Space Cartesian

Usually, two conformations are regarded as geometrically different if their minimized RMS deviation is equal to or larger than 0.3 A in Cartesian space RMSx z)> or 30 " in torsion angle space (RMSja). respectively. 

For each combination of atoms i.j, k, and I, c is defined by Eq. (29), where X , y,. and Zj are the coordinates of atom j in Cartesian space defined in such a way that atom i is at position (0, 0, 0), atomj lies on the positive side of the x-axis, and atom k lies on the xy-plaiic and has a positive y-coordinate. On the right-hand side of Eq. (29), the numerator represents the volume of a rectangular prism with edges % , y ., and Zi, while the denominator is proportional to the surface of the same solid. If X . y ., or 2 has a very small absolute value, the set of four atoms is deviating only slightly from an achiral situation. This is reflected in c, which would then take a small absolute value the value of c is conformation-dependent because it is a function of the 3D atomic coordinates. 

HyperChem allows the visualization of two-dimensional contour plots for a certain number of variables, fh esc contour plots show the values of a spatial variable (a property f(x,y,z) in normal th rce-dimensional Cartesian space ) on a plane that is parallel to the screen. To obtain these contour plots the user needs to specify ... 

One way to describe the conformation of a molecule other than by Cartesian or intern coordinates is in terms of the distances between all pairs of atoms. There are N(N - )/ interatomic distances in a molecule, which are most conveniently represented using a N X N S5munetric matrix. In such a matrix, the elements (i, j) and (j, i) contain the distant between atoms i and and the diagonal elements are all zero. Distance geometry explort conformational space by randomly generating many distance matrices, which are the converted into conformations in Cartesian space. The crucial feature about distance geometi (and the reason why it works) is that it is not possible to arbitrarily assign values to ti... 

Cartesian space coordinates V, y, z Condensed phase (solid or liquid) cd... 

The conformational distance does not have to be defined in Cartesian coordinates. Eor comparing polypeptide chains it is likely that similarity in dihedral angle space is more important than similarity in Cartesian space. Two conformations of a linear molecule separated by a single low barrier dihedral torsion in the middle of the molecule would still be considered similar on the basis of dihedral space distance but will probably be considered very different on the basis of their distance in Cartesian space. The RMS distance is dihedral angle space differs from Eq. (12) because it has to take into account the 2n periodicity of the torsion angle. 

The expressions for the screened potentials in Cartesian space are then obtained straightforwardly... 

The position of any point in three-dimensional cartesian space is denoted by the vector r with components v, y, z, so that... 

A set of complete orthonormal functions ipfx) of a single variable x may be regarded as the basis vectors of a linear vector space of either finite or infinite dimensions, depending on whether the complete set contains a finite or infinite number of members. The situation is analogous to three-dimensional cartesian space formed by three orthogonal unit vectors. In quantum mechanics we usually (see Section 7.2 for an exception) encounter complete sets with an infinite number of members and, therefore, are usually concerned with linear vector spaces of infinite dimensionality. Such a linear vector space is called a Hilbert space. The functions ffx) used as the basis vectors may constitute a discrete set or a continuous set. While a vector space composed of a discrete set of basis vectors is easier to visualize (even if the space is of infinite dimensionality) than one composed of a continuous set, there is no mathematical reason to exclude continuous basis vectors from the concept of Hilbert space. In Dirac notation, the basis vectors in Hilbert space are called ket vectors or just kets and are represented by the symbol tpi) or sometimes simply by /). These ket vectors determine a ket space. 

The harmonic oscillator may be generalized to three dimensions, in which case the particle is displaced from the origin in a general direction in cartesian space. The force constant is not necessarily the same in each of the three dimensions, so that the potential energy is... 

We next consider a three-dimensional cartesian space with axes rix, %, z-Each point in this -space with positive (but non-zero) integer values of rix, y. 

A vector x in three-dimensional cartesian space may be represented as a column matrix... 

A linear operator A in three-dimensional cartesian space may be represented as a 3X3 matrix A with elements ay. The expression y = x in matrix notation becomes... 

The vector concept may be extended to n-dimensional cartesian space, where we have n mutually orthogonal axes. Each vector x then has n components (xi, X2,...,... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

This means that the operators are mutually exclusive and that the operator is idempotent. Nevertheless, in three-dimensional Cartesian space the atoms do overlap, often even to a large extent. So they have no boundaries. 

Simulated Annealing by Molecular Dynamics Simulation in Cartesian Space... 

This is followed by the geometry, in either cartesian or internal coordinates. Each atom in the system is entered on a separate line. If internal coordinates are used, the atom s position is defined relative to other atoms in terms of a distance and two angles. If Cartesian coordinates are used, the position of each atom is defined rdative to some arbitary origin in Cartesian space. 

The first issue we discuss is the choice of degrees of freedom. We briefly review the literature pertaining to justification and implementation of performing molecular simulations in non-Cartesian space, most prominently torsional and rigid-body space. We lay out advantages of such a procedure and comment upon common implementation difficulties, with specific attention to the issue of consistency between the development and application of force field parameters for use in MC simulations. 

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