12553
J. Phys. Chem. 1993,97, 12553-12560
Time-Domain Information from Frequency- or Time-Resolved Experiments Using Maximum Entropy F. Remade+ The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel, and Dgpartement de Chimie, Universitg de LiPge. 86, 4000 LiPge, Belgium
R. D. Levine’ The Fritz Haber Research Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91 904, Israel Received: June 7. 1993’
A parametrization of the time evolution of intramolecular processes using characteristic times determined by
a maximum entropy formalism is discussed with examples. The approach is equally useful when the frequency spectrum is measured. The analytical aspects are discussed as well as the practical details of implementing the formalism. Special reference is made to systems exhibiting a multiplicity of time scale and to nonexponential time evolution. The physicochemical problems used as illustrations include photodissociation of a diatomic molecule, absorption spectroscopy, and the role of intermolecular perturbations.
1. Introduction Now that the time evolution of intramolecular processes can be (directly or indirectly) monitored, the notion of a “duration” must be given a quantitative meaning. This is particularly so for systems of realistic complexity where the time evolution is not simply exponential. In this paper we discuss an approach based on a maximum entropy (ME) formalism: an approach which is equally applicable to either frequency- or time-resolved experiments. The output of the approach is a parametrization of the frequency response of the system (or of its time domain Fourier transform) by one or more characteristic times. We discuss not only the formalism and its numerical implementation but also the different types of experiments to which it can be applied. Starting with frequency-resolved information, the squared Fourier transform of an absorption spectrum corresponds to the survival probability of an initial Franck-Condon localizedl-3 excitation. It has been extensively used to characterize the intramolecular dynamics and the different time regimes of the sampling of phase space in polyatomic molecules.+12 The spreading of the initial excitation in phase space can also be probed using resonant Raman excitation profiles13J4 which are related to the cross correlation function3J5of the initial excitation with another region of phase space than the Franck-Condon region initially accessed. Complementarily, the recent developments in ultra-short-time experiments16J7 require very often the computation of the power spectrum of the time-resolved data in order to extract information about the distribution of the spacings of the Hamiltonian and ultimately to determine the parameters of its potential energy surface.I8 The Fourier relation between the time and the frequency domains is also used in the analysis of time-resolved computational data (like those generated by wave packet propagation19 or by the running of ensembles of classical trajectories) in order to characterize the successive time scales of energy redistribution.20 What we have in mind by time information is the determination of the time scales of the energy redistribution (including dissociation21.22) which has been shown to proceed in a sequential manner.8JlJ2J3J4 A separation of time scales is expected to result from a tier structure25 of the Hamiltonian matrix, due to the Chercheur QualifiC, FNRS,Belgium. Abstract published in Aduonce ACS Abszracrs, November 1, 1993.
different orders of magnitude of the harmonic and anharmonic frequencies of the molecule together with the wide spread in the anharmonic couplings.1’,12.23-28
2. Introduction to the ME Spectrum We argue that the ME procedure is particularly well suited for extracting this time-domain information. There is a ME functional form of the ~ p e c t r u m ~ ~in- ~which l the Lagrange multipliers and their conjugate constraints are directly connected to time-domain information. In the ME analysis, the constraints used are the values of the Fourier transform of the spectrum for a given set of values t,. The constraints may either be computed by Fourier transform of the frequency-resolved data30J2 or be obtained directly from timeresolved experiments.I8 The functional form of the ME spectrum (which is equivalent to the one derived for the power spectrum of a stochastic proce~s~l-33) can be s h o ~ n ~ ~ to - 3 be 3 written as
where the (2M + 1) real constraints are the values, C(t,),of the Fourier transform of the spectrum at the times t,
c(t,)= c S ( w ) exp(-iwt,)
do
= J-rSME(o) exp(-iwt,) d o
= (exp(-iot,)
)
(2.2)
We discuss below and e l ~ e w h e r e ~how ~ . ’ ~the times t, are to be determined. However, we wish to emphasize already at this point that the characteristic times that we aim to determine are not numerically equal to the t,values nor are they 2M + 1 in number. Due to the real character of the spectrum and to the resulting time cross symmetry property
C(-t,) = C*(t,)
(2.3)
(so that h,= A,*), the ME spectrum (eq 2 . I ) can be written as
Q022-3654/93/2091-12553%04.QQ/Q0 1993 American Chemical Socicty
Remacle and Levine
12554 The Journal of Physical Chemistry. Vol. 97,No. 48, 1993
a
a square modulus34-35 M
where the Lagrange multipliers yr are related to the A, values by a bilinear transformation. We have recently proposed a numerical algorithm (the min-max alg0rithm32.3~)where the Lagrange multipliers yr and the values of the times t, are determined by a variational procedure. In the ME analytical form (2.4), the Lagrange multipliers yI govern the fluctuations of the spectral intensities while the distribution of the spacings is determined by the time values tr.32 The time-domain information is thus seen to be inherently built into the ME procedure. Moreover, the time information is compacted in the few (2M 1) real parameters which appear in the analytical form of the spectrum. Note also that SME(w) (eq 2.4) is written as the square modulus of an amplitude, a(o). When applied to resonant Raman excitation profiles,’3J4 the ME procedure gives direct access to the complex Raman amplitude (or polarizability) a’(@) and thus to the time cross correlation function of the initial state C’(t) without requiring the intermediate construction of a model potential.34 One interesting way of visualizing the time information is to use a mapping of the ME spectrum SME(w) in the z complex plane (z = exp(-iw6t), where 6t is a time interval small enough to span the entire range of the spectrum Aw, 6t = l/Au). Doing so, the denominator of eq 2.4 can be rewritten as the square modulus of a polynomial in the z variable:
+
-
z complex plaue
- 0
1
1.5
2
2.5
3
0
5
10
dh
15
20
25
t16t
where
Figure 1. Panel a: Mapping of the real frequency axis w into the complex plane. z = exp(-iwbt) with br = l/Au (see section 3.1 for more details). A one-root ME spectrum is plotted vs the frequency w in panel b. Its two poles ZI and 21 in the I complex plane are shown in panel a (bll = 0.90 and 91 = -2Aw). The square modulusof its corresponding complex decaying time exponential is plotted in panel c.
nr = tJ6t (2.6) Note that with this change of variable, the constraints C(t,) (eq 2.2) become the moments of the z variable:
principle,Mrather than using a discrete representation one could use an expansion of C(t)in a basis of orthogonal polynomials (the associated Laguerre polynomials would be particularly useful),
M
c(tr)= (z*’) C(tr)=
(2.7)
say” -A42
(2.11) r= 1
S(w) exp(-iwt,) dw
= (iAo)fS(z) z*’ dz
2M+ 1
(2.8)
Here the constraints are the 2M + 1 coefficients C,. We need to very strongly emphasize a cardinal point. It is not meant that the expansion (2.1 1) or its discrete analogue
C
(2.12)
and the contour integral is around the unit circle in the complex z plane, see Figure 1. The power in (2.7) is n, - 1 because of the Jacobian in changing from an w to a z variable, cf. (2.8). The polynomial,,:E 79%in eq 2.5 is of degree nM 5 K and can beexpressedintermsofitsKrootszk,sothat theMEspectrum becomes
Here, the highest power K is given by t M / 6 t = nM K
(2.10)
Some of the roots zk in (2.9) can be repeated more than once in which case we shall speak of a root of higher multiplicity. As will be discussed below, roots of higher multiplicity are related to nonexponential time dependence. In our work on the inversion of the Raman excitation profiles34 and the fluctuation of the spectral intensities32 as well as this paper, we use as constraints the values of the time autocorrelation function C ( t ) at 2M + 1 discrete points in time, eq 2.2. In
r
provide an accurate representation for C(t). The ME formalism will determine the entire C(t),from the 2M 1 values provided as constraints. Thevalues of theconstraints, taken by themselves, are definitely not meant to be sufficient to determine C ( t ) . A practical point and one that becomes obvious upon application is that one does not actually need to known C(t). Knowing the frequency spectrum (or the Raman profile, etc., as the case may be) is enough. In this paper, we discuss the extraction of the time information from the ME polynomial form of the spectrum. The ME form (2.9) has many interesting analytical properties which are discussed in section 3. In particular, it provides a mapping of the successive time regimes as a nested circles in the z complex plane. Some details on the computation of the roots from a ME fitting procedure can be found in section 4. Section 5 is an application to the frequency spectrum of a photodissociating diatomic molecule. In section 6,we discuss the separation of time scales in connection with the effects of a dissipative environment. Section 7 is devoted to concluding remarks.
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The Journal of Physical Chemistry, Vol. 97,No. 48, 1993 12555
Time-Domain Information Using Maximum Entropy
1 CME(n6t)= -i$SME(z) N6t
3. Analytical Properties
In this section we first discuss the case of a one-root spectrum. Examples of many-root frequency profiles are then presented, with special reference to the determination of time scales.
1 - lZl12 = -$SME(z) 2ai
3.1. The One-RootME Spectrum. In the case of a single-root
amplitude, the z polynomial in eq 2.5 is of degree one and the spectrum SME(z) is written
SMEW = l/IYo + Y1ZI2
z = exp(-iw6t) (3.2) The range of the spectrum Aw is defined by l/bt (in time unit-'). The range of the spectrum is thus mapped onto the circumference of the unit circle (121 = 1) in the z complex plane. In terms of its roots, the spectrum is written as SME(z) = l / [ ( z - z l ) ( z * - z I * ) ]
(3.3)
with z1 = -Yo/Y1 = IZll exp(i41)
(3.4)
The pole 21 is defined such that it is inside the unit circle (see Figure la) (lzlI< 1). The other pole il (which is the root of l / z - zI*,z* = l/z) is easily seen to be outside the unit circle
2, = l / z l * = (l/lzll) exp(i4,)
(3.5)
SME(z) is a periodic function of w , SME(z) 1 SME(e-iul'), with period Aw = 1/61. As will be discussed below, SME(z) is defined so as to reproduce the actual spectrum over its first period. It can be expanded in a discrete Fourier series
c[n] exp(+iwnat)
(3.6)
"'-.a
Jhj2Awl2
dw
The range of the spectrum Aw must be chosen so that S(o) = 0 for IwI > Aw (3.14) in order for the periodic spectrum SME(e-iw*') to coincide with the actual spectrum S(w) in the frequency range IwI < Aw. If such is not the case, there is a so called aliasing error33 due to the overlap of the different replicas of S(w). The value of the root zl depends on the range of the spectrum since Aw is mapped on the unit circle 121. In the one-root case, the condition on the value of the root to minimize the aliasing error is
In lZll
