System-bath coupling factorization

However, it is common practice to sample an isothermal isobaric ensemble NPT, constant pressure and constant temperature), which normally reflects standard laboratory conditions well. Similarly to temperature control, the system is coupled to an external bath with the desired target pressure Pq. By rescaling the dimensions of the periodic box and the atomic coordinates by the factor // at each integration step At according to Eq. (46), the volume of the box and the forces of the solvent molecules acting on the box walls are adjusted. 

Similarly, if the simulation system is coupled to an external bath of pressure Pq, the average pressure that we wish our system to attain, then we re-scale both the box dimensions and atomic coordinates at each integration step by the factor... 

In a molecular dynamics calculation, you can add a term to adjust the velocities, keeping the molecular system near a desired temperature. During a constant temperature simulation, velocities are scaled at each time step. This couples the system to a simulated heat bath at Tq, with a temperature relaxation time of "r. The velocities arc scaled bv a factor X. where... 

The simplest method that keeps the temperature of a system constant during an MD simulation is to rescale the velocities at each time step by a factor of (To/T) -, where T is the current instantaneous temperature [defined in Eq. (24)] and Tq is the desired temperamre. This method is commonly used in the equilibration phase of many MD simulations and has also been suggested as a means of performing constant temperature molecular dynamics [22]. A further refinement of the velocity-rescaling approach was proposed by Berendsen et al. [24], who used velocity rescaling to couple the system to a heat bath at a temperature Tq. Since heat coupling has a characteristic relaxation time, each velocity V is scaled by a factor X, defined as... 

The pressure can similarly be held (approximately) constant by coupling to a pressure bath . Instead of changing the velocities of the particles, the volume of the system is changed by scaling all coordinates by a factor closely related to that shown in... 

Recently, much attention has been paid to the investigation of the role of this interaction in relation to the calculations for adiabatic reactions. For steady-state nonadiabatic reactions where the initial thermal equilibrium is not disturbed by the reaction, the coupling constants describing the interaction with the thermal bath do not enter explicitly into the expressions for the transition probabilities. The role of the thermal bath in this case is reduced to that the activation factor is determined by the free energy in the transitional configuration, and for the calculation of the transition probabilities, it is sufficient to know the free energy surfaces of the system as functions of the coordinates of the reactive modes. 

In this section we present results using the two approaches described in the previous sections the Trotter factorized QCL (TQCL), and iterative linearized density matrix (ILDM) propagation schemes, to study the spin-boson model consisting of a two level system that is bi-linearly coupled to a bath with Mh harmonic modes. This popular model of a quantum system embedded in an environment is described by the following general hamiltonian ... 

The temperature may be adjusted by coupling the system to a thermal bath as described by Berendsen et al. In this method the atomic velocities are scaled (multiplied) by a factor... 

The quantitative agreement with simulations of this work is not found with existing models. For future developments the most important conclusion is that the influence of nonlinear coupling is accounted for by a correction factor that dep>ends only on the particle masses independent of the system energy. While a generalization of this factor will be required for polyatomics (. involves the masses of all three atoms for a diatomic in a bath and is not a property of atom-atom interactions), it indicates that a scaled generalized Langevin approach will have the necessary prerequisites for quantitative accuracy. 

This approach allows for a fully quantum mechanical treatment of the dynamics, avoiding the nse of quantum correction factors used to denote classical dynamical approaches, with the concession that the potential energy surface must be expanded, ignoring higher order nonlinearity in the mode coupling. The potential energy surface is expanded with respect to the normal coordinates of the system, q, and bath, 01, and their freqnencies up to third and fourth order nonlinear conpling ... 

The radical clock experiments with MMO from M. capsulatus (Bath) and M. trichosporium OB3b carried out in our laboratory indicate that there may not be a single mechanism operative for these enzyme systems. Instead, the mechanism may depend on factors such as the steric and energetic requirements of the substrate, as well as the temperature employed in the MMO hydroxylation reaction. Differences in the MMO systems from the two different organisms include their optimal hydroxylation temperatures as well as sequence variations in the coupling protein B from the two organisms. ... 

In order to correct the deviation of the actual system temperature from the prescribed Tq, Berendsen et al. introduced a proportional method in which the system is weakly coupled to an external heat bath, and at each time step, the velocities of the atoms in the system are multiplied by a scaling factor ... 

At the beginning of this chapter we considered an isolated system. When this system has the energy E then there are Q. E) different microstates with this energy. If the system is not isolated but coupled to an external heat bath with temperature T then it becomes a subsystem within this heat bath. The probability for the system to have the energy E is then determined by two factors, E) and, as we have just seen, exp[-/3 ]. Thus we have... 

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