SPECIFICITY OF LONDON-EISENSCHITZ-WANG FORCE 43 



dimensional one. It is assumed ihal the distributions of molecule types are 

 reasonably uniform in the W, intervals, and for the purpose of comparison it is 

 assumed that the mean scjuare deviations (X]s {'^'^s — (Ws)av)-)av of the two 

 distributions in Fig. 4 have the same value, i.e. that the two distributions have 

 rearrangement energies of the same size. 



The notion ''degree of specificity at separation R" refers to the degree of dis- 

 crimination of some particular molecular type I when confronted, each time in a 

 quadruplet fashion, with a manifold of types of molecules II. Degree of specific- 

 ity can be defined as the measure of the subset of types II discriminated against 

 when confronted with type I, divided by the measure of the total set of all 

 types II in the manifold. Generalizing this definition, one can form an average 

 over different molecules I by taking I as well as II from a given manifold of 

 molecular types, all of a similar category and size. Alternatively, one confronts 

 an average molecular type (I) of the manifold with all other types (II) of mole- 

 cules chosen at random from this same manifold (cf. Fig. 4). On the left in that 

 figure, in the many (two) dimensional case, the relative measure of the non- 

 discriminated subset (i.e. one minus the degree of specificity) is the ratio of the 

 number of points inside the hypersphere "c?-tant" (15a) (indicated by a circle 

 quadrant), to the total number of points. On the right, in the one dimensional 

 case, it is the ratio of the number of points inside the linear interval (I5a) to the 

 total number of points. The latter ratio is of the order of magnitude of the ratio 

 of the non-discriminated interval length ^/~2 to the length occupied by the 

 total set of points. The former ratio (left side of Fig. 4) is a product of several 

 (two) such fractions and may readily become much smaller. If the polarizabili- 

 ties are suthciently strong, and of diversified frequency, and if the separation 

 R is small, the relative measure of the non-discriminated subset becomes very 

 small, and that means a high degree of specificity, close to unity. As the actual 

 case (10c) involves oscillators oriented in three dimensions even the situation 

 of Fig. 3 involves an effectively many dimensional manifold "Ws^v . 



In simple words one can summarize the essential point: The question was; 

 why does a many (two) dimensional distribution show so much higher a de- 

 gree of specificity than a one dimensional distribution (cf. Fig. 4), even 

 though in both cases there may be, in the average, the same amounts of re- 

 arrangement free energies A4A1 n involved? The answer is this: In the one di- 

 mensional distribution (right side of Fig. 4) one might perhaps pick out five 

 (about equidistant) dots, \/2 apart from each other, so that no two out of 

 these five dots are closer together than the discrimination limit s/l. In the 

 many (two) dimensional distribution (left side of Fig. 4) one can pick out a 

 good many more dots such that no two of them are closer together than \/2. 

 This is so simply because it is a many dimensional distribution, even though 

 the average distances between pairs of dots is the same in both distributions 

 illustrated in Fig. 4. 



It is evident that the degree of specificity is highly sensitive to the values of 



