Atoms degrees of freedom

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

Figure 4 shows that initial elastic compression is characterized by oscillations of the volume. These volume oscillations are damped within about 5 oscillations in this case. The damping of volume oscillations occurs by transfer of energy from the strain degrees of freedom to the atomic degrees of freedom, and the atomic temperature can be seen to increase while this process occurs. 

Depending on the properties and systems of interest one can choose different theoretical approaches for such studies. When focusing on the mechanical properties of macroscopic samples, the precise arrangement of the atoms and electrons is often of only secondary interest (although they ultimately dictate the mechanical properties) and might therefore not be considered in the models. On the other hand, when, e.g., studying the electronic properties of semiconductors or the reactivity of specific molecules, one needs to include explicitly both electronic and atomic degrees of freedom in the models. 

In Chapter 1 it was noted that the number of vibrational modes of a molecule can be calculated by counting the degrees of freedom of the atoms (three per atom for X,Y and Z movement) and subtracting the degrees of freedom for motion of the molecule as a whole, three for its translation and (for nonlinear molecules) three for rotation. This was used in Section 5.2 to arrive at a reducible representation for the basis of nine atomic degrees of freedom for H2O, the classic C2V molecule. The characters for this representation were given in Table 5.1. We can now apply the reduction formula to identify the irreducible representations for the three vibrations of HjO. 

Table 5.5 The application of the reduction formula to the nine basis vector representation of H2O atomic degrees of freedom defined in Figure 5.2.

So, the irreducible representations for the rigid-body movement and rotation of H2O have now been identified and we can remove them from the total set of irreducible representations derived from the atomic degrees of freedom. This leaves us with... 

The basis set size is determined by the particular problem in hand. In the analysis of H2O in Section 5.2 we concentrated on the stretching vibrations of the O—H bonds. This is a problem which could be approached using a basis as simple as the two bond vectors. However, with that basis we would miss the bending mode that was identified by using the nine basis vectors that represent the full atomic degrees of freedom. 

Quantum mechanics is the theory that captures the particle-wave duality of matter. Quantum mechanics applies in the microscopic realm, that is, at length scales and at time scales relevant to subatomic particles like electrons and nuclei. It is the most successful physical theory it has been verified by every experiment performed to check its validity. It is iso the most counter-intuitive physical theory, since its premises are at variance with our everyday experience, which is based on macroscopic observations that obey the laws of classical physics. When the properties of physical objects (such as solids, clusters and molecules) are studied at a resolution at which the atomic degrees of freedom are explicitly involved, the use of quantum mechanics becomes necessary. 

If lattice vibrations and deformations are not considered, X is completely equivalent to the whole set of the atomic positions. If the validity of the Born-Oppenheimer approximation and of a classical approximation for the atomic degrees of freedom are assumed, then E,/(A c) can be regarded as a classical Hamiltonian for the alloy in study. Probably the functional dependence of Ef,i(X c) on the atomic degrees of freedom, X, is too much complicated for exact, even though approximate, statistical studies. My group is currently developing a mixed CEF-Monte Carlo scheme in which a Metropolis Monte Carlo algorithm is used to obtain ensemble... 

---