Partition function microcanonical ensemble

If Q (E) is differentiable in the ordinary sense the partition function of a generalized ensemble with m intensive parameters is the m-fold Laplace transform of the microcanonical partition function e, ... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

Boltzmann6 proposed that at the temperature T = 0, all thermal motion stops (except for zero-point vibration), and the entropy function S can be evaluated by a statistical function W, called the thermodynamic probability W (or, as we will learn in Section 5.2, the partition function Q for a microcanonical ensemble) ... 

Micro-canonical ensemble fiCE (each system has constant N, V, and U the walls between systems are rigid, impermeable, and adiabatic each system keeps its number of particles, volume, and energy, and it trades nothing with neighboring systems). The relevant partition function is the microcanonical partition function Cl ( N, V, U) ... 

Given an ensemble of static electric dipole moments of magnitude fi and random orientation in an external static electric field E, we can use the microcanonical ensemble partition function to compute the average moment... 

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. 

The partition function Z, which normalizes the density, is effectively a function of N, V and E it represents the number of microstates available under given conditions. As this ensemble is associated to constant particle number N, volume V and energy E, it is often referred to as the NVE-ensemble, and when we speak of NVE simulation, we mean simulation that is meant to preserve the microcanonical distribution this, most often, would be based on approximating Hamiltonian dynamics, e.g. using the Verlet method or another of the methods introduced in Chaps. 2 and 3, and assuming the ergodic property. For a discussion of alternative stochastic microcanonical methods see [126]. 

There are four main ensembles in statistical thermodynamics for which the independent variables are NVE (microcanonical), NVT (canonical), NpT (Gibbs or isothermal isobaric), and VT (grand canonical). The characteristic fnnetions provided in Equations 1.2 and 1.3 can be expressed in terms of a series of partition functions such that (Hill 1956)... 

Assuming a relationship of the form of Eq. (115), it is possible to derive the configurational partition function of the weak-coupling ensemble as a function of a ([109] see Appendix) The limiting cases a = 0 (tb — 0 canonical) and a = 1 (tb —> oo microcanonical) are reproduced. Note that the Haile-Gupta thermostat generates configurations with the same probability distribution as the Berendsen thermostat with a = 1 /2. 

The proof that the Nose thermostat samples a canonical ensemble of noicrostates, provided that g = Ndf + l (virtual-time sampling) ox g = Ndf (real-time sampling), is as follows [53]. The partition function of the microcanonical ensemble generated for the extended system using virtual-time sampling (i.e., using the natural time evolution of the extended system) reads... 

The criteria of comparison are (a) conservation of the total energy in a simulation in the microcanonical (NVE) ensemble (b) conseavation of the total momenrnm (c) quality of radial distribution functions extracted from an MD simulation in the canonical ensemble (d) scahng in terms of number of QM/MM partitions, M being the number of molecules in the transition region... 

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