How to understand the correspondence between practical and theoretical free energy computation?

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I am trying to understand free energy calculation in NWChem.

In NWChem documentation the free energy difference between two states is approximated as a sum of internal QM contribution and solvation free energy. I called this as 'practical free energy computation'.
\Delta W_{AB} \approx \Delta W_{AB}^{int} + \Delta W_{AB}^{solv}
and \Delta W_{AB}^{int} = E_{B}^{int} - E_{A}^{int}

However, in the paper JCP-2007.127.051102 written by Marat, proposed a thermodynamic cycle to perform free energy calculation. I called this as 'theoretical free energy computation'.
\Delta W_{AB} = ( \Delta W_{AA}^{DFT-ESP} - \Delta W_{BB}^{DFT-ESP} ) + \Delta W_{AB}^{ESP}

According to my understanding, \Delta W_{AB}^{solv} should correspond to \Delta W_{AB}^{ESP}. But, how to construct the correspondence between \Delta W_{AB}^{int} and ( \Delta W_{AA}^{DFT-ESP} - \Delta W_{BB}^{DFT-ESP} )  ? Is there some approximation between these?

Any suggestion is appreciated. Thanks.

Edited On 2:06:12 AM PST - Fri, Nov 15th 2013 by Wjb0920

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