C. REBER and J. LANDRY-HUM
2.43.2 LUMINESCENCE AND ABSORPTION SPECTRA FOR ELECTRONIC
The traditional view of an electronic absorption or luminescence transition is shown inFigure 1.
The Franck–Condon picture for an allowed transition relates the intensity of each vibronic band in a spectrum to the square of the overlap of vibrational eigenfunctions for potential energy surfaces representing the initial and final electronic states. Analytical expressions for these overlap integrals exist for a few special cases, such as one-dimensional harmonic potentials with identical force constants for the initial and final states of the electronic transition.15Anharmonic potential energy surfaces, such as those obtained in numerical form from electronic structure calculations, are harder to treat in the traditional framework and closed formula solutions generally do not exist. The time-dependent approach is mathematically equivalent to the time-independent Franck–Condon picture, but can be easily generalized to numerical potentials given as arrays of energy values along one or several normal coordinates. Important parameters that are directly
Energy
Normal coordinate Q
1.0x10-3 0.0
Absorption Luminescence
Absorbance
Luminescence intensity
E00
∆Q
Figure 1 Electronic absorption and luminescence transitions between initial and final states represented by harmonic potential energy curves. The solid and dotted spectra denote calculated luminescence and absorp- tion spectra, respectively. The electronic origin transition, E00, and the offset between potential energy
minima,Q, are indicated.
determined from spectra such as those shown inFigure 1 are the vibrational frequencies of the two potential energy curves and their energy offset, which is defined by the electronic origin transition denoted asE00inFigure 1. The most important parameter that can not be read directly from the spectra is the offset of the potential minima Q along the normal coordinate Q, illustrated in Figure 1. It is adjusted to lead to the best agreement between experimental and calculated spectra. In general, Qi values for i multiple coordinates Qi are easily determined from fit procedures and the effects of potential energy surfaces that are no longer harmonic can be analyzed, as illustrated by the examples inFigures 8and10.
Figure 2illustrates the approach to the time-dependent calculation of electronic spectra. In this example, we show the final state of the luminescence transition in Figure 1 and calculate the luminescence spectrum. In the time-dependent picture of an allowed electronic transition, the initial wavefunction, corresponding to the lowest-energy eigenfunction of the excited state in Figure 1, is placed onto the final state surface, where it is no longer an eigenfunction and evolves with time. This evolution of the time-dependent wavepacket (t) is obtained from solutions of the time-dependent Schro¨dinger equation, with given as a starting point. Figure 2 illustrates the time-dependent wavefunction at three times, corresponding to 0, 10, and 50 fs. At time 0, the wavepacketis identical to the eigenfunction of the excited state. After 10 fs, the wavepacket has moved away from its initial position, and after 50 fs, a time corresponding to approximately half the vibrational period, it is near the turning point. This example shows a special situation:
harmonic potentials with identical force constants for the initial and final states. The time- dependent wavefunction does therefore retain its Gaussian shape and it simply moves across the harmonic potential well. In general, the shape of the wavefunction changes significantly as it moves, as illustrated by published animations and wavepacket snapshots.16–18
The quantity that defines the shape of a spectrum is the autocorrelation function<|(t)>of the wavepacket moving on the final state potential energy surface.19 Its absolute value as a function of time is illustrated in the bottom panel of Figure 2. It corresponds to the overlap of the wavefunctionat time 0 with the wavepacket(t) at longer times and it obviously depends on the shape of the potential energy surface in the region explored by the wavepacket, as illustrated,
1.0 0.5 I<φIφ(t)>I 0.0
200 150
100 50
0
Time [fs]
5000
4000
3000
2000
1000
0 Energy [cm–1 ]
–6 –4 –2 0 2 4 6
Q (350 cm ) –1 10 fs
50 fs
φ=φ(0 fs) φ(10 fs)
φ(50 fs)
Luminescence
Figure 2 Time-dependent view of the luminescence transition inFigure 1. Only the potential energy curve of the final state is shown. Time-dependent wave functions are given for times of 0, 10, and 50 fs. The bottom panel shows the absolute value of the autocorrelation function, visualized as the overlap between the time-dependent wavefunction (t) and the wavefunction at time zero. The first recurrence at 95 fs occurs after a single vibrational period of the 350 cm1vibrational frequency used to define the harmonic potential energy curve.
Time-dependent Theory of Electronic Spectroscopy 561
for example, by the initial drop of the autocorrelation from its maximum value over the first few tens of fs. At longer times, the autocorrelation reaches a maximum after each vibrational period, as shown in Figure 2.
The absorption spectrum is given by the Fourier transform of the autocorrelation function as:2
:Iabsð!ị ẳC!
Z1
1
ei!thjð ịiet ðG2t2ỵ ðiE00=hịtịdt ð1ị
The luminescence spectrum is given by an equation closely related to Equation (1):2
Ilumð!ị ẳC!3 Z1
1
ei!thjðtịieðG2t2ỵ ðiE00=hịtịdt ð2ị
In both equations,!denotes the wave number abscissa of the calculated spectrum,E00is the energy of the electronic origin transition,Gis a phenomenological damping factor determining the width of each vibronic line in the calculated spectrum, and C is an adjustable scaling factor.
The resolution in most condensed-phase experimental spectra is inferior by orders of magnitude than the limit imposed by the Heisenberg uncertainty relation on the width of a spectroscopic transition. Most often, the resolution is determined by inhomogeneous broadening. Its effect can be included in the model inFigure 2via the damping factorG, typically treated as an adjustable parameter to reduce the autocorrelation at long times. The damping factor G determines the width of each vibronic band in Figure 1, and it should be chosen small enough not to influence the overall width of a spectrum. It also determines up to which time the autocorrelation function has to be calculated, as illustrated in Figure 3, where <|(t)> values insignificantly different from zero are calculated at times longer than approximately 450 fs.
Figure 3 shows the absolute values of the autocorrelation functions for three different offsets Q, defining three different initial positions for on the final state potential energy surface in Figure 2. The slope of the potential surface at the initial position determines the decrease of the autocorrelation function from its initial value of 1, and it depends on the offsetQbetween the minima of the potentials along the normal coordinate in Figures 1and 2. For an offsetQ of zero, the center of the wavepacket encounters a flat potential surface. No decrease of the absolute value of the autocorrelation is expected with time, as the overlap remains 1 at all times.
The slow decrease seen for the solid line in Figure 3is therefore caused by the damping factorG and the calculated spectrum is narrow. For an offsetQof 1, the decrease at short times is faster, due to the nonzero slope of the potential surface at Qẳ 1, and the calculated spectrum shows a large overall bandwidth. This trend is even more pronounced for the larger offset Qẳ3, the
1.0
0.5
0.0
I<φIφ(t)>I
500 400
300 200
100 0
Time [fs]
∆Q=0
∆Q=1
∆Q=3
Figure 3 Absolute values of the autocorrelation functions used to calculate the spectra in Figure 4 for different values ofQ, given in the Figure. The initial drop of the autocorrelation is related to the slope of the potential energy surface directly below the excited state potential minimum. Recurrences occur at each
vibrational period, independent of the value used forQ.
situation for which time-dependent wave functions are shown in Figure 2. Another quantity influencing the initial decrease is the vibrational frequency of the final state potential.16–18 It is often illustrative to inspect the initial decrease of the autocorrelation in order to analyze spectro- scopic effects or to compare spectra of related molecules.
The spectra calculated from the autocorrelation functions inFigure 3are given inFigure 4. The top panel shows absorption and luminescence spectra for an offsetQof zero, a situation that corresponds to many spin–flip transitions in the spectra of transition metal complexes. Absorp- tion and luminescence spectra are shown in Figure 4a and they consist of a single line with a maximum atE00, the energy of the electronic origin. The spectra for a small offset Qẳ1 are shown inFigure 4b. A short vibronic progression with four easily discernible members is observed with an interval corresponding to the ground or excited state vibrational frequencies in lumines- cence and absorption spectra, respectively. A larger offset Qẳ3 leads to a longer vibronic progression consisting of at least twelve members and to the largest overall bandwidth of the three calculated spectra, shown inFigure 4c. A resolved progression in an experimental spectrum can therefore be directly analyzed with this model and leads to a quantitative estimate forQalong the normal coordinate. Such values are important for the characterization of metal–ligand bonding and structural changes in excited states and for excited-state dynamics relevant to transition metal photochemistry.1,20 Representative examples for Q values determined from fits of calculated spectra to experimental data are given inTable 1. It is interesting to note that similarQvalues are observed for many different types of transitions, emphasizing the need for detailed analyses of the spectra of interest. The transformation ofQvalues in dimensionless units to changes of bond lengths and angles has been published for many compounds1,21,22and the use of normal coordinate analysis calculations has also been illustrated in detail.22–24The spectroscopic signatures of even slight changes in the parameters describing the potential energy surfaces inFigure 1 are easily calculated and can be compared to variations of the experimental spectra. Recent examples of studies reporting such detailed effects and analyzing them with time-dependent theory include the pressure dependence of resolved vibronic structure in luminescence spectra.22,25
A special category of transition that has received significant attention, mainly because of the well resolved bands, is intraconfigurationald–d transitions. These involve initial and final states with identical electron configurations, and aQvalue of zero is expected, leading to luminescence and absorption spectra as shown in Figure 4a. In experimental spectra, short progressions, similar to those illustrated inFigure 4b, are sometimes observed. In these cases, the offsetQis
34x103 32
30 28
26
Wave number [cm–1] (b)
(a)
(c)
∆Q = 0
∆Q = 1
∆Q = 3
Figure 4 Luminescence and absorption spectra calculated for different offsets Q between potential minima along the normal coordinate in Figure 1 are shown as solid and dotted lines, respectively. The interval between members of the vibronic progression corresponds to the vibrational energy of the final state, chosen as 350 cm1for the potential curves inFigures 1and2. Values of the offsetQare given for each
pair of spectra.
Time-dependent Theory of Electronic Spectroscopy 563
different from zero due to spin–orbit coupling to a nearby excited state with a large Qvalue.
This effect is illustrated in Figure 5 for the potential energy curves in the range of the lowest- energy intraconfigurational transition of the Ni(H2O)62þ complex, where two states of Egsym- metry interact.26The minimum of the harmonic (diabatic) final state of the intraconfigurational transition to the 1Egstate is at Qẳ0, directly above the minimum of the ground state potential energy surface. The anharmonic (adiabatic) potential has its minimum offset by an amount Q due to coupling with the 3T1g state, and a short progression is observed in the experimental spectrum. Table 1summarizes several compounds where nonzero Qvalues arise from coupling between states, and it is interesting to note that values almost as large as those for charge–transfer
Table 1 RepresentativeQvalues (Figures 1and5) in dimensionless units determined from absorption or luminescence spectra with time-dependent theory. Only the largest offsetsQare given for transitions where
more than four modes are involved.
Type of electronic transition
Coordination geometry, compound (host lattice for doped solids)
Normal coordinate (where indicated in the reference), vibrational frequency(cm1)
Q (dim.-less
units) References d–d
intraconfigurational (spin–flip) transition
Octahedral TiCl64
(MgCl2) (Ti–Cl),260 0.01a 56
VCl63(Cs2NaYCl6) (V–Cl), 291 0.02a 56
CrF63(K2NaAlF6) (Cr–F), 568 0.00a 56
CrF63 (Cr–F), 415 0.11 32
MnF63 (Mn–F), 595 0.00a 56
NiCl64(CsMgCl3) (Ni–Cl), 255 0.80a 26
trans-NiCl2(H2O)4 (Ni–OH2), 364 0.94a 26 Ni(H2O)62þ (Ni–OH2), 3970.88a 26 Tetrahedral
MnO43
(Li3PO4) (Mn–O), 800 0.00 57 MnO43
(Ba5(VO4)3Cl) (Mn–O), 754 0.00 57 Linear
NiO22 (Ni–O), 590 1.43a 56
d–d
interconfigurational transition
Octahedral
TiCl64(MgCl2) (Ti–Cl), 200 3.05a 56
VCl63
(Cs2NaYCl6) (V–Cl), 215 3.53a 56
CrF63
(K2NaAlF6) (Cr–F), 510 2.12a 56
CrF63 (Cr–F), 415 2.68 32
MnF63 (Mn–F), 490 2.49a 56
CrCl63(Cs2NaScCl6) a1g(Cr–Cl), 298 2.37b 22 eg(Cr–Cl), 236 2.41b
NiCl64(CsMgCl3) (Ni–Cl), 255 2.78a 26
trans-NiCl2(H2O)4 ( Ni–OH2), 364 4.11a 26 Ni(H2O)62þ ( Ni–OH2), 3974.11a 26 Tetrahedral
MnO43(Li3PO4) (Mn–O), 800 1.03a 57 MnO43(Ba5(VO4)3Cl) (Mn–O), 754 1.60a 57 Linear
NiO22 (Ni–O), 500 2.98a 56
Square planar
PtCl42 b1g(Pt–Cl), 304 2.25, 2.23 21
a1g(Pt–Cl), 329 2.58, 2.70 58
PdBr42 b1g(Pd–Br), 1671.73 59
a1g(Pd–Br), 1873.90
Pd(SCN)42 a1g(Pd–SCN), 280 3.74 60
b2g(NCSPd–SCN), 179
1.61
transitions have been determined. The time-dependent theoretical calculations allow a quantitative determination of these detailed effects through the comparison of experimental and calculated vibronic intensity distributions. The relative intensities of the transitions to the 3T1g and 1Eg
excited states inFigure 5can be calculated with the time-dependent approach, but the limitation to the anharmonic (adiabatic) approximation used above to determineQis no longer valid as wave packet amplitude transfer between potential energy surfaces becomes an important effect.2,3,26 Forbidden transitions often gain their main intensity through enabling modes, leading to electric dipole intensity for vibronic origins. In the literature, these vibronic origins and their progressions are usually treated with the same models as true electronic origins and their vibronic patterns. In a strict approach, the coordinate dependence of the transition dipole along the normal coordinate of the enabling mode has to be multiplied to the eigenfunction of the initial state in order to obtain the wavepacket.27This has been shown recently to lead to characteristic spectroscopic signatures that can not be analyzed with the traditional assumption of an allowed transition.28The time-dependent approach is easily extended to include such forbidden transitions.