J. Phys. Chem. 1987, 91, 6469-6478 centrations of nitroxyl in the two arms of the tube are nearly equal (about 0.3 mM), and a uniform line width of 3.6 G is observed throughout the sample. A slice through the image is shown in Figure 2D. The line width of the signal in Figure 2D is intermediate between those observed in Figure 2B,C. These results demonstrate that spectral-spatial EPR imaging provides both the line width of the EPR spectrum as a function
6469
of time and position in the sample and the spatial distribution of the signal. Thus, EPR imaging studies of transport process can be extended to cases in which the line width of the radical signal varies spatially.
Acknowledgment. This work was supported in part by N S F Grant CHE8421281.
FEATURE ARTICLE Semiclassical Methods in Muitlphoton Diatomic Spectroscopy: Beyond Perturbation Theory Andri D. Bandrauk* DZpartement de Chimie, Facult; des Sciences, UniversitZ de Sherbrooke, Sherbrooke, Quebec, Canada Jl K 2Rl
and Osman Atabek Laboratoire de Photophysique MolZculaire, UniversitZ de Paris-Sud, 91 405 Orsay, France (Received: May 8, 1987; In Final Form: July 8, 1987)
Perturbations between bound-bound and bound-continuum states have been previously treated successfully by semiclassical techniques, thus bridging weak and strong perturbation regimes. In the case of multiphoton transitions, use of the dressed molecule picture enables us to extend these semiclassical techniques to treat simultaneously radiative and nonradiative perturbations. Applications of the semiclassical method to diatomic multiphoton spectroscopy are shown to be particularly useful in identifying new molecular bound states induced by lasers of high intensities and in interpreting multiphoton transition amplitudes in the nonperturbative regime.
I. Introduction Electronic perturbations in diatomic molecules generally involve couplings between electronic states which are most important a t crossings or near crossings of the potential curves of these states. Such perturbations are usually termed nonradiative, with the crossing case called diabatic, as it involves the neglect of certain electronic couplings (interstate electronic, spin-orbit, etc.), or adiabatic, the case of avoided crossings.'S2 In the latter case, the interstate diabatic electronic couplings referred to above have been included, and the remaining couplings are of the nonadiabatic type, involving nuclear coordinate derivatives of adiabatic electronic wavefunctions. These nonadiabatic couplings are a consequence of the law of conservation of total momentum. This law is violated in the adiabatic approximation since the adiabatic electronic functions are calculated a t fixed nuclear positions, Le., a t zero nuclear Perturbations between bound states can be usually treated by diagonalizing finite-size Hamiltonian matrices in the diabatic or adiabatic representation.* The more difficult perturbations occur whenever continuum states intervene. For instance, bound-continuum perturbations arise always in problems of predissociation.'V2 For the latter type of problems, (1) Herzberg, G. Molecular Spectra and Molecular Structure-Spectra of Diatomic Molecules, 2nd ed.; Van Nostrand: New York, 1950; Vol. 1. (2) Lefebvre-Brion, H.; Field, R. W. Perturbations in the Spectra of Diatomic Molecules; Academic: Orlando, FL, 1986. (3) Smith, F. T. Phys. Rev. 1969, 179, 111. (4) Bandrauk, A. D.; Child, M. S . Mol. Phys. 1970, 29, 95. (5) Bandrauk, A. D.; Nguyen-Dang, T. T. J. Chem. Phys. 1985,83, 2840.
0022-365418712091-6469$01.50/0
a semiclassical scattering theory approach has proved to be very useful as a guide to the understanding and interpretation of the bound to continuum transitions and also has served as a practical computational too1496-8for any coupling strengths. With the advent of lasers of tunable frequencies and variable intensities, radiative perturbations in the presence of intense fields lead to situations where treatments going beyond perturbation theory are needed. Thus a t high intensities (strong radiative couplings), it is not possible to predict interesting features in the multiphoton cross sections from a knowledge of the field-free (unperturbed) molecular levels alone as is usually done in traditional (weak-field) spectroscopy. In discussing intense field multiphoton processes, it is the levels of the total system, molecule and photons, which have to be considered. Such a representation is called a dressed state representation and leads naturally to multistate curve crossings for resonant multiphoton processes and The interstate noncrossings for nonresonant (virtual) (6)
Child, M. S. J. Mol. Spectrosc. 1974, 53, 280; Mol. Phys. 1976, 32,
1495.
(7) Child, M. S.; Lefebvre, R. Mol. Phys. 1977, 34, 979. (8) Sink, M. L.; Bandrauk, A. D. J . Chem. Phys. 1977, 66, 5313. (9) Bandrauk, A. D.; Turcotte, G.; Lefebvre, R. J. Chem. Phys. 1982,76, 225. (10) Kroll, N.; Watson, K. M. Phys. Rev. A 1976, 23, 1018. (11) Voronin, A. I.; Samokhin, A. A. Sou. Phys. JETP (Engl. Trans/.) 1976, 43, 4. (12) Lau, A. M.F. Phys. Rev. A 1976, 13, 139. (13) George, T. F.; Z m e r m a n , I. H.; Yuang, J. M.; Laing, J. R.; Devries, P. L. Acc. Chem. Res. 1977, 10, 449. (14) Yuan, J. M.; George, T. F. J . Chem. Phys. 1978, 68, 3040.
0 1987 American Chemical Society
6470 The Journal of Physical Chemistry, Vol. 91, No. 26, 1987
Bandrauk and Atabek
” {VI ___-
W’
I
SA
7,’
I
-f) I
L
+
-
- n-t -
-
-
~
{”}
Figure 1. Diagrammic representation of the multicurve-crossing problem. T’ and T” symbolize the Stueckelberg transition matrices at a crossing point met on inward (e) or outward (+) nuclear motions. S’and S” are elastic phase propagation matrices; Jt represents the reflection matrix
for left turning points. couplings now arise from radiative field-induced couplings. Frequently, an important step in molecular multiphoton transitions involves transitions between molecular bound and continuum (dissociative) states. One thus encounters a situation similar to the predissociation case invoked in the previous paragraph. It is precisely in this dressed state representation that radiatiue and nonradiative perturbations can be treated simultaneously and for any coupling strengths. This enables one to span in principle the weak coupling (Fermi-golden rule) perturbative limit to the strong coupling, nonperturbative limit. We will endeavor to show in the next sections that a uniform semiclassical approach can be used to interpret results of quantum coupled equation approaches to radiative and nonradiative problems. It will be shown that semiclassical transition amplitudes can be entirely expressed, apart from certain phase factors, in terms of action angles involving only diabatic and adiabatic phase integrals of the nuclear motions on the corresponding diabatic or adiabatic electronic potential curves with or without the presence of the radiation fields. Bound-continuum transitions induced by intense radiative field become analogous to predissociation problems so that field-free molecular bound states become unstable in intense fields, acquiring finite lifetimes that are most easily interpreted in a semiclassical scattering approach. These finite-lifetime dressed states appear as laser-induced resonances in the scattering S matrix, calculated in the presence of the radiation field. 11. Semiclassical Scattering Matrix We consider a molecular problem involving an arbitrary number (M) of coupled open (continuum) or closed (bound) channels with potential curves presenting N crossings in a diabatic picture. Alternative semiclassical treatments for such a situation have been proposed in the Landau-Zener case and its generalizations by Bauer, Fisher, and Gilmore,21Sink and Bandrauk,8 Child and ~ ~ George and Laing.26 The Baer,22Wooley,23C ~ h e nE, ~u ~, and theory can be readily cast into a diagrammatic form,each linkage being reflected by a matrix product. These matrices describe local solutions for curve-crossing points, phase accumulations, and classical turning points, valid for any interstate coupling. The main limitation is the restriction to local two by two state analysis (15) (16) (17) (18)
2683.
Bandrauk, A. D.; Sink, M. L. Chem. Phys. Lett. 1978, 57, 569. Bandrauk, A. D.; Sink, M. L. J. Chem. Phys. 1981, 74, 1110. George, T. F. J . Phys. Chem. 1982,86, 10. Atabek, 0.;Lefebvre, R.; Jacon, M. J . Chem. Phys. 1980, 72, 2670,
(19) Kodama, K.; Bandrauk, A. D. Chem. Phys. 1981, 57, 461. (20) Bandrauk, A. D.; Turcotte, G. Collisions and HalfCollisions with Lasers; Rahman, N. K.; Guidotti, C., Eds.; Harwood Academic: New York, 1985; pp 351-371. (21) Bauer, E.; Fisher, E. R.; Gilmore, F. R. J . Chem. Phys. 1969, 51, 4173. (22) Baer, M.; Child, M. S. Mol. Phys. 1978, 36, 1449. (23) Wooley, A. M. Mol. Phys. 1971, 22, 607. (24) Cohen, J. S . Phys. Rev. A 1976, 23, 86. (25) Eu, B. C. J . Chem. Phys. 1970,52, 3902; J . Chem. Phys. 1972,56,
2507.
(26) Laing, J. R.; George, T. F. Phys. Rev. A 1977, 16, 1082.
Figure 2. Double predissociation or double photodissociation in dressed molecule representation of Ar2+.I6RIand R2 are the two crossing points. ell is the diabatic (unperturbed ground-state) bound-state action angle. OI2 and OI3 are the adiabatic action angles created by the diabatic or radiative perturbations VI, or VI, at R,and R,.
of each interaction region. Noncrossings can be also treated by the introduction of complex intersections as done by Miller.27 A general multistate curve-crossing problem which leads to a transition amplitude from open channel u to channel u via A4 diabatic channels presenting N crossings is viewed through the S matrix element S,,, for which a diagrammatic representation is that of Figure 1. The method relies on following changes in the amplitudes W” and W‘of ingoing and outgoing semiclassical wave functions. Referring the oscillatory exponentials to the last crossing point, the one for which RN is largest, the typical semiclassical wavefunction is
Although the crossings occur in a diabatic representation, it must be emphasized that the semiclassical wavenumbers k,(r) are a d i a b a t i ~ , ’ ~ *so* they ~ * ~ are ~ to be calculated by using adiabatic potentials. Thus, several adiabatic potentials may contribute to ki(r). A typical situation is depicted in Figure 2, where a curve crossing between diabatic channels i and j occurs at Rp. ki(r)is made up of the two adiabatic potentials V+(r)for r < Rp and V-(r) for r > Rp, with the definitions
V*(r) = 72[Vi(r) + V,(r)I f 72{[K(r)- V,(r)12+ 4 [ ~ ; t r ) i ) * / ~ (2) the convention being that K ( r ) > V,(r) for r > Rp and &,(r) is the diabatic coupling between channels i and j . Relations between the amplitudes W”and W’are obtained by matching $i(r) to the asymptotic parts of exact quantal solutions of appropriate model problems at curve crossings, for free propagation and at turning points. ( a ) Curve Crossings. The case of curve crossings at point Rp has been examined by several a ~ t h o r s . ~ * ~Transitions * ~ ~ - ~ * across the crossing point Rp can be conveniently expressed in terms of Stueckelberg connection matrices corrected by appropriate phase factors exptfix) obtainable from the model solutions. These connection matrices are symbolized by rectangular boxes noted T/ and TP)in Figure 1 and are displayed in increasing order of curve crossing points, Le., R 1 < ... < Rp < ... < R,. In the hypothesis according to which only two channels i and j are involved in the vicinity of the pth crossing point, Tp” is an M X M transition matrix containing unity on its diagonal and zero (27) (28) (29) (30) (31)
Miller, W. H. Ado. Chem. Phys. 1974, 25, 69. Crothers, D. S . F. Ann. Phys. 1971, 20, 405. Atabek, 0.; Lefebvre, R.; Jacon, M. J. Chem. Phys. 1984,81, 3876. Nikitine, E. E. Ado. Chem. Phys. 1970, 5 , 135. Miller, W. H.; George, T. F. J . Chem. Phys. 1972, 56, 5637.
The Journal of Physical Chemistry, Vol. 91, No. 26, 1987 6471
Feature Article elsewhere with the exception of four elements situated a t the intersections of columns and lines i a n d j labeled according to the convention of Figure 2:
TpI is the transposed transition matrix TT. The parameters A,
and xt are defined for arbitrary coupling by means of a parameter or,which for linear diabatic potentials vb3(r) = Fib>(‘ - Rp) and constant coupling V,, reduces to the well-known Landau-Zener expression30 w, = V J m U p A F i / )
(4)
with up the classical velocity at the turning point and Mi,= F, - F,. A more general expression for w, involves the adiabatic wavenumbers k*P(r) for the adiabatic potentials in eq 2: 27rwlj = Im
1 1 ; [kP(r) - k+P(r)] dr)
(5)
The integration limits r* are the complex crossing points for the adiabatic terms V*P(r) (Le., [[V,(r) - V,(r)I2 + 4Vit(r)l p 0). Finally, the expressions for A, and xp are kp = exP(-v/)
(6)
xp = arg I’(iwr,) - wi, In wr, + w, + a / 4 (7) In the weak (diabatic) coupling limit, xp -a/4 whereas in the strong (adiabatic) coupling limit xp Os4
--
(6) Elastic Propagation. Pure phase accumulation will occur for elastic propagation of nuclear waves between crossing points Rp and Rptrleading thus to phase accumulation or propagation matrices S/ and Si. These are symbolized by the lozenges in Figure 1 and have the diagonal form 0
exp(-id IP)
S,” =
[.
exP(-i4j9
3
(8)
ai = L > i ( r ) d r
Clearly some of the phase accumulations are meaningless in the case of left turning points being to the right of the innermost crossing R , (Le., a, > A!,). But since the integral (12) depends only on the lower and upper limits of integration, cancellations with phases originating from the propagation matrices S / , eq 8, will always lead to correct phase changes, even though from a computational point of view, diverging (tunneling) exponentials are met in the classically forbidden regions for the above cases. The connection between ingoing and outgoing amplitudes before the introduction of the effect of channels closed to the right is obtained as a product of transition, propagation, and reflection matrices in the order they appear in Figure 1: W’ = AW” (13) A = (TN’SN’)---(T;S2’)(Tl’JeT1’’)(S2/’T2”)- -(SN”TN”)
(14)
Equation 14 can be recast in the form of a product of three matrices which are diagonal, full, and again diagonal, respecti~ely:~~ A = (SN’-- - S i L)A(SN’-- -SiL) (15) where L is a diagonal matrix satisfying
L2 Je The full matrix A is now written as
A = (TNt--
(16)
- T I t ) ( T N t --?.,!)t
(17)
=
[
-( 1
(1
- X,2)’/2 exp[-i($i,P - $ j j ) ]
- X,2)’/2 exp[i(h/’ - q j j 1 1
1
(18)
The new phases 3, can now be recognized as diabatic angles
exp(-i@MP)
The phase d r for channel j is a phase integral given by
associated with the ith diabatic potential, and as adiabatic angles (9)
SpI is the transposed matrix ST.The entrance and exit channels ( i = u or v) are exceptions because they are analyzed at the right of the largest crossing points R, or R, they are involved in. For these channels, the phase integral (9) is modified to
N
N
associated with the adiabatic potential V+P(r) of eq 2. The elements of the new diagonal matrix (SN’-- -Sz’L) become expressible in terms of the diabatic angles, i.e.
(SN’-- -S2’L),, = exp(ill;,) where 0 is the step function that takes the value 1 or 0 for positive or negative argument. ( c ) Turning Points. The correct phase behavior at turning points is obtained by comparison with the asymptotic form of the regular solution to Airy’s equation for the linear potential.32 The linkage conditions on the left turning points (a, ( i = 1, M)) in the hypothesis where all channels are closed on the left are symbolized in Figure 1 by the circle corresponding to an M X M diagonal reflection matrix Jc:
The factor -i accounts for a phase change by -r f 2 due to the left turning points and the ais correspond to phase accumulation between the first crossing point R , and the ais: (32) Landau, L.D.; Lifshitz, E.M.Quuntum Mechanics; Pergamon: New York, 1965; Sections 46-50.
(21)
We now introduce closed channel linkage conditions to produce resonances as bound states coupled to continua. Assuming K to be the total number of channels labeled n, (j = 1, K ) , the right turning point b , gives rise to the bound state linkage condition4 from eq 11: W,,,” = - i exp(2iP,,,) W,,;
(22)
with
In order to calculate scattering amplitudes, we set the incoming entrance channel flux to unity, Le., W,,” = 1, whereas all other incoming fluxes are taken to be zero, W,” = 0, k = 1, M with k # or n,. This defines completely the problem, and the element of the scattering matrix we are looking for is simply the outgoing amplitude of the exit channel: s,, = w; (24) The repeated use of eq 22 and 13 together with the incoming flux conditions detailed above gives S,, as a sum of products of A
Bandrauk and Atabek
6472 The Journal of Physical Chemistry, Vol. 91, No. 26, 1987
matrix elements. The right turning point linkage conditions (28) introduce. new phases j3,eq 23, for the closed channels, which when added to the angles defined in eq 19 and 20, lead to new diabatic angles
+
4, = + I , + PI
indicators of -1 for the entrance channel and 0 for the others. Equations 13 and 32 now lead to an inhomogeneous linear system of 2 M equations where the unknowns are the amplitudes W
(25)
and new adiabatic angles
41 = $11+ P]
(26)
with
Quantizing these phase angles according to the Bohr-Sommerfeld rules gives energies associated to diabatic and adiabatic resonances as in the previous theory of p r e d i s ~ o c i a t i o n . ~ ~ ~ 111. Half-Collisions: Predissociation As is well-known, when a repulsive state crosses and interacts with a bound state of a diatomic molecule, the energy levels of the bound state are affected in two ways. First, these levels described by a wave function X,(r) undergo a transition to the continuum state +E(r) with a probability given in second-order perturbation theory by Y,,E
= (2../h)l(X,lVl+E)I2
(27)
This determines the width ,'I that the levels E, acquire and which is given in terms of a golden rule expression as
r v = al(xulVIC'E)IE=E,2
(28)
The second effect is a shift A, which occurs in the energy levels E, due to the interaction with the continuum A, = P P l d E
(34)
and I is the M X M unit matrix. We now examine the bound-free transition in a type I1 predissociation as depicted in Figure 2. Thus one can calculate the amplitude in the open channel 2 with the channel 1 closed. This leads, for arbitrary interaction strengths, to the following semiclassical expression for the single S-matrix element S 2 f i 8 (COS e,, u COS e,, exp[-i(O,, - ell)]] 2' 2 = (cos ell u cos e,, exp[i(~,,- o , ~ ) ] ] (35)
+ +
Complex resonance energies are obtained as zeros of the denominator of S22: cos + u cos 02, exp[i(02, - 011)1= 0 (36) where u = (1 - X Z ) / P
I (X"lV+dl* E, - E
where PP stands for the principal part of the integral, Le., E, # E. Both quantities can be obtained from the matrix element of the transition operator to second order in the perturbation V"9*33
T = VGOV, Go = ( E - H&'
(30)
so that
is essentially the ratio of the nondiagonal element of the Stueckelberg transition matrix T",eq 3, to the diagonal one. The limiting cases of weak interaction (u 0, X 1) and very strong a, X 0) are the most important in the interaction ( u spectroscopiccontent. Thus in the weak interaction (Fermi-golden rule) limit, the BohpSommerfeld quantization condition is applied to the diabatic angle ell which is linearly expanded in the vicinity of the unperturbed level energy E, as
- - - -
O,,(E) =
+
with ( x k)-I = PP(x-I) - inS(x). Analytic expressions for the matrix element (31) and thus for the energy widths and shifts (28) and (29) respectively have been obtained for various models.19-34,35 Direct numerical approaches for the perturbation expressions of the widths and shifts have also been used p r e v i o u ~ l y . ~ ~ - ~ ~ The more general approach, valid for any perturbation strength V, is to treat the predissociated bound state as a scattering reso n a n ~ e . ~This * ~ method is based on the determination of the complex poles of the scattering matrix where the imaginary part leads to the width of the resonance. The single-channel case has been recently reviewed from the semiclassical viewpoint.39 Recent extensions to multistate curve crossings have been presented for orbiting (shape) as well as Feshbach (predissociation) resonances.& In the previous section we have derived the general semiclassical multistate S matrix involving K bound states and M - K continua, which method depended on evaluating outgoing amplitudes, the Ws. The linkage conditions (22,23) and the flux conditions giving eq 24 can be summarized by the expression W'' = &W'+ (i = 1, M) (32) where the tr(s are either the right linkage phase factors for closed channels or zero for the open ones. As for the el's, these are flux (33) Watson, K. M.; Nuttall, J. Topics in Several Particle Dynamics; Holden-Day: San Francisco, 1967. (34) Sink, M. LI.; Bandrauk, A. D. Chem. Phys. 1978, 33, 205. (35) Bandrauk, A. D.; Laplante, J. P. J. Chem. Phys. 1976,65,2592,2602. (36) Atabek, 0.;Lefebvre, R. Chem. Phys. Lett. 1972, 17, 167. (37) Julienne, P. S.; Krauss, M. J . Mol. Spectrosc. 1975, 56, 270. (38) Ben-Aryeh, Y . J. Quant. Spectrosc. Radiat. Transfer 1976, 13, 1441. (39) Korsch, H. J. Lect. Notes Phys. 1986, 211, 217. (40) Korsch, H. J.; Mahlenhamp, R.; Thylwe, K. E. J. Phys. A 1986, 19, 2151.
(37)
(U
+ 1/2)a+
(2)
( E - E,)
+ ...
(38)
E"
and (&911/dE)Euis given in terms of the local energy spacing
To order u, one obtains for the energy shifts and widths A, = ( h w , / a ) u sin
cos
e,,
r, = ( h w l / a ) ~COS, ozl
(40)
-
(41)
In the reverse situation of a very strong interaction ( u m), the bound levels E,+ of the upper adiabatic potential curve V+(r), Figure 2, are the lowest order approximation for the resonances given by the Bohr-Sommerfeld quantization condition applied to the adiabatic angle e,,: e,,(E) = (u+
+ y2). + (../hw+)(~- .E,+) + ...
(42)
From (42) we obtain to order u-I A"+ = ( h w + / a ) u - ' sin 8,' cos ell
(43)
r,+= ( h w + / a ) ~ -COS, l e,,
(44)
where hw+ is the local adiabatic level spacing. It is to be noted that diabatic resonances, eq 40 and 41, are regulated by the = 012,whereas adiabatic resonances, eq 43 adiabatic phases and 44, are defined by the diabatic phases ell. Sharp or, rather, narrow resonances will arise whenever a diabatic or adiabatic phase is itself quantized to (n + l/z)a, resulting in vanishing small line widths. The pole condition, eq 36, can be extended to intermediate couplings by considering the case of quasiresonance between diabatic and adiabatic levels. This
The Journal of Physical Chemistry, Vol. 91, No. 26, 1987 6473
Feature Article
case has been discussed abundantly in the l i t e r a t ~ r e . 6Thus, ~ ~ ~ ~ ~ ~ground ~ ~ ~ state of Arz+ whereas Vz and V, are the two first excited states, zXg+ and 211g,which are both dissociative.) Multiphoton introducing the quantizing conditions (38) and (42) into the absorptions would correspond to further crossings which for Ar2+ S-matrix pole condition (36) results in intermediate coupling are expected to be negligible since the next excited states are high resonance energies E, and r, which are weighted averages of the in the UV region. For this molecule, the two continua in Figure diabatic and adiabatic energies E , and E:: 2 are accessible in the visible and near-UV, leading to a second E, + WE,+ curve crossing with potential Vz(r)+ ( n - l ) h q and the largest E, = (45) 1 + €U transition moment is p 1 2 ( r ) r / 2 ( p I z 10pi3).16The curve crossings illustrated in Figure 2 correspond to resonant transitions. TCZU(1 u) Since in the quantized version of the electromagnetic field one r, = h o ( 1 + € 4 3 (E,+ - E"IZ writes the electric field, for a cavity of length L,as48-50 with t = w / w + , Le., the ratio of diabatic and adiabatic local B(x,t) = ( h w / 2 ~ ) l / * [ c i + ( t ) e - l ~d(t)eik"] " (47) frequencies. It is to be noted that the diabatic and adiabatic where d+ and d are photon creation and annihilator operators for representations appear as natural bases for expanding intermediate the states In) of n photons coupling resonances, even though the two bases are not orthogonal to each other. This is in contrast to the Fano theory of contind+ln) = (n + l)'/'ln 1 ) , din) = nl/'ln - 1) (48) uum-bound-state configuration interaction which involves linear then both emissions and absorptions can occur simultaneously. combinations of many orthogonal states.43 The unperturbed states are molecule-field states I$,) In) correThe above formulas apply to a two-channel system, Le., a single sponding to eigenstates of the zero-order Hamiltonian curve crossing. Multiple continuum crossings of a single attractive potential have been treated in the same semiclassical approximation as described above and applied to predissociation in 02.8 wherehm)$,) = Vm(r)l$m)is the molecular eigenstate equation, Another interesting application of these semiclassical techniques and H J n ) = (n ' / , ) h w ( n ) is the photon equation. Thus in can be found in the problem of accidental predissociation, where radiative transitions from the ground state V , ( r )to the excited a bound state dissociates into a continuum via a coupling with states V2(r)and V,(r),resonant transitions lead to final potentials a quasidegenerate second bound state. This double curve crossing with n - 1 photons due to absorption (annihilation of a photon problem, with one crossing between two bound potentials and the via 6). On a total energy scale, these must cross the initial state other crossing between a bound and a dissociative potential, has with n photons, as illustrated in Figure 2, as a result of energy been treated semiclassically by Atabek et a1.44and has been used conservation, i.e. to explain anomalous isotope effects in NZ+45and BeH46which manifest such two-step predissociation processes. As indicated E = Vi + nho = V2,3 + (n - l ) h w (50) in the Introduction, explicit consideration of photon states in the Emission from the ground state can also occur by the creation description of radiative processes leads also naturally to multiple of a photon (d+ interaction), thus creating the nonresonant pocrossings between various molecular potentials. We now examine tentials Vz ( n 1)hw or V3 (n + l)hw, which are at an in the next section applications of the semiclassical formalism energy 2hw aboue the resonant states illustrated in Figure 2. In described above to treat radiative transitions to all orders of the zero field, these nonresonant states would contribute to the Lamb radiative interaction, since the semiclassical approximation is valid shift of the ground state, as a result of virtual emissions and for all coupling strengths. reabsorptions in vacuo.51 Neglecting these virtual transitions IV. Photodissociationof a Dressed Molecule amounts to invoking the rotating wave approximation (RWA)8! This approximation is valid whenever the radiative perturbation Photodissociation presents a great analogy with predissociation 3-E is much less than the energy separation of these virtual states when the dressed molecule picture of multiphoton processes is from the resonant states, i.e., 2hw in our case. The neglect of considered. One now has a situation where bound states are virtual transitions and resonant multiphoton transitions leaves the em-bedded in a continuum and interact with a radiative coupling states illustrated in Figure 2 to describe the photodissociation of ?.E, where i; is the electronic transition moment and E the laser Arz+ from its ground state. field amplitude. As in the case of predissociation, zero-order bound As a result of the above discussion, channel 1 undergoes a states undergo energy shifts and acquire a width and therefore C-type predissociation toward channel 3 with weak coupling even a finite lifetime in the presence of the radiation field. At weak in intense fields, whereas a C+-type predissociation toward channel fields, Fermi-golden rule and hence Franck-Condon factor ex2 will evolve from a weak coupling to strong coupling as the laser pressions for line widths may still be valid. However, for strong intensity is increased since plzgrows linearly with r for 2,+ fields, renormalization of the spectrum via multiphoton processes Z,+ transitions.16 The relevant information concerning the shifts will invalidate this expression and new bound states induced by and widths of the ground-state molecular levels in the presence the laser field itself will be of importance (seeref 20 and references of the field can be obtained from the poles of the semiclassical cited therein). A semiclassical version of the theory of predisscattering matrix described in the previous section, with the disociation valid for any field strength is thus a great help in understanding exact close-coupled channel calculations. abatic couplings at the crossings being replaced now by the radiative matrix elements As a simple example of photofragmentation in an intense laser for which field, we considered previously the molecule Ar2+ the relevant dressed potential surfaces are illustrated in Figure 2. As is seen,the photodissociation process in the dressed picture becomes a curve-crossing problem between the bound-state field potential Vl(r)+ nhw and the continuum V3(r)+ ( n - 1)hw. The where 2! is the electronic angular momentum projection on the interaction is mediate_d by the field through the radiative coupling internuclear axis, X is the field polarization with respect to labterm VI3(r)= iiI3(r)+E.(Vi is the electronic potential for the *ZU+ oratory axes, d , is the m = 0, f l component of the electronic transition moment pii along the molecular axis, and finally y is (41) Gordon, R. D.; Innes, K. K. J . Chem. Phys. 1979, 71, 2824. a field-dependent unit conversion factor for a field intensity Z in (42) Knockel, H.; Tiemann, E.; Zoglowek, D. J. Mol. Spectrosc. 1987,85, watts per square centimeter (we use the relation Z = c Eo2/4 T
-
+
-
+
+
+ +
+
-
16320947
225. (43) (44) 364. (45) (46) (47)
Fano, U. Phys. Rev. 1961, 124, 1866. Atabek, 0.; Lefebvre, R.; Requena, A. J . Mol. Spectrosc. 1980,82, Lorquet, A. J.; Lorquet, J. C. Chem. Phys. Lett. 1974, 26, 138. Lefebvre-Brion, H.; Colin, R. J . Mol. Spectrosc. 1977, 65, 33. Bandrauk, A. D.; Turcotte, G. J . Phys. Chem. 1983,87, 5098.
(48) Loudon, R. The Quantum Theory o f l i g h t ; Oxford University Press: Oxford, U.K., 1983. (49) Bandrauk, A. D. Mol. Phys. 1984, 28, 1259. (50) Goldin, E. Waues and Photons; Wiley: New York, 1982. (51) Nguyen-Dang, T. T.; Bandrauk, A. D. J. Chem. Phys. 1983,79,3256.
6474 The Journal of Physical Chemistry, Vol. 91, No. 26, 1987
Bandrauk and Atabek
where Eo is the maximum field amplitude): y = (1.17 X 10-3)(Z (W/cm2))'i2 (cm-'/au)
(52)
The Clebsch-Gordon factors in eq 51 fix the selection rules at A J = 0, f l and AM = 0, f l . In actual calculations, the S matrix for an initial J level becomes a multichannel calculation due to the selection rules over J and M. However for I: Z: transitions and z polarization, the M = 0 sublevel is most strongly coupled (this corresponds to the molecule parallel to the field). Hence for this configuration, one can perform a four-channel S-matrix calculation corresponding to the potentials in Figure 2 to examine the effects of large radiative interactions. Applying the methods described in the previous section, one obtains the semiclasical inelastic scattering matrix element S 3 2 as16*29
-
S32
0.5
15
1.0
I
( 10'0
2.0
w/cd
= 2ul'/2u21/2(1+ u2)1/2exp[-i(a2
+ a3)] x ( N / D ) exp[2i(8,2 - 4l)l (53)
N = cos 4' sin (813 - Oil) (1 + u2)-l cos 612 sin D = cos ell
ell e ~ p [ - i ( 8-~813 ~ + Sll)] (54)
+ uI cos 813 exp[i(OI3- Oil)] + u2 cos
4 2
exp[-i(42 - 4Jl ( 5 5 )
where the angles 6 are illustrated in Figure 2. The poles of this matrix element are given by the zeroes of the denominator D of eq 5 5 . cos O l 1 exp(i611) ul COS 813 exp(ieI3) + u2 cos 012exp(iO12)= 0 (56)
+
Clearly, this expression can be generalized to admit other resonant multiphoton processes that might occur, with terms similar in structure exhibited by the symmetry of eq 56. From this equation it is clear that a pole of S32 can be associated with a situation where all phases, diabatic (ell), and adiabatic (813, eI2),correspond to quasi-bound states, bringing all cosines close to zero. Introducing Taylor expansions for ell,el!, and 612 as in eq 38 and 42 in regions where diabatic and adiabatic levels are close together, we obtain from the real and imaginary parts of (56)16
E,
E, =
+ CtiuiEUz+ I
1+
C€,Ui
+ AEr
(57)
I
4a(E, - E,)' (hw)-'
+ ~ C ~ U , ( ~ U , +-)E,:)2 -~(E, I
The sum over i includes all crossings (e.g., i = 1 , 2 for the resonant process illustrated in Figure 2). As previously, E, is the diabatic (field-free) state energy, whereas the E,;'S correspond to the adiabatic states newly created by the laser field. h w and ha,+ are the corresponding local energy spacings, and e, is their ratio w/w,+. The background shift A E < not given by a simple semiclassical theory based on a linear potential approximation, results from strong nonlinear variation in the diabatic potential^.^,^^ For the Arz+ case discussed above, the radiative coupling at crossing R2 being much more important than at crossing R1, we can expect identifiable field-induced adiabatic states to occur at R2. With ell and 612 now being quantized, simplified expressions are obtained for the energy E, and width I ', of the exact dressed states of the system
15
30
45
60
75
90
0
Figure 3. (a) Behavior of r as a function of intensity I for the 22:y+, u = 8, J = 9.5,M = 0 level of Ar2+in a XeF laser (w = 28 328 cm-I). (b) Photodissociation angular distribution of the above level: (-) ( I = IO8 W/cm2) and (-.) ( I = 1.5 X 1Olo W/cm2) without AJ = 0; (- - -) ( I = 1.5 X 1O'O W/cm*) wtih AJ = 0, fl).
Comparing with the previous results derived above, one can readily identify the intermediate coupling predissociation shifts and widths in eq 45 and 46 augmented by widths and shifts of the single crossing R 1already obtained in eq 40 and 41. Two remarks are in order: (i) When one coupling is dominant (u2 >> u l ) , the other perturbations are reduced by a factor of 1 e2u2,which can be large; Le., all other resonant and nonresonant processes are made less important. (ii) Neglecting all these side processes, the semiclassical theory predicts that no matter what the strength of the interaction is, one should obtain sharp states induced by the laser field. Their energies result from a mixing of the energies of the original diabatic (unperturbed) bound states and the new adiabatic bound states supported by a new adiabatic potential, V+(r), Figure 2, formed by the radiatively induced avoided crossing at R2. In particular, no photoabsorption will occur at energies where diabatic states are accidentally degenerate with adiabatic states, as in this case one expects the new resonances E, to be very sharp; their widths r, a (E,' - E,) go to zero. Photodissociation under the above conditions can be more usefully envisaged as a predissociation induced by the laser field, in the sense that the fragmentation occurs through sharp resonances created by the field, which resonances are the true dressed states of the molecule-field system. Numerical comparisons between semiclassical and exact close-coupled equations have been presented for Arz+ for various field strengths for the M = 0 state, Le., parallel field-molecule configuration and for other M config~rations.l~~~~~~~~~~ Very sharp resonances for particular field-molecule orientations are indeed obtained from the quantum calculations when coincidences occur by varying the angular momentum J , in conformity with the semiclassical predictions. As shown by Bandrauk and T ~ r c o t t e , ~ ~ these field-induced resonances are responsible for nonstatistical photodissociation angular distributions in state-selected experiments, especially at very high intensities (I 1O'O W/cm2), where one would expect statistical angular distributions due to the considerable mixing of rovibrational states by the field in this
+
-
(52) Bandrauk, A. D.; Turcotte, G . J . Chem. Phys. 1982, 77, 3867. (53) Turcotte, G.; Bandrauk, A. D. Chem. Phys. Leu. 1983, 94, 175.
The Journal of Physical Chemistry, Vol. 91, No. 26, 1987 6475
Feature Article TABLE I: Line Widths I' at Various Energies E above the Crossing Point R 2(Figure 2) for a Field Intensity I = 10" W / c d and a Parallel Molecule-Field Configuration ( M = O)I6 u1
Elcm-1 1300 3600 5300
3.48 1.46 1.08
r,Oa/cm-' 308 113 70
r,SCb/cm-'
rl(cxaa)c
0.3 12 3.2
2 17 3
,I Fermi-golden rule values, eq 63. bSemiclassical values, eq 60. cExact values obtained from a phase shift analysis of S22according to a Fox-Goodwin integrator method16vs8
nonperturbative regime. This is illustrated in Figure 3a, where we show the behaviour of the laser-induced line width r for M = 0 as a function of intensity. The minimum in r can be shown to correspond to a quasicoincidence (degeneracy) of diabatic and adiabatic (field-induced) resonances/eq 60. The effect of this minimum in the photodissociation probability for the strongest coupling orientation ( M = 0) appears in Figure 3b as a clearly nonstatistical photodissociation angular distribution at I = 1.5 X 1Olo W/cm2. We remark that the above calculations were performed for a state-selected case, Le., u = 8, J = 9.5. Such selective vibrational pumping is now feasible by two-photon processes.54 Similarly, the plane of rotation of a molecule can ~~ also be selected by current high-power t e c h n i q ~ e s .Combination and application of these techniques to the Ar2+photodissociation in intense fields would be of interest to verify the nonstatistical distributions illustrated in Figure 3. Some of the resonance widths calculated for M = 0 at a field intensity I = 10" W/cm2 are displayed in Table I together with Fermi's golden rule expression for I'r:4
rO :)
-
= ( h w / . n ) ~cos2 l e13
(hw/24ul
(63)
where cos2 8 is replaced by its average 1/2. We remark first that Fermi's golden rule and hence the Franck-Condon factor approach to photodissociation are no more valid for such field intensities. A simple explanation can be found by considering the electronic Rabi frequency w12 = p 1 2 E o / hwhich for I = 10" W/cm2, p12 2 au gives from eq 52, w l 2 = 600 cm-l. In the language of the theory of predissociation; this coupling (radiative in the present case) corresponds to an intermediate coupling case since w l 2 o(diabatic), i.e., the diabatic local vibrational frequency. Clearly the radiative perturbation is nonperturbative, inducing strong mixing of bound and dissociative nuclear levels, which are conveniently described in eq 57 and 58 as combinations of diabatic and adiabatic levels. r r calculated from the semiclassical expression (60) agrees with the exact results at energies well above the crossing point (3600 and 5500 ad).Better agreement around the crossing point, especially at 1300 cm-I, could be obtained by a more elaborate version of the semiclassical theory, necessitating the calculation of phases in the complex r plane where adiabatic curves cross.26,31,56,57 As a final example of the application of semiclassical descriptions of half-collisions in terms of dressed states, we turn to the treatment of stimulated Raman scattering by Bandrauk et al.5840 This process is illustrated in Figure 4. A resonant continuum state Ic,nl - l,n2) of the dissociative potential U , ( R ) (e.g., the lXg+ continuum of Ar2+;previous section) is coupled strongly to the bound state Ib,nl - l,n2 1 ) of the potential U3(R) by stimulated emission of photons of frequency w2. The continuum is also perturbed by the photoexcitation from the initial bound state la,nl,nz) (e.g., ground state 2Zu+,previous section) via the
-
-
+
(54) Shimizu, F.; Shimizu, K.; Takuma, H. Chem. Phys. Lett. 1983, 102, 375. (55) DeVries, M. S.;Srdanov, V. I.; Hanrahan, C. P.; Martin, R. M. J. Chem. Phys. 1983, 78, 5582. (56) Bandrauk, A. D.; Miller, W. H. Mol. Phys. 1979, 38, 1893. (57) Baranyi, A. J . Phys. B 1979, Z2, 2841. (58) Bandrauk, A. D.; Gelinas, N. J . Comput. Chem. 1987, 8, 313. (59) Bandrauk, A. D.; Turcotte, G. J. Phys. Chem. 1985, 89, 3039. (60) Bandrauk, A. D.; Giroux, M.; Turcotte, G. J . Phys. Chem. 1985,89, 4413.
Figure 4. Dressed potentials VI and U, for intense stimulated emission of a photon w2 after weak photoexcitation by a photon in Ar2+. Potentials 1 and 2 would also cross at R, in a dressed picture. wIJ99@'
photon of frequency ol. Furthermore, the continuum states I C ) could in principle couple to higher energy channels via further transitions induced by the intense field of frequency w 2 . We therefore examine the effect of extraneous couplings of the continuum ic) on the stimulated emission to the bound state Ib) out of IC). The stimulated emission of photons of frequency w2 in the dressed picture corresponds therefore to the crossing at Rb between the molecular field potentials Ul (nl - l)hwl n2hw2and U2 + (nl - l)hwl (n2 + l)hw2. Another crossing occurs at R, between the dissociative potential Ul (nl - l)hwl + n2hw2and the other radiative channels coupled to the continuum states, as for instance the initial potential U2(R) n,hw, n2hw2. (In Figure 4, we show only the crossing between U3and Ul at Rb and the states of photon 2 for simplicity.) Assuming that both crossing R , and Rb are well separated and assuming U2is another continuum in order to mimic other photophysical processes emanating from IC) beside the stimulated emission, one can derive from the method described in section I1 the inelastic scattering matrix element S,2:58
+
+
+
+
+
+
S12= -2iu,(u, + 1)-' exp[i(P - e)] X cos 8 sin (a - p 8) ub cos (p X b ) sin ( a + X b ) cos 8 + ub exp[i(P + X b - e)] cos (p + X b ) (64)
+ +
+
I
where u,(b) are the coupling parameters (eq 37) defined for the crossing points R, and Rb;x's are defined in eq 7. We furthermore have the following phase integrals or angles: diabatic, 8 = 033; adiabatic, p = 823. The phase a! essentially corresponds to the phase of the Franck-Condon factor, sin a, between the nuclear waves propagating on the continuum potentials VI and U,. In the weak radiative coupling case, u,
