3. Obediencia como aceptar la vocación a ser llamado a participar en la vida divina 69
3.1. Criatura: aceptar vivir la autoapertura
3.1.1. Obediencia como llamada a responder a la iniciativa de otro
Knowing what each contribution to the ∆OD makes to the resulting TAS helps to assign spectral features to specific processes. For example, the very existence of pro- cesses themselves are very informative, e.g. if a GSB does not fully recover, it suggests the original molecule is not reforming on the timescale of the experiment. Quantita- tively TAS are typically analysed under two models, (i) simultaneous dynamics or (ii)
−200 0 200 0 1 2 3
m∆OD
−200 0 200 −200 0∆t
/ fs
200 −200 0 200 −200 0 200 350 400 450 500 550 600 650Wavelength / nm
75 90 105 120 135FWHM
/
fs
Figure 3.5 | An example of a cross correlation study as a function of wavelength; shown is
methanol photoexcited at 325 nm. The IRF is dependent on wavelength, although it typically remains around 80–120 fs.
(i) Most dynamical processes occurring during the relaxation of a molecule can be adequately described through first-order kinetics. This means a single exponential decay with some characteristiclifetime describes one process. Experimentally things are a bit more complicated because of nonlinear (polarisation) processes occurring around pump- probe overlap (time zero), e.g. multiphoton effects in the sample cell,382,384 referred
to as the Instrument Response Function (IRF). This is typically well modelled by a Gaussian function with a FWHM in the region of ∼100 fs which ultimately limits the time resolution of the experiment (Figure 3.5). Together, this means a general TAS under simultaneous dynamics can be modelled as the sum of n exponential decay functions with lifetimes τn, convoluted with a Gaussian IRF G(λ,∆t),
TASmodel(λ,∆t) = n X i G(λ,∆t)⊗Ai(λ)e −(∆t−t0) τi , (3.6)
where Ai(λ) is referred to as the Decay Associated Spectrum (DAS) for the correspond-
ing exponential decay function with lifetime τi, and t0 denotes time zero. Nonlinear
for a particular wavelength. This analysis can be extended to include all wavelengths of the TAS simultaneously, so-called global fitting, the result of which is the ‘average’ dynamics across the entire TAS. This is the modelling procedure used throughout this thesis; the basis of the global fitting routine was written in MATLAB,390 and can be
found in Dr Adam Chatterly’s PhD thesis.391 One potential issue with this analysis is
that the model intrinsically assumes that all dynamics start at the same time. If the processes within a relaxation mechanism are far removed from one another, i.e. occur on different timescales this assumption is adequate. However, for dynamical processes very close in magnitude this assumption breaks down and sequential dynamics become important.
Uncertainties at the 95% level are assigned to the determined lifetimes using either an Asymptotic Standard Error (ASE) technique, or Support Plane Analysis (SPA).392
In the latter, the ‘goodness-of-fit’, χ2, of the lifetimes reported (τ
i) which together
parametrise the raw TAS, is a global minimum with value χ2
min. The lifetimes used in
the global fit are varied in a systematic manner to sample the goodness of fit in the local parameter space surrounding this minimum, which return the values χ2(τ
1, τ2, . . . , τn).
The ratio χ2(τ1,τ2,...,τn)
χ2
min is then determined. The global minimum is given by
χ2(τ
1,τ2,...,τn)
χ2
min
= 1. For all fitting values which are not a global minimum, the ratio is always >1. A confidence interval at the N level (N = [0,1]; N = 0.95 throughout this thesis) is defined as χ2(τ 1, τ2, . . . , τn) χ2 min = 1 + p νF (N, p, ν), (3.7)
where p is the number of parameters in the global fit, ν is the number of degrees of freedom and F, is the inverse cumulative F-distribution function. Thus, the upper bound on the uncertainty for each varied parameter is the value which results in the largest deviation from the global fitted values whilst satisfying Equation 3.7.
ASE are simply the limit of SPA where only one lifetime (τ) is varied in a systematic manner whilst keeping all other lifetimes fixed. The global fit optimisation proceeds as before and a new goodness-of-fit is calculated denoted by χ2(τ). A 95% confidence
interval for the value of the lifetime τ is then defined by
χ2(τ) χ2 min = 1 + p νF −1(0.95, p, ν). (3.8)
This is then repeated for each lifetime.
(ii) Sequential dynamics use a kinetic model for the TAS where each process leads onto the next. This solves the problem of two or more dynamical processes being close
increases when one process may ‘branch’ into multiple processes, something that has to be known before fitting of the TAS can be achieved. Often these models are formed on chemical intuition or following ab initio electronic structure calculations to inform on the likely model.
Either of these fitting methods can model a TAS and thus extract the lifetimes of the processes occurring following photoexcitation of the molecule of interest. Choosing a method of fitting will depend on the situation; whilst simultaneous dynamics can reveal very useful information, if sequential dynamic fitting is possible, it is ultimately always more representative of the actual relaxation mechanism. This experiment and subsequent analysis can reveal processes with lifetimes as short as ∼100 fs. The issue remains in assigning lifetimes to physical processes e.g. IC, ISC, photoproduct forma- tion etc. With sensible assignment (sometimes easier said than done), the relaxation mechanism of a molecule and the timescale it occurs on may be deduced, which is essentially the subject of majority of the work presented in this thesis.
A final note regarding the assignment of the photophysical processes is that a num- ber of steady-state techniques are used throughout the analysis of the measurements reported in this thesis, such as UV-visible spectroscopy, infrared spectroscopy and nu- clear magnetic resonance. In particular, they are most often used to interpret long-lived components such as photoproducts given they appear static within the temporal reso- lution of TEAS and TVAS. However, there are no specialised customisation to the use of these techniques and thus details are given in the relevant methodology sections of appropriate chapters.