Weak field limit

The case of a weak external potential energy field is generally important, and the simple result is also suggestive. We augment the potential energy to include an external field, and work out that result for the chemical potential, 

This approximation uses the fundamental but elementary point that In (e ) (e) 

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In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

The inequality S S S B is sometimes called the weak-field limit because the external magnetic field B of a typical X-band spectrometer is sufficiently weak for... 

The zero-field spin Hamiltonian parameters, D and E, are assumed to be distributed according to a normal distribution with standard deviations oD and aE, which we will express as a percentage of the average values (D) and (E). -Strain itself is not expected to be of significance, because the shape of high-spin spectra in the weak-field limit is dominated by the zero-field interaction. 

Equation 12.3 is readily rewritten for any other half-integer spin. For integer spins we have one singulet (the ms = 0 level) and S doublets, but we do not bother writing down the modified equation, because spin counting for integer-spin systems in the weak-field limit is near-to-impossible (Hagen 2006). 

The ideas presented here use a weak UV pulse. One could also imagine using strong UV pulses, e.g. in one of the schemes presented by Sola et al. in this book, which alter the BO potentials by mixing them. These schemes usually involve three electronic states. This alters the discussion somewhat since even in the weak-field limit a trivial solution may be found in a pump-dump setup the wave packet is first excited to a repulsive state, where the (now fast) dynamics takes place and when the wave packet has reached the desired location it is dumped to the excited bound state. Exploring various three-state setups is a topic for future research. 

However, Fig. 10 also yields a very surprising result. The total population transferred to the 2 IIg state after the end of the pulse is much larger for a down-chirped laser although the ionization yield with this pulse is smaller. This is not due to population transfer to the ionic ground state since in the weak-field limit ionization does not (significantly) decrease the neutral state population. Rather a mechanism very similar to that in the experiments performed by Bardeen et al. [36] is responsible for this effect. To understand... 

In this weak-field limit the relationship between the yield of product and the optimal field is... 

The eigenvalue problem for the simple cos y potential of Eq. (4) can be solved easily by matrix diagonalization using a basis of free-rotor wave functions. For practical purposes, however, it is also useful to have approximate analytical expressions for the channel potentials V,(r). The latter can be constructed by suitable interpolation between perturbed free-rotor and perturbed harmonic oscillator eigenvalues in the anisotropic potential for large and small distances r, respectively. Analogous to the weak-field limit of the Stark effect, for linear closed-shell dipoles at large r, one has [7]... 

Note that, mathematically, representation (4.179) is not complete. Indeed, in a general case the exact amplitude of the mu mode must contain, along with the contribution c", an infinite set of terms E,"+2, E,"44, and so on. However, in a weak-field limit c < 1 the terms with higher powers are of minor importance so that the main contribution to the magnetization response signal filtered at the frequency no is proportional to c". 

In the weak field limit, the time evolution of wave packets in pump-probe experiments can be evaluated by second-order time-dependent perturbation theory. The second-order solution of Eq. (24) is expressed as... 

Consider the problem of wave packet control in a weak laser field. Here wave packet control refers to the creation of a wave packet at a given target position on a specific electronic potential energy surface at a selected time tf. For this purpose, a variational treatment is introduced. In the weak field limit, the wave packet can be calculated by first-order perturbation theory without the need to solve explicitly the time-dependent Schrodinger equation. In strong fields, where the perturbative treatment breaks down, the time-dependent Schrodinger equation must be explicitly taken into account, as will be discussed in later sections. 

Second, in the weak-field limit it is possible to obtain the transition 1 Eq. (12.36) directly by solving Eq. (12.45) in first-order perturbation theory is done by assuming that (TV, e E, n, — 1 ) always remains larger stl... 

Figure 12.2 Effect of a strong laser field on the line shape for dissociation of an intermediate level at four hv2 IR frequencies and at two intensities of the IR laser. The spectrum on the hy2 axis (left-hand side) is computed IBr absorption spectrum in weak-field limit, starting from 960cm l above the intermediate level (which is 16,333.03 cm-1 above ground vibrational level). We see that broadening of the 16,333.03 cm-1 line occurs for I = 109 W/cift2.s whenever the hv2 photon is in near resonance with a strong predissocating line. (Taken from Fig. 2, Ref [388].) If...

In the weak-field limit, the population and the phase of the initial levels can be assumed constant with time ... 

Equation (B15) is the main result of this Appendix. The way to pass from this equation to the weak field limit given by Harris and Cina [16] has been noted above. The derivation [51] of the current in Eq. (B21) complements the discussion of charge density given by Li and Percus [53]. 

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