SPECIFICITY OF LONDON-EISENSCHITZ-WANG FORCE 41 



According to the definition of Ws , one has for a particular molecule always 



-W|.+ii < -'^^Vi > 



and for the manifold of molecules the distribution of adjacent \Vs and Ws+i 

 are correlated. For 5 » h(ui)„,^^/2TrkT (where (a)/),nax is the highest oscillator 

 frequency with appreciable polarizability), W^ drops to negligible values. 



A more general molecular manifold than that of Fig. 3, a manifold of macro- 

 molecules of a particular size (or number of oscillators N) with specified separa- 

 tion R, can have a many-dimensional Ws distribution. By definition, the number 

 of parameters on which the manifold of W, vectors depends limits the dimen- 

 sionality of the manifold. The set of all 



W,(-^ < s < -\- x) 



is of course not a set of independent quantities. If N denotes the number of 

 oscillators which each molecular type possesses, there are at most 2N parameters 

 (the oscillator frequencies and their polarizabilities) on which the W^ depend. 

 The Ws distribution can therefore not span out more than 2N dimensions (con- 

 dition 1), but that may be a very large number. But the presence in a molecule, 

 of several oscillators which all have about the same frequency and orientation, 

 does not provide for more independent parameters than a single oscillator. 



In order to investigate the effective dimensionality of the Ws distribution 

 of the manifold of macromolecules, one should first of all define a limit of 

 discrimination, i.e. a limit for the magnitude of the rearrangement free energy 

 AiAi II (at intermolecular separation R) such that the rearrangement tendency 

 becomes dominant over Brownian motion. One can define it by (—AiAi n)discr = 

 kT or 



IZ (^^^ I - "^^^ ii)'}discr = {-2A4.4i u/kT} = 2 (15) 



or one can define the discrimination limit between an arbitrary molecule and an 

 average molecule, cf. the circles in Fig. 4. 



{X ^^'^ - <W.>Av)'}discr = 2 (15a) 



The right side of Fig. 4 pictures a strongly correlated distribution which one 

 may call "effectively" a one dimensional distribution. The left side of Fig. 4 

 refers to the special manifold of the type which was discussed along with Fig. 3. 

 Fig. 4 illustrates a distribution of certain practically uncorrelated abscissae 

 and ordinates (representing the W^ distribution), each of which shows a mean 

 square deviation larger than or even large compared with the discrimination 

 limit 2 (equation 15a). That this be the case one can regard as a necessary pre- 

 requisite for calling the manifold "effectively" a two dimensional one. The 



