Potential energy curves, coordination

Figure A3.12.5. A model reaction coordinate potential energy curve for a fluxional molecule. (Adapted from [30].)...

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. 

FIGURE 17.2 Illustration of the reaction coordinate for a reaction with a change in the electronic state, (a) Potential energy curves for the two electronic states of the system. (A) Avoided crossing that can be seen in single-detenninant calculations. 

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. 

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... 

The angular-dependent adiabatic potential energy curves of these complexes obtained by averaging over the intermolecular distance coordinate at each orientation and the corresponding probability distributions for the bound intermolecular vibrational levels calculated by McCoy and co-workers provide valuable insights into the geometries of the complexes associated with the observed transitions. The He - - IC1(X, v" = 0) and He + 1C1(B, v = 3) adiabatic potentials are shown in Fig. 3 [39]. The abscissa represents the angle, 9,... 

FIGURE 30. Potential energy curves for a neutral molecule M, and its radical cation M in the ground and first excited state (equilibrium distances with respect to an arbitrary coordinate q along which the three geometries differ). Note the shift in the M+ /(M+ ) energy difference AE on going from e/eq of M (AE = A/v from the PE spectrum of M) to qeq of M (AE corresponds to /-IM ,X from the EA spectrum of M+")... 

Figure 3.42 Potential-energy curves for benzene (17), comparing the total energy (-Etotai, solid curve) with the energy of the idealized single Lewis structure dotted curve) along a D3h distortion coordinate AR = R2,3 — R i that lowers the D6h symmetry to cyclohexatriene form.

Figure 4.61 Potential-energy curves for reactions inserting singlet (circles), triplet (triangles), and quintet (squares) Ti into H2. The reaction coordinate R is the distance between Ti and the midpoint of H2, and the zero of energy corresponds to R = oo for ground-state (triplet) Ti + H2.

Figure 4.67 depicts the potential-energy curve for reaction (4.102) along an adiabatic reaction coordinate (R = /Oimc) obtained by stepping along the H—CH3 stretching coordinate with full optimization of geometries at each step. As shown in Fig. 4.67, the reaction exhibits a substantial barrier ( 20.5 kcal mol-1) and overall exothermicity. 

In the general case R denotes a set of coordinates, and Ui(R) and Uf (R) are potential energy surfaces with a high dimension. However, the essential features can be understood from the simplest case, which is that of a diatomic molecule that loses one electron. Then Ui(R) is the potential energy curve for the ground state of the molecule, and Uf(R) that of the ion (see Fig. 19.2). If the ion is stable, which will be true for outer-sphere electron-transfer reactions, Uf(R) has a stable minimum, and its general shape will be similar to that of Ui(R). We can then apply the harmonic approximation to both states, so that the nuclear Hamiltonians Hi and Hf that correspond to Ui and Uf are sums of harmonic oscillator terms. To simplify the mathematics further, we make two additional assumptions ... 

Fig. 6-3S. Potential energy curves for water adsorption on metal surface in the states of molecules and hydrozjd radicals c = energy r = reaction coordinate solid curve = adsorption as water molecules and as partially dissociated hydroxj4 and hydrogen radicals broken curve = adsorption of completely dissociated oxygen and hydrogen radicals.

Figure 10.16 Potential energy curves for the transformation of dihydrogen-bonded complex 2t-2TFA to contact ion pair 3t-2TFA. The O-H distance is taken as a reaction coordinate of the transferring proton. Energies are given in kcal/mol. (Reproduced with permission from ref. 6.)...

This striking result can be qualitatively understood as related to CB DOS-influenced changes in the 02 anion lifetime [118]. For a diatomic molecule with R as the internuclear coordinate, a transient anion state is described in the fixed nuclei limit [123,124] by an energy and i -dependent complex potential Vo i R,E ) = Fd(2 ) + A( i)—l/2 T( i), where Va R) = a R) + is the potential energy curve of the discrete state, Vg(R) is the... 

Where AG is the activation energy of the process, and T are the Boltzmann constant and the absolute temperature, respectively, v is the nuclear frequency factor, and is the transmission coefficient, a parameter that expresses the probability of the system to evolve from the reactant to the product configuration once the crossing of the potential energy curves along the reaction coordinate has been reached (Fig. 17.5). 

In Figure 1 are shown potential energy-normal coordinate curves for the vaig mode for site A as Fe111 in the reactants (R) and Fe11 in the products (P). The coordinate is a normal coordinate of the molecule composed of an equally weighted linear combination of the displacement coordinates (q) for the six Fe—O local coordinates, Q = l/V6[q(Fe—0)i + g(Fe—0)2 +. .. + g(Fe—0)6] = V6[q(Fe—O)]. It has no connection with an electron transfer distance nor with the intersite separation between reactants. 

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