Potential surfaces perturbations

One final point should be noted. Theoretical discussions of electron transfer processes have focused almost entirely on outer-sphere processes. When we have an inner-sphere mechanism, or sufficient electronic interaction in a dynamically trapped mixed-valence complex to produce a large separation between upper and lower potential surfaces, the usual weak-interaction approach has to be abandoned. Thus a detailed knowledge of a potential surface which is not describable as an intersection surface of perturbed harmonic surfaces, for example, is required. For this purpose, detailed calculations will be required. The theory of these processes will be linked more... 

Finally, I refer back to the beginning of this paper, where the assumption of near-adiabaticity for electron transfers between ions of normal size in solution was mentioned. Almost all theoretical approaches which discuss the electron-phonon coupling in detail are, in fact, non-adiabatic, in which the perturbation Golden Rule approach to non-radiative transition is involved. What major differences will we expect from detailed calculations based on a truly adiabatic model—i.e., one in which only one potential surface is considered [Such an approach is, for example, essential for inner-sphere processes.] In work in my laboratory we have, as I have mentioned above,... 

This expression was derived by Bell (1978), who used Kramers theory to show that bond lifetime ean be shortened by an applied force in processes such as cell adhesion. Although Eq. (3.2) is quite useful, it is in practice limited, most notably by the fact that it assumes that xp is constant. Typically, measurements of force dependency are made under conditions in which force changes with time, and it is likely that the position of the transition state will move as the shape of the potential surface is perturbed by an applied force (Evans and Ritchie 1997 Hummer and Szabo 2003). Theoretical and empirical treatments of various cases have been put forth in the hterature, but they are outside the scope of this chapter and will not be reviewed here. 

Figure 2.8 is the energy schematic of the combined system. As the tip and the sample approach each other with a finite bias V, the potential 1/ in the barrier region becomes different from the potentials of the free tip and the free sample. To make perturbation calculations, we draw a separation surface between the tip and the sample, then define a pair of subsystems with potential surfaces Us and Un respectively. As we show later on, the exact position of the separation surface is not important. As shown in Fig. 2.8, we define the potentials of the individual systems to satisfy two conditions. First, the sum of the two potentials of the individual systems equals the potential of the combined system, that is. 

Fig. 2.8. Perturbation approach for quantum transmission, (a) A separation surface is drawn between the two subsystems. The precise location is not critical, (b) The potential surface of the system, (c) and (d) The potential surfaces of the subsystems. (Reproduced from Chen, 1991b, with permission.)...

Fig. 7.6. Perturbation theory of the attractive atomic force in STM. (a) The geometry of the system. A separation surface is drawn between the tip and the sample, (b) The potential of the coupled system, (c) The potential surface of the unperturbed Hamiltonian of the sample, Us, which may be different from the potential surface of the free sample, Uso, (d) The potential surface of the unperturbed Hamiltonian of the tip, Ut, which may be different from the potential surface of the free tip, U-m- The effect of the difference between the "free" tip (sample) potential and the "distorted" tip (sample) potential can be evaluated using the perturbation method see Chapter 2. (Reproduced from Chen, 1991b, with permission.)...

In Reference [35], numerical examples of perturbative Sq - S2 excitation and the S2 IC dynamics for the / -carotene are discussed, too. The absence of reliable potential surfaces for this system motivated the use of a minimal two-dimensional model [66], which utilizes a Morse potential in each dimension. All three electronic surfaces Sq, and S2 involved in this example assume the same 2D potential form however, these potentials are shifted to each other. More importantly, in Ref. [35], each potential has 396 bound states in each electronic state within this model, while additionally the S2 and electronic states are coupled by linear coupling. Thus, the Q-space and P-space, as introduced in the context of the QP-algorithm in Section 1.3.1, consist of the S2 and 5 bound states, respectively. 

Near the minimum of the ground electronic surface, the anharmonicities generally play the role of perturbations so that the spectra are regular and the first mechanism is weak. It should become more important when the potential surface deviates significantly from the parabolic shape. The second mechanism, by contrast, may have a very marked effect, as illustrated by the example of N02 [5, 6],... 

In fact, for tightly localized electron pairs, the dominant excitation level is the value of k nearest "vO.OlN (i.e., for about 200 electrons the double excitations in aggregate are more important than the SCF configuration and for 400 electrons quadruple excitations should dominate). Even for molecules with only 40 electrons quadruple and higher excitations must be considered in order to reproduce excitation energies (30) or potential surfaces to an accuracy of 0.1 eV. Thus, configuration interaction calculations for very large molecules are hopeless unless perturbation theory can be used to correct for unlinked cluster effects. 

The reaction between ammonia and methyl halides has been studied by using ab initio quantum-chemical methods.90 An examination of the stationary points in the reaction potential surface leads to a possible new interpretation of the detailed mechanism of this reaction in different media, hr the gas phase, the product is predicted to be a strongly hydrogen-bonded complex of alkylammonium and halide ions, in contrast to the observed formation of the free ions from reaction hr a polar solvent. Another research group has also studied the reaction between ammonia and methyl chloride.91 A quantitative analysis was made of the changes induced on the potential-energy surface by solvation and static uniform electric fields, with the help of different indexes. The indexes reveal that external perturbations yield transition states which are both electronically and structurally advanced as compared to the transition state in the gas phase. 

In principle, however, the intermolecular force problem has a compensating advantage. Because of the very small changes introduced by the interaction, at least until the repulsive part of the potential surface is reached, this is an ideal situation for the application of perturbation theory which has not, in general, been a particularly powerful tool in valence problems. Even so, except at large distance where the overlap of the wavefunctions of the two molecules is negligible, the application of perturbation theory raises some problems. In the main these were sorted out in the years around 1970, but work in this area continues and some of it is discussed in Section 2. 

Two theoretical techniques worthy of serious review here, perturbation and Green function methods, can be considered complementary. Perturbation methods can be employed in systems which deviate only slightly from regular shape (mostly from planar geometry, but also from other geometries). However, they can be used to treat both linear and nonlinear PB problems. Green function methods on the other hand are applicable to systems of arbitrary irregularity but are limited to low surface potential surfaces for which the use of the linear PB equation is permitted. Both methods are discussed here with reference to surfactant solutions which are a potentially rich source of nonideal surfaces whether these be solid-liquid interfaces with adsorbed surfactants or whether surfactant self-assembly itself creates the interface. 

---