Adiabatic limit

Ben]amin I, Barbara P F, Gertner B J and Hynes J T 1995 Nonequilibrium free energy functions, recombination dynamics, and vibrational relaxation of tjin acetonitrile molecular dynamics of charge flow in the electronically adiabatic limit J. Phys. Chem. 99 7557-67... 

We wish to prove that as the adiabatic limit is approached, the zeros of the component amplitude for the time-dependent ground state (TDGS, to be presently explained) are such that for an overwhelming number of zeros b, Imtr > 0 and for a fewer number of other zeros Imtj 1/A 2n/[Pg.116]

The present paper is organized as follows In a first step, the derivation of QCMD and related models is reviewed in the framework of the semiclassical approach, 2. This approach, however, does not reveal the close connection between the QCMD and BO models. For establishing this connection, the BO model is shown to be the adiabatic limit of both, QD and QCMD, 3. Since the BO model is well-known to fail at energy level crossings, we have to discuss the influence of such crossings on QCMD-like models, too. This is done by the means of a relatively simple test system for a specific type of such a crossing where non-adiabatic excitations take place, 4. Here, all models so far discussed fail. Finally, we suggest a modification of the QCMD system to overcome this failure. 

Thus, the time-dependent BO model describes the adiabatic limit of QCMD. If QCMD is a valid approximation of full QD for sufficiently small e, the BO model has to be the adiabatic limit of QD itself. Exactly this question has been addressed in different mathematical approaches, [8], [13], and [18]. We will follow Hagedorn [13] whose results are based on the product state assumption Eq. (2) for the initial state with a special choice concerning the dependence of 4> on e ... 

Fig. 2. The BO model is the adiabatic limit of full QD if energy level crossings do not appear. QCMD is connected to QD by the semiclassical approach if no caustics are present. Its adiabatic limit is again the BO solution, this time if the Hamiltonian H is smoothly diagonalizable. Thus, QCMD may be justified indirectly by the adiabatic limit excluding energy level crossings and other discontinuities of the spectral decomposition.

Thus, neither BO nor QCMD can describe the non-adiabatic excitation at the crossing. However, as studied in [7], there is yet another feature of the QCMD model that could turn out to be useful here and might help to include the non-adiabatic process. After the crossing the adiabatic limit of QCMD is, in a sense, not uniquely determined ... 

QCMD describes a coupling of the fast motions of a quantum particle to the slow motions of a classical particle. In order to classify the types of coupled motion we eventually have to deal with, we first analyze the case of an extremely heavy classical particle, i.e., the limit M —> oo or, better, m/M 0. In this adiabatic limit , the classical motion is so slow in comparison with the quantal motion that it cannot induce an excitation of the quantum system. That means, that the populations 6k t) = of the... 

In Fig. 1 the absorption spectra for a number of values of excitonic bandwidth B are depicted. The phonon energy Uq is chosen as energy unit there. The presented pictures correspond to three cases of relation between values of phonon and excitonic bandwidths - B < ujq, B = u)o, B > ujq- The first picture [B = 0.3) corresponds to the antiadiabatic limit B -C ljq), which can be handled with the small polaron theories [3]. The last picture(B = 10) represents the adiabatic limit (B wo), that fitted for the use of variation approaches [2]. The intermediate cases B=0.8 and B=1 can t be treated with these techniques. The overall behavior of spectra seems to be reasonable and... 

In the vibrational-adiabatic limit this formula reduces to the familiar form... 

Naturally, neither of these approximations is valid near the border between the two regions. Physically sensible are only such parameters, for which b < 1. Note that even for a low vibration frequency Q, the adiabatic limit may hold for large enough coupling parameter C (see the bill of the adiabatic approximation domain in fig. 30). This situation is referred to as strong-fiuctuation limit by [Benderskii et al. 1991a-c], and it actually takes place for heavy particle transfer, as described in the experimental section of this review. In the section 5 we shall describe how both the sudden and adiabatic limits may be viewed from a unique perspective. 

In the Keilson-Storer model of J-diffusion, non-adiabatic relaxation is assumed to extend to the whole energy spectrum of a rotator. Actually, for large J the relaxation becomes adiabatic. The considerable difference between the times appears in the adiabatic limit since xe = 00, while xj is defined by m-diffusion according to Eq. (1.12). As is seen from Eq. (1.5) and Eq. (1.6), both J- and m-diffusion are just approximations which hold for low- and high-excited rotational levels, respectively. In general 0 < xj/xE < 1 + y. 

The situation is more complicated in the adiabatic limit when this inequality is reversed. According to Eq. (4.36) and Eq. (4.4) the off-diagonal parts of T and y are different. To elucidate this difference and explore its consequences we shall examine the spectra of the four-level system passing from non-adiabatic to adiabatic broadening. 

In the purely non-adiabatic limit the phase (5.52) coincides with that calculated in [203] and for very long flights (rt b,v" v) or high energies (.E e) it reduces to what can be obtained from the approximation of rectilinear trajectories. However, there is no need for these simplifications. The SCS method enables us to account for the adiabaticity of collisions and consider the curvature of the particle trajectories. The only demerit is that this curvature is not subjected to anisotropic interaction and is not affected by transitions in the rotational spectrum of the molecule. 

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