Coordinates, atomic independent

An N-atom molecular system may he described by dX Cartesian coordinates. Six independent coordinates (five for linear molecules, three fora single atom) describe translation and rotation of the system as a whole. The remaining coordinates describe the nioleciiUir configuration and the internal structure. Whether you use molecular mechanics, quantum mechanics, or a specific computational method (AMBER, CXDO. etc.), yon can ask for the energy of the system at a specified configuration. This is called a single poin t calculation. ... 

Polyatomic molecules vibrate in a very complicated way, but, expressed in temis of their normal coordinates, atoms or groups of atoms vibrate sinusoidally in phase, with the same frequency. Each mode of motion functions as an independent hamionic oscillator and, provided certain selection rules are satisfied, contributes a band to the vibrational spectr um. There will be at least as many bands as there are degrees of freedom, but the frequencies of the normal coordinates will dominate the vibrational spectrum for simple molecules. An example is water, which has a pair of infrared absorption maxima centered at about 3780 cm and a single peak at about 1580 cm (nist webbook). 

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.)...

However, we should emphasize that the NBO/NRT concepts of hybridization, Lewis structure, and resonance differ in important respects from previous empirical usage of these terms. In earlier phases of valence theory it was seldom possible to determine, e.g., the atomic hybridization by independent theoretical or experimental procedures, and instead this term became a loosely coded synonym for the molecular topology. For example, a trigonally coordinated atom might be categorized as sp2-hybridized or an octahedrally coordinated atom as d2sp3-hybridized, with no supporting evidence for the accuracy of these labels as descriptors of actual... 

For atomic chemisorption, similar structural effects are found (see the middle panel of Figure 4.10). As for molecular chemisorption, low-coordinated atoms at steps bind adsorbates stronger and have lower barriers for dissociation than surfaces with high coordination numbers and lower d band centers. The d band model thus explains the many observations that steps form stronger chemisorption bonds than flat surfaces [1,20,24-28]. The finding that the correlation with the d band center is independent of the adsorbate illustrates the generality of the d band model. 

Therefore, any atom in a special position where it belongs to a mirror plane has two positional degrees of freedom. Only two of the three coordinates vary independently, whereas the third is restricted to either a constant value [e.g. position 4(e) in the space group Cmm2] or it is constrained to be proportional to one of the two free coordinates Figure 1.43). Similar to a general position, any special position on the mirror plane can accommodate many independent atoms provided they do not overlap. 

By contrast it may be shown that the solution of Eq. (18) is invariant to linear combination of the components of X. Therefore, determination of Cartesian coordinates or determination of internal coordinates should lead to the same result. It should be pointed out, however, that the principal axis conditions may be taken into account in different ways in Cartesian coordinate and internal coordinate calculations. We include the center-of-mass and product-of-inertia relations along with the moments when we fit Cartesian coordinates. On the other hand, internal coordinates are independent of the origin of the coordinate system. Thus, holding some Cartesian coordinates constant while others are varied is not the same as holding some internal coordinates constant while others are varied. This is true even if the same atomic positions are involved. 

F. V. Kekule interpreted benzene to be a cyclic entity in 1865. The concept of carbon as a tetrahedraUy, four-fold coordinated atom was presented independently by J. H. van t Hoff and J. A. Le Bel in 1874 and revolutionized the interpretation of the element s chemical activity (Figure 1.1). Since then, fundamental discoveries on this ubiquitous element multiplied. L. Mond and co-workers published the first metal carbonyls in 1890, and in 1891, E. G. Acheson for the first time achieved artificial graphite via intermediate silicon carbide (carborundum), which itself had been unknown then, too. 

One check that can be performed is to cross-validate the actual residue sequence as directly extracted from the atomic coordinates with independent references entries in the PDB will habitually have a SEQRES field that lists the expected sequence. This, however, may have been generated from the atoms automatically. For public domain structures, one can cross-reference to independent sources of sequence data such as UniProt [33,34]. Cross-referencing, however, only illuminates discrepancies it does not tell the modeler the correct sequence. 

In the crystal structure of kainosite-(Y), the Y (rare earth) atom and calcium atom independently occupy two crystallographic sites. Both cations are coordinated by... 

For an N-atom molecule there are 3N - 6 internal degrees of freedom, so there must be 3N - 6 independent vibrations and one needs as many independent coordinates for their description. Chemical intuition works in terms of chemical bonds, so that the most intuitive way of representing molecular vibrations is in terms of the variations in bond lengths, bond angles, and torsion angles. These are internal coordinates, [ i]. Cartesian coordinates are independent by definition, but finding a set of truly independent internal coordinates for a large polyatomic molecule is all but a trivial matter [1],... 

Fig. 4. In NAMD 2 forces are calculated not by force objects owned by individual patches, but rather by independent compute objects which depend on one or more patches for atomic coordinates. As suggested by shading in this illustration, a compute object need not reside on the same node as the patches upon which it depends.

Besides structure and substructure searches, Gmclin provides a special search strategy for coordiuation compouuds which is found in no other database the ligand search system, This superior search method gives access to coordination compounds from a completely different point of view it is possible to retrieve all coordination compounds with the same ligand environment, independently of the central atom or the empirical formula of the compound. 

The quaniity, (R). the sum of the electronic energy computed 111 a wave funciion calculation and the nuclear-nuclear coulomb interaciion .(R.R), constitutes a potential energy surface having 15X independent variables (the coordinates R j. The independent variables are the coordinates of the nuclei but having made the Born-Oppenheimer approximation, we can think of them as the coordinates of the atoms in a molecule. 

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