IXTEEMOLECULAR FORCES AND INTERACTION ENERGY 223 



the dipole center, cp = — Uq^i-JIcI^D-. When the molecules are freely ro- 

 tating one may take the average field of the dipole {F =\/ 2juld^), from 

 which the average potential energy due to the induced moment is 99 = 



— UojU'ld^D'-. Two identical dipoles will mutually polarize each other and 

 the potential energy will then be 9? = — 2aoju^ld^D^. The type equation 

 (p ^ — AaQi-i^jd^D- may be written in terms of kcal/mole: 



^ _ _ 14.4 A -^^ kcal/mole (6-31) 



d^D- 



cp = - 5.71 A ^^ kcal/mole (6-32) 



when Uq is in units of 10-^4 ml/molecule, R^ is in ml/mole, // is in debyes, 

 and d is in A. Such energies are usually quite small unless the dipole is 

 very near the molecule that is polarized. Two water molecules at a distance 

 apart of 5 A would involve an energy due to mutual induction of only 



— 0.009 kcal/mole, which may be compared to the — 0.384 kcal/mole 

 previously calculated for the potential energy of a water molecule at the 

 same distance from an ion. Even at 3 A separation in a mutually fixed 

 configuration the energy would be less than 0.1 kcal/mole. 



Electrokinetic Interactions (Dispersion or London Forces) 



Neutral nonpolar molecules or groups attract one another despite the 

 apparent lack of any charge localization. Such forces have been recognized 

 for a long time and often are the major contribution to van der Waal's 

 interaction. However, the theories of classic electrostatics were unable to 

 explain such attraction and the nature of such forces was unknown until 

 London in 1930 pointed out the relationship to optical dispersion and de- 

 rived expressions for these interactions on a quantum mechanical basis 

 (London, 1930, 1937). A symmetrical molecule will at any instant possess 

 a dipole moment due to fluctuations in the relative positions of the nuclei 

 and outer electrons; this instantaneous dipole will create a field that will 

 polarize an adjacent molecule. Likewise, the second molecule will possess 

 a fluctuating dipole which will polarize the first molecule. There will thus 

 arise a coupling between the oscillations of the electrons of the two mole- 

 cules so that the statistical distribution of electrons will continuously 

 favor attraction. There is not a single dipole in a molecule but many oscil- 

 lating dipoles; these dipoles of different moments must be integrated for 

 each molecule to arrive at the interaction energy. Since the London formu- 

 lation has been used in estimating interactions in protein and enzyme 

 systems, it will be presented first, following which some more recent sug- 

 gestions for improvement will be considered. 



