Classical degrees of freedom

For each degree of freedom, classical states within a small volume A/ij Aq- h merge into a single quantum state which cannot be fiirther distinguished on account of the uncertainty principle. For a system with /... 

Here we have treated tlie nuclear degrees of freedom classically as in tlie Marcus foniiulation [1]. 

One common approximation is to separate the nuclear and electronic degrees of freedom. Since the nuclei are considerably more massive than the electrons, it can be assumed that the electrons will respond mstantaneously to the nuclear coordinates. This approximation is called the Bom-Oppenlieimer or adiabatic approximation. It allows one to treat the nuclear coordinates as classical parameters. For most condensed matter systems, this assumption is highly accurate [11, 12]. 

This is a classical result valid only at high temperatures. At low temperatures, quantum mechanical attributes of a degree of freedom can partially or fully freeze it, thereby modifying or removing its contribution to U and... 

Consider a gas of N non-interacting diatomic molecules moving in a tln-ee-dimensional system of volume V. Classically, the motion of a diatomic molecule has six degrees of freedom—tln-ee translational degrees corresponding to the centre of mass motion, two more for the rotational motion about the centre of mass and one additional degree for the vibrational motion about the centre of mass. The equipartition law gives (... 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

The fluctuation dissipation theorem relates the dissipative part of the response fiinction (x") to the correlation of fluctuations (A, for any system in themial equilibrium. The left-hand side describes the dissipative behaviour of a many-body system all or part of the work done by the external forces is irreversibly distributed mto the infinitely many degrees of freedom of the themial system. The correlation fiinction on the right-hand side describes the maimer m which a fluctuation arising spontaneously in a system in themial equilibrium, even in the absence of external forces, may dissipate in time. In the classical limit, the fluctuation dissipation theorem becomes / /., w) = w). 

To generalize what we have just done to reactive and inelastic scattering, one needs to calculate numerically integrated trajectories for motions in many degrees of freedom. This is most convenient to develop in space-fixed Cartesian coordinates. In this case, the classical equations of motion (Hamilton s equations) are given... 

The classical mechanical RRKM k(E) takes a very simple fonn, if the internal degrees of freedom for the reactant and transition state are assumed to be hamionic oscillators. The classical sum of states for s harmonic oscillators is [16]... 

Meyer H D and Miller W H 1979 A classical analog for electronic degrees of freedom in nonadiabatic... 

To make connection between the spectra and tire ET process clearer, we note a simple model for tire lineshape that includes a classical and a high-frequency degree of freedom. In tliis case tire overall lineshape is... 

Unlike classical systems in which the Lagrangean is quadratic in the time derivatives of the degrees of freedom, the Lagrangeans of both quantum and fluid dynamics are linear in the time derivatives of the degrees of freedom. 

Solving the Eqs. (C.6-C.8,C.12,C.13) comprise what is known as the Ehrenfest dynamics method. This method has appealed under a number of names and derivations in the literatnre such as the classical path method, eilconal approximation, and hemiquantal dynamics. It has also been put to a number of different applications, often using an analytic PES for the electronic degrees of freedom, but splitting the nuclear degrees of freedom into quantum and classical parts. 

Since 5 is a function of all the intermediate coordinates, a large scale optimization problem is to be expected. For illustration purposes consider a molecular system of 100 degrees of freedom. To account for 1000 time points we need to optimize 5 as a function of 100,000 independent variables ( ). As a result, the use of a large time step is not only a computational benefit but is also a necessity for the proposed approach. The use of a small time step to obtain a trajectory with accuracy comparable to that of Molecular Dynamics is not practical for systems with more than a few degrees of freedom. Fbr small time steps, ordinary solution of classical trajectories is the method of choice. 

There are various approaches to the problem of coupling quantum degrees of freedom to classical degrees of freedom. The QCMD model is given by the following equations of motion ... 

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