3D space

Before the data can be visualised, ie displayed in a two or three-dimensional representation, the ultrasonic responses from the interior of the test-piece must be reconstructed from the raw ultrasonic data. The reconstruction process projects ultrasonic indications into 3D space. As well as reconstructing the entire ultrasonic data set within an acquisition file, it is possible to define an arbitrary sub-volume of the test object over which reconstruction will take place. The image resolution may also be defined by the user. Clearly, larger volumes or greater resolution will increase the computation time for both the reconstruction and visualisation processes. 

If compounds have the same topology (constitution) but different topography (geometry), they are called stereoisomers. The configuration expresses the different positions of atoms around stereocenters, stereoaxes, and stereoplanes in 3D space, e.g., chiral structures (enantiomers, diastereomers, atropisomers, helicenes, etc.), or cisftrans (Z/E) configuration. If it is possible to interconvert stereoisomers by a rotation around a C-C single bond, they are called conformers. 

Clearly, the next step is the handling of a molecule as a real object with a spatial extension in 3D space. Quite often this is also a mandatory step, because in most cases the 3D structure of a molecule is closely related to a large variety of physical, chemical, and biological properties. In addition, the fundamental importance of an unambiguous definition of stereochemistry becomes obvious, if the 3D structure of a molecule needs to be derived from its chemical graph. The moleofles of stereoisomeric compounds differ in their spatial features and often exhibit quite different properties. Therefore, stereochemical information should always be taken into ac-count if chiral atom centers are present in a chemical structure. 

Before we go into further detail on the handling of chemical structures in 3D space from the chemoinformatics point of view, it should be noted that there... 

Basically, two different methods arc commonly used for representing a chemical struchiive in 3D space. Both methods utilize different coordinate systems to describe the spatial arrangement of the atoms of a molecule under con.sidcration. The most common way is to choose a Cartesian coordinate system, i.e., to code the X-, y-, and z-coordinates of each atom, usually as floating point numbers, For each atom the Cartesian coordinates can be listed in a single row. giving consecutively the X-, )> , and z-valnc.s. Figure 2-90 illustrates this method for methane. 

The second method for representing a molecule in 3D space is to use internal coordinates such as bond lengths, bond angles, and torsion angles. Internal coordinates describe the spatial arrangement of the atoms relative to each other. Figure 2-91 illustratc.s thi.s for 1,2-dichlorocthanc. 

Spanned by tbc atoms 4, 2, and 1, and 2, 1, and 3 (tlic ry-planc), Except of the first three atoms, each atom is described by a set of three internal coordinates a distance from a previously defined atom, the bond angle formed by the atom with two previous atoms, and the torsion angle of the atom with three previous atoms. A total of 3/V - 6 internal coordinates, where N is the number of atoms in the molecule, is required to represent a chemical structure properly in 3D space. The number (,3N - 6) of internal coordinates also corresponds to the number of degrees of freedom of the molecule. 

Tabic 2-6 gives an overview on the most common file formats for chemical structure information and their respective possibilities of representing or coding the constitution, the configuration, i.c., the stereochemistry, and the 3D structure or conformation (see also Sections 2..3 and 2.4). Except for the Z-matrix, all the other file formats in Table 2-6 which are able to code 3D structure information arc using Cartesian coordinates to represent a compound in 3D space. 

In the next step, the suggested models are translated into 3D space by subsequently combining the templates. Again, each model is assessed and ranked according to various structural criteria, such as the geometric fit of the 3D templates and non-bonding interactions (steric clashes). If none of the solu-... 

In 4D-QSAR, a grid is used to determine the regions in 3D space responsible for binding. Nevertheless, neither a probe nor interaction energy is used. 

Since IR spectroscopy monitors the vibrations of atoms in a molecule in 3D space, information on the 3D arrangement of the atoms should somehow be contained in an IR spectrum. However, the relationships between the 3D structure and the IR spectrum are rather complex, so no general attempt has yet been successfiil in deriving the 3D structure of a molecule directly from the IR spectrum. 

The dimensionality of a potential energy surface depends on the number of degrees of freedom in a molecule. If Vp s is a function of two variables, then a plot of the potential energy surface represents a 3D space. 

TEM is still the most powerful technique to elucidate the dispersion of nano-filler in rubbery matrix. However, the conventional TEM projects three-dimensional (3D) body onto two-dimensional (2D) (x, y) plane, hence the structural information on the thickness direction (z-axis) is only obtained as an accumulated one. This lack of z-axis structure poses tricky problems in estimating 3D structure in the sample to result in more or less misleading interpretations of the structure. How to elucidate the dispersion of nano-fillers in 3D space from 2D images has not been solved until the advent of 3D-TEM technique, which combines TEM and computerized tomography technique to afford 3D structural images, incidentally called electrontomography . 

This is very different from the case with single bonds, which are freely rotating aU of the time. But a double bond is the result of overlapping p orbitals, and double bonds cannot freely rotate at room temperature (if you had trouble with this concept when you first learned it, you should review the bonding structure of a double bond in your textbook or notes). So there are two ways to arrange the atoms in space cis and trans. If you compare which atoms are connected to each other in each of the two possibilities, yon will notice that all of the atoms are connected in the same order. The difference is how they are connected in 3D space. This is why they are called stereoisomers (this type of isomerism stems from a difference of orientation in space— stereo ). 

Before we can talk about drawing Newman projections, we need to first review one aspect of drawing bond-line stractures that we did not cover in Chapter 1. To show how groups are positioned in 3D space, we often use wedges and dashes ... 

These two compounds are different from each other even though the atoms are connected in the same way. The difference between them comes from their positions in 3D space. Therefore, they are called stereoisomers ( stereo for space). More specifically, they are called enantiomers, because the two compounds are mirror images of each other and they are not superimposable. If we construct models of these two compounds, we see that they are not the same—i.e., they cannot be superimposed. 

And we indicated the position of the double bond with the numbering system. But then we saw that there are often two ways for the atoms of a double bond to connect to each other in 3D space. We saw a system for distinguishing these possibilities, using the terms cis and trans ... 

Methods that deduce a pharmacophore, an arrangement in 3D space of features that contribute or detract from binding and look for its presence in the database that is searched. This method places emphasis on features like hydrogen bond donors, hydrogen bond acceptors, acidic or basic units and hydrophobic fragments and opens the possibility of identifying unexpected scaffolds with required features (pharmacophore-based VS or PHBVS). 

The density here refers to the spatial coordinate, i.e. the concentration of the reaction product, and is not to be confused with the D(vx,vy,vz) in previous sections which refers to the center-of-mass velocity space. Laser spectroscopic detection methods in general measure the number of product particles within the detection volume rather than a flux, which is proportional to the reaction rate, emerging from it. Thus, products recoiling at low laboratory velocities will be detected more efficiently than those with higher velocities. The correction for this laboratory velocity-dependent detection efficiency is called a density-to-flux transformation.40 It is a 3D space- and time-resolved problem and is usually treated by a Monte Carlo simulation.41,42... 

Thus, a molecule can be characterized in terms of its potential hydrogen bonding, polar, hydrophobic and ionic interactions in 3D space. The size and the spatial distribution of these molecular interaction contours is translated into a quantitative scheme, the VolSurf descriptors, without the need to align the molecules in 3D space [8, 9] (Fig. 17.1). 

D-molecular descriptors, alignment-independent and based on molecular interaction, called GRIND have been developed. These are autocorrelation transforms that are independent of the orientation of the molecules in 3D space. The original descriptors can be extracted from the autocorrelation transform with the ALMOND program. The basic idea is to compress the information present in 3D maps into a few 2D numerical descriptors which are very simple to understand and interpret. 

Unlike crystals that are packed with identical unit cells in 3D space, aperiodic crystals lack such units. So far, aperiodic crystals include not only quasiperiodic crystals, but also crystals in which incommensurable modulations or intergrowth structures (or composites) occur [14], That is to say, quasiperiodicity is only one of the aperiodicities. So what is quasiperiodicity Simply speaking, a structure is classified to be quasiperiodic if it is aperiodic and exhibits self-similarity upon inflation and deflation by tau (x = 1.618, the golden mean). By this, one recognizes the fact that objects with perfect fivefold symmetry can exist in the 3D space however, no 3D space groups are available to build or to interpret such structures. 

These two measurements give a clear picture of the orientation of the dipole in 3D space and how the dipole moment of a molecule moves when different computational methods are used. 

During the last five decades, an alternative way of looking at the quantum theory of atoms, molecules, and solids in terms of the electron density in three-dimensional (3D) space, rather than the many-electron wave function in the multidimensional configuration space, has gained wide acceptance. The reasons for such popularity of the density-based quantum mechanics are the following ... 

The three main approaches based on the single-particle density are the density functional theory (DFT), quantum fluid dynamics (QFD), and studying the properties of a system through local quantities in 3D space. In this chapter, we present simple discussions on certain conceptual and methodological aspects of the single-particle density for details, the reader may consult the references listed at the end of this chapter. 

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