Grassmann binary

In this chapter we describe the unified generation of stereoisomers including conform-ers of a molecular structure [102,103,105]. This method has the potential to generate stereoisomers that cannot be described in terms of stereocenters, stereogenic double bonds or single bond rotations. Fundamentals such as the concept of a (partial) orientation function are discussed, and mathematical tools such as Radon partitions and binary Grassmann-Plucker relations are used to construct tests for abstract orientation functions. Some simple examples are treated in detail. 

First, an alternating, nontrivial (i.e. not constantly zero) function x from ri to +,-,0 fulfilling the binary Grassmann-Pliicker relations (defined below) is called a chirotope (of rank 4) over n= 0,..., n -1, according to Definition 3.5.3 in... 

As orientations are signs of determinants, it follows that the orientation function of any conformation fulfills the binary Grassmann Pliicker relations, introduced below (Herrmann Grassmann, German mathematician, 1809-1877, Julius Pliicker, German mathematician and physicist, 1801-1868). 

Definition (Binary Grassmann-PlUcker relation) if two quadruples of atoms and... 

Note that different binary Grassmann-Pliicker relations may lead to the seune condition. For example, the one corresponding to (2,3,0,1) and (6,5,1,0) leads to the same condition as the one discussed above. Test 4 checks all such criteria obtained by binary Grassmann Pliicker relations ... 

Definition (Chirotope) An alternating, not constantly zero function n —> +,0 fulfUlingthe binary Grassmann-Plucker relations... 

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