by developing two independent codes: one for the derivatives of the overlap integrals with respect to the centers of the functions, and another for derivatives of the potential (electrostatic field, field gradient, and so forth) at arbitrary points. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008     

Force related atomic multipoles in planar molecules. Derivation of atomic quadrupole and octupole moments

Atomic multipoles as defined by current methods generally do not account for forces in molecules that arise from external electrostatic fields. It is pointed out that such forces and the electrostatic potential that the molecule itself generates are both determined by the molecular multipolar tensors. The latter constitute therefore the fundamental molecular constants that determine the molecular electrostatics apart from polarization. In general the multipolar tensors include contributions from the atomic multipoles and their fluxes. In planar molecules, however, the perpendicular charge flux is zero by symmetry. This gives rise to a (previously introduced) formalism that extracts analytical, force-related, atomic multipoles from the molecular multipolar tensors. This formalism is extended in this work to include force-related (FR) atomic quadrupoles and octupoles in planar molecules. The properties of the FR atomic multipoles, including their perpendicular fluxes, are discussed and some formal theoretical and computational advantages that characterize them are indicated. As an example, the electrostatics of OCS, including the molecular electrostatic potential and the forces on the nuclei due to an external point charge, is discussed.     

Polarisable multipolar electrostatics from the machine learning method Kriging: an application to alanine

We present a polarisable multipolar interatomic electrostatic potential energy function for force fields and describe its application to the pilot molecule MeNH-Ala-COMe (AlaD). The total electrostatic energy associated with 1, 4 and higher interactions is partitioned into atomic contributions by application of quantum chemical topology (QCT). The exact atom–atom interaction is expressed in terms of atomic multipole moments. The machine learning method Kriging is used to model the dependence of these multipole moments on the conformation of the entire molecule. The resulting models are able to predict the QCT-partitioned multipole moments for arbitrary chemically relevant molecular geometries. The interaction energies between atoms are predicted for these geometries and compared to their true values. The computational expense of the procedure is compared to that of the point charge formalism.     

A new method to determine electrostatic potential around a macromolecule in solution from molecular wave functions

The three-dimensional reference interaction site model integral equation theory (3D-RISM) combined with the ab initio molecular orbital method (3D-RISM-SCF) is applied to a solvated macromolecular system. The solvation structure around a solute molecule is obtained from the 3D-RISM integral equation under the electrostatic potential of the solute molecule, calculated by the ab initio molecular orbital theory. The electrostatic potential should be calculated on each grid point in the three-dimensional real space. Therefore, the calculation of the electrostatic potential is the most time consuming part in this method. In this article, we propose a new procedure to save the computational cost for calculating the electrostatic potential and the solvated fock matrix. The strategy of this procedure is to evaluate the electrostatic potential and the solvated fock matrix in different ways, depending on the distance between solute and solvent. Inside the repulsive cores of solute atoms, it is possible to avoid the calculation of electrostatic potential and solvated Fock matrix by assuming the potential to be infinity. In the region sufficiently far from solute, they are evaluated classically by putting the effective point charge on each atom. In the intermediate region, the electrostatic potential is evaluated directly by integrating the molecular orbitals of the solute molecule. The electronic structure and the energy gradient of Methionine-Enkephalin and solvation structure are estimated by using this procedure in aqueous solution, and are compared with the results from other procedures. The results are compared also with those from the continuum model.     

Calculation of electrostatic interaction energies in molecular dimers from atomic multipole moments obtained by different methods of electron density partitioning

Accurate and fast evaluation of electrostatic interactions in molecular systems is still one of the most challenging tasks in the rapidly advancing field of macromolecular chemistry, including molecular recognition, protein modeling and drug design. One of the most convenient and accurate approaches is based on a Buckingham-type approximation that uses the multipole moment expansion of molecular/atomic charge distributions. In the mid-1980s it was shown that the pseudoatom model commonly used in experimental X-ray charge density studies can be easily combined with the Buckingham-type approach for calculation of electrostatic interactions, plus atom-atom potentials for evaluation of the total interaction energies in molecular systems. While many such studies have been reported, little attention has been paid to the accuracy of evaluation of the purely electrostatic interactions as errors may be absorbed in the semiempirical atom-atom potentials that have to be used to account for exchange repulsion and dispersion forces. This study is aimed at the evaluation of the accuracy of the calculation of electrostatic interaction energies with the Buckingham approach. To eliminate experimental uncertainties, the atomic moments are based on theoretical single-molecule electron densities calculated at various levels of theory. The electrostatic interaction energies for a total of 11 dimers of alpha-glycine, N-acetylglycine and L-(+)-lactic acid structures calculated according to Buckingham with pseudoatom, stockholder and atoms-in-molecules moments are compared with those evaluated with the Morokuma-Ziegler energy decomposition scheme. For alpha-glycine a comparison with direct "pixel-by-pixel" integration method, recently developed Gavezzotti, is also made. It is found that the theoretical pseudoatom moments combined with the Buckingham model do predict the correct relative electrostatic interactions energies, although the absolute interaction energies are underestimated in some cases. The good agreement between electrostatic interaction energies computed with Morokuma-Ziegler partitioning, Gavezzotti's method, and the Buckingham approach with atoms-in-molecules moments demonstrates that reliable and accurate evaluation of electrostatic interactions in molecular systems of considerable complexity is now feasible.     

Atomic multipoles and perpendicular electrostatic forces in diatomic and planar molecules

Current methods for assigning atomic multipoles focus on reproduction of the molecular electrostatic potential. Another aspect of electrostatic interaction, which is usually overlooked, is the forces that an external electric field exerts on the nuclei of a molecule. In a self-consistent theory, both the electrostatic potential and force should be accounted for. However, in general it is not easy to meet this requirement for the force. For planar molecules, though, a formal solution is available in terms of atomic multipoles that are extracted from the molecular multipolar tensors. These Force-Related (FR) atomic multipoles are discussed in detail for some typical diatomics and planar polyatomics, and are shown to provide a solid uniform framework for treating both aspects of the electrostatics. In contrast, the commonly used potential-derived charges (i.e., the atomic charges obtained by fitting the electrostatic potential) can yield large deviations with respect to electrostatic forces on the nuclei, even when the electrostatic potential is very well reproduced.     

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     

Electrostatic representations of molecular groups

The treatment of electrostatic interactions between molecular groups is based on quantum mechanical electron density distributions. A general method is suggested using tesseral harmonic multipole expansions and representing each set of point multipoles by a cluster of discrete charges small with respect to atomic dimensions. The procedure is carried out for the amino group with an electron density derived from the wave function of ammonia. We evaluate the optical rotation of two aminopyrrolidones as a test case. Expansions are compared for several approximate wave functions for ammonia.     

Electrostatic model for infrared intensities in a spectroscopically determined molecular mechanics force field

A new electrostatic model for the calculation of infrared intensities in molecular mechanics and molecular dynamics is presented. The model is based on atomic charges, atomic charge fluxes, and internal coordinate dipoles and their fluxes. The internal coordinate dipoles are used instead of atomic dipoles, thus simplifying the derivation of parameters. The model is designed to reproduce ab initio dipole derivatives, and the parameters can be obtained by (iterative) transformations from these, or by linear least squares fitting to them. A first application to linear alkanes has been made. For these molecules, the intensities can be predicted with an average accuracy of 30–40%. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 754–768, 1998     

An alternative multipolar expansion for intermolecular potential functions

We have derived a new multipolar expansion for intermolecular potential-energy functions with applications in molecular physics, theoretical chemistry, and mathematical physics. The new formulation employs a separation of radial and angular terms with a simple index structure that leads to computational efficiency and ease of physical interpretation. For the case of the Coulomb interaction, we compare the present formulation with two conventional multipole expansions: the Cartesian tensor and the irreducible spherical tensor expansions. The new formalism leads to efficient numerical algorithms that are useful for general applications beyond intermolecular potentials. In addition to the electrostatic Coulomb interaction, we illustrate the formalism with applications to special function theory and a bipolar expansion involved in potential theory.     

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.     

Fast tools for calculation of atomic charges well suited for drug design1

Two novel approaches to construct empirical schemes for partial atomic charge calculation were proposed. The charge schemes possess important benefits. First, they produce both topologically symmetrical and environment dependent charges. Second, they can be parameterised to reasonably reproduce ab initio molecular electrostatic potential (MEP), which guarantees their successful use in molecular modelling. To validate the approaches, the parameters of the proposed charge schemes were fitted to best reproduce MEP simultaneously on grids around a set of 227 diverse organic compounds. The residual errors in MEP reproduction due to calculated atomic charges were compared to those due to charges from known charge schemes.     

甲硫氨酸-脑啡肽的活性位点

用ＡＭ１方法中计算静电势的ＰＭＥＰ子程序了甲硫氨酸－脑啡肽的活性位点。通过计算得到了整个分子的三维空间静电势分布和由静电势导出的各原子的电荷分布。进一步分析,确定模型分子活性位点为酪氨酸残在的叔氨原子和苯酚基、苯丙氨酸残基的苯基及部分氧原子。     

Accurate molecular electrostatic potentials based on modified PRDDO/M wave functions. I. Electrostatic potential derived atomic charges

A new approach for the calculation of electrostatic potential derived atomic charges is presented. Based on molecular orbital calculations in the PRDDO/M approximation, the new parametrized electrostatic potential (PESP) method is parametrized against ab initio MP2/6-31G** calculations. For a data set of 820 atoms in 145 molecules containing H, C, N. O, F, P, S, Cl, and Br (including hypervalent species), the PESP method achieves a mean absolute error of 0.037 e⁻ with a correlation coefficient of 0.990. Unlike other approximate approaches, no scaling factor is required to improve the agreement between PESP charges and the underlying ab initio results. PESP calculations are an order of magnitude faster than the simplest ab initio calculation (STO-3G) on large molecules while achieving a level of accuracy that rivals much more elaborate ab initio methods. © 1997 by John Wiley & Sons, Inc. J Comput Chem 18: 955–969, 1997     

CoMFA Analysis of Biodegradability

Abstract

CoMFA (comparative molecular field analysis) has been used to correlate the biodegradability of several classes of compounds. The technique uses an atomic probe to detect the steric and electrostatic fields around a molecule. Good cross-validated correlations were obtained for some series (alcohols, carboxylic acids and linear alkyl benzene sulphonates). Correlations were weaker for esters and benzene sulphonates, and were non-existent for phenols. These results may reflect the unreliability of biodegradation data, but may also be a result of molecular misalignment in the CoMFA procedure. CoMFA appears to be a potentially very useful method for the prediction of biodegradability.     

Simulation of liquid imidazole using a high-rank quantum topological electrostatic potential

Rigid body molecular dynamics simulations were carried out on pure liquid imidazole at four different temperatures and at 1 atm. Imidazole, which is important both in life science and materials science, is one of the simplest molecules to possess both a lone pair and a π system. These two features are known to benefit from multipolar electrostatics. Here the electrostatic interaction is governed by atomic multipole moments obtained from topologically partitioned ab initio electron densities. The non-electrostatic terms are modeled with Lennard-Jones parameters adjusted to fit the experimental liquid density. All σ values are incrementally increased by one single scaling factor. We report on how the presence of multipolar electrostatics influences the local structure, dynamics and thermodynamics of the liquid compared to electrostatics by atomic point charges. The point charge force field exaggerates the number of π-stacked dimers in the liquid, and underestimates the number of hydrogen-bonded dimers. The effect of the temperature on the local structure of liquid imidazole was analysed using radial and spatial distribution functions.     

A combined electronegativity equalization and electrostatic potential fit method for the determination of atomic point charges

We report an approach for the determination of atomic monopoles of macromolecular systems using connectivity and geometry parameters alone. The method is appropriate also for the calculation of charge distributions based on the quantum mechanically determined wave function and does not suffer from the mathematical instability of other electrostatic potential fit methods.