J. Phys. Chem. 1991, 95, 9151-9158
-
dimers, So, can be excited ( 1 1) to the S', state, which deactivates by fluorescence (8, subsection E) and SI 'E ISC (9). Future publications will describe the dependence of the excited-state dynamics on the nature of the bridging group, the size of the aromatic moiety, the position of substitution (e.g., a vs j3),
9151
and the solvent polarity and viscosity.
Acknowledgment. This work was supported by the Department of Energy, Office of Basic Energy Sciences (DE-FGO282ER14024).
Vibronic Intensities in Electronic Spectra Involving Excited States with Double-Minimum Potential Surfaces Christian Reber and Jeffrey I. Zink* Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: May 17, 1991; In Final Form: June 28, 1991)
Electronic spectra involving excited states with double-minimum potential surfaces are calculated by using the split-operator technique for numerical integration of the time-dependent Schrixlinger equation, and the time-dependent theory of electronic spectroscopy. Absorption spectra to an excited state with a double-minimum potential surface contain unusual intensity distributions in the vibronic structure. Detailed spectra are calculated and trends are analyzed in terms of the width of the barrier and the width of the initial wave packet. Emission spectra from an excited state with the double-minimum surface to a harmonic ground state are calculated. Wave functions of the double-minimum surface are calculated and propagated on the ground-state surface. The doubling of the eigenvalues and the temperature effects on the spectra are calculated. The theory is applied to the spectroscopy of K2[PtC14]. In this molecule the active asymmetric bl, mode is represented by the double-minimum potential. A two-dimensional surface consisting of the b,, Pt-CI stretch and the totally symmetric Pt-CI stretch is constructed. Both the absorption and the emission spectra including the unusual spectroscopicfeatures of the MIME (missing mode effect) and the apparent 'energy gap" between the emission and the absorption spectra are calculated. The molecule has D2,, symmetry in its lowest excited electronic state with one pair of Pt-C1 bonds enlongated by 0.14 A and the other pair contracted by 0.02 A relative to the ground-state distances.
1. Introduction Transition-metal compounds provide a rich area for the spectroscopic study of the geometrical distortions that occur in excited electronic states.'4 In general, many normal modes are displaced, resulting in complicated and interesting spectra. Most frequently, the spectra can be interpreted in terms of a multidimensional potential surface in which the important vibrational coordinates are totally symmetric and whose eigenvalues are far from the dissociation limit. Under these conditions the harmonic oscillator approximation to the normal coordinates is good. A situation in which the harmonic oscillator approximation cannot be used occurs when a nontotally symmetric normal mode is active. By symmetry the slope of the potential surface for an asymmetric mode must be zero at the ground-state equilibrium internuclear geometry for a nondegenerate electronic state,s but the excited-state surface may have local minima or maxima elsewhere. The emission and absorption spectra resulting from electronic transitions between nonharmonic potential surfaces can now be accurately calculated as a result of recently developed time-dependent theoretical methods. The split-operator techniques of Kosloff6 and of Feit and Fleck' make it possible to calculate the wave packet dynamics for any potential surface. The eigenvalues (1) Zink, J. 1.; Kim Shin, K.-S. Molecular Distortions in Excited Electronic States Determined from Electronic and Resonance Raman Spectroscopy. In Advances in Photochemistry; Volman, D. H., Hammond, G. S.,Ncckers, D. C., Eds.; John Wiley & Sons: New York, 1991; Vol. 16. (2) Larson, L. J.; Zink, J. 1. Inorg. Chem. 1989, 28, 3519. (3) Yersin, H.;Otto, H.; Zink, J. 1.; Gliemann, G. J. Am. Chem. Soc. 1980, 102, 951. (4) Wilson, R. B.; Solomon, E. 1. Inorg. Chem. 1978, 17, 1729. (5) Sturge, M. D. Solid Srare Phys. 1967, 20, 91. ( 6 ) Kosloff, D.; Kosloff, R. J. Compur. Phys. 1983, 52, 35. (7) Feit, M.D.;Fleck, J. A.; Steiger, A. J. Compur. Phys. 1982,47, 412. For an introductory overview see: Tanner, J. J. J. Chem. Educ. 1990,67,917.
0022-3654/91/2095-9151$02.50/0
and eigenfunctions can also be calculated. In combination with the time-dependent theory of electronic spectroscopy,* emission and absorption spectra can be calculated. What are the spectroscopic ramifications of nonharmonic potential surfaces? Two important ones are examined in this paper. First, unusual intensity distributions in the vibronic structure in electronic spectra will be found which will result in unusual band shapes. These intensities can be far from the familiar Poisson distribution which results from harmonic potential surfaces. Of particular interest is the situation when the intensities near the energy origin of the band are extremely small, resulting in an "energy gap" between the onsets of the absorption and emission bands. Second, progressions in some or all quanta of the nontotally symmetric mode may be observed. The latter situation arises when the electronically excited molecule relaxes to local minima along the nontotally symmetric normal coordinate. This molecule will have a new point group (different from that of the molecule in the ground electronic state) in which the formerly asymmetric coordinate now belongs to the totally symmetric irreducible representation. In this paper we specifically examine the spectroscopic consequences of a nondegenerate nontotally symmetric normal mode which in a nondegenerate excited electronic state has local minima displaced from the ground-state geometry. The experimental example to be treated is the bl, mode of square-planar PtC142-. In the emission and absorption spectra of this molecule both the b,, and the a l gnormal modes are i m ~ r t a n t .The ~ two-dimensional potential surfaces that are appropriate for this example are shown in Figure 1. The ground-state surface is harmonic along both the algand b,, coordinates. The emitting state is represented (8) Heller, E. J. Arc. Chem. Res. 1981, 14, 368. (9) Preston, D. M.;Giintner, W.;Lechner, A.; Gliemann, G.; Zink. J. I. J . Am. Chem. Soc. 1988, 110, 5628.
0 1991 American Chemical Society
9152 The Journal of Physical Chemistry, Vol. 95, No. 23, 1991
Reber and Zink equation in two coordinates x and y is
+
where V2 = ij2/6x2 62/6y2and V(x,y) is the potential energy surface, The wave function at a time t + At is7 d(t+At) = exp( (%)VI)
):(i- 1
exp(-iAtV) exp
V2 X
~ ( X , Y J+ ) O [ W 3 1 (4)
[AI Figure 1. Two-dimensional potential surfaces for the ground state and the lowest energy excited state of K,[PtCI,]. The maxima of the absorption and emission spectra are indicated by the arrows labeled A and E, respectively. Qbig
by a double-minimum surface along the bl, coordinate and a displaced harmonic surface along the algcoordinate. Absorption and emission spectra are calculated from the wave packet dynamics, the eigenfunctions, and the eigenvalues of these potentials. These surfaces are then refined to simultaneously fit both the emission and absorption spectra of K2[PtC14]. The origin of the energy gap and the missing mode effect (MIME) in the spectra of this molecule are quantitatively evaluated and discussed. 2. Theory The time-dependent theory of electronic spectroscopy provides a powerful method of both calculating and interpreting the emission and absorption spectra, especially when nonharmonic potential surfaces are involved. Because the theory has been discussed in detail previously,1,2-8-1'only a brief discussion of the general aspects will be given here. The spectra are governed by the motion of a wave packet on the multidimensional potential surfaces. The initial wave packet, 4 = 4(t=O), is projected onto the potential surface corresponding to the final state (the excited electronic state for an absorption spectrum or the ground electronic state for an emission spectrum). The final surface is in general either displaced relative to the initial surface along normal vibrational coordinates or has a different functional form from that of the initial surface. The wave packet is in general not an eigenfunction of the final surface and develops in time according to the time-dependent Schriidinger equation. The absorption spectrum is given byl0*'l
I ( @ ) = Cw~+mexp(iwt)((414(t)) exp(-r2t2 + ( i E o / h ) t ) dt ] where Cis a constant and I ( w ) is the absorption cross section. The quantity (414(t))is the overlap of the initial wave packet, 4 = 4(t=O), with the time-dependent wave packet, $ ( t ) . r is a phenomenological Gaussian damping factor, and Eo is the energy of the origin of the electronic transition. The emission spectrum is given by a similar expression
where I(o)is the intensity in photons per unit volume per unit time at frequency of emitted radiation w. The time dependence of the wave packet evolving on a potential surface can be numerically determined by using the split-operator technique of Feit and Fleck.7 The time-dependent Schrijdinger (IO) Heller, E. J. J . Chem. Phys. 1978, 68, 3891. ( I I ) Heller, E. J. J . Chem. Phys. 1978, 68, 2066.
The dominant error term is third order in Af. The initial wave function 4(x,y,t) at t = 0 is known. It normally is the lowest energy eigenfunction of the initial state of the spectroscopic transition. The value of the wave function at incremental time intervals At is calculated by using eq 4 for each point in the (XJ) grid. The autocorrelation function is then calculated at each time interval, and the resulting ( + l $ ( t ) ) is Fourier transformed according to eq 1 or 2 to give the absorption or emission spectrum, respectively. The eigenfunctions \ki can be computed by numerically evaluating the integral^^,'^
where T i s the time encompassed by the calculation, w(t) is a Hanning window function, and Ei is the eigenvalue corresponding to qi. This method is a very general one for spectroscopic applications. For example, the vibrational energy levels for a particular electronic state representing the initial state in the electronic transition can readily be found by propagating a simple function such as a Gaussian on the relevant electronic-state potential surface and Fourier transforming the autocorrelation function according to eq 1. Then the eigenfunction corresponding to a particular energy level can be determined by using eq 5. This eigenfunction can then be propagated on the final state involved in the electronic transition in order to calculate the spectrum. When more than one normal mode is involved in the calculation, the assumption that the normal coordinates are not mixed or coupled in the final state of the transition may significantly reduce computing time. In this case the total overlap ( 9 ) 4 ( t ) )is the product of the overlaps of the k contributing normal modes
The assumption that no mixing occurs may be a good approximation. Thus, in the case of two normal modes, for example, the (&l&(t)) can be calculated for each dimension separately, the product of the two taken as given by eq 6, and the result Fourier transformed according to eq 1 or 2 to give the spectrum. The two most important choices that must be made in the numerical calculations are the size of the time steps and the size of the computational grid. The smaller the increment in the time steps and the smaller the spacing between the grid points, the greater the accuracy in the calculation. General criteria for initial choices have been published.6-' In the work reported here, the time increment is considered small enough and the total time large enough when the plot of the calculated spectrum does not distinguishably change when the increment is halved and the total time is doubled. A time increment At = 2 fs is usually suitable. The grid spacing is considered to be small enough when the plot of the calculated spectrum does not distinguishably change when the increment is halved. A grid of 256 points/A is generally suitable for the spectra calculated in this paper. The specific calculations of importance to the electronic spectra of K2[PtCl,] involve the use of a double-minimum potential in one of the coordinates. The double-minimum potential has several (1 2) Numerical eigenfunctions obtained by eq 5 agreed with analytical eigenfunctions for harmonic and Morse potentials and also to numerical eigenfunctions for the potential in eq 7 calculated with a time-independent method to at least five digits at each grid point (Blukis, U.; Howell, J. M.J . Chem. Educ. 1983, 60, 207).
Vibronic Intensities in Electronic Spectra specific properties that require further comment. The eigenvalues of the double-minimum potential come in pairs.I3 The gerade eigenfunction of each pair lies lower in energy than its ungerade partner. In the case of the electronic absorption transition into a double-minimum potential, the position of the initial wave packet at t = 0 is at the top of the barrier, because the slope of a potential along a non totally symmetric coordinate is required to be zero. The initial wave packet, corresponding to the lowest vibrational eigenfunction of the ground electronic state, is gerade. Thus, nonzero absorption intensity is found only for the gerade vibrational levels of the double-minimum potential. (If the initial wave packet were ungerade, nonzero intensity would be found only for the ungerade vibrational levels). In this special case, not all of the eigenvalues are observed because the transitions to all levels with odd parity eigenfunctions are forbidden. In order to find all the eigenvalues in a certain energy interval, a step that is necessary in order to find the eigenfunctions (eq 5), it is easiest to put a Gaussian wave packet on one side of the doubleminimum potential and carry out the propagation, although this computational stratagem does not have physical meaning in the spectroscopic problem. 3. Electronic Spectroscopy Involving a Double-Minimum Potential
In the following two sections we calculate the absorption and emission spectra involving a one-dimensional harmonic ground state and a double-minimum excited state potential surface. Many different double-minimum potential surfaces are used in the lite r a t ~ r e . ' ~ - 'We ~ chose the function proposed by Coon et al.Is A double-minimum potential is constructed by superimposing a harmonic potential and a Gaussian barrier:
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9153
a 6
Q [AI Figure 2. Absorption transition from a harmonic ground-state potential into a double-minimum excited-state potential. The wave packets on the excited-statesurface have hw = 300 cm-l (long dashes), 900 cm-' (solid line) and 3200 cm-' (short dashes). The resulting absorption spectra are shown in Figure 5.
+
V(Q) = !/zk,&2 A exp(-a2Qz) (7) where ken, the force constant corresponding to the harmonic term, is calculated as kcll = 47r2m(h ~from ) the ~ vibrational energy hw and the reduced mass m of the mode along configurational coordinate Q. The height and width of the Gaussian barrier are determined by the values of A and a, respectively. The potential minima corresponding to the distortion AQ along the nontotally symmetric mode Q are given by'$ AQ = In ( 2 ~ ~ A / k , ~ ~ ) ) ~ / ~(8) It is a convenient property of this potential that both the height and width of the barrier can be controlled independently to fit calculated to experimental spectra in order to determine the distortion AQ. The potential function in eq 7 has been shown to be a better model for the energy levels of NH3 and ND3 than other types of double-minimum pote11tia1s.I~ A discrimination between various types of double-minimum potentials requires highly resolved experimental spectra. The distortions AQ obtained from different potential functions are identical for practical purpose^.'^ The choice of a specific function is therefore not critical for the determination of AQ, one of the goals of this work (section 4 ) . The following values were used for the double-minimum potential in the next two sections: hoen = 200 cm-I, mass of the mode m = 35.453 g/mol, A = 3209 cm-I, a = 23.36 leading to a distortion AQ of 0.09 A and a barrier height of 3000 cm-I. Absorption Spectroscopy. In the absorption process, the wave packet is transferred vertically from the ground-state potential surface to the excited-state potential surface. The initial wave packet in the calculations to be discussed below is the product of a harmonic oscillator ground-state eigenfunction times a constant transition moment. This wave packet is placed on the top of the barrier of the double-minimum excited-state surface as shown in Figure 2. It is not an eigenfunction of this surface and develops with time according to the time-dependent Schradinger ( 13) Herzberg, G.Infrared and Raman Spectra of Polyatomic Molecules; Van Nostrand Reinhold: New York, 1945; p 220 ff. (14) Manning, M. F.J . Chem. Phys. 1935.3, 136. (15) Coon, J. B.; Naugle, N. W.; McKenzie, R. D. J . Mol. Spectrosc.
1966. 20. 107.
-1.0
0.0
-0 5
0.5
Q [AI Figure 3. Wave packet dynamics in the double-minimum excited-state potential shown in Figure 2. The time dependence of the initial wave packet with hw = 900 cm-' (solid line, Figure 2 ) is shown.
equation. The magnitude of the wave packet at several key times is shown in Figure 3 . At t = 0, the wave packet is a Gaussian. At short times, the wave packet bifurcates and the two pieces move away from the initial position. In addition, the wave packet develops a complicated structure. At t = 67 fs the pieces of the wave packet have reached the outer walls of the double-minimum potential and begin to move back to the origin. The much distorted wave packet returns to its initial position at about t = 100 fs and then moves away from the barrier again as illustrated at t = 116 fs and t = 133 fs. The time dependence of the absolute overlap 1(414(t))lis shown in Figure 4. The absorption spectra, given by the Fourier transforms of (414(r)),are shown in Figure 5 for all three initial wave packets in Figure 2. The most important features in the spectrum can be understood from the coarse features in the overlap. The overlap is equal to 1 at t = 0. At short times, the overlap rapidly decreases as the bifurcated wave packet moves away from its initial position. The width of the absorption spectrum is related to this initial decrease in overlap; if there were a simple, smooth decrease, the faster the decrease in overlap the broader the spectrum. At longer times the overlap increases to reach a maximum at about t = 100 fs. This recurrence in the overlap is related to the vibronic structure in the spectrum. If it were the only recurrence, the spacing in the frequency domain
9154
The Journal of Physical Chemistry. Vol. 95, No. 23, 1991 1
0
I
I
I
200
400 Time [fs]
600
800
Figure 4. Absolute overlap as a function of time calculated from wave functions as shown in Figure 3. Main figure: absolute overlap 1($1#(t))1 for the wave packet with hw = 900 cm-I. Inset: overlap at short times for the wave packets with hw = 300 cm-I (solid line) and 3200 cm" (dashed line).
0
8
-e P
9
4
25 Wavenumber [cm"] x10'
20
Figure 5. Calculated absorption spectra for the potentials shown in Figure 2. Vibrational frequency of the initial wave packet is (a) hw = 300 cm-I, (b) hw = 900 cm-l, and (c) hw = 3200 cm-I.
would be equal to c-l(tmumna)-l corresponding to 330 cm-l. The decrease in the magnitudes of the recurrences as a function of time is related to the phenomenological damping factor r. The value of r used in Figure 4 is I5 cm-I. The resolution of the vibronic structure in the spectrum is governed by l'. If r is much smaller than the spacing between the vibronic peaks, the individual peaks are well-resolved. If r is greater than the spacing, then only a broad envelope is observed. The details in the plot of the overlap versus time are complicated and rich. Unlike a Gaussian wave packet on a harmonic potential surface, the time-dependent wave packet does not remain Gaussian. The wave packet bifurcates and develops complicated nodal structure. In addition, the dynamics is more complicated. In a simple physical picture, part of the initial wave packet experiences a very small slope (the part at Q = 0 in Figure 2), part of it experiences a steep slope, and, depending on its width, part may even experience a slope of the opposite sign. These aspects result in a time-dependent overlap that does not necessarily have recurrences at equal time intervals and that does have local maxima and minima between the major recurrences as shown in Figure 4. The spectrum therefore does not necessarily have to contain one regularly spaced progression or a smooth trend in the intensities. The trends in the intensity distribution among the vibronic bands can be interpreted from trends in the wave packet dynamics in
Reber and Zink the time domain. It is helpful to consider two conceptual extremes in the relationship between the width of the initial wave packet relative to the width of the barrier in the double-minimum potential. First, if the initial wave packet were very narrow relative to the width of the barrier, the wave packet would bifurcate very slowly in time because most of the wave packet experiences a very small slope on the top of the relatively broad barrier. Thus, the overlap ( $ I O ( t ) ) would decrease slowly with time and the spectrum would have most of its intensity in a line at an energy corresponding to that of the top of the barrier at Q = 0, above the minimum of the ground-state potential surface. For extremely narrow initial wave packets, the width of the resulting spectrum is governed by the uncertainty principle. All the wave packets investigated for the spectroscopic applications presented in this work are too broad by orders of magnitude to have the uncertainty principle govern the calculated line widths. Second, if the initial wave packet consisted of two narrow spikes located at the minima of the double-minimum potential, ( $ ( $ ( t ) ) would again be almost constant and the spectrum would have most of its intensity in a single line at an energy corresponding to that of the minima of the double-minimum potential. From these conceptual extremes, the trends in the intensity distributions in the spectrum can be deduced from the widths of the initial wave packets. The narrower the initial wave packet, Le., the higher the frequency of the relevant mode in the ground electronic state, the greater the intensity at the energy that corresponds to the energy difference between the top of the barrier and the minimum of the ground-state potential. The broader the initial wave packet, the smaller the probability at the top of the barrier and the greater the intensity at the energy that corresponds to the energy difference between the minima of the excited-state potential surface and the minimum of the ground-state potential surface. The trends in the intensity distributions are more accurately described in terms of the time dependences of the overlaps shown in Figure 4. In all cases, the initial wave packet $ is placed on the top of the barrier. The overlap for a narrow wave packet, with most of its probability on the part of the potential surface where the slope is small, decreases slowly with time, and the spectrum has most of its intensity around the energy of the barrier. In contrast, the decrease in part of ($I$(t)) for a broad wave packet will be faster in time, but part decreases very slowly because a broad wave packet has appreciable probability on the part of the potential where the slope is steep as well as in the three regions of the potential surface where the slope is small. The result is a complicated spectrum. This behavior is illustrated in the inset of Figure 4: the overlap for the narrow wave packet (hw = 3200 cm-', dashed line) decreases slower than that for the broad wave packet ( h w = 300 cm-I, solid line). The three spectra shown in Figure 5 illustrate the intensity distributions. They are calculated from the motion of the three wave packets shown in Figure 2 on the double-minimum potential. The only difference between the three calculations is the width of the initial wave packet. The spectrum in Figure 5a is calculated for a transition from a harmonic ground state with a vibrational frequency of 300 cm-I to the double-minimum potential. The initial wave packet is broad relative to the width of the barrier as illustrated in Figure 2. The spectrum has large intensities in the vibronic bands corresponding to the eigenvalues at the bottom of the excited-state minima. In fact, the most intense peak is the lowest energy peak in the spectrum corresponding to the Eo transition. The intensity of the vibronic features decreases toward higher energy in the absorption spectrum up to the fourth peak and then again increases for the peaks near the energy of the top of the barrier. In this example, the eighth peak (corresponding to the first vibrational state at the top of the barrier) is not the most intense. The intensity of the peaks corresponding to vibrational levels above the top of the barrier decreases rapidly. The intensity distribution shown in this spectrum results from placing a broad initial wave function on the barrier. The spectrum in Figure 5b is calculated for a transition from a harmonic ground state with a vibrational frequency of 900 cm-I. The initial wave packet is narrower than that of the preceding
The Journal of Physical Chemistry, Vol. 95, No. 23, 1991 9155 30
25 -
m
0
-
5
5blVV
20-
-0.6 -0.4 -0.2 0 0
10 151
I
I
200
0-0.6
-0.4
-0.2
0.0
0.2
Q [AI
0.2
0.4
400
600
0
Q [AI Figure 6. Emission transition from a double-minimum excited-state
potential into a harmonic ground-state potential. The lowest vibrational levels of the excited state coincide on the energy scale of the figure; their eigenfunctions are offset for clarity. Both functions were propagated on the ground-state surface to obtain the emission spectra in Figure 8. example, and the intensities are correspondingly greater in the energy region of the top of the barrier. In this case, the two highest intensity vibronic peaks in the absorption spectrum are those corresponding to the levels immediately below and above the barrier, respectively. The intensity of the Eo peak is dramatically reduced from that in Figure 5a. The overall distribution accidentally appears to be similar to a Poisson distribution. The bottom spectrum is calculated for a transition from a ground state with a frequency of 3200 cm-'. The initial wave packet is narrow relative to the width of the barrier, and the largest intensity is found in the vibronic band corresponding to the energy level at the top of the barrier. The intensities of the vibronic bands from states near the bottom of the minima are low. In fact, the Eo band and the next two peaks to higher energy are barely observed in the spectrum. Much more of the intensity in the spectrum is found in the peaks corresponding to energy levels well above the top of the barrier. The origin of the apparent energy gap between experimental absorption and emission spectra is related to the width of the initial wave packet relative to that of the excited-state potential barrier. The spectrum in Figure 5c illustrates a specific example in which an energy gap might be observed. The Eo peak and the peak next to it are less than 2% of the intensity of the largest peak in the spectrum. In an experimental spectrum such weak transitions might be buried in the noise and might not be observable. The intensities of the low-energy peaks in a spectrum depend sensitively on both the specific potential surface and the initial wave packet and can be significantly smaller than those in the examples described here. The specific case of the potential surface and initial wave packet for [PtCI,]*- is discussed later. Emission Spectrosropy. In the emission process, the wave packet is transferred vertically from the excited-state potential surface to the ground-state potential surface as shown in Figure 6. In the spectra to be discussed below, the excited-state potential surface is the same double-minimum surface that was used in the calculations of the absorption spectra. Thus, the initial wave packet is a vibrational eigenfunction of the double-minimum surface times the transition moment. The vibrational eigenfunctions of the excited-state surface are calculated by using eq 5 . The two lowest energy eigenfunctions are shown in Figure 6. The two lowest energy eigenfunctions must both be considered when calculating the emission spectra because the two lowest vibrational energy levels are separated by only 0.4 cm-I. The eigenfunctions with eigenvalues lower in energy than the top of the barrier come in pairs. The lowest energy function is an even function in the coordinate space (gerade parity).13 The second member of the pair is higher in energy and is an odd function (ungerade parity). The choice of the initial wave packet to be
800
11
IO
Time [fs]
Figure 7. Absolute overlap as a function of time for the emission transition illustrated in Figure 6 . The inset shows the propagating wave function at selected times.
used to calculate the emission spectrum depends on the energy separation between the lowest vibrational levels and on the temperature that the calculation is intended to represent. When the energy separation is large and the temperature is very low, AE >> kT and only the lowest energy level is significantly populated. In this extreme the emission spectrum is calculated by propagating only the gerade eigenfunction on the ground-state potential surface. When the energy separation AE = kT, both the gerade and the ungerade states are appreciably populated and the spectrum is calculated by propagating each of the initial functions individually, weighting the intensities of the two resulting spectra by the relative thermal populations, and adding the result. Note that the energies of the apparent origins of the two spectra will be separated by the energy separation of the two initial states. In the following examples, only the limits when AE >> kT and AE
