Multidimensional energy surface

Indeed, before 2000, single conformers had been basically considered, although it was possible that the multiple torsional degrees of freedom of these molecules would produce a complex multidimensional energy surface, and therefore a variety of putative 3D geometries capable of interacting with the tubulin binding site [62, 63], Numerous research teams pursued different indirect approaches to the pharmacophore... 

To determine the deepest minimum on the multidimensional energy surface as a function of many structural parameters is a formidable mathematical task. Usually, simplifications and assumptions are... 

In a more general case a set of internal coordinates (see e.g. Ref. > and define a geometry and we are dealing with a multidimensional energy surface instead of the two dimensional exemplified above. The ground state geometry (or geometries) is (are) then defined as a minimum (minima) on the potential surface with respect to variation of all gi. 

Looking from this basis Into the future one sees as a most essential task the development and application of methods for calculating the multidimensional energy surfaces not only of simple but also of more complex molecules In their ground and excited states. Mo less Important, the "traffic rules" that determine which pathways are chosen going from (excited) starting material to the various (photo)products will have to be traced and worked out. 

The elements of the transition rate matrix A in the DRIS model are estimated from the multidimensional energy surface associated with the interdependent rotation of neighboring bonds u g Kramers rate theory, as described above. Accordingly, the probability of occurrence of a given isomeric state for a bond depends on the state of its first neighbors aloi the chain. Likewise, in the kinetic Ising model the transition rate Wi(Oi) of the i-th spin is assumed to be coupled to the state of its first iKighbors by the relation... 

Fig. 2 - Internal potential energy of isotactic P(S)3MP. In the parts A and B are reported the regions of the multidimensional energy surface corresponding both to a left--handed helix for the two different conformations of the lateral group observed in the crystalline state. In the part C the minimum energy region corresponding to a right-handed helix is reported. The energies, referred to the absolute minimum (part B e, - 180 and 0 6O ), are in Real (mol of CRU) " .

Besides the energy factors, defined by the closepacking principle, entropic factors guide the mode of packing of molecules. A molecule in a crystal tends to maintain part of its symmetry elements, provided that this does not cause a serious loss of density. As outlined by Kitajgorodskij, in a more symmetric position a molecule has a greater freedom of vibrations that is, the structure is more probable (entropy factor) because it occupies a wider potential well on the multidimensional energy surface [73]. 

The problem of finding the conformation of lowest energy is rather simple for a terminally blocked amino acid (such as a glycyl or alanyl residue) since it is a problem including only two variables (, 0) however, the location of the minimum becomes more difficult to determine even for a terminally blocked amino acid when rotations x are also included. A fortiori, when the molecule contains more than one amino acid, difficulties in finding the deepest minimum arise. As previously discussed (Section 4.1.1), the multidimensional energy surface has multiple minima. Since there is no procedure... 

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