Energy and hardness differences

Consider a system that is composed of several molecules that interact with each other. The total energy difference between an initial state, that will be considered here as the state when the interacting molecules are very far apart from each other, and any other state, when all molecules are close to each other, is given by 

In order to evaluate the electronic energy difference one can make use of the expression [15] 

Note that since the integral of the electronic density over the whole space is equal to the total number of electrons, and the integral of the fukui function over the whole space is equal to one, then pg(r) integrates to Ng, 

Equation (26) may be derived from Eqs. (1) and (3), the second order functional expansion [13,28,31,32] of F[p] in terms of its functional derivatives. 

by assuming that the core density of the system remains unchanged at any distance during the interaction, and that there is practically no overlap between the core densities of all the atoms that form part of the molecules that interact with each other, then one can show that if Ng(( N, the sum of the terms in ( Eg fg[pf ] - E(.(,re[Pi]) associated with the first two terms in the right hand side of Eq. (27) is approximately equal to - AVjsjn. If it is further assumed that the core terms difference related with the second functional derivatives of the kinetic and the exchange-correlation energies cancel each other, one finds, from Eq. (32), if the chemical potential remains constant, that 

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Gazquez JL (1997) Bond energies and hardness differences. J Phys Chem A 101... 

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