The Adiabatic and Harmonic Approximations

So far we have regarded the atoms merely as point masses with no structure. In reality we are, of course, dealing with nuclei and electrons, and the question arises to which extent the motion of the nuclei are independent of the motion of the electrons. The core electrons no doubt move rigidly with 

In the following, we briefly sketch some formal expressions of the main results of the adiabatic approximation. For a detailed discussion the reader is referred to [1.25-33]. We start with the total Hamiltonian for the crystal, using r as a collective symbol for the coordinates of the valence electrons and R similarly for the ionic coordinates. Thus we write 

Here X ( ) is a wave function for the entire system of electrons x ) is a function of all variables represented by r and contains the ionic coordinates R as parameters. Likewise, the electronic energy Eg(R) depends on the 

Substituting (1.6) in (1.4) and making use of (1.5), it may be shown that (1.6) is indeed a solution of (1.4), provided certain terms can be neglected and the ionic wave function ij (R) is chosen to satisfy (Problem 1.5.1) 

An elementary discussion of the neglected terms is given in [1.32]. Equation (1.8) is an equation for a wave function of the ions alone. The essential point is that for the ionic motion, an effective potential energy function 

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Thus, in order to treat the indirect damping of H-bond bridge beyond the adiabatic and harmonic approximations, it is necessary to propose new simple approaches. This is the aim of this section, which is dealing with two new approximate methods [71]. One of them, the non-Hermitean Diagonal Hamiltonians (NHDH) method uses non-Hermitean diagonal Hamiltonians. 

The other one, the Complex Energy Levels (CEL) method, directly postulates complex energy levels. As will be seen, these two methods lead to equivalent line shapes when both the adiabatic and harmonic approximations are simultaneously removed. 

Now, owing to the well-behaved character of the approached NHDH and CEL methods, it appears reasonable to apply them to more complex situations where, in the absence of Fermi resonance, the quantum indirect damping, and not only the direct damping [72], has to be treated beyond the adiabatic and harmonic approximations [71]. 

Figure 11. SDs within adiabatic and harmonic approximations. Comparison of the approached NHDH method (dots ydoJ = 0.585 Y° f2(jot = 0.30 ft) with the reference (grayed y° = 0.2 ft, y = 0.3 ft). Common parameters a° = 1, co° = 3000cm-1, ft = 100cm-1, T = 300 K.

Because of the agreement between the two approached NHDH and CEL methods for the situation dealing without Fermi resonance, but beyond adiabatic and harmonic approximations, it appears reasonable to apply these methods to many more complex physical situations involving Fermi resonances, beyond the... 

First, let us note that the adiabatic potentials and V [Eq. (67)], even in the lowest order (harmonic) approximation, depend on the difference of the angles 4>j- and t >c this is an essential difference with respect to triatomics where the adiabatic potentials depend only on the radial bending coordinate p. The foims of the functions V, Vt, and Vc are determined by the adiabatic potentials via the following relations... 

Specifically, the various papers working within both the adiabatic and the Condon approximations, and using the (frequent) assumption of harmonic vibrations, can still differ in how many and what type (optical, acoustic, or local) modes they consider and in how they approximate the four separate integrals on the right-hand side of Eq. (40). And the choice of modes applies to both the ground and the excited states (so does the choice of electronic wave functions, but this choice is implicit in the evaluation of the electronic integrals.) It is this choice regarding the two states that was emphasized in connection with Fig. 15 (Section 10b). It can be seen that even within the stated approximations (adiabatic, Condon, harmonic) there is an appreciable number of permutations and combinations. 

Quantum mechanically, resonance Raman cross-sections can be calculated by the following sum-over-states expression derived from second-order perturbation theory within the adiabatic, Born-Oppenheimer and harmonic approximations... 

In the simplest adiabatic case with an orbital singlet term, potential energy of the crystal lattice is parabolic with one minimum point. At low temperatures, vibrations of the lattice are localized at the bottom of this well, and as a rule, the so-called harmonic approximation applies. This corresponds to the so-called polaron effect and brings us to the concept of electrons coated with phonons. 

The harmonic approximation is typically a good approximation for low vibrational levels. The adiabatic approximation is often valid for high vibrational levels and even energies in the continuum above the dissociation limit. Both harmonic and adiabatic approximations are expected to fail when the separation between electronic energy curves is small compared to differences between vibrational levels. 

Since the inner direct product of eq. (11-3) is the only one which contains the totally symmetric representation Als, only the 1B2u and 3Blu curves will mix within the harmonic approximation. In the adiabatic approximation, however, certain vibrational motions may destroy the point group symmetry and allow the 1BZu state to mix with the 3B2u and 3Elu states. 

The suppression effect is most pronounced in the adiabatic (low-frequency limit). A typical zero-level curve p(i 0) (see Fig- 4.28) may be, although roughly but reasonably, divided into three characteristic parts the steep ascend with the noise strength (NIR branch), the bend (NIR-FIR crossover), and the noise-independent saturation (FIR branch). To evaluate the parameters of the suppression resonance, namely, the positions of the branchoff points c0/i and the saturation values pt-(oo) of the zero-level curves for particular harmonics, we have obtained simple but rather accurate approximate expressions. 

This has the form of a double-well oscillator coupled to a transverse harmonic mode. The adiabatic approximation was discussed in great detail from a number of quantum-mechanical calculations, and it was shown how the two-dimensional problem could be reduced to a one-dimensional model with an effective potential where the barrier top is lowered and a third well is created at the center as more energy is pumped into the transverse mode. From this change in the reactive potential follows a marked increase in the reaction rate. Classical trajectory calculations were also performed to identify certain specifically quanta effects. For the higher energies, both classical and quantum calculations give parallel results. 

A basic means of modelling approximate reaction paths is the adiabatic mapping or coordinate driving approach [123,149]. The energy of the system is calculated by minimizing the energy at a series of fixed (or restrained, e.g. by harmonic forces) values of a reaction coordinate, which may be the distance between two atoms, for example. More extensive and complex combinations of geometrical variables can be chosen. This approach is only valid if one... 

The problem has not been resolved analytically. Thirunamachandran and I showed that in special cases answers can be given. If we suppose that both electronic and vibrational motions are represented as simple harmonic vibrations, and the coupling between them given a sufficiently simple form, then the full Hamiltonian can be solved exactly to find energies and eigenfunctions. These exact solutions can be compared with those found in the adiabatic approximation with non-adiabatic corrections. 

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