The potential energy curve

The electronic energy of a molecule is defined as zero when all electrons and nuclei are infinitely far apart and at rest. When discussing the properties of the molecule it is, however, often more convenient to define the energy as zero when all bonds are broken, that is when the constituent atoms are infinitely far apart and in their ground states. We therefore define the potential energy of the molecule as the difference between its electronic energy and the sum of the electronic energies of the atoms  

If the adiabatic approximation is valid, we may draw a potential energy curve for a diatomic molecule that shows how the potential energy depends on a single variable, the internuclear distance R. See Fig. 4.2. 

---

The fact that the separated-atom and united-atom limits involve several crossings in the OCD can be used to explain barriers in the potential energy curves of such diatomic molecules which occur at short intemuclear distances. It should be noted that the Silicon... 

A transition structure is the molecular species that corresponds to the top of the potential energy curve in a simple, one-dimensional, reaction coordinate diagram. The energy of this species is needed in order to determine the energy barrier to reaction and thus the reaction rate. A general rule of thumb is that reactions with a barrier of 21 kcal/mol or less will proceed readily at room temperature. The geometry of a transition structure is also an important piece of information for describing the reaction mechanism. 

If the system can only be modeled feasibly by molecular mechanics, use the potential energy curve-crossing technique or a force held with transition-structure atom types. 

The reason that does not change with isotopic substitution is that it refers to the bond length at the minimum of the potential energy curve (see Figure 1.13), and this curve, whether it refers to the harmonic oscillator approximation (Section 1.3.6) or an anharmonic oscillator (to be discussed in Section 6.1.3.2), does not change with isotopic substitution. Flowever, the vibrational energy levels within the potential energy curve, and therefore tq, are affected by isotopic substitution this is illustrated by the mass-dependence of the vibration frequency demonstrated by Equation (1.68). 

The dissociation energy is unaffected by isotopic substitution because the potential energy curve, and therefore the force constant, is not affected by the number of neutrons in the nucleus. However, the vibrational energy levels are changed by the mass dependence of 03 (proportional to where /r is the reduced mass) resulting in Dq being isotope-... 

The potential energy curve in Figure 6.4 is a two-dimensional plot, one dimension for the potential energy V and a second for the vibrational coordinate r. For a polyatomic molecule, with 3N — 6 (non-linear) or 3iV — 5 (linear) normal vibrations, it requires a [(3N — 6) - - 1]-or [(3A 5) -F 1]-dimensional surface to illustrate the variation of V with all the normal coordinates. Such a surface is known as a hypersurface and clearly cannot be illustrated in diagrammatic form. What we can do is take a section of the surface in two dimensions, corresponding to V and each of the normal coordinates in turn, thereby producing a potential energy curve for each normal coordinate. 

Figure 6.38 General shapes of the potential energy curves for the vibrations (a) v- and (b) V3 of CO2...

It might be supposed that, since the potential energy curve for V2 is of a similar shape to that in Figure 6.38(a), if we excite the molecule with sufficiently high energy it will eventually dissociate, losing six hydrogen atoms in the process ... 

However, because of the avoided crossing of the potential energy curves the wave functions of Vq and Fi are mixed, very strongly at r = 6.93 A and less strongly on either side. Consequently, when the wave packet reaches the high r limit of the vibrational level there is a chance that the wave function will take on sufficient of the character of Na + 1 that neutral sodium (or iodine) atoms may be detected. 

Such diagrams make clear the difference between an intermediate and a transition state. An intermediate lies in a depression on the potential energy curve. Thus, it will have a finite lifetime. The actual lifetime will depend on the depth of the depression. A shallow depression implies a low activation energy for the subsequent step, and therefore a short lifetime. The deeper the depression, the longer is the lifetime of the intermediate. The situation at a transition state is quite different. It has only fleeting existence and represents an energy maximum on the reaction path. 

Sketch an approximate potential energy diagram similar to that shown in Figures 3.4 and 3.7 for rotation about the carbon-carbon bond in 2,2-dimethylpropane. Does the form of the potential energy curve of 2,2-dimethylpropane more closely resemble that of ethane or that of butane ... 

What are the energies for each species Plot the general shape of the potential energy curve for this reaction. 

Now consider the disubstituted molecule CH2FCH2CI. The potential energy curve has a new feature. There are three minima, corresponding to the three conformations of Figure 1.23. A potential energy curve is shown in Figure 1.24. 

You will find the detailed solution of the electronic Schrddinger equation for H2" in any rigorous and old-fashioned quantum mechanics text (such as EWK), together with the potential energy curve. If you are particularly interested in the method of solution, the key reference is Bates, Lodsham and Stewart (1953). Even for such a simple molecule, solution of the electronic Schrddinger equation is far from easy and the problem has to be solved numerically. Burrau (1927) introduced the so-called elliptic coordinates... 

If we want to calculate the potential energy curve, then we have to change the intemuclear separation and rework the electronic problem at the new A-B distance, as in the H2 calculation. Once again, should we be so interested, the nuclear problem can be studied by solving the appropriate nuclear Schrodinger equation. This is a full quantum-mechanical equation, not to be confused with the MM treatment. 

Finally we have to remember to add on the nuclear repulsion and, if we repeat the calculation for a range of values of the internuclear separation, we arrive at the potential energy curves shown in Figure 4.3 for the ground-state (singlet)... 

At the other end of the spectrum are the ab initio ( from first principles ) methods, such as the calculations already discussed for H2 in Chapter 4. I am not trying to imply that these calculations are correct in any strict sense, although we would hope that the results would bear some relation to reality. An ab initio HF calculation of the potential energy curve for a diatomic Aj will generally give incorrect dissociation products, and so cannot possibly be right in the absolute sense. The phrase ab initio simply means that we have started with a certain Hamiltonian and a set of basis functions, and then done all the intermediate calculations with full rigour and no appeal to experiment. 

In Chapter 4, we considered a simple LCAO treatment of dihydrogen and calculated the potential energy curve reproduced in Figure 11.2. The LCAOs we deduced correspond to HF-LCAO MOs for a minimal basis set. 

Let us now consider the same species of molecule situated in a particular solvent and dissociated into a pair of ions. The potential-energy curve will be similar but will have a much shallower minimum, as in Fig. 86, because in a medium of high dielectric constant the electrostatic attraction is much weaker. Let the dissociation energy in solution be denoted by D, in contrast to the larger Dvac, the value in a vacuum. 

We may fix our attention on the minimum of the potential-energy curve in Fig. 7 and ask how much higher the lowest vibrational level will lie. This energy gap between the potential minimum and the lowest vibrational level is equal to the vibrational energy of the molecule at the absolute zero of temperature and is known as the zero-point energy of... 

If e is now decreased, with the chain in the extended state, the dumbbell nevertheless stays in the stretched state where the potential is the lowest. The transition back to the coiled state occurs only when there is a single minimum on the potential energy curve, i.e. at et = 0.15. Since the critical strain-rate for the stretch-to-coil transition (esc) is much below the corresponding value for the coil-to-stretch transition (eca), the chain stretching phenomenon shows hysteresis (Fig. 11). 

FIGURE 1 The strength of a bond is a measure of the depth of the well in the potential energy curve the stiffness—which governs the vibrational frequency—is determined by the steepness with which the potential energy rises as the bond is stretched or compressed. 

---