State, -non-stationary

Bala, R, Lesyng, B., McCammon, J.A. Extended Hellmann-Feynman theorem for non-stationary states and its application in quantum-classical molecular dynamics simulations. Chem. Phys. Lett. 219 (1994) 259-266. 

This spatial distribution is not stationary but evolves in time. So in this ease, one has a wavefunetion that is not a pure eigenstate of the Hamiltonian (one says that E is a superposition state or a non-stationary state) whose average energy remains eonstant (E=E2,i ap + El 2 bp) but whose spatial distribution ehanges with time. 

For an initiator concentration which is constant at [l]o, the non-stationary-state radical concentration varies with time according to the following expression ... 

When results are compared for polymerization experiments carried out at different frequencies of blinking, it is found that the rate depends on that frequency. To see how this comes about, we must examine the variation of radical concentration under non-stationary-state conditions. This consideration dictates the choice of photoinitiated polymerization, since in the latter it is almost possible to turn on or off—with the blink of a light—the source of free radicals. The qualifying almost in the previous sentence is actually the focus of our attention, since a short but finite amount of time is required for the radical concentration to reach [M-] and a short but finite amount of time is required for it to drop back to zero after the light goes out. 

The superpositioning of experimental and theoretical curves to evaluate a characteristic time is reminiscent of the time-tefnperature superpositioning described in Sec. 4.10. This parallel is even more apparent if the theoretical curve is drawn on a logarithmic scale, in which case the distance by which the curve has to be shifted measures log r. Note that the limiting values of the ordinate in Fig. 6.6 correspond to the limits described in Eqs. (6.46) and (6.47). Because this method effectively averages over both the buildup and the decay phases of radical concentration, it affords an experimentally less demanding method for the determination of r than alternative methods which utilize either the buildup or the decay portions of the non-stationary-state free-radical concentration. 

The term three-phase fluidization, in this chapter, is taken as a system consisting of a gas, liquid, and solid phase, wherein the solid phase is in a non-stationary state, and includes three-phase slurry bubble columns, three-phase fluidized beds, and three-phase flotation columns, but excludes three-phase fixed bed systems. The individual phases in three-phase fluidization systems can be reactants, products, catalysts, or inert. For example, in the hydrotreating of light gas oils, the solid phase is catalyst, and the liquid and gas phases are either reactants or products in the bleaching of paper pulp, the solid phase is both reactant and product, and the gas phase is a reactant while the liquid phase is inert in anaerobic fermentation, the gas phase results from the biological activity, the liquid phase is product, and the solid is either a biological carrier or the microorganism itself. 

In our method reproducible non-stationary states are effected as follows the low stationary-state rate of an autoxidizing hydrocarbon is decreased by a factor of 2 to 5 by adding an appropriately small amount of inhibitor. Under the conditions outlined below, the time required to establish the new stationary state at the inhibited rate is not immeasurably small, as it would be in conventional measurements, but of the order of 100-300 sec. With sufficiently sensitive apparatus a number of determinations of the decreasing velocity can be made, which delineate the course of the non-steady state. Similarly a non-steady state with an increasing velocity can be realized by introducing a small amount of initiator. 

In order to estimate kinetic constants for elementary processes in template polymerization two general approaches can be applied. The first is based on the generalized kinetic model for radical-initiated template polymerizations published by Tan and Alberda van Ekenstein. The second is based on the direct measurement of the polymerization rate in a non-stationary state by rotating sector procedure or by post-effect in photopolymerization. The first approach involves partial absorption of the monomer on the template. Polymerization proceeds according to zip mechanism (with propagation rate constant kp i) in the sequences filled with the monomer, and according to pick up mechanism (with rate constant kp n) at the sites in which monomer is outside the template and can be connected by the macroradical placed onto template. This mechanism can be illustrated by the following scheme ... 

In the literature on elimination reactions, it is found that mechanistic conclusions are quite frequently made on the basis of the values of the activation energy. This is a dubious practice, especially when the data have been obtained by the gas chromatographic (pulse-flow) technique, i.e. when there is a non-stationary state on the catalyst surface, and on the basis of supposed first-order kinetics. 

The experiments discussed above were all carried out with total pressures below 10-4 Torr. However, Hori and Schmidt (187) have also reported non-stationary state experiments for total pressures of approximately 1 Torr in which the temperature of a Pt wire immersed in a CO—02 mixture was suddenly increased to a new value within a second. The rate of C02 production relaxed to a steady-state value characteristic of the higher temperature with three different characteristic relaxation times that are temperature dependent and vary between 3 and 100 seconds between 600 and 1500 K. The extremely long relaxation time compared with the inverse gas phase collision rate rule out an explanation based on changes within the chemisorption layer since this would require unreasonably small sticking coefficients or reaction probabilities of less than 10-6. The authors attribute the relaxation times to characteristic changes of surface multilayers composed of Pt, CO, and O. The effects are due to phases that are only formed at high pressures and, therefore, cannot be compared to the other experiments described here. 

The coupling of nuclear and electronic motions in electronic transitions may then happen through Z //<(Q) if induced by light, or through Hjf) and Gw if induced by nuclear displacements. The nuclear motion functions can be obtained by additional expansion in a basis of functions of nuclear coordinates, or by numerical solutions on a grid of points in the space of nuclear positions. The second approach is specially suitable for non-stationary states, and is briefly described. 

To apply described above non-perturbative method to the non-radiative transitions caused by the quadratic non-diagonal vibronic interaction qfli, one needs to find the phonon correlation functions Du(t, r) in the initial (non-stationary) state 11,0) for l > 0. To this end, we use the equation of motion xu + (ojxu + Veuqt j = 0 il, l > 0 / V l1) the integral form of it for t > 0 reads... 

The general time-dependent solutions to Eq. (1.10) are denoted as non-stationary states. They can be expanded in terms of the eigenstates n(R) of the Hamiltonian... 

Fig. 7. Positronium in the ground state with the wave function W enters a target and interacts at point 1 with an atom of the target. After the interaction, a non-stationary (e+e ) state with the wave function moves in the target and interacts at point 2 with another atom creating a new non-stationary state 2 After the last interaction, a non-stationary (e+e )-system moves to the vacuum and transforms into a set of stationary states ground and exited states of Ane and free (e+e )-pairs...

In this case the probability of the passage of an atom through a layer of matter becomes greater than the one that follows from the usual exponential dependence. This phenomenon, superpenetration of ultrarelativistic A.2e, allows for measurement of the time of conversion of a non-stationary state of e+e, formed in the target, to stationary states and to verify the form of the Lorentz transformations for the time [8]. The theory of superpenetration has been formulated in [9,10,11]. A quantitative calculation shows that even for a film thickness L = 2.5A the deviation from an exponential absorption law reaches 100%. 

The choice of a single function from either set (36) or (37) does not permit such a useful physical interpretation, and may indeed lead to difficulties as the internuclear distance is varied. Thus if one chooses just the perfectly paired function from the set (36), as -R-> 00 one finds each N atom is described by a curious non-stationary state - the so-called valence state of the atom, about which there has been so much discussion in the literature.18 The choice of the set of functions (36) in which orbitals participating in a bond are directly coupled to each other is just the VB theory as proposed by Slater and Pauling,19 whereas the set (37) formed from atoms in specific L-S coupled states corresponds to the spin-valence theory employed by Heitler.20... 

Bengough, W. J., and H. W. Melville A thermocouple method of folloving the non-stationary state of chemical reactions. 2. The evaluation of velocity coefficients and energies of activation for the propagation and termination reactions for the initial and later stages of the polymerization of vinyl acetate. Proc. Roy. Soc. A 230, 429 (1955). 

With growing viscosity, the diffusion rate of propagating macromolecules decreases, the probability of their effective collision is diminished, and thus also the rate of bimolecular termination. As the rate of initiation is not appreciably affected by increasing viscosity, in a certain phase some radical polymerizations pass from a stationary to a non-stationary state in which the number of radicals increases. The polymerization is appreciably accelerated. This situation is called the gel effect or the Norrish-Tromsdorff effect [55]. As the change in the rate of termination contributes considerably to the gel effect, it will be discussed in detail in Chap. 6, Sect. 1.3. 

---