Distributed Multipole Research Articles

A classical model for three-body interactions in aqueous ionic systems.

We present a classical induction model to evaluate the three-body ion-water-water (I-W-W) and water-water-water (W-W-W) interactions in aqueous ionic systems. The classical description of the induction energy is based on electrostatic distributed multipoles up to hexadecapole and distributed polarizabilities up to quadrupole-quadrupole on the O and H atoms of water. The monatomic ions were described by a point charge and a dipole-dipole polarizability, while for the polyatomic ions, distributed multipoles up to hexadecapole and distributed polarizabilities up to quadrupole-quadrupole were used. The accuracy of the classical model is benchmarked against an accurate dataset of 936 (I-W-W) and 2184 (W-W-W) three-body terms for 13 different monatomic and polyatomic cation and anion systems. The classical model shows excellent agreement with the reference second order Moller-Plesset and coupled-cluster single double and perturbative triple [CCSD(T)] three-body energies. The Root-Mean-Square-Errors (RMSEs) for monatomic cations, monatomic anions, and polyatomic ions were 0.29, 0.25, and 0.12 kcal/mol, respectively. The corresponding RMSE for 1744 CCSD(T)/aVTZ three-body (W-W-W) energies, used to train MB-pol, was 0.12 kcal/mol. The accuracy of the proposed classical model demonstrates that the three-body term for aqueous ionic systems can be accurately modeled classically. This approach provides a fast, efficient, and as-accurate path toward modeling the three-body term in aqueous ionic systems that is fully transferable across systems with different ions without the need to fit to tens of thousands of ab initio calculations for each ion to extend existing many-body force fields to interactions between water and ions.

Substituent-Modulated Affinities of Halobenzene Derivatives to the HIV-1 Integrase Recognition Site. Analyses of the Interaction Energies by Parallel Quantum Chemical and Polarizable Molecular Mechanics

The C-X bond of halobenzenes (X = Cl, Br) has a dual character, its electron density being depleted in its prolongation and built-up on its sides. We have recently considered three protein or nucleic acid recognition sites of halobenzenes and quantified the energy gains that either electron-attracting substituents or electron-donating ones contribute due to such a character (El Hage et al., paper in revision). Nonadditivity was found to impact the total interaction energies. We focus here on one recognition site, that of the HIV-1 integrase, in which the halobenzene ring of the drug elvitegravir is sandwiched between a guanine and a cytosine base. We perform energy-decomposition analyses of the ab initio quantum-chemistry (QC) binding energies of the parent halobenzene ring and its derivatives with this G-C base pair. In these complexes, the nonadditivity of ΔE could be traced back mostly to the polarization contribution Epol. In view of large-scale applications to the entirety of the complex formed between the integrase, the viral DNA, and the whole drug, the analyses were performed in parallel with a polarizable molecular mechanics method, SIBFA. This method could faithfully reproduce most features of the QC energies. This is due to its use of QC-derived distributed multipoles and polarizabilities, which enable us to account for both nonisotropy and nonadditivity.

Control system design and input shape for orientation of spherical wheel motor

This paper presents control system design of a multi degrees-of-freedom (DOF) spherical wheel motor (SWM) in a class of ball-joint-like direct drive actuators to control orientation of the shaft. Three controllers (model based open-loop (OL), two closed-loop (CL) controllers) based on a push-pull torque model have been developed from rotor dynamics and magnetic field model referred to here as Distributed Multipole (DMP) model which provides accurate torque computation. The model based OL controller along with three control input shapes has been examined for the inclination control. Their results offer physical intuition, practical effectiveness, and also demonstrate the accuracy of magnetic field and torque computation. Then, two feedback controllers, a PD controller with and without the observer, have been developed for regulating its rotor inclination and experimentally evaluated against the OL controller. Finally, the performance on each controller has been compared to show the effect of the controllers on transient response. The experimental results verify control system design and demonstrate the motion capability of the SWM. While the experimental results illustrate the ability to control, they also reveal constraints and limitations of the controllers and provide insights for future design of control systems for the SWM.

Design of multi-DOF electromagnetic actuators using distributed multipole models and image method

Design and control of many electromagnetic actuators involve solving three dimensional (3D) magnetic fields of permanent magnets and/or electromagnets in the presence of magnetic conducting surfaces. This paper extends the distributed multipole (DMP) method, which offers compact but precise analysis for three dimensional magnetic fields in closed form, to account for the effects of magnetic conducting boundaries using the image method on the torque generated by electromagnetic actuators. We validate the proposed method referred to here as DMP-Image method by comparing the calculated torques against results computed by a finite element method (FEM). Wh ile two methods agree to within 5% in maximum torque, the DMP-image method takes less than 1% of the FEM computation time. Finally, we demonstrate the DMP-Image method to design a spherical motor in a class of multi-DOF actuators. While developed in the context of the multi-DOF actuators, the modeling methods presented in this paper are applicable to design of other PM-based actuators.

Two-DOF magnetic orientation sensor using distributed multipole models for spherical wheel motor

This paper presents a new method for measuring a two degree-of-freedom (DOF) orientation of a permanent magnet (PM) based system using magnetic field measurements. The method exploits distributed multipole (DMP) modeling method to accurately predict a magnetic field, and provides a rational basis to inversely solve for the orientation of the PM from measured data. The PM-based magnetic sensor along with the ability to characterize the magnetic field in real-time offers advantages in sensing and control such as contact-free measurements eliminating frictional wears commonly encountered in existing designs with a combination of single-axis encoders, and high-speed sampling rate thus offering a higher bandwidth than methods based on imaging sensors. This paper demonstrates the efficient method capable of measuring the orientation of the PM by implementing it on a spherical wheel motor (SWM), where the two-DOF orientation is measured. Sensor performance has been studied both analytically and experimentally to validate the DMP-based sensor model. The results can offer valuable insights for optimizing contact-free sensor designs.

Multipole electrostatics in hydration free energy calculations

Hydration free energy (HFE) is generally used for evaluating molecular solubility, which is an important property for pharmaceutical and chemical engineering processes. Accurately predicting HFE is also recognized as one fundamental capability of molecular mechanics force field. Here, we present a systematic investigation on HFE calculations with AMOEBA polarizable force field at various parameterization and simulation conditions. The HFEs of seven small organic molecules have been obtained alchemically using the Bennett Acceptance Ratio method. We have compared two approaches to derive the atomic multipoles from quantum mechanical calculations: one directly from the new distributed multipole analysis and the other involving fitting to the electrostatic potential around the molecules. Wave functions solved at the MP2 level with four basis sets (6-311G*, 6-311++G(2d,2p), cc-pVTZ, and aug-cc-pVTZ) are used to derive the atomic multipoles. HFEs from all four basis sets show a reasonable agreement with experimental data (root mean square error 0.63 kcal/mol for aug-cc-pVTZ). We conclude that aug-cc-pVTZ gives the best performance when used with AMOEBA, and 6-311++G(2d,2p) is comparable but more efficient for larger systems. The results suggest that the inclusion of diffuse basis functions is important for capturing intermolecular interactions. The effect of long-range correction to van der Waals interaction on the hydration free energies is about 0.1 kcal/mol when the cutoff is 12Å, and increases linearly with the number of atoms in the solute/ligand. In addition, we also discussed the results from a hybrid approach that combines polarizable solute with fixed-charge water in the HFE calculation.

The Rankwise Distributed Multipole Analysis (RWDMA) of the Electrostatic Field of Large Biomolecules

Electrostatic interactions play an essential role in many molecular processes in living organisms. However, given the large size of the macromolecules typically involved in such processes, the accurate representation of the electrostatic potential is difficult to achieve in simple and computationally efficient ways. Among the methods used to reduce the complexity of such models, the multipolar expansions provide a systematic method to separate essential features of the electrostatic field according to spatial scale.

Distributed Multipole Model for Design of Permanent-Magnet-Based Actuators

This paper presents a general method for deriving a closed-form solution for precise calculation of the magnetic field around a permanent magnet (PM) or an electromagnet (EM). The method, referred here as distributed multipole (DMP) modeling, inherits many advantages of the dipole model originally conceptualized in the context of physics, but provides an effective means to account for the shape and magnetization of the physical magnet. Three examples are given to illustrate the procedure of developing a DMP model, which derives an appropriate set of distributed dipoles from a limited set of known field points, for a general cylindrical PM, a customized PM, and a multilayer coil. The DMP modeling method has been validated by comparing simulated fields and calculated forces against data obtained experimentally and numerically; the comparisons show excellent agreement. Finally, we illustrate how the closed-form DMP models can offer an inexpensive means to visualize the effect of the EM fields on the leakage and unexpected flux paths, which have significant influences on the magnetic torque of a spherical motor.

Use of X-ray Charge Densities in the Calculation of Intermolecular Interactions and Lattice Energies: Application to Glycylglycine,dl-Histidine, anddl-Proline and Comparison with Theory

Experimental X-ray charge densities from low-temperature data are used in the evaluation of the intermolecular interactions and lattice energies of crystals of glycylglycine, dl-histidine, and dl-proline. The X-ray analysis leads to a set of atom-centered distributed multipoles, from which electrostatic interactions are calculated. Nonempirical exp-6 atom−atom potentials are used to calculate the smaller contributions of van der Waals interactions. For comparison, parallel theoretical calculations are performed on the molecular dimers (B3LYP) and the periodic crystals (Periodic Hartree−Fock, PHF). The dimer interactions show good agreement with experimental values, except for the strongest interactions in the glycylglycine crystal. The experimental charge density results correlate well with those based on the PHF calculations, but quantitative agreement for the interaction energies is only obtained after application of a scaling factor of ∼0.76 to the PHF values. The discrepancy is attributed to the well-...

Toward Improved Force Fields. 2. Effective Distributed Multipoles

This is the second of a series of papers discussing the possibility of separating and accurately calculating electrostatic and polarization energies in simulations using classical force fields. A method is described for determining a set of effective distributed multipoles which have significantly improved convergence properties in evaluating the electrostatic interaction energy between molecules. These fitted multipoles are derived to reproduce the electrostatic potential and its derivatives as calculated from a full distributed multipole analysis. The method is based on previous work on the determination of multipole-fitted charges (Ferenczy, G. G. J. Comput. Chem. 1991, 12, 913; Chipot et al. J. Phys. Chem. 1993, 97, 6628) and does not involve the use of a numerical grid. In applications on model systems, fitted charges and dipoles are able to reproduce both the interaction energy and the optimized geometry obtained from a full distributed multipole analysis. Potential-derived charges, however, result ...

Multipole-based calculation of the polarization energy

A novel method for calculating the polarization energy of a biomolecular complex is developed. It is based on direct calculation of the interaction between the unperturbed charge distribution of one molecule and the perturbed charge distribution of the other. This “multipole-based polarization energy” is shown to include charge transfer. The method can alternatively incorporate Gaussian multipoles in place of the usual distributed point multipoles. This definition of the polarization energy is also used as the basis of a partitioning scheme for the total supermolecule interaction energy. Results are presented for the formamide/formaldehyde complex.

A perturbation theory study of adlayer CO on NaCl(100)

We have constructed a potential energy function to describe the interaction of CO molecules with each other and with the (100) surface of NaCl. It uses distributed multipoles to describe the electrostatic interactions, distributed polarizabilities to describe induction, anisotropic dispersion coefficients taken from work by Rijks and Wormer, and anisotropic atom–atom repulsion terms derived from ab initio intermolecular perturbation theory. We have also investigated the structure and charge distribution of the NaCl surface. We find that an isolated CO molecule on the NaCl(100) surface adopts a position with the CO axis perpendicular to the surface and the C atom over a surface Na+ ion, but in monolayers of CO the molecules are tilted by about 28°. There are two 1×1 and two 2×1 structures, all with very similar energies; a further 1×1 structure with the molecules perpendicular to the surface is an index-2 stationary point with energy only 32 cm−1 above the lowest-energy structure. The results are consistent with the experimental infrared spectra and with the measured enthalpy of adsorption.

Applications of realistic electrostatic modelling to molecules in complexes, solids and proteins

The forces between molecules reflect their charge distributions, and thus the ab initio wavefunctions of the isolated molecules can be used to predict the intermolecular potential. The electrostatic forces can be readily represented by sets of atomic multipoles, and such distributed multipole models have been widely used for accurate studies of the electrostatic interactions of organic molecules. These models automatically incorporate the effects of lone pair and π electron density, short range inductive and other environmental effects on the intermolecular interactions, in a way that corresponds to chemical intuition. Studies using these realistic electrostatic models have provided many insights into the structures of van der Waals complexes, molecular crystal structures and protein–ligand interactions, as well as pointing the way forward to reliable model intermolecular potentials for future simulation studies.

A new distributed multipole procedure for linear molecules

A point multipole can be represented using multipoles at different positions by dividing it into a number of fractions and moving one to each new position. It is shown that there exists a choice for these fractions which is optimal and unique within certain constraints. Distributed multipoles for any linear molecule can readily be obtained in this way. Results for HF show that convergence of the electrostatic energy is faster than using the conventional DMA procedure, with less dependence of the multipoles on basis set.

Central or distributed multipole moments? Electrostatic models of aromatic dimers

Multipole moments up to rank six are calculated at the SCF level for sym-triazine and 1,3,5-trifluorobenzene. Staggered face-to-face configurations of the homodimers are stabilized by both electrostatic and dispersion interactions. Low-order central multipole models fail to show the electrostatic attraction; high-order central multipole expansions are strongly divergent. Distributed multipoles derived either by fitting to ab initio central multipoles, or by analysis of the electron density, predict the correct orientational minimum and give reasonable estimates of electrostatic interaction energies.