Topic

# Cartesian coordinate system

About: Cartesian coordinate system is a research topic. Over the lifetime, 6714 publications have been published within this topic receiving 136351 citations. The topic is also known as: cartesian coordinates.

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TL;DR: In this paper, a numerical algorithm integrating the 3N Cartesian equations of motion of a system of N points subject to holonomic constraints is formulated, and the relations of constraint remain perfectly fulfilled at each step of the trajectory despite the approximate character of numerical integration.

18,394 citations

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TL;DR: Analysis of extended molecular dynamics simulations of lysozyme in vacuo and in aqueous solution reveals that it is possible to separate the configurational space into two subspace: an “essential” subspace containing only a few degrees of freedom and the remaining space in which the motion has a narrow Gaussian distribution and which can be considered as “physically constrained.”

Abstract: Analysis of extended molecular dynamics (MD) simulations of lysozyme in vacuo and in aqueous solution reveals that it is possible to separate the configurational space into two subspaces: (1) an "essential" subspace containing only a few degrees of freedom in which anharmonic motion occurs that comprises most of the positional fluctuations; and (2) the remaining space in which the motion has a narrow Gaussian distribution and which can be considered as "physically constrained." If overall translation and rotation are eliminated, the two spaces can be constructed by a simple linear transformation in Cartesian coordinate space, which remains valid over several hundred picoseconds. The transformation follows from the covariance matrix of the positional deviations. The essential degrees of freedom seem to describe motions which are relevant for the function of the protein, while the physically constrained subspace merely describes irrelevant local fluctuations. The near-constraint behavior of the latter subspace allows the separation of equations of motion and promises the possibility of investigating independently the essential space and performing dynamic simulations only in this reduced space.

2,896 citations

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TL;DR: In this paper, a redundant internal coordinate system for molecular geometries is constructed from all bonds, all valence angles between bonded atoms, and all dihedral angles between pairs of atoms.

Abstract: 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.

2,577 citations

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Bell Labs

^{1}TL;DR: The results of two kinds of test applications of a computer program for multidimensional scaling on the basis of essentially nonmetric data are reported to measures of interstimulus similarity and confusability obtained from some actual psychological experiments.

Abstract: A computer program is described that is designed to reconstruct the metric configuration of a set of points in Euclidean space on the basis of essentially nonmetric information about that configuration. A minimum set of Cartesian coordinates for the points is determined when the only available information specifies for each pair of those points—not the distance between them—but some unknown, fixed monotonic function of that distance. The program is proposed as a tool for reductively analyzing several types of psychological data, particularly measures of interstimulus similarity or confusability, by making explicit the multidimensional structure underlying such data.

2,461 citations

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TL;DR: A combination of variable-metric second-order update schemes and the DIIS method for both geometry and Hartree-Fock wave function optimization is described and a recursive procedure for updating large Hessians is presented.

Abstract: A combination of variable-metric second-order update schemes and the DIIS method for both geometry and Hartree-Fock wave function optimization is described. A recursive procedure for updating large Hessians is presented. The performances of geometry optimizations with respect to the choice of the coordinate system (symmetry-adapted, internal, and Cartesian coordinates), the initial nuclear Hessian, and the optimization procedure have been investigated by a series of benchmark molecules. Formulas for the generation of initial nuclear Hessians are given

1,864 citations