C. REBER and J. LANDRY-HUM
2.43.3 LUMINESCENCE AND ABSORPTION SPECTRA FOR ELECTRONIC
FROM ELECTRONIC STRUCTURE CALCULATIONS AND BETWEEN MULTIDIMENSIONAL POTENTIAL ENERGY SURFACES
In this section, we present two model cases where spectra are calculated from potential energy surfaces obtained from electronic structure calculations. The first example is the Ni(H2O)62þ
Table 1 continued
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 Pt(SCN)42 a1g(Pt–SCN),
300
4.86 17
b2g(NCSPt–SCN), 151
1.67 LMCT transition Octahedral
trans-OsO2(oxalate)22 sym(Os=O), 701 2.10a 44
MLCT transition trans-W(N2)2(1,2-bis 65 0.92 61
(diphenylphosphino) ethane)2
125 1.34
(W–P), 176 1.09
(W–N2), 450 0.97
(W–N), 524 1.02 (N–N), 2000 0.35
MMCT transition [(CO)5Cr0-CN- 2 states 62
OsIII(NH3)5] (CF3SO3)2
(Cr–CN), 396 2.20, 1.00 (Os–NC), 488 1.30, 2.50 (C–N), 2100 0.17, 0.27 LLCT transition Square planar
Ni(1,2-maleonitrile) (Ni–N), 273 1.33 63
(biacetylbisaniline) (Ni–S), 330 0.62
(C–S,C–CN), 520 1.05 (C–N), 1542 1.05
Pd(1,2-maleonitrile) (Pd–N), 258 1.32 63
(biacetylbisaniline) (Pd–S), 360 0.71
(C–S,C–CN), 520 0.89 (C–N), 1550 1.82
a Converted from A˚ units given in the reference to dimensionless units withQẳ0:1722Qẵ˚A ffiffiffiffiffiffiffi p!m
where!denotes the vibrational frequency in wave number (cm1) units and m the mass of the ligand in g/mole. b Calculated from the Huang–Rhys parameter S given in the reference withQẳ2 ffiffiffi
pS .
Time-dependent Theory of Electronic Spectroscopy 565
complex, one of the coordination compounds whose absorption spectrum is shown in many textbooks. Ab initiocalculations using the CASSCF and CASPT2 methods were carried out for a series of points along the Ni–OH2 totally symmetric stretching coordinate and a harmonic potential curve along this coordinate was defined for the ground state and for each triplet excited state from the calculated energies.29These potential energy curves were then used to calculate the absorption spectrum from Equation (1) without any adjustable parameters. The Q value for the3T1g(3F) excited state obtained from this calculation is 3.11, lower by approximately 25% than the value of 4.11 obtained from fitting harmonic model potentials and leading to the best agreement between experiment and calculation shown in Figure 8 of Chapter 2.22, illustrating the difficulty of calculating precise Qvalues from first principles even for simple coordination compounds. Nevertheless,Figure 6shows that the overall agreement is remarkably good and that a quantitative link between electronic structure calculations and experimental absorption spectra can be established with the time-dependent approach described here. Calculated spectra of similar good quality can be expected for many other compounds of the first-row transition metals, but detailed effects can not be calculated exactly, as computational methods are most often based on the Born–Oppenheimer approximation and energetically close electronic states can lead to effects where this approximation is no longer valid. Examples for such discrepancies are the middle band in the spectrum of Ni(H2O)62þ
, where the interaction between the two excited Egstates illustrated in Figure 5 leads to the double maximum in the experimental spectrum shown in Figure 6 and to an unusual series of resolved peaks in the low-temperature single-crystal spectrum of the trans-NiCl2(H2O)4 complex in Figure 7 of Chapter 2.22.26 The distinct interference dips in the absorption spectra of octahedral chromium(III) complexes, sometimes called Fano antires- onances, are another illustrative example of an intricate spectroscopic effect that can be analyzed in a straightforward way with the time-dependent approach.3,30–32
Potential energy surfaces for second- and third-row metals are challenging for modern compu- tational methods, as illustrated by the analysis of the luminescence spectrum of trans- ReO2(ethylenediamine)2þ
, a well-studied compound for which the detailed experimental spectra indicate a large Q value along the normal coordinate of the high-frequency symmetric OẳReẳO stretching mode.33,34 The solid line inFigure 7 shows a calculated potential energy curve along this coordinate obtained with an advanced DFT method.35 It is compared to the harmonic potential curve defined by the ground state Raman frequency of 880 cm1and shown as a dotted line. The anharmonicity of the solid potential energy curve is obvious, and anharmonic potential surfaces need to be routinely used for the analysis of spectra for compounds of 4dand 5d
25x103
20
15
10
Energy [cm–1 ]
8 6 4 2 0 –2 –4
Normal coordinate Q
∆Q Eg(1Eg)
Eg(3T1g, 3F)
Figure 5 Potential energy curves for the final state of an intraconfigurational absorption transition in proximity to an excited state arising from a different electron configuration. The potential energy curves and labels of electronic states (Oh point group) apply to Ni(H2O)62þ.26 The ground state potential energy minimum is at Qẳ0, given as a vertical dotted line. The offset between the minima of the harmonic (diabatic, dotted line) Eg(1Eg) state and the ground state is zero. Spin–orbit coupling leads to the anharmonic (adiabatic, solid line) curves and to a nonzero offsetQ, given by the horizontal bar. The Eg(3T1g) excited
state has a large offset, as given inTable 1.
metals. The spectra calculated from the potential energy surfaces inFigure 7are compared to the experimental spectrum inFigure 8. For these calculations, the excited state vibrational frequency was set to 780 cm1, a value determined from the absorption spectrum, the electronic origin E00
was set to 14, 410 cm1, and a value of 1.90 was determined forQin order to best reproduce the intensity distribution and bandwidth of the experimental spectrum. The calculated spectrum obtained from the anharmonic potential inFigure 7is closer to the experimental result than the calculation using the harmonic potential, but a significant discrepancy remains, most likely due to the neglected spin–orbit coupling in the DFT calculation. Other time-dependent studies in this area include high-quality calculations of potential energy surfaces and excited state dynamics involving multiple states, work that has led to detailed insight on the photochemical properties of several first and second row transition metal compounds,36including, for example, ruthenium(II) complexes with -diimine ligands.37,38
30x103 25
20 15
10
Wave number [cm–1] 8
6
4
2
0 ε [M–1 cm–1 ]
Experiment Calculation
3T1g(3P)
3T2g 3T1g(3F)
3T1g(3F), 1Eg
Figure 6 Experimental spectrum of Ni(H2O)62þ
in aqueous solution, compared to the spectrum calculated from potential energy curves obtained fromab initiocalculations. The calculated spectrum does not involve
adjustable parameters.
40x103
30
20
10
0 Energy [cm–1 ]
15 10 5
0 –5
Normal coordinate QO = Re = O
Figure 7 Potential energy curve along the symmetric OẳReẳO coordinate of trans- ReO2(ethylenediamine)2þ
from DFT calculations, shown as dots and interpolated solid line. The dotted line shows a harmonic curve calculated from the observed vibrational frequency of 880 cm1.
Time-dependent Theory of Electronic Spectroscopy 567
The time-dependent approach is efficient to calculate spectra for transitions between multi- dimensional potentials.39 In the absence of coupling between coordinates, autocorrelation func- tions are calculated separately along each normal coordinate, and the total autocorrelation is obtained as theproductof the individual autocorrelation functions obtained for coordinates with nonzero values ofQ.1It is obvious from the solid line inFigure 3that coordinates along which the wavepacket explores a flat area of the potential curve, such as the example for Qẳ0 in Figures 3 and 4, do not change the product and can therefore be neglected. Resolved vibronic structure involving many coordinates has been fully analyzed with this approach.40 Illustrative examples are compiled inTable 1. Detailed insight has also been obtained from spectra that are not well enough resolved to show each vibronic line, a phenomenon denoted as the missing mode effect (MIME).39,41,42 These analyses of experimental spectra have allowed the identification of normal coordinates with the largest Qvalues, modes of relevance for photochemical observa- tions.20,39 Electronic structure calculations need therefore only be carried out along a few select coordinates, an important shortcut over the calculation of full multidimensional potential surfaces.
If the coordinates are coupled, the total autocorrelation is no longer a product of the auto- correlation along each individual coordinate, and multidimensional wave packets and potential energy surfaces must be used. The storage requirements of wave packets on multidimensional coordinate grids become prohibitive if more than two coordinates are involved, and efficient computational methods have been developed to solve this problem.43,44 Some of the effects of coupled coordinates are seen from the two-dimensional potential energy surfaces shown inFigure 9.
They allow easy visual detection of coupling from cross-sections parallel to the normal coord- inates. If these cross-sections show a variation other than a simple shift along the energy axis, coupling is present. Such a variation is obvious from the comparison of cross-sections along the left-hand and right-hand vertical axes in Figure 9b. The top half of the left-hand axis defines a wider curve than the top half of the right-hand vertical axis, a difference that indicates coupled coordinates. Analytical equations for potential energy surfaces have coupling terms that are easily identified, such as the well-known Q1Q2product of two coordinates, which leads to a Duschinsky rotation.45,46
An example where spectroscopic manifestations of coupling between coordinates are observed is the low-temperature, single-crystal luminescence spectrum of trans-ReO2(vinylimidazole)4þ
in Figure 10.34 The overall shape of this spectrum is similar to that of trans- ReO2(ethylenediamine)2þ
in Figure 8, but vibronic peaks corresponding to two different modes are resolved. A spectrum calculated from the two-dimensional harmonic potential energy surface
Luminescence intensity
16x103 15
14 13
12 11
10
Wave number [cm–1] Experiment
Calculations
Figure 8 Experimental luminescence spectrum of crystalline trans-ReO2(ethylenediamine)2Cl at room temperature. The calculated spectra were obtained from the one-dimensional potential energy curve
calculated by DFT and from the harmonic curve, both shown inFigure 7.
–6 –4 –2 0 2 4 6
QO = Re = O
–4 0 4
Q Re-Imidazole
–4 0 4
Q Re-Imidazole
(a) (b)
Figure 9 Two-dimensional potential energy surfaces for trans-ReO2(vinylimidazole)4Cl along the OẳReẳO coordinate and a low-frequency Re–N(imidazole) mode. Elliptical contours are observed for the harmonic potential (top), the bottom potential includes coupling between normal modes and electronic
states and is flattened in the area below the excited state minimum, indicated by the dot.
Luminescence intensity
15x103 14
13 12
11 10
Wave number [cm–1] Experiment
Calc. with coupled coordinates
Calc. with uncoupled coordinates
Figure 10 Comparison of the experimental low-temperature, single-crystal luminescence spectrum oftrans- ReO2(vinylimidazole)4Cl with calculated spectra using a harmonic potential energy surface for the ground state and a potential energy surface derived including the coupling between the ground state and excited states of identical symmetry, giving rise to coupled coordinates. The potential energy surfaces used to
calculate this spectrum are shown inFigure 9.
Time-dependent Theory of Electronic Spectroscopy 569
in Figure 9 leads to a good overall agreement with the experimental spectrum, but it fails to reproduce the vibronic intensity distributions given by the sloped lines. The calculated spectrum arising from the two-dimensional potential energy surface with coupled coordinates inFigure 9b shows that the vibronic intensity distributions emphasized by the sloping lines in Figure 10are a manifestation of coupled coordinates. The origin of the coupling is the interaction between ground and excited states and only coordinates with nonzero offsets Qcan be involved. The comparison of absolute autocorrelation functions in Figure 11 provides a link between the potential energy surfaces in Figure 9and the calculated spectra inFigure 10. The initial drop in Figure 11a is less rapid for the potential with coupled coordinates in Figure 9b, where the dot indicating the emitting state potential minimum and the position of is in a slightly flatter area than on the harmonic surface inFigure 9a. This difference in the decrease of the autocorrelation function at short times and the differences at longer times, illustrated in Figure 11b and by the difference trace inFigure 11c, are all contributing to the variation of vibronic intensities given by the sloping lines and allow a detailed characterization of the physical origin of coupling between coordinates.34This example illustrates the intuitive understanding of detailed spectroscopic effects that can be gained from the time-dependent approach. Intervalence bands in mixed-valence compounds are a particularly intricate category of transitions involving potential energy surfaces with coupled normal coordinates that have recently been analyzed with time-dependent theory.47,48
1.0
0.5
0.0
|<φ|φ(t)>|
40 30
20 10
0
Time [fs]
1.0
0.5
0.0
|<φ|φ(t)>|
0.1
∆(|<φ|φ(t)>|) 0.0
200 150
100 50
0
Time [fs]
(a)
(b)
(c)
Figure 11 Autocorrelation functions used to calculate the spectra inFigure 10from the potential energy surfaces in Figure 9. Dotted and solid lines correspond to autocorrelation functions obtained from the potential surfaces inFigures 9a (harmonic potential energy surface)and 9b (potential energy surface with coupled coordinates), respectively. a) short-time comparison over the time interval for the initial decrease of the autocorrelation function; b) comparison over a time interval corresponding to several vibrational
periods; c) difference trace between the autocorrelation functions in b.