- Email: [email protected]
Nonclassical Franck-Condon processes
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
29 January 1993
Nonclassical Franck-Condon processes E.J. Heller a,b and Doug Beck a a Departmentof Chemistry, Universityof Washington,Seattle, WA98195, USA b Department of Physics. Universityof Washington.Seattle, WA98195, USA Received 23 October 1992; in final form 16 November 1992
Franck-Condon factors govern many important molecular processes, including electronic spectroscopy and radiationless transitions. A better understanding of the propensities for exciting various kinds of motion in nonclassical, nonverticaltransitionsis necessary to attachtherightsignificance to certain experiments. Picosecond and femtosecond laser pulse experiments in the wings of absorption bands are possible which may reveal some of the new effects discussed here.
1. Introduction Radiative and radiationless transitions depend on vibrational overlap factors (Franck-Condon factors) involving eigenstates from two different BornOppenheimer potential energy surfaces. The FranckCondon factors provide propensity rules which spectroscopists take almost for granted. For example, if a bending mode is the most displaced coordinate in one electronic state relative to another, it is assumed that states with bending excitation will be produced in an electronic transition between the Bom-Oppenheimer potential energy surfaces. We shall distinguish two regimes for Franck-Condon factors: classical and nonclassical. The classical regime corresponds to the realm of favorable conditions for the Franck-Condon overlap. The nonclassical regime of small Franck-Condon factors and violations of the “usual intuition” is the subject of this Letter. An absorption spectrum for a bound state going to a steep part of an excited electronic state is shown in fig. 1. As the incident light frequency is increased, we proceed from the “red” wing, through the band center, to the “blue” wing. Drawing a horizontal line between the two classical turning points of the “donor” or initial surface at the quantum energy of the donor state marks the classically allowed region of the initial wavefunction. This surface is progressively raised by increasing #MI.When the surfaces cross in the
350
Fig.1. The construction of the nonclassical wings of an absorption band from the crossing of the donor and acceptor BomOppenheimer potential energy surfaces.
0009.2614/93/$ 06.00 8 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume202.number 5
CHEMICALPHYSICSLETTERS
classically allowed region, the overlap is good and the spectrum is near its maximum. When they cross outside the classically allowed region, the overlap is poor and corresponds to the nonclassical wings. This distinction extends to any number of dimensions. Loosely speaking, when the surfaces cross in classically allowed regions, the electronic state change can occur without any instantaneous change or jump in position or momentum. Otherwise, in the nonclassical wings, a jump is required. A common situation has the relative horizontal displacement of the two surfaces small. There is no red wing in absorption, and the blue wing starts just above the strong “O-O” transition. This case has interesting experimental implications, because by tuning to the blue of the O-Oline the nonclassical region is reached in the blue wing, yet other excited potential energy surfaces may be comfortably remote. Alternatively, one could populate the ground vibrational level of the excited potential energy surface and examine the emission to the red of the O-Oline, either via spontaneous emission or SEP [ 11. As discussed below, perhaps the most exciting prospects are for pump-probe experiments [ 2-41, The issue is very similar for radiationless transitions, except that one cannot change the energy gap. Very often, this gap places the system in question in the nonclassical regime. The Franck-Condon factors in the wings will be small, but in the blue wing there are two possibilities to explain that residual which remains: Either it results from an incomplete oscillatory cancellation in the region of large amplitude, or it could come from the overlapping region of the tails of the two states. This distinction leads to important effects in two or more dimensions. If the tail region is the major contributor, the integral is coming from a region not within the classically allowed region in coordinate space; we call this a “position jump”. If the major
part of the (small) integral comes from the region of large amplitude, the integral may be viewed as arising from the coordinate region where the Gaussianlike initial (donor) state is large, but the rapid oscillations of the final (acceptor) state imply a sudden change of momentum as the transition from donor to acceptor is made. We call this a momentum jump case. The remainder of the Letter is devoted to
29 January 1993
justifying this picture and discussing its implications.
2. Position versus momentum jumps In a subsequent fuller account of this work, we will present four or five different viewpoints on the issue of position and momentum jumps in Franck-Condon factors, all leading to the same conclusions about the utility of the picture and the circumstances under which each occurs. Here we will give a truncated, more intuitive account. Consider the Golden Rule expression for the rate of the electronic transition, k= ;
( V2>P(&) 3
(1)
wherep(&,) is the density of states at energy E,, and ( I”) is the mean squared matrix element. Making the Condon approximation, we write this as k= 2 viTr[G(E,-H,)]d){dl]
(2)
where 1d) is the donor wavefunction, I’: is the constant coupling strength, and Pd= I d) (dl is the density matrix for the donor wavefunction. We simplify this further here by approximating eq. (2) as k= ;
v:Sdpdl6[E,-H.@,q)lp~@,q).
(3)
Eq. (3) says that the rate k is proportional to the integral of the Wigner density of the donor, p? @, q) , over the energy hypersurface 6[ E0 - H, (p, q) ] of the acceptor. The essence of our work is to ask where on the acceptor energy surface the bulk of the rate integral comes from, This is a matter of investigating the integrund of eq. (3 ). By understanding the phase space structure of the donor wavefunction and the energy hypersurface of the acceptor Hamiltonian, we can come to an understanding of the way amplitude appears on the acceptor potential energy surface. The rate k is proportional to the average squared FranckCondon factor; it is a non state specific quantity. By interpreting the integrand of eq. (3), we are putting back the specificity in a very useful way, by saying 351
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
where (and with what velocity) the amplitude “lands” on the acceptor surface. Considered as a Wigner phase space distribution, the donor wavefunction is a smooth distribution peaked at zero momentum and in coordinate at the minimum of the donor potential energy surface. The bulk of such a phase space density has too low an energy on the acceptor potential energy surface to be germane to the donor-+acceptor transition. (This transition has to be quasi-degenerate.) To find the relevant regions in phase space we must examine those parts of the “tail” of the Wigner density which have the right energy on the acceptor potential energy surface. The inequivalent treatment of the donor and acceptor wavefunctions requires justification. (The donor is given a full fledged Wigner density, the acceptor is treated as a classical energy hypersurface. ) This was done much earlier in a treatment of the rate of radiative transitions, where eq. (3) was used directly, in the classical and non-classical regime [ 5 1. Although the results were very satisfactory, and an improvement over the venerable reflection approximation, we did not attempt at that time to go deep into the nonclassical regime, nor did we analyze the rate in terms of its phase space composition. A subsequent paper [6] will do more to justify the inequivalent treatment, but the results shown below are encouraging. We examine a one-dimensional system in fig. 2 to get started. Two cases are shown corresponding to blue wing absorption (or red wing emission). On the left are the wavefunctions, and on the right the phase space pictures are seen. The Wigner phase space distribution for the donor state is smeared distribution, and the quasi-degenerate contour interval on the acceptor potential energy surface is the parabolic curve. In the top case, the donor wavefunction is narrow, and the oscillations of the acceptor wavefunction are not completely effective at killing the integral. At the same time, the tail of the donor wavefunction is very smaI1 near the tail of the acceptor wavefunction. There is no doubt that the overlap comes from the region where the donor wavefunction is large. This is a momentum jump case, because the amplitude leaving the donor wavefunction may be thought of as appearing near the donor wavefunction, with considerable momentum on the repulsive acceptor po352
29 January 1993
Momentum jump
Positionjump
Fig. 2. Two blue absorption wing cases, a momentum jumpand a position jump, are shown in coordinate and phase space representations.
tential. The phase space picture reflects the increased momentum uncertainty and decreased position uncertainty associated with the narrow coordinate space distribution. It is clear that a shift in momentum at (nearly) constant position gives the shortest path between the two distributions. At the bottom in fig. 2 we see a much wider donor wavefunction, centered as in the previous case. A glance at the wavefunction plots shows there is now good overlap of the tails, and much more complete cancellation of the integral in the oscillatory region. The integral is coming mostly from the region of the tails, and the phase space plot shows that a position jump (again at zero momentum) gives the shortest path between the two states. We have just seen that position jumps and momentum jumps can compete to be the major contribution to the Franck-Condon integral in the nonclassical regime. In several dimensions, each position and each momentum becomes a competitor. The winning coordinates can be quite surprising. Thus,
Volume202,number 5
CHEMICALPHYSICSLETTERS
for nonclassical transitions, the usual Franck-Condon propensity rules mentioned at the beginning can be very misleading. For example, the mode with the largest equilibrium position displacement need not be excited in a nonclassical Franck-Condon transition. Some examples of this kind are well known. Radiationless transitions in aromatic hydrocarbons can be dominated by the carbon-hydrogen stretching motion (they are the accepting modes on the accepting surface) even though the carbon-carbon backbone is much more displaced.
3. Pulsed transient “nonclassical” wavepacket A pulsed laser (typically of 50 fs to a few picoseconds in duration) may prepare a nonstationary wavepacket on the acceptor potential energy surface. We want to create a “nonclassical” wavepacket by tuning the laser frequency to the blue of the absorption band (or analogously the red wing in emission) and adjusting the pulse duration so that an appropriate group of states is excited. The population in each of the eigenstates will then be governed by two factors: the Franck-Condon factor and the energy density of the pulse at that frequency, which is controlled by the pulse duration and center frequency. There is an associated transient wavepacket which can be defined as I$,,,>=
1 exp[-(G-fi~0)2/2a21 n
x In> *
(4)
Clearly, 1q$_,,~ ) is a nonstationary wavepacket which can be prepared in the laboratory. At this juncture we mention a connection with the effort to control chemical reactions with special laser pulses [ 7 1. The pulse we need is rather primitive; we only require a short pulse with a center frequency in the blue wing. On the other hand, we are the mercy of the Franck-Condon factors and our goal here is indeed not control, but rather understanding of the factors which make the nonclassical wavepacket what it is. The fascinating question is: what is the nature of I&,#)‘! That is, is it a more or less localized state and if so, what modes or better what part of phase
29 January 1993
space is populated? Even if the nonclassical wavepacket is not created literally in the laboratory, it is necessary to know its form in order to interpret the spectrum in the blue wing in C.W.absorption, or for example the red SEP spectra for acetylene [ 11. The spectrum
is the result of the initial
nonclassical
wavepacket and its dynamics on the accepting surface. We focus attention on one issue out of many which should be addressed. Consider two identical, separable Morse oscillators. They may represent two equivalent C-H bond stretches in the same molecule, for example. Both the donor and the acceptor potential energy surface are Morse potentials but the equilibrium positions, well depths, and effective force constants will vary between the donor and acceptor states. Even in this simplified donor+acceptor transition, several interesting scenarios are possible: - Either position jumps or momentum jumps can be dominant. - The C-H bonds may share the energy in the acceptor nonclassical wavepacket equally, or one may be excited much more than the other. - The two previous items, each with two extremes, combine to make four scenarios.
4. Sample calculations In what follows we shall see two of the four scenarios arise with slight adjustments of the donor potential only; the acceptor potential will remain the same. (Indeed it is only the differences in the two potentials that matter.) We will examine the cases with two tools: first, we create the nonclassical Franck-Condon wavepacket for a given (blue) band center detuning (or equivalently the energy gap, or red detuning in emission) and transform limited pulse bandwidth. By inspecting the location and nodal structure of the nonclassical pulsed wavepackets, the various possibilities for position and momentum jumps become apparent. Second, we plot the phase space density plot of the function d[‘% -Ha@, 4) bd(P,
4)
>
(5)
where G[E-H,(p, g)] is the energy hypersurface (“energy contour”) of the accepting Bom-Oppenheimer Hamiltonian, andpd@, q) is the Wigner den353
Volume202.number5
29January1993
CHEMICAL PHYSICS LETTERS
sity of the donor wavefunction. The Wigner method is fast and applicable to several nonseparable degrees of freedom, making it possible to explore the effects of small changes in the potential energy surfaces rather easily. The parameters for the two cases are as follows. Both have the acceptor potential energy surface
+D,[l-exp(-0)12,
(6)
with Da=20,&=0.2. The kinetic energy is simply T=f(p;+p;)
.
(7)
The two donor potential energy surfaces are of the form
(8) The parameters for the two examples’ are listed in table 1. The examples nicely illustrate competition for the role of accepting mode, or more properly accepting coordinate. The donor potential in the first example is tighter and has its equilibrium shifted to larger distance, as compared to the accepting potential. Both differences tend to favor momentum jumps. The donor potential in the second example is looser and has its equilibrium shifted to smaller distance, as compared to the accepting potential. In this case, both differences tend to favor position jumps. These trends are exactly what is seen in figs. 3 and 4. Both figures show the accepting potential contours, with the contour for E= 16 darkened; this is the average final energy on the acceptor potential energy surface. This is the center energy E. of the Gaussian energy envelope, eq. (4). It is also the energy E. of the classical energy surface (see eq. (5) ) on which the Wigner density of the donor wavefunction is evaluated.
Fig.3. The acceptorpotentialenergysurfacecontours(E= 16 darkened)togetherwiththe pulse excited wavepacket and the phase space density of the Wigner function of the donor ground state, pd at E= 16. The latter is plotted as a phase space distribution, as a series of line segments representing the momentum distribution at selected positions. The direction of a line segment shows the direction of the momentum, and its length is proportional to the magnitude ofp,, as a function of momentum for the given position, not the momentum itself. Note that the wavepacket and the phase space distribution agree on the predominantly independent bond momentum jump in this case (the wavepacket is a linear combination of two symmetrically related wavepackets, each with energy mostly in one bond.
The phase space density of the Wigner function pd(p, Q) is plotted as a series of line segments representing the momentum distribution at selected positions. The direction of a line segment shows the direction of the momentum, and its length is proportional to the magnitude of pd as a function of momentum for the given position, not the momentum itself. In case 1, fig. 3, note that the wavepacket and the phase space distribution agree on the predominantly
Table1
354
case
Dd
Ad
xo=Yo
Eo
u
1 (momentum jump, unshared) 2 (position jump, shared)
28 17
0.20 0.25
0.55 -0.5
16 16
2.24 1.73
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
Fig. 4. Same as fig. 3, for case 2. The wavepacket and the phase space distribution agree on the correlated bond position jump (the wavepacket has both bonds sharing the energy simultaneously).
independent bond momentum jump in this case (the wavepacket is a linear combination of two symmetrically related wavepackets, each with energy mostly on one bond). Further, the energy is stored mostly as momentum, as seen from the nodal structure of the wavepacket and the fact that the wavepacket is very small near the classical contour V(x, JJ)=E,. In case 2, fig. 4, the wavepacket and the phase space distribution agree on the correlated bond position jump (the wavepacket has both bonds sharing the energy simultaneously). The energy is stored mostly as potential energy, as seen from the fact that the wavepacket is very near the classical contour v(x, v) =E,. The wavepacket has energy uncertainty, given by the energy width of the envelope used. There is a reciprocal time uncertainty implied; one effect of this is that the pulse prepared wavepacket has some amplitude and nodes leading away from the region in phase space where the amplitude is largest according to the Wigner phase space density. The pulsed prepared wavepackets and the Wigner phase space density agr:e remarkably well. This raises the hope that predictions of the acceptor coordinates in general circumstances will be quite feasible.
29 January 1993
The wavepacket is quite sensitive to the relative shape of the potentials. This should ultimately be a blessing. Our simple results have already revealed a case where the electronic energy shows up primarily in one of two nominally equivalent bond modes (case l), and another case where the energy is shared equally (case 2). One case (case 1) was primarily a momentum jump, and the other, case 2, a position jump. Direct or indirect experiments depend critically on the nature of the nonclassical wavepacket In direct measurement, the effects of putting energy into very specific modes (and indeed with specific initial momenta or positions) can have a dramatic effect; e.g. the breaking of a specific bond. In a pulsed laser experiment, the accepting modes can dramatically change as the energy gap is increased by tuning the center frequency to the blue. Pump-probe experiments [2-41 would be ideal. By exciting just a part of the blue wing of an absorption band, the nonclassical Franck-Condon wavepacket is literally produced in real time, and its potentially bizarre character will appear as changes in the probe transitions with time and detuning. Many existing experiments could benefit from an understanding of the effects discussed here. For example, given the extensive work on formaldehyde S, +SO predissociation [ 8 1, it would be good to know whether the hydrogens are “launched” on the So potential energy surface with equal or unequal energies (this is the question treated here for a model potential). The same can be said for the acetylene SEP experiments [ 11. We hope to study these questions in the near future.
Acknowledgement This research was supported by the National Science Foundation under grant number CHE-9014555.
References [ 1] E. Abramson, R.W. Field, D. Imre, K.K. Innes and J.L. Kinsey, J. Chem. Phys. 83 (1985) 453; K. Yamanouchi, N. Ikeda, S. Tsuchiya, D.M. Jonas, J.K.
355
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
Lundberg, G.W. Adamson and R.W. Field, J. Chem. Phys. 95 (1991) 6330. [2] M. Dantus, R.M. Bowman, J.S. Baskin and A.H. Zcwail, Chem. Phys. Letters 159 (1989)406. [3]N.F. Schcrer, R.J. Carbon, M. Alexander, M. Du, A.J. Rugiero, V. Romero-Rochin, J.A. Cina, G.R. Fleming and S.A. Rice, J. Chem. Phys. 95 (1991) 1487.
356
29 January 1993
[4] R.A. Mathies, C.H. Brito Cruz, W.T. Pollard andC.V. Shank, Science 240 (1988) 777. [5] E.J. Heller, J. Chem. Phys. 68 (1978) 2066. [6] D. Beck and E.J. Heller, to be submitted. [ 7 ] R. Kosloff, S.A.Rice, P. Gaspard, S. Tersigniand D.J. Tannor, Chem. Phys. 139 (1989) 201. [8] W.F. Polik, D.R. Guyer, W.H. Miller and C.B. Moore, J. Chem. Phys. 92 (1990) 3471, and references therein.
CHEMICAL PHYSICS LETTERS
29 January 1993
Nonclassical Franck-Condon processes E.J. Heller a,b and Doug Beck a a Departmentof Chemistry, Universityof Washington,Seattle, WA98195, USA b Department of Physics. Universityof Washington.Seattle, WA98195, USA Received 23 October 1992; in final form 16 November 1992
Franck-Condon factors govern many important molecular processes, including electronic spectroscopy and radiationless transitions. A better understanding of the propensities for exciting various kinds of motion in nonclassical, nonverticaltransitionsis necessary to attachtherightsignificance to certain experiments. Picosecond and femtosecond laser pulse experiments in the wings of absorption bands are possible which may reveal some of the new effects discussed here.
1. Introduction Radiative and radiationless transitions depend on vibrational overlap factors (Franck-Condon factors) involving eigenstates from two different BornOppenheimer potential energy surfaces. The FranckCondon factors provide propensity rules which spectroscopists take almost for granted. For example, if a bending mode is the most displaced coordinate in one electronic state relative to another, it is assumed that states with bending excitation will be produced in an electronic transition between the Bom-Oppenheimer potential energy surfaces. We shall distinguish two regimes for Franck-Condon factors: classical and nonclassical. The classical regime corresponds to the realm of favorable conditions for the Franck-Condon overlap. The nonclassical regime of small Franck-Condon factors and violations of the “usual intuition” is the subject of this Letter. An absorption spectrum for a bound state going to a steep part of an excited electronic state is shown in fig. 1. As the incident light frequency is increased, we proceed from the “red” wing, through the band center, to the “blue” wing. Drawing a horizontal line between the two classical turning points of the “donor” or initial surface at the quantum energy of the donor state marks the classically allowed region of the initial wavefunction. This surface is progressively raised by increasing #MI.When the surfaces cross in the
350
Fig.1. The construction of the nonclassical wings of an absorption band from the crossing of the donor and acceptor BomOppenheimer potential energy surfaces.
0009.2614/93/$ 06.00 8 1993 Elsevier Science Publishers B.V. All rights reserved.
Volume202.number 5
CHEMICALPHYSICSLETTERS
classically allowed region, the overlap is good and the spectrum is near its maximum. When they cross outside the classically allowed region, the overlap is poor and corresponds to the nonclassical wings. This distinction extends to any number of dimensions. Loosely speaking, when the surfaces cross in classically allowed regions, the electronic state change can occur without any instantaneous change or jump in position or momentum. Otherwise, in the nonclassical wings, a jump is required. A common situation has the relative horizontal displacement of the two surfaces small. There is no red wing in absorption, and the blue wing starts just above the strong “O-O” transition. This case has interesting experimental implications, because by tuning to the blue of the O-Oline the nonclassical region is reached in the blue wing, yet other excited potential energy surfaces may be comfortably remote. Alternatively, one could populate the ground vibrational level of the excited potential energy surface and examine the emission to the red of the O-Oline, either via spontaneous emission or SEP [ 11. As discussed below, perhaps the most exciting prospects are for pump-probe experiments [ 2-41, The issue is very similar for radiationless transitions, except that one cannot change the energy gap. Very often, this gap places the system in question in the nonclassical regime. The Franck-Condon factors in the wings will be small, but in the blue wing there are two possibilities to explain that residual which remains: Either it results from an incomplete oscillatory cancellation in the region of large amplitude, or it could come from the overlapping region of the tails of the two states. This distinction leads to important effects in two or more dimensions. If the tail region is the major contributor, the integral is coming from a region not within the classically allowed region in coordinate space; we call this a “position jump”. If the major
part of the (small) integral comes from the region of large amplitude, the integral may be viewed as arising from the coordinate region where the Gaussianlike initial (donor) state is large, but the rapid oscillations of the final (acceptor) state imply a sudden change of momentum as the transition from donor to acceptor is made. We call this a momentum jump case. The remainder of the Letter is devoted to
29 January 1993
justifying this picture and discussing its implications.
2. Position versus momentum jumps In a subsequent fuller account of this work, we will present four or five different viewpoints on the issue of position and momentum jumps in Franck-Condon factors, all leading to the same conclusions about the utility of the picture and the circumstances under which each occurs. Here we will give a truncated, more intuitive account. Consider the Golden Rule expression for the rate of the electronic transition, k= ;
( V2>P(&) 3
(1)
wherep(&,) is the density of states at energy E,, and ( I”) is the mean squared matrix element. Making the Condon approximation, we write this as k= 2 viTr[G(E,-H,)]d){dl]
(2)
where 1d) is the donor wavefunction, I’: is the constant coupling strength, and Pd= I d) (dl is the density matrix for the donor wavefunction. We simplify this further here by approximating eq. (2) as k= ;
v:Sdpdl6[E,-H.@,q)lp~@,q).
(3)
Eq. (3) says that the rate k is proportional to the integral of the Wigner density of the donor, p? @, q) , over the energy hypersurface 6[ E0 - H, (p, q) ] of the acceptor. The essence of our work is to ask where on the acceptor energy surface the bulk of the rate integral comes from, This is a matter of investigating the integrund of eq. (3 ). By understanding the phase space structure of the donor wavefunction and the energy hypersurface of the acceptor Hamiltonian, we can come to an understanding of the way amplitude appears on the acceptor potential energy surface. The rate k is proportional to the average squared FranckCondon factor; it is a non state specific quantity. By interpreting the integrand of eq. (3), we are putting back the specificity in a very useful way, by saying 351
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
where (and with what velocity) the amplitude “lands” on the acceptor surface. Considered as a Wigner phase space distribution, the donor wavefunction is a smooth distribution peaked at zero momentum and in coordinate at the minimum of the donor potential energy surface. The bulk of such a phase space density has too low an energy on the acceptor potential energy surface to be germane to the donor-+acceptor transition. (This transition has to be quasi-degenerate.) To find the relevant regions in phase space we must examine those parts of the “tail” of the Wigner density which have the right energy on the acceptor potential energy surface. The inequivalent treatment of the donor and acceptor wavefunctions requires justification. (The donor is given a full fledged Wigner density, the acceptor is treated as a classical energy hypersurface. ) This was done much earlier in a treatment of the rate of radiative transitions, where eq. (3) was used directly, in the classical and non-classical regime [ 5 1. Although the results were very satisfactory, and an improvement over the venerable reflection approximation, we did not attempt at that time to go deep into the nonclassical regime, nor did we analyze the rate in terms of its phase space composition. A subsequent paper [6] will do more to justify the inequivalent treatment, but the results shown below are encouraging. We examine a one-dimensional system in fig. 2 to get started. Two cases are shown corresponding to blue wing absorption (or red wing emission). On the left are the wavefunctions, and on the right the phase space pictures are seen. The Wigner phase space distribution for the donor state is smeared distribution, and the quasi-degenerate contour interval on the acceptor potential energy surface is the parabolic curve. In the top case, the donor wavefunction is narrow, and the oscillations of the acceptor wavefunction are not completely effective at killing the integral. At the same time, the tail of the donor wavefunction is very smaI1 near the tail of the acceptor wavefunction. There is no doubt that the overlap comes from the region where the donor wavefunction is large. This is a momentum jump case, because the amplitude leaving the donor wavefunction may be thought of as appearing near the donor wavefunction, with considerable momentum on the repulsive acceptor po352
29 January 1993
Momentum jump
Positionjump
Fig. 2. Two blue absorption wing cases, a momentum jumpand a position jump, are shown in coordinate and phase space representations.
tential. The phase space picture reflects the increased momentum uncertainty and decreased position uncertainty associated with the narrow coordinate space distribution. It is clear that a shift in momentum at (nearly) constant position gives the shortest path between the two distributions. At the bottom in fig. 2 we see a much wider donor wavefunction, centered as in the previous case. A glance at the wavefunction plots shows there is now good overlap of the tails, and much more complete cancellation of the integral in the oscillatory region. The integral is coming mostly from the region of the tails, and the phase space plot shows that a position jump (again at zero momentum) gives the shortest path between the two states. We have just seen that position jumps and momentum jumps can compete to be the major contribution to the Franck-Condon integral in the nonclassical regime. In several dimensions, each position and each momentum becomes a competitor. The winning coordinates can be quite surprising. Thus,
Volume202,number 5
CHEMICALPHYSICSLETTERS
for nonclassical transitions, the usual Franck-Condon propensity rules mentioned at the beginning can be very misleading. For example, the mode with the largest equilibrium position displacement need not be excited in a nonclassical Franck-Condon transition. Some examples of this kind are well known. Radiationless transitions in aromatic hydrocarbons can be dominated by the carbon-hydrogen stretching motion (they are the accepting modes on the accepting surface) even though the carbon-carbon backbone is much more displaced.
3. Pulsed transient “nonclassical” wavepacket A pulsed laser (typically of 50 fs to a few picoseconds in duration) may prepare a nonstationary wavepacket on the acceptor potential energy surface. We want to create a “nonclassical” wavepacket by tuning the laser frequency to the blue of the absorption band (or analogously the red wing in emission) and adjusting the pulse duration so that an appropriate group of states is excited. The population in each of the eigenstates will then be governed by two factors: the Franck-Condon factor and the energy density of the pulse at that frequency, which is controlled by the pulse duration and center frequency. There is an associated transient wavepacket which can be defined as I$,,,>=
1 exp[-(G-fi~0)2/2a21 n
x In>
(4)
Clearly, 1q$_,,~ ) is a nonstationary wavepacket which can be prepared in the laboratory. At this juncture we mention a connection with the effort to control chemical reactions with special laser pulses [ 7 1. The pulse we need is rather primitive; we only require a short pulse with a center frequency in the blue wing. On the other hand, we are the mercy of the Franck-Condon factors and our goal here is indeed not control, but rather understanding of the factors which make the nonclassical wavepacket what it is. The fascinating question is: what is the nature of I&,#)‘! That is, is it a more or less localized state and if so, what modes or better what part of phase
29 January 1993
space is populated? Even if the nonclassical wavepacket is not created literally in the laboratory, it is necessary to know its form in order to interpret the spectrum in the blue wing in C.W.absorption, or for example the red SEP spectra for acetylene [ 11. The spectrum
is the result of the initial
nonclassical
wavepacket and its dynamics on the accepting surface. We focus attention on one issue out of many which should be addressed. Consider two identical, separable Morse oscillators. They may represent two equivalent C-H bond stretches in the same molecule, for example. Both the donor and the acceptor potential energy surface are Morse potentials but the equilibrium positions, well depths, and effective force constants will vary between the donor and acceptor states. Even in this simplified donor+acceptor transition, several interesting scenarios are possible: - Either position jumps or momentum jumps can be dominant. - The C-H bonds may share the energy in the acceptor nonclassical wavepacket equally, or one may be excited much more than the other. - The two previous items, each with two extremes, combine to make four scenarios.
4. Sample calculations In what follows we shall see two of the four scenarios arise with slight adjustments of the donor potential only; the acceptor potential will remain the same. (Indeed it is only the differences in the two potentials that matter.) We will examine the cases with two tools: first, we create the nonclassical Franck-Condon wavepacket for a given (blue) band center detuning (or equivalently the energy gap, or red detuning in emission) and transform limited pulse bandwidth. By inspecting the location and nodal structure of the nonclassical pulsed wavepackets, the various possibilities for position and momentum jumps become apparent. Second, we plot the phase space density plot of the function d[‘% -Ha@, 4) bd(P,
4)
>
(5)
where G[E-H,(p, g)] is the energy hypersurface (“energy contour”) of the accepting Bom-Oppenheimer Hamiltonian, andpd@, q) is the Wigner den353
Volume202.number5
29January1993
CHEMICAL PHYSICS LETTERS
sity of the donor wavefunction. The Wigner method is fast and applicable to several nonseparable degrees of freedom, making it possible to explore the effects of small changes in the potential energy surfaces rather easily. The parameters for the two cases are as follows. Both have the acceptor potential energy surface
+D,[l-exp(-0)12,
(6)
with Da=20,&=0.2. The kinetic energy is simply T=f(p;+p;)
.
(7)
The two donor potential energy surfaces are of the form
(8) The parameters for the two examples’ are listed in table 1. The examples nicely illustrate competition for the role of accepting mode, or more properly accepting coordinate. The donor potential in the first example is tighter and has its equilibrium shifted to larger distance, as compared to the accepting potential. Both differences tend to favor momentum jumps. The donor potential in the second example is looser and has its equilibrium shifted to smaller distance, as compared to the accepting potential. In this case, both differences tend to favor position jumps. These trends are exactly what is seen in figs. 3 and 4. Both figures show the accepting potential contours, with the contour for E= 16 darkened; this is the average final energy on the acceptor potential energy surface. This is the center energy E. of the Gaussian energy envelope, eq. (4). It is also the energy E. of the classical energy surface (see eq. (5) ) on which the Wigner density of the donor wavefunction is evaluated.
Fig.3. The acceptorpotentialenergysurfacecontours(E= 16 darkened)togetherwiththe pulse excited wavepacket and the phase space density of the Wigner function of the donor ground state, pd at E= 16. The latter is plotted as a phase space distribution, as a series of line segments representing the momentum distribution at selected positions. The direction of a line segment shows the direction of the momentum, and its length is proportional to the magnitude ofp,, as a function of momentum for the given position, not the momentum itself. Note that the wavepacket and the phase space distribution agree on the predominantly independent bond momentum jump in this case (the wavepacket is a linear combination of two symmetrically related wavepackets, each with energy mostly in one bond.
The phase space density of the Wigner function pd(p, Q) is plotted as a series of line segments representing the momentum distribution at selected positions. The direction of a line segment shows the direction of the momentum, and its length is proportional to the magnitude of pd as a function of momentum for the given position, not the momentum itself. In case 1, fig. 3, note that the wavepacket and the phase space distribution agree on the predominantly
Table1
354
case
Dd
Ad
xo=Yo
Eo
u
1 (momentum jump, unshared) 2 (position jump, shared)
28 17
0.20 0.25
0.55 -0.5
16 16
2.24 1.73
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
Fig. 4. Same as fig. 3, for case 2. The wavepacket and the phase space distribution agree on the correlated bond position jump (the wavepacket has both bonds sharing the energy simultaneously).
independent bond momentum jump in this case (the wavepacket is a linear combination of two symmetrically related wavepackets, each with energy mostly on one bond). Further, the energy is stored mostly as momentum, as seen from the nodal structure of the wavepacket and the fact that the wavepacket is very small near the classical contour V(x, JJ)=E,. In case 2, fig. 4, the wavepacket and the phase space distribution agree on the correlated bond position jump (the wavepacket has both bonds sharing the energy simultaneously). The energy is stored mostly as potential energy, as seen from the fact that the wavepacket is very near the classical contour v(x, v) =E,. The wavepacket has energy uncertainty, given by the energy width of the envelope used. There is a reciprocal time uncertainty implied; one effect of this is that the pulse prepared wavepacket has some amplitude and nodes leading away from the region in phase space where the amplitude is largest according to the Wigner phase space density. The pulsed prepared wavepackets and the Wigner phase space density agr:e remarkably well. This raises the hope that predictions of the acceptor coordinates in general circumstances will be quite feasible.
29 January 1993
The wavepacket is quite sensitive to the relative shape of the potentials. This should ultimately be a blessing. Our simple results have already revealed a case where the electronic energy shows up primarily in one of two nominally equivalent bond modes (case l), and another case where the energy is shared equally (case 2). One case (case 1) was primarily a momentum jump, and the other, case 2, a position jump. Direct or indirect experiments depend critically on the nature of the nonclassical wavepacket In direct measurement, the effects of putting energy into very specific modes (and indeed with specific initial momenta or positions) can have a dramatic effect; e.g. the breaking of a specific bond. In a pulsed laser experiment, the accepting modes can dramatically change as the energy gap is increased by tuning the center frequency to the blue. Pump-probe experiments [2-41 would be ideal. By exciting just a part of the blue wing of an absorption band, the nonclassical Franck-Condon wavepacket is literally produced in real time, and its potentially bizarre character will appear as changes in the probe transitions with time and detuning. Many existing experiments could benefit from an understanding of the effects discussed here. For example, given the extensive work on formaldehyde S, +SO predissociation [ 8 1, it would be good to know whether the hydrogens are “launched” on the So potential energy surface with equal or unequal energies (this is the question treated here for a model potential). The same can be said for the acetylene SEP experiments [ 11. We hope to study these questions in the near future.
Acknowledgement This research was supported by the National Science Foundation under grant number CHE-9014555.
References [ 1] E. Abramson, R.W. Field, D. Imre, K.K. Innes and J.L. Kinsey, J. Chem. Phys. 83 (1985) 453; K. Yamanouchi, N. Ikeda, S. Tsuchiya, D.M. Jonas, J.K.
355
Volume 202, number 5
CHEMICAL PHYSICS LETTERS
Lundberg, G.W. Adamson and R.W. Field, J. Chem. Phys. 95 (1991) 6330. [2] M. Dantus, R.M. Bowman, J.S. Baskin and A.H. Zcwail, Chem. Phys. Letters 159 (1989)406. [3]N.F. Schcrer, R.J. Carbon, M. Alexander, M. Du, A.J. Rugiero, V. Romero-Rochin, J.A. Cina, G.R. Fleming and S.A. Rice, J. Chem. Phys. 95 (1991) 1487.
356
29 January 1993
[4] R.A. Mathies, C.H. Brito Cruz, W.T. Pollard andC.V. Shank, Science 240 (1988) 777. [5] E.J. Heller, J. Chem. Phys. 68 (1978) 2066. [6] D. Beck and E.J. Heller, to be submitted. [ 7 ] R. Kosloff, S.A.Rice, P. Gaspard, S. Tersigniand D.J. Tannor, Chem. Phys. 139 (1989) 201. [8] W.F. Polik, D.R. Guyer, W.H. Miller and C.B. Moore, J. Chem. Phys. 92 (1990) 3471, and references therein.