Cartesian coordinates, atomic positions using

Atomic Positions Using Cartesian Coordinates and Corresponding Wyckoff Letters... 

Instead of using a Z-matrix, one can specify the nuclear positions using Cartesian coordinates for each atom. For example, for CO2, if we take the z axis as the molecular axis and guess a bond length of 1.22 A, the geometry specification is... 

FIGURE 4.1 Matrices can be used for describing chemical structures in different fashions. The adjacency matrix (a) of thionyl chloride shows whether are bonded to each other. The distance matrix (b) describes the number of bonds between two elements of a structure. The Cartesian distance matrix (c) contains the real three-dimensional (Euclidean) distances between atoms calculated from Cartesian coordinates of the atom positions. The bond path distance matrix (d) contains the sum of bond length between two atoms and is, in contrast to the Cartesian matrix, independent of the conformation of the molecule. 

Molecules containing N atoms require 3N Cartesian coordinates to specify the locations of all the nuclei. For convenience, we may use three coordinates to locate the center of mass of the molecule, and three more coordinates to orient a chosen axis passing through the molecule. Only two are needed to describe the orientation if the molecu-le is diatomic. The remaining 3N - 6 coordinates are used to. sped.fy the relative posit 0.ns of the atorns. The potential energy V (r )... 

This section displays positioning of the atoms in the molecule used by the program internally, in Cartesian coordinates. This orientation is chosen for maximum calculation efficiency, and corresponds to placing the center of nuclear charge for the molecule at the origin. Most molecular properties are reported with respect to the standard orientation. Note that this orientation usually does not correspond to the one used in the input molecule specification the latter is printed earlier in the output as the Z-matrix orientation. ... 

Carrying out a simulation, which we view as the readiest source of data, does require coordinate choices. Thus, we do need some notation for coordinates, and we use Pn genetically to denote the configuration of a molecule of n atoms, including translational, orientational, and conformational positioning see Fig. 9.3. This notation suggests that Cartesian coordinates of each atom would be satisfactory, in principle, but does not require any specific choice. 

For a quantitative description of molecular geometries (i.e. the fixing of the relative positions of the atomic nuclei) one usually has the choice between two possibilities Cartesian or internal coordinates. Within a force field, the potential energy depends on the internal coordinates in a relatively simple manner, whereas the relationship with the Cartesian nuclear coordinates is more complicated. However, in the calculations described here, Cartesian coordinates are always used, since they offer a number of computational advantages which will be commented on later (Sections 2.3. and 3.). In the following we only wish to say a few words about torsion angles, since it is these parameters that are most important for conformational analysis, a topic often forming the core of force field calculations. 

The minimum amount of information needed to specify a crystal structure is the unit cell type, that is, cubic, tetragonal, and so on, the unit cell parameters, and the positions of all of the atoms in the unit cell. The atomic contents of the unit cell are a simple multiple, Z, of the composition of the material. The value of Z is equal to the number of formula units of the solid in the unit cell. Atom positions are expressed in terms of three coordinates, x, y, and z. These are taken as fractions of a, b, and c, the unit cell sides, say and j. The x, y, and z coordinates are plotted with respect to the unit cell axes, not to a Cartesian set of axes. The space group describes the symmetry of the unit cell, and although it is not mandatory when specifying a structure, its use considerably shortens the list of atomic positions that must be specified in order to built the structure. 

Once the protein interaction pattern is translated from Cartesian coordinates into distances from the reactive center of the enzyme and the structure of the ligand has been described with similar fingerprints, both sets of descriptors can be compared [25]. The hydrophobic complementarity, the complementarity of charges and H-bonds for the protein and the substrates are all computed using Carbo similarity indices [26]. The prediction of the site of metabolism (either in CYP or in UGT) is based on the hypothesis that the distance between the reactive center on the protein (iron atom in the heme group or the phosphorous atom in UDP) and the interaction points in the protein cavity (GRID-MIF) should correlate to the distance between the reactive center of the molecule (i.e. positions of hydrogen atoms and heteroatoms) and the position of the different atom types in the molecule [27]. 

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. 

Plane of symmetry. If a plane can be placed in space such that for every atom of the molecule not in the plane there is an identical atom (which is to say, the same atomic number and isotope) on the other side of the plane at equal distance from it (i.e., a mirror image ), the molecule is said to possess a plane of symmetry. The Greek letter o is often used to represent both the plane of symmetry and the operation of mirror reflection that it performs. An example of a molecule possessing a plane of symmetry is methylcyclobutane, as illustrated in Figure B.l. Note that a planar molecule always has at least one ct, since tire plane of tire molecule satisfies the above symmetry criterion in a trivial way (the set of reflected atoms is the empty set). Note also that if we choose a Cartesian coordinate system in such a way tliat two of the Cartesian axes lie in the symmetry plane, say x and y, then for every atom found at position (x,y,z) where z there must be an identical atom at position (x,y,—z). 

It is often useful to transform from simple Cartesian coordinates to other sets of coordinates when we study collision processes including chemical reactions. In a collision process, it is obvious that the relative positions of the reactants are relevant and not the absolute positions as given by the simple Cartesian coordinates. It is therefore customary to change from simple Cartesian coordinates to a set describing the relative motions of the atoms and the overall motion of the atoms. For the latter motion the center-of-mass motion is usually chosen. In the following we will describe a general method of transformation from Cartesian coordinates to internal coordinates and determine its effect on the expression for the kinetic energy. 

If we place the nucleus of the hydrogen atom at the origin of a set of Cartesian coordinates, the position of the electron would be given by x, y, and z, as shown in Fig. 2.1.1. However, the solution of the Schrodinger equation for this system becomes intractable if it is done in Cartesian coordinates. Instead, this problem is solved using polar spherical coordinates r, 6, and (f>, which are also shown in Fig. 2.1.1. These two sets of coordinates are related by ... 

Typke has introduced the rs-fit method [7] where Kraitchman s basic principles are retained. A system of equations is set up for all available isotopomers of a parent (not necessarily singly substituted) and is solved by least-squares methods for the Cartesian coordinates (referred to the PAS of the parent) of all atomic positions that have been substituted on at least one of the isotopomers The positions of unsubstituted atoms need not be known and cannot be determined. The method is presented here with two recent improvements true derivatives are used for the Jacobian matrix X, and the problem of the observations and theircovariances, which is rather elaborate, is fully worked out. The equations are always given for the general asymmetric rotor, noting that simplifications occur in more symmetric situations, e.g. for linear molecules, which could nonetheless be treated within the framework presented. 

As we move from one-electron to many-electron atoms, both the Schrodinger equation and its solutions become increasingly complicated. The simplest many-electron atom, helium (He), has two electrons and a nuclear charge of +2e. The positions of the two electrons in a helium atom can be described using two sets of Cartesian coordinates, (xi, yi, Zi) and (xi, yz, Zz), relative to the same origin. The wave function tf depends on all six of these variables if = (x, y, Zu Xz, yz Zz)-... 

Distance matrices may also be calculated for real three-dimensional (Euclidean) distances between atoms using the Cartesian coordinates of the atom positions (Figure 4.1c). These matrices allow the calculation of descriptors that account for the shape and conformation of atoms. If conformation is not required or desired, the bond... 

---