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Why does the middle band in the absorption spectrum of Ni(H_{2}O)_{6}^{2+} have two maxima?Myriam Triest, Guillaume Bussière, Hugo Bélisle andChristian Reber* Département de chimie, Université de Montréal, Montréal QC H3C 3J7, Canada AbstractIntroductionThe absorption spectra of octahedral complexes of nickel(II) are often used to illustrate basic aspects of metalligand bonding in coordination compounds. A number of practical guides for the analysis of such spectra in order to determine crystal field parameters or the equivalent quantities in the angular overlap molecular orbital model have been published. (1, 2) The band maxima of the spin allowed transitions are used to determine 10Dq and B and the spectrochemical series can be established from a comparison of absorption maxima for a series of compounds. (3) The goal of this work is to extend the traditional crystal field analysis and to give the undergraduate student an opportunity to explore quantitative models linking point group symmetry, electronic states and potential energy surfaces. Octahedral nickel complexes are well suited for this purpose, and group theory provides a full description of their electronic states in the energy range of the absorption spectra. (4) The concept of molecules moving on one or several potential energy surfaces is important for many areas of chemistry; examples include detailed reaction dynamics (5) electron transfer processes, (6) where multiple potential energy surfaces similar to those presented in the following, are essential. An aspect of the absorption spectra of some octahedral nickel(II) complexes that is discussed only in a very cursory manner in the standard texts concerns the shape of the absorption bands, leading to the following questions from our undergraduate students: The neglect of this spectroscopic observation in the conventional crystal field approach leaves a taste of lax analysis of experimental data, and we have not found a rigorous literature illustration of its physical origin suitable for undergraduate students. Several inorganic textbooks offer diverging discussions on the subject, as documented by an extensive set of references in a recent publication. (7) A very short assessment is given by Cotton and Wilkinson, (8) assigning the bandshape to two energetically close excited states. We present in the following a full quantitative model that will allow us to calculate the band shape and characterize the two states. The double maximum is a consequence of interacting electronic states with potential energy surfaces crossing in the FranckCondon region of the absorption spectrum. Our model was developed for and rigorously tested against highly resolved polarized lowtemperature absorption spectra of a series of nickel(II) complexes with octahedral, tetragonal and trigonal point group symmetry (7) and it is applied here to less resolved solution spectra. Even without a detailed understanding of the underlying theory, undergraduate students can use the Quicktime animations to explore the processes that define bandshapes. Experimental ResultsFigure 1. Absorption spectra showing all three spin allowed crystal field bands of a Ni(H_{2}O)_{6}^{2+} (upper trace, 0.101 M in aqueous solution) and Ni(NH_{3})_{6}^{2+} (lower, 0.315 M in aqueous NH_{3} solution) . Band assignments are given as illustrated in the TanabeSugano diagram in Figure 2. Figure 1 presents absorption spectra of two octahedral nickel(II) complexes measured in our undergraduate coordination chemistry laboratory on a Varian Cary 5E spectrometer. The three spin allowed bands are clearly visible for both compounds and their maxima are used to determine 10Dq and B. The triplet excited states are easily assigned with a TanabeSugano diagram for the d^{8} electron configuration shown in Figure 2. Some absorption bands (most notably the highest energy ^{3}A_{2g} > ^{3}T_{1g}(^{3}P) bands) have only a single maximum for octahedral complexes of nickel(II), others, especially the "middle" ^{3}A_{2g} > ^{3}T_{1g}(^{3}F) band in spectra of nickel(II) complexes with halide or oxygen ligands, show a more complicated band shape with two maxima. This can be clearly seen in the spectrum of Ni(H_{2}O)_{6}^{2+} . Shoulders are sometimes observed in the lowest energy ^{3}A_{2g} > ^{3}T_{2g} band of complexes with nitrogen donor ligands. (9) A comparison of the experimental spectra with the TanabeSugano diagram reveals that double maxima are observed for situations in the neighborhood of a crossing of the lowestenergy singlet excited state with the first or second triplet excited states. This situation will be investigated in the following. Figure 2. TanabeSugano diagram for octahedral nickel(II) complexes calculated with a C/B ratio of 4. The bands observed in Figure 1 are shown as dotted lines for both complexes. The crystal field parameters 10Dq and B for Ni(H_{2}O)_{6}^{2+} are 9100 cm^{1} and 950 cm^{1}, respectively. Theoretical BackgroundThe three spinallowed electronic transitions observed for nickel(II) complexes in Figure 1 are very broad, indicating that the molecules undergo important structural changes in the course of the electronic transition. This observation is easily rationalized with the overall bonding or antibonding character of the electron configurations from which the states arise: excited electronic states with a higher population of the santibonding e_{g} orbitals are expected to have longer Ni^{2+}  ligand bond lengths than the ground state. These bond elongations result in broad absorption bands. In contrast, transitions to excited states arising from the same electron configuration as the ground state, with almost unchanged Ni^{2+}  ligand bond lengths, lead to narrow bands in the absorption spectrum. Often both types of bands can be observed experimentally. (10) The traditional quantitative approach to analyze band shapes in absorption spectra involves the FranckCondon principle and overlap integrals of vibrational eigenfunctions, an approach based on the timeindependent theory of spectroscopic transitions. This approach is easily applied to harmonic potential energy surfaces, but it becomes less obvious for situations with nonharmonic potentials and multiple states that have to be considered simultaneously, as is the case for the spectra in Figure 1. Modern microcomputers have made an alternative, timedependent approach feasible. In this approach, the evolution of the initialstate eigenfunction on the potential energy surface of the final state is calculated, providing an intuitively appealing picture of the behavior of a molecule that absorbs light. This approach is compared to the standard FranckCondon theory in a recent electronic publication in this journal. (11) The most important quantity for the calculation of an absorption spectrum is the timedependent autocorrelation function <ff(t)>, corresponding to the overlap of the wavefunction at time zero (f) with the wavefunctions that evolve on the potential energy surfaces (f(t)). The Fouriertransform of the autocorrelation to the frequency domain is the calculated spectrum. The theoretical equations have been described in detail. (12, 13) Numerical algorithms to solve the timedependent Schrödinger equation and to calculate the timedependent wavefunction on potential surfaces have also been described in this journal before. (14) The first Quicktime animation illustrates the timedependent approach to an absorption transition between onedimensional harmonic potential energy surfaces. The vibrational motion of the wavefunction f(t) is easily followed in Figure 3. The absolute value of the autocorrelation <ff(t)> is shown as a yellow trace in Figure 3. It decreases as the wavefunction f(t) moves away from its initial position f, then reaches a maximum after each vibrational period when returning to its starting position. All autocorrelation functions in the following are multiplied by a phenomenological damping factor that decreases their value with increasing time, leading to maxima in Figure 3 that are becoming smaller with time. The damping factor chosen for Figure 3 determines the width of each peak in the highly resolved calculated spectrum in Figure 4. A larger value leads to the calculated band without resolution, also included in Figure 4, representing a typical solution spectrum. It is obvious from this figure that the damping factor does not change the total width or overall envelope of the absorption spectrum, quantities that are entirely determined by the potential energy surfaces. The simple model involving only one excited state potential energy surface is appropriate for all absorption bands showing a single maximum in Figure 1. (Download and play the animation, 2.0 M.) Figure 3. Animation of an electronic absorption transition between two harmonic surfaces. The top panel shows the timedependent wavefunction f(t) on the potential energy surfaces of the excited state, the bottom panel shows the absolute value of the autocorrelation function <ff(t)> as a function of time (yellow trace). This quantity corresponds to the overlap of the timedependent wavefunction f(t) with the wavefunction f at 0 fs, shown as a thin red line. The time interval between frames is 1 fs. Figure 4. Calculated absorption spectra from the model in Figure 3. This spectrum was obtained by a Fourier transform of the autocorrelation in Figure 3. The resolved spectrum shows the vibronic structure of the band, the unresolved spectrum reproduces a typical solution absorption spectrum, such as the examples in Figure 1. Model calculations for Ni(H_{2}O)_{6}^{2+} and Ni(NH_{3})_{6}^{2+}In this section, we focus on models analogous to those in Figure 3, but adapted to the situations for the "middle" absorption bands of Ni(H_{2}O)_{6}^{2+} and Ni(NH_{3})_{6}^{2+} , shown in Figure 1. The values of 10Dq, B and C/B obtained from the band maxima(3, 7) allow us to calculate the energies of all crystal field excited states of the complexes held "frozen" at their ground state equilibrium geometry. The calculations therefore give not only triplet but also singlet excited state energies, and we notice that for Ni(H_{2}O)_{6}^{2+} the ^{3} T_{1g}(^{3}F) state is close to the lowestenergy singlet electronic state, ^{1}E_{g}. The energy separation of these two states is much larger for Ni(NH_{3})_{6}^{2+} . The crystal field energies can be used to place the complexes along the horizontal axis of Figure 2. A more detailed view of the electronic states of interest is given in Figure 5a. Spinorbit coupling separates the ^{3}T_{1g} state into A_{1g} , E_{g} , T_{1g} and T_{2g} levels. A second E_{g} level arises from the ^{1}E_{g} excited state. The avoided crossing between the two E_{g} levels is caused by an offdiagonal matrix element, given in eq. 1, which depends on l, the spin orbit coupling constant. The literature value of l is 270 cm^{1} for Ni(H_{2}O)_{6}^{2+} .(15) Spinorbit coupling is not expected to influence metalligand bond lengths and therefore all potential energy surfaces arising from ^{3}T_{1g} have their minima at the same position along Q. Molecular structures in excited states are often different from the ground state, leading to minima of excited state potentials that do not coincide with the ground state potential minimum. The minimum of the ^{3}T_{1g} excited state is at a larger value of Q than the minima of the ground state and the ^{1}E_{g} excited state, a direct consequence of the increased population of the antibonding e_{g} orbitals in the triplet excited state. The ^{3}A_{2g} ground state and the lowestenergy singlet excited state arise from the same electron configuration and are therefore expected to have very similar bond lengths, leading to potential minima at the same value of Q, as shown in Figure 5b. Of particular interest is the avoided crossing between the two E_{g} states. We describe their coupled potentials V_{Eg} as: (1) These potential energy surfaces are shown in Figure 5b for Ni(H_{2}O)_{6}^{2+} . The abscissa Q denotes the totally symmetric breathing mode of the octahedral complex. The origin of this axis is at the ground state equilibrium geometry. Experimental vibrational energies were used to calculate the force constants k_{S} and k_{T} of the harmonic potentials for the E_{g} states arising from the singlet (subscript S) and triplet (subscript T) states, respectively. Their offsets E_{S} and E_{T} along the energy axis and the offset DQ along Q for the E_{g} level arising from the triplet excited were adjusted to fit the experimental spectrum. (7) The absorption spectrum in the region of the middle band of Ni(H_{2}O)_{6}^{2+} is a superposition of several transitions. The transitions to the A_{1g} , T_{1g} and T_{2g} states correspond exactly to the situation illustrated in Figures 3 and 4. The spectra for these transitions can therefore be easily calculated with the model in Figure 3. The situation is more complicated for the two potential energy surfaces describing the coupled E_{g} states arising from ^{3}T_{1g} and ^{1}E_{g} , respectively. These states are coupled by the offdiagonal matrix element in eq. 1 and their potential energy surfaces have to be considered simultaneously in order to understand the observed absorption spectrum. The effects of coupling influence the absorption spectrum and lead to the double maximum measured for Ni(H_{2}O)_{6}^{2+} . The calculation of absorption spectra for these coupled surfaces is outlined in the following. Figure 5. a) Calculated energy levels in the region of the ^{3}T_{1g}, ^{1}E_{g} crossing. b) Coupled potential energy surfaces for the E_{g} (^{1}E_{g} ) and E_{g} (^{3}T_{1g} ) electronic states of Ni(H_{2} O)_{6} ^{2+}. The potential energy surfaces for the coupled states are shown in Figure 5b. The first panel of this figure shows harmonic surfaces for the singlet (blue) and triplet (red) excited states, not including any coupling. The second panel shows the surfaces that include the coupling, the adiabatic surfaces. It is important to realize that the singlet and triplet character of these surfaces changes along Q, as illustrated by the color change from red to blue. The comparison of the first two panels shows directly which character dominates the adiabatic surface for each value of Q. The full (combined) description of the model for two coupled electronic states is shown in the third panel of Figure 5b, and we will use this model for our calculations. The calculation of the spectrum involves an allowed electronic transition to the ^{3}T_{1g} excited state and a forbidden transition to the ^{1}E_{g} excited state. We assume for simplicity that all transitions are parity allowed by the same enabling mode of ungerade parity, not explicitly included in the model presented here. A full discussion of the intensity mechanisms is given elsewhere. (7) The model for Ni(H_{2}O)_{6}^{2+} is shown in Figure 6. At t=0 the molecule is described by the red wavefunction on the triplet surface. The amplitude on the singlet surface, shown in blue, is exactly zero because the transition to the singlet state is spinforbidden. The evolution of the wavefunction and autocorrelation can be followed by playing the movie. It is obvious that there is very important amplitude transfer between the red and blue surfaces even at very short times. This is easily analyzed by following the evolution of the wavefunction frame by frame. We used 1 fs time intervals in the production of all Quicktime animations. At 0 fs, all amplitude is on the red surface. Approximately 510 fs later, the blue surface has gained significant amplitude, an effect that leads to a steep drop of the autocorrelation function. At 18 fs, some of this amplitude has returned to the red surface, leading to a bump in the autocorrelation, which is not observed if only one potential energy surface is involved (Figure 3). The autocorrelation then drops to zero as wavefunction amplitude moves away from the initial region and only very little amplitude transfer occurs, as most amplitude is far from the region where the potential energy surfaces cross or change color. Important amplitude transfer occurs again after approximately one vibrational period, when the wavefunctions are again approaching this region. (Download and play the animation, 2.2 M.) Figure 6. Time dependent wavefunctions f(t) on the coupled surfaces of Ni(H_{2}O)_{6}^{2+}. The bottom panel (yellow trace) shows the absolute value of the autocorrelation <ff(t)>. The calculated absorption spectrum for this model is shown in Figure 7. The unresolved calculated spectrum consists of two bands with different maxima, the resolved calculated spectrum shows a much more complicated pattern of lines than the calculated spectrum in Figure 4. From the potential energy surfaces in Figure 5 we note that the maximum at higher energy qualitatively corresponds to levels in the narrow upper adiabatic surface. The broader band at lower energy corresponds to transitions to levels within the broad lower energy adiabatic surface. Model calculations on highly resolved single crystal spectra have confirmed this observation. (7) We note from the nature of the adiabatic surfaces that it is impossible to attribute meaningful singlet or triplet labels to each of the two absorption maxima, a finding that is emphasized by the varying color along each adiabatic surface in the FranckCondon region. The intensity of the higher energy band is much too high for a formally spinforbidden transition, but its width is relatively small, as expected for a transition to the undisplaced singlet state. The intensity of the lower energy band is too low for a spinallowed transition, but the band is broad, as expected for a transition to the triplet surface which is displaced along the Q axis by DQ. This calculated spectrum shows that it is important to note which aspects of the experimental spectra have been used to assign labels of excited states to observed bands. Figure 7. a) Calculated absorption spectrum for the transition to the two coupled E_{g} states of Ni(H_{2}O)_{6}^{2+}. b) Calculated absorption spectrum to all levels of the ^{3}T_{1g}^{1}E_{g} manifold shown in Figure 5a. The yellow trace shows the experimental spectrum for comparison. The total calculated spectrum is the sum of the spectra obtained for the transitions to each of the excited states in Figure 5a. The calculated spectra were normalized and then multiplied by the degeneracy of the final state. A degeneracy of 2 was used for the spectrum arising from the coupled E_{g} states, because intensity only arises from the E_{g} (^{3}T_{1g} ) level. The sum of all the calculated absorption spectra was then scaled to the experimental molar absorptivity. The agreement between calculated and experimental spectra is very good and supports our model. The double maximum observed for Ni(H_{2}O)_{6}^{2+} arises from the coupled E_{g} states because their potential energy surfaces cross in a region that is very close to the starting position of the wavefunction at 0 fs. This leads to important amplitude transfer at very short times, a new maximum in the autocorrelation and two maxima in the calculated absorption spectrum.
The third situation illustrated is the middle absorption band of the Ni(NH_{3})_{6}^{2+} complex. The potential energy surfaces for this complex are shown in Figure 8. They are again defined by eq. 1. The offdiagonal coupling element between the E_{g} states is identical in magnitude to the one for Ni(H_{2}O)_{6}^{2+} . The stronger ligand field in the hexammine complex leads to a shift of the triplet excited state to higher energy, placing the color change of the adiabatic surfaces at a different location along Q than for Ni(H_{2}O)_{6}^{2+} in Figure 5b. Figure 8. Coupled potential energy surfaces for Ni(NH_{3})_{6}^{2+}. The time evolution of the wavefunction is shown in Figure 9. It is obvious that much less amplitude transfer occurs at short times than for Ni(H_{2}O)_{6}^{2+} in Figure 6. The autocorrelation therefore does not show an additional maximum and the calculated spectrum in Figure 10 resembles the spectrum obtained from the model involving only one excited state potential surface in Figure 3. It is much easier to attribute singlet and triplet electronic labels to the spectrum in Figure 10 than to the spectrum in Figure 7. We notice that the calculated intensity for the spinforbidden transition to the singlet state is entirely borrowed from the allowed triplet transition. This is the typical situation for transitions to all singlet states of octahedral nickel complexes: their intensity is very low because in the vast majority of cases there is no state of identical symmetry available to "steal" wavefunction amplitude. These transitions are therefore not easily observed in absorption spectra of solutions. (Download and play the animation, 2.2 M.) Figure 9. Time dependent wavefunction f(t) on the coupled surfaces of Ni(NH_{3})_{6}^{2+}, including the absolute value of the autocorrelation <ff(t)> in the bottom panel. The timedependent approach applied here to the middle band in the absorption spectra of Ni(H_{2}O)_{6}^{2+} and Ni(NH_{3})_{6}^{2+} provides a quantitative illustration of the observed bandshapes in solution absorption spectra. It links the traditional crystal field symmetry analysis to the dynamic behavior of molecules on potential surfaces, illustrating the important insight provided by electronic spectroscopy. Figure 10. Absorption spectrum for the transitions to the two coupled E_{g} states of Ni(NH_{3})_{6}^{2+} calculated from the potential surfaces in Figure 9. AcknowledgmentThis work was made possible by research grants from the Natural Sciences and Engineering Research Council (Canada) and by an intramural teaching grant from the Université de Montréal. We thank Véronique Nadeau and Lori Jinbachian for measuring the spectra in Figure 1 during their inorganic laboratory course. References










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