J. Phys. Chern. 1081, 85,2535-2542
it is a combination of the dipolar and the hydrogenbonding interaction which jointly plays the complex role in determiing the energy barrier for the molecular reorientation and for the process of viscous flow.
Summary and Conclusion We have studied the reorientational process of Me2S0 in pure liquid and in aqueous solutions by depolarized Rayleigh scattering. The depolarized Rayleigh scattering spectra of aqueous solution reflect mainly the orientation fluctuation of Me2S0 molecules; the contribution of scattering from water molecules is found to be negligible. From the spectral line width measurements we have obtained T~~ as a function of temperature in solution of various Me2S0 concentrations. The activation energy for reorientation of MezSOin pure liquid is found to be greater than that in solution. In the pure Me2S0 liquid, ~b,, is found to follow the SED equation with the hydrodynamic volume for reorientation being considerably smaller than the molecular volume. This result suggests that despite the strong intermolecular interaction slip boundary conditions are appropriate for describing the reorientation process of the pure Me2S0 liquid. In aqueous solutions, T~ values are found not to follow the SED equation and the data suggest the presence of a nonhydrodynamic effect. The nonhydrodynamic effect is found to be significantly greater in solutions with low Me2S0 concentration. This result lends support to the conclusion derived from the neutron- and X-ray-scattering studies that in aqueous solutions of low M e a 0 content, the M e a 0 molecules tend
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to regidify the water structure. The water structure, however, is found to break down at high Me2S0 concentration. From the T~ vs. q/T plot we have obtained the hydrodynamic vofume for the Me2S0 reorientation as a function of Me2S0 concentration. The hydrodynamic volume of Me2S0 in aqueous solution is found to be quite similar to that found in the pure liquid. This result suggests that in spite of the Me2SO/H20complex formation in solutions, the lifetimes of Me2SO/H20complexes are s, which corresponds to the characteristic shorter than time in Rayleigh scattering. Thus, our result is in apparent disagreement with the conclusion previously derived from the NMR relaxation time study that the Me2S0 molecule and H20 molecules in aqueous solution reorient together. The discrepancy between the two results is believed to be due to the invalid approximation introduced in the analysis of NMR relaxation time data. The present work illustrates the usefulness of the depolarized scattering for the study of the molecular reorientation process and interaction in the liquid state. Rayleigh scattering is a direct technique and is free from complicated assumptions which are needed in the analysis of experimental data obtained in other techniques. Acknowledgment. Acknowledgment is made to the donors of Petroleum Research Fund, administered by the American Chemical Society, for support of this research. Travel support to D.H.C. by the Danish Natural Science Research Council to Salt Lake City is also acknowledged.
Laser Intensity Induced Nonradiatlve Processes in Molecules Claude Needham and William Rhodes” Institute of Molecular Biophysics and Department of Chemistry, FlorMa State University, Tallahassee, Florida 32306 (Received: February 18, 198 1; In Final Form: April 28, 198 1)
The effects of the coupling strength (Le., the Rabi frequency) of a coherent radiation mode interacting with a multilevel molecule are considered. A doorway-state basis is used in which the radiative doorway state, carrying all of the radiative interaction strength from the ground (initial)state, is coupled by intramolecular (nonradiative) interaction to other excited states. The resulting coupling scheme involves an effective Hamiltonian formulation in an extended rotating basis. Quantitative results from the theory are obtained by computer simulation. It is shown how variation of the laser coupling strength can modify the dynamics of nonradiative transitions, thereby producing (1) decoupling of radiationless decay, (2) enhancement of radiationless decay, (3) selectivity of photophysical and photochemical processes, or (4)laser-induced isolation of states, depending on the conditions of the system. The case of two states coupled through a common manifold of states and the conditions for biexponential decay are also considered.
Introduction The theory of nonlinear optical effectsin multilevel systems has been severely limited by the intractable mathematics involved. The two-level system, on the other hand, has been well studied and gelds for arbitrarily intense excitation within the dipole approximation and rotating wave appr~ximation.l-~
Although there has been some success with detailed solutions for three-level systems,6 success with truly systems has been sporadic.”6 In this research we study multilevel coupling models with the primary objective of discerning laser-molecule coupling dynamics qualitatively different from the oscillation-dissipation characteristics of two-level systems. We
(1) J. H. Eberly and P. Lambropoulos, “Multiphoton Processes”, Wiley, New York, 1978. (2) J. I. Steinfeld, “Molecules and Radiation”, Harper and Row, New York, 1974. (3) J. D. Macomber, “The Dynamics of Spectroscopic Transitions”, Wiley-Interscience, New York, 1975.
(4) L. Allen and J. H. Eberly, “Optical Resonance and Two-Level Atoms”, Wiley, New York, 1975. (5) J. V. Moloney and W. J. Meath, Mol. Phys., 30, 171 (1976). (6) J. R. Ackerhalt, P. L. Knight, and J. H. Eberly, Phys. Reu. Lett., 30, 456 (1973). (7) M. Quack, J. Chern. Phys., 69, 1282 (1978). (8) S. Mukamel, J. Chern. Phys., 71, 2012 (1979).
0022-3654/81/2085-2535$01.25/0
0 1981 American Chemical Society
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have found several types of behavior qualitatively distinct from that found in two-level systems and dissipative systems of conventional spectroscopy. These behaviors can be grouped under the two general headings of (1)intensity selection of nonradiative coupling and (2) optical selection of isolated states. Intensity-dependent selection of nonradiative coupling is descriptive of the general class of phenomena in which the nonradiative transitions out of the radiative doorway state display a dependence on the radiative excitation intensity into that doorway state. The phenomena of this type discussed within this paper are radiative decoupling, radiative-induced decay, and intensity-dependent photophysics. Optical selection of isolated states is descriptive of the general phenomenon in which the excitation frequency is manipulated to partially decouple both the ground state and a degenerate nonradiative state from the radiative doorway state. This can give rise to a nonzero population density for these two states in the long-time limit. In this paper we present a basis transformation that analytically demonstrates the nature of this isolated state and specifies the required conditions for its selection. In the study of multilevel coupling schemes, it is nearly always essential that approximations be used. Herein we utilize a theoretical approach that requires only standard assumptions such as the rotating wave approximation (RWA): the Wigner-Weisskopf approximation (WWA): and restriction to excitation by a single-frequency pulse. The latter limitation to single-frequencyexcitation can be eased by use of averaging techniques. With these approximations a wide variety of coupling models may be transformed into an extended rotating basis yielding a time-independent symmetric Hamiltonian. In this rotating basis the problem is readily solved by computer simulation methods. The experimental situation is defined in the following manner. At t = 0 a coherent monochromatic excitation source of arbitrary intensity is suddenly turned on. This source irradiates the molecular system for the duration of the experiment which maintains full coherence of the molecule-field interaction. This coherent driving field is treated theoretically by the methods of semiclassical quantum mechanics,’O whereby the driving field is treated classically and the molecule is fully quantized. When photon scattering probabilities are studied, however, the scattered photon field is treated quantum mechanically as well. The molecule is defined by a set of discrete levels coupled by radiative and nonradiative interaction potentials. One limit of our approach is the exclusion of coupling schemes in which two states are coupled by both radiative and nonradiative interaction. The coupling of the set of discrete states to the radiative and nonradiative dissipative continua is approximated by the inclusion of an imaginary part in the expectation value for the state energies in the zeroth-order basis. In this paper we utilize several basis sets and corresponding Hamiltonian matrixes for the molecule. Initially the molecule is described by a zeroth-order basis typically a spectroscopic channel basis” (SCB), consisting of a sequence of states each coupled to its nearest neighbors and possibly to a dissipative continuum. The Hamiltonian in (9)V.Weisskopf and E. Wigner, 2.Phys., 63,54(1930);65,18(1930); V. Weisskopf, ibid., 85,451 (1933). (10)M. Sargent 111, M. 0. Scully, W. E. Lamb, Jr., “Laser Physics”, Addison-Wesley, Reading, MA, 1974. (11)A. R. Ziv and W. Rhodes, J. Chem. Phys., 65,4895 (1976);R. Cable and W. Rhodes, ibid., 73,4736 (1980).
Needham and Rhodes
this basis is a time-dependent nonsymmetric effective Hamiltonian. The next basis is an extended rotating basis generated by a time-dependent transformation of the zeroth-order basis. The Hamiltonian in this basis is a time-independent, symmetric, effective Hamiltonian. In the calculation of transition probability matrix elements, we use the eigenstates of this Hamiltonian and the corresponding eigenenergies. In the analytic solution of optical selection of isolated states, we transform the extended rotating basis (before diagonalization) into a third basis. In this basis, under well-defined conditions, a state will be isolated by virtue of zero coupling with the rest of the basis set. Because the essential condition for nonlinear optical effects is oscillation of the microscopic (single isolated molecule) population inversion, we will calculate this quantity and neglect ensemble effects. We feel that it is important to understand the effects of the multilevel character of the molecule before probing ensemble effects of multilevel systems. The required quantities for the calculation of the microscopic population inversion are the matrix elements of the time-developmentoperator. Our method of calculating these matrix elements consists of two steps. First, after the appropriate approximations (RWA, WWA, singlefrequency excitation) are made, the zeroth-order effective Hamiltonian is transformed into an extended rotating basis yielding a time-independent symmetric Hamiltonian. Then, given a detailed (numerical) description of the molecular coupling, this Hamiltonian is diagonalized by computer methods. The time-development operator matrix elements may then be solved directly by expansion in this eigenbasi~.~J~-’~ Since this method does not yield analytic results, it is necessary to study the temporal dynamics of the time-development operator matrix elements by numerical search procedures. After development of the theoretical background in the next section, we present results of these numerical search procedures and discuss the nonlinear optical dynamics found in our computer simulations of laser excitation in multilevel systems. In a final section we explore the effects of multiple pathways on transition dynamics in both the weak and strong excitation limit. This will have application to such problems as the “breakdown” of the WWA, multiphoton excitation through multiple pathways and biexponential decay.
Theoretical Background The use of effective Hamiltonians has facilitated study of model coupling schemes involving large manifolds of ~ t a t e s . ’ ~ JWhen ~ an effective Hamiltonian is used, the coupling of a discrete state to a large manifold (dissipative quasi-continuum) is approximated by the inclusion of an imaginary part to the self-energy of the state. The main consequences of this approximation are that the state will decay irreversibly into the continuum and coupling of two states through a common continuum is assumed to be zero. If the time of interest (observation time) is short compared to the reciprocal of the average manifold spacing and the energy separation between the discrete states is large (12)B. Carmeli, I. Schek, A. Nitzan, and J. Jortner, J.Chem. Phys., 72,1928 (1980). (13)J. Jortner and J. Kommandeur, Chem. Phys., 28, 273 (1978). (14)P.W.Milonni and J. H. Eberly, J.Chem. Phys., 68,1602(1978). (15)J. R. Ackerhalt, J. H. Eberly, and B. W. Shore, Phys. Rev. A, 19, 248 (1979). (16)J. M. Delory and C. Tric, Chem. Phys., 3,54 (1974). (17)A. R. Ziv, J. Chem. Phys., 68,152 (1978). (18)D.F. Heller, M. L. Elert, and W. M. Gelbart, J.Chem. Phys., 69, 4061 (1978).
The Journal of Physical Chemistry, Vol. 85, No. 17, 198 1 2537
Laser Intensity Induced Nonradiative Processes
Scheme I A
A physical situation describable by the coupling Scheme IB is multiphoton excitation. The transformation matrix needed in this case is given by
WO
(::
6
~ ( t ) =o e-iwat
WO
1
compared to their respective decay widths, then the above consequences are not unduly restrictive. The time-dependent Schrodinger equation may be written as ( h 1)
In the semiclassical representation with use of effective Hamiltonians, " ( t ) is complex and explicitly time dependent. We use a time-dependent transformation to transform the zeroth-order effective Hamiltonian into an extended rotating basis. = A(t)lW)) (2) (3)
H = A-l(t) H'(t) A ( t ) - iA-'(t) d A ( t ) / d t
(4)
In this rotating basis the Hamiltonian has no explicit time dependence, thus allowing simple integration of the time-dependent Schrodinger equation and yielding the familiar exponential form for U(t,t,).
U(t,to)l$(to)) = e-ix(t-to)I$@,)) (5) By the use of the eigenenergiesand expansion coefficients for the eigenstates of the transformed effective Hamiltonian, the time-development operator can be written in the form I$(t))=
( alU(t,to)Ib) = CCagCEbe-i(Orir~/2)(t-to) (6) E
Here {c&] are the coefficients for the expansion of Ib) in terms of the H eigenstates {It)),having eigenvalues utiy /2, and similarly for (ai. bwo generic classes of Hamiltonians that can be transformed into a time-independent form (as in eq 4) correspond to parts A and B of coupling scheme I. Scheme I, A and B, is a pictorial representation of interaction potentials in a zeroth-order basis. The vertical axis represents relative energy. The horizontal axis has no significance. One possible physical situation that would be describable by Scheme IA is laser excitation (frequency 0,) into a manifold of molecular eigenstates. The transformation matrix that will remove the time dependence in this case is seen by inspection to be (7)
0
o
)
e-izwat
(8)
When these physically distinct coupling schemes are transformed into a time-independent basis, the resulting coupling schemes are of the same archetypical form (see Scheme 11). The distinction arises in the identification of the physical significance of the coupling potentials. Note that Scheme I1 is a pictorial representation of an extended rotating basis in which the energies of the excited states have been lowered relative to the ground state. By inspection of the transformation matrixes, eq 7 and 8, the restriction to monochromatic excitation is understood. If there were a frequency distribution in the excitation pulse, then a simple transformation matrix such as in eq 7 and 8 could not produce a time-independent Hamiltonian. When the Hamiltonian is symmetric ( v b , = V&),the discrete eigenstates are orthogonal and normalizable. In what follows we apply our computer simulations to the problem of nonlinear optical excitation of multilevel systems as in Scheme IA. However, because we are studying the problem in an extended rotating basis, the results may be applied to any physical situation having a coupling scheme transformable into the archetypical form, Scheme 11. These two generic classes of coupling schemes may be combined to generate a wide variety of coupling schemes. The sole requirement is that no two states be coupled by both radiative and nonradiative couplings. This restriction is to allow successful transformation of the zeroth-order effective Hamiltonian into a time-independent form. The zeroth-order basis that we will find useful for our discussion in the next section is the spectroscopic channel basis (SCB).ll For a three-level system the SCB consists of a ground state IO), dipole coupled to the radiative doorway state, IR). The doorway state is coupled by intramolecular potentials to a nonradiative state If). The states IR) and If) are coupled to radiative and nonradiative dissipative continua, approximated here by their decay widths yR and yf. This is shown pictorially in Scheme IA and in the transformed basis of Scheme 11. The name SCB is suggestive of the way in which these states are coupled forming a chain. It is not meant to imply that probability is necessarily channeled in a sequential manner through the chain.
Methods A simplified flow chart for our computer simulations is as follows: (1)the original Hamiltonian calculated in the extended rotating basis is read into the computer or taken off tape; (2) the Hamiltonian is diagonalized, and eigenvectors are obtained by subroutines CORTH and COMQR2; l9 (3) the eigenvectors are normalized; (4) eigenvector matrix (C) and eigenvalue matrix (A) are checked by computation of HC - CX, CC-l, and C-W; (5)I(alU(t)lb)I2(as per eq 6) is calculated at various time points (19) G. Goos and S.Hartmanis, Eds., "Lecture Notes in Computer Science", Springer-Verlag, New York.
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The Journal of Physical Chemistty, Vol. 85, No. 17, 1981
Needham and Rhodes
and stored for plotting or tabulation. Step 5 is the most expensive part of our software, making up 8690% of our cost. Future plans call for using a small desk-top calculator to calculate the transition probability matrix elements using eigenvalues and eigenvectors obtained and tested by the larger computers required by CORTH and COMQR2. Intensity-Dependent Selection of Nonradiative Coupling Intensity-dependent selection of nonradiative decay is the descriptive title for a general class of phenomena in which nonradiative transitions out of the doorway state display a dependence on the radiative excitation intensity. This feature is theoretically observed as either a change in the optical nutation decay rate or the time-integrated probability of transitions into a particular nonradiative state. This general class of behavior can be divided into three phenomenologically different categories: (1)radiative decoupling; (2) radiatively induced decay; and (3) intensity-dependent photophysics. Radiative Decoupling. Radiative decoupling20is manifested as a decrease in the rate of decay of optical nutation as the radiative excitation intensity is increased. Let us review briefly the dynamics of an isolated twostate system driven by a strong excitation source. The empirical definition of strong-excitation is that the Rabi frequency, equal to twice the molecule-laser interaction intensity, is greater than the excited-state decay half-width. When this criterion is met, the system displays temporal dynamics characterized by oscillation and dissipation. The probability of being in either the ground or excited state oscillates with ever decreasing maximum value. The oscillation rate is given by the Rabi frequency, and the decay in peak amplitude is exponential with a rate given by one-half the expected fluorescence decay width. This may be considered a trivial form of intensity-dependent decay with a step function dependence on the excitation intensity. Once the strong coupling limit has been reached, the decay of the system is independent of further intensity increases. The effect of increasing excitation intensity past this original limit is merely to shorten the oscillation period. The simplest system that we have found to possess the possibility of displaying more complicated temporal dynamics and intensity dependence is a three-level system. To study the phenomenon of radiative decoupling, consider Schemes IA and I1 with YR = 0 and q = wp With the dissipative decay directly out of the doorway state set equal to zero, any decay of that state is by indirect coupling through the nonradiative state. In this case the decay of optical nutation originates from transitions through the nonradiative state If) into its dissipative continuum. It is because of this indirect coupling that the possibility of radiative decoupling exists. Figure 1shows the probability of being in the doorway state as a function of time, I(RlU(t,to)10)12, assuming the initial state at t = 0 to be the ground state IO). Comparison of these two plots dramatically demonstrates the effect that increasing the radiative interaction has on the optical nutation decay rate. To gather an intuitive understanding of this phenomenon, we transform the basis of Scheme I1 into a new basis by prediagonalization of the radiative interaction. This yields the two new states I&), both linear combinations of the ground and radiative doorway state. Because of the doorway-state component in each state, they are both (20) Paolo Grigolini, private discussions.
.6O1
0
.20 TIME X IO"
Figure 1. Time dependence of the radiative state, showing radiative decoupiing of radiationlessdecay (according to Schemes I1 and I11 of the text). Upper curve: V , = 4; VI = 5; yR = 0; y, = 20. Lower curve: V, = 40; VI = 5; yR = 0; y, = 20 (all quantities given in standardizable units; dimensions, cm-').
Scheme I11
coupled through nonradiative interaction to the state If) (see Scheme 111). In Scheme I1 the energy separation between I+) and I-) is governed by the excitation intensity. As the excitation intensity increases, the energy separation between the prediagonalized states also increases. For wo = wR this energy separation is simply 2V,. As this energy separation increases, the superposition states are effectively decoupled from the damped state If). This gives rise to the decrease in the optical nutation decay rate. Note, however, that the period of optical nutation, T ~also , decreases as V, increases according to the formula 70 = r/(cV,), where V, is in unita of wavenumbers and c is the speed of light. Transitions out of the doorway state directly into a dissipative continuum (as in the picket-fence model)21are not affected by the radiative intensity. Only those transitions through a secondary doorway state, such as If), can be decoupled. When dissipative transitions directly out of IR) are reintroduced, they may overshadow this effect. In order to observe radiative decoupling, it is necessary that YR
