Reversible RESPA algorithm

One step of the multiple timestepping algorithm is then implemented as follows Reversible RESPA Algorithm 

Multiple timestepping is widely used for efficiency purposes in systems with expensive force computations, but it should be used cautiously. To illustrate the potential dangers of multiple timestepping, one need only consider its application to a linear one-DOF model problem 

The frequency is therefore S2 = Vl + In the limit of small inner step, we solve 

Both of these matrices have unit determinant for any h, so the product of the eigenvalues of W/, is 1. On the other hand, a short calculation shows that 

For a given h, the eigenvalues of Wh are either complex conjugates or they are real and reciprocal to one another. If they e complex conjugates with nonzero imaginary part, then we have A1A2 = AiAi = 1, so Ai = 1 and the two eigenvalues are clearly distinct and on the unit circle in the complex plane. 

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An excellent illustration of the power of the reversible RESPA algorithm is the study of HIV-Fprotease that is described in [30]. Here, the authors studied the HIV-I protease complexed to an inhibitor, Saquinovir, in explicit water (5041 water molecules). The simulation box had a volume of 28125 A and contained 18354 atoms. 

The new reversible-RESPA [25] exploits the Trotter factorization [27] to make the algorithm reversible. Re-writing the equations of motion, for a system of N particles having coordinates q. .. qn and momentapi... p, to incorporate both the fast and slow components of the force 3uelds [28] ... 

Reversible Reference System Propagator Algorithms (r-RESPA) 299... 

Reversible Reference System Propagator Algorithms (r-RESPA) 307 Thus the propagator in Eq. (27) produces the following dynamics algorithm ... 

Since many systems of interest in chemistry have intrinsic multiple time scales it is important to use integrators that deal efficiently with the multiple time scale problem. Since our multiple time step algorithm, the so-called reversible Reference System Propagator Algorithm (r-RESPA) [17, 24, 18, 26] is time reversible and symplectic, they are very useful in combination with HMC for constant temperature simulations of large protein systems. 

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