Structural isomorphisms

Redundant, isomorphic structures have to be eliminated by the computer before it produces a result. The determination of whether structures are isomorphic or not stems from a mathematical operation called permutation the structures are isomorphic if they can be interconverted by permutation (Eq. (6) see Section 2.8.7). The permutation P3 is identical to P2 if a mathematical operation (P ) is applied. This procedure is described in the example using atom 4 of P3 (compare Figure 2-40, third line). In permutation P3 atom 4 takes the place of atom 5 of the reference structure but place 5 in P2. To replace atom 4 in P2 at position 5, both have to be interchanged, which is expressed by writing the number 4 at the position of 5 in P. Applying this to all the other substituents, the result is a new permutation P which is identical to P]. 

With the chemical structure of PbTX-1 finally known and coordinates for the molecule available from the dimethyl acetal structure, we wanted to return to the natural product crystal structure. From the similarities in unit cells, we assumed that the structures were nearly isomorphous. Structures that are isomorphous are crystallographically similar in all respects, except where they differ chemically. The difference between the derivative structure in space group C2 and the natural product structure in P2. (a subgroup of C2) was that the C-centering translational symmetry was obeyed by most, but not all atoms in the natural product crystal. We proceeded from the beginning with direct methods, using the known orientation of the PbTX-1 dimethyl acetal skeleton (assuming isomorphism) to estimate phase... 

In the four isomorphous structures the different guest components are located in a similar crystalline environment. Since the species involved are of nearly identical... 

Powder diffraction can confirm whether two similar compounds, where one metal substitutes for another for instance, have an isomorphous structure. 

Fig. 4. Directed rooted trees of the Kekule patterns of coronene. Note their isomorphic structures and the similarity in their local structures...

Common superscripts indicate isomorphous or nearly isomorphous structures. 

In addition to being coordinatively highly unsaturated, small clusters also are dominated by their surface properties. For example, a 19 atom cluster may have at most one or two interior atoms. Even a 100 atom cluster has only about 28 interior (bulklike ) atoms. Thus the surface sensitive properties become increasingly important as the size becomes smaller. Clusters do not possess the long range periodicity one would have with a bulk crystal. Thus we expect that each small cluster will be a unique entity, and that its ground state structure (or structures if different isomorphic structures are nearly isoenergetic) will depend sensitively on size and likely will not be simply a subunit of the bulk lattice. 

Vijayalakshmi J, Padmanabhan KP, Mann KG, Tulinsky A. The isomorphous structures of prethrombin2, hirugen-, and PPACK-thrombin changes accompanying activation and exosite binding to thrombin. Protein Sci 1994 3 2254-2271. 

When crystals are noncentrosymmetric, the situation becomes more complicated because there are two possible phases angles that are derived for each reflection for data from one isomorphous pair. Therefore, in order to derive the phase angle, more than one isomorphous pair is required. This problem was first tackled by Johannes M. Bijvoet, Cornells Bokhoven and Jean C. Schoone, who solved the noncentrosymmetric isomorphous structures of strychnine sulfate and selenate. ... 

For structure isomorphism problems, the atom-atom correspondence of the two isomorphic structures is calculated from a pair of starting atoms. One starting atom is chosen from the first structure in such a way that it shares the fewest identical EV values with other atoms of the same structure. The other starting atom is chosen from the second structure so that... 

The three methanesulfonyl derivatives, sulfur, selenium, and tellurium dimethanethiosulfonates, are isomorphous. Structure analyses have been carried out, starting with tellurium dimethanethiosulfonate and using tellurium as a heavy atom in the first stages. So far, the structure of the tellurium compound has been published in detail 128), and an electron density projection along the short crystal axis of the trisulfide (98) has shown the unbranched sulfur chain structure. 

Because of the isomorphous structures of the four compounds and their phase instabilities, they are an interesting set of compounds for detailed lattice-dynamic calculations. However, despite their relative simplicity with respect to other metal azides, their structure, with eight atoms per primitive unit cell, presents a formidable calculational problem. With compounds of this complexity it is imperative that dispersion-curve data be available to test lattice-dynamic models, and, thus far, this has been possible only for KN3. 

Water confined between two hydrophilic objects forms an interesting and important system with unique properties. Sometimes the amount of water molecules confined between two hydrophilic surfaces is rather small, consisting only of a few tens (or even fewer) of water layers. In such cases, the stmctural arrangement of water molecules can become quite distorted compared with the bulk stmcture. Also, the order imposed by each hydrophilic surface can propagate inwards and counter each other. Water s ability to form many isomorphic structures is tested in such a situation, as we discuss below. 

In the isomorphous structure of benzophenone and biphenyl complexes, the 3 cyd molecules are associated in head to head dimers all secondary hydroxyl ends of the two partners are connected by hydrogen bonds. These dimers are stacked along the c axis to produce channels... 

Crystals are described as being isomorphous with one another if the spatial arrangement of the atoms is the same, despite the chemical composition differing. Isomorphous structures are commonly foimd in transition metal salts, whereby the difference in sfrucfures between many of the first-row transition metals is only marginal due to size difference of the M + species. The intermolecular interactions within isomorphous crystals are therefore the same and it can reasonably be assumed that the maimer of their growth and assembly is also identical. In such... 

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