Eigenstates of Thiophosgene Near the Dissociation Threshold: Deviations From Ergodicity

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Article

Eigenstates of Thiophosgene Near the DissociationThreshold - Deviations From Ergodicity

Srihari KeshavamurthyJ. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp4033386 • Publication Date (Web): 07 May 2013

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Eigenstates of Thiophosgene Near the DissociationThreshold - Deviations From Ergodicity

Srihari Keshavamurthy∗

Department of Chemistry, Indian Institute of Technology, Kanpur, Uttar Pradesh 208016, India

E-mail: [email protected]: +91 512 2597043. Fax: +91 512 2597436

KEYWORDS: Highly excited eigenstates, Thio-phosgene, Quantum ergodicity, Vibrational energyflow, Eigenstate thermalization, Level-velocities,State space, Intensity-velocity correlation, Non-statistical

∗To whom correspondence should be addressed

AbstractA subset of the highly excited eigenstates of thio-phosgene (SCCl2) near the dissociation thresholdare analyzed using sensitive measures of quantumergodicity. We find several localized eigenstates,suggesting that the intramolecular vibrational en-ergy flow dynamics is nonstatistical even at suchhigh levels of excitations. The results are consis-tent with recent observations of sharp spectral fea-tures in the stimulated emission spectra of SCCl2.

IntroductionThe statistical Rice-Ramsperger-Kassel-Marcus(RRKM)1 theory of reaction rates occupies a cen-tral place in the field of reaction dynamics bothfor its elegance and simplicity. Indeed, within theRRKM approximation reaction rates can be cal-culated2 irrespective of the intricate intramolec-ular dynamics that happen prior to the reaction.In recent years, however, the appearance of sev-eral examples3–9 of intrinsically non-RRKM reac-tions have rekindled an old question10,11 - whenis a system expected to be in the RRKM regime?From one perspective, answering the question re-quires understanding the nature of the moleculareigenstates near reaction thresholds. In particu-lar, the situation wherein all the eigenstates aresufficiently delocalized seems ideal for the valid-ity of the statistical approximation. On the otherhand, existence of localized eigenstates at the re-action threshold would suggest strong deviationsfrom the RRKM regime. Consequently, there have

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been many studies12–20 aiming to characterize thenature of highly excited eigenstates in differentsystems. Recent reviews21,22 highlight the re-lationship between eigenstate assignments basedon classical phase space structures and deviationsfrom the RRKM regime.

The above arguments, more precisely, are con-nected to a question that has confounded re-searchers for nearly a century - what constitutesa quantum analog of the classical ergodic hy-pothesis in an isolated many body quantum sys-tem? Progress towards answering this question hasmainly come from semiclassical analysis of quan-tum chaotic models,23 giving rise to the notionsof weak quantum ergodicity24,25 and quantumunique ergodicity.26 Recently, an approach thathas attracted substantial attention is the so calledeigenstate thermalization hypothesis (ETH).27 Ac-cording to ETH, for an isolated quantum sys-tem governed by the Hamiltonian H with eigen-states |α〉, the eigenstate expectation value Vαα ≡〈α|V |α〉 of an observable V changes slowly andsmoothly with the state.27–29 Specifically,

〈α|V |β 〉 ≡Vαβ = V (Eα)δαβ +O(h( f−1)/2) (1)

with the O(h0) leading term of V (Eα) being theclassical microcanonical average24,25

〈V 〉mc(Eα) =

∫V (q,p)δ [H(q,p)−Eα ]dqdp∫

δ [H(q,p)−Eα ]dqdp(2)

In the above equation, f is the number of degreesof freedom and V (q,p) is the classical symbol cor-responding to the quantum V . Furthermore, con-sider the long-time average of the time-dependentexpectation value 〈 j(t)|V | j(t)〉 ≡ 〈V (t)〉

V ≡ limt→∞〈V (t)〉= ∑

α

|C jα |2Vαα (3)

in some initial nonstationary state | j〉=∑α C jα |α〉with mean energy E j. Since ETH implies that Vαα

is approximately constant in an appropriately cho-sen energy window of width ∆E one can write

V = 〈V 〉mc(E j)

≡ 1N∆E

|E j−Eα |<∆E

∑α

Vαα (4)

with N∆E being the number of eigenstates withinthe energy window. These predictions of ETH,along with the effect of finite state space, havebeen tested in a variety of systems like interact-ing hard core bosons on lattices30,31 and one-dimensional interacting spin chains.33

The issue of whether all classically noninte-grable quantum systems obey ETH is still underdebate.34 In particular, many body Anderson lo-calization can lead to systems showing both ther-malized and localized phases.35 Recent studies36

show that ETH can still hold as long as the vari-ous perturbations act homogeneously i.e., do nothave any specific selection rules, leading to ergod-icity over the entire relevant Hilbert space. In-terestingly, the phenomenon of intramolecular en-ergy flow, which is at the heart of RRKM, has in-timate connections37,38 to Anderson localizationin terms of the state space model.39 In the statespace model two different classes of initial statescalled as the edge and interior have different mech-anisms for exploring the quantum state space withthe dynamics of the former class being dominatedby dynamical tunneling.40 Moreover, the dynam-ics of the initial states as determined by the vari-ous anharmonic resonances is typically inhomoge-neous,41 suggesting that ETH might be genericallyviolated in effective Hamiltonian models describ-ing intramolecular energy flow. In an early work40

Leitner and Wolynes provided a criterion for quan-tum ergodicity and have argued that facile energyflow throughout the state space is possible if theinterior states are extended. Thus, assuming quan-tum effects such as dynamical tunneling42–44 canefficiently couple40 the edge states to the interiorstates, one expects that violation of ETH for theinterior states signals the system being in the non-RRKM regime.

Clearly, as evident from Eq. 1 and Eq. 4, the cen-tral quantities of concern are the various eigenstateexpectation values Vαα . There is also a close con-nection between the Vαα and the parametric evolu-tion of energy levels with changing system param-eters, also known as level-velocities.45,46 The lit-erature on level-velocities and curvatures as toolsto characterize quantum chaos is quite extensiveand we refer to only a select few studies here.47

Three very early studies, however, are worth men-tioning. The first one is a study48 of the highly

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excited states of the Henon-Heiles and BarbanisHamiltonians using the level-velocity approach byWeissman and Jortner. The second study49 on thelocal ergodicity probes in Henon-Heiles system byRamachandran and Kay in fact is an early exam-ple of ETH. The third study50 by Nordholm andRice puts forward a definition of quantum ergod-icity which also is very close to the spirit of ETHand it is interesting to compare the arguments lead-ing up to their definition with those provided inthe more recent work31 by Rigol and Srednicki.Later studies have used the eigenstate expecta-tion values to identify special classes of localizedeigenstates in systems with several degrees of free-dom. For instance, eigenstate expectation valueshave been utilized before to dynamically assignthe highly excited states of several systems.20 Veryrecently,51 using the technique of parametric levelmotion, Hiller et al. have identified robust states ofultracold bosons in tilted optical lattices coexistingwith extensively delocalized states. Existence ofsuch robust few-body bosonic states and dynami-cally assignable states in molecules are indicativeof a lack of statisticality in the system. As an ex-ample, in a recent work Leitner and Gruebele haveshown52 that significant corrections to the RRKMrates for conformational reactions can be associ-ated with different classes of eigenstates that existnear the reaction threshold. In the molecular con-text, the eigenstate viewpoint is naturally relatedto the nature of the intramolecular vibrational en-ergy redistribution (IVR) dynamics at the energiesof interest. Thus, Jacobson and Field have usedthe time-dependent expectation values of the an-harmonic resonances to understand the origins ofunusually simple stretch-bend dynamics in acety-lene.53

In this work the ETH perspective is utilized toinvestigate the nature of the eigenstates of thio-phosgene (SCCl2) near its dissociation energy (≈600 THz). The motivation for this study comesfrom the recent work54 by Chowdary and Grue-bele wherein they observed sharp assignable fea-tures near and above the dissociation energy inthe stimulated emission pumping spectra. Previ-ous works55–57 have focused on the nature of theeigenstate and IVR dynamics near the thresholdfor onset of IVR (≈ 240 THz) and have already es-tablished the existence of different classes of local-

ized eigenstates and hence nonstatistical IVR dy-namics. Employing a model spectroscopic Hamil-tonian it is shown that localized states do persistpersist at energies close to the dissociation energy.Moreover, it is shown that there are special initialstates that are robust despite the increased densityof states and strong anharmonic couplings.

Model HamiltonianThe Hamiltonian used for the current study is afairly accurate effective Hamiltonian for SCCl2constructed56 by Sibert and Gruebele using canon-ical Van Vleck perturbation theory and can be ex-pressed as H = H0 +Vres with

H0 = ∑i

ωi

(vi +

12

)+∑

i jxi j

(vi +

12

)(v j +

12

)+ ∑

i jkxi jk

(vi +

12

)(v j +

12

)(vk +

12

)(5)

being the zeroth-order anharmonic Hamiltonianand the various anharmonic resonances, couplingthe zeroth-order states, are contained in

Vres = k526a†2a5a†

6 + k156a1a†5a†

6

+ k125a†1a†

2a25 + k36a2

3a†26

+ k261a†1a2a2

6 + k231a†1a2a2

3 + c.c (6)

In the above expression, the operators ak and a†k are

lowering and raising operators for the kth mode re-spectively with [ak,a

†k ] = 1, in analogy to the usual

harmonic oscillator operators. Note that there areother much weaker resonances in the Hamiltonianwhich are ignored in the current study. The aboveHamiltonian has effectively three degrees of free-dom due to the existence of three, approximately,conserved quantities or polyads

K = v1 + v2 + v3

L = 2v1 + v3 + v5 + v6

M = v4 (7)

Therefore H is block diagonal and in the presentwork the focus is on (K,L,M) = (13,25,14) blockwhich has a total of 1365 eigenstates spanningan energy range of about (18099,20841) cm−1.

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Note that Chowdary and Gruebele used the aboveHamiltonian to perform an extensive study of thevibrational state space and concluded58 that about1 in 103 zeroth-order states are localized near thedissociation energy. In what follows, for conve-nience, we will refer to the resonances by the high-lighting the modes involved. Thus, the first term inEq. 6 will be referred to as the 526-resonance.

The classical Hamiltonian corresponding toEq. 6 can be expressed55 in terms of action-anglevariables as H(J,ψψψ) = H0(J)+Vres(J,ψψψ) where

H0(J,ψψψ) = C+ ∑i=1,3

ϖiJi +∑i j

αi jJiJ j

+ ∑i jk

βi jkJiJ jJk (8)

is the zeroth-order Hamiltonian and the resonantperturbations given by

Vres(J,ψψψ) = v156(J;Kc,Lc)cosψ1

+ v526(J;Kc,Lc)cosψ2

+ v125(J;Kc,Lc)cos(ψ1 +ψ2)

+ v36(J;Kc,Lc)cos2ψ3

+ v231(J;Kc,Lc)cos(ψ1−ψ2−2ψ3)

+ v261(J;Kc,Lc)cos(ψ1−ψ2) (9)

In the above equations the actions (Kc = K +3/2,Lc = L + 5/2,Mc = M + 1/2) are con-served, being the classical analogs of the quan-tum polyads, and hence H(J,ψψψ) is ignorable inthe conjugate angles (ψ4,ψ5,ψ6). The parame-ters (C,ϖϖϖ ,ααα,βββ ) of the reduced Hamiltonian (notgiven here) are related to the original parametersvia the canonical transformation and the readeris referred to a previous work for details.55 It isimportant to note that the overlap of nonlinear res-onances in Eq. 9 render the classical system non-integrable with the possibility of extensive chaosin the classical phase space.

Results and discussionsWe begin this section by computing two mea-sures59 for eigenstate delocalization that are well

100

101

102

L

18750 19500 20250E , cm-1

0

2

4S

Figure 1: Participation ratio (top panel) and Shan-non entropy (bottom panel) for the eigenstatesof SCCl2 belonging to the polyad (K,L,M) =(13,25,14). Both measures are computed in thezeroth-order number basis. The region indicatedby gray rectangle comprises of the most delocal-ized eigenstates.

known. The first quantity is the participation ratio

Lα =

(∑n|〈n|α〉|4

)−1

≡(

∑n

p2nα

)−1

(10)

which is a measure of the total number of ba-sis states {|n〉} that participate in making up theeigenstate |α〉. A second quantity is the informa-tion or Shannon entropy of an eigenstate given bythe expression

Sα =−∑n

pnα ln pnα (11)

Note that the maximal59 Sα ≈ ln(0.48N) for agaussian orthogonal ensemble which has the valueof∼ 6.5 since in our case N = 1365. It is also use-ful to observe that in Eq. 4 if we choose V = |n〉〈n|then V ≡ σn = ∑α |Cnα |4 is the dilution factorassociated with |n〉 and assuming ETH one hasσn = N−1

∆E .In Figure 1 the measures Lα and Sα are shown

for all the eigenstates computed in the basis of the

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zeroth-order states i.e., eigenstates of H0. As ex-pected both measures agree and indicate localizedstates at the low and high end of the spectrum. Themiddle part (19250,19750) cm−1 of the spectrumhas states exhibiting large values for both the mea-sures. However, note that Sα in particular does notbecome maximal even in this complicated energyrange. Thus, although one expects that even at thishigh energy the eigenstates are not maximally de-localized, the origin of such partial localization arenot apparent from the results.

In order to gain more detailed insights we com-pute the eigenstate expectation values of the var-ious resonance operators in Eq. 6 and comparethem to the classical microcanonical average com-puted in the (J,ψψψ) reduced representation usingthe expression

〈V 〉mc(E) =∫

V (J,ψψψ)δ [H(J,ψψψ)−E]dJdψψψ∫δ [H(J,ψψψ)−E]dJdψψψ

(12)The results, shown in Figure 2, clearly showconsiderable fluctuations of the Vαα about the〈Vmc〉(E) and imply the existence of several lo-calized eigenstates. Note that strongly localizedstates are seen, in agreement with Figure 1, at boththe low and high energy regions of the polyad.However, now it is possible to ascribe the low andhigh energy localized states with the 36-resonanceand a combination of the 156 and 526 anharmonicresonances respectively. Particularly striking isthe 36-resonance case at low energies exhibitingrather large quantum expectation values. In fact,following previous works,20 one can analyze thesingle integrable 36-nonlinear resonance term inEq. 9

v36(J;Kc,Lc)cos2ψ3≡ 2J3(L1c−K2c−J3)cos2ψ3(13)

with L1c ≡ Lc− J1 and K2c ≡ Kc− J2 being con-stants. The analysis yields the extremal values forthe quantum expectations as

〈α|a3a3a†6a†

6 + c.c|α〉=±12(L1c−K2c)

2 (14)

with ± corresponding to ψ3 = 0 and ψ3 = π/2respectively. Decent agreement of the above es-

timate with the computed values for the first fewstates shown in Figure 2 suggests that even around∼ 540 THz above ground state the classical phasespace of SCCl2 has a large regular region capa-ble of supporting several regular quantum states.Using previously established techniques20,55 it ispossible to provide dynamical assignments for anumber of states at the low and high end of thepolyad. However, we do not attempt this here. In-stead we focus on the (19250,19750) cm−1 part ofthe polyad shown in Figure 1 in order to check theextent of deviation from ETH in this region.

Deviation from ergodicityThe results in Figure 2 show the strong fluctua-tions about the classical microcanonical averagesuggesting deviations from ergodicity. However,as suggested by Pechukas,45 the diagonal matrixelements Vαα are expected to have quantum fluctu-ations which are of the same order of magnitude asthe off-diagonal matrix elements Vαβ . It is there-fore interesting to see if the deviations seen in thecomplicated (19250,19750) cm−1 region in Fig-ure 2 are essentially the expected quantum fluctu-ations. As discussed by Feingold and Peres thecomputation of the quantum fluctuations in Vαα israther involved and related to the classical auto-correlation of the relevant observables.29 One ap-proach is to model the distribution of |Vαβ |2 by aGaussian as in the earlier work.29 However, in thiswork we take a more stringent constraint and as-sociate the extent of quantum fluctuation of Vαα

with the largest off-diagonal matrix element |Vαβ |.Typically, the largest such element is found to beclose to the diagonal.

In Figure 3 we show the complicated region ofthe polyad with the expected quantum fluctuationsin Vαα at specific points. It is clear from Figure 3that despite the near constant values for the variousVαα there are several states (highlighted by boxes)that indeed deviate strongly from the microcanoni-cal average. Interestingly, many of these states arenot easily identifiable from Figure 1 since both Lα

and Sα are not capable of identifying localizationin the phase space. Hence, Figure 3 clearly estab-lishes the power of the approach based on eigen-

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-30

-15

0

15

-15

0

15

30

-15

0

15

30

18000 19000 20000 21000

E , cm-1

-150

0

150

V

-10

0

10

-20

0

20

40

156 526 125

36 231 261

Figure 2: Quantum eigenstate expectation values (circles) of the resonance operators (indicated on top ofthe panel) versus the classical microcanonical average (red line). The x-axis has the same scale in all plotsand the Vαα are dimensionless.

state expectation values. Note that different res-onance operators reveal different localized eigen-states in a complementary fashion. For instance,a localized state with large negative expectationvalue for the 36-resonance in Figure 3 is not easilyidentifiable from any of the other expectation val-ues. Although not shown here, we have furtherconfirmed the predictions of Figure 3 by study-ing the semiclassical angle space representation55

〈ψψψ|α〉 ≡ ∑n cnα exp(in ·ψψψ) of the eigenstates.The ETH result in Eq. 4 essentially depends on

the lack of correlation between the intensities pnα

and the expectation values Vαα . Thus, are the re-sults in Figure 3 contrary to the ETH expectations?In this context it is interesting to note that a sensi-tive measure for violations from ergodicity calledas the intensity-velocity correlator was introducedearlier by Tomsovic60 and has been applied to sev-eral paradigmatic systems.61 This measure corre-lates the intensity pzα ≡ |〈z|α〉|2 of some probestate |z〉 and the parametric evolution of the eigen-

values (level velocities) as

Cz(λ ) =1

σzσE

⟨pzα

∂Eα(λ )

∂λ

⟩α∈∆E

≡

⟨pzα

˜∂Eα(λ )

∂λ

⟩α∈∆E

(15)

where λ is a Hamiltonian parameter of interest. Inthe second line of Eq. 15 the quantities are scaledto unit variance and the level-velocities are zerocentered to remove any net drift. The averagingabove is done over a window ∆E consisting ofN∆E states with σ2

z and σ2E being the local vari-

ances of the intensities and level-velocities respec-tively. As defined Eq. 15 is expected to be large foreigenstates exhibiting common localization char-acteristics with levels moving in the same directionwith changing parameter value. Moreover, one canshow that in the random matrix theory (RMT) limitthe correlator vanishes for every |z〉 up to fluctua-tions depending on the number of states within theenergy window i.e., Cz(λ ) = 0±N−1/2

∆E . In thiswork, however, we will use the covariance instead

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-20

-10

0

10

-10

0

10

-10

0

10

19250 19500 19750E , cm-1

-100

0

100

V

-15

0

15

-20

0

20

40

156 526 125

36 231 261

Figure 3: Same as in Figure 2 but focusing on the complicated (19250,19750) cm−1 region of the polyad.The expected quantum fluctuations in Vαα are also indicated at specific points. A few of the eigenstateslying outside the fluctuations, and hence partially localized, are highlighted by blue boxes.

of the correlator since σz and σE are also influ-enced by the classical phase space structures. Con-sequently, interpretation of Cz(λ ) needs to be donewith some care.61 Moreover, using the covarianceis closer to the spirit of Eq. 4 and suffices for ourwork. In addition, note that an earlier work62 ofHeller on the rate of exploration of phase spaceutilizes the strength function, which is the Fouriertransform of the autocorrelation 〈z(0)|z(t)〉, as thecentral quantity. The Cz(λ ) is essentially the para-metric response of the strength function60 and, asseen below, connects very naturally to the ETHperspective.

To make contact with ETH we choose the pa-rameter in Eq. 15 as one of the resonance strengthsin Eq. 6, say k36, and the initial state as the zeroth-order number state |n〉. Then, using the Hellman-Feynman theorem, the covariance can be writtendown as

Covn(k36) =⟨

pnα(V(36)αα −V (36)

∆E )⟩

α∈∆E(16)

where V (36)∆E is the window average of V (36)

αα . Sim-

ilarly, one has measures for every anharmonic res-onance in the Hamiltonian. Thus, as the expec-tations are zero-centered, ETH as in Eq. 4 im-plies that Covn(k36) = 0 for every choice of |n〉.On the contrary, nonzero values for Covn yielda quantitative measure of the deviations from er-godicity. We now choose two zeroth-order states|5,8,15,14,0,0〉 and |6,7,13,14,0,0〉 with ener-gies E0

n = 19388.3 and 19574.7 cm−1 respectivelyin the range appropriate to Figure 3 and com-pute the intensity-velocity covariance for each res-onance operator. The energy windows are centeredat their respective E0

n with widths ∆E = 2δE0n de-

termined by the energy uncertainty δE0n (≈ 50

cm−1 for both states). Within the selected windowthere are about 80 eigenstates and the respectiveclassical microcanonical expectations are nearlyconstant (cf. Figure 3(B)). Hence, the classicalphase space is not expected to change much overthe averaging window. The results are shown inFigure 4 and clearly indicate deviations from er-godicity. As expected, the 36-resonance plays animportant role for both the states. However, local-ization due to the 526 and the 125 resonances are

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also observed. The dominance of the 36-resonancecan be understood from the parametric evolutionof the eigenvalues shown in Figure 4 with varyingresonance strength. Several states exhibiting lin-ear parametric motion (“solitonic" states45,46) canbe seen amidst a sea of avoided crossings. Suchsolitonic states are the robust localized states, alsoseen exhibiting large fluctuations in Figure 3, re-sulting in the strong deviations from ergodicity asmeasured by the covariance.

A final remark is in order at this stage regard-ing the choice of the two specific zeroth-orderstates chosen in Figure 4. In their recent work54,58

Chowdary and Gruebele associated the sharp SEPfeatures (∼ 600 THz) with zeroth-order states ofthe form |n1,n2,n3,n4,0,0〉with n3 being typicallyrather small (≈ 0 as seen in a later work58) in com-parison to the other mode occupancies. Althoughthe states chosen here have the same form, then3 occupancies being large makes them stronglysusceptible to the 36-resonance and hence frag-mented in the SEP spectra. However, such statesare fragmented much less compared to the statesat similar energies but with occupancy in the n5and n6 modes. For example, the zeroth order state|3,6,9,14,4,6〉with E0

n ≈ 19389.1 cm−1 is dilutedby nearly an order of magnitude more as com-pared to the state |5,8,15,14,0,0〉. Thus, the re-sults in Figure 4 showing considerable localizationat these high energies are compatible with the find-ings of Chowdary and Gruebele.54,58

ConclusionsIn this work we have shown that several eigen-states of SCCl2 exhibit localization even nearthe dissociation threshold. Such ‘nonthermalized’states are identified using sensitive and quantita-tive measures for deviations from ergodicity. Con-sequently, the dynamics at these energies shouldexhibit deviations from RRKM predictions. How-ever, as pointed out in an earlier work,58 observingthe deviations from RRKM depends on whetherthe optically accessible states have sufficient over-laps with the partially localized eigenstates or not.In other words, experimentally prepared initial

0

0.1

0.2

Cov

n

19400 19600E , cm-1

0

0.1

pn

19350 19425

-5

0<V>mc

-0.8-0.6k36, cm-1

19350

19400

E , cm-1

156 526 125 36 261 231

(A) (B)

Figure 4: Intensity-velocity covariance (cf.Eq. 16) for two zeroth-order states |n〉 =|5,8,15,14,0,0〉 (black) and |6,7,13,14,0,0〉(red). The resonant operators chosen to com-pute the covariances are indicated. (A) Inten-sity spectra of the two states showing the frac-tionation due to resonances. (B) Classical mi-crocanonical averages of the resonances varyvery little over the energy window correspond-ing to the |5,8,15,14,0,0〉 state. Right panelshows the parametric evolution of the eigenvalues(level-velocities) upon varying the 36-resonancestrength. Actual resonance strength is indicated bya vertical red line.

states which have significant overlaps with par-tially localized eigenstates are bound to have non-statistical dynamics since the eigenstates encodethe infinite time energy flow dynamics of the sys-tem. Although the converse statement i.e., validityof ETH implies RRKM is yet to be established, theimportant studies50 by Nordholm and Rice doeslead us to believe that it is reasonable to expect so.It is also relevant to note that the present work isan example of a system wherein the integrabilitybreaking perturbations do have specific selectionrules and hence testing the ETH in a more com-plex situation.36

The techniques used here are very general andnot limited by the dimensionality of the system.The intensity-velocity correlator used in this studyis not only capable of identifying localized statesbut also singles out the relevant perturbations that

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lead to localization. Note that the eigenstate ex-pectation values play a key role in both ETHand the matrix fluctuation-dissipation (MFD) ap-proach63 of Gruebele. Hence, a better understand-ing of the connection between the two should beuseful in gaining fundamental classical-quantumcorrespondence insights into the dynamics andcontrol of IVR. Finally, it would be of some in-terest to determine the exact nature of the rela-tion between ETH on one hand, and the Logan-Wolynes transition criterion,37 and the Leitner-Wolynes quantum ergodicity criterion,40 whichincludes the dynamical tunneling effects, on theother hand.

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Eigenstates of Thiophosgene Near the Dissociation Threshold: Deviations From Ergodicity