Ensemble classic limit

In a classical limit of the Schiodinger equation, the evolution of the nuclear wave function can be rewritten as an ensemble of pseudoparticles evolving under Newton s equations of motion... 

While most derivations focus on the equation of motion, an equally important aspect of the MFT method is the correct representation of the quantum-mechanical initial state. It is well known that the classical limit of quantum dynamics in general is represented by an ensemble of classical orbits [23, 24, 26, 204]. Hence it is not appropriate to use a single classical trajectory, but it is necessary to average over many trajectories, the initial conditions of which are chosen to mimic the quantum nature of the initial state of the classically treated subsystem. Interestingly, it turns out that several misconceptions concerning the theory and performance of the MFT method are rooted in the assumption of a single classical trajectory. 

Therefore, in the classic limit, thermal averages in the grand mixed isostress isostrain ensemble may be cast as... 

The link to the molecular level of description is provided by statistical thermodynamics whore our focus in Chapter 2 will be on specialized statistical physical ensembles designed spc cifically few capturing features that make confined fluids distinct among other soft condensed matter systems. We develop statistical thermodynamics from a quantum-mechanical femndation, which has at its core the existence of a discrete spectrum of energj eigenstates of the Hamiltonian operator. However, we quickly turn to the classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framework for the subsequent discussion because of the region of thermodynamic state space in which most confined fluids exist. 

In section 1.2, we introduced the quantum mechanical partition function in the T, V, N ensemble. In most applications of statistical thermodynamics to problems in chemistry and biochemistry, the classical limit of the quantum mechanical partition function is used. In this section, we present the so-called classical canonical partition function. 

Proton transfer in the protonated water trimer has been studied extensively with transition path sampling using empirical and ab initio models [10,15], In these studies, shooting moves were implemented by using momentum displacements dp only. Since in the classical limit an isolated cluster evolves at constant energy E according to Newton s equation of motion, the simulations were carried out in the microcanonical ensemble, that is, p x) oc d[E — H(x)]. Furthermore, the dynamics conserves the total linear momentum P and the total angular momentum L. Thus the complete distribution of initial conditions is... 

While most derivations focus on the equation of motion, an equally important aspect of the MFT method is the correct representation of the quantrnn-mechanical initial state. It is well known that the classical limit of quantmn djmamics in general is represented by an ensemble of classical orbits. Hence it is not appropriate to use a single classical trajectory... 

The present article presents an introduction to the path integral formulation of quantum dynamics and quantum statistical mechanics along with numerical procedures useful in these areas and in electronic structure theory. Section 2 describes the path integral formulation of the quantum mechanical propagator and its relation to the more conventional Schrddinger description. That section also derives the classical limit and discusses the connection with equilibrium properties in the canonical ensemble, Numerical techniques are described in Section 3. Selective chemical applications of the path integral approach are presented in Section 4 and Section 5 concludes. 

A system of N particles under thermodynamic constraints of constant volume and temperature is described by the canonical ensemble partition function Q. In the classical limit for a three-dimensional system Q is given by... 

There are 2 temis in the sum since each site has two configurations with spin eitlier up or down. Since the number of sites N is fmite, the PF is analytic and the critical exponents are classical, unless the themiodynamic limit N oo) is considered. This allows for the possibility of non-classical exponents and ensures that the results for different ensembles are equivalent. The characteristic themiodynamic equation for the variables N, H and T is... 

Free energy calculations rely on the following thermodynamic perturbation theory [6-8]. Consider a system A described by the energy function = 17 + T. 17 = 17 (r ) is the potential energy, which depends on the coordinates = (Fi, r, , r ), and T is the kinetic energy, which (in a Cartesian coordinate system) depends on the velocities v. For concreteness, the system could be made up of a biomolecule in solution. We limit ourselves (mostly) to a classical mechanical description for simplicity and reasons of space. In the canonical thermodynamic ensemble (constant N, volume V, temperature T), the classical partition function Z is proportional to the configurational integral Q, which in a Cartesian coordinate system is... 

Here Tq are coordinates in a reference volume Vq and r = potential energy of Ar crystals has been computed [288] as well as lattice constants, thermal expansion coefficients, and isotope effects in other Lennard-Jones solids. In Fig. 4 we show the kinetic and potential energy of an Ar crystal in the canonical ensemble versus temperature for different values of P we note that in the classical hmit (P = 1) the low temperature specific heat does not decrease to zero however, with increasing P values the quantum limit is approached. In Fig. 5 the isotope effect on the lattice constant (at / = 0) in a Lennard-Jones system with parameters suitable for Ne atoms is presented, and a comparison with experimental data is made. Please note that in a classical system no isotope effect can be observed, x "" and the deviations between simulations and experiments are mainly caused by non-optimized potential parameters. 

Limitation to ensembles that allow exchange of energy, but not of matter, with their environment is unnecessarily restrictive and unrealistic. What is required is an ensemble for which the particle numbers, Nj also appear as random variables. As pointed out before, the probability that a system has variable particle numbers N and occurs in a mechanical state (p, q) can not be interpreted as a classical phase density. In quantum statistics the situation is different. Because of second quantization the grand canonical ensemble, like the microcanonical and canonical ensembles, can be represented by means of a density operator in Hilbert space. 

It is the arrangement and symmetry of the ensemble of the atomic nuclei in the molecule that is considered to be the geometry and the symmetry of the molecule. The molecules are finite structures with at least one singular point in their symmetry description and, accordingly, point groups are applicable to them. There is no inherent limitation on the available symmetries for molecules whereas severe restrictions apply to the symmetries of crystals, at least in classical crystallography. 

The mixed quantum classical description of EET can be achieved in using Eq. (49) together with the electronic ground-state classical path version of Eq. (50). As already indicated this approach is valid for any ratio between the excitonic coupling and the exciton vibrational interaction. If an ensemble average has been taken appropriately we may also expect the manifestation of electronic excitation energy dissipation and coherence decay, however, always in the limit of an infinite temperature approach. 

A different approach simulates the thermodynamic parameters of a finite spin system by using Monte Carlo statistics. Both classical spin and quantum spin systems of very large dimension can be simulated, and Monte Carlo many-body simulations are especially suited to fit a spin ensemble with defined interaction energies to match experimental data. In the case of classical spins, the simulations involve solving the equations of motion governing the orientations of the individual unit vectors, coupled to a heat reservoir, that take the form of coupled deterministic nonlinear differential equations.23 Quantum Monte Carlo involves the direct representation of many-body effects in a wavefunction. Note that quantum Monte Carlo simulations are inherently limited in that spin-frustrated systems can only be described at high temperatures.24... 

We have so far limited ourselves to a classical description, the natural requirement for which is the condition /, /" —> oo. In order that the description is valid for any angular momentum value, it is necessary to employ the quantum mechanical approach. We presume that the reader is acquainted with the density matrix (or the statistic operator) introduced into quantum mechanics for finding the mean values of the observables averaged over the particle ensemble. Under the conditions and symmetry of excitation considered here one must simply pass from the prob-... 

Figure 52 displays a quantum-classical comparison of energy diffusion K = 3.5, in terms of the dimensionless scaled energy averaged over the quantum and classical ensemble, denoted by Eg = l )x /2Tp and Ec = (L )/2, respectively. The effective Planck constant x is chosen to be 0.1, a value far from the semiclassical limit but relatively small compared to the area of the transporting islands shown in Fig. 51. Furthermore, this value of x ensures that...

---