J. Phys. Chem. 1988, 92, 5398-5404
5398
Stark Effect on Dynamics and Spectroscopy of Isolated Molecules S. H. Lin,* A. Boeglin, Department of Chemistry, Arizona State University, Tempe, Arizona 85287- 1604
H. L. Dai, Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104
and E. W. Schlag Institute for Physical and Theoretical Chemistry, Technical University of Munich, 8046 Garching, West Germany (Received: February 25, 1988; In Final Form: June 13, 1988)
In this paper, we report the density matrix treatment of the Stark effect on fluorescence spectra and dynamics of isolated excited molecules. We also present some experimental results on the Stark level crossing spectroscopy of formaldehyde. Some numerical calculations are carried out to demonstrate the theoretical results.
1. Introduction Recently, the electric field effect on fluorescence and lifetimes of formaldehyde has been reported.'+ It is observed that the Stark effect can affect the lifetimes and fluorescence intensity and that the plot of fluorescence intensity and lifetime versus electric field strength will exhibit dips due to the electric field induced reson a n c e ~ . ~In this paper, we treat the Stark level crossing spectroscopy (SLCS) and the electric field effect on the lifetimes of collision-free molecules; we also report some experimental results on SLCS of formaldehyde. Weisshaar and Moore's2 have measured the lifetimes of a number of H 2 C 0 4; rotational levels under the influence of a uniform external electric field (0-4.6 kV/cm) in the collision-free condition. Several types of behavior have been observed. For excitation of the most rapidly decaying single rotational levels (SRL), long-lived fluorescence decay components gradually grow in as the field strength is increased to a few kilovolts per centimeter. The decay becomes nonexponential. Examples for which faster decay components grow in with increasing electric field have also been found. But the most striking case is the behavior of the rQ1(13) (J' = 13, K' = 2) level. The zero-field fluorescence decay is a single exponential with lifetime 880 ns; no change is observed for fields up to 2270 V/cm; in the narrow range 2270-2600 V/cm, about 60% of the amplitude rapidly shifts to faster components whose lifetimes are -250 ns. Above 2900 V/cm, the decay again becomes a single exponential with lifetime 710 ns, 20% faster than the zero-field lifetime. Guyer et aL4 have measured the lifetime of Ooo(M=O) l l I ( M = l ) in 4l S1 D 2 C 0 in the jet condition under the influence of an electric field. Lifetime data points have been collected at equal voltage increments (4-10 V) over 20-kV scans. The plot of lifetime versus electric field exhibits a spectrum due to the Stark effect induced resonances. Dai et aL3 have also studied the Stark effect on 4l S1 H2C0 at a total energy near 28 300 cm-'. They have performed the measurements of the electric field effect on the fluorescence intensity and on the fluorescence decay curves. They have observed two pairs of resonances; for the 202rotational level, fluorescence = 2 resonance and at intensity dipped at 2.4 kV/cm for the 4.6 kV/cm for the IW = 1 resonance. Similarly for the 220level, the 1W = 2 and = 1 resonances were observed at 0.4 and 0.9 kV/cm, respectively. For 202,quantum beat was detected at the
-
1w
Iw
(1) Weisshaar, J. C.; Moore, C. B. J . Chem. Phys. 1980, 72, 2895. (2) Weisshaar, J. C.; Moore, C. B. J . Chem. Phys. 1980, 72, 5415. (3) Dai, H.L.; Field, R. W.; Kinsey, J. L. J. Chem. Phys. 1985, 82, 1606. (4) Guyer, D. R.; Polik, W. F.; Moore, C. B. J . Chem. Phys. 1986, 84, 6519. ( 5 ) Schlag, E. W.; Henke, W. E.; Lin, S. H. In?.Reu. Phys. Chem. 1982, 2, 43.
0022-3654/88/2092-5398$01.50/0
1 4= 2 resonance, but no quantum beat was observed for the lw = 1 resonance and in the 220fluorescence.
The present paper is organized as follows. In section 2, we present the density matrix treatment of the Stark level crossing spectroscopy of isolated molecules. In section 3, the Stark effect on the dynamics of isolated molecules is treated; the behaviors of decay curves and lifetimes of molecules affected by the applied field are discussed. In section 4,we present some SLCS experimental results of formaldehyde and demonstrate the application of the theoretical results to analyze the SLCS experimental data. 2. Theory: Spectroscopy We consider the model shown in Figure 1. The {a] states represent the excited state manifold that can be pumped from the ground state g. The (m)states denote the group of states coupled to the {a)states and the continuum by the interaction H' and the interaction U, respectively. In previous papers: it has been shown that by eliminating the states of the irrelevant degrees of freedom and continuum the generalized master equations (GME) can be expressed as dpnn i + -C(vnkPkn - Pnkvkn) dt h k ~
+ gRifPkk'
+ xkr % k k
=0 (2- 1)
dPh dt
- + (iwkn
i
+ rk)Pkn + xx(vkkfpk'n - pWvkfn) + k'
REpYn, = 0 (2-2) k'n'
+
where P = 6 H',D represents the pumping operator, r: and represent the relaxation rate constant and the dephasing constant, respectively, and R Y are defined by
+
RZF = SnntJrnrnJ Srnrn,Jnnt* J,,
(2-3)
=
( 1 / h 2 ) L m d 7 Curnc(t)
~ c m l ( t-
r ) exp[-iJ:yti
wcrnr(t,)]
(2-4) The summation in eq 2-4 is over the continuum states. Notice that the diagonal elements pnn describe the population of the system, while the off-diagonal elements plol describe the phase (or coherence) of the system. Let us first consider a simple model. That is, there is only one level m in the (m)manifold and only one level a in the (a)manifold. In the Stark level crossing spectroscopy, an electric field is applied (6) Boeglin, A.;Fain, B.; Lin, S . H. J . Chem. Phys. 1986,84,4838. Lin, S. H.; Boeglin, A,; Lin, S. M. J . Photochem. 1987, 39, 173. Voltz, R.; Villaeys, A. A,; Boeglin, A,; Lin, S. H. Laser Chem. 1983, 2, 253.
0 1988 American Chemical Societv
The Journal of Physical Chemistry, Vol. 92, No. 19, 1988 5399
Stark Effect on Isolated Molecules
2
h Im (Dga(-w)
2
Pag(w))
-h Im
(”amPma)
+ r%aa = 0 (2-15)
2
:’
h Im
1
(H;,pma)
+ (R:: + r;:)p,,
=0
(2-16)
Figure 1. General model for the Stark effect.
to tune levels a and m; whenever the a and m levels cross due to the Stark effect, the fluorescence intensity from the a level decreases. In other words, the zeroth-order level m is a dark state. In this case, the GMEs are given by dPaa
i
dt + h(DagPga - @gal
i
+ h(”arnPma
- PamH‘ma) +
r::Paa= o
(2-5)
In other words, substituting (2-17)-(2-19) into (2-15) and (2-16), we can solve for paa and pmm (see Appendix A). As can be seen from (2-17)-(2-19), the contributions from the pmg(w) terms in (2-17) and (2-18) represent the correction terms. Thus, for the case in which these terms are neglected (Le., secular approximation or isolated line approximation), we obtain
where A’ consists of the original interaction H o wchout th_e presence of electric field and the Stark interaction H:. H : = -,%F, where T; is the dipole operator and F is the applied electric field. In dms, dmg, and w’,~,, the Stark shifts have been included. In other words, by changing the electric field strength, ofrna varies equals zero. and the dip in the fluorescence appears when dma From eq 2-4 we obtain
where P means that the principal value should be taken. The above equation indicates that the real part of J,, is related to the dissociation rate constant, while the imaginary part is related to the level shift of the m level. Notice that 27r (2-11) R:: = 2 Re J,, = --CIUmelZ 6(w,,) h* e R:: = J,,;
R;; = J,,
(2-12)
where Re means that the real value should be taken. In other words, RE: represents the dissociation rate constant of the m level. Suppose that the pumping-from g to a is accomplished by an optical means; in this case, D can be expressed as B(t) = b(w)e-i‘u
+ B(-w)e’fu
(2-13)
where w is the optical frequency. Using the rotating wave approximation (RWA)
where
It is to be noted that rE, I?:;, and R:: consist of imaginary parts and real parts; the imaginary parts represents the level shifts, and it is to be understood that the imaginary parts are included in dag and dm.The real parts play the role of dephasing. Notice that, for example’
where, for example, I’: represents the total decay rate constant denotes the lifetime of the of the m level; in other words, (I’:)-l m level. In eq 2-22, l?E:(d) denotes the pure dephasing due to elastic processes associated with the m and a levels; for isolated molecules, the pure dephasing is frequently ignored. Equation 2-20 indicates that by tuning the applied electric field, we can see that at the Stark resonance, i.e., wlma = 0, paa will exhibit a dip. In the density matrix method, the dynamics of the phase of the system is taken into account, and thus the dephasing (or damping) constant is properly described (see eq 2-22). In using the Schri5dinger approach (e.g., the time-dependent perturbation method or Green’s function method), the pure dephasing I’z:(d) is missing and the dephasing constant is given as r: =
Pag(t) = pag(w)e-i‘w;Pmg(t) = Pmg(w)e-ifu (2-14) we obtain the steady-state solution of (2-5)-(2-9) as
(7) Lin, S.H.; Eyring, H. Proc. Natl. Acad. Sci. U.S.A. 1977, 74, 3623. Fain, B.;Lin, S . H.Surf: Sci. 1984, 147, 497.
The Journal of Physical Chemistry, Vol. 92, No. 19, 1988
5400 I
Lin et al. A more general case is discussed in Appendix B. The approximate steady-state GMEs are given by7J2
’
Waghaa - P,) + Ckaabaa - Pats,) + Ckambaa -Pmm) + a‘
m
cr%p:Pkk = o (2-26) k 0 (2-27)
-1
n
-0 5
no I \ I l TIH’,,,~~~ h
+
+ :r
-
rg
(3-5B) m’2m
Ck r g d k + REgpmm + C ‘ (R:‘Pmmt m‘
(3-6)
+ R:$Pmlm)
=0 (3-14)
(3-7) The expression for p,(t) given by eq 3-4 is very useful in analyzing experimental results. It should be noted that depending on the LO’, l’zi, and Rzi, the model described above magnitude of HTm, may exhibit the quantum beat. From (3-1)-(3-4) we can see that the quantum beat cannot be observed if the dephasing rate constant r: + RR is much larger than the coupling matrix element lH’,,,al/h; that is, in this case, the quantum beat disappears in the time scale of (I’g+ Z?:;)-l. In particular, for the weak coupling case (Le., IH’J < A’,), the quantum beat cannot be observed. For the intermediate and strong coupling cases, the quantum beat can be observed provided that the energy levels are sparse; the nonvanishing initial coherence (or phase) pam(0)will enhance the observation of quantum beats. The two-level model for treating quantum beats in molecules has been reported by a number of w o r k e r ~ ; ~ more J ~ J ~recently
The higher order terms have been neglected in obtaining (3-13) and (3-14). In the formaldehyde case, the states in the (m] manifold have much shorter lifetimes compared with those in the (a) manifold. In this case, we have dPaa
+ Ckaabaa - PataO + CkmaPaa +EPaa =0 dt rn
(3-15)
We can study the interference between the two states, say, a and a’ in the (a] manifold: (3-16) dpaa/dt - kaatpalal + k’apaa = 0 dpazat/dt - kaaipaa+ ki,pa,a,= 0
(3-17)
~
(1 3) Lendi, K. Chem. Phys. 1980,46, 179. (14) Description of the experimental procedure and more detailed experimental SLCS results will be published by Ritter and Dai.
Lin et al.
5402 The Journal of Physical Chemistry, Vol. 92, No. 19, 1988
0
2
1
6
I\
I )
, lL4l.l Figure 3. Stark effect on decay curves: (a) off resonance, (b) resonance. 11\11
( 1 1 1 1 1 IO.,
Electric Field Strength (kV/cm)
Figure 5. Stark level crossing spectroscopy of formaldehyde at the SI 4l l l olevel. The points before and after 19 kV/cm were taken in two
separate runs of experiments.
I
IL
0
2
I
I
I
4 6 8 IO 12 14 16 le E l e c t r i c Field Strength (kV/cm)
I
22
Figure 6. Comparison between experimental and theoretical SLCS re-
sults of formaldehyde.
total fluorescence intensity. The solid line corresponds to the case in which the system is prepared coherently at t = 0. As is to be expected in this case, the quantum beat is observed. The broken line corresponds to the case in which the system is prepared incoherently to the a level; in this case, no beat is observed. 4. Discussion
where
+ k\,) + l/z[(k\ - kit)’ + 4kaatz]1/2 (3-20) A2 = 1 / 2 ( / ~ ’ ~+ k’ar)- yz[(k’a- k’,,)’ + 4kaa,2]I/2 (3-21)
X I = 72(k\
In Figure 3, we show the Stark effect on the dynamics of an isolated molecule by changing the electric field (i.e., changing the energies). For this purpose, we calculate the population decays for the model consisting of one a level and one m level by choosing Hrma= cm-’,::’I = los s-l, and RRR = lo8 s-l. The broken line corresponds to the off-resonance case of dma = 0.01 cm-I; the population of the a level exhibits an exponential decay with the lifetime of 10 ps. In Figure 3, the solid line corresponds to the resonance case; in the microsecond time scale the population of the a level again exhibits an exponential decay, but the lifetime in this case is 4 ps, Le., shorter than that of the off-resonance case. It is important to notice that in the case of weak coupling and ::?I >>:?I as shown in Figure 3, the steady-state approximation can be applied to pma,and in this case eq 3-13 and 3-14 can be used. Next, we demonstrate the quantum beat by using the typical physical constants for formaldehyde (see Figure 4 ) . We consider the model where there are level a and level a’ (daa = cm-I) in the (a) manifold which couple to a single m level (uam= 0). The plots shown in Figure 4 represent the time evolution of the
In the previous sections we have presented the theoretical treatment of the Stark effect on the dynamics and spectroscopy of isolated molecules. In Figure 5, we present the experimental SLCS results of the SI 4l lol level in H2C0.I2 Total fluorescence intensity data were collected by using a gated integrator and boxcar averager. The gate used for the intensity measurements was 10 ps for lo, which had a 2.7-ps maximum lifetime. Thus, more than 99% of the total fluorescence in time was collected. The polarization of the laser light was set parallel to the electric field direction. In this case, since the intensity of the A(q(lM1 - IM’j) = 0 transitions is proportional to only the IM1 = 1 sublevel of the S, 1 level was excited. The total fluorescence intensity spectrum of Figure 5 displays the change in fluorescence decay rates induced by the mixing of the So = 1 and predominantly J = 1 level(s) into the excited S,level. Procedurewise, the fluorescence excitation spectrum near the desirable transition was first recorded at zero-field strength and assigned. A spectral range of f l cm-I around the desirable transition was then scanned at a series of different applied electric field strengths. At the end of these scans, another zero-field spectrum was taken to ensure that the intensity of the transition under investigation was not affected by long-term laser power drift or formaldehyde pressure change. We apply the theoretical results given in section 2 to fit part of the SLCS of Figure 5 . Here eq 2-23 has been used. The fit to the data is shown in Figure 6. It should be noted that from the analysis of SLCS experimental results one can determine only the relative magnitude of the
1w2,
Iw
Stark Effect on Isolated Molecules
The Journal of Physical Chemistry, Vol. 92, No. 19, 1988 5403
coupling matrix elements and rate constants. From the first dip shown in Figure 6 we obtain = 2.3 x 104
R:;(i)/r::
(4-1)
wma(i)/r;; = 80
(4-2)
where I’: represents the radiative rate constant. Here it is assumed that only the dissociation rate constant RE contributes (Le., F E , the intramolecular vibrational relaxation rate constant, is negligible). Similarly from the second dip we obtain ~::(2)/r::
= 1.0 x 104
L
(4-3)
Hga(2)/r:: = 25
(4-4)
and from the third dip we obtain ~;:(3)/r:;
= 1.2 x 104
(4-5)
~ ; ~ ( 3 ) / r : : = 22
(4-6)
The radiative rate constant r:: for the SI 4l state(2)of H 2 C 0 is 3.2 X los s-l. Using this value, we can determine Rgg and ”,,,a given by (4-1)-(4-6) as RRE(1) = 7.4 X lo9 s-I; Hrma(l)= 0.85 X
cm-’
(4-7)
R::(2) = 3.2 X lo9 s-*;
cm-I
(4-8)
lo9 s-l; Hga(3) = 0.23 X 10-3 cm-I
(4-9)
R::(3) = 3.8
X
H;,(2)
= 0.27 X
It should be noted that the fit of the theory to the experimental data shown in Figure 6 and the results given in (4-1)-(4-9) obtained from the fitting should be received with reservations due to the fact that the experimental data points are not numerous enough to accurately define the resonance positions and widths. Our purpose here is to show how to apply the theory to analyze the experimental data. From the above discussion we can see that the Stark effect fluorescence spectroscopy can provide the dip positions in terms of the electric field strength but only the relative magnitude of coupling matrix elements and dissociation and relaxation rate constants. On the other hand, the Stark effect lifetime measurement can provide the dip positions in terms of the electric field strength, coupling matrix elements, and dissociation and relaxation rate constants. It should be noted that in eq 2-23 the term urma includes the Stark effect on the a and m levels. The Stark level shift of the energy levels in the (a) manifold (SI in H,CO) can easily be determined by using the perturbation method or by diagonalization of the Hamiltonian matrix. However, due to the fact that the energy levels in the (m] manifold (So in H,CO) are highly excited and dense, for determination of the Stark level shifts of these energy levels it is necessary to diagonalize these levels with respect to the Stark interaction and the rotation-vibration couplings. It should be noted that the dependence of the dipole moments on vibrational coordinates will also have some effect on the calculation of the Stark level shifts. It has not been included in this work and should be taken into account in a more refined calculation. In conclusion, in this paper we have applied the density matrix formalism to treat the Stark effect on the dynamics and spectroscopy of isolated molecules. In this treatment, the dephasing of the system is properly taken into account, and thus one can safely use this theory to analyze the shapes of the Stark effect dynamics dips and spectroscopic dips to obtain the coupling matrix elements and dissociation and/or relaxation rate constants.
Substituting eq A-1 into eq 2-17 and 2-18 yields
From eq 2-15, 2-16, A-3, and A-14 we obtain pmm(RE$
+
+ k’am) - Paak”am = KamgP,
(A-6)
and
where
2 K’amg = -IH’am121Dag12Im h4
[
vmg
i(wrag- w )
+ r;f
]
(A-10)
Acknowledgment. S.H.L. and A.B. thank N S F for supporting this work. H.L.D. acknowledges partial support for this work from the Department of Energy (DE-FG02-86ER13584). We thank a referee for helpful suggestions. Appendix A To determine paa and pmm, we first rewrite p,,(w) as
I I
(A-11)
5404
The Journal of Physical Chemistry, Vol. 92, No. 19, 1988
Lin et al.
“l.”(
W’g, = Wag 1 - - Vmg(r)+f”-(r))
+
where, for example,j’-(r) andYme(i)represent the real part and the imaginary part of f6,,respectively.
We can solve eq A-16 and A-17 to obtain paa and pmm:
Paa
(A-14)
=
Pmm
=
R E
+
k”am :’I
+ ktamPaa
+
:R:
+
Kamg :?I
+ kimPgg (A- 15)
These expressions are useful to check an approximate result such as eq 2-20. Notice that k’,,, k’B,, KamgrKimg, Wig, and W’Bg can be rewritten as
Appendix B For the general model described by Figure 1, the GMEs for treating level crossing spectroscopy are given by
