Rotated Morse potential

Rotated Morse Potential Wall and Porter introduced the idea of generating global PESs for A - - BC AB - -C reactions by rotating a Morse potential, equation (15), about a fixed point (located in the direction of separated atoms), which smoothly goes from the BC reactant valley... 

In order to evaluate the above expression, solutions were found for the Schrodinger equation using the Morse potential for rotational quantum number i not equal to zero ... 

A number of other analytic potential functions such as Rotated-Morse-Curve-Spline (RMCS), Bond-Energy-Bond-Order (BEBO) have been used for fitting of ab initio surfaces. 

For typical values of p, re and V, encountered in molecules, Eq. (l.ll) is an excellent approximation to the exact solution (better than l part in 109). The Morse potential is the simplest member of a family of potentials that give rise to a vibrational spectrum of the functional form E(v) = coc(v +1/2) -a>exe(v +1/2)2. This is quite realistic at lower levels of excitation. The vibrational spectrum does not however suffice, by itself, to specify the potential uniquely. The dependence of the eigenvalues on the rotational state is therefore important. For / 0 (as well as for the / = 0) the energy eigenvalues are given by... 

The MEP for dissociation of the dimer is defined by the condition that at every intermolecular distance R, 6X and 02 take the values so as to minimize the energy the potential along this MEP is modelec by a Morse potential with De = 4.3-4.5 kcal/mol and a = 1.45-1.55 A. The onedimensional barrier for interconversion and the corresponding correlation between d2 and 0, are presented in Figure 8.11a. The concerted rotation... 

Kupperman et a/.223-228 compared transition-state, classical, and quantum mechanical thermal rate constants using the SSMK surface.112 In ab initio potential energy surface calculations, the potential energy is known only as a list of values at selected geometries of the system. It becomes necessary, then, if trajectory calculations are to be made, to fit the calculated points to a smooth and continuous map . In their calculations, Kupperman et al. fit the collinear SSMK surface by the rotating Morse function procedure of Wall and Porter.227... 

Namely, the circular orbit (l = n — 1), which is a rotating state with a nodeless radial wavefunction, corresponds to a vibrational quantum number v = 0, and the next-to-circular single-node state (l = n — 2) corresponds to v = 1,.... This theoretical possibility of large-/ circular orbits behaving like bound states in a Morse potential seems to have no other natural manifestation than in the present case of metastable exotic helium. This situation is presented in Fig. 2, where the potential as well as the wavefunctions are shown. 

Morse curve with the coefficients depending on two angular variables one (a) defines the deviation of the molecule from linearity, while the other (v) defines the angle of rotation of the Morse potential around a fixed point on the atom-atom-atom dissociation plateau (Ru is the distance between this point and the minimum of the Morse curve) ... 

In spite of the flexibility introduced into equation (62) by using 2D splines to describe DM(a, v), a comparison of classical trajectories run on a model potential with those run on a rotated Morse-cubic spline fitted to that model does not show good point-by-point agreement (see ref. 54 and references therein). 

Berend and Benson [49] have compared experimental rotational relaxation rates for / -H2-/ -H2, />-H2-He, and o-D2-He with their classical theory utilizing the two-dimensional model of Figure 3.5, with atom-centered Morse potentials. Their conclusion that H2-H2 collisions are more effective than are H2-He collisions differs from the predictions of Roberts and is not confirmed experimentally however, there is fair agreement with regards to both magnitude and temperature dependence, particularly for the H2-H2 case. 

The microwave experiment studies rotational structure at a given vibrational level. The spectra are analyzed in terms of rotational models of various symmetries. The vibration of a diatomic molecule is, for instance, approximated by a Morse potential and the rotational frequencies are related to a molecular moment of inertia. For a rigid classical diatomic molecule the moment of inertia I = nr2 and the equilibrium bond length may be calculated from the known reduced mass and the measured moment, assuming zero centrifugal distortion. 

Determine v (v ) from your spectrum and then r" - from this expression. Compare the resultant value of r with the literature value of 0.3025 nm obtained by analysis of the rotational structure of the electronic spectrum. Include the Morse potential curve for the upper state on your plot for the X state and comment on the differences in the various parameters determined for the two states. 

The value of r Is obtained from that for gaseous BaBrgC ) with r (BaBr)/r (BaBrg) = 0.96. This value for the ratio Is calculated from bond lengths (4) for several other alkaline earth halide systems. Two other estimates (5, 6) of r agree with the adopted value to within 0.05 A. The rotational constant Is calculated from the estimated value for r. The value of Is obtained from a Morse potential function. 

The rotational constant is adjusted for the natural Isotopic abundances of lead and chlorine. The value for r is obtained from the adopted value. The results of an earlier rotational analysis of the same bands by Rao and Rao (13) are in reasonable agreement with our adopted values. The value for is estimated from the other constants assuming a Morse potential function. 

Jordan ( ) predicted five electronic levels (X E, A a, B e, D n, B e) based on the reported value (2) of 29560 cm" for the C n level (this level is designated A II by Herzberg (2)). Several qualitative spectroscopic investigations of PH(g) have been reported and are in general accord with the predictions of Jordan (1 ). Ishaq and Pearse (3 ) reported the values of the rotational constant B and the fundamental vibrational frequency The value of is calculated from the Morse potential function. The... 

A broken-path model considered by Middleton and Wyatt (1972) consists of dividing the surface into several regions and introducing local Cartesian coordinates s, p in each of them. These coordinates are chosen to lie close to the curvilinear coordinates and rotate from region to region. In each region V(s,p) is assumed separable in Tt(s) + V2(p), with Vl a linear potential and V2 a Morse potential. Hence, the wavefunction is locally of the form... 

For more realistic models of the diatom potential, for instance a Morse function, the dynamics is more complicated. For a nonrotating Morse potential, the necessary analytical equations have been worked out (8). For a rotating diatom, however, the problem is more complex. The best approach in this case is one based on the concept of action-angle variables for periodic motions (1). Porter et al. (19) give one recipe. More recently, Eaker (20) has proposed a method based on a Fourier analysis (21) of the motion. 

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