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SURVEY OF CORRELATION PROPERTIES OF POLYATOMIC MOLECULES VIBRATIONAL ENERGY LEVELS USING F T. ANALYSIS R. JOST and M. LOMBARDI Service National des Champs Intenses (C.N.R.S.) BP 166 X, 38042 Grenoble C~dex, FRANCE. and Laboratoire de Spectrom~trie Physique (U.S.T.M.G.) BP 87, 38402 Saint Martin d'H~res C~dex, FRANCE.

I.

Introduction In the l a s t few years molecular spectroscopists have begun to

study the highly excited v i b r a t i o n a l l e v e l s of polyatomic molecules. In t h i s high energy regime v i b r a t i o n a l quantum numbers can no longer be i n t r i n s i c a l l y assigned (in contrast with v i b r a t i o n a l l e v e l s at low energy). One can only characterize these l e v e l s by t h e i r c o r r e l a t i o n p r o p e r t i e s ( 1 ) , we shall consider : - Short range c o r r e l a t i o n s which are characterized by the Next Neighbor D i s t r i b u t i o n , (N.N.D.). These c o r r e l a t i o n s range from a POISSON (random or uncorrelated spectra) to a WIGNER d i s t r i b u t i o n (which shows " l e v e l r e p u l s i o n " ) . -

~3(L)

Long range c o r r e l a t i o n s are characterized by t h e ~ 2(L) and function. They describe the behavior which ranges from an un-

correlated spectra (POISSON s t a t i s t i c ) to a spectra with "spectral rigidity". In nuclear physics, spectra obtained many years ago by low ener gy neutron scattering show the phenomena of " l e v e l repulsion" and "spectral r i g i d i t y " (1). These results have stimulated t h e o r e t i c a l developments such as Random Matrix Theory (R.M.T.) (2) and the Gaussian Orthogonal Ensemble (G.O.E.)

(2). In the f i e l d of molecular

physics the t h e o r e t i c a l predictions have preceeded experimental res u l t s . Many c l a s s i c a l t r a j e c t o r y calculations of two-dimensional

sys-

tems show features of " i r r e g u l a r " behaviour (3). These c l a s s i c a l t r a j e c t o r i e s have also been calculated with the potential energy surfaces of the e l e c t r o n i c ground state of t r i a t o m i c molecules. All these calculations suggest a smooth change of the dynamics from regul a r to i r r e g u l a r behavior as the energy exceeds a certain threshold. At an intermediate energy regime, the phase space is embedded with both regular and i r r e g u l a r regions. Quantum calculations with models

73

of ( u s u a l l y ) two degrees of freedom, produce eigenvalues which d i s play strong c o r r e l a t i o n s p r o p e r t i e s ( 4 ) . Very few quantum c a l c u l a t i o n s have employed the r e a l i s t i c p o t e n t i a l surfaces of polyatomics, due to the p r o h i b i t i v e size of the matrix to be diagonalized. Even c a l c u l a t i o n s concerning the simplest polyatomic, t r i a t o m i c molecules involve three v i b r a t i o n a l degrees of freedom. Comparison of the c l a s s i c a l and the quantum c a l c u l a t i o n s for two dimensional systems, show a q u a l i t a t i v e agreement, i . e . ,

the c o r r e l a t i o n properties of

eigenvalues evolve from POISSON to G.O.E. in the same energy range where the c l a s s i c a l t r a j e c t o r i e s display a t r a n s i t i o n from "regular" to " i r r e g u l a r " behavior (5).

Up to now, there is no q u a n t i t a t i v e re-

l a t i o n s h i p between these c l a s s i c a l and quantum d e s c r i p t i o n s . The main i n t e r e s t in molecular physics f o r these problems arise from the need to understand the phenomenon of v i b r a t i o n a l energy red i s t r i b u t i o n , which is the basis of the usual R.R.K.M. theory f o r unimolecular reactions. This theory supposes complete intramolecular t h e r m a l i s a t i o n of v i b r a t i o n a l energy before reaction (the i s o l a t e d molecule acts as i t s own "thermal bath"). The basis of t h i s theory is believed to be that at relevant e x c i t a t i o n energy a l l

classical tra-

j e c t o r i e s are ergodic. A few c l a s s i c a l t r a j e c t o r i e s addressing d i r e c t l y t h i s problem f o r two-dimensional systems have been made recently(6). In t h i s paper we w i l l

summarize in chapter V the experimental

r e s u l t s obtain to date concerning the s t a t i s t i c a l c o r r e l a t i o n propert i e s of the v i b r a t i o n a l energy l e v e l s of polyatomics. F i r s t we present, in chapter I f ,

a general

review of the properties of v i b r a t i o -

nal energy l e v e l s and, in chapter I l l ,

t h e i r r e l a t i o n s h i p with expe

rimental molecular spectra. In chapter IV the Fourier Transform method is presented as a tool to displays c o r r e l a t i o n s in experimental spectra. II.

General considerations on the v i b r a t i o n s of polyatomic molecules. In t h i s section, we shall only consider molecules w i t h o u t rota-

t i o n ( s e ~ J I I I ) . The number of v i b r a t i o n a l degrees of freedom, i . e . the number of v i b r a t i o n a l modes for non l i n e a r species is N = 3n-6 ( f o r n ~ 3 ) , where n is the number of atoms (diatomics have only one v i b r a t i o n a l mode and they are not considered here). For each e l e c t r o nic state, the bottom of the N dimensional p o t e n t i a l energy surface can be approximated by N harmonic o s c i l l a t o r s . This means that at "low" energy, the v i b r a t i o n a l l e v e l s are well characterized by the

74 normal modes d e s c r i p t i o n , except f o r the occurence of an occasional Fermi resonance. When we consider l e v e l s at high energies, E, the density of states

increases as ~ ( E ) ~ [ E / ( w ~ ] N-I

whereC~J,t~is the

geometric mean of the frequencies. T y p i c a l l y , the magnitude o f , ~ > i s of the order of 1000 cm-1 (0.12 eV) and the height of the well ranges from 104 cm- I to few 104 cm- I

(a few eV) f o r the ground state of sta-

ble molecules. Consequently, there is a

very rapid increase in the

v i b r a t i o n a l density of states with energy and with the number of atoms. The coupling terms in the Hamiltonian, l i k e higher order cross terms in the p o t e n t i a l or k i n e t i c energy operators induce an increasing number of Fermi resonance as the density of states becomes l a r g e r . The absence of a complete set of spectroscopic constants, or accurate ab i n i t i o

p o t e n t i a l energy surfaces, precludes any precise

c a l c u l a t i o n of eigenvalues at high v i b r a t i o n a l energies. Nevertheless a model of o s c i l l a t o r s coupled with cubic and q u a r t i c cross terms in the p o t e n t i a l , with a harmonic basis set can be considered. This model y i e l d s a very sparse matrix. The v a l i d i t y of such a model stems from the spectroscopy of the lower excited v i b r a t i o n a l l e v e l s which shows a very rapid decrease of matrix elements with increasing the order of the cross terms. The size of the cubic coupling matrix elements may be as large as a few hundred cm-1 in the p a r t i c u l a r case of stretch-bend C-H coupling but, most of the matrix elements are much smaller: 0.1 to 10 cm- I

f o r cubic and q u a r t i c coupling terms.

We ex-

pect from rough c a l c u l a t i o n s on random matrix (7) that the mechanism of overlapped Fermi resonances plays a c r u c i a l role in terms of statistical

properties when the density of states becomes l a r g e r than

1 to 100 l e v e l s per cm-1, depending on the molecule. The e f f e c t of these coupling on v i b r a t i o n a l energy r e d i s t r i b u t i o n has been studiedby techniques which depend mainly on the phenomenon of mixing of wavefunctions (observed via p r o b a b i l i t y of t r a n s i t i o n s ) , not on energy l e v e l p o s i t i o n s . The various v i b r a t i o n a l modes which are mixed at a given energy emit d i f f e r e n t spectra, which may be used to label them (at l e a s t q u a l i t a t i v e l y ) (8).

Some more recent work use picosecond t i -

me resolved spectroscopy to study these c o u p l i n g s ( 9 ) . The conclusion of such recent studies is t h a t one can f i n d lack of v i b r a t i o n a l red i s t r i b u t i o n at s u r p r i s i n g l y high energy in large molecules, increasing the i n t e r e s t to f i n d other measures of these coupling on r e l e vant molecules and at relevant energies. On one hand, the f a s t convergence of the polynomial expansion in harmonic coordinates may be no longer v a l i d f o r large amplitude v i b r a t i o n a l motion. A c o n t r a r i o ,

75 for a large molecule, a great amount of v i b r a t i o n a l energy, enough to dissociate i t

for example, can be attained by d i s t r i b u t i n g the energy

over the large number of modes available with only very few (0, 1, 2) quanta in each mode. This kind of v i b r a t i o n a l l e v e l , named "combination l e v e l " , which correspond to very harmonic motion at large energy, i s in fact the vast majority of states present at energies i n t e r e s t i n g f o r chemical studies l i k e photochemistry, intermolecular v i b r a t i o n a l r e l a x a t i o n , and multiphoton e x c i t a t i o n . But the minority (in terms of numbers) of very anharmonic states which concentrate the energy on one given mode ( f o r example a local mode which leads to dissociation) play probably an important role in chemical process l i k e d i s s o c i a t i o n , isomerization, fluorescence decay, e t c . . . Ill.

Experimental technique on molecular spectra. According to usual ideas coming from previous work of nuclear

p h y s i c i s t s , in order to Study the s t a t i s t i c a l correlations between levels, it

is necessary to consider only those l e v e l s which have the

same good quantum numbers (spin, p a r i t y and t o t a l angular momentum). Thus the spectroscopic method must be able to sort these l e v e l s by t h e i r quantum numbers in order to avoid the superposition of sets (see chapter IV). In addition, the spectrum must contain nearly a l l the l e v e l s in a given energy range (no "missing" l e v e l s ) . The experimental signal to noise r a t i o must be large enough to l e t a l l the l e vels appear with s u f f i c i e n t i n t e n s i t y . The spectrum should y i e l d a complete set of l e v e l s , therefore the resolution should be s u f f i c i e n t to avoid overlapping l i n e s (no blended l i n e s ) . Only gas phase laser spectroscopy is able to produce molecular data at high v i b r a t i o n a l e x c i t a t i o n with a t y p i c a l resolution of MHz ( f o r c.w. laser) to GHz ( f o r pulsed laser) required for s t a t i s t i c a l analysis. Nevertheless, most of experimental molecular spectra do not sat i s f y the requirement cited above. I t is why, a new technique of analysis, the Fourier Transform described in chapter IV has been evolved to deal with experimental spectra which suffers imperfection. This new technique, which takes f u l l advantage of the fact that large (but imperfect) stretches of l e v e l s can be obtain, c l e a r l y opens new avenues to analyze the large amount of data which can be obtained in molecular physics in chemically i n t e r e s t i n g s i t u a t i o n s . The c l a s s i c a l spectroscopic techniques relevant to study the v i b r a t i o n s of polyatomic molecules are summarized as follows :

76 A. Infrared spectroscopy ( v i b r a t i o n a l e x c i t a t i o n ) . This technique has produced a great deal of information about the fundamental v i b r a t i o n a l frequencies and the overtones of CH and OH stretches in the ground state.

However i n f r a r e d studies at high

v i b r a t i o n a l e x c i t a t i o n are r e s t r i c t e d to mainly these types of overtones and then do not allow to observe the numerous "combination" l e vels since the t r a n s i t i o n s occur between l e v e l s on the same p o t e n t i a l surface. Furthermore, r o t a t i o n a l congestion occurs and precludes to observed dense sets of l e v e l s (see B, below). B. E x c i t a t i o n spectroscopy ( e l e c t r o n i c e x c i t a t i o n ) . In t h i s technique, t r a n s i t i o n s from the v i b r a t i o n l e s s e l e c t r o nic ground state (SO) to the ( f i r s t ) excited state ($I) allow to study the r o v i b r o n i c l e v e l s in the excited state. But Franck Condom factors do not permit the observation of every v i b r a t i o n a l level because many of them are too small. Furthermore, as in I.R. spectra, r o t a t i o n a l congestion occurs : at room temperature, there is a large number of r o t a t i o n a l l e v e l s populated in the molecule as the r o t a t i o n a l constants range from 0.1 cm-1 to a few cm-1 (10 -4 - 10-3 eV). Even though the selection rules (~ J = O, ~ 1) l i m i t the number of r o t a t i o n a l t r a n s i t i o n s , the number of r o t a t i o n a l l i n e s appearing in the spectra is p r e t t y large ( t y p i c a l l y 102 to 103 per band). These l i n e s are spread over a region of about 100 cm-1 ( ~ 1 0 -2 eV). This spread is much l a r g e r than the mean spacing of v i b r a t i o n a l l e v e l s we want to study (see above) and consequently precludes the observation of every v i b r a t i o n a l l e v e l . The use of a supersonic j e t

(a free expansion of a mixture of a

c a r r i e r gas ( H e , A r . . . ) with the molecule of i n t e r e s t ) cools the rotat i o n a l temperature to about 1K f o r a few ~,~sec. The corresponding rot a t i o n a l spectrum is much less congested as few r o t a t i o n a l l e v e l s are populated in the v i b r a t i o n l e s s ground e l e c t r o n i c state. But, even w i t h a supersonic j e t ,

the p o s s i b i l i t y of the overlap of r o v i b r o n i c

bands remains (see NO2 r e s u l t s below). A f u r t h e r spectral s e l e c t i v i t y is possible with an a d d i t i o n a l spectroscopic step : double resonance technique, such as Stimulated Emission Pumping (S.E.P.) or Microwave-Optical Double Resonance (MODR) and also Anticrossing Spectroscopy

(A.S.) are possible techniques

f o r producing spectra corresponding to a simple set of good quantum numbers. For example, in S.E.P., a t r a n s i t i o n from the v i b r a t i o n l e s s

77 ground state populate a single rovibronic level (J = 0 for example), in the excited state. Then a second t r a n s i t i o n , by stimulated emission, t r a n s f e r population, according to well defined selection rules, down to one (or very few) r o t a t i o n a l l e v e l s of high v i b r a t i o n a l l e vels of the ground state. The spectral s i m p l i c i t y achieved by t h i s method allows to consider each l i n e (or set of very few l i n e s ) of the spectrum as a v i b r a t i o n a l eigenvalue with a known angular momentum. We shall present three examples of results mainly obtained by double resonance method in NO2, Acetylene (C2H2) and Methylglyoxal (CH3-O-C-C-H-O). A very related problem is the coupling between r o t a t i o n a l and v i b r a t i o n a l degrees of freedom. Two examples concerning NO2 and H2CO w i l l be discussed at the end of the NO2 chapter. Beforehand, in chapt e r IV, we present a new method of analysis of the c o r r e l a t i o n properties of spectra : the Fourier transform. IV. The Fourier Transform : a new method to analyse the c o r r e l a t i o n properties of spectra. The s t a t i s t i c a l method l i k e N.N.D. and A 3 have been developped in order to analyse the a v a i l a b l e data in nuclear physics, i . e . a rel a t i v e l y small set of l e v e l s ( t y p i c a l l y 50 per nucleus). These l e v e l s c o n s t i t u t e a high q u a l i t y data set with respect to signal to noise, resolution and spectral p u r i t y (single J ~assignment). Then, these data have been analysed with the N.N.D. and ~ 3 in terms of s t i c k spectrum. In a recent paper, L. Leviandier

et al (10) introduce the

Fourier Transform (F.T.) as a tool to measure s t a t i s t i c a l c o r r e l a t i o n properties able to t r e a t a noisy, poorly resolved and s p e c t r a l l y impure spectrum. In t h i s method, the raw spectrum is Fourier t r a n s f o r med, without e x t r a c t i n g a s t i c k spectrum, to obtain a function C(t). The c o r r e l a t i o n properties can be determined from the smoothed, ensemble averaged, square of the modulus of C ( t ) , i . e . | C ( t ) 1 2. Consider a spectrum composed of l i n e s with the l i n e shape L ( f ) . The amplitude of l i n e s is assumed to be the product of a d e t e r m i n i s t i c envelope AE by a a stochastic function AS of the position. Then, I C(t)l 2 contains two components as sketched in f i g . i . a) The " f a s t component", which is the square of the F.T. of the envelope AE, is proportional to N2, i . e . the square of the nomber of l e v e l s . This " f a s t component" gives a very large peak at the o r i g i n . For example, i t

is a (sin N t / t ) 2 function when AE is a rectangle function.

78

IT.El 2 POISSON

Sticks

II11 llU I1 II1"

!

o

1

t/p

o

1

t/p

o

1

t/p

o G.O.E.

Sticks

I lll l

:

G.O.E. goussian lines

finite length correlations Sticks

o

t~/~

1

t/e

Fig. i : Typical spectra and their corresponding smoothed (or ensemble averaged) square of Fourier Transform I F.T I 2. The "fast component" appears near the origin. Unsmoothed]F.Tl 2 displays 100 % fluctions as shown in figure 4 (down). t/~ is a dimensionless variable, where~ is the density of lines.

79 b) The "slow" component is composed of the F.T. of the i n d i v i d u a l lineshapes L ( f ) .

I t s amplitude is proportional to N.

Furthermore, i f

there are c o r r e l a t i o n s in the spectrum, the sha-

pe of I C(t) I 2 changes a f t e r the f a s t component : a " c o r r e l a t i o n hole" appears. This can be explained i n the f o l l o w i n g way : the random pos i t i o n of l i n e s , f i ,

are described by one and two level c o r r e l a t i o n

functions : Rl(f) = d

which is the level density, and R 2 ( f l , f 2)

which is the j o i n t p r o b a b i l i t y that there i s a level at f l

and ano-

ther level at f2- The two l e v e l s c o r r e l a t i o n s can be expressed as R 2 ( f l , f 2) = R l ( f l ) . R ( f 2 ) . ( 1

Y 2 ( d .f l , d . f 2 ) )

where Y2 is the two l e v e l s

c l u s t e r f u n c t i o n given by Metha(2). I f Y2 is not zero, i . e .

if

corre-

l a t i o n s e x i s t , the "slow component" is m u l t i p l i e d by [1 - b 2 ( t ) ] , where b2(t) is the F.T. of Y2. The shape of [1 b 2 ( t ) ] is shown on f i g u r e 1 f o r two l i m i t i n g cases : POISSON (b2(t) = O) and G.O.E. The decrease of the amplitude of 1 - b2(t) near the o r i g i n ( f o r ~ / ~ 1 )

is

c a l l e d a " c o r r e l a t i o n hole". The shape of t h i s "hole" depends ~n the p a r t i c u l a r system and experimental s i t u a t i o n . Furthermore, f o r any random spectrum (Poisson or G.O.E), the F.T. of a single spectrum is 100 % randomly modulated with a frequency of the order of the r e c i procal of the width of the envelope. This random modulation is mathem a t i c a l l y analogous to the speckle phenomenon f a m i l i a r to laser users and i t

can be reduced e i t h e r by an ensemble averaging or by smoothing

of the spectrum or both. The r e l a t i o n s h i p between spectra and i t s I C(t) I 2

is shown f o r

t y p i c a l examples in f i g u r e 1. Some comments are necessary in order to better understand the meaning of the F.T. of a spectrum and e s p a c i a l l y why c o r r e l a t i o n s properties can been seen even when the spectrum is "bad" (see above). F i r s t t h i s F.T. method, looks f a m i l i a r f o r molecular p h y s i c i s t s dealing w i h t the theory of r a d i a t i o n l e s s t r a n s i t i o n s in "intermediate" molecules. Consider phenomena l i k e intramolecular t r a n s f e r of energy between a single v i b r a t i o n a l non s t a t i o n a r y state which can be d i r e c t l y excited by a pulse of l i g h t (with a laser) and a dense manif o l d of "dark"

(i.e.

f o r which there are no o p t i c a l t r a n s i t i o n s )

v i b r a t i o n a l states. This dense set may belongs to the same e l e c t r o n i c state (pure v i b r a t i o n a l energy t r a n s f e r ) or an other e l e c t r o n i c states with the same e l e c t r o n i c spin ( i n t e r n a l conversion to the ground

80 state) or with d i f f e r e n t spin m u l t i p l i c i t y (intersystem crossing between s i n g l e t and t r i p l e t ) . The "doorway state" which carry o s c i l l a t o r strength is " d i l u t e d " by coupling between a l o t of neighbouring

sta-

tes, giving a spectrum composed of a l o t of l i n e s whose amplitude are the product of a d e t e r m i n i s t i c envelope AE which represente the smoothed energy dependence of the amount of the doorway state contained in each the molecular eigenstate, and a stochastic component because the coupling can be considered as random. The time evolution of the p h y s i c a l system is then gouverned by two components. A s u f f i c i e n t l y short burst of l i g h t (a picosecond laser pulse) excite the pure (non s t a t i o n a r y ) doorway state which is in fact a coherent superposition of a stretch of stationary molecular eigenstates. All these eigenstates s t a r t to radiate in phase, but, since they have d i f f e r e n t s f r e quencies, they dephase in a time of the order of the reciprocal of the width of AE, giving the " f a s t component" After t h i s time the mol e c u l a r eigenstates radiate incoherently giving the slow component whose time constant is the reciprocal of the width of i n d i v i d u a l spectral l i n e ( l i f e t i m e ) . The i n t e r e s t i n g point is the r e l a t i o n s h i p between the c o r r e l a t i o n s properties of eigenstates and t h e i r s corresponding time evolution. This question has been addressed by Delory and Tric as early as in 1974(11) but up to now, no example of correl a t i o n hole has ever been observed in time resolved experiments. Only biexponential decay corresponding

to a POISSON s t a t i s t i c s have been

reported. The speckle "noise" on the slow component, which can also be called many l e v e l s quantum beats, seems to have been overlooked in time resolved experiments because the fluorescence decay is smoothed, due to poor time resolution and/or superposition of several decay corresponding

to simultaneous observation of d i f f e r e n t r o t a t i o n a l

levels. As second comment, i t

is i n t e r s t i n g to notice the r e l a t i o n s h i p

between the c o r r e l a t i o n hole of the F.T. and theZ2(L) function used as standard t e s t of long range c o r r e l a t i o n s . The shape of the c o r r e l a t i o n hole can be numerically related with the shape of the~2(L) function. For example, the ordinate and the slope at the o r i g i n of the slow component is related with the POISSON and G.O.E. contribution of the~2(L) function for large L. We conclude that the c o r r e l a t i o n hole of the F.T. of a spectrum is mainly a measure of long range s t a t i s t i c a l c o r r e l a t i o n s properties of t h i s spectrum. This explains why the F.T. is r e l a t i v e l y i n s e n s i t i v e

81 to lack of r e s o l u t i o n in the spectrum p e r t i e s are l o s t ,

: long range c o r r e l a t i o n s pro-

in p r i n c i p l e , only i f

r e s o l u t i o n i s worse than the

considered range of these p r o p e r t i e s . By c o n t r a s t N.N.D. i n f o r m a t i o n s are r a p i d l y l o s t as soon as the r e s o l u t i o n is worse than the average level

spacing. As t h i r d

comment,

the s u p e r p o s i t i o n

spectra

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mations

about correlations

containing allows

correlations with

w h i c h the c o r r e l a t i o n s only

At t h i s pendent reasons inside

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good quantum n u m b e r s ) .

to measure t h e of

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number o f corresponding

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larger was t o

avoiding

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with

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gives

The c o r r e l a t i o n

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where t c i s

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"bad"

t h a n a few t i m e s f o u n d an i n t e g r a l the step o f

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G.O.E.

time

which t c is

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ba-

as soon as of

t c.

method t o measure

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spectrum from the e x p e r i m e n t a l

w h i c h c a n n o t be done when t h e r e s o l u t i o n

As a c o n c l u s i o n , correlation

is

point

range p r o p e r t i e s ,

spectrum)

a strong

t c can be measured even f o r

The c r u c i a l long

for

disappears

the s u p e r p o s i t i o n

properties.

(see f i g .

We emphasize the d i f f e r e n c e of

is

much s m a l l e r

shape t o F . T .

stretch

point

(mean s p a c i n g ) . coupling

O. T h i s

This

techniques

repulsion,

hole to the time

o n l y between t = ~ and t c, t c ~ t o , for vibrational redistribution.

the

that

by m e a s u r i n g t h e r a t i o

comment,

i.e.,

method

p r o f o u n d and i n t e r e s t i n g

are r e m a i n i n g

range o f t h e c o r r e l a t i o n s

sically

level

method e n a b l e s

the c o r r e l a t i o n

As f o u r t h finite

or N.N.D.

like

may be due t o

the F.T.

subsets,

racteristic

F.T.

G.O.E.

example when t h e r e

density

coupling",

why t h i s

a stretch

independent G.O.E.'s.

we s h o u l d remark

mean t h a t

the onset of

the

The same i n f o r -

one F . T .

o f modes o f a m o l e c u l e b u t n o t c o u p l i n g

As a r e s u l t to

of

properties,

levels

if

impure s p e c t r u m w h i c h can be

autocorrelation

two i n d e p e n d e n t

narrower.

This explain even i n

as the s u p e r p o s i t i o n

to be c o n t r a s t e d

with

hole m times

can be r e c o v e r e d

m t i m e s more l e v e l s .

to d i s p l a y

considered is

a correlation

o f m i n d e p e n d e n t G.O.E.

i s worse or compa-

spacing of lines. the

"correlation

hole"

is

the s i g n a t u r e

of

the

82 Examples of F.T. of experimental spectrum are given below. V. Experimental r e s u l t s , A. E x c i t a t i o n Spectrum and Microwave Double Resonance of NO2 In 1975 Smalley et ai(12) have obtained the e x c i t a t i o n spectrum of NO2, r o t a t i o n a l l y cooled (TR = 3 K) in a supersonic j e t in the region of 6708 A - 5708 A, i . e . from 14900 cm-1 up to 17500 cm-1 above the v i b r a t i o n l e s s e l e c t r o n i c ground state. They observed 140 v i b r a t i o n a l bands. The strong non adiabatic i n t e r a c t i o n between 2A1 and 2B2 e x p l a i n the main features of the observed spectrum(13). We discuss here the s t a t i s t i c a l analysis done by Hailer et al(14) on t h i s data set f o r which they found a Wigner s t a t i s t i c s f o r N.N.D. F i r s t ,

in the

range of 14900-16580 cm-1, there are 20 "hot bands" i d e n t i f i e d among the 83 bands observed ("hot bands" o r i g i n a t e from excited v i b r a t i o n a l l e v e l ( s ) of the e l e c t r o n i c ground s t a t e ) .

In the remaining region of

the spectrum (16580-17500 cm-1) there are probably numerous hot bands; thus, there is a t o t a l of the order of 35 "hot bands" among the 140 observed bands. Second, each band consists of 10 to 50 r o t a t i o n a l l i nes which spread over 10 cm-1. The mean spacing between the bands is 19 cm-1, thus many bands overlap and give a high p r o b a b i l i t y t h a t two bands are considered as one. This e f f e c t would produce a spurious " l e v e l r e p u l s i o n " . For example, in reference (12), there are many more l i n e s in bands number 113 and 22 than in other bands (see bands number

115 and 95 f o r instance). This is a strong i n d i c a t i o n that

bands number 113 and 22 are composed of at l e a s t two overlapped v i b r a t i o n a l bands. As a conclusion the v a l i d i t y of the s t a t i s t i c a l anal y s i s done by H a l l e r spurious

et

and m i s s i n g

necessity

to

obtain

al

on NO2 i s

levels.

doubtful

Furthermore,

a pure data set

because t h e r e

this

analysis

and c o n s e q u e n t l y t o

are t o o many

demonstrates the use d o u b l e

resonance techniques. Recently,

Lehmann and Coy(15)

optical

double resonance technique,

optical

transition

to high excited

17100 cm -1 above t h e v i b r a t i o n l e s s Specific

rotational

ve r o t a t i o n a l

levels

transition

in

have o b t a i n e d , vibrational

(J = 7 t o

i0)

if

transition

the a l l o w e d r o t a t i o n a l

were o b s e r v e d .

Hardwick(16)

level

are s e l e c t e d

the v i b r a t i o n l e s s is

only

level

ground s t a t e

number o f o b s e r v e d t r a n s i t i o n s level

by a m i c r o w a v e -

a v e r y dense spectrum o f NO2.

a factor

at

An

16800 cm - I

to

was e x c i t e d . with

a microwa-

2A I ground s t a t e .

of 8 greater

The

than expected

t o any B2 symmetry v i b r o n i c

has s u g g e s t e d t h a t

t h e numerous

83

Transitions from 10(0,10) level of NO2 C3

1678.0

1684.0

1690.0

1696.0

1702.0

1708.0

• 101

CM-1 Fourier Transform of the 10(0,10) level of NO2

t~3

~>'~. rm~

CL~ V3

o~ c~

0.00000

0.6 196

1.56392

2.04588

2.7278,3

5.40979

1/Cm-1

:ig. 2 : Stick experimental spectrum of M.O.D.R. of NO2.and the corresponding, partly smoothed I F.TI 2. There is no correlation hole near the origil

84 forbidden t r a n s i t i o n s observed are related to quantum e r g o d i c i t y in these h i g h l y excited v i b r a t i o n a l l e v e l s . The number of observed l i n e s i s a f a c t o r of 3 less than the number predicted i f les wer broken as predicted by Hardwick.

all

selection ru-

However, the analysis of

these spectra by the F.T. method, as described above, displays no c o r r e l a t i o n s (17) as shown of f i g .

2. This breaking of r o t a t i o n a l

s e l e c t i o n rules may also be explained by a molecular axis switching instead of v i b r a t i o n a l coupling. B. S.E.P.

spectra on formaldehyde H2CO

Very s i m i l a r r e s u l t s have been obtained by the S.E.P.

technique

at MIT by H.L. DAI et al (18) on H2CO : the only l i n e s which appeared in the spectra at low values of the r o t a t i o n a l quantum number (J ~ 3) are those expected. At higher J and Ka r o t a t i o n a l quantum numbers, the spectra r a p i d l y become more complex and the observed level densit i e s at J = 10, K = 2 are several times larger than the known t o t a l density of v i b r a t i o n a l l e v e l s . This increase in the density of accessible v i b r a t i o n a l l e v e l s was the r e s u l t of a r o t a t i o n induced mixing of the anharmonic v i b r a t i o n a l basis f u n c t i o n ( C o r i o l i s coup l i n g ) which compromised the "goodness" of both v i b r a t i o n a l and Ka quantum numbers. A f u r t h e r analysis of these r e s u l t s , with a c r i t e r i o n f o r chaos developped by H e l l e r et al (19) shows t h a t the i n creasing complexity in the spectrum with increasing angular momentum corresponds

nonetheless with decreasingly chaotic behavior. This can

be understood in terms of an a v a i l a b l e phase space volume t h a t expands more r a p i d l y than the occupied phase space volume as J increases. In conclusion one can study the "pure" v i b r a t i o n a l coupling by considering J = 0 l e v e l s and study r o t a t i o n a l - v i b r a t i o n a l coupling ( C o r i o l i s ) by studying higher J (and K) r o t a t i o n a l l e v e l s . C. Stimulated Emission Pumping (S.E.P.) spectra of acetylene. In 1983, Abramson et al(20) have observed,

by S.E.P., very high

v i b r a t i o n a l energy l e v e l s (at 27900 cm-1) of the ground e l e c t r o n i c state of acetylene. A pulsed dye laser (the PUMP) excites (at 45300 cm- I ) state.

a s p e c i f i c r o v i b r o n i c level of the 1A excited e l e c t r o n i c

A second pulsed dye laser (the PUMP) stimulates t r a n s i t i o n s

85 down to high v i b r a t i o n a l o f the t o t a l frequency behavior.

the same J v a l u e s . (N.N.D.)

levels which

analysis

kest lines

The a n a l y s i s

from a s t i c k follow

spectrum,

the o b s e r v a t i o n

an e s t i m a t i o n

that

analysis

the ~ 3

with

the levels

shows t h a t

either

due to the l i m i t e d The a n a l y s i s

statistics Further

75 % and 6 % o f m i s s i n g l e v e l s

analysis for

that

i n each clump. resolution

of the i n t e n s i t y

are m i s s i n g

indicates

between a d j a -

or because the wea-

Porter-Thomas

70 % o f the l e v e l s

50 l i -

t h e r e are m i s s i n g

o f too c l o s e l e v e l s

to a m o d i f i e d

are m i s s i n g .

to a

of the spectrum shows

a Wigner d i s t r i b u t i o n

are hidden i n the n o i s e .

distribution~according

a transition

of the s p a c i n g s

representation

by Mukamel e t a l ( 2 1 )

i n the s t i c k forbids

i n resonance w i t h

are o r g a n i z e d i n clump of t y p i c a l l y

they approximatively

Further

is

A decrease

i s observed when the

They observe a new t y p e of v i b r a t i o n a l

The t r a n s i t i o n s

cent lines that

level.

of the i x g ground s t a t e .

of the upper l e v e l

of the PUMP l a s e r

1Xg h i g h v i b r a t i o n nes w i t h

levels

fluorescence

distribution

! On t h e o t h e r hand,

o n l y about 15 % o f

by Sundberg e t al (22)

intensity

gives

distribution

gives

analysis

andS3 statistics respectively. C l e a r l y more e x p e r i m e n t s , e s p e c i a l l y at higher resolution, are r e q u i r e d i n o r d e r to a v o i d m i s s i n g l e v e l s . But the F o u r i e r

Transform

(see c h a p t e r

as d i s p l a y e d

in figure

the original

spectrum

presentation

of t h e a c e t y l e n e

jective

27894

analysis,

3 shows t h a t

a strong correlation

. The s t r o n g c o r r e l a t i o n spectrum,

is confirmed

27892

energy

IV) of the o r i g i n a l

cm -1

0

exists

in

found i n the s t i c k

re-

which m i g h t be due to a sub-

by the F o u r i e r

27890

spectrum

I

Transform technique.

i'/9 _

Fig. 3 : S.E.P. Acetylene spectrum near 27 900 cm-1 and the corresponding smoothed, Fourier Transform ( I F.TI 2) which displays a correlation hole. A hole appears also on the I F . T ] 2 of the corresponding stick spectrum.

B6 D. Anticrossing spectra of glyoxal C2H202 and methylglyoxal. In the s i n g l e t - t r i p l e t anticrossing technique, a (strong) magnetic f i e l d is applied in order to Zeeman tune t r i p l e t l e v e l s into resonance with a single r o v i b r a t i o n a l level of a s i n g l e t e l e c t r o n i c s t a t e . The anticrossing i s a l o r e n t z i a n , which is detected as a decrease in the fluorescence of the excited s i n g l e t . A strong magnetic f i e l d , provided by the Service National des Champs Intenses (S.N.C.I.) (Grenoble,

France) allows one to obtain many anticrossings (and v i -

brational l e v e l s ) . Our B i t t e r c o i l gives a f i e l d up to 8 Tesla, which allows the observation of numerous t r i p l e t v i b r a t i o n a l l e v e l s w i t h i n 7 cm-1 of the s e l e c t i v e l y laser excited s i n g l e t level of S1. The ground e l e c t r o n i c state (So ) of glyoxal (or methylglyoxal) is a sing l e t , and the f i r s t first

t r i p l e t s t a t e , T1, is at lower energy than the

s i n g l e t excited state S1. At the energy of the v i b r a t i o n l e s s

l e v e l of $1, the corresponding

v i b r a t i o n a l density of states of

t r i p l e t s , T1, is ~- 1 l e v e l per cm-1 for g l y o x a l , a n d S 2 0 l e v e l s per cm-1 for methylglyoxal (see f i g .

4). A supersonic j e t of glyoxal i s

locates within the 100 mm bore diameter of the B i t t e r c o i l . A single r o t a t i o n a l (N = O) v i b r a t i o n a l level of S1 is s e l e c t i v e l y excited by c.w. ring dye laser. In contrast with an optical or microwave spectrum, a l l of the anticrossings have the same amplitude (a decrease of up to 50 % of the fluorescence i n t e n s i t y ) . The only "missing" a n t i crossings are due to magnetic f i e l d inhomogeneity broadening (1 Gauss : 0.7 MHz), natural linewidth (23), and overlapping anticrossings. Consequently, we expect only a few missing l e v e l s in the a n t i crossing spectrum. For glyoxal, the width of the anticrossings varies over three orders of magnitude, f r o m ~ 1 gauss to mm103 gauss. These widths are related to the magnitude of the s i n g l e t - t r i p l e t s p i n - o r b i t vibronic coupling. The e x c i t a t i o n of the r o t a t i o n l e s s N = 0 s i n g l e t l e v e l s avoid the

superposition of several anticrossing spectra which cor-

respond to d i f f e r e n t Zeeman sublevels. According to the f i r s t

order

s i n g l e t - t r i p l e t anticrossing selection rules due to s p i n - o r b i t vibronic coupling(24), only t r i p l e t

(S = 1) r o t a t i o n a l level with N = 1,

MN = ~ 1 , Ms = ~ 1 (and K = 0 or K = 1 according to the v i b r a t i o n a l symmetries) can anticross the N = 0 excited s i n g l e t . The d i s t r i b u t i o n of the widths (as contrasted with the d i s t r i b u t i o n of l i n e i n t e n s i t i e s in the spectra considered previously) r e s u l t s in a more complex shape for the "slow component" of the F.T. of the spectrum.

87

(b (-

bq

GJ 0

3O J

0

magnetic

field

Testa

tie

!

0

1

5

t/~

Fig. 4 : 0 to 8 Tesla anticrossing spectrum of Methylglyoxal and the unsmoothed I F.TI of this spectrum. The arches are due to hyperfine doublets of triplets levels which are separated by 28 Gauss, The smoothed I F'TI near the origin displays a correlation hole for t/~ ~ 0.05.

88

The

F.T. of the a n t i c r o s s i n g spectrum of glyoxal does not d i s -

play any " c o r r e l a t i o n hole", we conclude t h a t there is no s i g n i f i c a n t c o r r e l a t i o n property between the v i b r a t i o n a l l e v e l s of the T1 e l e c t r o nic state at 3500 cm-1. At t h i s energy, above the v i b r a t i o n l e s s level the v i b r a t i o n a l density of states is 1.5 l e v e l s per cm-1 per symmetry. In c o n t r a s t , s i g n i f i c a n t c o r r e l a t i o n properties are observed between v i b r a t i o n a l l e v e l s of the TI e l e c t r o n i c state of methylglyoxal at 3000 cm-1 above the v i b r a t i o n l e s s l e v e l . The 0 to 8 Tesla a n t i c r o s s i n g spectrum of methylglyoxal (CH3-CO-CHO) of f i g u r e 4 looks l i k e noise ! However ail

the features

of t h i s spectrum are reproducible. There are approximately 5000 a n t i crossings in t h i s spectrum d e t a i l l e d below : -

The F.T. of t h i s spectrum displays arches ( f i g .

4) which are

due to hyperf~ne doublets ( s p l i t t i n g of 28 gauss), induce by Fermi contact i n t e r a c t i o n . These arches correspond to d e t e r m i n i s t i c propert i e s of the spectrum. - Furthermore, the Ms = + 1 and Ms = - 1 a n t i c r o s s i n g spectra are superposed. - We do not know a l l

the a n t i c r o s s i n g s e l e c t i o n rules f o r methyl-

glyoxal but, assuming the same kind of i n t e r a c t i o n

(spin-orbit-

v i b r o n i c ) as in g l y o x a l , the observed a n t i c r o s s i n g i s composed of both N = 1, K = 0 and N = 1, K = 1 components. These component may be uncoupled or coupled. - The two v i b r a t i o n a l symmetry classes of the Cs point-group of methylglyoxal (or the classes of the corresponding G6 molecular symmetry group) (25) may also give two superposed a n t i c r o s s i n g spectra. As a r e s u l t , between four and sixteen sets of independent symmetry are contained i n our a n t i c r o s s i n g spectrum. A " c o r r e l a t i o n hole" appears in the smoothed F.T. of the a n t i crossing spectrum ( f i g u r e 4) at O ~ t ~ 0.05. The estimated shape of the "slow component" i s also drawn. The width of the very narrow " c o r r e l a t i o n hole", 0.05,

(instead of 1 for one G.O.E. spectrum,

see

above) i s due to the superposition of several "pure" a n t i c r o s s i n g spectra, as explained in chapter IV : the F.T. of a spectrum composed of m " c o r r e l a t e d " spectra displays a " c o r r e l a t i o n hole '! which is diminished in width by a f a c t o r of m. Furthemore,

a d e t a i l l e d inspection of the F.T. near the o r i g i n

shows that the c o r r e l a t i o n s disappear f o r large L (L ~.r I00) or conversely f o r small times t ~ 0 , 0 1

( i n u n i t of reciprocal mean spacing).

89 Nevertheless, the existence of the " c o r r e l a t i o n hole" d e f i n i t i vely shows s i g n i f i c a n t c o r r e l a t i o n s properties in t r i p l e t v i b r a t i o n a l l e v e l s of methylglyoxal.

Summary and general conclusions. The range of the number of degrees of freedom, the available range of density of states, and a v a r i a t i o n in the coupling strength with energy make polyatomic molecules a f e r t i l e t e s t i n g ground for the study of c o r r e l a t i o n properties in physical systems. We are in the early age of the experimental study of the s t a t i s t i c a l properties of these v i b r a t i o n a l l e v e l s . Only a few experimental molecular spectra display s i g n i f i c a n t c o r r e l a t i o n properties. According to the discussion presented in chapters I to I l l ,

only two step

state to state process, l i k e o p t i c a l - o p t i c a l , 'or microwave-optical or o p t i c a l - a ~ t i c r o s s i n g , are able to produce spectra with the spectral p u r i t y required for the analysis of s t a t i s t i c a l c o r r e l a t i o n properties. The understanding of these c o r r e l a t i o n properties may provide new insignts for molecular dynamics such as intramolecular v i b r a t i o nal r e d i s t r i b u t i o n and quantum chaos. The new technique of F.T. described in section IV is a crucial progress in analysis because i t enables one to study large amounts of data at chemically relevant energies and complexities. The existence of f i n i t e range of c o r r e l a t i o n properties (and the p o s s i b i l i t y of measuring very long streches of l e v e l s in molecul a r physics) is a new feature not contained in G.O.E., which, by construction, have an " i n f i n i t e " range of c o r r e l a t i o n properties. REFERENCES (1) T.A. BRODY, J. FLORES, J.B. FRENCH, P.A. MELLO, A. PANDEY and S.S. WONG. Rev. Mod. Phys. 5__33 385 (1981) and references cited w i t h i n . (2) M.L. MEHTA, Random Matrix (Academic, New-York 1967) (3) G.E. POWELL and J.C. PERCIVAL J. Phys. A : Math. Gen. 12 n°11 2053 (1979) (4) a. O. BOHIGAS, M.J. GIANNONI and C. SCHMIT, Phys. Rev. Lett. 52 1 (1984) b. E. HALLER, H. KDPI~I~L and L.S. CEDERBAUM Phys. Rev. Lett. 62 1665 (1984) c. T.H. SELIGMAN, J.~M. VERBAARSCHOT and M.R. ZIRNBAUER Phys. Rev. Lett. 63 215 (1984)

90 (5) T.H. SELIGMAN, J.J.M. VERBAARSCHOT and M.R. ZIRNBAUER J. Phys. A : Math. Gen. 18 2751 (1985) (6) J. KATO, J. Chem. Phys. 82

3020 (1985)

(7) R. JOST, Second Conf. on Quantum Chaos, Mexico 1986. (8) i . C.S. PARMENTER, J. Phys. Chem. 86 1735 (1982) 2. B. FOURMANN, C. JOUVET, A. TRAM~'I~, J.M. LE BARS and P. MILLIE J. Chem. Phys. 92 25 (1985) (9) P. FELKER and A. ZEWAIL J. Chem. Phys. 82 2961, 2975, 2994, 3003 (1985) (10) L. LEVIANDIER, M. LOMBARDI, R. JOST, J.P. PIQUE Accepted in Phys. Rev. L e t t . (1986) (11) J.M. DELORY and C. TRIC, Chem. Phys. 3, 54 (1974) (12) R.E. SMALLEY, L. WHARTON and D.H. LEVY J. Chem. Phys. 63, n°11, 4977 (1975) (13) E. HALLER, H. KDPPEL and L.S. CEDERBAUM J. Mol. Spectros. 111, 377 (1985) (14) E. HALLER, H. KOPPEL and L.S. CEDERBAUM, Chem. Phys. Letters 101 215 (1983) (15) K.K. LEHMANN and S.L. COY, J. Chem. Phys. 83 3290 (1985) (16) J.L. HARDWICK, J. Mol. Spectrosc.

109 85 (1985)

(17) K.K. LEHMANN, S.L. COY, M. LOMBARDI and J.P. PIQUE to be published (18) H.L. DAI, R.W. FIELD and J.L. KINSEY, J. Chem. Phys. 82, n°4, 2161 (1985) (19) M.I. DAVIS and E.J. HELLER J. Chem. Phys. 80 5036 (1984) and cited paper. (20) E. ABRAMSON, R.W. FIELD, D. IMBRE, K.K. INNES and J.L. KINSEY J. Chem. Phys. 80 2298 (1984) (21) S. MUKAMEL, J. SUE and A. PANDEY Chem. Phys. Lett. 105 134 (1984) (22) R.L. SUNDBERG, E. ABRAMSON, J.L. KINSEY and R.W. FIELD J. Chem. Phys. 83 466 (1985) (23) J. DEROUARD, R. JOST and M. LOMBARDI Journal de Physique Lettres 37, L 135 (1976) (24) P. DUPRE, R. JOST and M. LOMBARDI Chem. Phys. 91 355 (1984) (25) P.R. BUNKER in Vibrational Spectra and Structure Vol. 3 J.R. DURIG ed. (Dekker, New-York), 1975 and in Molecular Symmetry and Spectroscopy, Acad. Press, New-York 1979