Utilization of shape data in molecular mechanics using a potential based on spherical harmonic surfaces

This article introduces a novel potential function that allows the use of topographical information in molecular modeling. Quantitative shape data are provided by techniques such as electron-microscopy–based three-dimensional image reconstruction for large macromolecular assemblies. Such data can provide important constraints for molecular mechanics. We represent topographical data by spherical harmonic surfaces, first used by Max and Getzoff²¹ for displaying molecular surfaces. A simple harmonic potential is used to constrain atoms within these spherical harmonic surfaces. This potential was implemented in the yammp molecular mechanics package.²⁷ Implementation details and results of several test cases are discussed here. © 1994 by John Wiley & Sons, Inc.     

Toward high throughput 3D virtual screening using spherical harmonic surface representations

Searching chemical databases for possible drug leads is often one of the main activities conducted during the early stages of a drug development project. This article shows that spherical harmonic molecular shape representations provide a powerful way to search and cluster small-molecule databases rapidly and accurately. Our clustering results show that chemically meaningful clusters may be obtained using only low order spherical harmonic expansions. Our database search results show that using low order spherical harmonic shape-based correlation techniques could provide a practical and efficient way to search very large 3D molecular databases, hence leading to a useful new approach for high throughput 3D virtual screening. The approach described is currently being extended to allow the rapid search and comparison of arbitrary combinations of molecular surface properties.     

The ellipsoidal Gaussian basis in molecular orbital theory

The ellipsoidal Gaussian basis function used in a minimal valence atomic orbital representation is compared with the double-zeta spherical Gaussian basis orbital representation for some seventeen molecules made up of first row atoms and hydrogen. Except for acetylene the double-zeta basis gives consistently better total electronic energies and generally better property values than the optimized ellipsoidal single zeta basis. Difference molecular density contour maps comparing the two basis sets, as well as other one-electron property values, indicate that the ellipsoidal basis exaggerates the transfer of charge from the atomic regions to the interatomic and lone pair regions of molecules. Apparently, the forced complete elliptization of the valence atomic orbital in the single-zeta representation does not allow the basis set sufficient flexibility to simultaneously represent both the basically spherical atomic part of these orbitals and the non-spherical molecular bond formation. Other properties and aspects of the ellipsoidal Gaussian basis are also discussed.     

Evaluation of integrals over STOs on different centers and the complementary convergence characteristics of ellipsoidal‐coordinate and zeta‐function expansions

One‐electron integrals over three centers and two‐electron integrals over two centers, involving Slater‐type orbitals (STOs), can be evaluated using either an infinite expansion for 1/r₁₂ within an ellipsoidal‐coordinate system or by employing a one‐center expansion in spherical‐harmonic and zeta‐function products. It is shown that the convergence characteristics of both methods are complimentary and that they must both be used if STOs are to be used as basis functions in ab initio calculations. To date, reports dealing with STO integration strategies have dealt exclusively with one method or the other. While the ellipsoidal method is faster, it does not always converge to a satisfactory degree of precision. The zeta‐function method, however, offers reliability at the expense of speed. Both procedures are described and the results of some sample calculation presented. Possible applications for the procedures are also discussed. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 71: 1–13, 1999     

New ab initio potential energy surface for the (HOCO+-He) van der Waals complex

A three-dimensional potential energy surface has been calculated for the ground electronic state of the HOCO+-He system. The calculations were performed at the coupled electron pair approximation level with an extended basis set which ensures a balance between accuracy and feasability. The validity of the method and of the basis set was tested through calculations of the polarizability of the He atom and of the spectroscopic constants of the HOCO+ ion. The calculated potential energy surface has been fitted to a spherical harmonic expansion to facilitate calculations of rotational excitation of HOCO+ by collisions with He.     

Approximate analytical expressions for the electrical potential between two planar, cylindrical, and spherical surfaces

Approximate analytical expressions for the electrical potential of planar, cylindrical, and spherical surfaces are derived for the case in which the dispersion medium contains counterions only. On the basis of the results for single surfaces, those for two identical surfaces can be derived. The curvature effect of a surface on the electrical potential distribution can be neglected when the order of its radius exceeds approximately 100 times the thickness of the corresponding double layer. If this effect needs to be considered, it can be taken into account by multiplying a correction function by the electrical potential of a planar surface. The electrical potential at the center between two derived surfaces is readily applicable to the evaluation of the electrostatic force per unit area between two surfaces, or the osmotic pressure. For the same set of parameters, the magnitudes of the osmotic pressure for various types of surfaces rank as follows: planar surface > cylindrical surfaces > spherical surfaces.     

Calculation of molecular integrals for systems confined by hard spherical walls: Use of the single‐center expansion of floating spherical gaussians

Assuming a gaussian basis set representation of atomic and molecular wave functions, the single‐center expansion of off‐centered spherical gaussian orbitals is exploited to calculate the one and two‐electron integrals for multielectronic atoms and molecules confined within hard spherical walls. As a validating test, the ground‐state energy of a helium atom positioned off‐center in a spherical box is calculated by applying the simplest form of the floating spherical gaussian orbital (FSGO) scheme, i.e., the use of a primitive basis set consisting of a single FSGO per electron pair. Comparison with corresponding recent accurate calculations gives supporting evidence of the adequacy of the method for its application to more elaborate gaussian‐type basis set representations for confined atoms and molecules. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 83: 271–278, 2001     

Molecular Simulations using Spherical Harmonics

Computer-aided drug design is to develop a chemical that binds to a target macromolecule known to play a key role in a disease state. In recognition of ligands by their protein receptors, molecular surfaces are often used because they represent the in-teracting part of molecules and they should reflex the comple-mentarity between ligand and receptor. However, assessing the surface complementarity by searching all relative position of two surfaces is often computationally expensive. The comple-mentarity of lobe-hole is very important in protein-ligand inter-actions. Spherical harmonic models based on expansions of spherical harmonic functions were used as a f‘mgerprint to ap-proximate the binding cavity and the ligand, respectively. This defines a new way to identify the complementarity between lobes and holes. The advantage of this method is that two spherical harmonic surfaces to be compared can be defined sep-arately. This method can be used as a filter to eliminate candi-dates among a large number of conformations, and it will speed up the docking procedure. Therefore, it is possible to select complementary ligands or complementary conformations of a ligand and the macromolecules, by comparing their fingerprints previously stored in a database.     

Pair correlation functions and a free energy functional for the nematic phase

In this paper we have presented the calculation of pair correlation functions in a nematic phase for a model of spherical particles with the long-range anisotropic interaction from the mean spherical approximation (MSA) and the Percus-Yevick (PY) integral equation theories. The results found from the MSA theory have been compared with those found analytically by Holovko and Sokolovska [J. Mol. Liq. 82, 161 (1999)]. A free energy functional which involves both the symmetry conserving and symmetry broken parts of the direct pair correlation function has been used to study the properties of the nematic phase. We have also examined the possibility of constructing a free energy functional with the direct pair correlation function which includes only the principal order parameter of the ordered phase and found that the resulting functional gives results that are in good agreement with the original functional. The isotropic-nematic transition has been located using the grand thermodynamic potential. The PY theory has been found to give a nematic phase with pair correlation function harmonic coefficients having all the desired features. In a nematic phase the harmonic coefficient of the total pair correlation function h(x1,x2) connected with the correlations of the director transverse fluctuations should develop a long-range tail. This feature has been found in both the MSA and PY theories.     

Ab initio calculation of the NH(3sigma-)-NH(3sigma-) interaction potentials in the quintet, triplet, and singlet states

We present the ab initio potential-energy surfaces of the NH-NH complex that correlate with two NH molecules in their 3sigma- electronic ground state. Three distinct potential-energy surfaces, split by exchange interactions, correspond to the coupling of the S(A) = 1 and S(B) = 1 electronic spins of the monomers to dimer states with S = 0, 1, and 2. Exploratory calculations on the quintet (S = 2), triplet (S = 1), and singlet (S = 0) states and their exchange splittings were performed with the valence bond self-consistent-field method that explicitly accounts for the nonorthogonality of the orbitals on different monomers. The potential surface of the quintet state, which can be described by a single Slater determinant reference function, was calculated at the coupled cluster level with single and double excitations and noniterative treatment of the triples. The triplet and singlet states require multiconfiguration reference wave functions and the exchange splittings between the three potential surfaces were calculated with the complete active space self-consistent-field method supplemented with perturbative configuration interaction calculations of second and third orders. Full potential-energy surfaces were computed as a function of the four intermolecular Jacobi coordinates, with an aug-cc-pVTZ basis on the N and H atoms and bond functions at the midpoint of the intermolecular vector R. An analytical representation of these potentials was given by expanding their dependence on the molecular orientations in coupled spherical harmonics, and representing the dependence of the expansion coefficients on the intermolecular distance R by the reproducing kernel Hilbert space method. The quintet surface has a van der Waals minimum of depth D(e) = 675 cm(-1) at R(e) = 6.6a0 for a linear geometry with the two NH electric dipoles aligned. The singlet and triplet surfaces show similar, slightly deeper, van der Waals wells, but when R is decreased the weakly bound NH dimer with S = 0 and S = 1 converts into the chemically bound N2H2 diimide (also called diazene) molecule with only a small energy barrier to overcome.     

New translation method for STOs and its application to calculation of overlap integrals

A new translation method for Slater‐type orbitals (STOs) is proposed involving exact translation of the regular solid harmonic part of the orbital followed by the series expansion of the residual spherical part in powers of the radial variable. The method is positively tested in the case of the overlap integral, showing good rate of convergence and great numerical stability under wide changes in the relevant molecular parameters. ©1999 John Wiley & Sons, Inc. Int J Quant Chem 73: 333–340, 1999     

Surface diffusion as a rate-enhancing mechanism in macromolecular association: The effectiveness of one- and two-dimensional surface sliding

The effective rate for a macromolecular association can be increased if the molecules can be held together loosely by nonspecific surface forces while they search for the specific reactive sites. In this way steric constraints can be relieved. In extreme cases, a specific interaction site can be surrounded by extensive surfaces either essentially linear (e.g. a DNA site) or two-dimensional (e.g. a membrane-bound receptor). If ligands can bind nonspecifically and search these surfaces in a lower-dimensional diffusion process, very substantial rate enhancements can be achieved under suitable conditions. The ranges of concentration and affinity of such nonspecific surfaces that can produce rate enhancements are derived and discussed. It is found that under the expected conditions in the living cell, such rate enhancements will be fairly modest. Nonetheless, nonspecific surface diffusion may play an important role.     

Multi-dimensional spectroscopy of polymer films; surface and interfacial interactions

Behavior of macromolecules near surfaces and interfaces of polymeric thin films and coatings may play a vital role in numerous applications. Therefore, understanding of molecular level processes responsible for durability, adhesion, and many other macroscopic processes is of a particular importance. This presentation will focus on stratification processes in multi-component polymeric films, with particular emphasis to polymer-surfactant interactions in latexes, responsiveness of individual components during coalescence of water-borne polyurethanes, and behavior of thermoplastic olefins (TPO). The presence of macromolecular arrangements and interactions among various components near the film-air (F-A) and the film-substrate (F-S) interfaces can be effectively monitored using attenuated total reflectance (ATR) and step-scan photoacoustic (SS-PA) Fourier transform infrared (FT-IR) spectroscopy. Both approaches are capable of obtaining information from various surface depths and complement each other if one seeks molecular level information from 0 – 150 μm into the film. If one combines ATR and PA information with IR and/or Raman surface imaging, it is possible to obtain a 3-dimensional representation of polymeric films.     

Potential energy surfaces for small alcohol dimers. II. Propanol, isopropanol, t-butanol, and sec-butanol

Potential energy landscapes for homogeneous dimers of propanol, isopropanol, tert-butanol, and sec-butanol were obtained using 735 counterpoise-corrected energies at the MP2/6-311+G(2df,2pd) level. The landscapes were sampled at 15 dimer separation distances for different relative monomer geometries, or routes, given in terms of the yaw, pitch, and roll of one monomer relative to the other and the spherical angles between the two monomer centers (taken as the C atom attached to the O). The resultant individual energy surfaces and their complex topographies were also regressed using a site-site pair potential model using a modified Morse potential that provides a mathematically simple representation of the landscapes suitable for use in molecular simulations. Generalized Morse parameters were also obtained for this model from a composite regression of these energy landscapes and those previously reported for methanol and ethanol. The quality of fit for all these energy landscapes suggests that these site parameters have transferability for possible use on other alcohols.     

Molecular density functional theory of solvation: from polar solvents to water

A classical density functional theory approach to solvation in molecular solvent is presented. The solvation properties of an arbitrary solute in a given solvent, both described by a molecular force field, can be obtained by minimization of a position and orientation-dependent free-energy density functional. In the homogeneous reference fluid approximation, limited to two-body correlations, the unknown excess term of the functional approximated by the angular-dependent direct correlation function of the pure solvent. We show that this function can be extracted from a preliminary MD simulation of the pure solvent by computing the angular-dependent pair distribution function and solving subsequently the molecular Ornstein-Zernike equation using a discrete angular representation. The corresponding functional can then be minimized in the presence of an arbitrary solute on a three-dimensional cubic grid for positions and Gauss-Legendre angular grid for orientations to provide the solvation structure and free-energy. This two-step procedure is proved to be much more efficient than direct molecular dynamics simulations combined to thermodynamic integration schemes. The approach is shown to be relevant and accurate for prototype polar solvents such as the Stockmayer solvent or acetonitrile. For water, although correct for neutral or moderately charged solute, it tends to underestimate the tetrahedral solvation structure around H-bonded solutes, such as spherical ions. This can be corrected by introducing suitable three-body correlation terms that restore both an accurate hydration structure and a satisfactory energetics.     

Molecular integrals over the gauge-including atomic orbitals. II. The Breit-Pauli interaction

Each accompanying coordinate expansion (ACE) formula is derived for each of the orbit-orbit interaction, the spin-orbit coupling, the spin-spin coupling, and the contact interaction integrals over the gauge-including atomic orbitals (GIAOs) by the use of the solid harmonic gradient (SHG) operator. Each ACE formula is the general formula derived at the first time for each of the above molecular integrals over GIAOs. These molecular integrals are arising in the Breit-Pauli two-electron interaction for a relativistic calculation. We may conclude that we can derive a certain ACE formula for any kind of molecular integral over solid harmonic Gaussian-type orbitals by using the SHG operator. The present ACE formulas will be useful, for example, for a calculation of a molecule in a uniform magnetic field, for a relativistic calculation, and so on, with the GIAO as a basis function.     

Weak polyelectrolytes tethered to surfaces: Effect of geometry,acid–base equilibrium and electrical permittivity

The structural and thermodynamical properties of weak polyelectrolytes end-tethered to surfaces of arbitrary geometry are studied using a molecular theory. The theory is based on writing down the free energy functional of the system including all the basic interactions and the explicit acid–base equilibrium for the chargeable groups of the polymer. The theory explicitly includes the size, shape, conformations, and charge distribution of all the molecular species. The electrostatic interactions include a density-dependent dielectric function, modeled with the Maxwell–Garnett mixing formula, to account for the composition-dependent permittivity. The minimization of the free energy leads to the distribution of all molecular species and their dependence on bulk pH and salt concentration. We apply the theory to polymer chains end-tethered to planar, cylindrical, and spherical surfaces. The radius of the curved surfaces is small to enhance the curvature effect. We find that when the grafting surfaces are uncharged, the approximation of a constant dielectric function works very well for both structural and thermodynamic properties. The structure of weak polyelectrolytes tethered on cylindrical and spherical surfaces is different from that of polymers tethered on planar surfaces due to the available volume as a function of the distance from the surface. Specifically, the degree of dissociation increases with increasing curvature of the surface. This is a manifestation of the coupling between the local density of protons, counterions, and polymer segments. The results can be interpreted in terms of the local Le Chatelier principle for the acid–base equilibrium, with proper account of the three local contributions: counterions, protons, and chargeable groups. We find that one can achieve local changes of pH between one to two units within 1–2 nm. The thickness of the tethered layers as a function of bulk pH shows a large increase when the pH is equal to the bulk pK. However, the variation with salt concentration is different for the different geometries. The largest swelling is found for cylindrical surfaces. The predictions from scaling theories of a maximum in the thickness of the film as a function of salt concentration is found for planar films, but not for curved surfaces. Finally, the interactions between cylinders with tethered polyelectrolytes is very different from the equivalent planar surfaces. These results are important for the interpretation of force measurements with nanoscale AFM tips. The implications of the results for the rational design of responsive tethered polymer layers is discussed together with the limitations of the theoretical approach. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2638–2662, 2006     

GPU-accelerated atom and dynamic bond visualization using hyperballs: a unified algorithm for balls, sticks, and hyperboloids

Ray casting on graphics processing units (GPUs) opens new possibilities for molecular visualization. We describe the implementation and calculation of diverse molecular representations such as licorice, ball-and-stick, space-filling van der Waals spheres, and approximated solvent-accessible surfaces using GPUs. We introduce HyperBalls, an improved ball-and-stick representation replacing tubes, linking the atom spheres by hyperboloids that can smoothly connect them. This type of depiction is particularly useful to represent dynamic phenomena, such as the evolution of noncovalent bonds. It is furthermore well suited to represent coarse-grained models and spring networks. All these representations can be defined by a single general algebraic equation that is adapted for the ray-casting technique and is well suited for execution on the GPU. Using GPU capabilities, this implementation can routinely, accurately, and interactively render molecules ranging from a few atoms up to huge macromolecular assemblies with more than 500,000 particles. In simple cases, based only on spheres, we have been able to display up to two million atoms smoothly.     

Semiclassical approach to collision-induced emission in the presence of intense laser radiation: An aspect in the study of cooperative chemical and optical pumping

Collion-induced emission in molecular systems in an intense laser field is studied using the semiclassical approach, with a view towards cooperative chemical and optical pumping in laser production. The formalism is developed with the electronic-field representation, which treats collision and radiative interaction on the same footing. Electronic-field surfaces can be regarded as forming spectra for spontaneous emission; and particular emission events can be accounted for by propagating classical trajectories on emission electronic-field surfaces. Pre-emission loss from the excited state is dealt with by propagating classical trajectories on a loss surface along a complex contour of emission branch points. This loss surface is derived on the basis of localized radiative couplings between electronic-field states and provides a framework to treat the general problem of discrete state-continuum interactions. The formalism is applied to a two-state, collinear exponential model to compute S-matrix elements and transition probabilities between asymptotic states.