Linear Response Theory The General Framework

The dielectric response of a solvated protein to a perturbing charge, such as a redox electron or a titrating proton, is related to the equilibrium fluctuations of the unperturbed system through linear response theory [49, 50]. In the spirit of free energy 

fp is the field due to pp P is a polarization density, which is related to the electric field E produced by the remaining charge density p E = -47tP. Finally, a dot represents the dot product between functions, / fp(r) P(r)dr, where the integration is over all space. We also introduce  

The constant of unity in the parentheses on the right can be dropped. Expanding the dot product on the right, we have 

The spatial dependencies have been made explicit in the last equation. Finally, we see that  

If AU( 1) has Gaussian fluctuations, this means that the free energy is also a parabolic function of AU(1) itself. AU( 1) is known as the energy gap [52-54] we will denote it by p 

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