3198
J. Phys. Chem. 1985,89, 3198-3201
Eigenmode Analysis of Vibrational and Rotational Energy Relaxation in Nonlinear Systems Kenneth Haug and Donald G. Truhlar* Department of Chemistry, University of Minnesota, Minneapolis, Minnesota 55455 (Received: May 22, 1985)
We calculate the relaxation rate constant for a second-order system by eigenanalysis of a master equation that is linearized by expansion about equilibrium. The results are in excellent agreement with numerical integration of the nonlinear master equation even for cases where the second-order terms change the relaxation rate by over an order of magnitude, and the observed relaxation rate is 2-3 orders of magnitude larger than its value close to equilibrium. The eigenanalysis is more efficient than numerical integration and provides additional insights into the relaxation mechanisms.
Introduction First-order and pseudo-first-order systems are governed by linear master equations which can be solved by a matrix eigenanalysis technique.’S2 Relaxation in these linear systems can be interpreted in terms of eigenmodes of relaxation, each associated with a particular time constant which is the eigenvalue for that node.^^^ This eigenanalysis gives useful physical insight into complicated relaxation systems and provides a rigorous analytic framework for their interpretation. Second-order systems are governed by nonlinear master equations, and for this reason they have been more difficult to study. Although standard linearization processes can be applied to nonlinear master equation^,^ these procedures have only recently been shown to be useful for numerical calculations.6 The usual approach to solving nonlinear master equations is numerical integration, which is a general technique but often very time consuming and not as illuminating as eigenanalysis. In the present communication we show that matrix eigenanalysis techniques can also be applied for both quantitative and qualitative interpretations of nonlinear relaxation processes, such as vibrational and rotational energy transfer in nondilute molecular gases. For the system chosen for the present applications, as well as for the previous applications to bimolecular reaction rates,6 the more efficient and physically more insightful linearized eigenanalysis approach yields observable rate constants in good quantitative agreement with the results of the numerical integration procedure, even for highly nonlinear systems.
The Relaxing System To compare the study of relaxation rates by eigenanalysis and numerial integration techniques we choose as a benchmark the system of para-hydrogen in the presence of helium over the H2 mole fraction range of 0 to 1 at T = 500 K. This system has previously been studied by Moise and Pritchard using numerical integrati~n.~Ten para-H2 states are included, with (v,J) = (O,O), (021, (0,4), (0,6),(0,8), (OJO), (LO), (1,2), ( 1 4 , and (1,6), where u and J denote vibrational and rotational quantum numbers. The state-to-state rate constants are the same as those by Moise and P r i t ~ h a r d . In ~ particular we include both first-order processes of the type H2(vi,Ji) + M H ~ ( v I J I+) M (R1) where M denotes H2 in any state or the diluent and the stateto-state rate constant is assumed to be the same for all M, and +
~~
(1)I. Oppenheim, K.E. Shuler, and G. H. Weiss, A h . Mol. Relaxation Process, 1, 13 (1967). (2)H.Haken, ‘Synergetics: An Introduction”, 3rd ed, Springer-Verlag, Berlin, 1983,pp 92-94. (3) H. 0. Pritchard and N. I. Labib, Can. J. Chem., 54, 329 (1975). (4) H.0.Pritchard, “The Quantum Theory of Unimolecular Reactions”, Cambridge University Press, London, 1984. (5) N.S.Snider and J. Ross, J. Chem. Phys., 44, 1087 (1966). (6) C.Lim and D.G. Truhlar, Chem. Phys. Let?., 114,253(1984);C.Lim and D. G. Truhlar, to be submitted for publication. (7)A. Moise and H. 0. Pritchard, Can. J . Chem., 59, 1277 (1981).
0022-3654/85/2089-3 198$01S O / O
second-order processes of the type H 2 ( ~ i J i )+ H,(Vj,J,i)
+
H ~ ( v I J /+) H,(um,Jm)
(R2)
The rate constants for (Rl) are k,~, and those for (R2) are kiplm. The master equation for the H2-He mixture can then be written as d - ni = C(kl+[M]nl - ki.+[M]ni) dt I C [( 1 + 6ij)klm,ijnlnm - (1 6ij)kiplmninj], i = 1, 10 (1)
+
+
jlm m>l
In this equation, [MI is the total concentration, which equals 3.5 x ioi9 ~ m - ~ . The first sum on the right-hand side of eq 1 corresponds to the master equation in the limit of H2 infinitely dilute in M, and it is pseudo first order. The nonlinearity is present in the second sum, which accounts for the second-order processes. If we divide each side of eq 1 by the total H2concentration and define xi as the fractional population of H2 state i xi = n i / Z n i = ni/[H2] i
(2)
and XH,as the mole fraction of H2
XH,= Cni/ [MI = W z J /[MI I then eq 1 can be rewritten as d - xi = C(k/+X,- ki4,Xi) 1 dt x H , C [(I ~ i , ) k , ~ - . i , x I-~ (1 ,
(3)
+
JIm
+
+ 6ij)kip~mxixj~,i = 1, 10
m>/
(4) where we have defined k = k[M] for all the rate constants k. In this form it is clear that the second-order processes enter this master equation in direct proportion to the mole fraction of H2 in the mixture. Complete details of the rate constants used in the master equation are given in the Appendix, which is included in the supplementary material for this Letter.
Methods We solve the coupled set of equations, eq 4, by two methods. First we use a standard predictor-corrector method for stiff systems, namely the EPISODE program,s to numerically integrate the equations for a given initial condition. (We use the backward-differentiation formula option and the chord method with the Jacobian generated internally by finite differences. A local error bound of 1104 and an initial step size of SlO-I4 s are used.) This directly gives the time evolution of all the populations and we calculate the relaxation rate coefficient as in ref 7, namely (8) A. C. Hindmarsh and G. D. Byrne, Lawrence Livermore Laboratory Report UCID-30112,1975.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985 3199
Letters X = -d(ln h E ) / d t
where hE(t) = E ( t ) - E,, E ( t ) is the average internal energy of H2 at time t , and E , is the Boltzmann equilibrium value of E . Typically X is a decreasing function o f t , but it may exhibit one or more quasisteady regimes. For the second method of solution the master equation is linearized about equilibrium and symmetrized by defining a scaled vector w of displacements from equilibrium, with components wj = ( X i - xt)/(xf)1/2
3.5
(5)
(6)
where x: is the value of xiat equilibrium. If the master equation is expanded to first order in the wi, it becomes d -& 0 = -sw (7)
3.0
-
d
I
vv) 0
Z
2.0
where S is a matrix; eq 7 is linear. The eigenvalues A, and eigenvectors x, of S are defined by
sx,
=
X,x,
(8)
Because the rate constants satisfy microscopic reversibility, the matrix S is symmetric, the eigenvalues are real, and the eigenvectors may be taken to satisfy
x:xu = 6,u
2.5
. I
x
(9)
Equation 8 is solved by the EISPACK system? Since the displaced system always returns to equilibrium, the eigenvalue spectrum is positive semidefinite with one zero eigenvalue. We order the eigenvaluesas 0 = A,, < XI < X2 < ... < AN-, where N is the number of states included (10 in the present example). The solution of eq 7 is
1.5 0
1
.5
Mole fraction o f H2 Figure 1. Comparison of X from the numerical integration procedure to XI from the eigenanalysis as a function of XH2,the mole fraction of H2. The symbols are XI and the curve is a fit to A.
N- 1
wi
=
, X I.8 -4J
(10)
P'O
or N- 1
xi(t) = $ + (xF)1/2C21g,xi,e-A~' ,=I
(1 1)
where 93, is a constant of integration. Equation 11 shows that the pth normalized eigenmode of relaxation has the components
mi, = . N r ( X : ) l / 2 X i ,
(12)
with the normalization constant given by N
.N, = ( Cexi;)-'/2 i- 1
(13)
Since the total number of H2molecules is conserved in this system
I
.
,
0.5
N
Cmi,,=O, p = 1 , 2 ,..., N - 1
i- I
(14)
Writing eq 11 in terms of the normalized relaxation eigenmodes gives N- 1
x i ( t ) = A$
+ ,=I
b,mi,e-xJ
(15)
where b, = S,/N,.
Results Three sets of initial fractional populations of the H2states were considered and will be denoted as cases A, B, and C . In each case, all states are first populated according to a Boltzmann distribution at the ambient temperature, 500 K, and then 0.0622% of the ground state, (O,O), is promoted to one particular excited state: (1,O) for case A, (0,lO) for case B, and (0,4) for case C . These promotions increase the fractional populations of these excited states to 2.01 X lo4, 1.99 X lo4, and 9.98 X which are (9) B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, N. C. Klema, and C. B. Moler, 'Lecture Notes in Computer Science", Vol. 6, G. Goos and J. Hartmanis, Eds.,Springer-Verlag,Berlin, 1976.
.
,
0.0
(E(t)-Eeq) / (E(01 -Eeq) Figure 2. Common logarithms of the relaxation time in s-l computed by numerical integration and eq 5 as functions of the ratio of the deviation of the energy at a given time t from the equilibrium value to the initial value of this deviation. Three initial conditions, cases A, B, and C, are shown. We begin the curves at 0.98 of the abscissa to exclude the transient regime. Just outside the right axis are given the common logarithms of A,, in s-I for p = 1 , 2, 3 from the eigenanalysis.
higher than the equilibrium values by factors of 100, 1023, and 1.002, r.espectively. For case A, the quasisteady values of X as calculated by the numerical integration procedure and the values of XI as obtained by eigenanalysis are shown in Figure 1, where they are plotted against mole fraction of H,;the two sets of results agree with each other to three significant figures, and they also completely reproduce the results of ref 7. Figure 2 shows the time-dependent values of X for XH2= 1 for all three cases. The left of the plot corresponds to t = 0 and the right to t a. Results are not shown for values of the abscissa greater than 0.98 because this includes the transient period during which A oscillates widely. For comparison to the curves obtained
-
3200
The Journal of Physical Chemistry, Vol. 89, No. 15, 1985
by numerical integration, the lowest three nonzero eigenvalues are shown at the right-hand side of the figure. The lowest, solid curve in Figure 2 corresponds to case A. In this case, after the transient period, all of the energy is relaxed with X = 3.03 X IO5 s-I. It can be seen that the quasisteady value of X for case A is in excellent agreement with XI, as already discussed in conjunction with Figure 1. The middle and upper curves in Figure 2 correspond to cases B and C. It can be seen that, in each case, one eigenmode dominates the relaxation spectrum and the quasisteady values of the relaxation times agree very well with the eigenvalues of the linearized master equation. This is true even though the quasisteady states in these cases do not persist all the way to equilibrium; in particular, if A,, denotes the value of X near equilibrium (Aq = Xl), then (A - Xeq)/Xeq is not small for these quasisteady states. In case B most of the energy relaxes according to mode 2, but the last 2l/,% relaxes according to mode 1. For case C the quasisteady state corresponding to mode 3 persists even longer, and the beginning of the decrease in X is only just barely visible at the right side of the plot. When the abscissa is IO4, X has still only decreased to 46% of X3; this decreases log [A(&)] to 7.32, which would be visible on the plot except that long before this point the curve has disappeared into the right axis. We have also used the eigenvalue-eigenvector solution to reproduce the entire time dependence of X(t) obtained by numerical integration. For example, for case B we obtained the 3,by taking an inner product with the initial conditions under the assumption that the linearized problem represents the system even at early times. With this assumption, X ( t ) calculated by eq 5 using the populations of eq 15 agrees with X(t) computed from the populations obtained by numerical integration to at least three significant figures for all times. In additional calculations not presented here, when xi - x: is allowed to be larger than it is here (where it is 1 2 X we find larger deviations of X ( t ) from the values predicted by eigenanalysis. We note that only for XH, = 0 is the master equation linear and that the three lowest relaxation times at XHz= 1 are greater than those for XH2= 0 by factors of 1.78 (II. = l ) , 16.1 (K = 2), and 44.5 ( p = 3). Therefore the validity of the linearization extends to highly nonlinear master equations. This validity is not dependent on the nearly linear dependence of XI on XH2,as we have also (in separate calculations not presented here) used linearization to reproduce numerical integration results for a system where A, is very nonlinear in XH2,An advantage of eigenanalysis is that the computational efficiency is much greater. When the calculations are performed with a VAX 11/780 computer with floating point accelerator, the three integrations to generate Figure 3 take about 6.0 X l o 1 min CPU time while the linearized eigenanalysis takes 0.7 min CPU time, an improvement of nearly two orders of magnitude. We stress though that the intrepretative advantages of eigenmode analysis for analyzing the characteristics of the modes of relaxation, as discussed below, are at least as important as the advantage in computational efficiency. It is probably useful to emphasize that the present eigenanalysis results are based on including all first-order terms in the Taylor expansion of the master equation about equilibrium. In contrast, Pritchard et a1.I0 have shown that a simple linearization based on replacing some of the ni by n; is in error by a factor of about 4 even for small XH2 (XH, = 0.02). As mentioned above, and as utilized very elegantly and informatively by Pritchard and co-workers for linear system^,^^^^" the eigenanalysis also yields the eigenmodes of relaxation. Figure 3 presents the normalized eigenmodes corresponding to XI, A,, and X,. The eigenmodes are presented for H2 mole fractions of 0 and 1. It is clear that mode 1 is primarily a vibrational relaxation, and it is reasonable that it dominates the decay when the (1,O) state is excited, as we observed in case A. Although X,(XH,=l) N 1.8Xl(XH,=O), the figure shows that eigenmode character does (10) H. 0. Pritchard, N. I. Labib, and A. Lakshmi, Can. J . Chem., 57, 1115 (1979). (11) H. 0. Pritchard, Can. J . Chem., 54, 2372 (1976).
Letters 2
1
-.a
h.8
0
3
h.8
0
.8
0
Components o f eigenmodes Figure 3. The components of three normalized relaxation eigenmodes, mi?, from eq 12 for = 1, 2, 3 as labeled at the top of the figure. The solid lines are for the limit as the mole fraction of H2tends to zero. The dotted lines are for the mole fraction of H2equal to unity.
r I I
0
1
.
.
.
.
.
.
.
.
.
2
Energy c u t o f f l103c~-il Figure 4. The lowest nonzero eigenvalue XI for X,, = 1 as a function of the energy cutoff c used in the calculation of not change very much as X,, is varied. In contrast, the characters of the second and third modes change dramatically as the mole fraction of H2 increases. It will be very interesting in later work to examine this kind of character change in detail and to understand its causes in the input rate constants and its effects on the observables in various experimental situations, both for the present systems and also for more complicated systems where characterizing the relaxation pathways by eigenanalysis may provide improved understanding of coupled-mode effects. All results presented so far are based on the state-to-state rate constants of Moise and P r i t ~ h a r d which ,~ involve a cutoff parameter c of lo00 m-'. Rate constants for second-order processes are set equal to zero if the internal energy change exceeds The value of c has a large effect on the calculated relaxation rates, as shown in Figure 4, for which it is varied. The results in the figure are encouraging, however, in indicating that the vibrational
J . Phys. Chem. 1985,89, 3201-3202
3201
relaxation time is reasonably insensitive to variations in c in the vicinity of 1000 cm-I. This plateau behavior indicates that the choice of 1000 cm-' is a physically reasonable value.
added computational efficiency and physical insight offered by the eigenvalue techniques, we believe they will prove to be useful tools to study nonlinear systems.
Conclusion
Acknowledgment. The authors are grateful to H. 0.Pritchard for providing numerical values of the state-to-state rate constants used in ref 7. This work was supported in part by the National Science Foundation under Grant No. CHE83-17944. Registry No. H2, 1333-74-0.
We have shown that eigenanalysis of a linearized master equation can be used to obtain relaxation rates in agreement with those obtained by numerical integration, even for a system where the non-first-order terms in the original master equation have a large effect. We anticipate that other systems with important V-V energy transfer terms can be treated with the same efficiency and insight that simpler systems involving an oscillator dilute in an inert gas have been amenable to previously. Because of the
Supplementary Material Available: An appendix containing rate constant data (3 pages). Ordering information is given on any current masthead page.
IVR in Isolated Molecules with Nearby Electronic States J. L. Knee, L. R. Khundkar, and A. H. Zewail* Arthur Amos Noyes Laboratory of Chemical Physics,+California Institute of Technology, Pasadena, California 91 125 (Received: May 28, 1985)
In this Letter, we report direct picosecond measurements of decay rates in a beam of isoquinoline, excited to different vibrational states in the second (S,) excited electronic state (m*). IVR and IER are discussed in relation to the proximity of the S2 electronic state to a nearby S,(na*) electronic state, and to internal conversion to the ground state.
By analogy with IVR (intramolecular vibrational-energy redistribution), IER is the process which describes intramolecular electronic-energy redistribution. IVR occurs among the vibrational states of a molecule on a single potential energy surface and is a result of anharmonic and/or Coriolis interactions. IER, on the other hand, involves coupling between electronic states and is mediated by the vibrational states in both electronic manifolds. In molecules with nearby electronic states IER is expected to be important and to play a major role in the nonradiative decay of these molecules. In this Letter we report direct picosecond measurements of decay rates in a beam of 2-azanaphthalene (isoquinoline), excited to different vibrational states in the second (S,) excited electronic state (?r?r*). This Sz state is located above S1 (nr*) with an energy separation of only 1 100 cm-l. To obtain decay rates for different vibrational excitations in S2,we used a picosecond pumpprobe arrangement in conjunction with a molecular beam apparatus.' The (tunable) pump excites the molecule to the different levels of the 3 100-8, absorption (m* state)2 and the probe monitors the population decay of the excited state using picosecond/mass spectrometry with a resolution of -2 ps.'s3 The beam conditions in these experiments are as follows: carrier gas, He; stagnation pressure, 20 psi; and the sample temperature is 50 O C . In all these experiments the two laser pulses interact with the molecular beam 9 cm downstream from the skimmer, and 10 cm from the nozzle. Figure 1 shows the experimental results obtained when exciting isoquinoline to the 732- and the 1400-cm-' levels in S2(these are the excess energies in S2and they correspond to 1832 and 2500 cm-' of excess energy above the lowest S1 (m*)state. We shall use excess energy here to mean Sz excess energy and add to it the 1100-cm-l energy separation whenever desired). Our measurements at zero excess energy and at 510 cm-' show longer decay time constants (subnanosecond and 60 ps, re~pectively).~The decays measured for the 732- and 1400-cm-I bands can be fit to
-
-
* Camille
& Henry Dreyfus Foundation Teacher-Scholar. Contribution No. 7196.
0022- 36 54/ 8 5/ 208 9-3 20 1$0 1.50/ 0
single exponentials with T = 24.7 f 0.4 ps and T = 7.9 f 0.4 ps, respectively. Because of the proximity of SIand S2there is a near-resonance vibronic coupling of levels in Sl(n?r*) with the vibrationless level of S,(lS,O)) through an out-of-plane vibration(s). This coupling (known as Herzberg-Teller coupling) has been analyzed thoroughly for isoquinoline by Fischer and Knight5 and Fischer and Naaman.6 Spectroscopic manifestations of this coupling in fluorescence and absorption spectra are evident, and careful analysisSof the spectra provide details of the coupling mechanism. Furthermore, because of this proximity effect, Lim and his coworkers' suggested that internal conversion from S2(a7r*)will be enhanced as a result of this coupling with S,. This is because SI(n?r*) is distorted and the out-of-plane mode responsible for the SI-S2coupling will be a good accepting mode (due to the large geometry change) for internal conversion to So. Indeed, this is evident from lifetime measurements made by Felker and Zewail:* the lifetime of the S2(?r7r*)origin of isolated isoquinoline (beam condition) is subnanosecond which is atypical of mr* states; complexing isoquinoline with water or alcohol (in beams) shifts S,(n?r*)to higher energy and the lifetime of S2becomes typical (1) J. L. Knee, F. E. Doany, and A. H. Zewail, J . Chem. Phys., 82, 1042 (1985). (2) L. M. Logan and I. G. Ross, Acro Phys. Polon., 34, 721 (1968) (3) J. L. Knee, L. R. Khundkar, and A. H. Zewail, J . Chem. Phys., 82, 4715 (1985). (4) The lifetime of the S2origin is -380 ps (see ref 8), and our results (0.6 0.3 ns) here are consistent with this finding. The measurements reported here for the 510-cm-' excitation give 7 = 65 f 10 ps. There appears to be
*
a fast decay component on the transient for the 510-cm" excitation, which could be due to fast IVR from this level to other levels in S2. The assignment of the 510-cm" level as due to a fundamental or an overtone of a low-frequency mode is unclear at the moment. ( 5 ) G. Fischer and A. E. W. Knight, Chem. Phys., 17, 327 (1976). (6) G . Fischer and R. Naaman, Chem. Phys., 12, 367 (1976). (7) W. A. Wassam, Jr., and E. C. Lim, J . Chem. Phys., 68,433 (1978); B. E. Forch, S. Okajima, and E. C. Lim, Chem. Phys. Lerr., 108.311 (1984). (8) P. M. Felker and A. H. Zewail, Chem. Phys. Left., 94,448,454 (1983); J . Chem. Phys., 78, 5266 (1983).
0 19 8 5 American Chemical Society
