Morse curve

Comparison of the simple harmonic potential (Hooke s law) with the Morse curve. 

An undesirable side-effect of an expansion that includes just a quadratic and a cubic term (as is employed in MM2) is that, far from the reference value, the cubic fimction passes through a maximum. This can lead to a catastrophic lengthening of bonds (Figure 4.6). One way to nci iimmodate this problem is to use the cubic contribution only when the structure is ,utficiently close to its equilibrium geometry and is well inside the true potential well. MM3 also includes a quartic term this eliminates the inversion problem and leads to an t". . 11 better description of the Morse curve. 

Fig. 20.16 Potential energy against distance curves Morse curves), (a) No potential dilTerence (p.z.c.), (b) at the equilibrium potential when / = / and the heights of the energy barrier are the same for both reactions, but p.z.c W potential made more negative than E q and (d) potential made more positive than E. The p.z.c. has been taken as zero potential, and A, and h,. are the heights of the potential barriersj or the anodic and cathodic reactions, respectively / is the rate of the cathodic reaction and / the rate of the anodic reaction (after Bockris...

Verwey and Hamaker (10) have modified the Morse curve to take into account the approach of two charged bodies, as shown in Figure 4. Here, as one moves outward toward increasing distance of separation, an electrostatic repulsion is encountered because the charges are similar. A secondary minimum is then encountered as a result of the concentration of counter ions around each charged particle. The shallowness of the secondary minimum shows that the deflocculated system is metastable. The importance of the Verwey and Hamaker concept lies in its ability to show graphically the correlation between the secondary minimum and the metastable position. A, Figure 1. 

Explicit forms for the potential energy in the terms Hi and Hf have been proposed by Saveant [1993], who has developed a semiclassical version, along the lines of the Marcus theory, and applied it successfully to several reactions. In his model, the potential curve for the reactants is a Morse curve, and that for the products is the repulsive branch of a Morse curve ... 

This time period is too short for a change in geometry to occur (molecular vibrations are much slower). Hence the initially formed excited state must have the same geometry as the ground state. This is illustrated in Figure 1.2 for a simple diatomic molecule. The curves shown in this figure are called Morse curves and represent the relative energy of the diatomic system as a... 

Thermodynamics. Microscopic reversibility 120 The Morse curve model 123... 

Fig. 1 Morse curve modeling of the contribution of bond-breaking to the dynamics of dissociative electron transfer...

The kinetics of the electron transfer reaction leading to the homolytically dissociating primary radical is also a question of interest. It may be modeled using the Morse curve for the reactant and the Morse curve shown in Fig. 10 representing the homolytic dissociation of the primary radical. This point will be discussed in detail in Section 5. 

There is thus an apparent continuity between the kinetics of an electron transfer leading to a stable product and a dissociative electron transfer. The reason for this continuity is the use of a Morse curve to model the stretching of a bond in a stable product in the first case and the use of a Morse curve also to model a weak charge-dipole interaction in the second case. We will come back later to the distinction between stepwise and concerted mechanisms in the framework of this continuity of kinetic behavior. 

A semi-classical treatment171-175 of the model depicted in Fig. 15, based on the Morse curve theory of thermal dissociative electron transfer described earlier, allows the prediction of the quantum yield as a function of the electronic matrix coupling element, H.54 The various states to be considered in the region where the zero-order potential energy curves cross each other are shown in the insert of Fig. 15. The treatment of the whole kinetics leads to the expression of the complete quenching fragmentation quantum yield, oc, given in equation (61)... 

Figure 9.12 Morse curve of the diatomic molecule X2 in the ground state...

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. 

Look at Figure 9.13, which now shows two Morse curves. The lower curve is that of the molecule in the ground state (in fact, it... 

The deep energy minimum in a Morse curve is often called an energy well. ... 

Following photon absorption, an electron from the HOMO of X2 is excited from the ground to the first excited state. The electronic excitation that occurs on photon absorption is represented on the figure by an arrow from the lower (ground state) Morse curve to the higher (excited state) curve. The time required for excitation of the electron is very short, at about 10-16 s. By contrast, because the atomic nuclei are so much more massive than the electron, any movement of the nuclei occurs only some time after photo-excitation of the electron - a safe estimate is that nuclear motion occurs only after about 10-8 s, which is 108 times slower. 

The Bom-Oppenheimer approximation has a further serious consequence. At the instant of excitation, the length of the bond in the excited state species is the same as that within the ground state. While close inspection readily convinces us that the two Morse curves are very similar and have the same shape, it is important to recognize that the equilibrium bond lengths differ. 

The molecule in its excited state rearranges after the photon is absorbed, and rearranges its bond lengths and angles until it has reached its new minimum energy, i.e. attained a structure corresponding to the minimum in the upper Morse curve. 

The multiplicity of excitations possible are shown more clearly in Figure 9.16, in which the Morse curves have been omitted for clarity. Initially, the electron resides in a (quantized) vibrational energy level on the ground-state Morse curve. This is the case for electrons on the far left of Figure 9.16, where the initial vibrational level is v" = 0. When the electron is photo-excited, it is excited vertically (because of the Franck-Condon principle) and enters one of the vibrational levels in the first excited state. The only vibrational level it cannot enter is the one with the same vibrational quantum number, so the electron cannot photo-excite from v" = 0 to v = 0, but must go to v = 1 or, if the energy of the photon is sufficient, to v = 1, v = 2, or an even higher vibrational state. 

We describe the vibrational levels in the lower (ground-state) Morse curve with the quantum number v", the lowermost vibrational level being v" = 0. The vibrational states in the upper (excited-state) Morse curve are described by the quantum number v. ... 

An electron excites from a vibrational energy level in the lower, ground-state Morse curve, to a vibrational level in the excited-state Morse curve. 

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