Cartesian coordinate space

These difficulties have led to a revival of work on internal coordinate approaches, and to date several such techniques have been reported based on methods of rigid-body dynamics [8,19,34-37] and the Lagrange-Hamilton formalism [38-42]. These methods often have little in common in their analytical formulations, but they all may be reasonably referred to as internal coordinate molecular dynamics (ICMD) to underline their main distinction from conventional MD They all consider molecular motion in the space of generalized internal coordinates rather than in the usual Cartesian coordinate space. Their main goal is to compute long-duration macromolecular trajectories with acceptable accuracy but at a lower cost than Cartesian coordinate MD with bond length constraints. This task mrned out to be more complicated than it seemed initially. 

II. NORMAL MODE ANALYSIS IN CARTESIAN COORDINATE SPACE... 

The set of unit vectors of dimension n defines an n-dimensional rectangular (or Cartesian) coordinate space 5 . Such a coordinate space S" can be thought of as being constructed from n base vectors of unit length which originate from a common point and which are mutually perpendicular. Hence, a coordinate space is a vector space which is used as a reference frame for representing other vector spaces. It is not uncommon that the dimension of a coordinate space (i.e. the number of mutually perpendicular base vectors of unit length) exceeds the dimension of the vector space that is embedded in it. In that case the latter is said to be a subspace of the former. For example, the basis of 5 is ... 

Now, to demonstrate the difference between partial derivative and total derivative, first, consider the left-hand side of Figure 2. This figure is an oval container which has holes on its sides. As shown, because of these holes, a mass can enter and leave the container. Let O represent any value of the property of the mass and let the container move in any direction. As the mass enters and leaves the container, it will be carrying with it its property, so the value of O will vary with distance as the container moves. 4> is then said to be a function of distance or space. If the Cartesian coordinate space, xyz, is chosen, then O is a function of x, y, and z. In addition, it must also be a function of time, t. In mathematical symbols. 

N. B. Ulyanov Z. Du T. L. James, Refinement of Nucleic Acid Structures with Residual Dipolar Coupling Restraints in Cartesian Coordinate Space. In Modern Magnetic Resonance G. A. Webb, Ed. Springer Netherlands, 2006 pp 665-670. 

Figure 1. A resonant quasiperlodic orbit displayed in Cartesian coordinate space (R,x,y) at E=0.9 eV with y 17 . The total ener-...

Ferguson and Raber s RIPS (Random Incremental Pulse Search) program also performs its stochastic search in Cartesian coordinate space. Two modes of operation are possible, the first to locate just the global minimum energy conformation and the second to perform a complete search of the conformational space. In the hunt for the global energy minimum, the lowest energy structure found to... 

When defining the PES one can use a number of different coordinate systems to describe the geometry of the system being studied the simplest of these being Cartesian coordinates. In Cartesian coordinate space, the position of each atom in the molecule (or cluster, etc.) is given by three spatial coordinates, x, y, and z. For a system having atoms there will be a set of Cartesian coordinates, x,. A similar coordinate... 

Vector d represents the normal mode p in internal coordinate space. It can be transformed to Cartesian coordinate space according to Eq. (21) ... 

The charge terms expressed in Cartesian coordinate space are very difBcult to inter] et. Besides, as we shall later see, its elements will carry some contributions frmn the compensatory rotation of polar molecules stipulated by the Edcart-Sayvetz conditions. In the Decius proach, on the other hand, the consideration of diese rotational effects is explicit and, as result, both the effective chaiges and charge fluxes determined are purely intramolecular quantities reflecting the charge reorganization effects acconqiaiiying vibrational distortions. 

Since molecules with higher symmetiy are only considered, i.e. other than diese wiA molecular point group C2h, C2, C, C and Ci, symmetry coordinates may be constructed either in internal or Cartesian coordinate space. The number of Cartesian symmetry coordinates associated with die equivalent set of atoms a is of interest to the treatment here. It is given by the expression... 

Most of the calculations are performed on the basis of Cartesian displacement coordinates. A curvilinear transformation from internal to Cartesian coordinate space is carried out to yield... 

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