J. Phys. Chem. lQ83, 87,3025-3027
3025
Laser-Induced Isomerization of Trimethylenimine D. C. Clary' and J. P. Henshaw Department of Chemistry, UnlversHy of Manchester Institute of Sclence and Technology, Manchester M60 100, U.K. (Received: May 20, 1983)
Quantum calculations on a double minimum potential representing the ring puckering motion in trimethylenimine suggest that laser-induced isomerization is likely to occur for realistic laser intensities. Furthermore, measurements of intense laser absorption spectra might enable the number of local minima in a potential function to be determined.
I. Introduction In recent years, there has been considerable research on the interaction of intense laser radiation with molecules. Multiphoton excitation, in which a molecule can absorb many photons from a single-frequency infrared laser, has become an important new area.' Furthermore, intense infrared laser radiation has been used to excite vibrational overtone transitions,2 enhance chemical rea~tivity,~ and induce molecular isomerization? Many theories have been developed in the field of laser-molecule excitation and these have been recently reviewed.6 Here we present a theoretical study on the laser-induced isomerization of trimethylenimine (CH,),NH (also called azetidine)
a
b
The far-infrared spectrum of this molecule has been measured6 in the range 25-300 cm-' and explained in terms of energy levels derived from a double-well potential energy function in the ring puckering coordinate. Molecular eigenfunctions localized in one of these wells in the potential correspond to isomer a while those in the other well correspond to isomer b. Our aim is to examine whether tunneling through the isomerization barrier can be forced by laser radiation with a realistic intensity. This theoretical analysis is relevant to the recent experimental studies on laser-induced isomerization.* A recent theoretical study of isomerization in trimethylenimine has also been reported by Cribb and coworkers.' Furthermore, there have been several other theoretical papers on the dynamics of molecular isomerization.8 None of these studies, however, has considered isomerization induced by intense laser radiation. A stimulus to the present work was also made in a recent paper by Robiette and co-workers? They showed that the observed6infrared spectrum of trimethylenimine could also (1) King, D. S. Adu. Chem. Phys. 1982, 50, 105. (2) Douglas, D. J.; Moore, C. B. Chem. Phys. Lett. 1978, 57, 485. (3) hhfold, M. N. R.; Hancock, G. In 'Gas Kinetics and Energy
Transfer";h h m o r e P. G., Donovan, R. J. Ed.; Royal Society of Chemistry: London 1981, Specialist Periodical Report, p 73. (4) Reddy, K. V.; Berry, M. J. Faraday Discuss. Chem. SOC.1979,67, 188. Glatt, 1.; Yogev, A. Chem. Phys. Lett. 1981, 77, 228. Hall, R. B.; Kaldor, A. J. Chem. Phys. 1979, 70, 4027. Buechele, J. L.; Weitz, E.; Lewis, F. D. J. Amer. Chem. SOC.1981, 103, 3588. Benmair, R. M. J.; Yogev, A. Chem. Phys. Lett. 1983,95, 72. (5) Quack, M. Adv. Chem. Phys., 1982,50, 395. (6) Carreira, L. A.; Lord, R. C. J. Chem. Phys. 1969, 51, 2735. (7) Cribb, P. H.; Nordholm, S.; Hush, N. S. Chem. Phys. 1982,69,259. (8) Gray, S. K.; Miller, W. H.; Yamaguchi, Y.; Schaefer 111, H. F. J. Chem. Phys. 1980,73,2733. Chistoffel, K. M.; Bowman, Ibid. 1981, 74, 5057. Bunker, D. L.; Hase, W. L. Ibid. 1973,59,4621. (9) Robiette, A. G.; Borgers, T. R.; S t r a w , H. L. Mol. Phys. 1981,42, 1519.
O022-3654183/2087-3O25$0 1 .SO10
be explained in terms of the energy levels calculated with a single minimum potential energy function. Ab initio computations also provide some evidence for a single minimum potential.'O Findings such as these draw attention to the well-known problem, arising for many organic ring molecules, as to whether molecular conformations are governed by potentials with single, double, or many local minima.'l We propose that one way to test if a coordinate motion has a single or double minimum potential is to observe the absorption spectrum for intense laser radiation and compare this with the spectrum for radiation of normal intensity. If the potential has more than one minimum, then transitions between energy levels corresponding to wave functions localized in different wells in the potential might be observed for radiation with high intensity, but not for low intensity. Thus, a second aim of the present study is to calculate and compare the intense laser radiation absorption spectra corresponding to the proposed singleg and double6 minimum potentials for trimethylenimine and examine whether any differences occur for realistic laser intensities. To calculate the absorption spectra we used a quantum-dynamical method which is exact for a given Hamiltonian. The time-dependent Schrodinger equation is solved for the one-dimensional ring puckering motion interacting with intense laser radiation described by the semiclassical electric dipole approximation. The technique is our own adaption12 of a method described by Leasure and co-workers13which exploits Floquet theory.I4 Section 2 discusses the single and double minimum potential energy functions proposed for trimethylenimine and presents details of our theoretical approach for calculating the probabilities for laser-induced transitions between energy levels of these potential functions. The computed intense laser spectra are reported and discussed in section 3. Conclusions are in section 4.
2. Method a. Potential Energy Surfaces. The effective Hamiltonian governing the ring puckering motion in trimethylenimine is
Bo = --h2 d2 + V(X)
2m dx2 where m is an effective reduced mass6s9and x is the ring puckering coordinate which is the displacement of the line (10) Skancke, P. N.; Fogarasi, G.;Boggs,J. E. J. Mol. Struct. 1980,62, 259. Cremer, D.; Dorofeeva, 0. V.; Mastryukov, V. S. J. Mol. Struct. 1981, 75, 225. (11) Lister, D. G.; MacDonald, J. N.; Owen, N. L. "Internal Rotation and Inversion";Academic: London, 1978. (12) Clary, D. C. Mol. Phys. 1982, 46, 1099. (13) Leasure, S. C.; Milfeld, K. F.; Wyatt, R. E. J . Chem. Phys. 1981, 74, 6197. (14) Shirley, J. H. Phys. Reu. B 1965, 138, 979.
0 1983 American Chemical Society
3026
The Journal of Physical Chemistty, Vol. 87, No. 16, 1983
Letters
TABLE I: Transition Energies and Dipole Moment Matrix Elements for Single-Minimum Potential transition
1'0 1'2 3'5 2'3 3+4 4+5 a
energy/ cm -
dipole moment matrix element"/D
208.2 184.8 167.3 148.6 85.3 82.0
1.642 (-1) 2.390 (-1) -1.844 (-1) 2.974 (-1) 3.675 (-1) 4.527 (-1)
0 . Y)
I = 1 MW cm-2
0.25
0.20
0 15
-3
-
Numbers in parentheses are powers of 10.
4
0.10
TABLE 11: Transition Energies and Dipole Moment Matrix Elements for Double Minimum Potential" transition
energy/ cm-
dipole moment matrix elementb/D
0'2 1'3 3'5 2+4 1+ 2 ( C ) 3'4 o + 1 (C) 4'5 2 + 3 (C)
202.4 185.7 164.0 152.5 118.0 84.8 84.4 79.2 67.7
1.632 (-1) 1.656 (-1) 1.474 (-1) 2.154 (-1) -1.743 (-2) 2.671 (-1) 2.155 (--3) 4.533 (-1) -1.143 (-1)
0.05
OW
w l cm-1 Flgwe 1. Absorption spectrum for double-minimum potential with laser Intensity 1 MW cm-*.
" Transitions denoted C are cross-well transitions. Numbers in parentheses are powers of ten.
joining the two opposite carbon atoms from the line joining the end carbon and nitrogen atoms. The single and double minimum potentials are both of the form V(x) = Vzx2
+ v3x3 + v4x4
(2)
and the constants V,, V,, and V4 are all given in ref 9. For the double minimum potential, V , is negative and both V, and V , are positive while, for the single minimum potential, V3is negative and both V , and V4 are positive. In the double minimum potential,6 the barrier to ring inversion is 441 cm-' from the bottom of the deepest well. The eigenfunctions were computed by using a large basis set of 80 harmonic oscillators with all integrals computed analytically. Tables I and I1 present the calculated transition energies and their quantum number assignments. These agree well with those reported in ref 9. Note that transitions labeled 0 1, 1 2, and 2 3 for the double-minimum potential are all "cross-well" transitions with initial and final eigenfunctions localized in different wells of the potential and thus correspond to isomerization transitions. b. Calculation of Transition Probabilties. The Hamiltonian operator for the ring puckering motion interacting with the laser radiation, described by the semiclassical electric dipole approximation, is Ei = l'iO- p ( x ) €0 cos (ut) (3)
- -
-
where p ( x ) is the dipole moment operator for the ring puckering motion, eo is the electric field amplitude of the radiation, w is the radiation frequency, and t is the time. The time-dependent Schrodinger equation is solved by using the wave function expansion N
(4)
where p represents the initial quantum state, a ( t ) are coefficients,and ( x , ( x ) ) are the eigenfunctions of With this expansion, the a,(t) coefficients are evaluated numerically12over the first optical cycle of the laser field up to t = 2.rr/w. Diagonalization of the matrix with elements a,,(t) at this value of t gives the Floquet characteristic
8.
eigenfun~tions.'~These are then used13 to compute the long-time average transition probabilities Pp4for a transition between energy levels p and q. We have adapted the Magnus appr~ximationl~ to develop a numerical methodI2 for determining the a,,(t) coefficients which is particularly efficient when results for a large number of laser frequencies are needed since most of the numerical work can be done at the first frequency. Use of this numerical method is essential for the present study since, for each spectrum, calculations had to be performed for 15000 different frequencies, with equal spacings of 0.01 cm-', to produce the absorption spectra discussed in section 3. The calculations were performed with six eigenfunctions in the expansion of eq 4. The dipole moment operator p ( x ) is not known for trimethylenimine and we used the simple form based on the function used in previous intense laser calculations on H P 3 (5) p ( x ) = do + d,x where do = 1.82 D and dl = 0.79 D %-'. Tables I and I1 also show the dipole matrix elements ( x p l p I x q ) . It is seen that the magnitude of these are relatively small for cross-well transitions in the double-minimum potential. Note that the dl parameter used in eq 5 is not a particularly large value for a typical diatomic molecule. To obtain the absorption spectra for a given laser frequency and intensity, we computed the average number of photons absorbed for an initial state p from N Pp,(E, - E p ) Pp = (6)
,=c'
ho
where E , is the energy of level q. The P p were then Boltzmann averaged for a temperature of 300 K. These Boltzmann-averaged,average number of photons absorbed for a given frequency w are denoted A(w) and, in the next section, the "absorption spectrum" refers to a plot of A ( w ) against w for a fixed laser intensity I. 3. Results and Discussion Figures 1and 2 present the calculated absorption spectra of trimethylenimine, described by the double-minimum potential function, for the laser intensities 1MW cm-, and 10 kW cm-,, respectively. Analogous results for the single-minimum potential are presented in Figures 3 and 4. The frequency range is 65-215 cm-'. (15) Pechukas, P.; Light, J. C. J. Chem. Phys. 1966, 44, 3897
The Journal of phvsical Chemistry, Vol. 87, No. 16, 1983 3027
Letters 0.10
1
0.m
.
0.M
-
0.25
-
I=10 kWcm-2 0.1
-
-
I
3
4 0.10
0.05
. -
wl cm-1 Flgw 2. Absorption spectrwn for doubleminimum potentlal with laser intensity 10 kW cm-2.
Intensity 10 kW cm-2.
duced efficiently for the case of the double-minimum potential. Furthermore the results suggest that the possibility of a potential function having more than one local minimum can be examined by measuring the absorption spectra for increasing laser intensities and observing whether new absorption bands correspondingto cross-well transitions appear at higher laser intensities.
0.Jo
0.25
0.XI
Flgurr 4. Absorption spectrum for singie-minlmum potentlal with laser
I = 1M W cme2
0 . IS I
3 I
a
0.10
0.05
I )(I)
F w e 3. Absorption spectrum for slngie-minlmum potential with laser intensity 1 MW cm-'.
The computations for both single- and double-minimum potentials give strong absorption peaks at frequencies close to 208, 184,167, 148,85,and 82 cm-'. These transitions correspond to those listed in Table I for the single-minimum potential and in Table I1 for non-cross-well transitions on the double-minimum potential. However, at I = 1 MW cm-2, the spectrum for the double-minimum potential also shows extra intense bands not observed for the single-minimumpotential. These bands, at 118.0and 67.7 cm-l, are the cross-well transitions, summarized in Table 11,which correspond to isomerization. At I = 10 kW cm-2, the relative intensities of these cross-well transitions are diminished (compare Figures 1 and 2) and, at laser intensities much less than this, the cross-well transitions cannot be detected. The computations clearly demonstrate that, at realistic laser intensities (MW cm-2), the isomerization can be in-
4. Conclusions Quantum calculations on the intense laser absorption spectrum of the ring puckering coordinate in trimethylenimine have been reported. The computations demonstrate that isomerization can be induced, with realistic laser intensities, for a double minimum potential that is typical for many organic molecules.'' Furthermore, the results suggest that the measurement of intense laser absorption spectra might be a practical new method for determining whether a certain molecular coordinate has a potential function with more than one well. For trimethylenimine, an intense laser source in the farinfrared range of 65-220 cm-' will be needed. Recently, lasers have been developedlein the far-infrared range with intensities of 1 MW cm-2. Even if intense laser experiments on trimethylenimine are not possible at the present time for technical reasons, there w i l l be many other organic molecules for which such experiments might be practical and of interest. The approach clearly has great promise in providing new and valuable information on intramolecular potential energy surfaces. Acknowledgment. J.P.H. thanks the Science and Engineering Research Council for a Research Studentship. We have also benefited from useful discussions with Mr. R. J. Price. Registry No. Trimethylenimine, 503-29-7. (16) Woskoboinikow, P.; Mulligan, W. J.; Erickson, R. IEEE J. Quont. Electron. 1983, QE-19,4.
