Interatomic distance matrices

The atom-mapping method provides a quantitative measure of the similarity between a pair of 3D molecules, A and B, that are represented by their interatomic distance matrices. The measure is calculated in two stages. First, the geometric environment of each atom in A is compared with the corresponding environment of each atom in B to determine the similarity between each possible pair of atoms. The resulting interatomic similarities are then used to identify pairs of geometrically related atoms, and these equivalences allow the calculation of the overall intermolecular similarity. 

This poster reports work to date on a project to evaluate techniques for measuring the degree of similarity between pairs of 3>D chemical structures represented by interatomic distance matrices. The four techniques that have been tested use the distance information in very different ways and have very different computational requirements. Preliminary experiments with two small data-sets, for which both structural and biological activity data are available, suggest that the most cost-effective technique is based on a mapping procedure that tries to match pairs of atoms, one from each of the molecules that are being compared, that have neighboiuing atoms at approximately the same distances. 

I conformation is illustrated schematically in Figure 9.15. The interatomic distance matrix his conformation is ... 

The ETMC is essentially an interatomic distance matrix (Fig. 3.47), with the diagonal elements containing an electronic structural parameter (atomic charge, polarizability, HOMO energy, etc.). Off-diagonal elements for two atoms that are chemically bonded are used to store information regarding the bond (bond order, polarizability, etc.). Matrices for active compounds in a series are then searched for common features that are not shared by inactive compounds. The successful examples cited are predominately for small, relatively rigid structures where the conformational parameter does not confuse the analysis. 

Fig. 2.33. Interatomic distance matrix (according to Ooi and Nishikawa, 1973) for elastase (lower left) and chymotrypsin (upper right). Black areas correspond to separation less than 15 A dotted areas to separation greater than 30 A. The plot clearly indicates (1) similarities in both structures, (2) the organization of the molecules into two distinct domains (which correspond to the two regions of high density of black areas) (from Sawyer et al, 1978). Black areas correspond to separation less than 15 A dotted areas to separation greater than 30 A.

One way to describe the conformation of a molecule other than by Cartesian or intern coordinates is in terms of the distances between all pairs of atoms. There are N(N - )/ interatomic distances in a molecule, which are most conveniently represented using a N X N S5munetric matrix. In such a matrix, the elements (i, j) and (j, i) contain the distant between atoms i and and the diagonal elements are all zero. Distance geometry explort conformational space by randomly generating many distance matrices, which are the converted into conformations in Cartesian space. The crucial feature about distance geometi (and the reason why it works) is that it is not possible to arbitrarily assign values to ti... 

We can now proceed to the generation of conformations. First, random values are assigne to all the interatomic distances between the upper and lower bounds to give a trial distam matrix. This distance matrix is now subjected to a process called embedding, in which tl distance space representation of the conformation is converted to a set of atomic Cartesic coordinates by performing a series of matrix operations. We calculate the metric matrix, each of whose elements (i, j) is equal to the scalar product of the vectors from the orig to atoms i and j ... 

Suppose interatomic distances are now randomly assigned between the lower and uppi bounds to give the following distance matrix ... 

A molecular dynamics force field is a convenient compilation of these data (see Chapter 2). The data may be used in a much simplified fonn (e.g., in the case of metric matrix distance geometry, all data are converted into lower and upper bounds on interatomic distances, which all have the same weight). Similar to the use of energy parameters in X-ray crystallography, the parameters need not reflect the dynamic behavior of the molecule. The force constants are chosen to avoid distortions of the molecule when experimental restraints are applied. Thus, the force constants on bond angle and planarity are a factor of 10-100 higher than in standard molecular dynamics force fields. Likewise, a detailed description of electrostatic and van der Waals interactions is not necessary and may not even be beneficial in calculating NMR strucmres. 

The second step concerns distance selection and metrization. Bound smoothing only reduces the possible intervals for interatomic distances from the original bounds. However, the embedding algorithm demands a specific distance for every atom pair in the molecule. These distances are chosen randomly within the interval, from either a uniform or an estimated distribution [48,49], to generate a trial distance matrix. Unifonn distance distributions seem to provide better sampling for very sparse data sets [48]. 

The metric matrix is the matrix of all scalar products of position vectors of the atoms when the geometric center is placed in the origin. By application of the law of cosines, this matrix can be obtained from distance information only. Because it is invariant against rotation but not translation, the distances to the geometric center have to be calculated from the interatomic distances (see Fig. 3). The matrix allows the calculation of coordinates from distances in a single step, provided that all A atom(A atom l)/2 interatomic distances are known. 

Linus Pauling, "The Nature of the Chemical Bond. III. The Transition from One Extreme Bond Type to Another," JACS 54 (1932) 981003 Linus Pauling, "Interatomic Distances in Covalent Molecules and Resonance between Two or More Lewis Electronic Structures," Proc.NAS 18 (1932) 293297 Linus Pauling, "The Calculation of Matrix Element for the Lewis Electronic Structure of Molecules,"... 

Suppose we approach M atoms of an element having an unfilled outer shell, disposed in a lattice. The point may be made clear if we suppose to approach hydrogen atoms (outer shell 1 s ). Equations (11) and (12) would predict the broadening of the electron state in a half-filled s-band, which should therefore allow metallic behaviour. Apparently, this would happen for any inter-atomic distance a and, therefore also at infinite distance. What would change is, of course, the bandwidth, which is determined by matrix elements dependent on the interatomic distance a but the metallic behaviour, depending essentially on the fact that the electrons have available energy states within the band, should occur also at distances where the atoms may well be supposed to be isolated. 

STOs have a number of features that make them attractive. The orbital has the correct exponential decay with increasing r, the angular component is hydrogenic, and the Is orbital has, as it should, a cusp at the nucleus (i.e., it is not smooth). More importantly, from a practical point of view, overlap integrals between two STOs as a function of interatomic distance are readily computed (Mulliken Rieke and Orloff 1949 Bishop 1966). Thus, in contrast to simple Huckel theory, overlap matrix elements in EHT are not assumed to be equal to the Kronecker delta, but are directly computed in every instance. 

Internal coordinate systems include normal coordinates which are symmetry adapted and used in spectroscopy, and coordinate systems based on interatomic distances ( bond lengths ), three-center angles ( valence angles ) and four-center angles ( torsion angles ). In the latter case a Z-matrix of the form shown in Table 3.1 defines the structure of a molecule. The input and output files of nearly all molecular mechanics programs are in cartesian coordinates. 

This is the simplest possible mechanistic model of the PES, derived from an approximate treatment of energy according to eq. (3.69). The FA type of treatment implies that the geminal amplitude-related ES V eqs. (2.78) and (2.81) are fixed at their invariant values eq. (3.7). This corresponds clearly to a simplified situation where all bonds are single ones. Within such a picture, the dependence of the energy on the interatomic distance reduces to that of the matrix elements of the underlying QM (MINDO/3 or NDDO) semiempirical Hamiltonian. 

The numerical value of the coefficient C2 is -0.20734 for the equilibrium interatomic distance in methane. The above form of the HO transformation matrix is perfectly confirmed by our numerical experiments performed within the FA picture even for very large distortions, which is a consequence of the mathematical structure of the hybridization manifold described above. The numerical data show that the linear response estimate performs very well up to improbably large distortions (the deviation from the linear response estimate is smaller than 0.25% for the distortion of 0.3 rad (about 17°)). 

Distance Geometry Distance geometry is a well-known technique from the area of structure determination via NMR technology.137,138 Instead of describing a molecule by coordinates in Euclidean space, it is described by a so-called distance matrix containing all interatomic distances in the molecule. Based on distance matrices, a set of allowable conformations can be described in a comprehensive form from a distance interval for each atom pair. 

Since about 15 years, with the advent of more and more powerfull computers and appropriate softwares, it is possible to develop also atomistic models for the diffusion of small penetrants in polymeric matrices. In principle the development of this computational approach starts from very elementary physico-chemical data - called also first-principles - on the penetrant polymer system. The dimensions of the atoms, the interatomic distances and molecular chain angles, the potential fields acting on the atoms and molecules and other local parameters are used to generate a polymer structure, to insert the penetrant molecules in its free-volumes and then to simulate the motion of these penetrant molecules in the polymer matrix. Determining the size and rate of these motions makes it possible to calculate the diffusion coefficient and characterize the diffusional mechanism. 

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