J. Phys. Chem. 1981, 85, 2159-2163
2159
Classical Trajectory Studies of Multiphoton and Overtone Absorption of HF Kurt M. Chrlstoffel and Joel M. Bowman* Department of Chemistry, Illinols Institute of Technology, Chicago, Iiiinois 606 16 (Received: April 24, 198 1; I n Final Form: May 27, 198 1)
Multiphoton and overtone absorption of intense laser radiation by HF is studied by classical trajectory simulation. The degree of multiphoton absorption and emission is studied as a function of the initial vibrational state of HF and analyzed in terms of the small signal power spectrum of HF. Both linear and nonlinear electrical dipole moments are considered. The probability for the first overtone transition is calculated as a function of time up to 1 ps for both a linear and nonlinear electrical dipole moment.
I. Introduction The interest of experimentalists and theoreticians alike has been aroused in recent years by the development of high-powered lasers which can be used to induce molecular processes previously termed forbidden.’ Of these processes laser-induced absorption of multiple infrared photons to cause bond rupture and subsequent chemistry in polyatomic molecules and use of visible wavelength lasers to pump vibrational overtones as a means of inducing isomerization are of particular interest to chemists. The laser-induced multiphoton absorption (MPA) of small model molecules has been extensively studied by using classical trajectory techniques.2-8 Until recently the only quantal treatments of the MPA process were studies of a model one-dimensional Morse oscillator in a laser field.2i9 Walker and Preston in their study of this model system found that the average quantal and classical descriptions of MPA are in good agreement, at least on the short time scale they considered.2 More recently, Noid et a1.8found “encouraging agreement between the classical and quantum simulation of the infrared multiphoton process’’ for a model bent triatomic molecule exposed to one or two laser fields for longer periods of time. Classical trajectory studies of MPA have indicated the existence of an incubation period which may be related to a transition in the nature of the classical motion from quasiperiodic to s t o ~ h a s t i c .Schematics ~ of molecular energy eigenvalue spectra have traditionally been divided into three regimes: discrete states, quasicontinuum, and continuum. The incubation period cited in the classical trajectory studies has been attributed to the time necessary to pass through the bottleneck in the discrete section of the spectrum which arises when attempts are made to pump transitions of decreasing frequency with light of constant frequency. Classical trajectory studies have shown that a second laser tuned far to the red of the fundamental absorption provides an effective way of loosening this discrete ~tranglehold.~,~ In diatomic molecules this discrete stranglehold should be particularly severe. Yet classical trajectory studies of model diatomics in one or two laser fields have shown that multiphoton dissociation of diatomics may be possible. In this paper we will use the classical trajectory method to divide the energy spectrum of a diatomic molecule into three regimes characterized by their response to laser stimulation and to illuminate the dynamics of absorption in each regime. The phenomenon of vibrational overtone absorption also has been the focus of recent theoretica11@14and experimental13-lsinvestigations. CH stretch overtones have been *Alfred P. Sloan Fellow 0022-3654/81/2085-2159$01.25/0
used to induce isomerization of methyl i s o ~ y a n i d e ~and ~J~*~ allyl isocyanide21 by Berry and co-workers. Experimentally, evidence for single, local mode overtone excitation has been given.13-19 The stationary state local mode description for water1°J2and CH stretches in benzene13J4has been given good theoretical support, as well. Nagy and Hase have presented some classical trajectory evidence that CH in benzene, once highly excited, retains its energy for at least 2 ps.’l However, to our knowledge no classical or quantum dynamical investigations similar to those described above for the MPA process have been reported for the vibrational overtone absorption process. Standard textbook treatments of vibrational spectroscopy treat molecular vibrations as oscillators which are nearly harmonic if their degree of excitation is sufficiently low. At these energies the displacement from equilibrium is small and so the molecular dipole moment is justifiably treated as a linear function of the internuclear separation. This standard harmonic oscillator-linear dipole moment treatment of molecular vibrations results in the standard selection “rules”, An = f l . Overtone transitions, An > 1, become possible through the effects of “mechanical” anharmonicity, i.e., the presence of nonquadratic terms in the vibrational potential or through effects of “electrical” (1) (a) R. V. Ambartsumian and V. S. Letokhov in “Chemical and Biochemical Applications of Lasers”, Vol. 3, C. B. Moore, Ed., Academic Press, New York, 1977 Chapter 2. (b) N. Bloembergen and E. Yablonovitch, Phys. Today, 23 (May 1978). (2) R. B. Walker and R. K. Preston, J . Chem. Phys., 67,2017 (1977). (3) D. W. Noid, M. L. Koszykowski, R. A. Marcus. and J. D. McDonald. Chem. Phvs. Lett.. 51. 540 (1977). (4) K. D. Hahsel, Chem. Phys.‘Lett., 57, 619 (1978). (5) D. W. Noid and J. R. Stine, Chem. Phys. Lett., 65, 153 (1979). (6) J. R. Stine and D. W. Noid, Opt. Commun., 31, 161 (1979). (7) D. Poppe, Chem. Phys., 45, 371 (1980). (8) D. W. Noid, C. Bottcher, and M. L. Koszykowski, Chem. Phys. Lett.. 72. 397 (1980). (9) (ai S. Leasure and R. E. Wyatt, Chem. Phys. Lett., 61,625 (1979); (b) Opt. Eng., 19, 46 (1980). (10) M. Elert, P. Stannard, and W. Gelbart, J. Chem. Phys., 67,5495 (1 ~ 977) - - .,. .
(11) P. J. Nagy and W. L. Hase, Chern. Phys. Lett., 54, 73 (1978). (12) H. S. Moller and 0. Sonnich Mortensen, Chem. Phys. Lett., 66, 539 (1979). (13) B. R. Henry, Acc. Chem. Res., 10, 207 (1977). (14) A. Albrecht in “Advances in Laser Chemistry”, A. H. Zewail, Ed., Springer Series in Chemical Physics 3, Springer-Verlag, Berlin, 1978, p 235, and references therein. (15) K. V. Reddy, R. G. Bray, and M. J. Berry in ref 14, p 48. (16) R. G. Bray and M. J. Berry, J . Chem. Phys., 71, 4909 (1979). (17) J. W. Perry and A. H. Zewail, J . Chem. Phys., 70, 582 (1979). (18) D. D. Smith and A. H. Zewail, J . Chem. Phys., 71, 540 (1979). (19) K. V. Reddy and M. J. Berry, Chem. Phys. Lett., 52, 111(1972). 67, (20) K. V. Reddy and M. J. Berry, Faraday Discuss. Chem. SOC., 188 (1979). (21) K. V. Reddy and M. J. Berry, Chern. Phys. Lett., 66,223 (1979).
0 1981 American Chemical Society
2160
The Journal of Physical Chemistry, Vol. 85, No. 15, 1981
Letters
TABLE I: Potential Energy and Dipole Moment Data for H F
D = 6.124 eV m = 0.950 amu number of bound states = 24
= 1.174 (bohr radius)-’ R, = 1.7329 bohr radius
oi
p , ( R ) = a t bR a = 1.1492 [eV/(bohr r a d i ~ s )‘I’ ~]
b = 1.5529 (eV.bohr radius)”’
p c , ( R )= a t bR + cR’ a = 2.1393 [eV/(bohr c = - 1.1242 [eV/(bohr radius)]
b = 5.4742 (eV.Lohr radius)”’
p 3 ( R )= aR exp(-bR4) a = 2.369 (eV.bohr radius)llz
b = 0.0064 (bohr radius)-4
anharmonicity, i.e., nonlinearity of the dipole moment function, or both. One of the goals of this paper will be to use classical trajectories to determine the relative importance of these two effects for the HF molecule. In this paper classical trajectory methods are used to illuminate the dynamics of the features of laser-induced absorption described above. Section I1 is devoted to a description of the model studied and of the computational details. Section I11 presents our results and a discussion of their importance in adding to the understanding of laser-induced processes. 11. Model a n d Computational Methods The molecule studied is the nonrotating Morse oscillator model of HF described previously by Walker and Prestona2 The intense laser radiation is described as usual by a large amplitude sinusoidal electric field. The field strength used for all results reported in this paper is 0.96 V/bohr radius. The molecule and radiation field are assumed to interact solely through the classical dipole interaction pE. Three different fits to ab initio data for the dipole moment of HF22were used at different times in this study. The parameters describing the H F model and the functional forms of, and parameters for, the three dipole moments are given in Table I. Two of the dipole moment functions, pl and pz, and the H F potential energy are plotted as functions of the internuclear separation in Figure 1. p3 is the same dipole moment as used by Stine and Noid6 and a plot of p3 vs. internuclear separation is given in Figure 1 of ref 6. The molecule’s Hamiltonian is then
H(P,R,t)= Ho(P,R) + /L.ECOS Ut
(1)
where
P2 Ho(P,R)= - + D[1 - exp(-a(R - R0)1]2 2m The two Hamilton’s equations for the molecule
(2)
are integrated numerically for an ensemble of initial conditions. To obtain the ensemble of initial conditions we simply integrate the Hamilton equations for the unperturbed Morse oscillator with the desired initial energy through one vibrational period. A suitable ensemble of initial values of R can be obtained by sampling R at uniform time steps throughout the vibrational period. The numerical integrations are performed by using a standard Adams-Moulton predictor/corrector integrator which employs a Runge-Kutta-Gill initiator. A time step of 4.84 X 1 O - l ’ ~ was used for all trajectories. Back-integration restored initial conditions to 1part in lo5for trajectories run for 1 ps. ~~~~
~
~
(22) T. H. Dunning, unpublished results.
7 W >
w CT
z W B
+ z
W
I
04 0
2 w _J
0
0
g
F$,F (bohr)
Flgure 1. Potentlal energy and dipole moments (p,, p2, and ab initio data22(0))as a function of the internuclear separation of the diatomlc molecule.
111. Results a n d Discussion A. Multiphoton Absorption. In our first study of laser-induced processes we calculated the short time absorption (emission) behavior of H F in a laser field as a function of the initial vibrational excitation. Batches of 24 trajectories were run at each of the energies corresponding to the first eleven quantal vibrational states (Le., with n = 0-10). The frequency of the driving field is 3922 cm-’, a frequency red-shifted from the n = 0 fundamental frequency by about 170 cm-’. Two sets of trajectories were run at each energy, one set employing pl for the dipole moment and the other using k 3 for the dipole moment, to isolate the effects of mechanical and electrical anharmonicities. All trajectories were integrated for 100 optical cycles, i.e., about 0.85 ps, where the optical period is r = 2aw. For each trajectory i in an ensemble of N (= 24) trajectories, the energy E i ( t )was recorded every 25 time steps. The average total energy E(t) was calculated at each recording time according to
E(t) =
1
c E;(t) N
1=1
(4)
For each ensemble of trajectories the average energy absorbed was calculated as
where T is the total duration of the trajectories and Eno is the initial vibrational energy corresponding to the quantal vibrational state In). The integral in eq 5 was calculated by using a standard Bode’s rule quadrature formula. In Figure 2 we have shown E ( t ) for the ensembles of trajectories initiated with energies corresponding to quantal states with n = 1,3,5,8and using p3 as the dipole
The Journal of Physical Chemistry, Vol. 85, No. 15, 1981 2161
Letters 0.040
0.100
0.036
0.096
-2 0.032
0.092
n=5
-M 2
t
c
IW
0.028
n=l
0.080
112 psec I
0.024 0.0
1.0
1
I
2.0
3.0
4.0
0.084
0.064
0.160
0.056
0.152
0.048
0.144
n=8
0 YI)
2
--5 t
c
IW
0.040
I 0032 0.0
1.0
112 psec
psec
2 .o
3.0
t/104 (a.t.u.)
4.0
0.128 0.0
I
I .o
I
2.0 t/104(a.t.u.)
I
3.0
4.0
Flgure 2. € ( T ) given by eq 4 for ensembles of trajectories initiated with energies corresponding to vibrational quantum numbers n = 1, 3, 5,
a.
function. For n = 1, clearly there is significant net absorption even on this subpicosecond time scale. This is easily understandable since our laser is tuned to the red of the fundamental vibrational transition and would nearly match the field-shifted 1 2 one-photon absorption. For n = 3 there is a marked loss of energy in this time scale. We might attribute this to the fact that the laser frequency more nearly matches the frequency for stimulated emission from this state than it matches the 3 4 absorption frequency. For n = 5 a “ringing” behavior is exhibited which results in a small net loss of energy. For n = 8 a pronounced absorption is exhibited which continues to increase with time. For n greater than 8 the absorption increases even faster with time and some dissociation occurs within 1 ps. In Figure 3 we plot a(t) as a function of initial vibrational excitation n. The results shown in Figure 3 reflect the considerations made in our discussion of Figure 2. We see that for n = 0, 1 significant absorption occurs as might
-
-
- -
be expected since the laser frequency nearly matches the frequencies of the one-quanta transitions 0 1, 1 2. Trajectories initiated with energies corresponding to the quantal states n = 2-6 exhibit an average loss in energy on this short time scale (i.e., stimulated emission occurs); if followed for longer times it is expected that these trajectories would eventually show a net absorption. For n I 7 we again see significant absorption, in fact for these initial energies several trajectories reach the dissociation energy in less than 100 optical cycles. How do we account for this behavior? First notice that the qualitative results differ little regardless of the form of the dipole moment. For n I 7 the nonlinear dipole moment function does enhance the absorption, however. This suggests that mechanical rather than electrical anharmonicity plays the major role in this behavior. We can investigate the role of mechanical anharmonicity in a straightforward manner. The unperturbed motion of the oscillator with energy E; as described by the displacement
2162
The Journal of Physical Chemistty, Vol. 85,No. 15, 1981
Letters 1.00 7
0.03
n=O 0.02
-., ---2
0.01
N
.-
X
0
0.50
L
0.00
c
IW
a
-0.01
-003 -Oo2
L 0
2
4
6
8
1
0.00
ttp----
I I
n=9
0
n Flgure 3. Average energy absorbed, A& T) of eq 5, for ensembles of trajectories initiated with energies corresponding to vibrational quantum numbers n = 0-10: (0)data obtained by using linear dipole approximation; (A)data obtained by using dipole moment of Stine and Noid.’
variable X ( t , n ) (= [R(t,n)- R,]) can be expanded in a Fourier series as
2ait sin - +
0.25
i.2
& n=17
2ait cos -
7,
where r, is the period of the unperturbed vibrational motion with energy E,O and the expansion coefficients are given as usual by
2ait Xi,2 = Jrndt X ( t ; n )sin Tn
2nit
= I r n d tX ( t ; n )cos 0
Tn
By an appropriate choice of boundary conditions, X(t=O) = 0, (6) is reduced to a Fourier sine series. Since this choice of boundary conditions results in the vanishing of all X;, we will suppress the superscripts (and for convenience the subscript n) in the following discussion. The infinite sum in (6) is truncated at an upper limit, m = 12, large enough that ~ X 1 2 ~ / 1) 7 the vibrational motion has become highly anharmonic. This is reflected in the X , by a significant increase in the ratio IX21/lXll. For example, for n = 1 ~ X 2 ~ /=~0.055; X l ~ however, for n = 7 this ratio has increased to 0.318. For n > 7 this ratio moves even nearer to unity. The magnitudes of the other X , (i > 2 ) have correspondingly increased also. For these high values of n we expect, therefore, that the oscillator would now be capable of absorbing a range of frequencies near 2w (in addition to the fundamental absorption). It is just at these values of n where X 2 is becoming comparable to X I that wd, the frequency of the laser radiation, approaches 20. We
w ( s e i ’ )x 1 0 - l ~ Flgure 4. Squares of the Fourier expansion coefficients 1X,I2 vs. frequency for Vibrational energies corresponding to quantum numbers n=: 0, 9, 17. The arrow indicates the frequency of the laser.
therefore feel that the sharp rise in absorption for energies greater than or equal to ET0can be accounted for by the onset of significant overtone absorption. Figure 4 which shows the normalized X,2 vs. frequency for three values of n, n = 0,9, 17, helps to clarify some of these points. The X;2 essentially represent the classical small signal power spectrum of this Morse oscillator. We expect that changes which occur in this spectrum as a function of the oscillator’s total energy will be indicative of similar changes in the true power-broadened and field-shifted spectra. Two gross features are readily discernible in Figure 4. First, the number of significant lines in the spectrum increases as a function of the total energy. This reflects the effect of increasing anharmonicity with increasing energy for a Morse oscillator. For n = 0, only a single fundamental absorption is discernible; for n = 17, five lines are seen on the scale of this figure. Secondly, the frequencies of these lines are significantly red shifted as the total energy increases. For example, for n = 9, the first overtone lies nearest Od, the driving frequency of the laser; for n = 17, the third overtone lies nearest wd. Recent e ~ p e r i m e n t a l ~and ~ p ~t h~e,o~r ~e t i ~ a lwork ~ , ~ has indicated that IR multiphoton dissociation can be induced by using much lower laser intensities when a laser of near-resonant frequency is supplemented by a second lower frequency laser. Figure 2 clearly indicates the existence of an absorption stranglehold for H F beyond the first 2 or 3 quanta of vibrational energy. Clearly if H F is exposed (23) (a) R. V. Ambartsumian, Y. A. Gorokhov, V. S. Letokhov, G. N. Makarov, A. A. Puretzkii, and N. P. Furzikov, Jetp Lett. (Engl. Trawl.), 23,194 (1976). (b) R. V. Ambartsumian, N. P. Furzikov, Y. A. Gorokhov, V. S. Letokhov, G. N. Makarov, and A. A. Puretzkii, Opt. Commun., 18, 517 (1976). (24) M. C. Gower and T. K. Gustafson, Opt. Commun., 23, 69 (1977).
J. Phys. Chem. 1981, 85. 2163-2165
-
to a second laser with a frequency near the resonant frequency of the 3 4 vibrational transition, this absorption stranglehold will be loosened. Much lower intensities can be used in the two-laser situation because each laser will only be required for pumping a limited number of nearresonant transitions. These transitions will initially be between adjacent vibrational energy levels; as the energy of the molecule increases these lasers will be nearly resonant with overtone transitions. B. Overtone Absorption. In the second part of this study we simulated continuous wave overtone excitation of HF using classical trajectories. In particular we sought to determine the relative importance of mechanical and electrical anharmonicity in promoting overtone absorption in our model HF molecule in a subpicosecond time scale. We considered only the first and strongest vibrational overtone transition, n = 0 to n = 2. Twenty-four trajectories were run for each of two forms of the dipole moment, pl,a linear form, and pz, a linear plus quadratic form. The driving frequency used was the field-free resonant frequency of the n = 0 to n = 2 transition, 7757.8 cm-l. Trajectories were followed for 1 ps. We obtained the time-dependent transition probability for the n = 0 to n = 2 transition as follows. First, the vibrational energy of the diatomic molecule was recorded every 25 time steps during the course of the integrations. A quantum number was assigned to each of these energies by a standard histogram method.25 If aE(n,n+l) = E(n+l) - E ( n ) is the quantum mechanical energy difference between vibrational states In 1) and In) and if E,] is a recorded classical energy, we assign the quantum number n if E(n)- 1/2AE(n-l,n) 5 EC1< E(n) + l/*AE(n, n+1). In particular we are interested in n = 2. Second, if we denote by N 2 ( t )the number of trajectories assigned the quantum number 2 a t time t , then the transition probability is given by
+
where N is the total number of trajectories, presently 24. A histogram approximation to PnE2(t)is obtained by considering finite values of At. (25) J. M. Bowman and A. Kuppermann, Chem. Phys. Lett., 12, 1 (1971).
2163
03
02
c
1,
N
a= 0.1
I psec
0
400 6 t (arbitrary units)
800
Figure 5. Histogram representation of Pn=&t ) for overtone excitation when y , (shown in Figure 1) is used as the dipole moment.
We found that Pn,l(t) and Pn=2(t) were zero for all times considered (up to 1ps) for the linear approximation to the dipole moment. For the quadratic approximation to the dipole moment P,,2(t) is shown in Figure 5 as a histogram where At of eq 8 is 25 integration steps. After an initial short time transient peak absorption, the absorption probability attains a fairly steady value of 0.12 f 0.04. From these results we can conclude that overtone excitation of an oscillator can occur in a subpicosecond time scale and that nonlinearity of the dipole moment function is critical for this to happen. To understand the second conclusion we note that WE contains a term proportional to RHF2(t)E(t). If R H F ( tis) given approximately by RHFO cos (ut + $), where w is the natural harmonic frequency, then Rm2(t)is (Rmo)2[1 + cos (2wt 2 $ ) ] / 2 . The cos 2wt term is evidently the source of a “resonant” absorption under the influence of a driving field, E ( t ) ,of frequency
+
2w.
An immediate speculation based on these limited results is the following. Overtone transitions of local mode bonds which have substantial ionic character can be pumped on a picosecond time scale. The basis for this, of course, is that bonds with strong covalent/ionic character (or more generally, a multiconfiguration ground electronic state) will have nonlinear dipole moment functions. Acknowledgment. This work was supported in part by the National Science Foundation.
Rate Constants for the Reaction of N,(A3ZU+,Y = 0, 1, and 2) with O2 M. P. Iannuzzi and F. Kaufman” Department of Chemistry, University of Pittsburgh, Pittsburgh, Pennsylvania 15260 (Received: April 30, 198 1)
The title reaction was studied at 298 K in a fast flow system by use of laser-induced fluorescence (LIF) detection of N2(A)at band heads of the first positive system to monitor the u = 0, 1, and 2 states. From a decrease of LIF intensity with increasing [O,] for fixed reaction times, applying an experimentally measured correction factor for the development of laminar flow, we obtained rate constants of (2.5 f 0.4) X 10-l2,(3.9 f 0.6) X and (4.3 i 0.7) X cm3 molecule-l s-l. These results are compared with published data and the reaction mechanism is discussed.
The lowest electronically excited state of nitrogen, N2(A3&+), has a long radiative lifetime’ (Vegard-Kaplan
-
emission, T 2 s ) , but is highly reactive. It plays an important role in the normal and perturbed upper atmo-
0022-3654/81/2085-2163$01.25/00 1981 American Chemical Society
