INTERMOLECULAR FORCES 87 



of the riA electrons in A, then MaCoIc) is the transition dipole moment of A con- 

 nected with the jump from the ground state to the k-th state: 



Va (ok) = n.,e j i/'A*(o)rii/'A(k) dndr.- • -drn^ (3) 



The energy recjuired for the molecule to go from the ground state to the k-th 

 state is EA(k) — Ea(o). Similar transition dipoles can be defined for molecule 

 B. Then if A and B were small molecules, the dispersion energy of interaction 

 between A and B would be 



£ ^ _ 2 y^ MA^(ok)MB^(ok') 



'" 3R«b ife (EA(k) - Ea(o)) + (EB(k') - Eb(o)) ^^^ 



Usually there is only one principal charge transfer transition so that it is only 

 necessary to consider a single excited state k for molecule A and a single excited 

 state k' for molecule B. Unfortunately, Eq. (4) does not hold for large mole- 

 cules except at extremely large separations. London (1942) showed that for large 

 molecules the notion of transition dipoles is no longer useful. Instead one must 

 consider the exact spatial configuration of the "transition charge distribution", 



PA(ok) = UAe Ji/'A(o)*^A(k) dr.- • -drn, (5) 



Here PA(ok) is a function of position within the molecule. There are zones in 

 which pA(ok) is positive and other zones in which it is negative depending 

 on the signs of the wave functions i/'a(o) and i^A(k). These zones are bounded 

 by the nodal surfaces of these two wave functions. Thus the i-th zone has the 

 transition charge 



(i 

 Ca 



'''(ok) = [ PA(ok)dr (6) 



•'i-th zone 



and its centroid is located at the position 



'^(ok) = [ rpA(ok)dr/eA^'^(ok) (7) 



•'i-th zone 



R 



These zones may be called transition monopoles. The energy of dispersion of 

 large molecules is then 



"Z e/'\ok)e^'"\ok')/ I RA^''(ok) - RB''"(ok') | > 

 Edis = - ^'"'" .^ .. . ^ ^ ., ,. i (8) 



(EA(k) - Ea(o)) + (EB(k') - Eb(o)) 



Here |Ra ' (ok) — Rb ' (ok') | is the distance between the i-th monopole in A 

 and the i'-th monopole in B. From Eq. (8), we see that the dispersion energy 

 of two large molecules can become very large when two transition monopoles 

 come close together. 



