Internal Coordinate System Research Articles

The quasi-independent curvilinear coordinate approximation for geometry optimization.

This paper presents an efficient alternative to well established algorithms for molecular geometry optimization. This approach exploits the approximate decoupling of molecular energetics in a curvilinear internal coordinate system, allowing separation of the 3N-dimensional optimization problem into an O(N) set of quasi-independent one-dimensional problems. Each uncoupled optimization is developed by a weighted least squares fit of energy gradients in the internal coordinate system followed by extrapolation. In construction of the weights, only an implicit dependence on topologically connected internal coordinates is present. This new approach is competitive with the best internal coordinate geometry optimization algorithms in the literature and works well for large biological problems with complicated hydrogen bond networks and ligand binding motifs.

A new CT method for measuring cup orientation after total hip arthroplasty: a study of 10 patients.

It is difficult to assess the orientation of the acetabular component on routine radiographs. We present a method for determining the spatial orientation of the acetabular component after total hip arthroplasty (THA) using computed tomography. Two CT-scans, 10 min apart, were obtained from each of 10 patients after THA. Using locally developed software, two independent examiners measured the orientation of the acetabular component in relation to the pelvis. The measurements were repeated after one week. To be independent of the patient position during scanning, the method involved two steps. Firstly, a 3D volumetric image of the pelvis was brought into a standard pelvic orientation, then the orientation of the acetabular component was measured. The orientation of the acetabular component was expressed as operative anteversion and inclination relative to an internal pelvic reference coordinate system. To evaluate precision, we compared measurements across pairs of CT volumes between observers and trials. Mean absolute interobserver angle error was 2.3 degrees for anteversion (range 0-6.6 degrees), and 1.1 degrees for inclination (range 0-4.6 degrees). For interobserver measurements, the precision, defined as one standard deviation, was 2.9 degrees for anteversion, and 1.5 degrees for inclination. A Student's t-test showed that the overall differences between the examiners, trials, and cases were not significant. Data were normally distributed and were not dependent on examiner or trial. We conclude that the implant angles of the acetabular component in relation to the pelvis could be detected repeatedly using CT, independently of patient positioning.

A new program complex EFOLD for molecular modeling: the use of a flexible weighting coefficient scheme and the standard valence geometry in modeling the enzyme active sites

An algorithm for the representation of biopolymer structures in an internal coordinate system (so-called structure regularization) by minimizing the target function with a flexible weighting coefficient scheme using three components that determine the reliability of deviations of each atom was proposed. For the structure regularization, an algorithm for taking into account the temperature factor was suggested for the first time. It was shown by the example of the aspartyl protease rhizopuspepsin that the representation in the internal coordinate system may result in an accurate reproduction of the structural details of separate molecule fragments, such as the active site region of the enzyme. This algorithm was realized as one of the modules of our EFOLD program complex. The English version of the paper.

Internally defined distances in 3D-quantitative structure-activity relationships.

A new type of 3D-QSAR descriptors is introduced. For each molecule under consideration an internal coordinate system is defined relative to molecular points, such as positions of atoms in the molecule or centers of mass or certain substructures. From the origin of this system distances to the solvent accessible surface are calculated at defined spherical coordinate angles, theta and phi. The distances represent steric features, while the molecular electrostatic potentials at the intersection points with the surface represent the electrostatic contributions. The approach is called IDA (internal distances analysis). Matrices obtained by varying the spherical coordinate angles by fixed increments are correlated with the biological activity by partial least squares (PLS). The descriptors, tested with the benchmark steroids and an also well characterized benzodiazepine data set, turn out to be highly predictive. Additionally, they share the advantage of grid-based methods that the obtained models can be visualized, and thus be directly used in a rational drug design approach.

Three-dimensional computerised atlas of the rat brain stem precerebellar system: approaches for mapping, visualization, and comparison of spatial distribution data.

Comparisons of microscopical neuroanatomic data from different experiments and investigators are typically hampered by the use of different section planes and dissimilar techniques for data documentation. We have developed a framework for visualization and comparison of section-based, spatial distribution data, in brain stem nuclei. This framework provides opportunities for harmonized data presentation in neuroinformatics databases. Three-dimensional computerized reconstructions of the rat brain stem and precerebellar nuclei served as a basis for establishing internal coordinate systems for the pontine nuclei and the precerebellar divisions of the sensory trigeminal nuclei. Coordinate based diagrams were used for presentation of experimental data (spatial distribution of labelled neurons and axonal plexuses) from standard angles of view. Each nuclear coordinate system was based on a cuboid bounding box with a defined orientation. The bounding box was size-adjusted to touch cyto- and myeloarchitectonically defined boundaries of the individual nuclei, or easily identifiable nearby landmarks. We exemplify the use of these internal coordinate systems with dual retrograde neural tracing data from pontocerebellar and trigeminocerebellar systems. The new experimental data were combined, in the same coordinate based diagrams, with previously published data made available via a neuroinformatics data repository (www.nesys.uio.no/Database, see also www.cerebellum.org). Three-dimensional atlasing, internal nuclear coordinate systems, and consistent formats for presentation of neuroanatomic data in web-based data repositories, offer new opportunities for efficient analysis and re-analysis of neuroanatomic data.

Determination of a Potential Energy Surface for CO2Using Generalized Internal Vibrational Coordinates

A potential energy surface for CO2is determined from experimental data using generalized internal vibrational coordinates. These coordinates are defined as two arbitrary distances and the angle between them and depend on two external parameters, which can be properly optimized. An optimal generalized internal coordinate system is obtained for CO2by minimizing unconverged vibrational energies with respect to the external parameters. The optimal coordinates are shown to be superior to previously derived normal hyperspherical coordinates for this molecule. A nonlinear least-squares fit of the potential energy surface of CO2to observed vibrational frequencies is made by using the optimal internal coordinates and fully variational calculations. The potential function is represented by a fourth-order Morse–cosine expansion and its quality is checked by computing highly excited vibrational transition frequencies which were not included in the fit.

Cardiac phase contrast gradient echo MRI: measurement of myocardial wall motion in healthy volunteers and patients.

A number of methods have been proposed for the noninvasive measurement of myocardial wall motion. The paper describes a strategy for assessing myocardial motion based on the sensitivity of the phase of the MR-signal to motion using a breath-hold phase contrast technique. A motion-sensitized and a motion-compensated MR-signal are measured during successive scans. The difference between the two MR-signals is used to calculate myocardial velocity in all three spatial dimensions. Postprocessing includes the transformation of the measured velocities into an internal coordinate system of the left ventricle. Also various presentation modes and further processing of the received velocity information are provided including calculation of global motion parameters. We examined 20 patients suffering from myocardial infarction. The overall left ventricular motion can be characterized by appropriate parameters describing the rotation and contraction or expansion, respectively. Regional motional disturbances are visualized using parametric images. Contrary to the highly consistent interindividual data in normal volunteers, patients showed significant localized motion deficits.

Implementation of the IMOMM methodology for performing combined QM/MM molecular dynamics simulations and frequency calculations

A technique for implementing the integrated molecular orbital and molecular mechanics (IMOMM) methodology developed by Maseras and Morokuma that is used to perform combined quantum mechanics/molecular mechanics (QM/MM) molecular dynamics simulations, frequency calculations and simulations of macromolecules including explicit solvent is presented. Although the IMOMM methodology is generalized to any coordinate system, the implementation first described by Maseras and Morokuma requires that the QM and MM gradients be transformed into internal coordinates before they are added together. This coordinate transformation can be cumbersome for macromolecular systems and can become ill-defined during the course of a molecular dynamics simulation. We describe an implementation of the IMOMM method in which the QM and MM gradients are combined in the cartesian coordinate system, thereby avoiding potential problems associated with using the internal coordinate system. The implementation can be used to perform combined QM/MM molecular dynamics simulations and frequency calculations within the IMOMM framework. Finally, we have examined the applicability of thermochemical data derived from IMOMM framework. Finally, we have examined the applicability of thermochemical data derived from IMOMM frequency calculations.

Methods for geometry optimization of large molecules. I. An O(N2) algorithm for solving systems of linear equations for the transformation of coordinates and forces

The most recent methods in quantum chemical geometry optimization use the computed energy and its first derivatives with an approximate second derivative matrix. The performance of the optimization process depends highly on the choice of the coordinate system. In most cases the optimization is carried out in a complete internal coordinate system using the derivatives computed with respect to Cartesian coordinates. The computational bottlenecks for this process are the transformation of the derivatives into the internal coordinate system, the transformation of the resulting step back to Cartesian coordinates, and the evaluation of the Newton–Raphson or rational function optimization (RFO) step. The corresponding systems of linear equations occur as sequences of the form yi=Mixi, where Mi can be regarded as a perturbation of the previous symmetric matrix Mi−1. They are normally solved via diagonalization of symmetric real matrices requiring O(N3) operations. The current study is focused on a special approach to solving these sequential systems of linear equations using a method based on the update of the inverse of the symmetric matrix Mi. For convergence, this algorithm requires a number of O(N2) operations with an O(N3) factor for only the first calculation. The method is generalized to include redundant (singular) systems. The application of the algorithm to coordinate transformations in large molecular geometry optimization is discussed.

Mode-specific energy analysis for rotating-vibrating triatomic molecules in classical trajectory simulation

A method for the mode-specific energy analysis in a classical trajectory calculation is developed. The pure rotational energy is evaluated by invoking the Eckart condition. To evaluate the vibrational energy in each normal mode, the vibrational velocity is divided into two parts, the angular motion part and the angular motion free part, and the latter is analyzed with the Cartesian and internal coordinate systems. The potential energy of each normal mode is also evaluated in the two coordinate systems. A simple algorithm to include some anharmonicity correction is presented. Sample calculations with nonreacting triatomic molecules, H2O and HCN, show that the internal coordinate system is more adequate than the Cartesian, especially for the linear molecule HCN. An excellent result is obtained for the product (CHO+) of a reaction, suggesting that the present method is adequate for the mode-specific energy analysis of classical trajectory results.

A new N-body potential and basis set for adiabatic and non-adiabatic variational energy calculations

A new functional form for multi-body expansions of potential energy surfaces and basis functions for correlated adiabatic and fully non-adiabatic variational energy calculations is presented. N-body explicitly correlated Gaussians with pre-multiplying factors consisting of products of powers of internal distance coordinates are utilized in a dual role to analytically represent isotropic potentials and energy eigen-functions in the same internal coordinate system. Practical aspects of this new methodology are presented. The ideas and methods are prototyped and illustrated with two simple diatomic examples; the Morse potential and an accurate H2 potential for which essentially exact results are obtained for vibrational energy levels.

Using redundant internal coordinates to optimize equilibrium geometries and transition states

A redundant internal coordinate system for optimizing molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between bonded atoms. Redundancies are removed by using the generalized inverse of the G matrix; constraints can be added by using an appropriate projector. For minimizations, redundant internal coordinates provide substantial improvements in optimization efficiency over Cartesian and nonredundant internal coordinates, especially for flexible and polycyclic systems. Transition structure searches are also improved when redundant coordinates are used and when the initial steps are guided by the quadratic synchronous transit approach. © 1996 by John Wiley & Sons, Inc.

Fermi resonances and local modes in stibine, SbH3: A Fourier interferometric and laser photoacoustic study of the overtone spectrum

The third stretching overtone region of a natural sample of stibine, SbH3, has been studied with high resolution infrared spectroscopy and the fifth and the sixth overtone region with Ti:Sapphire ring laser intracavity photoacoustic spectroscopy. The third overtone consists of a local mode pair of bands (400A1/E) which have been rotationally assigned both for 121SbH3 and 123SbH3 with a vibration-rotation model based on rectilinear normal coordinates. The vibrational dependencies of the model parameters are explained well with a simple block diagonal vibrational model. An extension of the standard vibration-rotation model is used to show that the upper state rotational energy level structures of both isotopic species are close to the rotational structure of an asymmetric rotor. High resolution laser spectrum of the fifth overtone consisting of a local mode pair of bands (600A1/E) shows severe perturbations in the upper state rotational structure. The (510A1/E) and (700A1/E) bands have been recorded with low resolution. All experimentally known vibration-rotation band origins of 121SbH3 have been reproduced well with a curvilinear internal valence coordinate system based Fermi resonance local mode model. The potential energy surface obtained agrees well with recent ab initio results.

Ab initio generalized valence force field for zeolite modelling. 1. Siliceous zeolites

Abstract A new potential permitting the molecular dynamics study of the structural and dynamical properties of siliceous zeolites is presented and tested. It is based on a quantum-chemical derivation of the Cartesian force constants which are transformed into an internal coordinate system and corrected for systematic errors. The calculated vibrational spectra for three siliceous zeolites (sodalite, faujasite and silicalite) show a very satisfactory agreement with the experimental ones and the force field is very similar to a potential recently derived from experimental data.

Mean-field energies of spin-flux phases.

It is suggested that conventional spin-density-wave states for an interacting many-electron system in a two-dimensional square lattice are energetically unstable to the formation of spin flux. Spin flux corresponds to a rotation of the internal coordinate system of the electron as it traverses a closed loop. The flux is dynamically generated by the electromagnetic interaction in the many-body system. This is illustrated by including the nearest-neighbor Coulomb repulsion between electrons in the Hubbard model. Using a Grassmann field theory, we derive the mean-field equations for spin flux and local-moment ordering. For a wide range of Hubbard model couplings U/t and doping \ensuremath{\delta} we find that the generation of spin flux leads to a lowering of the mean-field local-moment amplitude. This suppression of the local moment offsets the additional quantum zero-point energy associated with spin flux and leads to an overall reduction in the total many-electron energy. When the spin-flux vortex filaments are endowed with a quantum dynamics, chiral symmetry breaking is possible. We show that under certain conditions, this leads to a further reduction of local-moment amplitude and many-electron energy.

An extension of the rigorous base-unit oriented description of nucleic acid structures.

Our proposed description for DNA base/base-pair structures (1), though rigorous, does not satisfy some of the requirements as established at the Cambridge Workshop (2). Here, we propose a revised description for base/base-unit structures of nucleic acids. This new description is as rigorous and satisfies all the requirements (2). Following the original approach, the moment-of-inertia frame is still the choice of the internal coordinate system for a base/base-unit. The revised description has the minimum number of parameters (i.e., six parameters per rigid body) in the set. Besides regular Watson-Crick type of helices (e.g., A-DNA, A-RNA, B-DNA, Z-DNA, etc.), the revised description also works for non-Watson-Crick, multiple stranded molecules (e.g., triplex, quadruplex, etc.) as well as parallel stranded molecules.

Analysis of orientation autocorrelation and cross-correlation functions for polyethylene in the inclusion complex with perhydrotriphenylene

The first and second orientation autocorrelation functions have been computed and analysed for polyethylene in the inclusion complex with perhydrotriphenylene at 223, 300, and 373 K. Two coordinate systems have been used-an internal coordinate system defined by the polyethylene chain and an external coordinate system defined by the channel. The orientation autocorrelation functions are denoted by M 1,int (t), M 2,int (t), M 1,ext (t), and M 2,ext (t). Cross-correlation functions have also been computed. The trajectories that provide the raw data are those reported by Zhan and Mattice for n-tetracontane in a channel formed by 90 molecules of perhydrotriphenylene. The results show the dynamics of the internal and external motions occur on different time scales which can be conveniently separated

Triatom: programs for the calculation of ro-vibrational spectra of triatomic molecules

The TRIATOM program suite calculates energy levels, wavefunctions, and where appropriate, dipole transition moments and spectra, for rotating and vibrating triatomic molecules. Potential energy, and where necessary, dipole surfaces must be provided. The programs use an exact (within the Born-Oppenheimer approximation) Hamiltonian, offer a choice of several body-fixed, internal coordinate systems based on two distances and an included angle and employ basis function expansions of orthogonal polynomials. The calculations are variational, and rotational excitation is treated using an efficient two-step algorithm. Constituent programs are TRIATOM which solves the vibrational problem and also performs the first step for rotationally excited systems. SELECT which optionally preselects basis functions for TRIATOM. ROTLEVD which performs the second step for rotationally excited states. DIPOLE computes either line or band transition intensities. SPECTRA uses the data generated by the other programs to give simulated absorption or emission spectra.

Conformational sampling by a general linearized embedding algorithm

AbstractLinearized embedding is a variant on the usual distance geometry methods for finding atomic Cartesian coordinates given constraints on interatomic distances. Instead of dealing primarily with the matrix of interatomic distances, linearized embedding concentrates on properties of the metric matrix, the matrix of inner products between pairs of vectors defining local coordinate systems within the molecule. We developed a pair of general computer programs that first convert a given arbitrary conformation of any covalent molecule from atomic Cartesian coordinates representation to internal local coordinate systems enforcing rigid valence geometry and then generate a random sampling of conformers in terms of atomic Cartesian coordinates that satisfy the rigid local geometry and a given list of interatomic distance constraints. We studied the sampling properties of this linearized embedding algorithm vs. a standard metric matrix embedding program, DGEOM, on cyclohexane, cycloheptane, and a cyclic pentapeptide. Linearized embedding always produces exactly correct bond lengths, bond angles, planarities, and chiralities; it runs at least two times faster per structure generated, and is successful as much as four times as often at refining these structures to full agreement with the constraints. It samples the full range of allowed conformations broadly, although not perfectly uniformly. Because local geometry is rigid, linearized embedding's sampling in terms of torsion angles is more restricted than that of DGEOM, but it finds in some instances conformations missed by DGEOM. © 1992 by John Wiley & Sons, Inc.

A theoretical study of conformational transition in polymethylene by means of a continuous model

Changes of parameters characterizing intramolecular ordering in a polymethylene chain during its transition from coiled to compact conformation have been studied by means of Monte Carlo simulation; a realistic model of the polymer chain with a continuous spectrum of conformations was employed. The interactions of distant chain units was described by the Lennard-Jones potential. The order tensor was evaluated in the internal (molecularly invariant) coordinate system in dependence on the thermodynamic quality of the solvent for a molecule containing 61 bonds; the variation in the distribution of the number of intramolecular contacts and the character of fluctuations of the radius of gyration were elucidated. The obtained data are confronted with results of known analytical models.