Systems, reference

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

Assuming a hard sphere reference system with the pressure given by... 

The most conunon choice for a reference system is one with hard cores (e.g. hard spheres or hard spheroidal particles) whose equilibrium properties are necessarily independent of temperature. Although exact results are lacking in tluee dimensions, excellent approximations for the free energy and pair correlation fiinctions of hard spheres are now available to make the calculations feasible. 

The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

The high-temperatiire expansion, truncated at first order, reduces to van der Waals equation, when the reference system is a fluid of hard spheres. 

Truncation at the first-order temi is justified when the higher-order tenns can be neglected. Wlien pe higher-order tenns small. One choice exploits the fact that a, which is the mean value of the perturbation over the reference system, provides a strict upper bound for the free energy. This is the basis of a variational approach [78, 79] in which the reference system is approximated as hard spheres, whose diameters are chosen to minimize the upper bound for the free energy. The diameter depends on the temperature as well as the density. The method was applied successfiilly to Lennard-Jones fluids, and a small correction for the softness of the repulsive part of the interaction, which differs from hard spheres, was added to improve the results. 

This implies, with the indicated choice of hard sphere diameter d, that the compressibilities of tlie reference system and the equivalent of the hard sphere system are the same. 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

S. J. Stuart, R. Zhou, and B. J. Berne. Molecular dynamics with multiple time scales The selection of efficient reference system propagators. J. Chem. Phys., 105 1426-1436, 1996. 

R. Zhou and B. J. Berne. A new molecular dynamics method combining the reference system propagator algorithm with a fast multipole method for simulating proteins and other complex systems. J. Phys. Chem., 103 9444-9459, 1995. 

Linearized Reference System We first formulate the following linearized Langevin system at some reference point (e.g., X", AT" + V") ... 

Molecular Dynamics in Systems with Multiple Time Scales Reference System Propagator Algorithms... 

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

The system defined by the Liouvillian is called the reference system. Now applying the Trotter factorization to the propagator exp iLs + Fi arising from this subdivision gives the new propagator,[17]... 

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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