Coupling between normal coordinates in the ground and excited states of coordination compounds. Electronic spectroscopy and theoretical models

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Abstract

Effects of coupling between normal coordinates can be analyzed from absorption and emission spectra. The spectroscopic phenomena reviewed here arise from interactions between electronic states with different equilibrium structures, a situation that leads to coupled coordinates. They have received some attention in the past for special categories of coordination compounds, such as Jahn-Teller active high-symmetry transition metal complexes and mixed valence compounds. We use model systems outside these established categories and illustrate the spectroscopic effects with calculations involving two normal coordinates and two electronic states. These models show directly how coordinate coupling arises as a consequence of the non-crossing rule for states of identical symmetry and they are easily generalized to other molecular systems with energetically close electronic states. Electronic structure calculations are presented to further illustrate the effects of coupling. The examples analyzed are based on experimental results reported for trans-dioxo complexes of rhenium(V) and chromium(III) fluorides.

Introduction

Traditionally, molecular electronic states are described by a single potential energy surface defined along the normal coordinates of the molecule. In this approximation, electronic transitions in absorption or luminescence spectra involve two potential energy surfaces, corresponding to the initial and final states of the transition. Interactions with other states are neglected or assumed to simply lead to changes of the overall intensity of bands in electronic spectra. An example is the intensity increase of a weak band corresponding to a forbidden transition in an absorption spectrum through coupling with a nearby allowed transition [1]. Other spectroscopic manifestations of interactions between states on vibronic bandshapes have received less attention. A few situations have been analyzed with coupled potential energy curves defined along a single normal coordinate. These one-dimensional models can be used to characterize some effects of coupling between electronic states, but obviously no coupling between normal coordinates can be examined with this approach [1], [2]. The validity and limitations of one-dimensional models have been explored in detail for several compounds using single-crystal absorption and luminescence spectroscopy at low temperature or under high external pressure [1], [2], [3], [4], [5], [6], [7], [8]. The goal of this overview is to examine interacting states defined along two normal coordinates in order to illustrate aspects of vibronic spectra beyond the one-dimensional models. The interaction between these electronic states leads to an intrinsic coupling between normal coordinates. In addition, we present results of electronic structure calculations that also indicate the presence of coupling between normal coordinates.

A significant number of recent publications, mainly using advanced computational tools, have led to new insight on coupled coordinates [9], [10], [11], [12]. Dynamic studies aimed at small polyatomic molecules and photochemical product distributions [13], [14] have been reported, and a few analyses of coupled coordinates through spectroscopy and theoretical calculations of spectra have been published [15], [16], [17]. The vast majority of these studies investigate molecules that are not coordination compounds, a somewhat surprising fact, as transition metal complexes with their rich electronic structure and significant spin–orbit coupling between states are highly susceptible to effects arising from interacting electronic states. Exceptions to this situation are Jahn-Teller systems [18] and intervalence compounds [17], [19]; both recently reviewed in depth.

Coupling between normal coordinates can arise as a consequence of interacting electronic states. We illustrate such effects with the simplest possible model: two electronic states described by potential energy surfaces along two normal coordinates. Effects such as spin–orbit coupling and configuration interaction can lead to distinct bandshapes and vibronic structure in the absorption and luminescence spectra of many coordination compounds and organometallic molecules. We present calculations for two specific situations, illustrated schematically with the one-dimensional cross sections along a single normal coordinate Qi in Fig. 1. All parameters defining the models in Fig. 1 are given in Section 2. The first situation of interest involves the electronic ground state interacting with an excited electronic state, as shown in Fig. 1a. The coupling between the states leads to a ground state potential energy surface that is flattened in the region below the minimum of the excited state potential surface. The bandshape and vibronic structure in the luminescence spectrum are affected by this coupling. The model in Fig. 1a is inspired by the experimental spectra of trans-dioxo complexes of rhenium(V) and osmium(VI) [20], [21], [22], [23], [24], [25]. Instead of carrying out the calculation of a spectrum with the full ensemble of diabatic and adiabatic potential energy surfaces, it is sufficient in this case to include only the adiabatic ground state potential surface, corresponding to the lower solid curve in Fig. 1a, as the final state of the transition in the calculation of a spectrum. The mixing between the states is small and the crossing between their potential energy surfaces occurs far from the Franck–Condon region of the luminescence transition, shown as a vertical arrow in Fig. 1a. Nevertheless, the coupling between states has to be considered, because it significantly alters the shape of the ground state potential energy surface, as illustrated by the difference between the adiabatic (solid line) and diabatic (harmonic, dotted line) ground state potentials in Fig. 1a. Detailed density functional calculations are used as an alternative approach to characterize a trans-dioxo rhenium(V) complex, qualitatively confirming the presence of coupled normal coordinates. This example illustrates the importance of coupling along normal coordinates on the ground state potential energy surface arising through interactions with an energetically well-separated excited state, an effect that is usually neglected in studies of transition metal compounds.

More dramatic effects of coupled coordinates are observed for interacting electronic states that are close in energy. This situation typically occurs for excited states of transition metal compounds and can be analyzed from absorption spectroscopy, as schematically illustrated in Fig. 1b. Strong mixing occurs between such states and the full set of diabatic and adiabatic potential energy surfaces has to be used for an adequate description of the final state of the electronic absorption transition, shown as a vertical arrow in Fig. 1b. The crossing of the diabatic potential energy surfaces occurs near the Franck–Condon transition, and the two coupled excited states have an important influence on the absorption spectrum [2]. In the literature, the overlapping lowest spin-allowed and -forbidden transitions in chromium(III) and vanadium(II) complexes with the d3 electron configuration are the most common examples of absorption spectra showing effects of coupling between states and coordinates [2], [26], [27], [28], [29], [30], [31], [32]. Such effects are expected to occur for many other compounds if states arising from different electron configurations are close in energy, as illustrated schematically for the excited states labeled 1 and 2 in Fig. 1b. These labels were chosen to be coherent with an earlier review on spectroscopic effects arising from coupled one-dimensional potential curves [1]; for a transition metal complex with the d3 electron configuration they correspond to the lowest doublet and quartet excited states, respectively. The transition from the quartet ground state to the doublet excited state is assumed to borrow its intensity entirely from the allowed transition to the quartet excited state.

Even in the absence of well resolved vibronic structure in an experimental spectrum, features such as interference dips [2] can be observed and allow us to analyze the coupling between electronic states and normal coordinates. The interference dip in molecular absorption spectra has often been denoted as a Fano antiresonance [28]. An exact quantitative relationship illustrating the crucial differences between Fano's approach for atomic transitions and molecular spectra has only recently been published [33]. Our calculations show that the shape of the interference dip in absorption spectra reflects the influence of coupled coordinates [26], an effect that can be observed even in absorption spectra without resolved vibronic structure.

The spectroscopic effects arising from the situations illustrated in Fig. 1 with two normal coordinates Qi (i=1, 2) are distinctly different from those that can be obtained with one-dimensional models involving only a single coordinate [1]. These effects are most easily seen in cases where the vibrational frequencies of the two modes are significantly different, as this leads to well-separated vibronic transitions that can be distinguished in experimental spectra.

Section snippets

Coupling between coordinates in the electronic ground state. Band envelopes and vibronic structure in luminescence spectra

This section is inspired by the detailed vibronic structure observed for trans-dioxo complexes of rhenium(V), such as trans-ReO2(ethylenediamine)2+ [20], [21], [22], [23], [24], [25]. The vertical energy difference between the ground and lowest-energy excited state in these compounds is on the order of 9000–20 000 cm−1. The detailed analysis of the spectra shows that coupling effects can be observed despite the considerable energy separations between electronic states, especially for third row

Coupling between coordinates in excited states. Interference dips in absorption spectra

Excited states that are close in energy can give rise to easily discernible spectroscopic manifestations of coupling. The example discussed in the following involves interference dips [2] in absorption spectra, arising from the situation schematically illustrated in Fig. 1b. In contrast to Section 2, the limitation to a single adiabatic surface is no longer an acceptable approximation. We need the full set of potentials defined by the matrix in Eq. (1) and have to include wavefunction amplitude

Summary and conclusions

The simple potential energy surfaces defined by Eq. (1) can be used to explore spectroscopic manifestations of coupling between normal coordinates. The origin of the coupling term for these model cases is straightforward and depends on vibrational frequencies and non-zero offsets of potential minima along the normal coordinates. This model illustrates the importance of coupled normal coordinates in transition metal compounds with resolved vibronic spectra that show non-replica patterns and in

Acknowledgements

We thank Dr Myriam Triest, Benoit Cromp, Professor Tucker Carrington (Université de Montréal), John K. Grey, Professor I.S. Butler (McGill University), Dr Ralph Schenker, Professor Hans U. Güdel (Universität Bern) and Professor Jeffrey I. Zink (University of California, Los Angeles) for collaborations and helpful discussions. Financial support from the Natural Sciences and Engineering Research Council (Canada) is gratefully acknowledged.

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