Potential energy surfaces, calculation Porter-Karplus

Fig. 6.4 Plot of reaction probability vs. initial translational energy for the H + HH = HH + H reaction for a certain empirical potential energy surface (the Porter-Karplus surface). Curves (reading down) are shown for the path shown as PP in Fig. 6.3a. (marked Marcus-Coltrin), the exact quantum mechanical result for the Porter-Karplus surface (marked Exact QM), the usual TST result calculated for the MEP, QQ (Fig. 6.3a) (The data are from Marcus, R. A. and Coltrin, M. E., J. Chem. Phys. 67, 2609 (1977))...

Fig. 1. Fourier coefficients for the LEPS (closed circles) and Porter-Karplus (open circles) potential energy surfaces calculated from equation (5). The uncertainty in each coefficient was computed from equation (6) and is shown explicitly for the Porter-Karplus coefficients. The noise in the LEPS results equals the size of the point. Note that for n = 8, 11, and 16 the points lie on top of each other.

Since the early work mentioned above, H3 calculations have proceeded along the two lines, semiempirical or ab initio. Semiempirical potential energy surfaces have been produced by Sato, Porter and Karplus, Cashion and Herschbach, and Salomon. ... 

In the early 1930 s, Eyring and his co-workers made some preliminary studies of the trajectories of systems on potential-energy surfaces, but not much progress could be made until the development of high-speed computers. There has recently been a revival of interest in this field, now known as molecular dynamics. In particular, Karplus, Porter, and Sharma have carried out calculations on the H -b Hg system, using what appears to be a very reliable potential-energy surface. The calculations are quasi-classical in nature the vibrational and rotational states in the Hg molecule are quantized, but the course of the collision is treated classically. 

The purely classical calculations of KARPLUS et al./54/, using the Karplus-Porter potential energy surface, show that the threshold of the relative translation energy E is only somewhat greater than... 

There is also a CS calculation for the reactions H + H2(v = 1) e- H2(v ) + H V = 0,1 on the Porter-Karplus potential energy surface,[39]. However, due to numerical problems, no fully converged results were obtained. Again, this lack of convergence may affect the absolute values but is expected to yield more reliable values for branching ratios. Here we find that the classical 1(1,0) values are 2 whereas both the CS and RIOSA yield values of 4. 

In this section we show the effects of resonances on calculated reaction probabilities for several collinear systems with model potential energy surfaces. For H + H2 we consider the scaled SSMK surface and surface no. 2 of Porter and Karplus. For F 4 H2 and isotopic analogs we consider surface V of Muckerman. For H + FH we consider a surface that differs from the Muckerman V surface in only one parameter. This surface has a very low classical barrier height, 1.75 kcal/mol further details are given elsewhere. 

We have used this method to compute 1(9) for the reaction H + H2 H2 + H. This is a well-studied system, and some calculations of the differential cross section have already been made. In particular, Ma3me and Toennies have computed 1(9) using the histogram method at a relative translational energy of = 0.65 eV for the Porter-Karplus no. 2 and potential energy surfaces. In... 

Walker and Wyatt (1972) have also performed a distorted-wave calculation for H + H2, based on the Porter-Karplus surface. They constructed reactant and product distortion potentials assuming adiabatic vibrational motion in each case, and obtained numerical solutions for the relative motions. Their results show that by choosing adequate potential parameters it is possible to reproduce the threshold behaviour, but that probabilities grow above unity soon after the threshold energy. 

Figure 2 Reaction probability for the collinear H + H2 reaction on the Porter-Karplus potential surface from a microcanonical classical trajectory calculation (CLDYN) and microcanonical classical transition state theory (CLTST) as a function of total energy above the barrier height (1 eV = 23.06 kcal/mole).

Porter and Karplus [19] constructed a LEPS potential for H + H2 including overlap and three-center terms in order to evaluate the energies of nonlinear configurations more realistically. Kuntz et al. [313] employed a modified LEPS function in a detailed investigation of metathetical reactions involving three atoms. Three adjustable parameters were included, instead of just S2. This provided a more flexible potential, and it was possible to vary the nature of the potential surface quite considerably. Other potentials based on pairwise interactions have been used for calculations where AB is ionic [72-74, 306],... 

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