Mechanisms of f–f hypersensitive transition intensities of lanthanide trihalide molecules: a spin–orbit configuration interaction study

Abstract

The optical properties of intra-4f^N transitions (f–f transitions) in lanthanide compounds are usually insensitive to the surrounding environment due to the shielding effect of the outer 5s and 5p electrons. However, there are exceptional transitions, the so-called hypersensitive transitions, whose oscillator strengths change sensitively to a small change of the surrounding environment. The mechanism of the hypersensitive transitions was explained mostly with the dynamic-coupling (DC) model. In this study, the oscillator strengths of hypersensitive transitions in lanthanide trihalides (LnX₃; Ln = Pr, Nd, Pm, Sm, Eu, Tb, Dy, Ho, Er, Tm; X = Cl, Br, I) were calculated by the multi-reference spin–orbit configuration interaction (CI) method, and the origin of the hypersensitive transition intensities was examined. To compare the intensities derived from the DC model and from the ab initio CI computations, we evaluated two Judd–Ofelt intensity parameters: τ ₂(dc) by the DC model and τ ₂(ab) by the CI computations. Although these two parameters showed similar overall behaviors, their Ln dependences were different, suggesting the involvement of other mechanism(s) in τ ₂(ab). Close examination of the spatial distributions of the transition densities and the integrand of the transition dipole moments (TDMs) suggested that the Judd–Ofelt theory contributions were also involved in τ ₂(ab) with the opposite sign relative to the TDMs with the DC model in all the hypersensitive transitions of LnX₃. Moreover, the different Ln dependences in τ ₂(dc) and τ ₂(ab) were related to the different amount of the mixing of ligand-to-metal charge transfer configurations into the dominant 4f^N configurations, especially for Eu and Tb.

Appendix: Definition of weight of LMCT configurations γ

In this appendix, we describe the detailed definition of γ. The N _occ × (N _act + N _vir) rectangular block transition density matrix T is translated to “diagonalized” rectangular matrix by using the “corresponding orbital” [77] or “natural transition orbital” [78, 79] method as follows,

$$\left[ {{\mathbf{U}}^{{\mathbf{ + }}} {\mathbf{TV}}} \right]_{ij} = \lambda_{i} \delta_{ij}$$

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where ${\mathbf{U}} = ({\mathbf{u}}_{1} ,{\mathbf{u}}_{2} , \ldots ,{\mathbf{u}}_{{N_{occ} }} )$ and ${\mathbf{V}} = ({\mathbf{v}}_{1} ,{\mathbf{v}}_{2} , \ldots ,{\mathbf{v}}_{{N_{act} + N_{vir} }} )$ are the unitary matrices determined by solving the following eigenvalue equations,

$$\begin{gathered} ({\mathbf{TT}}^{ + } ){\mathbf{u}}_{i} = \lambda_{i}^{2} {\mathbf{u}}_{i} \,\,\,\,\,(i = 1, \ldots ,N_{occ} ) \hfill \\ ({\mathbf{T}}^{ + } {\mathbf{T}}){\mathbf{v}}_{i} = \lambda_{i}^{2} {\mathbf{v}}_{i} \,\,\;(i = 1, \ldots ,N_{act} + N_{vir} ) \hfill \\ \end{gathered}$$

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Here, the new set of the occupied and (active + virtual) orbitals is obtained by using the unitary transformations, and a pair of the orbitals whose matrix elements with $({\mathbf{U}}^{ + } {\mathbf{TV}})_{ii} = \lambda_{i}$ are called the i-th hole-particle pair orbitals. The importance of a particular hole-particle transition to the overall rectangular matrix T is reflected in the magnitude of the associated eigenvalue λ ² _i .

Next, we define a quantity of charge transfer from occupied orbitals of X₃ in Ψ _I to active and virtual orbitals of Ln in Ψ _F . This quantity, called γ, is evaluated by averaging the change of the Mulliken charge population on Ln $\Delta p_{i}^{Ln}$ with the associated weight factor of λ ² _i as follows,

$$\gamma = \frac{{\sum\nolimits_{i}^{{N_{occ} }} {\lambda_{i}^{2} \Delta p_{i}^{Ln} } }}{{\sum\nolimits_{i}^{{N_{occ} }} {\lambda_{i}^{2} } }}$$

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This parameter γ of LnX₃ takes a positive value because the amount of mixing of LMCT CSFs is much larger than that of MLCT CSFs. This weight of λ ² _i is a “diagonalized” transition density matrix element and has the information of the products of the CI coefficients, i.e., those for the reference CSFs in Ψ _I times and those for one electron excitation CSFs in Ψ _F . With these weight factors, parameter γ represents the component of LMCT mixing in the final state of the target transition.