# (p.229) Appendices

# (p.229) Appendices

# Appendix A: Diabatic representation

According to the usual adiabatic theory (Born–Oppenheimer (BO) approximation; Section 2.2), the electronic Hamiltonian ${\stackrel{\u02c6}{H}}_{e}={\stackrel{\u02c6}{T}}_{r}+V(\mathit{\text{r}}\mathbf{,}\mathit{\text{R}}\mathbf{)}$ is diagonal in the representation defined by eqn. (2.11). The analysis is based on the usual stationary Schrödinger equation (2.7). Equation (2.8) determines the electronic terms, ɛ_{m}(** R**). In quantum chemistry the dependence ɛ

_{m}(

**) is “visualized” as the potential-energy surface.**

*R*Assume that the energy terms have a structure as shown in Fig. A.1 (solid lines). The lowest term contains two regions. One of them (L) corresponds to the bound ionic state, and the second region (R) to the continuum spectrum.

Another example is the case when the potential has two close minima (double-well structure); Fig. A.2. The problem of propagation L→R can be solved with use of the usual method with matching at the boundaries, and so on. However, one can employ a different method: the so-called diabatic representation. Let us introduce a new potential, $\tilde{V}(\mathit{\text{r}}\mathbf{,}\mathit{\text{R}}\mathbf{)}$, so that the effect of the substitution $V\to \tilde{V}$ in the electronic equation (2.8) is to change the adiabatic terms to diabatic; that is, to the picture of crossing terms ${\tilde{\mathrm{\epsilon}}}_{1}$ and ${\tilde{\mathrm{\epsilon}}}_{2}$. In addition, the analysis in the diabatic representation is based not on the stationary but on the time-dependent Schrödinger equation. In other words, the L → R transfer can be described as a quantum transition between the terms 1 and 2. The total wavefunctions are sought in the form (see eqn. (2.39)):

*ϕ*

_{1,v 1}(

**) and**

*R**ϕ*

_{2,v 2}(

**) are not orthogonal, since they belong to different energy terms.**

*R*The approach is similar to that employed by Bardeen (1961) to describe the tunneling effect (see also, for example, Kane, 1969; Reittu, 1995). Instead of the usual method based on the stationary Schrödinger equation, Bardeen analyzed tunneling as a time-dependent phenomenon, and, more specifically, as a quantum transition between two (initial and final) states. The tunneling Hamiltonian formalism (Harrison, 1961; Cohen *et al*., 1962) appears to be a powerful tool in solid-state physics, and especially in the physics of superconductivity.

(p.230)
Assume that *a*(0) = 1 and *b*(0) = 0. Then the probability of the 1–2 transfer is described by eqn. (2.40), where ${H}_{el;1,2}=\int d\mathit{\text{R}}\cdot \mathbf{L}\mathbf{(}\mathit{\text{R}}\mathbf{)}{\mathrm{\varphi}}_{\mathbf{2}}^{\mathbf{\ast}}\mathbf{(}\mathit{\text{R}}\mathbf{)}{\mathrm{\varphi}}_{\mathbf{1}}\mathbf{(}\mathit{\text{R}}\mathbf{)}$; $L(\mathit{\text{R}}\mathbf{)}\mathbf{=}\int \mathbf{d}\mathit{\text{r}}\cdot {\mathrm{\psi}}_{\mathbf{2}}^{\mathbf{\ast}}\mathbf{(}\mathit{\text{r}}\mathbf{,}\mathit{\text{R}}\mathbf{)}{\stackrel{\mathbf{\u02c6}}{\mathbf{H}}}_{\mathbf{e}\mathbf{l}}{\mathrm{\psi}}_{\mathbf{1}}\mathbf{(}\mathit{\text{r}}\mathbf{,}\mathit{\text{R}}\mathbf{)}$.

The function *L*(** R**) depends slowly on

**. The main contribution to the matrix element comes from the region of the overlap of nuclear wavefunctions. As a result, one can write:**

*R**L*

_{0}=

*L*(

*R*_{0}),

*R*

_{0}is the crossing point, and

(p.231) is the Franck–Condon factor. As noted previously, it is essential that the nuclear wavefunctions ${\tilde{\mathrm{\varphi}}}_{2}$ and ${\tilde{\mathrm{\varphi}}}_{1}$ are not orthogonal.

The scale of the electronic wavefunction ${\tilde{\mathrm{\psi}}}_{i}(\mathit{r}\mathbf{,}\mathit{R}\mathbf{)}$ is much larger than that for the nuclear function ${\tilde{\mathrm{\varphi}}}_{i}(\mathit{R}\mathbf{)}$, which is localized in the region δ*R* ~ *a*, where *a* is the vibrational amplitude.

As noted previously, it is essential that, unlike the usual BO approximation, the operator ${\stackrel{\u02c6}{H}}_{el}$ has non-diagonal matrix elements. Such an element describes the transition (in our case, tunneling) between the terms.

The diabatic representation is a convenient tool in molecular spectroscopy and solid-state physics. It is employed to describe the polaronic state when each ion is characterized by two close minima (Section 2.62), the isotope effect (Chapters 9 and 10), and the quasi-resonant states of nanoclusters (Chapters 15).

# Appendix B: Dynamic Jahn–Teller effect

As we know, a system containing degenerate electronic states is unstable. The degeneracy is removed via the Jahn–Teller (JT) effect, and as a result, one can observe the splitting of initially degenerate energy levels. One should distinguish static and dynamic JT effects.

The static JT effect (Fig. A.3) is caused by electron–lattice interaction. This interaction leads to the so-called JT static distortion, so that the ions move to different equilibrium positions. As a result, we are dealing with ionic configuration which has a lower symmetry, which leads to removal of the initial degeneracy. The scale of the distortion is determined by the interplay of the electron–lattice interaction and the elastic-energy terms. The shift is energetically favorable (static distortion energy) and as a result, distorted configuration corresponds to a new stable state. It is essential that the distortion energy exceed the zero-point energy of the corresponding vibrational mode. Indeed, only in this case can the system be permanently distorted (the static JT effect).

However, if this is not the case, then we are dealing with dynamic picture (the dynamic JT effect; see, for example, Salem, 1966), in which the system displays a dynamic interconversion between different distorted forms; that is, it oscillates between energy-equivalent distortions. There could be an intermediate case which can be treated as an interplay of the static and dynamic JT effects (see Fig. A.3).

(p.232) The so-called “tunneling splitting” is a typical example of the dynamic JT effect (see Bersuker, 2006). The apical oxygen in the cuprates also displays such an effect, and the same is true for the oxygen ion in manganites (Section 12.5.3 and Fig. 12.8). For both these cases the dynamic JT effect is manifested in the peculiar isotopic dependence. We call it the “polaronic isotope effect” to stress the fact that the electronic and lattice degrees of freedom are not separable, contrary to the usual adiabatic scenario. This is a key ingredient of the dynamic polaron picture.

Shape deformation upon filling the energy shell in metallic nanoclusters is a manifestation of the Jahn–Teller effect (Section 15.1 and Fig. 15.3). At small filling of the unoccupied shell, the cluster acquires the prolate shape (static JT effect). However, near the half-shell filling the picture is more complicated and does not seem so straightforward as plotted on Fig. 15.3. This can be seen from the adsorption spectra (Borggreen, 1993). Indeed, as is known (see, for example, the review by Kresin V. V., 1992) the adsorption spectrum peaks at the cluster’s plasmon frequency; and correspondingly, one can observe two peaks for the prolate configuration. For this configuration the intensity of the peak corresponding to the long axis should be twice as small relative to another peak, because the peak for the shorter axis is doubly degenerate. However, experimentally the intensities of both peaks near the half-shell filling are almost equal. This can be explained (Kresin and Friedel, 2011) by the dynamic JT effect. Indeed, the energy levels for the prolate and oblate configurations are becoming close near half-filling (see, for example, Ekardt and Penzar, 1991), and we are dealing with quantum oscillations between these two configurations. These oscillations represent the manifestation of the dynamic JT effect which is dominant in this region.

Note that the diabatic representation (Section 2.6.2 and Appendix A) is a very convenient tool for describing the dynamic JT effect.