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Intramolecular dynamics: time evolution of superposition states in the regular and irregular spectrum
Volume
72, number
3
CHEMICAL
INTRtMOLECULAR
DYNAMICS.
PHYSICS
15 June
LETTERS
TIME EVOLUTION
OF SUPERPOSITION
1980
STATES
THEREGULARANDlRRECLJLARSPECTRUM*
IN
Paul BRUMER
*
Dcporrrncm o/Clmmstr~ UHIIersq Tororzro Otzrano Canada
o/ Tororrro
and hloshe SHAPIRO Deportrnerrr of Chemtcal Plr~sacs, I~‘er:rnarrnlt~strr~rtc of Scrence RehoI or, Israel Recened
IO March
1980
The ttme e\olutton an we
ti(r) IS compared in two systems. one utth a regular of P(r) = 1(9(O)I 0 (r))l’ IS found to be the same tn both the degrees of freedom
ofuavefuncttons
rrre@ar spectrum The behavior to rhe couplm;
betaeen
1. Introduction Colhston-free adrabatrc mtramolecular dynamics m polyatomrc molecules IS currently the subJ2ct of mtense theoretical and experimental study [ 1.2]. a conssquence of interest m selective laser-induced chemlstry [3]. What IS required for apphcarrons m this area IS an understandmg of the rate and extent of mtramoIzcular energy transfer and theu dependence on molecular propertres The nature of vtbratronal dynanucs m non-integrable classtcal mcchantcal systems is now hnown to change m character with mcreasmg energy [ 1.31 In partrcular, the dynamics at low energy IS regular and penodrc whereas it IS erratic at hrgh energies The former regun2 IS that normally consrdered m spectroscopy (e-g local and normal modes) whereas the latter ts presumed rzlated to statrstrcal behavior m ummolecular drssocratron The relatronshrp of this change in classtcal bchavror to expected features of the true quantum system has been the subject of much drscussron [-1,5]. In * Supported, ofToronto. v A P Sloan
tn part, Foundatron
b) an internal Fellou
grant
from
the Unwerstty
energy spectrum and one \rtth systems. P(r) decal IS tnsensl-
parttcular, two types of quantum behavior are anttcrpated [S] The regular regtme, predominant at low energies, IS characterrzed by wavefunctrons wrth ordered nodal patterns and an ergenvalue spectrum with a drstrrbutron of ad!acent level spacmgs peaked at zeror. Th2 Irregular quantum regme, exrstent at hrgher energres, 1s characterized by wavefunctrons wnh mghly Irregular nodal patterns and an erpenvalue spectrum with an adJacent level spacmg drstrrbutron peaked at nonzero values. Additronal differences ar2 summarrzed by Percrval [5]. Systems whtch are regular at all energies may be readily constructed (2 g. two uncoupled anharmomc oscillators) whereas f2w reahstrc systems wrth irregular spectra are known. Recently, however. McDonald and Kaufman [6] have solved the Schrodmger equatron for the stadmm problem, i e. a particle m a two-drmensronal box 1~1th an impenetrable boundary shaped l&e a stadrum They confirmed that the ergenvalue spectrum of this system drsplays the two expected features of an Irregular system mentioned above. The classrcal mechamcs of thrs system has also been studled L ’ Thl~ ISnot the case for the non-generic oscdlators
See Berry
and Tabor
[S]
case of harmontc
Volume [7]
72, number
CHEMICAL
3
and the dynamics
is known to be extremely
PHYSICS
errat-
15 June
LETTERS
1980
2. I. Afodel sysretns
1C. In this letter we compare the time evolution of a superposItIon of eigenstates in a regular uncoupled anharmonic osclLlator system to that m the Irregular stadium problem. The naive expectation that the initial rate of decay differs substanrlally in these two cases, due to the degree of couphng between the modes, IS shown to be unfounded. Tlus result is II~accord with
anaiytlc predlctlons but the avalabtity of u-regular spectrum elgenvalues now makes a numerical demonstratlon of this result possible In addition, we demonstrate that the long-time behavior in both systems is quahtatlvely mdlstmgulshable, being composed of recurrences of seemmgly arbitrary spacmgs and intensities. Numerical computations and results are discussed in section 2. The important unphcations of this result for future studies of regular versus Irregular quantum mtramolecular dynamics IS discussed m sectlon 3.
2. Computations We wish to study the decay of an mltlal state G(O) where spontaneous emlsslon IS assumed to be a neghgable perturbatron and where experunental conditions render the state preparation and subsequent evolutron mdependent. Denotmg the exact bound elgenstates of energy El by 1~7,) we have (in au) lJ/(O))=
c c,E-,A
A total of 233 stadium momentum eigenvalues Ki, conslstmg of those in ref. [6] plus additional eigenvalues to Km, = 79.18, were kindly provided by McDonald and Kaufman. The mass nr and energy zero E were adjusted to provide an energy spectrum Ei = !? k;/2m - E, over the same range of that of the uncoupled oscdlator system discussed below (resultant E, = 0.48144 au, nz = 65 11.2 1 au). The elgenvalue spectrum for the uncoupled oscdlator system was obtamed by meshing the eigenvalues of two typlcal anharmonic oscillators, i.e. E(nl,
tz2) = [-0 2275
-4x
10-~(tzl
+ [-0.2912 -4.6
+4.56X
+f)Z
+ 1.3 x 10-f+,
+ 5.93 X 10-3(n2
X 10-5(t~2 +$)’
10-3(tz,
++) f$)3]
+f)
+ 2.5 X lo-6(tz,
+$)3]_
The mdlcated parameter choice, corresponding to oscdIators with harmonic frequencies of 1000 cm-l and 1300 cm-l, yielded 228 ergenvalues over the energy range between -0.28824 au and 1.295 X lo-’ au. This is to be compared to the 3-33 stadium eigenvalues over the same range. Distributions of adjacent eigenvalue spacmgs in the two systems were considerably different wrth the stadium spectrum showing a broader distribution of spacings and the oscillator system bemg sharply peaked at a small but nonzero value (=I5 X 10m4 au)_
I
I $I (f)> =
C I
2.2. Tnne evolution c, exp (-UT, t)l EI ).
The probablhty P(E) of observmg at tune t 1s given by
P(f)=
$J (0) m the system
l<~(O)l~(t))l~ = C[c,[’ exp(-iE,t) I I
=,cm Ic,,l’lc,12
* See. e g , the study of the Barbarus system
2
Rice [4] _
’
1
cos[(Enl-EJt].
?he computation of P(t) IS therefore only the system eigenvalues *_
and
of superposition
states
(1)
(2) sunple,
requiring
by Nordholm
The dependence of P(t) on the regular versus irregular nature of the eigenvalue spectrum is of particular mterest. For this reason it is advantageous to compare the evolution of stadium and oscillator +(O) of similar character, i e. wtth simdar distrtbutions of coefficients Ci- In most cases studied oscillator and stadium P(E) were compared with the uutial cz selected accordmg to a gaussian dlstrrbution of width 26 and center E,_ That is, + = exp[-c@,
-Ec)2/A2]iL,
(3)
with c,’ values less than 0.1 of the maximum neglected. The quantity iy = ln 2 and L is fiied such that the cf so obtained are normalized to one. 529
Volume
7 2. number
CHEMICAL
3
PHYSICS
The short-tune behavtor of P(t) for an arbitrary c, drstnbutton may be analytrcally predicted by standard means [8]. In particular, one expects that for tunes t < p,(E), where p,(E) IS the average denstty of states m the energy regrme encompassed by G(O), that P(r)
=
C
cf eup(--iE,
I I
r)
2 I
IJ
1
[c(E)J’e\p(--rEt)dE
=
-
,
t < P,(E)-
(4)
Here [c(E)]’
= p(E)
exp[--a(E
- E,)“/JX]/Z,
where p(E)
IS the density of states at energy
2 =
exp [- a(E - E,)2/S]
f
p(E)
Ifp(E) ISslowly mtegral giving
varymg rt can be removed
E and
from the
[c(E)jZ=(l/~)(~/lr)‘lae~p[--(E-Ec)’/~’].
Evaluatmg eq (4) for the gaussran drstnbutron gives P(t) = exp(- tZJz/%), a result, vahd for t
and s)srem
E
Stadmm R
are quite close throughout_
Additional
calcula-
OscdJator r,/cWc)
P,(E)
A
t,le(catc)
~~0
4 4 6 8 7 9 11 18 25
b) b) 400 405 413 399 409 400 410
232 154 510 57-l 591 774 880 1448 7-195
IO 10 11 14 9 11 11
505 361 408 438 335 376
817 755 817 9’2 771 843
40’
939
-0063
10
391
760
6
371
819
530
1980
features for J = 0 033 aua)
-0280 -0.149 -07-18 -0 187 -0.156 -0 115 -0.094
a) AU numbers x 10-17 s b, No gaussian
15 June
lattvely narrow drstnbutron (A = 0.003 au). Columns 2-4 provide, for the stadium case, the number of states N in $J(0), the calculated time tile (talc) at whrch P(t) = l/e, and the local p,(E). Sunilar data for the uncoupled oscthator system are provided in columns 5-7. The quantity tile (cak) IS seen, for all cases where gaussian fall-off [i.e. P(t) a exp(- of”), wrth constant /3] IS observed, to be srmlar and to be m good agreement with the value predrcted for gaussian cf dntnbut:ons, i.e. t lle = (2o~)tI~/A = 392.5 au. The two cases where gaussian decay 1s not observed are those for which c llc > p,(E), IX. where system recurrences are expected to mterfere with the uutral decay. Slm~lar results were obtamed for other EC, A cases. Studies of the long-tune behavior of P(t) for the stadium and oscrllator cases wrth A = 0.003 were also carried out. At all energies other than those 111the harmonic regtme of the oscrhators one finds, for both stadrum and oscillator cases wtth file > p,(E), long-time behavror which is characterrzed by sequences of recurrences with no partrcular pattern in intenstty or location. Results typical of those obtamed are shown in fig. 1 for E, = -0.094 au, A = 0.003 au over a tune period = 10 p,_(E). Neither the stadium nor oscillator case drsplays an identrfiable signature. Results typical of those obtamed with larger A are shown in fig. 2 whrch shows the short-tune P(t) for the oscillator and stadtum cases wtth EC= -0.07 au and A = 0.03 au Although the two G(O) contam different numbers of states (89 for the stadmm and 148 for the oscdlator) reflectmg differences m p,(E),the resultant P(t)
Table J rl,e(calc)
-0032
LE’ITERS
are III atomic
fall-off
units
1 au of tune
= 2 41889
Flp
l_
P(f) versus c for uncoupled
anharmonlc oscillator sysfor Ec = - 0 094 au,
tem (- - -) and stadium system ( -) A = 0 003 au
Volume 72, number 3
CHEMICAL
PHYSICS
t(ou)
Fig 2 Short-ttme P(r) verstis f for uncoupled anharmomc os) forEc= cdlator system (- - -) and stadturn system <-0 07 au, P = 0 03 au Scahng the osctllator spectrum (as described tn text) ytelds results tndntmguahable on thrs scale from the stsdtum results
Irons w2re performed for thrs case with a scaled oscillator spectrum, r e using E’(ir t, 12,) = AE(tz 1, n,), EL =AEc, wrth A adJusted to allow 89 states rn the uutral ~JJ(0) The resultant mrtial decay of P(r) was found to be virtually rdenticaf to the stadrum P(f)_ Excellent agreement between file (talc) and file = 33.33 IS observed. The long-trme behavror (f to IOp,(E) = 8000 au) of P(r) for these stadrum and oscrllator cases are compared in fig 3. Several drfferences are readiIy noted. Ii
I
II
I
OCU
Cl02 -Z a
0
006
0G-z
LETTERS
15 June 1980
Frrst, the stadium P(t) is generally irregular over all t > 80 au whereas the oscdlator P(t) shows isolated peaks for t < 4000 au with almost regularly spac2C peaks beyond. Desprte these drfferences both P(t) remain small throughout [P(t) < 0.071 and no obvious correlation between peak locatron, mtensity, and systern recurrence time is observed. In particular, in both cases the mtramolecular dynamics, from the P(t) view point, is comprised of very weak passage into and out of the initral state G(O) for times t > 80 au. Sumlar studies were carried out with a lirruted nurnber of other types of initial cr distnbutions. Once agam in cases where tile
3. Summary and comments Studies of the time evolutron of P(t) for simifar 9 (0) in a regular and rrregu!ar systsm show no 2ss2w tial drfferences in behavtor. Results for the initial decay are in accord wrth simple analytic arguments and long-time behavior shows weak peaks in P(t)_ Although only a restricted set of initial .cf distriiutions were studred, further studies using a host of other drstnbuttons appears unwarranted. In particular, our results suggest that rf there IS a difference in the behavior of P(t) for regular and Irregular systems it lizs in how the system responds to the external mfhrence creatmg G(O). if the two systems respond differently to the preparation apparatus then the resultant e(O), and hence f’(t), will differ. This may be the case for preparation by optrcal excrtation in which the different nodal patterns m the wavefunctions of the hvo systems lead to expected differences m oscillator strengths. Thus, further comparrsons of irregular and regular intramolecular dynamics requue explicit consrderation of the preparation step. Such work is currently m progress [9] (se2 also ref. [lo]).
00s
References J
0
I
2
3
6
7
3
Ftg 3. P(t) versus f for (a) stadmm system and (b) scaled u = I 739) osctllator system Here E, = -0 07 au. ZI = 0 03 au and both tmttal $ (0) contatn 89 energy etgenstates. Note small ordtnate scale.
111M.V.
Berry, in. Topms tn nonlinear mechanics, eb S. Jorna (AJP, New York, 1978); P. Brtuner, Advan Chem Phys , to be published; hf. Tabor, Advan. Chem. Phys , to be published; R D. Levme, Advan. Cbem. Phys, to be published_
VoIume [7]
72. number
3
CHEhfICAL
R-G. Bray and bf_J. Berry, J Chem. Phys 71 (1979) 4909, and references therem [3 J S A. Rice, m. Advances m laser chematry, ed. A Zewell (Sprmger, Berlm, 1978). [4] K S J Nordholm and S A. Rice, J Chem. Phys 61 (1974) 203,768;62 (1973) 157; E. Heller, Chem Phys. Lerters 60 (1979) 338; bf V. Berry, Phil Trans Roy. Sot 287 (1977) 137. [S] I C Percival, Advan Chem. Phys 36 (1977) 1, G hf Zaslavsky, Zh Eksp Teor. FIZ. 73 (1977) 2089, hi V Berry and hf Tabor, Proc. Roy. Sot A356 (1977) 375: R hi. Stmltt. N C Handy and W H hfdler. J. Chem Phys 71 (1979) 3311
532
PHYSICS
LEMERS
15 June
1980
[6 j S W. LicDonald and A N Kaufman, Phys Rev Letters 42 (1979) 1189. [7] G. Benettmand J -iU Strelcyn, Phqs Rev. Al7 (1978) 773 [S] J Jortner, S A Rice and R hf Hochstrasser, Advan Photochem 7 (1969) 149, KE. Freed. Chem Phys Letters 42 (1976) 600. [91 hf Shapuo and P Brumer. work m progress [iOJ E J Heiler, J.Chem Phys 72 (1980) 1337,and reie:ences therem. R A. hfarcus. m Proceedmgs of the 3rd Internatronal Congress on Quantum Chemlsfry, ed B Pullman (Reldel, Dordrrcht, 1980)
72, number
3
CHEMICAL
INTRtMOLECULAR
DYNAMICS.
PHYSICS
15 June
LETTERS
TIME EVOLUTION
OF SUPERPOSITION
1980
STATES
THEREGULARANDlRRECLJLARSPECTRUM*
IN
Paul BRUMER
*
Dcporrrncm o/Clmmstr~ UHIIersq Tororzro Otzrano Canada
o/ Tororrro
and hloshe SHAPIRO Deportrnerrr of Chemtcal Plr~sacs, I~‘er:rnarrnlt~strr~rtc of Scrence RehoI or, Israel Recened
IO March
1980
The ttme e\olutton an we
ti(r) IS compared in two systems. one utth a regular of P(r) = 1(9(O)I 0 (r))l’ IS found to be the same tn both the degrees of freedom
ofuavefuncttons
rrre@ar spectrum The behavior to rhe couplm;
betaeen
1. Introduction Colhston-free adrabatrc mtramolecular dynamics m polyatomrc molecules IS currently the subJ2ct of mtense theoretical and experimental study [ 1.2]. a conssquence of interest m selective laser-induced chemlstry [3]. What IS required for apphcarrons m this area IS an understandmg of the rate and extent of mtramoIzcular energy transfer and theu dependence on molecular propertres The nature of vtbratronal dynanucs m non-integrable classtcal mcchantcal systems is now hnown to change m character with mcreasmg energy [ 1.31 In partrcular, the dynamics at low energy IS regular and penodrc whereas it IS erratic at hrgh energies The former regun2 IS that normally consrdered m spectroscopy (e-g local and normal modes) whereas the latter ts presumed rzlated to statrstrcal behavior m ummolecular drssocratron The relatronshrp of this change in classtcal bchavror to expected features of the true quantum system has been the subject of much drscussron [-1,5]. In * Supported, ofToronto. v A P Sloan
tn part, Foundatron
b) an internal Fellou
grant
from
the Unwerstty
energy spectrum and one \rtth systems. P(r) decal IS tnsensl-
parttcular, two types of quantum behavior are anttcrpated [S] The regular regtme, predominant at low energies, IS characterrzed by wavefunctrons wrth ordered nodal patterns and an ergenvalue spectrum with a drstrrbutron of ad!acent level spacmgs peaked at zeror. Th2 Irregular quantum regme, exrstent at hrgher energres, 1s characterized by wavefunctrons wnh mghly Irregular nodal patterns and an erpenvalue spectrum with an adJacent level spacmg drstrrbutron peaked at nonzero values. Additronal differences ar2 summarrzed by Percrval [5]. Systems whtch are regular at all energies may be readily constructed (2 g. two uncoupled anharmomc oscillators) whereas f2w reahstrc systems wrth irregular spectra are known. Recently, however. McDonald and Kaufman [6] have solved the Schrodmger equatron for the stadmm problem, i e. a particle m a two-drmensronal box 1~1th an impenetrable boundary shaped l&e a stadrum They confirmed that the ergenvalue spectrum of this system drsplays the two expected features of an Irregular system mentioned above. The classrcal mechamcs of thrs system has also been studled L ’ Thl~ ISnot the case for the non-generic oscdlators
See Berry
and Tabor
[S]
case of harmontc
Volume [7]
72, number
CHEMICAL
3
and the dynamics
is known to be extremely
PHYSICS
errat-
15 June
LETTERS
1980
2. I. Afodel sysretns
1C. In this letter we compare the time evolution of a superposItIon of eigenstates in a regular uncoupled anharmonic osclLlator system to that m the Irregular stadium problem. The naive expectation that the initial rate of decay differs substanrlally in these two cases, due to the degree of couphng between the modes, IS shown to be unfounded. Tlus result is II~accord with
anaiytlc predlctlons but the avalabtity of u-regular spectrum elgenvalues now makes a numerical demonstratlon of this result possible In addition, we demonstrate that the long-time behavior in both systems is quahtatlvely mdlstmgulshable, being composed of recurrences of seemmgly arbitrary spacmgs and intensities. Numerical computations and results are discussed in section 2. The important unphcations of this result for future studies of regular versus Irregular quantum mtramolecular dynamics IS discussed m sectlon 3.
2. Computations We wish to study the decay of an mltlal state G(O) where spontaneous emlsslon IS assumed to be a neghgable perturbatron and where experunental conditions render the state preparation and subsequent evolutron mdependent. Denotmg the exact bound elgenstates of energy El by 1~7,) we have (in au) lJ/(O))=
c c,E-,A
A total of 233 stadium momentum eigenvalues Ki, conslstmg of those in ref. [6] plus additional eigenvalues to Km, = 79.18, were kindly provided by McDonald and Kaufman. The mass nr and energy zero E were adjusted to provide an energy spectrum Ei = !? k;/2m - E, over the same range of that of the uncoupled oscdlator system discussed below (resultant E, = 0.48144 au, nz = 65 11.2 1 au). The elgenvalue spectrum for the uncoupled oscdlator system was obtamed by meshing the eigenvalues of two typlcal anharmonic oscillators, i.e. E(nl,
tz2) = [-0 2275
-4x
10-~(tzl
+ [-0.2912 -4.6
+4.56X
+f)Z
+ 1.3 x 10-f+,
+ 5.93 X 10-3(n2
X 10-5(t~2 +$)’
10-3(tz,
++) f$)3]
+f)
+ 2.5 X lo-6(tz,
+$)3]_
The mdlcated parameter choice, corresponding to oscdIators with harmonic frequencies of 1000 cm-l and 1300 cm-l, yielded 228 ergenvalues over the energy range between -0.28824 au and 1.295 X lo-’ au. This is to be compared to the 3-33 stadium eigenvalues over the same range. Distributions of adjacent eigenvalue spacmgs in the two systems were considerably different wrth the stadium spectrum showing a broader distribution of spacings and the oscillator system bemg sharply peaked at a small but nonzero value (=I5 X 10m4 au)_
I
I $I (f)> =
C I
2.2. Tnne evolution c, exp (-UT, t)l EI ).
The probablhty P(E) of observmg at tune t 1s given by
P(f)=
$J (0) m the system
l<~(O)l~(t))l~ = C[c,[’ exp(-iE,t) I I
=,cm Ic,,l’lc,12
* See. e g , the study of the Barbarus system
2
Rice [4] _
’
1
cos[(Enl-EJt].
?he computation of P(t) IS therefore only the system eigenvalues *_
and
of superposition
states
(1)
(2) sunple,
requiring
by Nordholm
The dependence of P(t) on the regular versus irregular nature of the eigenvalue spectrum is of particular mterest. For this reason it is advantageous to compare the evolution of stadium and oscillator +(O) of similar character, i e. wtth simdar distrtbutions of coefficients Ci- In most cases studied oscillator and stadium P(E) were compared with the uutial cz selected accordmg to a gaussian dlstrrbution of width 26 and center E,_ That is, + = exp[-c@,
-Ec)2/A2]iL,
(3)
with c,’ values less than 0.1 of the maximum neglected. The quantity iy = ln 2 and L is fiied such that the cf so obtained are normalized to one. 529
Volume
7 2. number
CHEMICAL
3
PHYSICS
The short-tune behavtor of P(t) for an arbitrary c, drstnbutton may be analytrcally predicted by standard means [8]. In particular, one expects that for tunes t < p,(E), where p,(E) IS the average denstty of states m the energy regrme encompassed by G(O), that P(r)
=
C
cf eup(--iE,
I I
r)
2 I
IJ
1
[c(E)J’e\p(--rEt)dE
=
-
,
t < P,(E)-
(4)
Here [c(E)]’
= p(E)
exp[--a(E
- E,)“/JX]/Z,
where p(E)
IS the density of states at energy
2 =
exp [- a(E - E,)2/S]
f
p(E)
Ifp(E) ISslowly mtegral giving
varymg rt can be removed
E and
from the
[c(E)jZ=(l/~)(~/lr)‘lae~p[--(E-Ec)’/~’].
Evaluatmg eq (4) for the gaussran drstnbutron gives P(t) = exp(- tZJz/%), a result, vahd for t
and s)srem
E
Stadmm R
are quite close throughout_
Additional
calcula-
OscdJator r,/cWc)
P,(E)
A
t,le(catc)
~~0
4 4 6 8 7 9 11 18 25
b) b) 400 405 413 399 409 400 410
232 154 510 57-l 591 774 880 1448 7-195
IO 10 11 14 9 11 11
505 361 408 438 335 376
817 755 817 9’2 771 843
40’
939
-0063
10
391
760
6
371
819
530
1980
features for J = 0 033 aua)
-0280 -0.149 -07-18 -0 187 -0.156 -0 115 -0.094
a) AU numbers x 10-17 s b, No gaussian
15 June
lattvely narrow drstnbutron (A = 0.003 au). Columns 2-4 provide, for the stadium case, the number of states N in $J(0), the calculated time tile (talc) at whrch P(t) = l/e, and the local p,(E). Sunilar data for the uncoupled oscthator system are provided in columns 5-7. The quantity tile (cak) IS seen, for all cases where gaussian fall-off [i.e. P(t) a exp(- of”), wrth constant /3] IS observed, to be srmlar and to be m good agreement with the value predrcted for gaussian cf dntnbut:ons, i.e. t lle = (2o~)tI~/A = 392.5 au. The two cases where gaussian decay 1s not observed are those for which c llc > p,(E), IX. where system recurrences are expected to mterfere with the uutral decay. Slm~lar results were obtamed for other EC, A cases. Studies of the long-tune behavior of P(t) for the stadium and oscrllator cases wrth A = 0.003 were also carried out. At all energies other than those 111the harmonic regtme of the oscrhators one finds, for both stadrum and oscillator cases wtth file > p,(E), long-time behavror which is characterrzed by sequences of recurrences with no partrcular pattern in intenstty or location. Results typical of those obtamed are shown in fig. 1 for E, = -0.094 au, A = 0.003 au over a tune period = 10 p,_(E). Neither the stadium nor oscillator case drsplays an identrfiable signature. Results typical of those obtamed with larger A are shown in fig. 2 whrch shows the short-tune P(t) for the oscillator and stadtum cases wtth EC= -0.07 au and A = 0.03 au Although the two G(O) contam different numbers of states (89 for the stadmm and 148 for the oscdlator) reflectmg differences m p,(E),the resultant P(t)
Table J rl,e(calc)
-0032
LE’ITERS
are III atomic
fall-off
units
1 au of tune
= 2 41889
Flp
l_
P(f) versus c for uncoupled
anharmonlc oscillator sysfor Ec = - 0 094 au,
tem (- - -) and stadium system ( -) A = 0 003 au
Volume 72, number 3
CHEMICAL
PHYSICS
t(ou)
Fig 2 Short-ttme P(r) verstis f for uncoupled anharmomc os) forEc= cdlator system (- - -) and stadturn system <-0 07 au, P = 0 03 au Scahng the osctllator spectrum (as described tn text) ytelds results tndntmguahable on thrs scale from the stsdtum results
Irons w2re performed for thrs case with a scaled oscillator spectrum, r e using E’(ir t, 12,) = AE(tz 1, n,), EL =AEc, wrth A adJusted to allow 89 states rn the uutral ~JJ(0) The resultant mrtial decay of P(r) was found to be virtually rdenticaf to the stadrum P(f)_ Excellent agreement between file (talc) and file = 33.33 IS observed. The long-trme behavror (f to IOp,(E) = 8000 au) of P(r) for these stadrum and oscrllator cases are compared in fig 3. Several drfferences are readiIy noted. Ii
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15 June 1980
Frrst, the stadium P(t) is generally irregular over all t > 80 au whereas the oscdlator P(t) shows isolated peaks for t < 4000 au with almost regularly spac2C peaks beyond. Desprte these drfferences both P(t) remain small throughout [P(t) < 0.071 and no obvious correlation between peak locatron, mtensity, and systern recurrence time is observed. In particular, in both cases the mtramolecular dynamics, from the P(t) view point, is comprised of very weak passage into and out of the initral state G(O) for times t > 80 au. Sumlar studies were carried out with a lirruted nurnber of other types of initial cr distnbutions. Once agam in cases where tile
3. Summary and comments Studies of the time evolutron of P(t) for simifar 9 (0) in a regular and rrregu!ar systsm show no 2ss2w tial drfferences in behavtor. Results for the initial decay are in accord wrth simple analytic arguments and long-time behavior shows weak peaks in P(t)_ Although only a restricted set of initial .cf distriiutions were studred, further studies using a host of other drstnbuttons appears unwarranted. In particular, our results suggest that rf there IS a difference in the behavior of P(t) for regular and Irregular systems it lizs in how the system responds to the external mfhrence creatmg G(O). if the two systems respond differently to the preparation apparatus then the resultant e(O), and hence f’(t), will differ. This may be the case for preparation by optrcal excrtation in which the different nodal patterns m the wavefunctions of the hvo systems lead to expected differences m oscillator strengths. Thus, further comparrsons of irregular and regular intramolecular dynamics requue explicit consrderation of the preparation step. Such work is currently m progress [9] (se2 also ref. [lo]).
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References J
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Ftg 3. P(t) versus f for (a) stadmm system and (b) scaled u = I 739) osctllator system Here E, = -0 07 au. ZI = 0 03 au and both tmttal $ (0) contatn 89 energy etgenstates. Note small ordtnate scale.
111M.V.
Berry, in. Topms tn nonlinear mechanics, eb S. Jorna (AJP, New York, 1978); P. Brtuner, Advan Chem Phys , to be published; hf. Tabor, Advan. Chem. Phys , to be published; R D. Levme, Advan. Cbem. Phys, to be published_
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R-G. Bray and bf_J. Berry, J Chem. Phys 71 (1979) 4909, and references therem [3 J S A. Rice, m. Advances m laser chematry, ed. A Zewell (Sprmger, Berlm, 1978). [4] K S J Nordholm and S A. Rice, J Chem. Phys 61 (1974) 203,768;62 (1973) 157; E. Heller, Chem Phys. Lerters 60 (1979) 338; bf V. Berry, Phil Trans Roy. Sot 287 (1977) 137. [S] I C Percival, Advan Chem. Phys 36 (1977) 1, G hf Zaslavsky, Zh Eksp Teor. FIZ. 73 (1977) 2089, hi V Berry and hf Tabor, Proc. Roy. Sot A356 (1977) 375: R hi. Stmltt. N C Handy and W H hfdler. J. Chem Phys 71 (1979) 3311
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[6 j S W. LicDonald and A N Kaufman, Phys Rev Letters 42 (1979) 1189. [7] G. Benettmand J -iU Strelcyn, Phqs Rev. Al7 (1978) 773 [S] J Jortner, S A Rice and R hf Hochstrasser, Advan Photochem 7 (1969) 149, KE. Freed. Chem Phys Letters 42 (1976) 600. [91 hf Shapuo and P Brumer. work m progress [iOJ E J Heiler, J.Chem Phys 72 (1980) 1337,and reie:ences therem. R A. hfarcus. m Proceedmgs of the 3rd Internatronal Congress on Quantum Chemlsfry, ed B Pullman (Reldel, Dordrrcht, 1980)