26 February 1999

Ž .Chemical Physics Letters 301 1999 217–222

Multifractal analysis for the eigencoefficients of the eigenstates ofhighly excited vibration

Jin Yu, Songtao Li 1, Guozhen Wu ) ,2

Department of Physics, Tsinghua UniÕersity, Beijing 100084, PR China

Received 6 October 1998; in final form 28 December 1998

Abstract

The multifractal nature of the eigencoefficients of the eigenstates of the highly excited vibration of H O, D O and H S2 2 2

is probed. The Hamiltonian includes the two stretches and the bend with anharmonic interaction and multiple resonances. Itis shown that there are eigenstates possessing multifractal structure but not all. As the polyad number increases, largerpercentages of eigenstates become multifractal. The self-similar structure that exists in the distribution of the eigencoeffi-cients of the eigenstates as displayed on the zero-order state energy axis is also demonstrated. q 1999 Elsevier Science B.V.All rights reserved.

1. Introduction

Fractal interpretation for the natural objects hasw xnow been well accepted 1,2 . This interpretation is

not only limited to geometric aspects; it can beextended to more abstract realms such as probability,

w xstatistics 2 . Its application in physical phenomenaw xsuch as percolation and phase transition 3 is notice-

able. The observation of fractal structure in molecu-w xlar spectrum 4 , the eigenstates in disordered sys-

w xtems 5 , quasi-energy states in intense polychro-w xmatic fields 6 as well as in electronic wave func-

w xtions in model solids 7,8 is important since thisshows that in the quantum systems, there can bestructures that are fractal. The recent reports of the

) Corresponding author. E-mail: wgz-dmp@tsinghua.edu.cn1 Currently, at the Department of Automation, Tsinghua Uni-

versity.2 Researcher of the Center for Advanced Study, Tsinghua

University.

w x w xfractal 9 and multifractal 10 structures of highlyexcited vibration imply that fractal concept can be auseful tool for studying the characteristics of thecomplicated dynamical structure of highly excitedvibration. This is important since experiments suchas stimulated emission pumping and dispersed fluo-

Ž w x.rescence techniques see, e.g., Ref. 11 have nowmade it possible to reach the realm of highly excited

Žvibration and many of its peculiar properties such asw x w x .assignment 12 , chaos 13 , only to name a few

which are related to its multi-resonances and nonlin-w xear effect 14 are of current concern and remain yet

to be probed. Indeed, fractal is another importantfacet for it. In this work, the multifractal structure forthe eigencoefficients of the eigenstates of tri-atomicmolecules H O, D O, H S whose multi-mode2 2 2

Hamiltonian is of an algebraic form with coefficientselucidated from the fit to the experimental spectraldata will be demonstrated. The purpose is to showthat multifractal structure does exist in the eigencoef-ficients of some, but not all, of the eigenstates of

0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved.Ž .PII: S0009-2614 99 00028-7

( )J. Yu et al.rChemical Physics Letters 301 1999 217–222218

highly excited vibration with a realistic Hamilto-nian. Hence, multifractal is not trivial for an eigen-state. Furthermore, the self-similar structure that ex-ists in the distribution of the eigencoefficients of theeigenstates as displayed on the zero-order state en-ergy axis will also be demonstrated.

2. Multifractal for the eigenstates of highly ex-cited vibration

2.1. Multifractal

Fractal is a measure to characterize an objectwhich often possesses self-similar structure. This isreflected in the following relation:

A L fLd f , 1Ž . Ž .where L is the measuring length scale comparativeto the object, A is the measure and d is thef

characteristic of the measure, known as the dimen-sion of the object. When d is not an integer,f

contrary to the usual Euclidean case, the object iscalled fractal and d the fractal dimension. Thisf

happens mostly often where there is a self-similarstructure. Suppose now a support which can be a lineor a fractal with the generator segments of differentlength r . On its every segment there is an associatedi

probability p and the distribution of the probabilitiesi

shows, instead, a distribution of fractal dimensions.w xThe situation is called multifractal 15 . Multifractal

can be described by two scaling exponents, one forthe supporting fractal, t and the other for the proba-bility distribution, q via defining a partition functionas shown below:

G q , t , r sÝ pqrt , 2Ž . Ž .i i

Ž .where t q is defined by requiring G q , t q , r sŽ .Ž .Ž . q1. As r sd is constant, let x q be Ý p , then:i i

x q sdyt . 3Ž . Ž .Ž .The slope of the linear plot of log x q against

y1 Ž . Ž .log d will offer us t q . t q is related to theŽ . Ž Ž .generalized dimension D q for the support D q ,

as qs0, is equal to the aforementioned fractal.dimension of the support as:

t q s 1yq D q , 4Ž . Ž . Ž . Ž .

with q the moment order which is not necessarily aninteger. Another convenient description of a multi-fractal is by the so-called ‘strength of singularity’Ž .f a . The singularity exponent a is defined via:

p sd a , 5Ž .i

a denotes the density distribution due to the associ-ated probabilities and f is the corresponding fractal

Ž .dimension for the distribution. f a is, in fact, themultifractal spectrum. Thus, depending on a whichrepresents a given probability, there is the fractaldimension for the support of that probability. Notethat the subsets with the given probability that corre-spond to a given f are scattered all over the supportand are intimately intertwined with those of otherdimensions. The number of the subsets that containthe probability represented by a will vary with d

like dyf Ža .. Therefore the number of the subsets ofŽ . yf Ža . Ž Ž .size d and type a , is r a d da with r a a

. Ž . q Ž .regular function . As x q is Ý p d is constantiŽ .with 5 one has:

x q s r a dyf Ža .qqa da . 6Ž . Ž . Ž .HFor small values of d the integral above is domi-nated by the minimum of the possible exponents.

Ž . Ž . Ž .With 3 , f a and t q can be found related toeach other via the Legendre transform as shown

Ž . w xbelow. As long as t q is obtained, one has 16 :

a q sydrd q t q 7Ž . Ž . Ž .and

f a q st q qqa q 8Ž . Ž . Ž . Ž .Ž .and finally, the spectrum f a . This relationship

also reflects a deep connection with the thermody-namic formalism of equilibrium statistical mechanics

Ž .where t q and q are the conjugate thermodynamicŽ .variables to f a and a .

2.2. The multifractal analysis

If one can succeed in building a vibrational eigen-state c from the zero-order state f with well-de-i

Žfined quantum numbers or actions in the classical.language , i.e.,

csÝC f , 9Ž .i i

( )J. Yu et al.rChemical Physics Letters 301 1999 217–222 219

Table 1The coefficients for the algebraic Hamiltonian

v v X X X X K K Ks b ss bb st sb st DD sbb

H O 3890.6 1645.2 y82.1 y16.2 y13.2 y21.0 y42.7 y0.1 y14.52

D O 2836.6 1204.6 y44.1 y7.0 y10.5 y13.1 y54.6 y0.0 y5.82

H S 2731.9 1225.6 y48.4 y8.9 y2.9 y21.9 y7.7 y0.2 y19.92

See text for definitions. Units are cmy1.

< < 2then C is the ‘probability’ for c on f with thei i

zero-order state energy ´ . The support then can bei

considered as the energy axis as demonstrated in theSection 3. For those points on the energy axis notcorresponding to the energies of the zero-order states,their probabilities are set as zero. This forms thebeginning of the possible multifractal analysis of theeigenstates.

In our cases of the highly excited vibration of< :H O, D O and H S, f is considered as n , n , n2 2 2 i s t b

with n , n the quantum numbers of the two stretchess tŽand n of the bend. Hereafter, subscripts s, t and bb

.refer to the stretches and bend, respectively. Theenergy of the zero-order state is taken as:

v n qn q1 qv n q1r2Ž . Ž .s s t b b

2 2qX n q1r2 q n q1r2Ž . Ž .ss s t

2qX n q1r2 qX n q1r2 n q1r2Ž . Ž . Ž .bb b st s t

qX n qn q1 n q1r2 , 10Ž . Ž . Ž .sb s t b

where v and X denote the Morse frequency andanharmonicity, respectively. The full Hamiltonianrequires the resonances as shown below:

K aqa qh.c. qK aqaqa a qh.c.Ž . Ž .st s t DD s s t t

qK aqa a qaqa a qh.c. 11Ž .Ž .sbb s b b t b b

where K and K terms depict the resonancesst DD

between the two stretches and K term the Fermisbb

resonance between the stretches and the bend. Theseresonances are in the second quantization languagewith aq and a the creation and destruction opera-a a

tors. The coefficients are listed in Table 1 from thew xfit with the experimental spectral levels 17 by the

w xMarquardt method 18 with standard deviation nomore than 10 cmy1, mostly down to 5 or 1 cmy1.The energy levels for fitting range from the ground

Ž y1 .level for H O, its energy is 1589.5 cm up to n ,2 sŽ 3n or n r2 equal to 7 with energy around 25=10t b

y1 .cm . These are the experimental data availabletoday. The details of the fit will be discussed sepa-

w xrately in another paper 19 . The diagonalization ofthe full Hamiltonian in the zero-order state space< :n ,n , n is straight. For this Hamiltonian, thes t b

quantum number, n qn qn r2, which is calleds t b

the polyad number P, is conserved. The size of thebasis set is 120. This comes from the fact that there

Ž .Ž .are Pq1 Pq2 r2 states for each P and forPs0 to 7. Therefore, we will speak of eigenstatescorresponding to a polyad number. The energy rangefor the validity of the Hamiltonian, in fact, dependson the experimental data available from which itscoefficients were elucidated. However, this shouldnot affect much for the following multifractal analy-sis. Though we will still employ the coefficientsdetermined in the lower energy range for those levels

Ž . Ž . Ž .Fig. 1. The eigenenergy patterns of: a H O; b D O; and c2 2

H S, with the polyad number 30.2

( )J. Yu et al.rChemical Physics Letters 301 1999 217–222220

Ž . Ž . y1 Ž .Fig. 2. The multifractal quantities a log x q rlog d with qs2. The line shows the linear regression with slope t 2 sy0.724. TheŽ . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .fit correlation factor is 0.984. b t q . c D q . d a q . e f q . f f a for the 194th eigenstate as counted from the lowest one with

energy 62014 cmy1 of H O. The polyad number is 20. See text for details.2

Ž .of much higher energy range, the qualitative resultsand conclusions reached as shown below should bestill preserved if one day the higher experimentalenergy levels are available and the coefficients forthe Hamiltonian can be more accurately determinedthereby since the analysis does not rely on the exactvalues of the coefficients.

With the elucidated C for each eigenstate, thei

following qth moment of the set of box distributionon the energy axis can be calculated:

x q s B q , 12Ž . Ž .Ý kk

where B is the sum of the probabilities in box kk

with an interval d , i.e.,

< < 2B s C . 13Ž .Ýk k ji

Index j shows those zero-order states falling in boxk. In practical calculation, d ranges around 400–1000

y1 < < 2 y12cm and those C -10 are set to zero toi

avoid numerical overflow for very negative q. IfŽ . y1linearity relationship between log x q and log d

Ž .exists, then the slope t q shows the exponent forthe multifractal. The criterion for the linearity is that

( )J. Yu et al.rChemical Physics Letters 301 1999 217–222 221

Žthe correlation factor is not -0.95 for q ap-.proaches "20, the correlation factor is not -0.90 .

3. Calculational results

Fig. 1 shows the eigenenergies of H O, D O and2 2

H S on the energy axis with the quantum number2

n qn qn r2, called the polyad number being 30.s t b

For this polyad number, there are 496 eigenstates.Among them, not all possess the multifractal charac-ter. For H O system, there are 176 eigenstates pos-2

sessing this character. For D O and H S, there are2 2

143 and 159 eigenstates, respectively. Hence, multi-fractal is not trivial for an eigenstate. For these threesystems, it is seen that the eigenstates with multifrac-tal are mostly in the region of higher density ofstates. Fig. 2 shows the multifractal quantities

Ž . y1 Ž . Ž . Ž . Ž . Ž .log x q rlog d , t q , D q , a q , f q , f a ofthe 194th eigenstate as counted from the lowest oneof the H O system with Ps20, taken as an exam-2

ple. Fig. 3 shows the locations of the eigenstates inthe action space that possess this multifractal prop-erty for the H O system with Ps30. The actions or2

quantum numbers for the eigenstates are simply as-signed by those of the zero-order states which are inone-to-one correspondence with the eigenstates withboth arranged in the sequence of energy. This assign-ment is not so much of physical consideration. In-

Fig. 3. The distribution of the eigenstates that possess multifractalproperty in the action space for H O of polyad number 30. See2

text for details.

Fig. 4. The self-similar structure in the distribution of the eigenco-efficients on the zero-order state energy axis of the 157th eigen-state of H S with polyad number 30. The units are cmy1. The2

dashed lines show the profiles of the magnitudes of the eigencoef-ficients for better viewing the self-similar structure.

stead, it is technically convenient since the shift of astate energy due to resonances is much less ascompared with the energy of its zero-order state. Fig.3 shows that those eigenstates with multifractal aremostly with closer n , n and n , i.e., they are thes t b

interior states in the action space corresponding to apolyad number. It is also noted that as the polyadnumber is larger, larger percentages of eigenstatesmay possess this multifractal property.

( )J. Yu et al.rChemical Physics Letters 301 1999 217–222222

Fractal in quite often the reflection of self-similar-ity. Indeed, we see that for an eigenstate, its distribu-

< < 2tion of C on the zero-order state energy axisi

sometimes shows that there are ‘bumps’ within‘bumps’: the character of self-similarity. This isshown in Fig. 4 for the 157th eigenstate of the H S2

system with polyad number 30, taken as an examplefor demonstration.

4. Concluding remarks

In this work, with a realistic algebraic Hamilto-nian whose coefficients were elucidated from the fitto the experimental spectral observation, the multi-fractal nature of the eigencoefficients of the eigen-states of the highly excited vibration of H O, D O2 2

and H S is probed. The Hamiltonian includes two2

stretches and one bend with anharmonic interactionand multiple resonances. It is shown that there areeigenstates with multifractal structure but not all.Hence, multifractal is not trivial for an eigenstate. Asthe vibrational system is of larger polyad numbers,larger percentages of the eigenstates will possess thismultifractal property.

This multifractal nature of the eigenstates of thehighly excited vibration indeed deserves our atten-tion. As known, highly excited states possess pecu-

Ž .liar properties such as chaos due to their multi-di-mensionality and prominent nonlinear effect. Multi-fractal is indeed their another important facet. An-other important observation is the self-similar struc-ture in the distribution of the eigencoefficients of theeigenstates as displayed on the zero-order state en-ergy axis. Evidently, the eigencoefficients possessthe geometric interpretation such as multifractal andself-similarity in addition to the traditional interpreta-tion of probability. The idea of eigencoefficientsoriginates from the quantum theory. Their geometric

interpretation of multifractal and self-similarity maybe related to the very essence of the wave functionsof the highly excited systems. This is indeed animportant topic for future study.

Acknowledgements

This work was supported by a fund from theNational Natural Science Foundation of China. GWwould also like to thank the Center for AdvancedStudy at Tsinghua University for its support.

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