Morse function diatomic molecules

Another way of looking at the Morse curve in Figure 9.12 is to say it represents the energy E (as y ) of the two atoms of X as a function of their bond length r (as V). The two atoms of X form a simple diatomic molecule in its ground state, i.e. before it absorbs a photon of light. 

The potentiul energy for the nuclear motion of a diatomic molecule is closely approximated hy the Morse function... 

Since all photochemical reactions require the absorption of a photon, the result is that the reactant molecule is raised to a higher energy level. The outcome of this process depends on the nature of the upper and lower electronic states of the molecule. Four types of absorption behaviour are possible and we will first illustrate these by referring to Morse curves for the simple, diatomic, molecules. Although the potential energy of a complex molecule as a function of its molecular geometry is not a simple two-dimensional curve but a complex multidimensional surface, the conclusions arrived at by the use of Morse curves are instructive. 

Fig. 3.1 Born-Oppenheimer vibrational potentials for a diatomic molecule corresponding to the CH fragment. The experimentally realistic anharmonic potential (solid line) is accurately described by the Morse function Vmorse = De[l — exp(a(r — r0)]2 with De = 397kJ/mol, a = 2A and ro = 1.086 A (A = Angstrom = 10 10m). Near the origin the BO potential is adequately approximated by the harmonic oscillator (Hooke s Law) function (dashed line), Vharm osc = f(r — ro)2/2. The harmonic oscillator force constant f = 2a2De...

FIGURE 3.2 (a) Vibration of diatomic molecule, HC1, (b) potential energy of an ideal harmonic oscillator, and (c) an anharmonic oscillator described by the Morse function. 

The initial exploration in this unit requires the students to compare the trajectories calculated for several different energies for both Morse oscillator and harmonic oscillator approximations of a specific diatomic molecule. Each pair of students is given parameters for a different molecule. The students explore the influence of initial conditions and of the parameters of the potential on the vibrational motion. The differences are visualized in several ways. The velocity and position as a function of time are plotted in Figure 2 for an energy approximately 50% of the Morse Oscillator dissociation energy. The potential, kinetic and total energy as a function of time are plotted for the same parameters in Figure 3. 

The shapes of the absorption band associated with the intensities of vibrational transitions, are sensitive functions of the equilibrium bond length, about which approximately harmonic vibrational oscillations occur. Potential energy curves for a diatomic molecule (Figure 4.2), are commonly represented by Morse equation,... 

Fig. VII-1.—A curve representing the electronic energy of a diatomic molecule as a function of the distance between the nuclei. The zero for energy is the energy of the separated atoms. The minimum of the curve corresponds to the equilibrium value of the internuclear distance. The curve shown, which approximates closely the observed electronic energy curves for many states of diatomic molecules, corresponds to the Morse function.

A. simple function that gives a close approximation to the electronic energy curve for many states of diatomic molecules is the Morse function. This function is... 

Figure 1.1 Morse curve characterizing the energy of the molecule as a function of the distance R that separates the atoms of a diatomic molecule such as hydrogen. At a distance equal to Re, which corresponds to point 0, the molecule is in its most stable position, and so its energy is called the molecular equilibrium energy and expressed as Ee. Stretching or compressing the bond yields an increase in energy. The number of bound levels is finite. Dq is the dissociation energy and De the dissociation minimum energy. The horizontal lines correspond to the vibrational levels.

If this Morse function is used to represent any single bond, not necessarily of a diatomic molecule, the constant a calculated from the harmonic force constant may not be entirely appropriate, and especially not over the entire range of r. Before deriving multiple-bond properties from the single-bond curve it is therefore useful to optimize the Morse constant empirically to improve the match between calculated and observed single-bond values of De and re. 

The usefulness of this function, like that of the Morse function for diatomic molecules, depends upon its applicability to observational data and upon its convenience. Unfortunately it does not possess the virtue of simplicity, a substantial deficiency from the standpoint of convenience. Whether its applicability to data will stimulate general acceptance of the function remains to be seen. Nevertheless, the goal of discovering an empirical potential function which expresses the energetics of the vibrational degrees of freedom of the H bond is one of considerable interest and value. (See also 989, 2070, 1792, 112, 1951, 1078.)... 

A considerable contribution to the theoretical study of isoelectronic diatomic molecules was made by Laurenzi (1969, 1972, 1976, 1981), who obtained the equations describing the behavior of Re, De, and k2 as functions of nuclear charges in both cases of homo- and heteronuclear diatomic molecules. In particular, Laurenzi (1976, 1981) solved exactly these equations for one-electron diatomics with some empirical diatomic potential functions (Morse and Varshni III). 

Fio. 34-2.—a typical function u(r) for tronic energy values Un and the a diatomic molecule (Morse function).. . ... 

Fig. 6.44. Morse curve (Equation 6.5) calculated by using constants Z), = 38.8 kcal mol and p = 13.87 A derived (least-squares method see text) to fit the bond length reactivity data in the way suggested by Figure 6.43. The dashed curve shows the shallower potential function calculated for the gas-phase dissociation into the component atoms of a diatomic molecule joined by a bond with similar properties to that of the C-OR bond of a 2-alkoxytetrahydropyran derived from an alcohol ROH of 15.5...

The interaction energy represents the binding energy of a diatomic molecule. It is usually known as a function of inter-nuclear distance from an experimentally determined equation, as, for example, a Morse curve. If we have some notion as to the relative proportions of and a/3, we can estimate all the coulomb and exchange integrals for the pairs and substitute into an expression such as Eq. (98) to obtain E for more complex systems. By judicious variation of the so-called coulomb fraction one can get a fair agreement of E with experimental determinations thereof. The coulomb fractions ordinarily need not be varied more than from about 1/10 to 3/10, which is close to the theoretical calculation for hydrogen atoms. 

The function of interatomic distance r proposed by Morse (Phys. Rev. 1929, 34, 57) to represent the interatomic energy E in a diatomic molecule is... 

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