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Numeric atomic coordinates

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. [Pg.424]

Applications of the theory described in Section III.A.2 to malonaldehyde with use of the high level ab initio quantum chemical methods are reported below [94,95]. The first necessary step is to define 21 internal coordinates of this nine-atom molecule. The nine atoms are numerated as shown in Fig. 12 and the Cartesian coordinates x, in the body-fixed frame of reference (BF) i where n= 1,2,... 9 numerates the atoms are introduced. This BF frame is defined by the two conditions. First, the origin is put at the center of mass of the molecule. [Pg.122]

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... [Pg.146]

The order parameter can be defined in two different ways. It can be either a function of atomic coordinates or just a parameter in the Hamiltonian. Examples of both types of order parameters are given in Sect. 2.8.1 in Chap. 2 and illustrated in Fig. 2.5. This distinction is theoretically important. In the first case, the order parameter is, in effect, a generalized coordinate, the evolution of which can be described by Newton s equations of motion. For example, in an association reaction between two molecules, we may choose as order parameter the distance between the two molecules. Ideally, we often would like to consider a reaction coordinate which measures the progress of a reaction. However, in many cases this coordinate is difficult to define, usually because it cannot be defined analytically and its numerical calculation is time consuming. This reaction coordinate is therefore often approximated by simpler order parameters. [Pg.119]

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. [Pg.177]

The ligands in coordination names are listed in alphabetical order before the central atom name. Numerical prefixes that indicate the number of ligands are not considered in determining the ligand alphabetical order, e.g. tricarbonyl/iydrido(tributylphosphine)cobalt. [Pg.120]

The general numerical approach employs rigid body motions and least-squares fitting. Given two sets of points xt,i = 1,2,..., N and yui= 1,2,..., N (here xt and yt are vectors specifying atomic coordinates), find the best rigid motion of the points y h) such that the sum of the squares of the deviations E xt — Yt 2 is a minimum. [Pg.316]

Crystal-structure determinations provide atomic coordinates of proteins, nucleic acids, and viruses. Computational studies of these data — using both purely-numerical techniques and interactive graphics — seek the principles of structure, dynamics, function and evolution of living systems at the molecular level. [Pg.146]

Individuals are the units upon which natural evolution operates, and also the unit manipulated in an EA, in which each individual is correlated with a distinct solution to the problem being studied. These individuals may be a direct representation of the solutions themselves in numeric or symbolic form, a list of atomic coordinates, for example, or they may instead be a coded form of that solution. Individuals are processed using evolution-like operations, the role of which is to gradually transform them from initial randomly chosen, and probably poor, solutions into optimum solutions. [Pg.12]

There are numerous examples of bond-length modifications when local environments of the atoms involved in the bond change. The usual trend is a contraction when the atom coordination Z decreases. The table of ionic radii given in Ref. 231 illustrates this effect. An increase in Z is always associated with an increase of the ionic radius, and since first neighbour interatomic distances are assumed to be the sum of two ionic radii, both quantities follow the same trends. [Pg.59]

Now let us ask what is needed to numerically evaluate this expression for the Fourier transform. The diffraction vector s = k — ko, as well as X, are experimental variables that are chosen, and the Zj are known for each atom as well. The only remaining variables are the xj, and these can be generated from the atomic coordinates xj, yj, zj. Thus all we really need to compute the resultant waves making up the diffraction pattern, for any array of scattering points, are their relative positions in space. [Pg.97]

As we have seen before, the vibrational spectra of molecules require the calculation of the second derivatives of the energy with respect to the atom coordinates about their equilibrium positions. Many DFT program codes can perform the calculation of the dynamical matrix either using analytical second derivatives or numerical differences. As a result their output includes a set of eigenvectors and vibrational frequencies which are the input to programs that calculate INS spectra ( 5.3). [Pg.174]

It may also be desirable to store the atomic coordinates read from these files. The purpose of parsing the coordinates from the file and putting them into a separate column is to enable use of the coordinates from within the database. If the column is properly defined as a numeric or float column, this will also ensure that the coordinates are proper numbers. If there is no need for atomic coordinates, it is not necessary to create a column for these. Later sections of this chapter will discuss ways in which these atomic coordinates might be used in SQL functions. [Pg.125]

Distinct from these are automatic methods that directly transform 2D input information on atoms, bonds, and stereochemistry into 3D atomic coordinates. The automatic methods are classified into rule-based and data-based, fragment-based, conformational analysis, and numerical methods (Fig. 5). These classes of methods overlap more or less with each other and belong more or less to the domain of automatic 3D structure generation ... [Pg.158]

Until 10-15 years ago it was common practice to publish a short note followed later by the full paper which superseded the earlier note. Nowadays, multiple publication of a study is still widespread, but very often the two (or more) papers eomplement eaeh other, rather than one superseding the other. In the early days of CSD construction, superseded entries were deleted from the database. Now all entries are retained. This means that a refcode family may contain two (or more) identical data sets if the author has published or deposited the same set of atomic coordinates in more than one article. Such duplicates should be eliminated in any numerical analysis project which involves the calculation of statistical measures. [Pg.101]

A more systematic way of improving on the trial structure is to use the least-square method. In this method the parameters of the system (such as the atomic coordinates and Debye-Waller factors) are altered by a numerical algorithm in the direction toward minimizing the sum... [Pg.99]


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See also in sourсe #XX -- [ Pg.125 ]




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