34 YOS, BADE AND JEHLE 



•sents actual macromolecules by oscillator sets, these sets will usually show 

 such a wide distribution of frequencies that neither the classical nor the quan- 

 tum limit results can serve as an adequate basis for the discussion of specificity. 



Previous to knowing the work of Hamaker (1937) and de Boer (1936), the 

 present procedure was followed which calculates the many oscillator case and 

 covers the entire range of frequencies. This procedure serves to define and 

 estimate the degree of specificity of the London force. 



This method provides a way of discussing the number of independent param- 

 eters upon which the rearrangement free energy effectively depends, and the 

 degree of specificity. Specific interactions involve discrimination between a very 

 large number of different molecular partners even though only moderate 

 spreads in the interaction free energy occur. In the present theory this phe- 

 nomenon is simply a consequence of the multidimensionality of the situation 

 (Jehle, 1950). 



The oscillator scheme is very convenient but not necessary. The calculations 

 can also be carried out on the basis of a general level scheme (Yos, 1956). That 

 general calculation readily permits inclusion of anharmonicities, permanent 

 electrical moments and quadrupole interactions. 



Free Energy Change for a Pair 



Let the molecules I and II have Xi and Nu oscillators, respectively. The 

 normal modes of the combined system I-II have the frequencies o^i/lir which 

 depend on the intermolecular distance R. The partition function and the free 

 energy of the pair are 



00 00 f Ni+Nii "\ 



^ = E • • Z exp - E (ui + y2)hc^i/kT\ 



"1=0 nA-i+A'ii=0 I, ? = 1 J 



n 



exp {—h(joi/2kT\ ^ exp {—uihui/kT} 



ni 



= n !2 sinh {hw,/2kT)\~' (2) 



;=i 



A = kT E Ir. [2 sinh {hu>i/2kT)] 

 1=1 



\ /,2 2 \ 00 / >,2 2 



1 , f fi cjoi \ , •'^r^ , / , , n 0)1 



= ^rE In'^J + Eln 1 + 



f 2 \4k''Ty t^i \ 4F72 5V 



(3) 



This form is suitable for expansion in powers of the intermolecular interaction. 

 This last expansion is convergent for all positive h^0i)r/4k^T". With properly 

 normalized normal coordinates, the potential energy matrix V^ has the eigen- 

 values >'2<^r. Let K be the matrix (^-/2^"^")V with eigenvalues F/ = h-cor/4k-T\ 



