EJ = Beff J2 ,
EV = E - EJ ,
where Beff is an effective rotational constant. Of course, it is more generally valid to consider the energy transfer in terms of the changes in the energy E and total angular momentum and the above equations supply a simple conversion.
In an initial study, the energy transfer in microcanonically excited SO2 colliding with a thermal Ar atom was monitored. For low rotational energies, the magnitude of <[[Delta]]E> increased with EJ. However, at high rotational energies, the period of rotation becomes smaller than the collision time (adiabatic limit), energy transfer becomes inefficient and |<[[Delta]]E>| becomes smaller. Also noteworthy, is that the rotational energy transfer and vibrational energy transfer go through an inversion of sign at roughly the thermal average and microcanonical average rotational energy, respectively. This suggests that, in this system, rotational energy transfer goes largely by a R<->T pathway and vibrational energy transfer is mediated by a V<->R pathway.
In a more detailed study, the behaviour of rovibrationally resolved (EV,EJ) initial states was investigated. Some significant dependences are noted:
1) -<[[Delta]]EJ> increases with EJ but decrease with EV
2) -<[[Delta]]EV> increases with EV
3) -<[[Delta]]EV> increases with EJ initially, but then decreases with EJ.
These results are consistent with but more general than the trends observed in a recent trajectory study by Brown and Miller.
Inspection of the data, reveals a pronounced dependence of the energy transfer on the initial vibrational and rotational state (EV,EJ). For a vibrationally hot and rotationally cold ensemble, there are many V -> R,T transitions; a significant portion of the vibrational energy is converted into rotational energy. The reverse occurs for a vibrationally cold but rotationally hot ensemble; rotational energy is converted into vibrational energy. For an ensemble with both rotationally hot vibrations and rotations there are many V -> R and R -> V transitions in addition to the predominant V,R -> T transitions. At thermal energies, the inelastic transitions are smaller in magnitude and much more symmetrically distributed.
An indication of the average behaviour of an ensemble of molecules with average state (EV,EJ) can be obtained from a flow field representation. For example, an initially highly vibrationally excited but rotationally cold ensemble is expected to heat up rotationally as it vibrationally cools. At some point both rotations and vibrations cool until thermal equilibrium is achieved. This 2D temporal evolution of the average ro-vibrational state (EV,EJ) is not addressed by the usual 1D-form of the master equation, but requires a 2D form of the master equation either in terms of the evolution of a probability density p(t;EV,EJ). More correctly, the master equation should be phrased in terms of energy and angular momentum resolved quantities.
The solution of this master equation not only requires detailed information on P(E',J';E,J) (or k(E',J';E,J)) but also requires a full description of the E,J-resolved decay rate constant k(E,J). The determination of k(E,J) is of considerable interest but is beyond the scope of this review. The work described above has yielded some interesting information on the behaviour of the lower moments of P(E,J;E',J') but a detailed treatment at the master equation level requires a full description of the energy transfer kernel.
In the more general case of 2D energy transfer kernels P(E',J';E,J) (or P(EV',EJ';EV,EJ)) the situation is even more complex and demanding than in the usual 1D P(E',E) representation. In principle a 2D histogram in (EV,EJ)-space is required. Scatterplots of final states in (EV,EJ)-space can be viewed as projections of these 2D histograms. Typical of the multidimensional forms employed for fitting P(EV',EJ';EV,EJ) were the "VJ model"
P(EV',EJ';EV,EJ) = F(1,N) exp(- F((|[[Delta]]EV|),aV) - F((|[[Delta]]EJ|),aJ))
the "EJ model"
P(EV',EJ';EV,EJ) = F(1,N) exp(- F((|[[Delta]]E|),a) - F((|[[Delta]]EJ|),aJ))
and the "VJE model"
P(EV',EJ';EV,EJ) = F(1,N) exp(- F((|[[Delta]]EV|),aV) - F((|[[Delta]]EJ|),aJ) - F((|[[Delta]]E|),a))
where the fitting parameters {aV, aJ, a} are {[[alpha]]V, [[alpha]]J, [[alpha]]} for down collisions ([[Delta]]EV<=0, [[Delta]]EJ<=0, [[Delta]]E<=0) and {[[beta]]V, [[beta]]J, [[beta]]} for up collisions ([[Delta]]EV>0, [[Delta]]EJ>0, [[Delta]]E>0) in vibrational, rotational and total energy, respectively. These fitting parameters are connected by detailed balance. As in the 1D case, fitting was performed to focus on inelastic collisions but proved problematic: 1) because of the high number of dimensions and inadequately fine bin size; 2) the fitting was often nonunique for P(E,J,E',J') because the data was not sensitive to the choice of parameters in the model under certain conditions. In principle, it may be necessary to fit a whole grid of data for P(EV',EJ';EV,EJ) simultaneously to produce a satisfactory model; 3) the single exponential form (in each energy component) was insufficiently flexible to adapt to the different collision efficiencies. However, a more complex functional form, e.g., two exponentials for each energy component is likely to involve even more severe technical problems (e.g., twice as many fitting parameters) that must be overcome.
Despite these difficulties, some qualitative behaviour regarding the shape of P(EV',EJ';EV,EJ) and the nature of energetic constraints can be gained by comparing the (EV-EJ)-scatterplots with schematic contour plots of the above 2D models. It appears that the simulation results correspond to a mode somewhere between the VJ and EJ models and perhaps most closely to the more flexible VJE model. It is clear that there is much scope for further research into resolving the functional form of P(EV',EJ';EV,EJ).