Cartesian atomic coordinates

We shall not discuss all the numerous energy minimisation procedures which have been worked out and described in the literature but choose only the two most important techniques for detailed discussion the steepest descent process and the Newton-Raphson procedure. A combination of these two techniques gives satisfactory results in almost all cases of practical interest. Other procedures are described elsewhere (1, 2). For energy minimisation the use of Cartesian atomic coordinates is more favourable than that of internal coordinates, since for an arbitrary molecule it is much more convenient to derive all independent and dependent internal coordinates (on which the potential energy depends) from an easily obtainable set of independent Cartesian coordinates, than to evaluate the dependent internal coordinates from a set of independent ones. Furthermore for our purposes the use of Cartesian coordinates is also advantageous for the calculation of vibrational frequencies (Section 3.3.). The disadvantage, that the potential energy is related to Cartesian coordinates in a more complex fashion than to internals, is less serious. 

One of the first computer programs written to obtain the Cartesian atomic coordinates referred to the PAS of the parent by means of a least-squares fit to the inertial or planar moments of a number of isotopomers (also multiply substituted) appears to have been the program STRFIT coded by Schwendeman [6]. It is a versatile r0-type program incorporating many useful features, it is not a rs-fit program in the sense in which this term is used in this paper. 

Owing to the fact that the mass-weighted and pure Cartesian atomic coordinates are related by ... 

ICART specifies the type of input of cartesian atomic coordinates. The ICART codes are as follows ... 

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. 

Figure 5 Time dependence of RMSD of atomic coordinates from canonical A- and B-DNA forms m two trajectories of a partially hydrated dodecamer duplex. The A and B (A and B coiTespond to A and B forms) trajectories started from the same state and were computed with internal and Cartesian coordinates as independent variables, respectively. (From Ref. 54.)...

Cartesian and cylindrical polar atomic coordinates of the structural repeating unit of 31 polysaccharide helices are provided in Tables A1 to A31. Errors, if any, in the original publications have been corrected. The coordinates of hydrogen atoms are given in a majority of structures. If missing, they are not available in the references cited in Table I. Each table caption contains the structure number and polymer name assigned in Table I. Refer to Table II for its chemical repeating unit. Cartesian (x, y, z) and cylindrical (r, , z) coordinates are related by x r cost ), y = r sin<(> and z is the same in both systems. 

Vibrational spectroscopy is of utmost importance in many areas of chemical research and the application of electronic structure methods for the calculation of harmonic frequencies has been of great value for the interpretation of complex experimental spectra. Numerous unusual molecules have been identified by comparison of computed and observed frequencies. Another standard use of harmonic frequencies in first principles computations is the derivation of thermochemical and kinetic data by statistical thermodynamics for which the frequencies are an important ingredient (see, e. g., Hehre et al. 1986). The theoretical evaluation of harmonic vibrational frequencies is efficiently done in modem programs by evaluation of analytic second derivatives of the total energy with respect to cartesian coordinates (see, e. g., Johnson and Frisch, 1994, for the corresponding DFT implementation and Stratman etal., 1997, for further developments). Alternatively, if the second derivatives are not available analytically, they are obtained by numerical differentiation of analytic first derivatives (i. e., by evaluating gradient differences obtained after finite displacements of atomic coordinates). In the past two decades, most of these calculations have been carried... 

The effect of the symmetry operations on the Cartesian displacement coordinates of the two hydrogen atoms in die water molecule. The sharp ( ) indicates the inversion of a coordinate axis, resulting in a change in handedness of the Cartesian coordinate system. 

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. 

Take an N-atomic molecule with the nuclei each at their equilibrium internuclear position. Establish a Cartesian x, y, z coordinate system for each of the nuclei such that, for Xj with i = 1,.., 3N, xi is the Cartesian x displacement of nucleus 1, x2 is the Cartesian y displacement coordinate for nucleus 1, X3 is the Cartesian z displacement coordinate for nucleus 1,..., x3N is the Cartesian z displacement for nucleus N. Use of one or another quantum chemistry program yields a set of force constants I ij in Cartesian displacement coordinates... 

Symmetry Types of the Normal Modes. For this nonlinear four-atomic molecule there are 3(4) -6 = 6 genuine internal vibrations. Using a set of three Cartesian displacement coordinates on each atom, we obtain the following representation of the group C3l, ... 

In VFF the molecular vibrations are considered in terms of internal coordinates qs (s = 1..3N — 6, where N is the number of atoms), which describe the deformation of the molecule with respect to its equilibrium geometry. The advantage of using internal coordinates instead of Cartesian displacements is that the translational and rotational motions of the molecule are excluded explicitly from the very beginning of the vibrational analysis. The set of internal coordinates q = qs is related to the set of Cartesian atomic displacements x = Wi by means of the Wilson s B-matrix [1] q = Bx. In the harmonic approximation the B-matrix depends only on the equilibrium geometry of the molecule. 

In a collision process, it is the relative position of the atoms that matters, not the absolute positions, when external fields are excluded, and the potential energy E will depend on the distances between atoms rather than on the absolute positions. It will therefore be natural to change from absolute Cartesian position coordinates to a set that describes the overall motion of the system (e.g., the center-of-mass motion for the entire system) and the relative motions of the atoms in a laboratory fixed coordinate system. This can be done in many ways as described in Appendix D, but often the so-called Jacobi coordinates are chosen in reactive scattering calculations because they are convenient to use. The details about their definition are described in Appendix D. The salient feature of these coordinates is that the kinetic energy remains diagonal in the momenta conjugated to the Jacobi coordinates, as it is when absolute position coordinates are used. 

Now, check the rules with a larger basis set, the Cartesian displacement coordinates of the atoms of HNNH (see Figure 4-8). Operation E leaves all the 12 vectors unchanged, so its character will be 12. C2 brings each atom into a different position so their vectors will also be shifted. This means that all vectors will have zero contribution to the character. The same applies to the inversion operation. Finally, as already worked out before, the horizontal reflection leaves all the x and y vectors unchanged and brings the four z vectors into their negative selves. The result is... 

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