Cartesian coordinates Crystals

The empirical pseiidopotential method can be illustrated by considering a specific semiconductor such as silicon. The crystal structure of Si is diamond. The structure is shown in figure Al.3.4. The lattice vectors and basis for a primitive cell have been defined in the section on crystal structures (ATS.4.1). In Cartesian coordinates, one can write G for the diamond structure as... 

The temperature factor (together with the Cartesian coordinates) is the result of the rcfincincnt procedure as specified by the REMARK 3 record. High values of the temperature factor suggest cither disorder (the corresponding atom occupied different positions in different molecules in the crystal) or thermal motion (vibration). Many visualisation programs (e.g., RasMol [134] and Chime [155]) have a special color scheme designated to show this property. 

One possible implementation of this piecewise adiabatic evolution could be a sequence of N birefringent crystals, each rotated by angle cpj with respect to the chosen Cartesian coordinate system. Figure 5.2 illustrates the general concept, with a sequence of shifts and thicknesses, chosen in this figure arbitrarily. ... 

The A00(r) term in Eq. (27) is by far the largest, but it does not remove the degeneracy of the ground state and only contributes a constant term which drops out when energy differences are computed. For this reason, it is customarily dropped from Vc in crystal field calculations. An alternative expression to that given in Eq. (27), using Cartesian coordinates in place of polar coordinates is... 

From crystallography, we obtain an image of the electron clouds that surround the molecules in the average unit cell in the crystal. We hope this image will allow us to locate all atoms in the unit cell. The location of an atom is usually given by a set of three-dimensional Cartesian coordinates, x, y, and z. One of the vertices (a lattice point or any other convenient point) is used as the origin of the unit cell s coordinate system and is assigned the coordinates x = 0, y = 0, and z = 0, usually written (0,0,0). See Fig. 2.4. 

In the above, ijk denote the Cartesian coordinates of a molecule, IJK those of a crystal (a unit cell), 7Vt is the number of molecules in a unit volume occupying each particular inequivalent site in the unit cell,/ is the number of inequivalent positions of a molecule in a unit cell, and Ng is the number of equivalent positions in a unit cell. The directional cosines are used to transform each of the molecular 0 components to those of the new coordinate system (bUK) and the contributions are summed. 

Finally, an interesting paper by Tokdemir and Nelson looks at irradiated inosine single crystals [87], The authors have used calculations on the anisotropic hyperfine couplings as an aid in identifying free radical structures. They find that the computed dipolar coupling eigenvectors correlate well with the experimental results. The input Cartesian coordinates used for the calculations were obtained from the crystallographic data. 

The hrst suffix of each tensor component gives the row and the second the column in which the component appears. The xis term, for example, measures the component of the polarization parallel to X2 (usually the y direction in a Cartesian coordinate system) when a field is applied parallel to X3 (the z direction). The susceptibility tensor must conform to any restrictions imposed by crystal symmetry, see Eqs. 6.5-6.9. 

In the following section the symmetry species of the thiourea molecule are denoted by lower-case letters, while that of the unit cell of the crystal is indicated by capital letters. The Cartesian coordinate axes of the free molecules are labeled u, v, and w, those of the crystal a, b, and c (see Figs. 2.7-8 and 2.7-9 Schrader et al., 1971). 

In order to establish a set of realistic aqueous solution conformations for the peptide Ala-Pro-Tyr, cartesian coordinates from the Brookhaven Protein Data Bank (PDB), the SCAN3D database [35], (a representative set of non-homologous high-resolution protein crystal structures selected from the PDB), and from the Cambridge Crystal Structure Database (CSD) were used. 

The Cartesian coordinate system is usually chosen such that the z axis and the yz plane are parallel to the c axis and the b axis of the crystal lattice, respectively. Then the elements of T are given by... 

FIGURE 11.4. From crystal coordinates to orthogonal (Cartesian) coordinates (.4 ... 

Although conventional Verlet-type molecular dynamics places restraints on bond lengths and bond angles, one could conceivably want to implement these restrictions as holo-nomic constraints. This is supported by the observation that the deviations from ideal bond lengths and bond angles are usually small in X-ray crystal structures. There are essentially two possible approaches to solve Newton s equations (Eq. 12) with holonomic constraints. The first involves a switch from Cartesian coordinates f) to generalized internal ones ft. Having thus redefined the system, one would solve equations of motion for the... 

The long partial bonds and abnormal hybridization of transition state structures preclude precise correspondence between the functional groups of an analogue and those of the transition state, but a range of plausible analogues can usually be derived by inspection of the transition state structure additional alternatives may be obtained from chemical or crystal structure files. Choice of the best analogues for synthesis is then based on a comparison of their molecular structures with that of the transition state. For this purpose, computed transition state structures and other points on the reaction pathway may conveniently be stored in a molecule library file, either as cartesian coordinates or as geometric variables it may also be convenient to store a connectivity matrix, since the presence of partial bonds in the transition state structure may not be self-evident from the interatomic distances. 

Unique atoms are randomly placed outside the forbidden zones in the unit cell, followed by automatic generation of their equivalent atoms based on symmetry operations. Placed atoms violating conditions (i) and (ii) above will be removed from further consideration otherwise, a data set containing unique T-atom coordinates will be saved. For distance calculation, the crystal coordinates are converted into Cartesian coordinates. 

If r t) is the time dependent atomic position vector, taken with respect to the origin of the crystallographic cell, it can be expressed as the sum of two terms, the molecular centre of mass, this is a time dependent vector, (0ext> that allows a description of the vibrations of the crystal, the phonons. The second term is the position vector of the atom, H(0int, given by a Cartesian coordinate system with its origin at the molecular centre of mass. 

A stress applied to a crystal results in a strain. A phenomenological description of the electron energy levels under elastic strain was developed by Bardeen and Shockley [12]. It is referred to as the deformation potential approximation (DPA), in which the one-electron Hamiltonian is developed in a Taylor s series of the strain components The perturbation is written in cartesian coordinates, for a linear order in strain, as ... 

Crystal structure information is stored in the CSD in the form of cell parameters and space group symmetry operations, these being in the computationally convenient form of rotation matrices and translation vectors. Appendix 12-1, Equations (5-6), shows how to obtain the cartesian coordinates of any atom in the crystal, using this information. Whenever elements of the space group S and of the molecular point group G coincide, an appropriate subgroup S/G (see Chapter 2) is used, so as to always have an Integral number of molecules in the asymmetric unit. 

A crystal moving in this liquid, if its density is different from that of the liquid, has an additional vertical component of velocity Vg. There is no horizontal component due to gravity. The components of crystal velocity can be written in polar and in Cartesian coordinates as follows ... 

This formalism has been used to adapt THBREL, a crystal structure relaxation program which has been widely used in modelling inorganic solids, to model rigid molecules whose electrostatic interactions are described by a DMA model (Willock et al., 1995). The program uses the Cartesian coordinates of the centres of mass of each molecule, rotations around the molecule fixed axes and the six strain matrix elements as variables, so the relaxation of the crystal structure to a minimum in the lattice energy is not restricted to specific space groups. 

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