Porter-Karplus potential energy

Eq. (3.8b) for the collinear H+H2 reaction on the Porter-Karplus potential energy surface. In the plot the saddle point height M is subtracted from l/(s). There are several different energy regions to distinguish. 

FIGURE 3. Reaction probability for collinear H+H2- H2+H on the Porter-Karplus potential energy surface. EQ denotes the exact quantum mechanical values, VAZC the results of the vibrationally adiabatic zero curvature approximation, and the points the results of the present SCP-IOS reaction path model. 

The RIOSA study on H + H2 was carried out on the Porter-Karplus potential energy surface [17]. The results for total cross sections from the ground state and the first excited state are shown in figs, (2) and (3a). The branching ratios T defined as ... 

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. 

Fig. 25. Contour diagram of the Porter-Karplus potential energy surface for the collinear H 4- H2 system in Delves coordinates The solid curves are equipotentials whose energies (with respect to the bottom of the isolated H2 well) are indicated at the lower right side of the figure. The dashed line is the minimum energy path and the cross along it is the saddle point. The polar coordinates p,a of a general point P in this configuration space are also indicated. The circular arcs centered at the origin are lines of constant p.

The classical threshold for any process is the energy below which there are no ordinary (i.e. real-valued) classical trajectories which lead to the process. For the collinear H + H2 reaction on the Porter Karplus potential surface, for example, the classical threshold is 0-20-0-21 eV collision energy quantum mechanically, of course, there is no absolute energetic threshold for the 0 - 0 reaction. 

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).

The first and second columns of the Tables give the reaction and potential energy surface used. Standard abbreviations are employed for the names of the potential surfaces. Thus. PK = Porter-Karplus potential surface No 2 for H+H2. LSTH = Liu-Siegbahn-Truhlar-Horowitz potential surface for H+Hg. YLL = Yates-Lester-LIu potential surface for H+Hp. LEPS = extended London-Eyring-Polanyi-Sato potential surface and DIM = diatomics-in-molecules potential surface. 

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))...

These experiments are important because they are performed on a reaction for which a priori calculations of V(rAB, rBC, rCA) are likely to have their best chance of success as only three electrons are involved. Even here the accurate computation of V, frequently termed the potential-energy hypersurface, is extremely difficult. Porter and Karplus [19] have determined a semiempirical hypersurface, and Karplus, Porter, and Sharma [20] have calculated classical trajectories across it. This type of computer experiment has been mentioned before and will be described in greater detail later. The objective of Karplus et al. was to calculate aR(E) and E0. Collisions were therefore simulated at selected values of E, with other collision parameters selected by Monte Carlo procedures, and the subsequent trajectories were calculated using the classical equations of motion. Above E0, oR was found to rise to a maximum value, of the same order of magnitude as the gas-kinetic cross section, and then gradually to decrease to greater energies. 

Figure 1. Adiabatic eigenenergies (r) (dashed lines) and effective potentials U (r) (full lines) for the collinear H+H2 reaction (,21) on the Porter-Karplus II surface (49) for n 0,l,2,3. The energies are relative to the dissociation energy into three free atoms.

HF + H and D + FD - DF + D on the low-barrier model potential of Muckerman, Schatz, and Kuppermann and the colllnear and three-dimensional H + H2 reactions on Porter-Karplus surface number 2. Finally we use an accurate potential energy surface for the three-dimensional H + H2 reaction to predict the energies of several series of observable resonances for a real system. 

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. 

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. 

Some years later, aided by considerably more rapid computers than available to Wall and co-workers, Karplus, Porter, and Sharma reinvestigated the exchange reaction between H2 and H [24]. As with the earlier work, the twelve classical equations of motion were solved. In addition, discrete quantum-mechanical vibrational and rotation states were included in the total energy so that the trajectories were examined as a function of the initial relative velocity of the atom and molecule and the rotational and vibrational quantum numbers j and v of the molecule. The more sophisticated potential energy surface of Porter and Karplus was used [7], and the impact parameter, orientation and momentum of the reactants, and vibration phase were selected at random from appropriate distribution functions. This Monte Carlo approach was used to examine 200-400 trajectories for each set of VyJ, and v. The reaction probability P can be written as... 

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... 

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. 

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