36 YOS, BADE AND JEHLE 



Specificity Theorem 



To proceed further, one must introduce the assumption that the inter- 

 molecular interaction It in ecjuation (4) can be written as a matrix product of 

 two matrices, one depending on molecule I only, the other on molecule II only: 

 11 = 1li1lii~. Dipole interactions clearly satisfy this assumption; ^i can be 

 written as a matrix with A^i rows and three columns, as in (12) below. Intro- 

 ducing the further abbreviations 



W.I ^ air(^vi + v,)~'^, , (8) 



which are three by three matrices, one can write 

 trt/o-=trI.^6I. ^L = tr (^ ^ D.-lt-^rV 



= 2 tr cui-i^iaiir-Uir^aiiiOir) (9) 



= 2 tr (airTJi-iaiiaiiri^ir^'Uii) = 2 tr (Woi Won ), 



tr UJ" = 2 tr (W. iW, „) = 2 tr (W, „W, i), (9) 



where 5 is any integer, positive, negative, or zero. Neglecting higher powers 

 of Us^ one can now express the ciuadruplet free energy difference in the form 



-AiA, u ^ -(A.4i I -f A.4„ II - 2A.4i n) 



+ 00 M 



= kT tr J^ - {W.iW^i -I- W^iiW.ii - 2W.iW.n} ^^^^^ 



-00 



= \kTj:,tr{{Wn - W.„)'} 



This expression is positive definite if (W^ i — Ws n) has real eigenvalues for 

 every s and any pair I-II, that is, if Ws i and Wg n are Hermitean matrices. 

 This provides a sufficient condition for 



A4.4i II < (10b) 



to hold in the limit of large distances, R. It can be shown (Ky Fan, 1951) that 

 the inec|uality holds also for the terms of higher order in the expansion (7). 



Since (sVI + Vj) is a symmetric matrix, Ws i and W^ n are by (8), sym- 

 metric matrices. The Hermitecity condition for W., i , Ws h thus implies that 

 all of the elements of W^ i and W s n are real. As (s'-tt-I + 'Ui)"^ has only real 

 matrix elements, and asIL = 'Ui'Uii~ is real, it follows that W^ i and W^ n will 

 be real if the elements of 'Ui andlln are either all purely imaginary or all real. 

 One is thus led to two suflicient conditions under either of which the quadruplet 

 inequality (10b) holds. 



