Europium Luminescence: Electronic Densities and Superdelocalizabilities for a Unique Adjustment of Theoretical Intensity Parameters

We advance the concept that the charge factors of the simple overlap model and the polarizabilities of Judd-Ofelt theory for the luminescence of europium complexes can be effectively and uniquely modeled by perturbation theory on the semiempirical electronic wave function of the complex. With only three adjustable constants, we introduce expressions that relate: (i) the charge factors to electronic densities, and (ii) the polarizabilities to superdelocalizabilities that we derived specifically for this purpose. The three constants are then adjusted iteratively until the calculated intensity parameters, corresponding to the 5D0→7F2 and 5D0→7F4 transitions, converge to the experimentally determined ones. This adjustment yields a single unique set of only three constants per complex and semiempirical model used. From these constants, we then define a binary outcome acceptance attribute for the adjustment, and show that when the adjustment is acceptable, the predicted geometry is, in average, closer to the experimental one. An important consequence is that the terms of the intensity parameters related to dynamic coupling and electric dipole mechanisms will be unique. Hence, the important energy transfer rates will also be unique, leading to a single predicted intensity parameter for the 5D0→7F6 transition.


Experimental Radiative Decay Rates from Emission Spectra
When ultraviolet light illuminates a suitable europium complex, its ligands absorb the photons as antennae. Subsequently, the ligands transfer the energy to the trivalent europium ion, which is excited mostly to its 5 D0 state. Once in its 5 D0 excited state, the trivalent europium ion undergoes both radiative and non-radiative decays in a competitive manner.
Radiatively, it goes from its 5 D0 state to any of the 7 FJ states, with J ranging from 0 to 6, emitting mainly orangish red light due to the intense 5 D0 to 7 F2 transition around 612nm ~614nm. The remainder of the energy dissipates non-radiatively. Of course, the more luminescent the complex, the less it will decay via non-radiative channels.
The experimental radiative decay rate, exp rad A , is the sum of the radiative decay rates of each of the possible transitions 5 D0→ 7 FJ , with J ranging from 0 to 6, which occur simultaneously: where the transition inside the square brackets serve to further index the quantity before it, in this case exp rad A .
Transitions 5 D0→ 7 FJ with J = 0, 3, and 5, are forbidden by three mechanisms: magnetic dipole, forced electric dipole and dynamic coupling. However, they usually appear -albeit with very low intensities -due to J-mixing effects. We will neglect them in the present work.   The transition 5 D0→ 7 F2 is called the hypersensitive transition 1 , which is well described by Judd-Ofelt theory and is remarkably responsive to the immediate chemical environment, namely the coordination polyhedron, around the metal ion, which is inscribed into it.
On the other hand, transitions 5 D0→ 7 F4 and 5 D0→ 7 F6 are less sensitive to the chemical environment neighboring the lanthanide ion, and are well described by Judd-Ofelt theory.
where is Planck-Dirac constant, e is the fundamental electric charge,  is the Lorentz local-field correction term given by  

Theoretical Radiative Decay Rates
The theoretical radiative decay rate for the forced electric dipole and magnetic dipole where J' is the total angular momentum quantum number of the emitting level, which in the case of europium is 5 D0 and therefore J' = 0. 57 is usually considered as being the difference between the barycenter of energy of the 5 D0 and 7 FJ levels (J = 2, 4, 6), which were determined by Carnall for the trivalent lanthanides in fluoride complexes 2 ; and Smd, the magnetic dipole strength constant, which, for the trivalent europium ion, is equal to 9.6 × 10 -42 esu 2 .cm 2 3 .
The theoretical intensity parameters calc Ω  (λ = 2, 4, 6) emerge from the Judd-Ofelt theory, and are proportional to the intensities of the 5 D0→ 7 F2, 5 D0→ 7 F4 and 5 D0→ 7 F6 transitions, respectively, in the emission spectrum of the complex. These parameters describe the coordination interaction between the lanthanide cation and the ligands, and are given according to the following expression: where the Btp terms are given by the following expression: where ed tp B  corresponds to the forced electric dipole contribution and dc tp B  corresponds to the dynamic coupling contribution, given by: where E is a constant, approximately given by the energy difference between the barycenters of the ground 4f n and first opposite parity excited state of configuration 4f (n- () f C f  is a Racah tensor operator of rank λ = 2, 4, and 6 whose value are -1.3660, 1.128, and -1.270, respectively, for any lanthanide; t p  is also a sum over coordinating atoms which further reflects the chemical environment; finally, t,λ+1 is a Kronecker delta symbol.
The odd rank ligand field parameters, in turn, are given by: where i runs over the ligand atoms, ρj and βi are defined according to the Simple Overlap Model (SOM) 6, 7 as and represent the correction introduced by SOM to the crystal field parameters of PCEM.
Indeed, the difference per directly coordinated ligand atom i, between the , () t pi SOM  of SOM, Eq. (S10) and , () In the equations above, ρ0 is a constant equal to 0.05 for any trivalent lanthanide ion, R0 is the smallest distance between the lanthanide ion and a directly coordinating atom of the ligand, Rj is the distance between the lanthanide ion and the directly coordinating atom j of the ligand. The plus sign in Eq. (S12) is used when the barycenter of the overlap region is The other odd rank parameter t p  , which further reflects the chemical environment, is given by: where i is the polarizability associated to the lanthanide-ligand atom bond. 12

Perturbation theory formalism on the semiempirical wavefunctions
Within the Sparkle Model, the trivalent lanthanide ion is represented by a potential due to the electrostatic point of charge +3e, superimposed to a spherical repulsive potential of the form exp(-r), in order to prevent the ligands from collapsing into the metal ion. Thus, in this model, the metal ion does not have any orbitals. The ligands, on the other hand, usually constitute a closed shell system described by the wavefunctions of a regular semiempirical model deformed by the potential that represents the metal ion. On the other hand, in the RM1 model for the lanthanides, the metal ion has a semiempirical basis set comprised of 5d, 6s, and 6p atomic orbitals. In both cases, the 4f electrons are considered part of the core, and are therefore taken care of implicitly. Accordingly, within these models, the restricted Hartree-Fock ground state wavefunction for the closed shell complex of 2n electrons is represented by a single Slater determinant: where the bars on top indicate a beta spin and i  are normalized molecular orbitals, which can be expressed in terms of a linear combination of semiempirical atomic orbitals   p as: where  runs over the NA atoms, p runs over the N atomic orbitals of atom  and  pi c is the linear coefficient of the semiempirical atomic orbital   p in the molecular orbital of order i.
The corresponding Hartree-Fock ground state energy is: where Hii is the core energy of the i th molecular orbital: where Zis the atomic number of atom , and r1 is the distance of electron 1 to the center of atom .
Applying Eq. (S16) in Eq. (S18), we obtain which can be partitioned as: where    and  can be defined as:   can be interpreted as the energy of an electron in the atomic orbitals of atom , being attracted by the nuclei of all atoms of the molecule. Likewise,  can be seen as the binding ability between atoms  and .
When either the Sparkle Model or the RM1 model for lanthanides calculations are usually carried out, we obtain the geometry of the complex in its minimum and in its electronic singlet ground state. The luminescence involves absorption of UV photons by their ligands, which go to an excited singlet state, and then, subsequently, go to a lower 14 energy triplet state. The absorbed energy is then transferred to the metal ion which goes to an excited state -in the case of the europium ion, to the 5 D0 state, while the ligands are back to their singlet ground state and the geometry of the complex is, for all practical purposes, the same as the initial one calculated for the complex. Now, what must occur is the decay of the metal ion from its excited state ( 5 D0 for europium(III)) to lower energy level ones. This decay may take two forms: radiatively (from 5 D0 to 7 FJ, J = 0 to 6 for europium) and nonradiatively. Each form is described by its corresponding decay rates, either Arad and Anrad. It is the radiative decay process, which is kinetically governed by which is the one that we are concerned with, via Eqs. (S4) and (S5). The almost essentially electrostatic coordinated bond between the ligands and the metal ion, can be viewed both as a perturbation on the ligands by the metal ion, and as a perturbation on the metal ion by the ligands, making luminescence possible. Likewise, the ligands perturb the metal ion, which makes the radiative process a phenomenon which occurs with the concurrence of the ligands, and more intensely so, with the partnership of their directly coordinated atoms. This cooperation is therefore important in the luminescence process as it is included into in the form of parameters gi and i, which reflect this partnership, as they affect the value of Arad. In this work, as stated in the main article, we model gi and i as resulting from a perturbation by the metal ion on the Sparkle Model or RM1 model for the lanthanides wavefunctions of the ligands.
We will now proceed with deriving the perturbation theory formulas in a way previously introduced by one of us 8 , which will describe the effect of the lanthanide trivalent ion on the coordinating atoms of the ligands as a perturbation.
Accordingly, assuming that (1,2,…,2n) is the zeroth order wavefunctions, from now on represented as As excited state wavefunctions to be considered in the perturbation formulae, we will only take into account those singlet states obtained from single excitations of the ground state determinant 8 , represented by The electronic energies associated with these functions can be obtained from: We will also assume that Koopmans theorem is strictly valid and we will therefore take as the energy difference between the ground state , an approximation that is accurate enough for our purposes. Now, assume that a perturbation H (1) , which can be described as a sum of one-electron operators, l, affects the system: The first order energy becomes In order to obtain the second order correction to the energy, it is necessary to obtain the first order correction to the wave function. From perturbation theory, where the sum in i runs over all occupied orbitals and the sum in a runs over all unoccupied ones.
It is easy to show that The first-order wavefunction then becomes: The second order energy is defined as  i polarizabilities from the semiempirical models We now proceed to further model the effect by the metal ion on the directly coordinated atoms of the ligands, as a second order effect. So, we start by assuming that the complex is composed of two systems: the ligands with their atoms in their positions, represented by L, and the metal ion represented by M, and that each of these two systems obeys their corresponding eigenvalue equations as: We now assume that no energy level of the ligands, . We now further assume that the metal ion will interact independently with each of the directly coordinating atoms of the ligands. Thus, assuming that any atomic orbital of the ligands can interact with any atomic orbital of the metal, this empirical perturbation can be represented 8 by: where  represents any atom belonging to the ligands;  represents the metal ion, and  represents any atoms which may belong to any of the two systems: ligands and metal.
As the perturbation defined above (Eq. (S44)) does not affect each of the two systems, L and M, taken independently, the first order correction to the energies, obtained from Eq.
(S29), can be easily proven to be zero: Accordingly, the first order correction to the wavefunctions of the ligands and of the metal are: The terms that possess integrals over levels of the same molecule are zero for the same reason that the first order correction to the energies are zero. As such, applying Eqs. (S47) and (S48) into Eq.(S33), we obtain By replacing the molecular orbitals by their respective expansion in terms of atomic orbitals, we now have The total interaction energy is the sum of both corrections At this point, we must introduce into the model a postulate that the metal ion is well defined and does not modify itself to adapt to the chemical environment -reflecting the experimental fact that it is relatively insensitive to the environment. So, we will first assume that the second order correction to the metal ion will be a constant, albeit different for each different complex, . Then, we will further assume that since the metal ion is a trivalent species, its dominant interaction with the ligands will be electrophilic, that is, with the occupied orbitals of the ligands. Therefore, we will restrict our sum over index i to the occupied orbitals of the ligands. Moreover, in the case of the Sparkle Model, since orbitals are non-existent, there will be no explicit orbital energy and no linear coefficients. We will thus postulate, as in the previous work by Simas 8  The total interaction energy will now be: As a generalization to an all valence electron method of the corresponding superdelocalizability of Fukui 9 , and as originally introduced by Simas 8  As we carried out research for this article, we also tried to use the superdelocalizability of Lewis 10 and of Simas and Brown 11 , and also the delocalizability of Schüürmann, but they all did not produce good fittings. So, we stayed with the superdelocalizability as defined by Eq. (S55) above.
Hence, we will use as the polarizability i in Eq. (S54), an expression homomorphic with Eq. (S54), a first degree polynomial: with the constants D and C being the same for all directly coordinated atoms i and adjusted in order to reproduce the various experimentally obtained exp   with λ = 2,4. Once again, please note that index i of Eq. (S57) above refers to a directly coordinated atom in agreement with the usual notation employed by lanthanide luminescence theory, and should not be confused with index i of the semiempirical perturbation theory, as in Eqs.