The perturbation approach

In this Chapter, we discuss a perturbation theory for STM, the modified Bardeen approach (MBA). The illustrate the concept of a perturbation approach, let us consider the following four regimes of interactions (zo denotes the microscopic tip-sample distance)  

When the tip-sample distance is large, for example, zo 100 A, the mutual interaction is negligible. By applying a large electrical field between them, field emission may occur. Those phenomena can be described as the interaction of one electrode with the electrical field, without considering any interactions from the other electrode. 

At intermediate distances, for example, 10 zo 100 A, a long-range interaction between the tip and the sample takes place. The wavefunctions of both the tip and the sample are distorted, and a van der Waals force arises. The van der Waals interaction follows a power law, with an order of magnitude of a few meV per atom. 

At extremely short distances, for example, zo 3 A, the repulsive force becomes dominant. It has a very steep distance dependence. The tip-sample distance is virtually determined by the short-ranged repulsive force. By pushing the tip farther toward the sample surface, the tip and sample deform accordingly. 

The polarization, or the van der Waals interaction, can be accounted for by a stationary-state perturbation theory, effectively and accurately. The exchange interaction or tunneling can be treated by time-dependent perturbation theory, following the method of Oppenheimer (1928) and Bardeen (1960). In this regime, the polarization interaction is still in effect. Therefore, to make an accurate description of the tunneling effect, both perturbations must be considered simultaneously. This is the essence of the MBA. 

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The first theoretical handling of the weak R-T combined with the spin-orbit coupling was carried out by Pople [71]. It represents a generalization of the perturbative approaches by Renner and PL-H. The basis functions are assumed as products of (42) with the eigenfunctions of the spin operator conesponding to values E = 1/2. The spin-orbit contribution to the model Hamiltonian was taken in the phenomenological form (16). It was assumed that both interactions are small compared to the bending vibrational frequency and that both the... 

The solution of equation 16 is a decreasing, simple exponential where = k ([A ] + [P ]) + k. The perturbation approach generates small deviations in concentrations that permit use of the linearized differential equation and is another instance of pseudo-first-order behavior. Measurements over a range of [A ] + [T ] allow the kineticist to plot against that quantity and determine / ftom the slope and from the intercept. 

Some methods of describing electron correlation are compared from the point of view of requirements for theoretical chemical models. The perturbation approach originally introduced by Mpller and Plesset, terminated at finite order, is found to satisfy most of these requirements. It is size consistent, that is applicable to an ensemble of isolated systems in an additive manner. On the other hand, it does not provide an upper bound for the electronic energy. ... 

Although eq. (10.103) for the propagator appears to involve the same effort as the perturbation approach (sum over all excited states, eq. (10.18)), the actual calculation of the propagator is somewhat different. Returning to the time representation of the polarization propagator, it may be written in terms of a commutator. 

The vibrational spectrum of 1,4-dioxin was studied at the MP2 and B3-LYP levels in combination with the 6-3IG basis set [98JST265]. The DPT results tend to be more accurate than those obtained by the perturbational approach. The half-chair conformation of 4//-1,3-dioxin 164 was found to be more stable than the corresponding conformations of 3,4-dihydro-1,2-dioxin 165,3,6-dihydro-1,2-dioxin 166, and of 2,3-dihydro-1,4-dioxin 167 (Scheme 114) [98JCC1064, 00JST145]. The calculations indicate that hyperconjugative orbital interactions contribute to its stability. 

Meyer and coworkers (18). In this case, we see from Figure 6 that the Weiner method predicts faster rates than the perturbation approach, the difference being about a factor of 5 at room temperature and more than an order of magnitude at low temperature. As might be expected, the two methods continue to diverge as increases. If gets so large that the transfer rate becomes comparable to v, the Landau-Zener correction ((5), eq 155) may be applied. c... 

The value of the dipole moment of LiH obtained in this work, 2.3140 a.u., is essentially identical to the experimental value, 2.314 0.001 [90]. Our calculations simulate experiment more closely than any previous calculations. The results also provide validation of the perturbation approach of Ref. 88, since the perturbation result, 2.317 a.u., is very close to our value. At the same time, our results are much more accurate than those of Ref. 57, the only other direct calculation of the LiH dipole moment. The value of the dipole moment of LiD, 2.3088, is also of good accuracy, compared to the experimental result, 2.309 0.001 [90]. Again, our result is much more accurate than that of Ref. 57. 

PESs for fluorine, hydrogen fluoride, and water using this ansatz showed promise [14]. In all cases the perturbative approach improved the accuracy considerably at a small increase of computational cost. Especially interesting is the possibility of linear scaling. 

The derivation of the transmission coefficients for a square barrier can be found in almost every textbook on elementary quantum mechanics (for example, Landau and Lifshitz 1977). However, the conventions and notations are not consistent. Figure 2.5 specifies the notations used in this book. To make it consistent with the perturbation approach later in this chapter, we take the reference point of energy at the vacuum level. 

Before closing this chapter, it is important to emphasize the context in which the transition rate expressions obtained here are most commonly used. The perturbative approach used in the above development gives rise to various contributions to the overall rate coefficient for transitions from an initial state i to a final state i these contributions include the electric dipole, magnetic dipole, and electric quadrupole first order tenns as well contributions arising from second (and higher) order terms in the perturbation solution. 

In the perturbative approach the first order (or higher order) expressions for the self-energy and the polarization operator are used. The other possibility is to summarize further the diagrams and obtain the self-consistent approximations (Figs. 18,19), which include, however, a new unknown function, called vertex function. We shall write these expressions analytically, including the Hartree-Fock part into unperturbed Green function Gq(1, 2). 

In this section we shall discuss an approach which is neither variational nor perturba-tional. This approach also has its origin in nuclear physics and was introduced to quantum chemistry by Sinanoglu47, It is based on a cluster expansion of the wave function. A systematic method for the calculation of cluster expansion components of the exact wave function was developed by C ek48 The characteristic feature of this approach is the expansion of the wave function as a linear combination of Slater determinants. Formally, this expansion is similar to the ordinary Cl expansion. The cluster expansion, however, gives us not only the physical insight of the correlation energy but it also shows the connections between the variational approaches (Cl) and the perturbational approaches (e.g. MB-RSPT). 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

In the following description, we consider a very simplified 3-state model, consisting of the purely LC Tj substate 3(rai )+1, the triplet MLCT substate 3(dir )+1, and the singlet MLCT state (d jt ). This model is chosen to illustrate the perturbational approach and certainly does not contain all relevant physics. The situation is schematically depicted in Fig. 13. 

Large amplitude motions and solvent librations cannot be described by the perturbative approach sketched above, but a classical treatment is usually sufficient. Then, the... 

We start by noting the basic difference between the perturbative approaches to the RHEG and the QED vacuum (discussed in Appendix A). As a consequence of the difference between the ground state 0 > of the RHEG and the homogeneous vacuum 0 > the fermion propagator... 

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