Alchemical Transformations

In the following we shall describe in details the theory of continuous alchemical transformations, with focus on the issues and technicalities regarding the implementation in molecular dynamics code using the Ewald method. As we will see, running a simulation using standard implementation of the Ewald methods of a system where atomic charges are varying, implies the insurgence of non trivial terms in the energy and forces that must be considered for producing correct trajectories. In a nutshell, Ewald resummations consists in adding and subtracting to the atomic point charges a spherical Gaussian charge distributions bearing the same charge, so that the electrostatic potential is split in a fast dying term (the Erfc term), due to the sum of the point charge and the neutralizing charge distribution and evaluated in the direct lattice, and in a slowly decaying term (the Erf term) due to the added Gaussian spherical distributions evaluated in the reciprocal lattice. Thanks to this trick, the conditionally convergent electrostatic energy sum is split in two absolutely convergent series. In standard implementations of the Ewald resummation technique, as we will see later on, the electrostatic potential at the atomic position ${\bf r}_i$ is actually not available with mixing of the interactions between alchemical and non alchemical species in the so-called Ewald reciprocal lattice contribution (i.e. the Erf part). The Smooth Particle Mesh Ewald method (see Chapter 3) makes no exception, with the additional complication that the atomic point charges (including the alchemical charges) are now smeared over nearby grid points to produce a regularly gridded charge distribution, to be evaluated using Fast Fourier Transform (FFT). Due to the extraordinary efficiency (see Figure 4.3), the Particle Mesh Ewald method is still an unrivaled methodology for the evaluation of electrostatic interactions in complex systems. Moreover, PME can be straightforwardly incorporated in fast multiple time step schemes producing extremely efficient algorithms for, e.g., systems of biological interest. For these reasons, it is therefore highly desirable to devise rigorous and efficient approaches to account for alchemical effects in a system treated with PME.