Free-molecular momentum calculations

Similar agreement with free-molecular momentum calculations for spherical particles has been obtained through study of Brownian diffusion in nonequilibrium gases as described by the Fokker-Planck equation [2.23,24]. 

Figure 10.7 Probability density of die centrifuged oxygen gas as a function of the molecular angle and the free propagation time, that is, die time elapsed since die molecules have been released from die centrifuge. The white dashed line (around 1.5 ps) marks the calculated trajectory of a dumbbell distribution rotating widi die classical rotational frequency of an oxygen molecule with an angular momentum of 39ft. Part of Fig. 4 in Ref. 39.

The computational efficiency of a FF approach also enables simulations of dynamical behavior—molecular dynamics (MD). In MD, the classical equations of motion for a system of N atoms are solved to generate a search in phase space, or trajectory, under specified thermodynamic conditions (e.g., constant temperature or constant pressure). The trajectory provides configurational and momentum information for each atom from which thermodynamic properties such as the free energy, or time-dependent properties such as diffusion coefficients, can be calculated. 

As in field-free calculations, there are very many basis sets that may be used in different circumstances. A totally uncoupled representation is often convenient because the matrix elements needed are usually quite simple. For example, for collisions of a molecule in a state with an unstructured atom we need basis functions to handle the molecular rotation n, electron spin s, and end-over-end angular momentum... 

All these basis sets are essentially optimized for the calculation of electronic energies and are therefore able to represent the operators included in the field-free electronic Hamiltonian reasonably well. However, in the calculation of molecular electromagnetic properties it is necessary also to represent other operators such as the electric dipole operator, the electronic angular momentum operator, the Fermi-contact operator and more. Most of these basis sets are a priori not optimized for this and have to be extended. 

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