Potential energy function collisions

The potential energy functions are expressed in terms of q, 0 = 1,2,. 6), which explicitly exhibits its independence of the coordinates of the center of mass. Again, since the momenta conjugate to coordinates 0/7. q, qg), i.e. Pi,p andp9 remains constants of motion during the entire collision, the term containing them in the Hamiltonian has been subtracted. 

The reduced collision integrals, which depend on the particular form of the potential-energy function, are usually found tabulated as a function of the reduced temperature. However, an approximate fit to the reduced collision integral is given as... 

Finally, the reduced collision integral is usually expressed in terms of a reduced temperature T. If the intermolecular potential energy function can be expressed in the form [178]... 

We note that the coordinate X directly reflects the distance between atoms one and two, whereas the coordinate X2 reflects a combination of both distances. Therefore, a knowledge of the two coordinates does not directly tell us what the distances are between the involved atoms. Also, for the potential energy function in a collinear collision, the natural variables will be the distances between atoms A and B and atoms B and C. These variables appear as the components along a new set of coordinate axes, if instead of a rectangular coordinate system we use a mass-weighted skewed angle coordinate system. 

In the classical trajectory approach, if a potential energy surface is available, one prescribes initial conditions for a particular trajectory. The initial variables are selected at random from distributions that are representative of the collisions process. The initial conditions and the potential energy function define a classical trajectory which can be obtained by numerical integration of the classical equations of motion. Then another set of initial variables is chosen and the procedure is repeated until a large number of trajectories simulating real collision events have been obtained. The reaction parameters can be obtained from the final conditions of the trajectories. Details of this technique are given by Bunker.29... 

As might be expected, the model leads to a great simplification over the calculations required for molecules with a continuous potential energy function, as it enables the analysis to be confined to binary collisions and permits the definition of a collision frequency. Because there is no molecular interaction between collisions, the velocity distributions of two colliding molecules may be assumed to be re-established by the time a second collision occurs between them. Thus a Maxwellian distribution around the local mass velocity may be postulated for the calculation of the mean frequency of collision and the average momentum and energy transported per collision in the nonuniform state of the liquid. 

In ideal circumstances, %(r) properly moderates the Coulomb potential to describe the interaction between ions and atoms at all separation distances. For large distances, %(r) should tend to zero, while for very small distances, %(r) should tend to unity. Such features allow a single interatomic potential energy function, (2.8), to describe the entire collision process. 

For molecules with central finite attractive and repulsive forces (Fig. 2-4c), we may take S v ) = n9 Xmm where b Xmin) is the impact parameter corresponding to a minimum angle of deflection selected as an arbitrary cutoff to prevent S Vr) from going to infinity as x goes to zero when classical collision theory is used. The specific dependence of h(Xmin) on will vary with the magnitude of the parameters s and t or a and b in the empirical potential-energy functions. A realistic calculation for this model, i.e., one which avoids an arbitrary cutoff Xmiw must be carried out quantum mechanically. 

Abstract, In this paper I give an overview of the current status of knowledge of the four-body potential energy function and dynamics of the HF dimer. The discussion of potential energy functions includes both single-center expansions and multi-site functions. The discussion of d)mamics includes both intramolecular processes of the van der Waals dimer and diatom-diatom energy transfer collisions. 

Diatomic molecules are a special case. Firstly, the dynamics of atomic collisions can usually be calculated accurately, and there are some inversion techniques that allow one to predict the potential from the experimental data the RKR method of analysing spectroscopic data is the most well known of these [1]. Secondly, the potential energy functions are one dimensional and even "complicated functions are simple compared with those of polyatomic molecules. 

It is not surprising that the two main classes of microscopic simulations have evolved quite independently. Aside from the obvious problem of calculating potential energy functions (surfaces), the greatest computational difficulty arises in treating systems with multiple time scales. Dynamical simulations within class A are feasible because the bulk properties of interest can be determined on a time scale corresponding to a computationally finite number of molecular collisions. When the most important events are rare on this time scale, one rapidly reaches the limits of feasibility for detailed molecular dynamics... 

Figure 1.5. Femtosecond spectroscopy of bimolecular collisions. The cartoon shown in (a illustrates how pump and probe pulses initiate and monitor the progress of H + COj->[HO. .. CO]->OH + CO collisions. The huild-up of OH product is recorded via the intensity of fluorescence excited hy the prohe laser as a function of pump-prohe time delay, as presented in (h). Potential energy curves governing the collision between excited Na atoms and Hj are given in (c) these show how the Na + H collision can proceed along two possible exit channels, leading either to formation of NaH + H or to Na + H by collisional energy exchange.

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