Indian Journal of Chemistry
Vo1.39A, Jan-March 2000, pp.307-3 l 5
Classical resonances and their quantum manifestations
Srihari Keshavamurthy
Department of Chemistry, Indian Institute of Technology, Kanpur 208 0 1 6, India Email : srihari @iitk.ac.in.
Received 12 October 1 999; accepted 15 December 1999
The quantum manifestations, in terms of resonant eigenstates, of a classical m:n resonance are discussed. Recent work demonstrated, numerically, that the slope of the classical resonance line as determined by the parameters of the spectroscopic Hamiltonian leads to progressions of quantum states along the resonance channel. The progressions comprise isomorphic states appearing with a specific periodicity. The periodicity is related to the slope of the classical resonance line as viewed in the corresponding discrete quantum number space. In this paper we provide a proof for the observed periodicity. These progressions, independent of the dimensionality of the system, demonstrate a detailed classical-quantum correspondence for resonant systems.
1. Introduction
Resonances play a central role in the phenomenon of energy flow among the modes of a coupled system. In the context of classical mechanics it is well- known I that resonances govern the stability of a system and are the seeds of chaos. For instance, presence of a resonance implies the existence of a seperatrix and the associated fixed points of alternating stability. The seperatrix is an indicator of vastly different types of dynamics that can manifest in a dynamical system. Presence of several resonances leads to stochastic regions around the sepera
trices which can overlap to create widespread stochastic
ity2. This viewpoint forms the basis for a simple estimate, known as the Chirikov criterion3, of stochasticity thresh
old in multiresonant systems. Consequently, consider
able efforts have been made4 to understand the effects of resonances in dynamical systems.
In the quantum context, resonances play an equally important role and have been implicated4-6 in the phe
nomenon of intramolecular vibrational energy redistri
bution (IVR) in molecules. The relevance of IVR to our understanding of re'action dynamics7.8 has led to a sus
tained effort by researchers in studying the spectrum, eigenstates and dynamics of resonant systems9,1O. The evidence for IVR comes8 from the study of molecular spectra at high energies which show that a description based on the equilibrium, normal mode - rigid rotor picture I I is inaccurate. Moreover, it was recognized that the experimental spectra corresponding to high excita
tions cannot be described with an effective spectroscopic Hamiltonian of the form (neglecting rotations)
Ho =
LW;
I('" + �).+
. _LL:X;;
I ,�.('" +�) (n; +�) +...
(1.1)with Xij representing the anharmonic corrections. Dy
namically, this implied the onset ofIVR with the zeroth
order states { Ini> } being strongly mixed. Thus, a satisfactory description of the high energy spectra en
tailed one to go beyond the attempts to correct the equation above to higher orders in {nj } . To obtain a reasonable description of the experimental spectrum it was necessary to introduce a coupling5 Hres /I between the different zeroth-order states { Ini>}. The form of Hm, /I in the semiclassical limit, corresponds to typical resonant terms in the classical Hamiltonian. For example, a Fermi resonant coupling between the symmetric stretch and bend modes of H20 is essential12 to reproduce the high energy vibrational spectrum of the molecule. In C2H2 one has to include a multimode resonant couplingl1 in order to account for the experimental spectruml4.
A crucial issue that arises from the experiments and the resulting resonant spectroscopic Hamiltonians is the nature of the energy spectrum, eigenstates and their assignments. In particular, from the standpoint of IVR, it is important to explore the quantum manifestations of the classical resonances. Two obvious candidates are the energy eigenvalues and the eigenstates of the resonant system. Considerable amount of work has been done in identifying signatures of resonances in the energy spec
trum6,15-22. A number of studies have associated avoided crossings in energy correlation diagrams with signatures of underlying resonances6,15. It was conjectured6,16 that
308 INDIAN J CHEM, SEC A, JAN-MARCH 2000
overlapping avoided crossings in the quantum system would correspond to overlapping resonances in the un
derlying classical system - a quantum Chirikov crite
rion.The conjecture has some supportl7.18 in the case of driven systemsl8, i.e., a zeroth- order time independent Hamiltonian perturbed by a periodic, time dependent potential. In contrast, it has been repeatedly established that states participating in an avoided crossing are not always classically resonantI9.21 . Alternative signatures of resonant behaviour in the energy spectrum have been suggested recently20.21 which are valid when the reso
nance widths are large enough to support a few quantum states. The conclusion from various studies on eigen
value correlations is that avoided crossings are not un
ambiguously related to classical resonant dynamics. The crux of the problem lies in the fact that at or near avoided crossings one does not have any idea of the competition between quantum and classical mechanisms of state mixing. In the absence of a clear, relative measure of the quantum and classical mixing the correspondence of avoided crossings to resonances is premature [see, how
ever, the second reference in (ref.22)] . There have been efforts2" however, to assign highly excited states by a diabatic correlation of the eigenvalues as one turns on or off the various resonances in a sequential fashion.
On the other hand, fewer attempts have been made from the standpoint of the eigenstates of the resonant system21.24-26. Although one does not have the benefit of dealing with observables, the eigenstate approach is crucial in identifying spectral patterns and hence possi
ble dynamical routes to IVR in the molecule. In this paper we are interested in capturing the distinctive fea
tures or patterns of the eigenstates which will identify them as being resonant. Since we are looking for the manifestation of classical resonances at the quantum eigenstate level, it is useful as well as relevant to make use of the underlying classical dynamics. There are two main advantages to this approach from the classical
quantum correspondence perspective. Firstly, the possi
bility to assign states in a dynamical fashion leads to insights in the dominant dynamical features of the mole
cule in a given energy regime. In particular, the dynami
cal nature of the highly excited eigenstates is very important from the standpoint of performing mode-spe
cific chemistry. Secondly, identifying and then quanti
fying intrinsically quantum mechanisms of IVR such as dynamical tunneling22 becomes feasible.
From the correspondence viewpoint it is important to know as to how the various classical invariant objects like periodic orbits, resonance channels and higher di-
mensional tori "scar" the quantum states. Scarring of quantum states by the underlying classiCal periodic or
bits has been a subject of intense study27. The scars of periodic orbits in resonant systems have been studied recently in both C2H2 (ref.25) and HOCI (ref.26). Such studies have led to a better understanding and a possible assignment of the quantum states. However, there are systems wherein periodic orbits are not sufficient21 and one needs the resonance channels as well as the higher dimensional tori to assign the eigenstates. The scarring of quantum states by resonance channels or zones is well known 10,24 and not entirely surprising. Nevertheless, ac
tive use of the resonance zones to understand and thus assign highly excited vibrational states has been a more recent developmentll . Essentially, all of the methods developed so farl3·28-JO are either inappropriate13·3o or difficult28,29 to apply/analyse for general multidimen
sional (three and higher) systems. The manifestations of classical resonances at the quantum state level have also been understood fairly well for low dimensional sys
tems. However, the application of these techniques to three or larger mode systems is either inappropriate or difficult to implement. Part of the difficulty lies in the fact that the classical dynamics of such systems is not as well developed as that of the low dimensional systems.
This makes the identification of the fingerprints of clas
sical resonance at the quantum state level, let alone the assignment of highly excited states, a difficult task.
The central problem to be solved is that of coming up with dimensionally independent techniques which faci Ii
tate the analysis of three or larger dimensional systems (a prerequisite for molecules and hence chemistry !). As a first step towards this objective, one asks for the details of the manifestations of classical resonances in the cor
responding quantum system. The hope is that it might be possible to identify signatures of a resonance which are robust and persist on increasing the dimensionality by the addition of other independent resonances. One po
tential candidate was identified in a recent study21 on H20. The classical-quantum correspondence for H20 revealed progressions of isomorphic quantum states, along resonance channels, with specific periodicities, The isomorphic nature of the states was demonstrated by computing the corresponding Husimi distribution func
tion31 and comparing with the appropriate classical Poin
care surface of section I . The study indicated that the scarring of eigenstates along a resonance channel was more intricate and had a specific pattern.
KESHAVAMURTHY : CLASSICAL RESONANCE AND QUANTUM MANIFESTATIONS 309
Exploitation of this feature led to the assignment2' of a surprisingly large number of states which included highly mixed states at very high energies. The possibility that such isomorphic states can aid in our understanding of the highly excited states and eventually, perhaps, IVR leads to the question as to whether such an observation holds for any arbitrary resonance. The answer, as shown in this paper, is affirmative and we provide a proof for the observed periodicity in an arbitrary, single m:n reso
nance case (Section III). In Section IV of this paper we provide an example in addition to discussing the regimes where the approach is useful and, preferably, comple
mentary to the existing techniques. In particular, we will argue that the progressions can lead to an understanding of the mixed eigenstates even in instances wherein other techniques 13.30 fail. Current work in progress in our group and future directions are outlined as well.
2. Hamiltonian for a resonant system
We are interested in characterizing the eigenstates of an effective spectroscopic Hamiltonian wherein two of the modes are involved in a m:n resonance. The roles of the rest of the nonresonant modes are understood fairly well from both classical and quantum mechanical per
spectives. Thus, without loss of generality, we will con
sider a 2-mode single resonance system. Multiresonant Hamiltonians, under suitable conditions, can be analysed using the methods developed for the single resonance case. Multimode resonances, important in their own right, will not conc.em us here, although results of this work can be extended to such situations.
A. Quantum Hamiltonian
The effective spectroscopic Hamiltonian for a single m:n resonance coupling has the form
"if = fIo + frnn , (2.1)
where Ho 1\ is the zeroth order Hamiltonian
1\ 1\
(2.2)
given in terms of the number operators (N" N2) for the
1\ 1\
zeroth order local modes with Nj == nj + 112. The nonlinear m:n resonant perturbation term is
(2.3)
where ak t and ak are the creation and destruction opera
tors for the two modes (k = 1 ,2) respectively. It is straightforward to check that the polyad5 operator 1\ 1\
pnm
=(n/m) N, + N2 commutes with the total Hamiltonian H 1\ and thus all of the eigenstates can be assigned. The operator
pmn
effectively decomposes the spectrum into blocks with each block labelled by an eigenvalue ofpmn.
The nature of the eigenstates can now be studied for a given polyad number.B. Classical limit Hamiltonian and the reduced phase space
The classical limit of H 1\ can be constructed via the standard correspondence32
(2.4)
where (h,ek) are the angle-action variables' for the clas
sical system. The resulting classical resonant Hamil
tonian has the form H(I,e) = Ho(I) + �(I, e) with
1\
Ho(l) = w,I, + WoJI2 + all I; + a22 Ii + "'2 hlz .
Ji"'"(I,O) =
fJ.JiFIf
cos(m8, -n(2) . (2.5b) (2.5a)The phase space counterpart to the polyad operator
pmn is Nmn = (n/m)/,+ h which commutes with H(I, e) in the Poisson bracket sense. As a result, Nmn is a constant of the motion and the system is integrable for any value of �. However, the phase space topology is quite differ
ent for �:;t 0 as compared to the � = 0 case.
The explicit reduction of the four- dimensional phase space to a two- dimensional one is achieved by consid
ering the generating function'
(2.6)
which generates the canonical transformation (I), e) -7 (I,Ja,,$,a) given by
a = 2' (mO, 1 -nO,) ; I, = '2 Jo ; m
(2.7a) (2.7b)
3 1 0 , INDIAN J CHEM, SEC A , JAN-MARCH 2000
It is clear that I can be identified with the polyad number
Nmn
since <j>, the variable conjugate to I, is cyclic.The zeroth-order Hamiltonian Ho can be written in the new variables as:
Ho(!, la) == C
-
. -21 (n + QI)la + If;' 4,
with
n = 1lW2
- 7nW
l ,(2.8)
(2.9a) (2.9b) (2.9c) (2.9d)
Thus, the original two-dimensional H(I, 8) is trans
formed to the one-dimensional Hamiltonian
(
21)"/2
H(I, Jo, ,p, a) = lIo(I. Jo) + (J
.r:/2
-; - JQ cos 2a, (2.1O)where
if
== (mm nn
2-(m+n»
1/2�. The energy and the polyad number provide two time independent constants of the motion and the system is integrable. Joyeux" has shown that for m + n :s; 4 it is possible to analytically determine action-angle variables appropriate for the torus quantization of the system.
3. Isomorphic resonant states
Previous work21 on the assignment of highly excited vibrational states of H20 molecule, using the resonance zones as a template, indicated the existence of sequences of quantum states with an approximate periodicity pro
gressing along the resonance zone. The periodicity arises by viewing the classical resonance center line in the corresponding quantum number (n/,n2) space. The slope of the classical resonance line will lead to intersections in the discrete (nl' n2) space. Although in a realistic system the periods of the intersections are nonintegraFI, we will assume otherwise for now. This amounts to assuming that the slope of the resonance line is integral (in general rational). The consequences of such an as
sumption are discussed in the next section. In this section we provide a proof for the existence of such isomorphic resonant eigenstates in the case of a general m:n single resonance Hamiltonian.
We start from the reduced Hamiltonian H(I,Ja.,<j>, a).
Assuming that the resonance coefficients are slowly varying functions of Ja, we can write H(I,Ja.,<j>,a) = E in the form:
J� - 2(J" + 2qcos2a. + DE -(; = 0 , (3.1)
where, � = y.1 (Q + Q I), q =
- A.
(Q + Q Irl, D = 2 (Q + Q I)-I, C = 2 C (Q + Q 1)-I , andA. �
the coefficient of the resonant term with Ja replaced by Ja i.e., the value at the resonance center. Quantization,4 of the reduced Hamiltonian is now performed with the choice
J <> -+ -z- + 1 . d dCY. . '
for the operator corresponding to J a.
This yields the Schrbdinger equation
cPlT! dlT!
-
da,2
-
2i( - 1) da. + (N-
2qcos2a)\lf=
0 ,(3.2)
(3.3)
with N==C-D E + 2 � . .. The nonlinear resonance results in a pendulum Hamiltonian operator which is also the paradigm for classical Chirikov analysis'. The Schrbdin
ger equation can be transformed to a Mathieu equation'S for the auxiliary function F(a) by substituting
'lira) = exp[i((
-
l)aJP(a) .Performing this transformation we obtain
cPF da2 + (a.
-
2q cos 2a)F = 0 ,(3.4)
(3.5)
where ay ==C -DE + �2, and v( ay,q) is the characteristic exponent or order of the solution. The range of v is related to the width of the classical resonance zone. It is well known from the theory of Mathieu equations that there exist Floquet solutions35 to the differential equation of the form
F.(a,) = exp(illa)P(a,) , (3.6)
with P( a) = P( a + n). Thus, the solution to the Schrbdin
ger equation can be written down as
KESHAVAMURTHY : CLASSICAL RESONANCE AND QUANTUM MANIFESTATIONS 3 1 1
'-
wv;[(a) = exp[i(v + ( - l)a]P(a) , (3.7)
where we have put the label I on the wavefunction to indicate that the solutions are for a given polyad number.
The boundary condition34.35 '¥v;I(a
+ IT) =
'¥vJ (a) has to be imposed on the wavefunction which leads to the relation v + S - 1=
2 n with n=
0, 1 ,2, ....The order v of the solution is fractional in generap5.
It is, however, convenient to think of v as labelling the possible states for a given value of the polyad number I.
At this stage, we explore the possibility of locating isomorphic states, i.e., states with the same v but at different value of I
=
I+
/). I. Such isomorphic states, described by the wavefunction '¥v;l + 6. /, have to satisfy the boundary condition '¥v:i+M(a +IT)
= '¥v:I+ M(a). This immediately results in the condition:exp[i(v + ( + b.( -1)7r] = 1 , (3.8)
where we have made use of the fact that S depends linearly on I. In general, although v also depends on I we will assume this dependence to be weak. This assump
tion is justified when the classical resonance width does not vary appreciably over the region of interest. How
ever, with v + S - 1 = 2n the condition for isomorphic states becomes:
exp(i7r6() = 1 . (3.9)
This is the central result of this section. The relation can be satisfied if /).s = 2
r
withr =
0, 1 ,2, .... Thus, one anticipates a sequence (or set) of isomorphic states (indexed by
r,
fixed v) which will have similar amplitude distributions in the (nl ,n2) zeroth- order basis but differ in the polyad to which they belong. We will call the state with r=0 as the parent state. The range ofr
depends on the parameters of the Hamiltonian. Many such sequences, characterized by different v, can exist. The total number of such sequences is a function of the strength of the resonant coupling.
Note that at this juncture we have a condition, equa
tion 3.9,which has to be satisfied for the existence of isomorphic states. However, it is not very clear as to where such states can be located in the (nl,n2) space.
Moreover, the classical-quantum correspondence for equation 3_9 is lacking. The issues concerned can be
sorted out by considering the classical condition for a m:n resonance. In fact, as shown below, such an analysis leads to the conclusion that r is intimately connected to the slope of the classical resonance line.
The classical resonance center line can be obtained I from demanding m:n commensurability between the zeroth-order frequencies,
aHo (!, 9) 8Ho (I, 9) m = n-=-'-.!..
OIl 812 (3. 10)
After some rearrangement the resonance line In I space can be written down as:
h = p (h -�),
where
_ nal2 - 2mall
p = ma12 - 2na22 ,
J? = nw2 -mwl
I - 2mau -nal2 '
(3_ 1 1)
(3.12a) (3_ I2b)
are the slope and II-axis intercept of the resonance center line respectively. It is straightforward to express the reduced Hamiltonian parameters (Q,y) in terms of the variables ( tJ ,n,f I) and one obtains the following rela
tions:
(3.13a) (3_I3b)
This allows us to determine S (tJ, n, I� ') and, In particular,
2 2 2
b.( = --b.I = -b.lt = -b.I2 _
mp + n m mp (3.14)
Thus, we see that r is related to the slope tJ of the classical resonance line by the relation (m tJ + n)
r =
/).I. Note that the classical analysis provides no indication of the range of v. This is not surprising since the range of v depends on the strength of the resonance and the classical analysis was based on the zeroth-order Hamil
tonian. A semiclassical estimate can be made for the range of v based on the width of the classical resonance zone. The important point, however, is that the integer r .
3 12 . . INDIAN J CHEM, SEC A, JAN-MARCH 2000
has a classical origin and provides a way to locate the isomorphic states. The quantum condition for isomor
phic states �
�
= 2r is combined with the classical expression for ��
to obtain the location of the isomorphic states in the (nl ,n2) space. The isomorphic states appear with a·polyad spacing of (m f.J + n). The existence of the isomorphic states is independent of the resonance strength since
�
is determined from the zeroth-order Hamiltonian.4.
Discussion and future directionsThe results of the previous section demonstrate the scarring of the quantum states by the classical reso
nances. It is important to point out that the periodicity f.J is a function of the parameters of the zeroth-order Hamil
tonian. For general m:n systems, it is quite possible for the many different sets of parameters to give rise to the same f.J but different JOI. The Hamiltonians with the same f.J then can be seen as belonging to a sort of equivalence class with respect to f.J . It would be inter
esting to study the classical-quantum correspondence for Hamiltonians belonging to a particular class.
The isomorphic sequences for I : I , 1 :2 and 2:3 reso
nances are clearly seen in the work of Roberts and J affe24 (Fig. 6). Here we will illustrate the resonant progressions using the following 2-mode Hamiltonian
(4.1)
as defined in section IIA, with al2 = O. Note that in the presence of both the 1 : 1 and the 1 :2 resonances, H is non integrable. The parameter values for the zeroth-order Hamiltonian are chosen as (01 = 1 .0,(02 = 0.8, al l = -0.04, and a22 = -0.02 in scaled units. With these values the slope f.J of the I : 1 and 1 :2 resonances are +2 and + I respectively. This is the simplest Hamiltonian to explore and analyse the existence of resonant progressions. For this particular choice of parameter values the intersec
tion of the two resonances in (I"h) space occurs at around ( 1 0.5, 18.0).
The eigenfunctions were generated by diagonaliz- ing H in the number basis Inl,n2). The eigenstates 1\
were obtained as a linear combination of the form Lnl,n2 Cnl ,n2 In"n2). In this paper we will represent the eigenstates graphically by plotting at every lattice point (nl ,n2) a circle with radius equal to 1C" I " d2 i.e., the square of the coefficient of the corresponding zeroth-order basis state in the eigenstate of interest. In some instances, for
Fig I - The resonant progressions for the 1 : 1 integrable subsystem are shown for plJ = 5 X 1 0.3. There are a total of eight eigenstates shown in this figure with each one constrained to lie along the II + 12= constant line (polyad number). The states in the progression are r = 0,1 , . . . , 7 with the periodicity f.J =
+2. The thick solid line is the I : I resonance center.
clarity of the figure, tre radii are uniformly scaled by a factor. In every figure showing the eigenstates the clas
sical (11'/2) space with the resonance center lines and the discrete quantum (nl,n2) space are superimposed.
In Fig. l we show a sequence of isomorphic states for the, classically integrable, 1 : I subsystem (�I I = 5 X 1 O-\�12 = 0) progressing along the resonance channel.
These correspond to the r ranging from 0 to 7. Note that every state is constrained to lie along the constant polyad,
1\ A
pl l == III + n2, line since [H , P I I ] =0 for �12 = O. The r=0 parent state is associated with a polyad value of 3. In accordance with the results from the previous section, we see the next isomorphic state
(r
= I ) coming at a polyad value of 6. In other words, the isomorphic states are spaced in polyad value by the quantity (m go + n) which is equal to 3 for a 1 : I resonance with slope go = 2. Clearly, the progressions come with a period of +2.In Fig.2, the isomorphic progression for the, classi
cally integrable, 1 :2 subsystem with �12 = 5 x 10-3 and
�I I = 0 is shown. The periodicity of + 1 is evident in the sequence corresponding to r = 0, ... ,6 i.e. , a total of seven states. In this case the states lie along the constant polyad,
12 1\ 1\
P == 2 III + n2, line since [ H , p12] = O. The isomorphic states are spaced in polyad value by 3 as expected for this case. There are many more families of such progressions for the larger resonant coupling values resulting in a
KESHAVAMURTHY : CLASSICAL RESONANCE AND QUANTUM MANIFESTATIONS 3 1 3
5 . 0
15
Fig. 2 - The resonant progressions for the 1 :2 integrable subsystem are shown for �J2 = 5 x 1 0-3. There are a total of seven states shown in this figure with each one constrained to lie along the 21J + h = constant line (polyad number). The states in the progression are r = 0, 1 ,,,., 6 with the periodicity f.J = + I . The thick solid line is the 1 :2 resonance center.
15
10
5
o o
. '//0 0
"j
0/�
· OJD
0;/
1
0 0 0. / . 0 0 • 0
// /
. / . . I . 0 0
/ /
/ . /
/
/.
/ :/
/.
5 10 15
Fig. 3 - An example of a highly mixed state for thc nonintegrable Hamiltonian with �l l = �l� = 5 X 1 0-3. There is only one state in this figure and it has no polyad constraint. There is, thus, extensive delocalization in the plane The dotted line is the I : I
resonance center and the dashed line is the 1 :2 resonance center.
m : n
It
Fig. 4 - A sketch for the general case of multi resonant 2-mode systems. The thick solid lines are the various resonance center lines. The dotted lines represent resonant progressions. R denotes resonance overlap regions wherein mixed states oc
cur. Note that away from the overlap region the states appear with the same periodicity.
larger range for v. Every one of the family of progres
sions respects the corresponding periodicity. Thus the entire set of resonant states for the integrable subsystems can be considered to comprise the various families of isomorphic resonant progressions.
The full Hamiltonian with both the resonant couplings having non-zero values is classically non-integrable.
Thus, highly mixed states are observed in the resonance overlap region. An example of a mixed state is shown in Fig.3 for the resonance strengths �ll = �12 = 5 X J O-3.
Although the quantum state is away from the resonance intersection, the classical resonance zones have over
lapped. Moreover, there is no polyad number and the only conserved quantity is the energy. This is the reason for the state amplitude to be smeared over the two dimensional plane in contrast to the integrable, single resonance cases discussed earlier. This state can be con
jectured to be a part of one of the isomorphic I : I se
quence (cf. Fig. I with r=6) and, perhaps, a different 1 :2 isomorphic sequence. A ciear way to confirm this con
jecture is to compute and compare the appropriate Hu
simi functions. This, more detailed, study is in progress.
It is, however, easy to see that the significant overlap of the two resonances has led to an extensive delocalizatioll of the eigenstate in the action plane.
3 1 4 INDIAN J CHEM, SEC A , JAN-MARCH 2000
A large number of the mixed states can be understood from the resonant progressions viewpoint. A sketch of the essential idea is shown in the Fig.4. The main as
sumptions are that the width of the resonances are not very large and that the Chirikov analysis is valid. The mixed states that arise in and around the region marked R can be analysed from the sequence of states progress
ing along the various resonance zones. Away from the resonance junctions R, states always appear with the same periodicity. Clearly, the mixed states in and around R can be understood from the corresponding parent state
(r
= 0). The assumptions involved are similar to those in the existing approachesl3Jo. However, an important distinction is that the progressions lead to insights in the resonance junction regions. In the regions wherein iso
lated resonances exist, one can define local polyad num
bers. In the approach of Kellman 13 the nature of resonant dynamics is explored for a given polyad number. The progressions, on the other hand, are exploring "isomor
phic" resonant dynamics at different polyad numbers.
Thus, although the polyad number ceases to be a good quantum number around the resonance junction, the resonant progressions provide avenues to understand the nature of eigenstates precisely in such non-trivial and interesting regions. A possible manifestation of such isomorphic dynamics can be found in the recent quan
tum-classical correspondence study, by Jacobson et aUS, of the highly excited C2H2 spectrum. The authors25 found that periodic orbits (actually 2-tori in the full space) in a mixed phase space region could be traced, through a chaotic phase space regime,. to the periodic orbits at lower energies, with nearly integrable dynamics.
Clearly, it would be useful to study the acetylene Hamil
tonian from the progressions point of view.
The main advantage of the proposed approach is the possibility of assigning "mixed" states. The periodicity go of a given resonance is independent of the dimension
ality of the system i.e., unaltered by addition of other independent resonances. Thus it is possible to extend this approach to true, multi mode systems. The disadvantages are twofold. Firstly, the need to be in the Chirikov3 regime might invalidate the approach in certain cases.
For example, in previous work21 on H20 it was observed that reducing the bend anharmonicity implied existence of nonchirikov regions. The corresponding quantum states were quite difficult to classify using the resonance template. This is not so surprising since go essentially involves the anharmonicities of the two modes partici
pating in the resonance. Thus reducing the anharmonic
ity in either mode could lead to a go either too small or
too large. Work is in progress in our group to investigate such systems. The second disadvantage has to do with the visualization. For example, in 3-mode systems, one has to choose appropriate projections of the resonance lines onto some two-dimensional I subspace. It is not quite clear as to how to choose such projections in a useful manner. Multimode resonances provide new chal
lenges as well. The resonant progressions have proved very useful in understanding the quantum states of 2- mode37, and quasi 2-mode systems21 . However, the real testing ground would be a true 3-mode system wherein important dynamical (both quantum and classical) phe
nomena, distinct from 2-mode cases, emerge 1 • It remains to be seen if the resonant progressions can facilitate our understanding of the fingerprints of such phenomena, as encoded in the quantum spectrum and states.
5. Acknowledgement
It is a pleasure to thank Greg Ezra for many enlight
ening discussions on this subject. Part of this work was done during a summer visit to the Raman Research Institute and I thank them for their hospitality. This work is supported by grants from the Department of Science and Technology and the Council of Scientific and Indus
trial Research.
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I Lichtenberg A J, & Lieberman M A, Regular and sloc/zaSlic motion (Springer-Verlag), 1 983.
2 Zaslavsky G M, Physics of chaos in Hamillonian systems, (Imperial College Press, London), 1 998 and references therein.
3 Chirikov B V, Phys Rep, 52 ( 1 979) 263.
4 See references in Section 4 of Uzer T, Phys Rep. 1 99 ( 1 99 1 );
Ezra G S, Adv Class Traj Meth, I ( 1 992) I and references therein.
5 Lehmann K K, Scoles G & Pate B H, Annu Rev Phys Chern, 45 ( 1 994) 241 ; Kellman M E, Annu Rev Phys Chern, 46 ( 1 995) 395.
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