Porter-Karplus surface

Collinear two-degree-of-freedom model with the Karplus-Porter surface by Pechukas et al. [ 143] (linear equilibrium geometry). 

This dissociative system, which represents the prototype system for chemical reaction dynamics, has been the object of many studies. Child et al. [143] have carried out a detailed analysis of the classical dynamics in a collinear model based on the Karplus-Porter surface. These authors have introduced the concept of PODS and first observed the subcritical antipitchfork bifurcation scenario in this system. 

The bottom of the exit channels is at -3194 cm-1 if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E = 1670 cm 1, which is followed by the subcritical antipitchfork bifurcation at Ea = 2633 cm 1. The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. 

Figure 19. Scattering resonances of 2F collinear models of the dissociation of H3 obtained by Manz et al. [143] on the Karplus-Porter surface and by Sadeghi and Skodje on a DMBE surface [132], The energy is defined with respect to the saddle point. The dashed lines mark the bifurcations of the transition to chaos 1, 5, and 3 are the numbers of shortest periodic orbits, or PODS, in each region.

P. Gaspard To answer the question by Prof. Poliak, we expect from our present knowledge that the periodic-orbit quantization of the H + H2 dissociative dynamics on the Karplus-Porter surface can be performed with the same theory as applied to Hgl2. 

Resonance energies for the collinear H+H2 reaction a comparison of one-dimensional model predictions in comparison to exact quantum mechanical results on the Porter-Karplus II surface (49). 

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.

Figure 1. Typical long-lived trajectory at E = 0.873 in H + H2 on the collinear Porter-Karplus potential surface, (a) log[d(r)/d(0)] versus t. (b) An eigenvalue of S(c) along the unperturbed trajectory yielding panel (a). Dashed curves denote a real S(t) eigenvalue, and solid curves denote an imaginary eigenvalue. Note the correspondence between exponential growth and times where the system encounters a real eigenvalue. (From Ref. 20.)...

A study of reactive H + H2 collisions has also been carried out by Wu and Levine (1971) using the close-coupling method and natural collision coordinates. They employed a Porter-Karplus (1964) surface and investigated the collision energy range between 9 and 35 kcal/mole. For the interaction they used... 

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. 

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. 

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).

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... 

H2 + H reaction according to quantum collision theory based on Porter-Karplus potential surface. Curves 1, 1 and correspond to colinear, complanar and... 

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. 

H. R. Mayne and J. P. Toennies, Quasiclassical cross sections for the H + H2(0,0) -> H + H2 reaction. Comparison of the Siegbahn-Liu-Truhlar-Horowitz and Porter-Karplus potential surfaces, J. Chem. Phys. 70 5314 (1979). 

QUASICLASSICAL DIFFERENTIAL CROSS SECTIONS FOR REACTIVE SCATTERING OF H + H2 ON LEPS AND PORTER-KARPLUS POTENTIAL SURFACES... 

Fig. 4. Raactive differential cross sections for three potential energy surfaces. The LEPS and KP (Karplus-Porter) curves are smoothed results computed from equation (9) the individual SLTH points were taken from reference 9.

The model used in this calculation is very similar to that described by Light and Altenberger-Siczek. The ground state surface was taken to be the Porter-Karplus potential surface for H + H2. 

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))...

We should also not leave this section without referring to the widely used Porter-Karplus surface for H3 which was constructed by semi-empirical valencebond methods and which agrees quite well with the best ab-initio surfaces (98). 

The H3 potential surface which has been most widely used in transition state,2X4 classical dynamical,213 215 and quantum mechanical dynamical216-218 calculations has been the semi-empirical surface of Porter and Karplus.22 Because of the thin potential barrier of the PK surface, one would expect a larger amount of quantum mechanical tunnelling to be predicted at room temperature (this has been found to be the case in calculations performed by Johnston219 on barriers in the very similar LEPS surfaces). However, Karplus et a/.213-216 compared classical and quantum mechanical calculations on the PK surface and found that reaction cross-sections for both are very similar, and therefore that the tunnelling effect in the Ha system is small. 

More recently, Yates and Lester230 fitted Liu s surface with a slightly modified form of the Porter-Karplus formulas after first fitting Liu s H2 potential to a simple Morse function. They then use the resulting surface to calculate the three-dimensional classical trajectory of the system. Their empirical fit very closely duplicates Liu s saddle-point properties. Reaction probabilities on this surface are compared with those on the PK surface. 

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],... 

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. 

Results using the Porter-Karplus surface show that reaction from m = 0 does not occur until the total energy (measured from H + H + H) is greater than the barrier height value of —4-351 eV, and becomes almost certain between — 4-2 eV and — 40 eV. Reaction probabilities from either m = 0 or 1 into n = 1 are larger than 05 between —3-85 and — 3-55 eV. 

An instructive description of the H + H2 reaction was provided by McCullough and Wyatt (1971a, b). They constructed a wavepacket and followed its time development on the Porter-Karplus surface. They introduced centre-of-mass coordinates appropriate for reactants and used these for all times. The wavefunction F at time t = n At was constructed, from the time-evolution operator U, in the form... 

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. ... 

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