Calculated geometries

Because geometry optimization is so much more time-consuming than a single geometry calculation, it is common to use different levels of theory for the optimization and computing hnal results. For example, an ah initio method with a moderate-size basis set and minimal correlation may be used for opti-... 

It is quite common to do the conformation search with a very fast method and to then optimize a collection of the lowest-energy conformers with a more accurate method. In some cases, single geometry calculations with more accurate methods are also performed. Solvent effects may also be important as discussed in Chapter 24. 

A distance geometry calculation consists of two major parts. In the first, the distances are checked for consistency, using a set of inequalities that distances have to satisfy (this part is called bound smoothing ) in the second, distances are chosen randomly within these bounds, and the so-called metric matrix (Mij) is calculated. Embedding then converts this matrix to three-dimensional coordinates, using methods akin to principal component analysis [40]. 

There are many extensive reviews on metric matrix distance geometry [41-44], some of which provide illustrative examples [45,46]. In total, we can distinguish five steps in a distance geometry calculation ... 

Figure 3 Flow of a distance geometry calculation. On the left is shown the development of the data on the right, the operations, d , is the distance between atoms / and j Z. , and Ujj are lower and upper bounds on the distance Z. and ZZj, are the smoothed bounds after application of the triangle inequality is the distance between atom / and the geometric center N is the number of atoms (Mj,) is the metric matrix is the positional vector of atom / 2, is the first eigenvector of (M ,) with eigenvalue Xf,. V , r- , and ate the y-, and -coordinates of atom /. (1-5 correspond to the numbered list on pg. 258.)...

Geometries Calculated bond distances are typically within 0.02A of experimental values and bond angles within 2°. Geometries for transition states are not available from experiment. Here it must be assumed that the calculations provide descriptions of similar quality to those for equilibrium geometries. 

Fig. 2. Geometries calculated (solid lines) and observed (bold dashed lines) for 1-propanol in its a-cyclodextrin adduct. G3 and G6 denote the numbers of glucopyranose units of a-cyclodextrin. H3 and H5 refer to the hydrogen atoms located inside of the cyclodextrin cavity. The hydrogen atoms for the observed geometry of 1-propanol are not shown, since their atomic coordinates have not been determined. The observed 1-propanol is twofold disordered, with site a occupied 80%, site b 20%. Interatomic distances are shown in bold italics on fine dashed lines (nm). Reproduced with permission from the Chemical Society of Japan...

A possible role for shape resonances has been postulated in a number of the photoionization studies mentioned above [52, 53, 57, 60], although it has to be noted that except for camphor [57], the evidence for the existence of the shape resonance is not definitive. (It also then remains an open question how any such resonances, inferred from fixed geometry calculations, would manifest themselves in practice in large, and sometimes floppy, molecules, such as these chiral species.)... 

Widmer H, Widmer A, Braun W. Extensive distance geometry calculations with different noe calibrations - new criteria for structure selection applied to sandostatin and BPTI. J Biomol NMR 1993 3 307-324. 

Pogozheva ID, Lomize AL, Mosberg HI. Opioid receptor three-dimensional structures from distance geometry calculations with hydrogen bonding constraints. Biophys J 1998 75 612-634. 

The information stored in the specimen database is sufficient to identify the particular specimen and the material from which it is made. Other parameters provide information on the orientation of the specimen and on its unit cell parameters. These latter parameters are used by the data collection tasks and the crystal geometry calculation function to determine diffraction angles, the angles between crystal planes, etc. The user can store information on several specimens in the specimen database, thus permitting him to easily remount and rerun a specimen after looking at the collected data. 

The differences between the geometries calculated with the two different sets BS1 and BS2 are very small, deviations in the bond lengths of 0.01 A as well as changes of torsional angles of up to 4° are observed. Energetics present however some differences between the two basis sets, as seen in Table 6. The agreement between experimental data and computational results improves with the larger basis set. 

Transition state geometry Calculated activation energy/kcal mol 1 ... 

Figure 3.2 The effect of glancing incidence geometry, calculated for a 150 nm lno.5i77Gao. 4823As film on a (001) InP substrate. The symmetric 004 and the grazing incidence 044 curves are shown...

Fig. 5. Display of the ELFfor various values of din the C2x, geometry, calculated with the TopMoD series of programs.

Molecular modeling, like all other technical disciplines, has its own jargon. Much of this is described in Appendix B (Common Terms and Acronyms), and only one aspect will be addressed here. This concerns specification of theoretical model used for property calculation together with theoretical model used for equilibrium (or transition-state) geometry calculation. 

---