Ensemble quantum-mechanical

The above derivation leads to the identification of the canonical ensemble density distribution. More generally, consider a system with volume V andA particles of type A, particles of type B, etc., such that N = Nj + Ag +. . ., and let the system be in themial equilibrium with a much larger heat reservoir at temperature T. Then if fis tlie system Hamiltonian, the canonical distribution is (quantum mechanically)... 

The T-P ensemble distribution is obtained in a maimer similar to the grand canonical distribution as (quantum mechanically)... 

A diagrannnatic approach that can unify the theory underlymg these many spectroscopies is presented. The most complete theoretical treatment is achieved by applying statistical quantum mechanics in the fonn of the time evolution of the light/matter density operator. (It is recoimnended that anyone interested in advanced study of this topic should familiarize themselves with density operator fonnalism [8, 9, 10, H and f2]. Most books on nonlinear optics [13,14, f5,16 and 17] and nonlinear optical spectroscopy [18,19] treat this in much detail.) Once the density operator is known at any time and position within a material, its matrix in the eigenstate basis set of the constituents (usually molecules) can be detennined. The ensemble averaged electrical polarization, P, is then obtained—tlie centrepiece of all spectroscopies based on the electric component of the EM field. 

In what is called BO MD, the nuclear wavepacket is simulated by a swarm of trajectories. We emphasize here that this does not necessarily mean that the nuclei are being treated classically. The difference is in the chosen initial conditions. A fully classical treatment takes the initial positions and momenta from a classical ensemble. The use of quantum mechanical distributions instead leads to a seraiclassical simulation. The important topic of choosing initial conditions is the subject of Section II.C. 

Z-matriccs arc commonly used as input to quantum mechanical ab initio and serai-empirical) calculations as they properly describe the spatial arrangement of the atoms of a molecule. Note that there is no explicit information on the connectivity present in the Z-matrix, as there is, c.g., in a connection table, but quantum mechanics derives the bonding and non-bonding intramolecular interactions from the molecular electronic wavefunction, starting from atomic wavefiinctions and a crude 3D structure. In contrast to that, most of the molecular mechanics packages require the initial molecular geometry as 3D Cartesian coordinates plus the connection table, as they have to assign appropriate force constants and potentials to each atom and each bond in order to relax and optimi-/e the molecular structure. Furthermore, Cartesian coordinates are preferable to internal coordinates if the spatial situations of ensembles of different molecules have to be compared. Of course, both representations are interconvertible. 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

E. Quantitative Aspects of Tq-S Mixing 1. The spin Hamiltonian and Tq-S mixing A basic problem in quantum mechanics is to relate the probability of an ensemble of particles being in one particular state at a particular time to the probability of their being in another state at some time later. The ensemble in this case is the population distribution of nuclear spin states. The time-dependent Schrodinger equation (14) allows such a calculation to be carried out. In equation (14) i/ (S,i) denotes the total... 

A basic theorem of quantum mechanics, which will be presented here without proof, is If a and commute, namely [a, / ] = 0, there exists an ensemble of functions that are eigenfunctions of both a and - and inversely. 

Where A F(z) is the free energy at z relative to that at the reactant state minimum zr, and the ensemble average < > is obtained by a quantum mechanical effective potential [15]. Note that the inherent nature of quantum mechanics is at odds with a potential of mean force as a function of a finite reaction coordinate. Nevertheless, the reaction coordinate function z[r] can be evaluated from the path centroids r, first recognized by Feynman and Flibbs as the most classical-like variable in quantum statistical mechanics and later explored by many researchers [14, 15]. 

Classical systems evolve from initial mechanical states that may, in principle, be known completely. The systems of classical ensembles have well-defined mechanical states and the initial system is always found in one of these states, although it is not known which one. In quantum mechanics however, the state of a system, including its initial state, is represented by a... 

I function which carries maximum information about that system. Definition of the -function itself, depends on a probability aggregate or quantum-mechanical ensemble. The mechanical state of the systems of this ensemble cannot be defined more precisely than by stating the -function. It follows that the same -function and hence the same mechanical state must be assumed for all systems of the quantum-mechanical ensemble. A second major difference between classical and quantum states is that the -function that describes the quantum-mechanical ensemble is not a probability density, but a probability amplitude. By comparison the probability density for coordinates q is... 

The previous discussion only applies when a -function for a system exists and this situation is described as a pure ensemble. It is a holistic ensemble that cannot be generated by a combination of other distinct ensembles. It is much more common to deal with systems for which maximum information about the initial state is not available in the form of a -function. As in the classical case it then becomes necessary to represent the initial state by means of a mixed ensemble of systems with distinct -functions, and hence in distinct quantum-mechanical states. 

The topic that is commonly referred to as statistical quantum mechanics deals with mixed ensembles only, although pure ensembles may be represented in the same formalism. There is an interesting difference with classical statistics arising here In classical mechanics maximum information about all subsystems is obtained as soon as maximum information about the total system is available. This statement is no longer valid in quantum mechanics. It may happen that the total system is represented by a pure ensemble and a subsystem thereof by a mixed ensemble. 

The quantum-mechanical equivalent of phase density is known as the density matrix or density operator. It is best understood in the case of a mixed ensemble whose systems are not all in the same quantum state, as for a pure ensemble. 

For conservative systems with time-independent Hamiltonian the density operator may be defined as a function of one or more quantum-mechanical operators A, i.e. g= tp( A). This definition implies that for statistical equilibrium of an ensemble of conservative systems, the density operator depends only on constants of the motion. The most important case is g= [Pg.463]

The major difference between classical and quantum mechanical ensembles arises from the symmetry properties of wave functions which is not an issue in classical systems. 

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. 

To summarize, the results presented for five representative examples of nonadiabatic dynamics demonstrate the ability of the MFT method to account for a qualitative description of the dynamics in case of processes involving two electronic states. The origin of the problems to describe the correct long-time relaxation dynamics as well as multi-state processes will be discussed in more detail in Section VI. Despite these problems, it is surprising how this simplest MQC method can describe complex nonadiabatic dynamics. Other related approximate methods such as the quantum-mechanical TDSCF approximation have been found to completely fail to account for the long-time behavior of the electronic dynamics (see Fig. 10). This is because the standard Hartree ansatz in the TDSCF approach neglects all correlations between the dynamical DoF, whereas the ensemble average performed in the MFT treatment accounts for the static correlation of the problem. 

Abstract The theoretical basis for the quantum time evolution of path integral centroid variables is described, as weU as the motivation for using these variables to study condensed phase quantum dynamics. The equihbrium centroid distribution is shown to be a well-defined distribution function in the canonical ensemble. A quantum mechanical quasi-density operator (QDO) can then be associated with each value of the distribution so that, upon the application of rigorous quantum mechanics, it can be used to provide an exact definition of both static and dynamical centroid variables. Various properties of the dynamical centroid variables can thus be defined and explored. Importantly, this perspective shows that the centroid constraint on the imaginary time paths introduces a non-stationarity in the equihbrium ensemble. This, in turn, can be proven to yield information on the correlations of spontaneous dynamical fluctuations. This exact formalism also leads to a derivation of Centroid Molecular Dynamics, as well as the basis for systematic improvements of that theory. 

Just as there is a fundamental function that characterizes the microscopic system in quantum mechanics, i.e., the wave function, so too in statistical mechanics there is a fundamental function having equivalent status, and this is called the partition function. For the canonical ensemble, it is written as... 

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