J. Phys. Chem. 1993,97, 12644-12649
12644
Selective Mode Excitations in the Electronic Ground States of H20, HDO,SiH2, and NH2 Wolfgang Gabriel and Pave1 Rosmus'lt Fachbereich Chemie der Universitat, 0 4 0 4 3 9 Frankfurt, Germany Received: June 18, 1993; In Final Form: September 28, 1993'
Based on ab initio three-dimensional functions of potential energy, electric dipole and transition moments, the rovibronic energy levels, eigenfunctions, and radiative transition probabilities have been calculated for several triatomic molecules (H20, HDO, SiH2, and the Renner-Teller system of NH2). Using ultrashort infrared picosecond laser pulses, the state-selective multiphoton excitations of t h e u l = 4 (symmetric stretching) vibrational eigenstate of these molecules as well as the corresponding diatomics OH, SiH, and N H have been investigated. The OH stretching mode in HDO and in H2O can be selectively populated. Strong anharmonic coupling (SiH2) or Renner-Teller coupling to another electronic state (NH2) may, however, considerably reduce the achievable state selectivity.
I. Introduction
11. Computational Approach
The possibility of mode-selective excitation and dissociation of selected bonds in polyatomic molecules using intense infrared (IR) laser pulses has been the subject of many experimental and theoretical investigations in the past Simplified harmonic or anharmonic models13J4 have been used to investigate the principles of ultrashort multiphoton IR excitations. For instance, Jakubetz, Manz, Schreier, Just, and T r i ~ c a l ~have -'~ employed the method of Paramonov and Sawas in the studies of the state-selectivevibrational excitation of OH induced by an IR picosecond laser pulse and the IR multiphoton excitation of nonrotating H2O. On the basis of a two-dimensional hindered rotor model, Hutchinson, Sibert, and Hynes19have analyzed the quantum dynamic flow of energy between coupled bonds in H2O. Recently, Amstrup and HenriksenZ0 have studied the twodimensional vibrational dynamics of HDO. To date, the problem of the mode-selectiveexcitations has been usually treated by using model dipole moment and potential energy functions of reduced dimensionality. In the present work, we have investigated state-selective excitations and three-dimensional vibrational dynamics in the electronic ground states of SiHz, H20 (including HDO), and NH2 by employing the ob initio potential energy and dipole moment functions, vibrational eigenfunctions, and vibrational transition dipole matrix elements. In all three nonrotating triatomics we have tried to selectively populate the fourth symmetric stretching level. In the case of H20 and HDO we can compare the results of previous model studies with the present more accurate approach. The second system, SiH2, exhibits a strong Fermi resonance in the levels lying below the v1 = 4 state and represents an example of a strong anharmonic perturbation in the multiphoton process. In the NH2 radical the level with four quanta in the symmetric stretching mode lies already above the barrier to linearity of the XZBl state and couples via the Renner-Teller effect to the upper A2A1 state. Since we were interested in the excitation of the AH stretching modes, we have also performed similar calculations for the diatomics OH, SiH, and N H in their electronic ground states. In the next section, we describe briefly the computational approach and the formalism used in the solution of the time-dependent SchrGdingerequation for various types of the pulse shape function. The results for the multiphoton excitation in the diatomic molecules OH, NH, and SiH are presented in section 111. In section IV the state-selective vibrational dynamics of the triatomic systems are analyzed.
The time-dependent computations required in our case the knowledge of the vibrationaleigenstatesand vibrationaltransition dipole matrix elements. This information has been obtained from variationalsolutionsof the nuclear motion problem21(or numerical solution for diatomics) taking into account the full dimensionality and the anharmonic or Renner-Teller coupling effects using the ab initio calculated potential energy and dipole moment functions for all molecules treated in this study (OH,22SiH,23NH,23H20,24 HDO,24SiH2,25 NH226J7). In previous works28329 we have shown that the vibrational band origins of the theoretical potentials reproduced theexperimentalvalues to within 10-30cm-1 or better (NH2). For water vapor, for instance, the dipole moment functions yielded the absolute values of the line strengths of the ro-vibrational transitions or integrated band intensities to within a few percent by comparison with experiment. The other dipole moment functions are of similar quality. In the time-dependent computationswe used a formalism based on the expansions of the time-dependent wave functions in the vibrational eigenstates and the interaction with the radiation field is taken into account by a time-dependent perturbation operator. The time-dependent Schriidinger equation can be written as
~~~
~
Visiting Fellow 1992-1993, Universitt de Marne la Vallte, France. Abstract published in Advance ACSAbstracrs, December 1, 1993.
0022-3654/93/2097- 12644%04.00/0
with the perturbation operator H'(Q,t):
The peak laser intensity is given by (4) The time-dependent wave function is expanded in the basis of the vibrational eigenstates:
@',O(Q,t)= @:(Q)eAERo'
(h=l)
(6)
The time-independent part
(7) of eq 1 has been solvedusing thevariational approach for triatomic molecules of Carter and Handy.21 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97,No. 48, 1993 12645
Excitations in Ground States of Triatomic Molecules
TABLE I: Investigated Laser Shape Functions
7. --- / = J 0.7 I
. . -,
exp(-y(t - r1)2) TABLE II: picosecond Laser Pulse Excitations of the Y = 4 Vibrational State of NH, SiH, and OH Dulse sham no En (aul w (cm-9 P"d X22- NH ~
4 4 2 2
a
~~
0.08 0.06
~
~~
2901 .O 2882.5
0.20 0.99
C
C
C
C
0.6 (1.5 0.4
4 4 2 2
1849.5 1861.0 3709.0 3709.5
0.27 0.90 0.67 0.92
4 4 2 2
X21&OH 0.145 0.125 0.150 0.145
3269.0 3279.0 6540.0 6541.0
0.65 0.99 0.82 0.98
1
I
I I
1
0.3 i 0.2
1
0.1
i
0.0
X2n, SiH 0.055 0.013 0.024 0.041
1 I
Q.>
0
t
3
100
200
300
400
500
600
700
800
900
lo00
Time [fa] Figure 1. State-selectivefour-photonexcitationof the u = 4 level of X2II OH (pulse parameters: €2, n = 4, w = 3279 cm-I, EO = 0.125 au). 10
09 08 07
06
Number of photons. Population probability at to = 1000 fs.
S 0.01 for Eo 5 0.13.
&' 0 5
The resulting set of coupled first-order differential equations
04
03
k ( t ) = Hc(t)
(8)
is solved numerically by a standard numerical method.30 The time-dependentpopulation probabilities have been calculatedfrom the projection of \k(Q,r) on the vibrational eigenstates @:(Q)
I
0 2 01
00
0
100
200
300
400
500
600
700
800
900
lo00
Time [fs]
(9) Numerical simplifications like the quasiresonant (QRA) or rotating-wave approximation (RWA) have not been considered. The range of validity of these techniques has been discussed in detail.31 At time t = 0 the molecule is assumed to be in its vibrational ground eigenstate. The spontaneous emission lifetimes of the excited molecular eigenstates, which lie on the millisecond time scale, have not been considered. In Table I the various types of investigated pulse shape functions are summarized. The squaresinusoidal (Q),triangular (tj), and Gaussian (€4) shapes have been taken from refs 8 and 16. The Gaussian shape (€5) used in ref 20 has been normalized in such a way that the energy is the same as for the rectangular, continuous wave (CW) shape function ( e l ) of length to. Throughout this paper n-photon absorption wt N w, = E,,,/n (10) will be classified according to the resonance conditions for the target vibrational state.I0J6 From many test calculations we can conclude that the population probabilities are converged with respect to the possible further improvements of the pulse parameters to within a few percent.
111. Multiphoton Excitation in the Electronic Ground States of OH,SiH, and NH
Using high-quality ab initio r e s ~ l t s for * ~the ~ ~dipole ~ moment functions of OH, SiH, and NH, the dipole matrix elements have been calculated. In the time-dependent calculations a basis set of the 15 lowest eigenstates has been employed. The results for the state-selective excitation of the v = 4 vibrational level are given in Table 11. The pulse parameters have been optimized with respect to maximum occupation of the target level at r =
Figure 2. State-selectivefour-photonexcitation of the u = 4 level of X3Z NH (pulse parameters: ~ 2 n, = 4, w = 2882.5 cm-', EO = 0.06 au).
1 ps. The basic mechanism of stepwise overtone excitations in OH has been discussed in detail by Jakubetz et al.I6 The vibrational transition probabilities in the OH radical crucially depend on the shape of the dipole moment function used. Due to the fact that the dipole moment function has a maximum close to the equilibriumgeometry of OH, the overtone transitions-even in low-lying vibrational levels-are more intense than the fundamental transitions. Jakubetz et a1.I6have used an empirical Mecke dipole moment function, which has a maximum shifted by about -0.6 bohr relative to the maximum determined from experiments32 and, therefore, yields incorrect transition probabilities. The problem of the OH dipole moment function has been discussed in detail in refs 22, 32, and 33. In Figure 1 the population probabilities P,(t) (cf. eq 9) of the six lowest-lying vibrational eigenstates of X211OH are displayed. In contrast to the results of Jakubetz et a1.,16 the v = 4 state has been excited much earlier after t = 300 fs. Similarly, also the state-selective excitation of the v = 4 state in the electronic ground states of N H and SiH has been investigated. The results are displayed in Figures 2 and 3. The time-dependent occupation probabilities are here more complex,since at t = 55MOO fs there is a significant recurrence of the initially prepared vibrational ground state. The dominanceof theu = 0- u = 1 excitation and the high occupation probability of the v = 3 state at t = 600-800 fs seems to be a general feature as Jakubetz et a1.I6 have pointed out. Due to the larger dipole matrix elements for the overtones in OH, the vibrational levels v = 3 and v = 4 are populated much earlier by comparison with SiH and NH. Furthermore, we found also that a sin2pulse shape yields higher selectivity than the CW pulse (cf. Table 11) and that n-photon transition to the target state v = n will be more efficient than a
12646 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 1.0 1 -
/
I
0.9
i
Gabriel and Rosmus
, 1
08
\ v=o
0.8
07
0.7 I
0.6 1
06
I
L' 0.5
1
0.4
I
\
03
I
02
02
0.1 I
no1 0
01
200
100
300
400
500
600
800
700
900
Figure 3. State-selective four-photon excitation of the u = 4 level of X2II SiH (pulse parameters: q, n = 4, w = 1861 cm-I, EO= 0.013 au).
TABLE III: Basis Sets of Vibrationel Eigeastates H20 HDO NH2 E,
59, 5 12077 16000
170 13965 20000
60,37 13946 18000
nl,
E(4,0,0)b
no
loo0
Time [fs]
0.8 O.g 0.7
a
pulse shape
n
EO(au) w (cm-I) X'AI H20
..
640)
4 4 2 2 2 2
0.04 0.0975 0.0925 0.125 0.1375 0.1275
4r)
4
Cl(t)
I
4 t ) Cl(l)
4
63W
cz(t) 4 t )
(2(d 4 t ) €2(1) Cl(t)
(2W
3485.0 3480.0 6950.0 6935.0 6935.0 6935.0
C
4 4 2 2
XZB1 NHz 0.05 3045.0 3010.0 0.06 0.135 6030.0 0.13 6030.0
0.210 0.237 0.527 0.145
700
800
900
loo0
I1
0.3 .
0.0
0
1 0 0 " 0 4 0 0 5 w 6 0 0 7 0 0 0 0 9 0 0 1 o o 0
Time [SS] Figure 5. State-selective four-photon excitation of the U I = 4 level of XIAI HzO (pulse parameters: (2, n = 4, w = 3480 cm-I, EO= 0.0975 au) invibrationalsymmetry bz [ l , (3,0,1); 2, (0,2,1); 3, (2,0,1);4,(4,0,1); 5 , (0,OJ)I.
C C
0.616 0.366
600
0.4 -
0.1
3945.0 3945.0
500
0.5 -
d
0.207 0.503 0.673 0.609 0.606 0.617
0.0125 0.02
400
0.6 .
o.a
2 2
300
."
p(4.0.01'
XLAlSiH2 C
2W
1.0
54,34 7892 loo00
TABLE I V Picosecond Laser Pulse Excitation of the 4vl State of H20, SiH2, and NH2
100
Time [fs] Figure 4. State-selective four-photon excitation of the U I = 4 level of XIA1 H20 (pulse parameters: t2, n = 4, w = 3480 cm-I, EO = 0.0975 au) in vibrational symmetry al.
SiH2
Number of vibrational states in symmetry al and bz. Vibrational energy (cm-I) of the U I = 4 target state. C Maximum energy (cm-I) of vibrational state.
0
0.03b 0.004b 0.02b
Population probabilityat to = 1000 fs. Including the Renner-Teller component A2Al. cP4,0,0 5 0.06 for EO I 0.13.
transition with lower number of photons. Using a CW pulse shape in a one-photon excitation, significant excitation of the target state has been achieved only with unrealisticlaser intensities higher than 1000 TW/cm2. For pulse shape functions t 3 and c4 the results are almost identical to those calculated with the sin2 shape.
IV. Multiphoton Excitations in the Electronic Ground States of H20, HDO, SiH2, and NH2 The details of the basis sets of three-dimensional eigenstates used for the triatomics are given in Table 111, and the optimized laser parameters are given in Table IV. As for the diatomics, the one-photon excitations require experimentally unrealistic laser intensities for a significant population of the target state. For triatomics, however, even the multiphoton processes studied have not yielded higher populations than about 7096of the target state (cf. Table IV). For example, the (4,0,0) stretching mode of H2O
can be populated only to about 50% via four-photon excitation (cf. Table IV). Slightly higher yield can be achieved using twophoton excitation. In Figure 4 the time-dependent population probabilities for the pure stretching levels are shown, whereas the populations of the stretching levels in vibrational symmetry b2 are displayed in Figure 5 . The main differences to the corresponding four-photon excitation of OH are the lower population of the target state and the recurrence of the initial state. Due to the fact that (4,0,0) and (3,0,1)levels are close in energy, they are populated simultaneously after t = 600 fs. In Figure 6, the three-dimensional wave packet I'k(Q,t)l2 (see eq 5 ) of H2O at t = 1 ps in symmetric stretching coordinate Ql = (1 /2)1/2(RI+ R2) and bending coordinate 8 is displayed. No bending excitation at the end of the pulse duration has been induced. Amstrup and Henriksenm have investigated selective fourphoton excitation of the 0-H and 0-D bond in HDO. For the dipole moment function they have also used an empirical Mecke type dipole moment function for which the first derivativesstrongly deviate from the experimentaV4 and ab initio24 values. Using our ab initio dipole moment functions for H20, we have investigated the four-photonexcitation again using the same pulse parameters as in ref 20, i.e., shape function €5, 21 = 100 fs, I,, = 50 TW/cm2, to = 500 fs, and fwhm = 50 fs. Our time-independentexpectationvalues of the 0-H and 0-D bond coordinate in HDO using three-dimensional wave functions for the first 16 eigenstates agreed with the data of ref 20 to within *l%.
I ::: P'O
V .
Nr > o
zo CO
90
80 I
.-
12648 The Journal of Physical Chemistry, Vol. 97, No. 48, 1993
Gabriel and Rosmus
- -
lo
09
0 25
1
0.8
02
h
-----
4:j
2
A8
Ill
\
0.7
,
i
06 9
e
3
a 0 15
a
v
0.5
04,
a” 01
03
I
02
0 05 00’
0
Time [fs] Figure 12. Time-dependent population probabilitiesof the lowest Fermi polyads ( u j = 0) in XIAISiHz (pulse parameters: cz, n = 4, w = E4,0,0/4 = 1973.6 cm-I, Eo = 0.02 au). In the time interval1 t = 400-800 fs the vibrational states (3,0,0) and (5,0,0) will be populated simultaneously. The results show that, using the two different laser shape functions studied, bondand state-selective excitation in HDO can indeed be induced. As an example with strong Fermi resonances, we have chosen the multiphoton excitation in the electronic ground state XIAl of SiH2. Due to very strong resonances in the lowest Fermi polyad 2u1 u2 = 2 (see ref 25 for details), no significant population of the (4,0,0) eigenstate using four-photon excitation was achieved. The optimized pulse parameters are shown in Table IV. In Figure 12 the population probabilities of the seven lowest Fermi polyads (2ul u2 = n, u3 = 0) are displayed. Contrary to previous experience, the two-photon C W laser yielded the best selectivity. Only additional investigations with other pulse shapes and sequences could provide the definite answer about the achievable selective mode excitations in this case. The time-dependent populations of the Fermi polyads are very complex due to strong anharmonic resonance effects. Only the polyad for n = 8, which comprises the target level u1 = 4, is significantly populated. In a series of additional calculations we found that a selective mode excitation can be accomplished with the present pulse shapes if the strong anharmonic coupling is turned off. In these computations we discarded all but one dominating coefficient in the expansion of the final vibrational eigenfunctions for the calculations of the dipole matrix elements. (For details of the composition of these wave functions cf. ref 21 .) Other pulse forms, like for instance chirped pulses,l2 could perhaps lead to higher selectivity. In order to analyze the coupling to other electronic states, the state-selective multiphoton excitation of the Renner-Teller molecule NHz has been studied. (All details of potential energy function calculations and extensive vibrational-rotational analysis will be published separately.26) Our target state (4,0,0)of the X ZB1 state lies close to the barrier to linearity. Since the X 2B1 state forms a Renner-Teller pair with the A 2Al, the nuclear motion problem close and above the barrier to linearity cannot be treated separately from the electronic motion. We have performed two different computations. In the first, only the electronic ground state has been taken into account, and in the second we have allowed for the Renner-Teller coupling. In this case the three lowest rotational states for J = 1 have been included. All ro-vibrational eigenstates for J = 0, 1 up to 19 000 cm-I have been used for the basis set (see eqs 5 and 6). The details of the calculation of the radiative transition probabilities in NH2 will be published ~ e p a r a t e l y . ~ ~ The results for the electronic ground state only are given in Table IV. A significant population of the (4,0,0)state has been achieved using the two-photon C W shape function. In Figure 13, the time-dependent population probabilities for optimized
+
+
0
100
200
300
400
500
600
700
800
900
lo00
Time [fs]
Figure 13. Four-photonexcitationoftheul= 4 levelof X *BINHz (pulse parameters: (2, n = 4, w = 3010 cm-l, EO = 0.06 au).
four-photon excitation are displayed. The initial state cannot be completely depopulated. In the second case, including the Renner-Teller coupling, for the same pulse parameters the timedependent population of the (4,0,0)state (calculated as a sum of occupation of all four ro-vibrational levels) is completely different: no significant population of the (4,0,0)level can be induced (cf. Table IV) due to very intense transitions into the Renner-Teller coupled bending levels of the 2A1 state.27 Other shape forms or pulse sequences could possibly yield better results.
V. Conclusions For H20, SiH2, and NH2 ab initio potential energy and dipole moment functions have been used to calculate the vibrational eigenstates and transition dipole matrix elements. This information allows to investigate the time-dependent perturbations by a radiation field. Several model calculations have been performed for different short laser pulses in order to investigate the possibility of state-selective preparation of excited vibrational states by multiphoton processes. It has been shown that bondand state-selective excitation in HDO can be achieved using different laser shape functions. The results are compared with previous studies of this type, in which less complete information about the potential energy and dipole moment functions have been employed. The anharmonic (SiH2) and Renner-Teller (NH2) coupling effects reduced considerably the selectivity of the mode excitations with the pulse forms used in the present work.
Acknowledgment. We thank Prof. W. Domcke for many stimulating discussions and his help. We also thank the referee for his critical comments. This work has been supported by the Deutsche Forschungsgemeinschaft and Fonds der Chemischen Industrie. References and Notes (1) Larsen, D. M.; Bloembergen, N. Opt. Commun. 1976, 17, 254. (2) Walker, R.B.; Preston, R. K. J . Chem. Phys. 1977, 67, 2017. (3) Cantrell, C. D.; Letokhov, V. S.;Makarov, A. A. In Coherent Nonlinear Optics, Recent Advances; Feld, M.S . , Letokhov, V. S., Eds.; Springer-Verlag: Berlin, 1980; p 165. (4) Carmeli, B.; Schek, I.; Nitzan, A.; Jortner, J. J . Chem. Phys. 1980, 72, 1928. (5) Brumer, P.; Shapiro, M.Chem. Phys. Lett. 1980, 72, 528. (6) Leasure, S. C.; Milfeld, K. F.;Wyatt, R. E. J . Chem. Phys. 1981, 74, 6197. (7) Taylor, R. D.; Brumer. P. Foraday Discuss. Chem. SOC.1983, 75, 117.
(8) Paramonov, G. K.; Sawa, V . A. Chem. Phys. Lett. 1984,107,394. (9) Milfeld, K.F.; Wyatt, R. E. J. Chem. Phys. 1985, 83, 1457. (10) Dolya, 2. E.;Nazarova, N. B.; Paramonov, G. K.; Sawa, V. A. Chem. Phys; Lett. 1988,145,499. (11) Shi, S.; Rabitz, H. Chem. Phys. 1989, 139, 185.
Excitations in Ground States of Triatomic Molecules (12) (13) (14) (15)
Chelkowski, S.;Bandrauk, A. D. Chem. Phys. Lett. 1991,186,264. Goggin, M. E.; Milonni, P. W. Phys. Rev. 1988, A37, 796. Marquardt. R.; Quack, M. J . Chem. Phys. 1989, 90,6320. Jakubetz, W.;Manz, J.;Schreier,H.-J. Chem.Phys.Lett. 1990,165,
100.
(16) Jakubetz, W.; Just, B.; Manz, J.; Schreier, H.-J. J . Phys. Chem. 1990, 94, 2294. (17) Just, 9.; Manz, J.; Trisca, I. Chem. Phys. Lett. 1992, 193, 423. (18) Just, B.; Manz. J.; Paramonov, G. K. Chem. Phys. Lett. 1992,193, 429. (19) Hutchinson, J. S.;Sibert, E. L.; Hynes, J. T. J . Chem. Phys. 1984, 81, 1314. (20) Amstrup, 9.; Henriksen, N. E.J. Chem. Phys. 1992, 97, 8285. (21) Carter, S.;Handy, N. C. Mol. Phys. 1984,52, 1367. (22) Werner, H.-J.; Rosmus, P.; Reinsch, E.-A. J. Chem. Phys. 1983,79, 905. (23) Meyer, W.; Rosmus, P. J. Chem. Phys. 1975,63, 2356. (24) Gabriel, W.; Reinsch, E.-A.; Rosmus, P.; Carter, S.;Handy, N. C. J . Chem. Phys. 1993, 99, 897.
The Journal of Physical Chemistry, Vol. 97, No. 48, 1993 12649 (25) Gabriel, W.; Rosmus, P.; Yamashita, K.; Morokuma, K.; Palmieri, P. Chem. Phys. 1993,174,45. (26) Gabriel, W.; Chambaud, G.; Rosmus, P.; Carter, S.;Handy. N. C. Mol. Phys., in press. (27) Gabriel, W.; Brommer, M.; Rosmus, P. To be published. (28) Senekowitsch,J.; Carter, S.;Zilch, A.; Werner, H.-J.; Handy, N. C.; Rosmus. P. J . Chem. Phvs. 1989. 90. 783. (29) 'Weis, B.; Carte;, S.;Rosmus, P.; Werner, H.-J.; Knowles, P. J. J. Chem. Phvs. 1989. 91. 2818. (30) Stoer, J.i Burlisch, R. Introduction to Numerical Analysis; Springer-Verlag: New York, 1980. (31) Quack, M.; Sutcliffe, E. J . Chem. Phys. 1985, 83, 3805. (32) Nelson, D. D.; Schiffman, A.; Nesbitt, D. J. J. Chem. Phys. 1989, 90, 5455. (33) Langhoff, S.R.; Bauschlicher, C. W.; Taylor, P. R. J . Chem. Phys. 1981,86, 6992. (34) Flaud, J.-M.; Camy-Peyret, C.; Toth. R. A. Water Vapor Line Parameters from Microwave to Medium Infrared; Pergamon: London, 1981.
