Small numbers

In small numbers, anodes have been made with iron (Fe), aluminium (Al), and titanium (Ti). 

The diffraction pattern consists of a small number of spots whose symmetry of arrangement is that of the surface grid of atoms (see Fig. IV-10). The pattern is due primarily to the first layer of atoms because of the small penetrating power of the low-energy electrons (or, in HEED, because of the grazing angle of incidence used) there may, however, be weak indications of scattering from a second or third layer. 

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

It was assumed that, apart from a vanishingly small number of exceptions, the initial conditions do not have an effect on these averages. However, since tire limitmg value of the time averages caimot be computed, an... 

Another example of a teclmique for detecting absorption of laser radiation in gaseous samples is to use multiphoton ionization with mtense pulses of light. Once a molecule has been electronically excited, the excited state may absorb one or more additional photons until it is ionized. The electrons can be measured as a current generated across the cell, or can be counted individually by an electron multiplier this can be a very sensitive technique for detecting a small number of molecules excited. 

This represents a large signal for a small number of atoms on the surface. With care, 10 monolayers of gold on silicon can be detected. 

In this approach [ ], the LCAO-MO coefficients are detemiined first via a smgle-configuration SCF calculation or an MCSCF calculation using a small number of CSFs. The Cj coefficients are subsequently detemiined by making the expectation value ( P // T ) / ( FIT ) stationary. 

Wall M R and Neuhauser D 1995 Extraction, through filter-diagonalization, of general quantum eigenvalues or classical normal mode frequencies from a small number of residues or a short-time segment of a signal. 

Off-lattice models enjoy a growing popularity. Again, a particle corresponds to a small number of atomistic repeat units... 

Studies of single molecules in cryogenic solids, while limited to a relatively small number of cliromophore/ matrix combinations (and small variations tliereupon), have covered a wide range of spectroscopic and dynamic processes [66,62]. 

A well-understood catalytic cycle is tliat of the Wilkinson alkene hydrogenation (figure C2.7.2) [2]. Like most catalytic cycles, tliat shown in figure C2.7.2 is complex, involving intennediate species in tire cycle (inside tire dashed line) and otlier species outside tire cycle and in dead-end patlis. Knowledge of all but a small number of catalytic cycles is only fragmentary because of tire complexity and because, if tire catalyst is active, tire cycle turns over rapidly and tire concentrations of tire intennediates are minute thus, tliese intennediates are often not even... 

The description of chemical reactions as trajectories in phase space requires that the concentrations of all chemical species be measured as a function of time, something that is rarely done in reaction kinetics studies. In addition, the underlying set of reaction intennediates is often unknown and the number of these may be very large. Usually, experimental data on the time variation of the concentration of a single chemical species or a small number of species are collected. (Some experiments focus on the simultaneous measurement of the concentrations of many chemical species and correlations in such data can be used to deduce the chemical mechanism [7].)... 

It remains to investigate the zeros of Cg t) arising from having divided out by. The position and number of these zeros depend only weakly on G, but depends markedly on the fomi that the time-dependent Hamiltonian H(x, () has. It can be shown that (again due to the smallness of ci,C2,...) these zeros are near the real axis. If the Hamiltonian can be represented by a small number of sinusoidal terms, then the number of fundamental roots will be small. However, in the t plane these will recur with a period characteristic of the periodicity of the Hamiltonian. These are relatively long periods compared to the recurrence period of the roots of the previous kind, which is characteristically shorter by a factor of... 

The curl condition given by Eq. (43) is in general not satisfied by the n x n matrix W (R ), if n does not span the full infinite basis set of adiabatic elechonic states and is huncated to include only a finite small number of these states. This tmncation is extremely convenient from a physical as well as computational point of view. In this case, since Eq. (42) does not have a solution, let us consider instead the equation obtained from it by replacing WC) (R t) by its longitudinal part... 

Quantum chemical methods, exemplified by CASSCF and other MCSCF methods, have now evolved to an extent where it is possible to routinely treat accurately the excited electronic states of molecules containing a number of atoms. Mixed nuclear dynamics, such as swarm of trajectory based surface hopping or Ehrenfest dynamics, or the Gaussian wavepacket based multiple spawning method, use an approximate representation of the nuclear wavepacket based on classical trajectories. They are thus able to use the infoiination from quantum chemistry calculations required for the propagation of the nuclei in the form of forces. These methods seem able to reproduce, at least qualitatively, the dynamics of non-adiabatic systems. Test calculations have now been run using duect dynamics, and these show that even a small number of trajectories is able to produce useful mechanistic infomiation about the photochemistry of a system. In some cases it is even possible to extract some quantitative information. 

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