Electrostatic long range multipole interactions, penetration effects

1 Electrostatic Long Range Multipole Interactions, Penetration Effects 

The first order, electrostatic, interaction energy is defined as  

The multipole operators are here defined as spherical tensors  

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic serie Still, for the long range the multipole approximation to can be quite 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic series . Still, for the long range the multipole approximation to AE can be quite accurate, if properly truncated (for instance, after the smallest term). For shorter distances, the penetration between the molecular charge clouds becomes significant, the screening of the nuclei by the electrons becomes incomplete even for neutral molecules, and the power law for AE 5 is modified by contributions which increase exponentially with decreasing R. These penetration contributions we define as  

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