Obtaining Thermodynamic Perturbation Formulas

Free energy calculations rely on a well-known thermodynamic perturbation theory [6, 21, 22], which is recalled in Chap. 2. We consider a molecular system, described by the potential energy function U(rN), which depends on the coordinates of the N atoms rN = (n, r2. r/v). The system could be a biomolecule in solution, for example. We limit ourselves to a classical mechanical description, for simplicity. Practical calculations always consider differences between two or more similar systems, such as a protein complexed with two different ligands. Therefore, we consider a change in the system, such that the potential energy function becomes  

The cumulants [26] are simple functions of the moments of the probability distribution of 5V-.C2 = (V- V))2),C3 = (V- V)f),C4 = ((]/-(]/))4) 3C22,etc. Truncation of the expansion at order two corresponds to a linear-response approximation (see later), and is equivalent to assuming V is Gaussian (with zero moments and cumulants beyond order two). To this order, the mean and width of the distribution determine the free energy to higher orders, the detailed shape of the distribution contributes. 

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To obtain thermodynamic perturbation or integration formulas for changing q, one must go back and forth between expressions of the configuration integral in Cartesian coordinates and in suitably chosen generalized coordinates [51]. This introduces Jacobian factors... 

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