Correlations in anticrossing spectra and scattering theory. Analytical aspects

Chemical Physics 146 (1990) 21-38 North-Holland

Correlations in anticrossing spectra and scattering theory. Analytical aspects T. Gubr and H.A. WeidenmCIIer Max-Planck-lnstitutftir Kernphysik, 6900 HeiaMerg, FRG Received 12 December 1989

Experimental results of anticrossing spectroscopy in molecules, in particular the correlation hole, are diissed in a theoretical model. The laser measurements are modelkd in terms of the scattering matrix formalism originally developed for compound nucleus scattering. Random matrix theory is used in the framework of this model. The correlation hole is analytically derived for small singlet-triplet coupling. In the case of the data on methylglyoxal this limit is realistic if the spectrum is indeed a superposition of several pure sequences as one cm conclude from the analysis of the measurements.

1. Introduction

Recent advances in high-resolution spectroscopy have yielded qualitatively new information on spectral fluctuation properties of highly excited molecules. Such data call for a suitable theoretical analysis. It is of particular interest to establish whether a given fluctuation pattern is or is not consistent with chaotic dynamics of the molecule. It is the purpose of the present paper to address this problem for a particular type of experiment - the observation of the resonance fluorescence yield of laser-excited molecules under the influence of an external magnetic field. Specifically we have in mind the data published in refs. [ 1,2] on the organic molecule methylglyoxal. Before defining our actual problem and our approach to it, we briefly recall the analysis of spectral fluctuation patterns in nuclear, atomic, and molecular physics. We do so in order to put the present investigation into proper perspective. Data on low-energy neutron scattering on heavy nuclei [ 31 provided the first evidence for the validity of a theoretical model using the Gaussian orthogonal ensemble (GOE); later investigations [ 4,5] confirmed this model and thereby the existence of fluctuation properties in nuclei which are typical for chaotic systems. The data obtained from nuclei suffer from two difficulties: (i) The investigation of short-range properties is hampered by the problem of missing levels. (ii) The limited length of the available level sequences makes it impossible to study long-range properties of the fluctuations in a convincing manner. (The recent analysis [ 6 ] of “Al constitutes a considerable improvement of the knowledge of the long-range properties in nuclei.) In atomic physics, the experimental situation is much better as longer sequences of well-resolved levels can be investigated. Early indications [ 7 ] of the validity of the GOE were later corroborated [ 81. The recent study [ 9 ] of the hydrogen atom in strong magnetic fields constitutes a spectacular improvement; it provided the fust example in a real physical system for the transition from regular to chaotic fluctuation properties, in particular concerning the long-range fluctuations. In molecular physics one expects an even larger variety of spectra than in atomic physics. Evidence for chaotic motion was first [ 10,111 obtained in NO* and later in acetylene [ 12,13 1. Additionally, in the case of acetylene applying a Fourier transform analysis of the SEP spectrum a “correlation hole” was found as a hint of chaotic dynamics [ 141. Long-range spectral fluctuation properties can be studied particularly well by anticrossing spectroscopy [ 15 ] with its high accuracy. The Grenoble group [ 1,2 ] performed a Fourier 0301-0104/90/$03.50

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

22

T. Guhr, HA. Weiaknmiilkr / Anticrossing spectroscopy

transform analysis of such data. The Fourier transform of the intensity autocorrelation function of the resonance fluorescence yield versus magnetic field strength in methylglyoxal (and other molecules) shows a “correlation hole” at short distances. This hole was interpreted [ 1,2 ] phenomenologically as evidence for chaotic fluctuation properties. In the present paper, we aim at a theoretical justification for this interpretation. We do so by modelling the triplet states in .methylglyoxal in terms of one (or several) Gaussian ensembles, and by describing the resonance fluorescence yield of the singlet state in terms of the formalism of scattering theory. This formalism was developed originally in the context of nuclear physics [ 16,17 1. In the present paper, we introduce the theoretical model and apply it to the laser measurements on methylglyoxal. We present analytical results, valid in the physically realistic regime of weak singlet-triplet coupling. Our results qualitatively confirm the phenomenological interpretation. The numerical results obtained from this model are presented and discussed elsewhere [ 18,191. In section 2 we summarize some results of anticrossing spectroscopy and formulate the problem. In section 3 the laser experiments are modelled in terms of the scattering formalism. For small singlet-triplet coupling we derive an analytical expression in section 4. In section 5 we discuss our results. 2. Anticrossing speetroscoPy In section 2.1 the anticrossing effect is explained. The experimental results which are important for our purposes are briefly summarized in section 2.2. In section 2.3 a phenomenological interpretation of these results is reported. 2.1. Anticrossing eflect Consider a singlet and triplet state, I s) and I t ) , of a molecule, with energies ES and E,, and suppose that in the absence of any external field, the mixing matrix element V,, between these two states is such that 1ES-E, 1z+ 1VI,,I. In a magnetic field B the degeneracy of the triplet state is lifted since the symmetry is cylindrical and the appropriate quantum number is the projection mJ of the total angular momentum J onto the symmetry axis. By a suitable choice of the magnetic field strength, the energy Et f B (with B in energy units) of the state with spin projection ms= + 1 can be made to coincide with that of the singlet state. Under these conditions the coupling between this one member of the triplet and the singlet state can no longer be ignored. Performing degenerate perturbation theory one finds that the energy difference of the new eigenstates I 1) and 12) isgivenby IE, -Ez I =,/(2Kt)2+

(Et +B-ES)2

,

(2.1)

which is obviously non-zero for all values of B. The resulting “level repulsion” gives rise to the anticrossing effect. Suppose that I s) and 1t ) are excited states of a molecule whose ground state is a singlet. Since dipole transitions are allowed among singlet states but forbidden between singlet and triplet states, only the de-excitation of I s) can contribute to the fluorescence yield. In the anticrossing region, I s) mixes significantly with It ) and so the fluorescence intensity, Z(B) , decreases. Measured as a function of B, this intensity is a Lorentzian given by [20,151

(2.2) where ya, y, and ylt are the radiative widths of I s) , I t ) and of the coherent superposition of I s) and I t ) , and 2 is the singlet excitation rate. Formula (2.2) is easily derived using the density matrix formalism in the basis I s) , I t > . The intensity Z(B) is proportional to the density matrix element p=, which is obtained by solving

T. Guhr, HA. Weiaknmiiller / Anticrossing spectroscopy

23

(2.3) where the matrix f contains the radiative widths, A the population constants and H is the full Hamiltonian including V,. The process is stationary, dp/dt= 0, so that (2.3) reduces to a simple algebraic equation. 2.2. Experimental results Precise measurements of anticrossing spectra have been performed by Lombardi and co-workers [ 15 ] at the Service National de Champs Intenses (SNCI) in Grenoble. For a fmed singlet state 1s) a large number of anticrossings was measured by shifting many triplet states 1t ) with a magnetic field B past the singlet state. In particular the molecule methylglyoxal was intensely studied. In methylglyoxal the lowest electronic configuration S,, has spin zero, the first excited (T,) spin one and the second excited ( S1) again spin zero. The laser populated singlet state 1s) belongs to the configuration S, and lies about 25000 cm-’ above the ground state of So. Due to the supersonic jet technique applied in the experiment, the pressure is so low that the contribution of collisions to the decay width yti is negligible. This is important for our model to be described in section 3. The magnetic field was varied continuously between 0 and 10 T with a homogeneity of ABIB= 10 --5. To populate I s ) Lombardi and co-workers used a dye laser producing 1020photons per second, so that there are 1O6to 10’ counts per bin in the detector. This means that the statistical error bars are so small that the tine structure of the spectra is reproducible. The resulting spectrum Z(B) is a complicated superposition of lines of the type (2.2). The purpose of the measurements is the investigation of the fluctuation properties. In order to remove the dependence on the level density, a dimensionless scale b= B/D is introduced where D is the mean level spacing. The resulting spectrum Z(b) is unfolded, i.e. it has mean level spacing unity everywhere. Now the absolute square of the Fourier transform c ( t ) of the spectrum is calculated, OD m

I40 12=

db, db2 exp[2ix(b, --m

-b,)t]Z(b,)Z(b,)

.

(2.4)

--03

This observable gives information about the fluctuations since the integrand is easily related to the autocorrelation function of the spectrum. Substituting p= bl + b2 and b= b, - b2 we find ]c(t)J*=$

5 dbexp(2inbt)

7 dpZ((p-b)/2)Z((p+b)/2).

(2.5)

Hence, I c( t ) I 2 is the Fourier transform of the autocorrelation function. Gf course, the number of anticrossings in the spectrum has to be sufficiently large in order to achieve meaningful results. Fig. 1 shows such an unfolded Fourier transform for methylglyoxal extracted by the Grenoble group [ 1,2 1. For small t one sees a considerable decrease of I c(t) 12. This is called the correlation hole. 2.3. Phenomenological interpretation The Grenoble group [ 1,2] presented a phenomenological interpretation of the correlation hole in terms of random matrix theory using arguments given by Mukamel et al. [ 121. The keystone of this interpretation is the assumption that the spectrum Z(B) may be regarded in the same way as the energy spectrum of a Hamiltonian system. In other words, one assumes that the magnetic field positions of the anticrossings underlie the same statistical laws as the energy eigenvalues of a Hamiltonian. Thus, the autocorrelation function of the unfolded anticrossing spectrum for a sufficiently large level sequence should be given by 6(b) + 1- Y2(b) where 6(b) is

T. Guhr, HA. WeiaknmGller / Anticrassing sperlroscopy

t

Fig. 1. The comlation hole. The Fourier transform of the autowrrelation function of the unfolded anticrossing spectrum in the case of methylglyoxalextracted by the Grenoble group. The theoretical result in the phenomenological model for an uncorrelated spectrum is plottedas solid line. The exponential decay is due to the radiative width of the levels. The gap behveen this line and the experimental data is the correlation hole. This fmre is taken from ref. [2].

the identical correlation. The function 1 - Y,(b), first introduced by Dyson and Mehta [ 2 11, is the probability density of finding two levels at the distance 6, Y,(b) is called the two-level cluster function. For completely uncorrelated spectra, i.e. Poisson spectra, one has Y,(b) = 0. For correlated spectra, the cluster function must reflect the Wigner repulsion, in particular l- Y,( 0) =O at b=O. If the correlation is modelled by a Gaussian ensemble, Y, ( b) is known analytically [ 2 I]. The Fourier transform yields 6( t ) + 1 - b2 ( t ) where Q) b*(t)=

1 Y*(r) exp(2ixrt)

dr

(2.6)

-0D is the two-level form factor [ 2 11. Obviously, bz (t) = 0 for a Poisson spectrum. In the case of a correlated spectrum, 1- b2 (t) approaches zero for small t. Hence, one is drawn to the conclusion that the correlation hole is just the two-level form factor of a correlated spectrum implying that the anticrossing spectrum of methylglyoxal displays chaotic dynamics. Here and in the remainder we use “chaotic” synonymous to “showing the statistical properties of the Gaussian ensembles”. Moreover, if the system has A4symmetries, i.e. the spectrum Z(B) is a non-interacting superposition of A4 spectra, one expects [ 1,2] that the two-level form factor b2( t) has to be replaced by bz (Mf). In the case of methylglyoxal, one can deduce from the position of the correlation hole in fig. 1 at tx 0.05 on the unfolded scale a number of Mz 20 symmetries. Although this number is not fully explained by the selection rules, we believe following the considerations in ref. [ 21 that the anticrossing spectrum of methylglyoxal is indeed a superposition of several, say about ten, pure spectra. The Fourier transform technique and thus the correlation hole are very useful for the investigation of longrange properties of the spectrum. This is easily seen by relating b2 ( t ) to the level number variance [ 5 ] L

z*(L)&21

Y,(r)(L-r)dr.

(2.7)

0

On the unfolded scale, an interval of length L contains L f Jm levels. For an uncorrelated Poisson spectrum one has J?(L) = L. If there are any correlations present, one has X2(L) CL. From eqs. (2.6) and (2.7 ) we find /rn+2(L)=l-b2(0).

(2.8)

The left-hand side is 1 for a Poisson spectrum and 0 for any correlated spectrum. Hence, the correlation hole should occur for correlated spectra. The influence of the line shapes on the Fourier transform of the autocorre-

T. Guhr,HA. WeidenmiUkr / Anticrossing spectroscopy

25

lation function has also been discussed by the Grenoble group [ 1,2 ] in the framework of this phenomenological interpretation.

3. Scattering model Assuming that the interpretation of the correlation hole proposed by the Grenoble group provides the correct physical picture we wish to put this phenomenological approach on a theoretically more rigorous basis. Our tool is the scattering theory developed in nuclear physics which is shortly sketched in section 3.1 In section 3.2 this formalism is applied to the methylglyoxal experiment. In section 3.3 the scattering matrix is related to the intensity measured in the experiment. Statistical assumptions are introduced in section 3.4. The problem of a finite versus an infinite number of anticrossings is discussed in section 3.5. 3.1. General aspects of scattering theory We present a brief summary of scattering theory. Our motivation for doing so will become obvious when we apply the formal expression of eqs. ( 3.1) and ( 3.2) below to the actual physical situation of methylglyoxal. We consider N+ 1 orthonormal bound states Ip), p=O, .... N and A channels. The channel states 1a; E), a= 1, .... A are real, and obey (a; E 1b; E’ ) = &,d( E - E’ ) where E denotes the total energy of the system. In the absence of direct reactions, the model Hamiltonian in the space of the states I ,u) and I a; E) has the form [ 16 ]

(3.1) where E, is the threshold energy in channel a. The first term involves the (N+ 1) dimensional matrix H with elements HLIcwhich describes the coupling of the bound states with each other. The second term contains the coupling matrix elements W,,, between bound states and channels. This term causes the bound states to acquire a width, and to become resonances in the continuum. The third term is assumed to be diagonal in the channels; transitions between different channels are, in the framework of eq. (3.1), possible only via intermediate population of the bound states ] p). The physical assumptions which are used in writing eq. (3.1) become transparent as one works out the scattering matrix S,,(E) pertaining to this model. This can be done algebraically [ 161; one finds (3.2) where the propagator G - ’ (E) is the inverse of G,,(E)=Ed,,,-H,,,+in:

c WorWov. (IOpcen

(3.3)

A channel a is open if E > c~. In this formulation, all threshold effects including the dependence of the W,, on energy have been neglected. The physical interpretation of eqs. (3.2) and (3.3) rests on the observation that S&E) has N+ 1 poles, corresponding to N+ 1 resonances. For N=O, the resonance has Breit-Wigner form, with W, denoting the partial width amplitudes, while for N8 1, eqs. (3.2) and (3.3) are (N+ 1)-level unitary generalizations of the Breit-Wigner formula. For Ct=, W& Q: D, the mean spacing between eigenvalues of H, the model describes isolated resonances. They can be made to overlap by increasing the ratios Xi=, W&,/D. We note that in eq. (3.2), we have not paid attention to the possible presence of elastic background phase shifts in the channels.

26

T. Guhr,HA. Weiaknmtikr/ Anticrossingspectroscopy

This was done because in the application to methylglyoxal, such phase shifts are negligible. 3.2. Application to the methylglyoxal experiment We turn to the application of this formalism to our problem. Our model is schematically shown in fig. 2. The molecule is excited from the ground state ISO) to the non-rotating singlet state 1s) . Thereafter only resonance fluorescence is measured. The “forbidden” transition from the groundstate 1to) of the triplet manifold to 1SO) is not seen in the experiment. Consequently we assume that there are two channels, a singlet and a triplet channel: The singlet (triplet ) channel consists of the molecule in the ground state ISO) ( It0 ) ) of the So (T, ) conflguration and a photon. We denote the corresponding kets by 1~0,E) and I tO, E) respectively. Here, E is the energy of the photon minus the threshold energy. Because of the selection rules for electromagnetic dipole transitions only the state (s) can be populated directly from the singlet channel. Following Mukamel et al. [ 12 1, we assume that the state Is) acts as a doorway to N triplet levels I tp), p= 1, .... N. The triplet levels I tp) in turn are coupled to the triplet channel I tO; E) . Because of the absence of collisions we do not need to include gamma transitions between the triplet levels I tp). Our model has the structure of eqs. (3.1)-(3.3). This is seen by identifying the two channels ISO;E) and I tO, E) with the channels 1a; E) ; the state Is) with the state IO) ; and the triplet states with the states I@), PL=1, .... N. The model has specific features which go beyond the general framework of eqs. (3.1)-(3.3): It is assumed that the coupling matrix elements W,, vanish if (i) a is the (SO;E) channel and p= 1, .... N denotes one of the triplet states; (ii) a is the I tO; E) channel and p= 0 denotes the singlet state. To exhibit more clearly the role of the doorway state 1s), we denote the matrix element W, with a the IsO; E) channel by m so that ys is the width of the singlet state; introduce the N-component vector Vwith elements I’,,, pu= 1, .... N to denote the coupling between Is) and the states I tp) ; and reserve the notation H for the N-dimensional Hamiltonian matrix that couples the triplet states I tp) with each other. In the case of a single triplet state (N= 1), the vector V reduces to a single element; this is the element V,, considered in section 2. Since there are only two channels, the S matrix (3.2) is a 2 x 2 matrix. We also account for an external magnetic field which is used to produce successive anticrossings. In order to describe these anticrossings, it is necessary to retain only those members of a triplet which are shifted under the action of the field, i.e. those with non-zero magnetic spin quantum number. The net effect of the magnetic field is then to split the given set of N triplet states into 2N non-degenerate states, N of which are shifted up by an energy + B, the other N being shifted down by -B. (In our notation we do not distinguish between the magnetic field and the energy shift due to the magnetic field.) These two sets of states are generated by identical matrices H. According to eq. (3.2) the singlet-singlet element of the scattering matrix, simply denoted by S( E, B), is given by S(E,B)=l-iy,[G-‘(E,B)],,

(3.4)

vqiz

-L

Ito>

Fig. 2. Scattering model for the laser experiment on methylglyoxal. Dynamic couplings are drawn as wavy lines.

21

T. Guhr,H.A. Weidenmiiller/ Anticrossingspectroscopy

where the propagator G- ’ (E, B) is a (2N+ 1) x (2N+ 1) matrix, E-E,

+iy,/2

(

v+

V+ E-H+iT,/2-B

V V

G(E, B) =

0 E-H+iT,/2+B

0

.

(3.5)

1,

Here, Es= Ho0 is the singlet state energy, B stands for the B-fold multiple of the unit matrix in N dimensions and the dyadic NX N matrix & (Jrtl ‘=

(k

(3.6)

*..Y/L )

1

contains the square roots of the N triplet radiative widths y,,,, i.e. the coupling between the triplet channel and the triplet states. Defining the 2N component vector and the 2Nx 2N matrix c( E, B) by

r

e(E,

B) =

E-H+ir,f2-B

0 E-H+ir,/2+B

0

(3.7)

we find from Cramer’s rule for the singlet-singlet matrix element of the propagator

[G-‘W,WIoo=

detG(E, B)

(3.8)

B).

detGtE

>

Laplace expansion yields for the denominator ‘-‘“det[(E-E,+iy,/2)6(E,

detG(E, B)=(E-E,+iy,/2)

.

B)-mt]

(3.9)

This gives in eq. (3.8) after some algebraic manipulations

tG-1’EyB)1,=E_E~iy/2de t(lZN-wt(E_Es+iys/i)2(E 6

3

I)

1

B)_wt

>

t2N

=E-E,+iy,/2

*-”

(E-E,+iy,/2)C?(E,

(3.10)

B)-mt

where 12Nis the 2NX 2N unit matrix. Inserting this result in eq. (3.4) we find for the scattering matrix element

St& B) - *= -i,/%

s

-‘&E

-s E :iy/2s

Li,, ,2

E_E

&

I

‘+c(E,B)-B(~~~+iy./2)-‘P+

‘E-Es!tiy,/2

&.

(3.11)

The first term on the right-hand side describes the scattering on the singlet doorway state without intermediate population of the triplet states. The second term describes the process singlet channel-singlet state-triplet statessinglet state-singlet channel under inclusion of the effect of the magnetic field and the possibility of multiple singlet-triplet transitions (the latter are contained in the propagator through the term P( E - E, + iy,/ 2 ) - ’ 8+ ) . The intuitively obvious formula ( 3.11) is the exact result of the model of fig. 2. and is the basis of our further work. 3.3. Scattering

matrix andjluorescence

intensity

For a fixed magnetic field, the intensity of the fluorescence light in an interval [E, E+ dE] of the photon energy is proportional to the cross section and thus, up to a normalization constant, given by the absolute square

28

T. Guhr, HA. Weidenmiiiler / Anticrossingspectroscopy

of the expression S( E, B) - 1. The total fluorescence intensity is found by integration over all photon energies,

IS(E,B)-112~,

(3.12)

-co

where 2: is the singlet excitation rate and Lt/Zxyf will turn out to be the appropriate normalization constant. Formula (3.12) allows for an indirect check of our scattering model since it must reproduce the result (2.2) derived within the density matrix formalism [ 20,15 ] when there is a single, isolated anticrossing. Such an isolated anticrossing corresponds to one of both possible values mslt 1 of the spin projection. Choosing one such value for m3, we see that the propagator G-r (E, B) reduces ‘to a 2 X2 matrix so that S(E, B) can easily be calculated from eq. (3.4). Using the residue theorem we perform the integral (3.12) and indeed find the result (2.2). The radiative width of the coherent superposition of the singlet and the triplet state is in our derivation fixed to yst= ( ys+ yJ /2 reflecting the assumed absence of collisions. As mentioned in section 2.2 this assumption is justified by the experimental technique. In the density matrix formalism, collisions are always allowed implying that ystmight be greater than (Ye+ 7,) /2. We conclude that in the absence of collisions the density and the scattering matrix formalisms are equivalent. The advantage of the scattering matrix formalism is, however, that overlapping anticrossings are easily dealt with. 3.4. Statistical assumptions Given the schematic model of fig. 2, eqs. (3.12) and (3.11) for the resonance fluorescence intensity are generally valid, irrespective of the detailed dynamical couplings embodied in the matrix H and the vector V. The only constraints on H and V derive from fundamental symmetries of the system: For time-reversal noninvariant systems, H is Hermitean and V complex, while for time-reversal invariant problems, H is real symmetric and V real. For actual calculations, however, it is necessary to specify ensembles, i.e. normalized probability density functions P(H) for the matrices H and p( V’) for the vectors V. Ergodicity then allows us to replace the interval average (the p integration in eq. (2.5 ) ) required for the calculation of the autocorrelation function by an ensemble average. We write the average of a function F(Z-Z,V) with respect to these ensembles as (3.13) where the differentials d [ I-I] and d [ V] denote the products of the differentials of all independent variables of Hand V. Without restrictions we may assume that all independent variables have zero mean value. In particular we introduce the 2nth moment of the statistical singlet-triplet coupling by the definition (3.14) Our fundamental demands on the probability density functions are: Firstly, invariance under transformations H+ VHV- ’ and V-+ VV where V is a unitary or orthogonal matrix according to the symmetry of H. Secondly, statistical independence of the various elements H,,” and V,,, respectively. These physically reasonable demands restrict the possible ensembles to Poisson and Gaussian ensembles [ 2 11. Hence, all statistically independent elements are also equivalently distributed. In addition, our assumptions imply that the radiative widths of the triplet states enter the averages only via Z z= 1ytr. We perform our calculations for the Gaussian orthogonal (GOE) and for the Gaussian unitary ensemble (GUE) corresponding to real symmetric and Hermitean Hamilton matrices, respectively. For later purposes we mention that one can write

T. Guhr, HA. Weidhmtikr

P(H) d[H] = j&j

/ Anticrossing spectroscopy

29

(3.15)

Z’E(~) d[Xl dp(U) 3

where X=diag(Xi, .... X,) is the diagonal matrix of the eigenvalues of H and PE( X) depends only on the eigenvalues. The differential dp( V) is the measure of the diagonalixing group of Z-Zand p(U) the corresponding volume, i.e. a constant. 3.5. Finite versus infinite number of anticrossings When we employ our statistical assumptions we necessarily have to consider the spectrum Z(B) in the limit of infinitely many anticrossings, i.e. we have to take the limit N+co. The experimental spectrum Zap(B), however, contains only a finite number of anticrossings, thus there is a cut-off field B,, so that Zexp(B)=Z(B)K(Bo,B),

K(B,,B)=l

if]B]
(3.16)

if]BJ >BO.

Such cut-offs influence in general the Fourier transform of the spectrum. Hence one might wonder whether the theoretically calculated 1c(t) I * for N+cc differs significantly from ]cexP(t) I*= 5 ZeXp(b)exp(2ixbt) db’ . I --oo

(3.17)

Here we have introduced the unfolded scale b = B/D as usual where D is the mean level spacing. We now show that the influence of the cut-off in our case is negligible. Indeed, the integral in eq. (3.17) is the Fourier transform of a product so that we may write according to the convolution theorem Ic=““(t)l*=

7

I -al

c(t’)k(b,,

t-t’)

dt’

*,

I

(3.18)

where c(t) is the Fourier transform of Z(b) and (3.19) is the Fourier transform of K( b,,, b) . The quantity b,,= B,/D is half the number of anticrossings measured in the experiment. The function k(bo, t) approximates the function s(t) very well if bOis larger than ten or so. We recall that the experimental data contain many more anticrossings; this is necessary already to define the unfolding procedure in a meaningful way. Hence, from eq. (3.18) we have always to a very good approximation )cexP(t) I * x )c( t) I * for non-zero t. The influence of the cut-off is negligible and we may take the limit N+a, in our calculation without restrictions.

4. Small singlet-triplet coupling In this section we discuss analytically the anticrossing effect for small coupling matrix elements between singlet and triplet states. For this purpose we expand the scattering matrix with respect to the dyadic matrix Wt. According to eq. (3.14) the corresponding scalar product I/+ Vis of the order NV& Since the mean level spacing for the Gaussian ensembles satisfies D- 1/fi we have NV:, *z /D*. In the experiment, the right-hand side of this relation is smaller than one if one considers pure sequences of levels all having the same quantum num-

30

T. Guhr, HA. Weidenmtiter / Anticrossing spectroscopy

bets. As discussed in section 2.3 the data on methylglyoxal’indicate that the experimental spectrum is a superposition of several, at most twenty, pure sequences, although this point is not yet fully understood. Hence we argue that the singlet-triplet coupling should not be compared with the total mean level spacing but with the mean level spacing of a pure sequence: About 5000 anticrossings were found [ 21 in the magnetic field strength interval between 0 and 8 T. Assuming ten symmetries one has about 500 anticrossings per pure sequence and hence a mean level spacing of about Dz220 MHz per pure sequence in frequency units. Since z % ( 100 MHz)’ is a typical value for the averaged squared singlet-triplet coupling [ 15,22 1, the relevant expansion parameter satisfies E/D2 x0.3. This makes the expansion meaningful. Unfortunately, the expansion in powers of VV requires the singlet-triplet coupling to be small compared to the singlet and triplet decay widths. In order to understand this point we consider the anticrossing formula (2.2) for one triplet state. Near the anticrossing, i.e. for E, Z!IBx Es, we may expand in powers of Vi* only if yi B 2 Vi, y,,( y; ’ + ye ’ ). As mentioned above we may replace in our case ys,by ( ya+ y,) /2. This yields the condition (4.1)

&*2lI/,I.

Since one knows from the experiment that y. x 7, we may write the condition for our treatment of the small coupling regime as ysz yt =9 2 1V,,1. The situation in the methylglyoxal experiment, however, is the other way around, the coupling I V., I is larger than the decay widths yI and y,. Hence, the purpose of the analytical discussion is not a quantitative description of the experiment, but a clarification of the mechanism of the anticrossing effect in our scattering model. Moreover, the analytical results serve as test case for the numerical simulations reported separately [ 18,19 1. For technical reasons we make a further simplification of the model: The matrix r, introduced in eq. (3.6) is replaced by yt1Nwhere yt is a number and 1Nis the NXN unit matrix. We expect of course that this replacement does not influence the anticrossing effect. In section 4.1 we consider the case of one symmetry of the system, i.e. the Hamiltonian Z-Zcannot be reduced further to a block-diagonal form. In section 4.2 we generalize our calculation to an arbitrary number of symmetries. 4.1. One symmetry Our goal is to calculate the Fourier transform of the autocorrelation function of the intensity Z(B) up to fourth order in the statistical singlet-triplet coupling, which is equivalent to fourth order in the vector V for a triplet Hamiltonian H which cannot be reduced to block-diagonal form. We expand the propagator in eq. ( 3.11) in a geometric series,

(4.2) This yields for the integrand in eq. ( 3.12)

lwE,m-112 2 =

(E-EsF2+y:/4

1

f n,m=O (E-E,+iy,lZ)*(E-Es-iy,/2)”

[ P+@‘(E,

B)8]“[

Hence we have up to fourth order m

Z(B)=Z0(B)+2ReZ2(B)+Z4(B),

JAB)=

2

(E-El;;+y:,4

dE ’

p+&‘*(E,

B)

n’” . (4.3)

T. Guhr, HA

I,(B)

=

2 #lo O3 27V: _-m(E$&y34

1 E-E,+&,2

I

2

31

Weidenmidler / Anticrossing spectroscopy

B+c-‘(E,

B)BdE.

(4.4)

The expression for Z,(B) is not given here since we show that Z,(B) does not contribute significantly to the final result. The integrals Zo(B) and Z2(B) can easily be evaluated with the help of the residue theorem, Z,(B)=

Z,(B)= -i’z

s,

P+G-‘(E,+iy,/2,

B)P=ZF(B)+Z:(B) ,

(4.5)

where . np

IN

zl(B)=-l~ I/tEs fB_H+i(y,+y,)/2

‘*

We took advantage of the fact that H has real eigenvalues. As we show in the following, expression (4.5) for Z2(B) reflects one of our main insights into the mechanism of the statistical anticrossing effect for small singlettriplet coupling: The relevant part of the spectrum contains the usual propagator of the Schriidinger equation except for the fact that the magnetic field replaces the energy. This indicates that the correlation hole is indeed a purely spectral fluctuation effect as far as the small coupling regime is concerned. The autocorrelation function, i.e. in our statistical model the ensemble average of the product I( B, )I( B,), is up to fourth order given by 2

I(B

0

+f

+2$Re[Z2(B,)+h(B2)]+

$

-[Z.,(B, +Z4(B2)]+4ReZ2(Bi)

ReZ2(B2) .

(4.7)

Only the last term depends on both magnetic fields. Consequently, upon performing the Fourier transform of the autocorrelation function with respect to both magnetic fields, every term of the right-hand side of eq. (4.7) except the last one containsa 6 function. Thus, in order to study the behaviour of the Fourier transform for nonzero t it is sufficient to consider the last term of the right-hand side of eq. (4.7). According to eq. (4.5) this term is ReZ,(B,) ReZ2(B2)=ReZ,(B,)

ReZ~(B2)+ReZ~(B,)ReZ~(B2)+ReZ~(B,)

+ReZ,+(B,) ReZg(B2),

ReZ:(B2) (4.8)

where the upper indices denote the sign of the spin projection. We calculate the first term of the sum in eq. (4.8) since the calculation of the other terms is similar. Introducing a infinitesimal imaginary increment ie we can write the function I? (B) as a convolution, 03 Zy(B)=ig

Yf

s

V’f

E

--cm

s

_,lr,+,

WB-B’,

2~st)do’

7

(4.9)

where L(B,2y,,)=

A-?& x B2+&

is a normalized Lorentzian. From now on we write yst= ( ys+ yt) /2 since no confusion with a collision-enhanced yst can occur. Moreover we omit the limit e--*0 in our notation. This yields for the first term of the right-hand side of eq. (4.8) (4.11) --oo

-CR

T. Guhr, HA. Weidmdkr

32

/

Antierasing sptWoscopy

where the function A (B, , B2 ) is the ensemble average of the propagator terms,

A(&,&)=

& S Id[Vlp(V)IdIHIP(H)ImVtE

I =24

n YsI

~d[XlP,W)

d[%(V&j

x Im( UV)t E,-RlNX+ie

s-

IN Vim Vt BINH+ir ES-B2-H+ic I-

~da((l)Im(WtE

s

V

_BINX+ic (UV I(4.12)

(“).

Here the properties of P(H) as discussed in section 3.4 have been used. We introduce new variables V’= (N; the Jacobian of the transformation is det U= 1. Since p( V) is assumed to be invariant under such transformations the whole integrand becomes independent of U. Thus the integral over the measure of U yields the volume p(U) of the diagonalizing group of H. This argument is of course general. The integrand would also become independent of U if we had used the full expression (3.11) for I S( E, B) - 11’ and not only the first relevant terms of the expansion ( 4.3 ) . Hence we have A(&,Bz)=

+ I ,g,

~d[Vlp(V)IV,121V.I’A,.(B,,B2)

9

(4.13)

where the function AtiIIy ( B1, B2) is independent of the coupling elements V,, and is given by .&(R,,&)=

$ I

d[Xl f’dx) Im ES-B,~X,,+ia1mES-B2~Xv+ic

d[X] Z-‘E(X)[6p6(ES-B,-X,)G(ES-B2

(4.14)

-X,)+(1-6,,)6(&-B,-X,)&E,-B,-X,)] .

The integrals over the 6 functions yield Dyson’s correlation functions R, and R2, i.e. the level density and the two-point correlations. Details are given in Mehta’s book on random matrices [ 2 11, chs. 5 and 6. We find

A;.(B,,B,)=6,.6(B,--B,)~R,(E,-B,)+(I--6,,) ’ R,(&-R,,&-&) N(N- 1)

3

(4.15)

so that the indices p, v only appear at the Kronecker symbol. Consequently the average with respect to p( V) in eq. (4.13 ) can easily calculated and yields NC, for S,, and N2Et 2 for 1-a,,. for the function A ( B1, B2) the surprisingly simple result A(R,,Rz)=

v’’ [%W, s

--B,)RI(E,-B,)+~~~~R~(E,-BI,E~-Bz)I

.

Collecting everything we find

(4.16)

This expression, however, and thus the convolution integrals (4.11) depend on the level density R, = 1/D. To remove this dependence we define the unfolded scales bl = B, ID, b2= B2/D and calculate Jy-(b,,b2)=

lim D2ReZ,(B,)ReZy(B2).

(4.17)

N-03

Introducing the new variables b; =B’JD, 65 = B;/D, the normalized decay widths rcs= y,/D, G,= y,,/D and the normalized moments r$ =E/D”’ we find from eq. (4.11) Jz-(b,,bz)=(&‘)2

1 db;L(b,-b;,2&) --oo

y db;L(b,-b;,2&)Z(b;,b;), -al

(4.18)

T. Guhr, HA. Weihmtiler

/ Anticrossing spectroscopy

33

where the function Z( br, &) can be evaluated using the fact that on the unfolded scale the level density is unity and that R2 yields the function 1 - Y2discussed in section 2.3 [ 2 11, lim D*A(B,,B2)= N+C.J

Z(b,,b2)=

{~6(b,-b2)+~*[1-Y*(b,-b2)1}.

zl

(4.19)

Note that this expression is independent of the singlet energy ES as one should expect on the unfolded scale. Of course, this is due to the translational invariance of generic fluctuations which is obvious in Z( b, , b2) = Z( b, - b2 ) . Hence introducing the new variables b= b, - b2, p= b, + b2 and b’ = bi -b;, p’ = bi + bi the integral in equation (4.18 ) becomes independent of p and the p’ integral can be evaluated, J,--(b,,

b,)=(d:)*

5 Z(b’)L(b’-b,

4x,,) db’ ,

(4.20)

--m so that.ZF-(b,, b2)=JF-(6). A similar calculation can be performed for the other terms on the right-hand side of eq. (4.8). In particular we find for the second term ar

JimmD2ReZ~(Z3,)ReZ~(Z?2)=(d~)*

J2+-(b,,b2)=

j Z(b’)L(b’-p,4&ddb’,

(4.21)

-03

so that .ZZ+ - ( b, , b2) = J2+- (p) . Hence this term depends on p but not on b and does not describe generic fluctuations. The physical reason is the opposite direction of the spin projections. The corresponding expressions for the third and fourth term on the right-hand side of eq. (4.8 ) are for symmetry reasons given by J2++ ( bl, b2) = JF-(b,, b2) and J,-+(b,, b2)=J2+-(b,, b2). The remaining task is to calculate the Fourier transform OD It(t) I*= I dbexp(2ixbt) -0D

lim D2Z(Z?,)Z(B2)

N-W

(4.22)

according to eq. (4.7 ). As mentioned above the Fourier transforms of the first three terms on the right-hand side of eq. (4.7) must contain the function s(t). The same is true for the contribution from J;-(p) and J,- + (p) since they do not depend on b. Thus the autocorrelation of the spectra with opposite spin projections does not yield a relevant contribution to Ic( t) I*. The Fourier transforms of the terms for equal spin projection are easily calculated since the integral in eq. (4.20) is a convolution. Using that exp ( - 4x15,I t I ) is the Fourier transform of the Lorentxian L (b, 4~~) and collecting everything we find (4.23) wheref( t) is some function which we have not worked out. For completeness we give the analytical results [ 2 1 ] for the two-level form factors b2 ( t ) . In the Poisson case we have b2 ( t ) = 0, for the GGE one finds b,(t)=1-2ltl+(tlln(2ltJ+l)

=-l+ltl

21t1+1 In21t1-1

and finally for the GUE

if{tl

(4.24)

T. Guhr, HA. Weidenmiiiier / Anticrossingspectroscopy

34

b,(t)=l-ItI

if Itl
=o

(4.25)

ifIt]>l.

The structure of the result (4.23) for small single-triplet coupling derived in the framework of the exact scattering theory confirms nicely the phenomenological interpretation of the Grenoble group discussed in section 2.3, including the exponential damping with the decay width &. It is worthwhile to emphasize that I c( t ) I ’ is sensitive to the probability distribution of the singlet-triplet coupling elements since the second and the fourth moment enter the final results. This observation coincides qualitatively with earlier numerical findings [ 18,19 1. For a Gaussian distribution the relation z

= (2n - 1 )!!z ” implies (4.26)

so that I c(t) I * displays the correlation hole but never approaches zero. Our result is shown in fig. 3 for a normalized decay width ~~~~0.03 in the Poisson and in the GGE case. The termf( I)S( t) is not plotted. 4.2. Arbitrarily many symmetries Molecules like methylglyoxal often are governed by (approximate) symmetries. In our case, this means that the triplet states decay into several classes of fixed symmetry. This is the case considered in the present section. We assume that the system has A4 symmetries, more precisely we assume that the triplet Hamiltonian H is reducible. Hence the Hamiltonian can be transformed to a block-diagonal matrix H=diag(H,, .... NM) where Hk, k= 1, .... Mare Nk x Nk matrices with elements H+. The corresponding diagonal matrices of the eigenvalues are denoted by X, with elements X,, and the diagonalizing matrices by U,. The totality of the X, and the U, form the block-diagonal matrices X and (I, respectively. The triplet states can be written as ]tkp), k= 1, .... A4, p= 1, ...) Nk. The total number of triplet states, i.e. the dimension of H, is given by N= ZZfSI Nk. The singlettriplet coupling vector V is composed of M vectors V, with elements V,,. The decay widths of the A4 triplet manifolds are ykt,k= 1, .... M and correspondingly we have ykrt- ( yS+ yh) /2. The differential joint probabilities P(H)d[H], PE(X) d[X] and p(V) d[ v] are k-fold products of differential probabilities Pk(Hk)d[HkJ, PEk(&) d [ X,] and pk( V,) d [ Vk], respectively, all probabilities being Gaussian distribution laws. In particular the average with respect to the functionspk( V,) yields moments E. Expanding with respect to a small singlet-triplet coupling does obviously not change the results of section 4.1 up to eq. (4.8) included, if we replace eqs. (4.6) and (4.9) by

1.0

Icltl12

0.5

0 0

1

2

t

3

Fig. 3. The Fourier transform It(t) I* of the autocorrelation function according to the analytical result (4.26) for a normalized decay width ~,,=0.03. The temt/(t)d( t) is not plotted, the other prefactors are normalized to unity. The thick line corresponds to the GCE case, the thin line to the Poisson case. The gap between these lines is the correlation hole.

T. Guhr, HA. Weidenmiikr

/ Anticwing

35

spectnzwopy

(4.27) --m

and similarly for Z2+(B). Again it is sufftcient to consider the contribution of parallel spin projections so that corresponding to eq. ( 4.11) , ReZ2(Bt)ReZ,(B,)=t~~)‘~

3 d& L(&-Z~\,~Y& . -02

5 dB~L(Bz-B;,2y,,)Aw(B;,B;), -co

(4.28)

where the function &( B,, B,) is given by

The independence of the integrand of the diagonalixing groups is shown as in section 4.1. Analogously to eq. (4.13) wefind &(&,

B2)=

(4.30)

JY:~~,“~,Id[Ylp(V)IV,,I”IV,.(‘~k~”(B,,B2),

where corresponding to eqs. (4.14 ) and (4.15 ) ~k&%r

B2)

=

j

d[Xl

4d(&-B2)

p/z(x)

+k,

[&d(&

-&

-xk,)

(‘%-&)+(1-&v)

’ Nk(Nk

Rk2tEs-&,Es-B2) -

1)

>

(4.31) Hence according to eq. (4.16) the average with respect to the probability density function p( V) yields &(Bl,

82)

=

;Is{S,[V~~6(8,-B,)Rk,(E,-B,)+~2Rk2(E,-B,,E.-B,)l s

(4.32)

+(l-S,)V$Rk,(E,-B,)~R,,(E,-B,)).

Note that the correlation functions Rkl and Rk2 have the index k since they depend on the parameters of the probability density function pEk(&). In particular we have A4mean level spacings Dk= 1/Rk, and a total mean level spacing D of the superposition which satisfies 1/D= IXf_, 1/Dk. Moreover we introduce fractional densities gk= DID,. Again we have to remove the dependence on the level density. We define b, = B, / D, b2= B21D and calculate Jy-(bl,b2)=

lim D2ReZ,-(B,)ReZ,-(B,).

(4.33)

N-03

Introducing the new variables b> B’JD, bi =B$/D, the normalized decay widths ~+y,lD, the normalized moments z = v’S,lD2” analogously to section 4.1 we find from eq. (4.28) Jz-(b,,b2)=(r#)2

E 7 db’iUb,-&, -0D

2Kkrl) 1 d& L(b,-b’,, --m

2~/&L(b’,,

b;) ,

tcIBt=y&D and

(4.34)

36

T. Guhr,HA. Weio!enmilller/ Anticrossingspectroscopy

where the function &( b,, b2) is corresponding to eq. (4.19) given by Z&b,, bd = )_m ~Z&(&,

4 1

(4.35) Introducing the new variables b=b, - b2, p= b, + b2 and 6’ = b’, -b$, p’ = b’, + 6; we find analogously to eq. (4.20) independently of p, J,-(b,,b2)=(%)l(”

,c,

7 db’L(bf-b,4~~~){vb,g,d(b)+~*g:Il--2(g*b)l}+2~~,~g*vhgr). 4

-

--oo (4.36) so thatJ,-(b,, b2)=Jy-(b). Again, only the functions J,- - (b, , b2 ) and J2++ ( b, , b2) for parallel spin projections yield relevant contributions to the Fourier transform in fourth order of the singlet-triplet coupling, lc(t)I*=

7 dbexp(2irrbt)~_~D2Z(Z?,)Z(E2) -co 2.u E,

gk

-~*b2(r/g,)lexP(-4nKb,Itl),

[vt

(4.37)

wheref( t) is some unimportant function as in section 4.1. Because of the Gaussian distribution we have z (2n- l)!!anand

=

thus *h4

-2

Ic(t)~*=f(f)S(t)+6

k&t k;,

[l-~bz(t/gk)lexP(--qffKkstItI).

(4.38)

g

This result confirms the phenomenological interpretation mentioned in section 2.3. If gk= 1/M and if & and ICKY are similar for all values of k the correlation hole becomes narrower by a factor M.

5. Discussion

The modelling of the laser experiments in terms of the scattering theory originally developed in nuclear physics provides a very useful tool for the study of the results of anticrossing spectroscopy. The knowledge of the scattering matrix is equivalent to the solution of the Lippmann-Schwinger equation of the scattering problem. Hence the scattering matrix formalism is a solid basis for further investigations. The equivalence of the density and the scattering matrix formalism was proved in the case of one isolated anticrossing. In general this is no surprise since both formalisms are derived from the Schrkiinger equation. In the scattering matrix formalism, however, overlapping resonances are easily treated. Moreover, the scattering matrix formalism itself does not force us to make any assumption about the Hamiltonian or the couplings. This is an enormous advantage compared to more phenomenological models. Gf course, in order to derive concrete results, we have to make assumptions about the Hamiltonian and the coupling. We used the Gaussian ensembles to study the fluctuation properties of anticrossing spectra. At first sight one might argue that the use of the Gaussian ensembles is very restrictive but this is not the case: The Gaussian ensembles are known to be the generic ensembles for systems with and without time-reversal invariance. Moreover, it is known that other ensembles, having the same symmetries with respect to time reversal as the Gaussian ones, yield very similar fluctuation properties [ 2 11. More-

T. Guhr, HA. Weidenmiiller / Anticmssingspectmscopy

37

over, it is known that even a weak Gaussian perturbation quickly tends to dominate the spectral fluctuation pattern [ 23,241. For all these reasons, we believe that the Gaussian ensembles are a very natural model in the case of methylglyoxal since the triplet manifold is sufftciently far away from the ground state. In the present paper we presented the analytical aspects of our scattering model. For the experimentally realistic situation of small singlet-triplet coupling we derived an analytical result for the correlation hole in the case of one and of arbitrarily many symmetries. Our results qualitatively confirm the phenomenological interpretation of the Grenoble group [ 1,2 1. After the integration over the photon energy the relevant part of the intensity function I(B) contains the usual propagator of the Schriidinger equation where the energy is replaced by the magnetic field. In this way the assumption that I(B) can be treated like an energy eigenvalue spectrum of a Hamiltonian system, which is fundamental for the phenomenological interpretation, is rigorously justified in the small coupling regime. Unfortunately, the regime of small singlet-triplet coupling can only be worked out under the further assumption fi w 2 I V, I which contradicts the experimental situation in methylglyoxal. Because of this assumption all anticrossings in our model have the same line shape, the Lorentzian L (B, 2y,,), with amplitudes proportional to the square of the couplings, I V: I ‘. In the methylglyoxal experiment, however, the width of the anticrossings does depend on V,, according to formula (2.2). Nevertheless, our analytical results are in surprisingly good agreement with the experimental data. This might be due to the fact that the condition fi w 2 I V, I is a worst case estimate. Mathematically it is a necessary condition at the anticrossing where the triplet energy (shifted by the magnetic field) coincides with the singlet energy. From the physical point of view, however, this condition might be too restrictive: Between two anticrossings the values of the decay widths do not matter since the difference between the singlet and the triplet energy is in any case large enough to guarantee convergence of the expansion. In other words, we actually need the condition & w 2 I V, I only at the anticrossings and we do not need it at other values of B. Consequently, it seems reasonable to conclude that the statistical anticrossing effect is indeed mainly of spectral origin: The positions of the anticrossings (as opposed to their individual line shapes) play the most important role. The present limitations of the analytical calculation do, of course, not affect the numerical simulations in which the scattering approach proves itself to be very useful [ 18,191, too. Our analytical insight that the distribution of the coupling elements influences the correlation hole was inspired by earlier numerical results which confirm and go beyond the phenomenological discussion of the Grenoble group

1121.

In concluding this paper, we wish to explain why our analytical approach is limited to the regime of small singlet-triplet coupling, and why it cannot at present be extended to a fulLfledged calculation of the autocorrelation function. Several years ago, Efetov [ 25 ] solved some problems in solid states physics, using a graded or supersymmetric technique. This method which is based on a delicate balance of commuting and anticommuting variables was also implemented in nuclear physics, allowing for a complete calculation of the fluctuations of the compound nucleus scattering matrix [ 17 1. Unfortunately this method cannot easily be used for the statistical anticrossing effect, for two reasons. Firstly the singlet-triplet coupling drastically disturbs the delicate balance of commuting and anticommuting variables. While this difficulty can be overcome, the second reason is even more serious: The observables evaluated with the graded technique have so far been mainly two-point functions, i.e. correlations of two propagators which differ only in the energy arguments. The anticrossing intensity I(B), however, is related to the absolute value of the scattering matrix and is thus already a two-point function. Hence the autocorrelation function involves a four-point function. Only in the small-coupling regime can it be reduced to a two-point function. The calculation of four-point functions is much more difficult than that of two-point functions. We have attempted this calculation. Our expression for the autocorrelation function for arbitrary singlet-triplet coupling and arbitrary decay widths is - after the ensemble average - a forty-dimensional integral which did not seem amenable to simplifications. For the reasons discussed above we feel, however, that our calculation for the experimentally realistic situation of small singlet-triplet coupling gives a sufficiently clear

38

T. Guhr, HA. Weidenmiilier / Anticrossing spectroscopy

insight into the origin of the statistical anticrossing effect. For quantitative purposes, our work has to be supplemented by numerical simulations [ 18,191.

Acknowledgement

We gratefully acknowledge stimulating discussions with U. Hartmann, R. Jost and M. Lombardi.

References [ I ] L. Leviandier, M. Lombardi, R. Jost and J.P. Pique, Phys. Rev. Letters 56 ( 1986) 2449. [ 2 ] R. Jost and M. Lombardi, Lecture Notes in Physics, Vol. 263 (Springer, Berlin, 1986) p. 72, and references therein. [ 31 G. Hackett, R. Werbin and J. Rainwater, Phys. Rev. C 17 (1978) 43. [4]T.A.Brody,J.Flores,J.B.French,P.A.Mello,A.PandeyandS.S.M.Won&Rev.ModPhys.53 (1981)385. [ 510. Bohigas and M.J. Gianonni, Lecture Notes in Physics, Vol. 209 (Springer, Berlin, 1984) p. I, and references therein. [6] G.E. Mitchell, E.G. Bilpuch, P.M. Endt and J.F. Shriner, Phys. Rev. Letters 61 (1988) 1473. [ 71 N. Rosenzweig and C.E. Porter, Phys. Rev. 126 (1960) 1698. [8] H.S. CamardaandP.D. Georgopulos,Phys. Rev. Letters 50 (1983) 492. [9] A. H&tie and D. Wintgen, Phys. Rev. A 39 (1989) 5642, and references therein. [lo] E. Haller, H. K6ppel and L.S. Cederbaum, Chem. Phys. Letters 101 (1983) 215. [ 111 Th. Zimmermann, H. Koppel, L.S. Cederbaum, G. Persch and W. Demtr&ler, Phys. Rev. Letters 6 1 ( 1988) 3, and references therein. [ 121 S. Mukamel, J. Sue and A. Pandey, Chem. Phys. Letters 105 ( 1984) 134. [ 131 E. Abramson, R.W. Field, D. Imre, K.K. IMCS and J.L. Kinsey, J. Chem. Phys. 83 (1985) 453. [ 14) J.P. Pique, Y. Chen, R.W. Field and J.L. Kinsey, Phys. Rev. Letters 58 (1987) 475. [IS] M. Lombardi, R. Jost, C. Michel and A. Tramer, Chem. Phys. 46 (1980) 273; 57 (1981) 341,355; P. Dupr6, M. Lombardi, R. Jost, C. Michel and A. Tramer, Chem. Phys. 82 ( 1983) 25; P. Dupr6, R. Jost and M. Lombardi, Chem. Phys. 9 1 (I 984) 355; E. Pebay-Peyroula, R. Jost, M. Lombardi and P. Dupr6, Chem. Phys. 102 ( 1986) 417; E. Pebay-Peyrotda, R. Jost, M. Lombardi and J.P. Pique, Chem. Phys. 106 ( 1986) 243. [ 161 C. Mahaux and H.A. Weidemutlller, Shell-Mode1 Approach to Nuclear Reactions (North-Holland, Amsterdam, 1969). [ 171 J.J.M. Verbaarschot, H.A. Weidenmtlller and M.R. Zimbauer, Phys. Rept. 129 ( 1985) 367, and references therein. [ 18 ] U. Hartmann, Doctoral Thesis, Universitit Heidelberg, FRG ( 1989 ) . [ 191 U. Hartmann and H.A. Weidenmtlller, in preparation. [20] H. Wieder and T.G. Eck, Phys. Rev. 153 (1967) 103. [ 2 11 M.L. Mehta, Random Matrices (Academic Press, New York, 1967). [ 221 M. Lombardi, private communication. [23]A.Pandey,Ann.Phys. (NY) 134 (1981) 110. [ 24) T. Guhr and H.A. Weidenmtlller, Ann. Phys. (NY) 193 ( 1989) 472. [25] K.B. Efetov, Advan. Phys. 32 (1983) 53.