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The rotational spectrum of a highly vibrationally mixed quantum state. II. The eigenstate-resolved spectroscopy analog to dynamic nuclear magnetic resonance spectroscopy
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32.The initial decay of the survival probability depends only on the direct coupling of the bright state to the bath states. This behavior can be shown by expanding the survival probability in a time series using the moments of the spectrum For a symmetric line shape with the energy of the bright state set to zero the survival probability is For a single bright state spectrum, the moments can be calculated from the expectation value of the corresponding power of the Hamiltonian, The initial decay is defined by the first two terms. The second moment is the sum of the squares of the matrix elements connecting the bright state to the bath. Because the second moment is the sum of the matrix elements, the total initial rate can be separated into the two contributions given in the text.
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37.This line width is chosen to add clarity to the figures in this paper. In our experiments, the linewidth is 300 kHz for saturation spectroscopy (Ref. 24) and 6 MHz for measurements using the Autler–Townes splitting technique (Ref. 14).
38.We choose to consider the rotational spectrum in the Type I frequency region so that we can compare the eigenstate rotational spectrum to the vibrational spectrum. Of course, there is also a spectrum in the frequency region for the Type II states. The results presented in this section are equally valid for spectroscopy in the Type II frequency range.
39.The definition is analogous to the use of this symbol to denote the spin–spin relaxation time in NMR spectroscopy. The definition is analogous to the spin-lattice relaxation time; A. Abragam, Principles of Nuclear Magnetism (Oxford University Press, New York, 1961).
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41.The decay rates are actually twice the value given by Eq. (6). The factor of 2 can be understood by comparison to relaxation kinetics techniques. A differential equation approach shows that when a reaction is perturbed away from equilibrium, it relaxes at the sum of the forward and reverse reaction rates. Eigenstate rotational spectroscopy is analogous to relaxation methods. The eigenstate is initially at “conformational equilibrium” because it is composed of both Type I and Type II basis states. Coherent rotational excitation takes the molecule away from equilibrium by causing structure relocalization. The decay of the total Type I probability [Eq. (8)] occurs at the sum of the forward and reverse isomerization rates. In these calculations, the forward and reverse rates are equal; M. J. Pilling and P. W. Seakins, Reaction Kinetics (Oxford University Press, New York, 1995), Chap. 8.
42.The determination of the initial state created by coherent excitation of the eigenstate rotational spectrum, Ψ(0), follows the same formulation that is used for the vibrational spectrum (Refs. 28 and 30). The initial state is a superposition of the eigenstates. The amplitude of each eigenstate in the coherent state is determined by the rotational transition moment for the eigenstate-to-eigenstate transition.
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46.The initial decay of the vibrational survival probabilities in Fig. 8(c) are the same for the reasons discussed in Ref. 32. The increase in the bath isomerization does affect the survival probability at longer times, however. The bath isomerization matrix elements contribute to the fourth moment of the spectrum. As a result, fast bath isomerization leads to a slowing of the decay as seen in Fig. 8(c). This effect is an example of motional narrowing;
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48.For example, direct vibrational transitions between vibrational states of two different conformations are generally much weaker than vibrational transitions that maintain conformational structure.
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