SPECIFICITY OF LONDON-EISENSCHITZ-WANG FORCE 35 



and let V be this diagonalized form of V. The matrix V can be written in the form 

 /V, \ /O at\ 



v = ( ] + { ^ ] = y^ + u = vM + v„-'u), (4) 



\0 VuJ \1l 0/ 



where '^ indicates the transposed matrix. Usually Foo and U do not commute. 

 The summation over / in the Helmholtz free energy (3) can be written as the 

 trace of the kT \], ecju. (3), of the diagonal matrix V and this trace is invariant 

 under the transformation which brings V into diagonal form. Thus 



A = kT tr l^]n V -{- J2 1" (I + V/sV) + I In 2I 



= ;^r tr|^ln[F.(I + V^'U)] 

 + Z In [(I + VjsV){l + (I + VjsVr'U/sV)] + I In 2) (5) 



s=l 



Now tr In x = tr In .v' = XI' ^'"^ •^"'' = ^'"^ II /(■^■'?) = 1" det .r, and also det (xy) = 

 det (x) det (y). Hence tr In {xy) = tr In .v + tr In y. Using the abbreviations 



V^-'U ^ Uo , (5VI + V^)-'U ^ Lh , (6) 



one can express the change in free energy as 



AA = A - A^ = kT tr l^ In (I + f/o) + E In (I + Us) . 



[Z s=l J 



Because of the form (4) of U, the trace of odd powers of Us vanishes, so that, 

 upon expansion in powers of Us this formula becomes 



AA =UTtr Z i-lUs' - ■■■]. (7) 



Actually the oscillators are distributed throughout the volume occupied by 

 each molecule. The assumption of an interaction matrix U (4) implies the 

 neglect of quadrupole and higher multipole terms in the expansion of the inter- 

 action in powers of the distance R between the centers of the molecules. This 

 assumption may be an admissible approximation for sufficiently large inter- 

 molecular separation R, but it is certainly a very crude assumption in view of 

 the fairly close approaches occurring in specific interactions. "Large" separa- 

 tions R are a necessary assumption insofar as the quadrupole terms would not 

 give simple general results as the dipole terms do. For such large separations R, 

 the terms UJ^ in the expansion (7) are presumably the only ones of importance, 

 and they make the free energy change vary as R~^. 



