32 YOS, BADE AND JEHLE 



of all the nearest neighbors to molecules of type i; because of (d), = 5Ni = 

 ^j hiij holds for all i. The total rearrangement free energy is 



y'2zliZ-j 5»,jA,4/j = , subtracting the preceding equation, 



= }''2^2iZl 5»,;(A.4,j — A.i,-,) = , interchanging dummy indices, 



= 3-^2 Z Z ^nji{^Aji - A.4y,) 



With buji = biiij and A/4ji = A,4,j , the rearrangement free energy becomes 

 ->i Z E ^nij{^Ai, + A.lyy - 2A/lo) 



i.e. it can again be expressed in terms of the quadruplet rearrangement free 

 energies. 



It is possible to determine the sign of the quantity A4.4i n under very general 

 circumstances, in which the assumptions (a), (b), (c), (d) above are not neces- 

 sarily valid (Yos, 1956, and the specificity theorem below). If these assumptions 

 are not made, the free energy change due to the rearrangement cannot be 

 evaluated so simply, and the above discussion can then serve only as a rough 

 guide to the interpretation of inequalities involving A4yli n . 



Some processes of crystallization present a similar situation. A crystal of a 

 globular molecule type I may be surrounded by a mixture of molecules I and 

 II, both of equal size and similar chemical constitution but of different polar- 

 izability constitution. An additional molecule I joining the crystal involves an 

 integer multiple of A4.4i u as free energy change. 



The Simplest Rearrangement Inequalities 



The London force is due to the interaction of polarizable molecules. In the 

 most elementary case one represents a simple molecule by a single isotropic 

 oscillator whose (circular) frequency may be denoted by co. 



In the classical limit, when the oscillator frequencies are very small, 



w « kT/h, 



the interaction of a pair of isotropic oscillators is 



A.Iiii = -3R-'kTaiau (la) 



This leads to the inequality 



A4/I1 II = -3R-'kT{ai - an)- < (lb) 



Here the free energy change A4.4i n depends only on one parameter, the polar- 

 izability difference. 



