Prepared states

Quantum mechanically, the time dependence of the initially prepared state of A is given by its wavefimc /("f), which may be detennined from the equation of motion... 

One-layer systems. One-layer systems might easily overcome most of the above-mentioned problems. Such materials show predominantly ionic conduction in the as-prepared state but behave as electrodes in that the concentration of the mobile component is increased and decreased by the charging process in the vicinity of the two electronic leads. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

Prepared State by a Time-Dependent Quantum Mechanical Method. 

Finally, in the semiclassical formulation introduced in Section VII, it is of interest to calculate the autocorrelation function of the initially prepared state... 

Figure 42. Diabatic population (a) and modulus of the autocorrelation function (b) of the initially prepared state for Model IVa. The full tine is the quantum result, and the dashed line depicts the semiclassical mapping result. The semiclassical data have been normalized. Panel (c) shows the norm of the semiclassical wave function.

Food sample preparation State of sample Preparative treatment... 

An issue of considerable interest is whether it would be possible to control or manipulate the photodissociation pathways of a particular molecule by initial vibrational preexcitation. A main concern in achieving this control is the difficulty in preparing states with vibrational energy that is localized in a specific mode that resembles the reaction coordinate. This is due to intramolecular vibrational... 

Furthermore, preservation of the character is assisted by the low density of states in the considered energy windows. The total vibrational state density, as calculated via the Stein-Rabinovitch extension of the Beyer-Swinehart algorithm [88], is about three states/cm, while that of states of Aj symmetry should be even lower. Clearly, this small density of states is insufficient to induce statistical energy flow out of the prepared states, conserving their character to a large extent. This is supported also by our finding that the action spectrum monitored at a delay of 50 ns between the SRS and UV beam resembles that monitored at 10 ns delay, implying... 

The Rydberg state which is optically prepared in a typical ZEKE experiment is usually directly coupled to the continuum [45c, 57]. Other considerations being absent, it should decay promptly, possibly with a stable, trapped component. The point is that the initially prepared state is also directly coupled to many other states, due both to external perturbations [37] and to intramolecular coupling [3b]. The conclusion that the initial state has two components, one that decays promptly and one that is trapped, is thus only valid in zero order (so-called golden rule limit). One needs to allow for the coupling terms represented by V and U. 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

We note that the separation into the three types of transitions (7), (2), and (2) is somewhat artificial. In fact, molecular collisions and transitions due to external fields are special examples of prepared states. Time evolution of a system described by a time-independent Hamiltonian does occur in general, unless the initial state of the system is described by a ket which is an eigenket to the complete Hamiltonian. 

First, we consider the case where the zero-order crossing is between 1B2u and 3Blu curves and where the initial state is pure singlet. The pure singlet Fj1B2 ,l ((2) Q 1B2u) is a prepared state which will evolve in time... 

Finally we consider intersystem crossing for the case of an initial prepared state Q, 1B2u[T] II). The zero-order crossing is taken to be between 1B2u and either [7] = 3Elu or [7] = aB2u curves. Following the procedure used to obtain (11-8) and (11-11), we obtain for the present case... 

Although the common features 1, 2, and 3 above were for certain idealized prepared states, it seems reasonable to expect that the actual prepared state appropriate to a given experiment may be similar enough that these same common features will be exhibited for the microscopic rate constants. These common features of the microscopic rate constant may in turn provide qualitative predictions for the averaged rate constant Arise- These qualitative predictions are expected to be characteristic of intersystem crossing due to a zero-order crossing. 

The time development of such a model system can be easily determined by the method of the Green s function. In Section XII-C, then, we present such a treatment, and obtain those results which allow us to directly predict the decay of our prepared state. 

A lot of thermodynamics makes use of the important concept of state function, which is a property with a value that depends only on the current state of the system and is independent of the manner in which the state was prepared. For example, a beaker containing 100 g of water at 25°C has the same temperature as 100 g of water that has been heated to 100°C and then allowed to cool to 25°C. Internal energy is also a state function so the internal energy of the beaker of water at 25°C is the same no matter what its history of preparation. State functions may be either intensive or extensive temperature is an intensive state function internal energy is an extensive state function. 

Ab initio MCHF calculations have been carried out (49) for the predissociative C2N2 (Cl]lu) state (see the appendix). The initially prepared state was found to remain linear. As a first approximation, the collinear dissociation has been studied thus neglecting bending vibrations and rotations. 

The basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

The first assumption, that phase space is populated statistically prior to reaction, implies that the ratio of activated complexes to reactants is obtained by the evaluation of the ratio between the respective volumes in phase space. If this assumption is not fulfilled, then the rate constant k(E, t) may depend on time and it will be different from rrkm(E). If, for example, the initial excitation is localized in the reaction coordinate, k(E,t) will be larger than A rrkm(A). However, when the initially prepared state has relaxed via IVR, the rate constant will coincide with the predictions of RRKM theory (provided the other assumptions of the theory are fulfilled). 

---