Energy Ensemble method

Finally, in Sect. 7.6, we have discussed how various free energy calculation methods can be applied to determine free energies of ensembles of pathways rather than ensembles of trajectories. In the transition path sampling framework such path free energies are related to the time correlation function from which rate constants can be extracted. Thus, free energy methods can be used to study the kinetics of rare transitions between stable states such as chemical reactions, phase transitions of condensed materials or biomolecular isomerizations. 

In spite of these potential concerns, the MEHMC method is expected to be a useful tool for many applications. One task for which it might be particularly well suited is to generate a canonical ensemble of representative configurations of a bio-molecular system quickly. Such an ensemble is needed, for example, to represent the initial conditions for the ensemble of trajectories used in fast-growth free energy perturbation methods such as the one suggested by Jarzynski s identity [104] (see also Chap. 5). 

In order to estimate the free energy many canonical simulations at different temperatures are necessary furthermore, it is often difEcult to define a suitable reference state with a known entropy Sq. Two alternatives can be followed to overcome these difficulties (i) expanded ensemble methods and (ii) multicanonical methods. 

Additionally, non-Hamiltonian d5mamics can be used in applications/ methodologies such as Path-integral MD, replica-exchange methods, variable transformation techniques, free energy dynamics methods, and other new applications. Generating these alternative statistical ensembles from simulation requires the use of extended systems or extended phase space [9]. In these systems, the simulations do not only include the N coordinate and momentum vectors that are needed to describe a classical Ai-particle system, but they also include a set of additional control or extended variables that are used to drive the fluctuations required by the ensemble of interest. 

B. Multistage Free-Energy Perturbation Methods Improving Sampling through Ensemble Selection... 

The phenomenon of strong association can be viewed as a type of chemical reaction. Indeed, a method that is entirely equivalent to RCMC was developed independently by Smith and Triska [10], and based on the ideas of chemical equilibrium. Smith and Triska call their method the reaction ensemble. We shall refer to both reactive canonical Monte Carlo and the reaction ensemble methods as RCMC, since they are in fact the same. Taking the view of chemical equilibria, we can very concisely write the equations that determine the equilibrium point of a system with n phases and C components. For a system at constant temperature T and pressure p, equilibrium is reached when the total Gibbs free energy G is minimized ... 

The free energy difference methods reviewed in this chapter, unless specified otherwise, are discussed for conditions of constant volume and constant temperature (NVT). The extension to ensembles of other types is straightforward.The classical canonical partition function is determined by the classical Hamiltonian 3 6(p, q ), describing the interactions of all N particles in the system in terms of the set of generalized coordinates and conjugated momenta p. For a system with N particles at temperature T, the canonical partition function can be written as... 

The isobaric-isothermic ensemble is characterized by a fixed number of atoms (N), a fixed pressure (P), and a fixed temperature (T). This method is applicable to periodic systems only. The unit cell vectors are allowed to change, and the pressure is adjusted by adjusting the volume (the size and shape of the unit cell). Several methods are available to control pressure. Those of Berendsen et al. (1984) and Anderson (1980) only vary the size of the unit cell, whereas that of Parrinello and Rahman (1982) allows both the cell volume and its shape to change. NPT is the ensemble of choice when the correct pressure, volume, and densities are important in the simulation. This ensemble can also be used during equilibration to achieve the desired temperature and pressure before changing to the constant-volume or constant-energy ensemble when data collection starts. 

The experimental data, also in gas phase, is from Ref. [41]. In the region below 7 eV, furan shows a series of Rydberg states over-imposed to a broad band. The nuclear-ensemble method provides a good qualitative prediction of the spectrum. The intensity and the shape of the broad band are in very good agreement with the experiment. The energy shift is caused by the electronic structure method (see Sect. 5.1), rather than by the spectrum simulation method itself. 

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