Modeling intramolecular energy transfer in lanthanide chelates: A critical review and recent advances

Chapter 310

Modeling intramolecular energy transfer in lanthanide chelates: A critical review and recent advances Albano N. Carneiro Netoa,*, Ercules E.S. Teotoniob, Gilberto F. de Sa´c, Hermi F. Britod, Janina Legendziewicze, Luı
s D. Carlosa, Maria Claudia F.C. Felintof, Paula Gawryszewskae, Renaldo T. Moura Jr.g, Ricardo L. Longoc, Wagner M. Faustinob and Oscar L. Maltac,* a Physics Department and CICECO—Aveiro Institute of Materials, University of Aveiro, Aveiro, Portugal b Department of Chemistry, Federal University of Paraı
ba, Joa˜o Pessoa, Paraı
ba, Brazil c Department of Fundamental Chemistry, Federal University of Pernambuco, Recife, Pernambuco, Brazil d Institute of Chemistry, University of Sa˜o Paulo, Sa˜o Paulo, Sa˜o Paulo, Brazil e Faculty of Chemistry, University of Wrocław, Wroclaw, Poland f Institute of Energy and Nuclear Research, Sa˜o Paulo, Sa˜o Paulo, Brazil g Department of Chemistry and Physics, Federal University of Paraı
ba, Areia, Paraı
ba, Brazil * Corresponding authors: e-mail: [email protected]; [email protected]

Chapter Outline 1 Introduction 2 Theoretical background 2.1 4f-4f Intensity parameters 2.2 Intramolecular energy transfer 2.3 Assigning the ligand state as donor or acceptor 2.4 Rate equations 3 Relevant photophysical properties 3.1 An important remark concerning intensity parameters

56 58 58 62 77 78 84

84

3.2 Experimental determination of the ligand donor states 3.3 Ligand-to-metal charge transfer (LMCT) states 3.4 Emission quantum yield 4 Modeling and calculations 4.1 Molecular structures 4.2 Excited states 4.3 Charge factors and effective polarizabilities 4.4 Computational programs 4.5 Modeling procedures

Handbook on the Physics and Chemistry of Rare Earths, Vol. 56. https://doi.org/10.1016/bs.hpcre.2019.08.001 © 2019 Elsevier B.V. All rights reserved.

87 89 93 98 100 109 111 114 116

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5 Selected cases 118 5.1 Energy transfer via singlet 119 5.2 Experimental energy transfer rates 125 5.3 Application of the theory: An example 128 6 Challenges and perspectives 132 Acknowledgments 134

Appendix A Radial integrals values Appendix B The ligand matrix element for the exchange mechanism References

134

135 138

1 Introduction In 1942 Weissman demonstrated that the strong red emission in certain europium coordination compounds originated from non-radiative intramolecular energy transfer from the organic ligands to the trivalent europium ion (Eu3+), under excitation wavelengths at which these ligands strongly absorbed radiation in the UV region [1]. This demonstration was an extraordinary achievement in the area of electronic spectroscopy of lanthanide coordination compounds with organic ligands, by showing that the 4f-4f luminescence could become much more efficient by exciting at the ligands instead of direct excitation at the intraconfigurational 4f levels of the lanthanide ion. The ligands (donors) may act as efficient sensitizers for the lanthanide (acceptor) luminescence through a process known as Intramolecular Energy Transfer (IET). This process is highly dependent on the characteristics of the ligands and lanthanide ion (Lnz+), namely on the composition of the coordination polyhedron, on the distance between the donor and acceptor states, and on the energy mismatch conditions (resonance conditions) between the energy levels of the ligands and of the lanthanide ion, which are considered as an independent system. In other words, the energy levels of the ligands are essentially localized on the organic part of the compound, whereas the 4f energy levels are highly localized on the lanthanide ion. Some years later (1946), F€ orster developed a theory for non-radiative energy transfer between molecules, thus an intermolecular energy transfer, based on the dipole-dipole interaction between the electronic densities of two molecular species [2], a mechanism known in the literature as F€orster (or Fluorescent) Resonant Energy Transfer (FRET). This theoretical approach was later transposed and expanded by Dexter in the early 1950s to solid state materials doped with lanthanide and transition metal ions, including the dipole-dipole and dipole-quadrupole mechanisms for the transfer rates and, mainly, a new mechanism for short-range donor-acceptor distances known as exchange or Dexter mechanism, often called through-bond IET [3]. Kushida, in the 1970s, developed detailed expressions specifically for nonradiative energy transfer rates between lanthanide ions taking into account the dipole-dipole, dipole-quadrupole and quadrupole-quadrupole mechanisms [4]. A relevant outcome of this treatment was the selection rules on the total angular momentum J and, moreover, these expressions involved, for the first time,

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the so-called Judd-Ofelt intensity parameters for the forced electric dipole 4f-4f transitions, the quadrupole-quadrupole mechanism being excepted. These approaches for energy transfer rates between lanthanide ions became highly useful in the description of several phenomena such as upconversion (discovered by Auzel and Feofilov in 1966 [5–7]), cross relaxation, energy migration, and transient luminescence [7]. However, since the elucidation by Weissman in 1942 of IET in lanthanide complexes with organic ligands, no detailed theoretical treatment on this subject has been developed for a long time, perhaps due to the complexity involving a molecular donor and a lanthanide ion acceptor that are of completely different nature, which imposes considerable theoretical and computational challenges, in contrast to the case of ion-ion energy transfer. Despites these challenges, a comprehensive theoretical approach started being developed in the mid-1990s [8–11]. This treatment of ligand-to-lanthanide transfer rates, including back energy transfer rates, continued to be improved by new developments and applications of computational methods, providing a useful tool to rationalize, understand and predict the luminescent properties of lanthanide complexes and materials [12–20]. In fact, this treatment has been quite successful because, in addition to estimating the transfer rates, selection rules, and transfer pathways, it could also be applied in the design and application of novel and efficient luminescent systems [1–3,21–35]. Nevertheless, several aspects in the description of IET in these luminescent systems need deeper clarification, since they are frequently confusing and/or misunderstood in the literature. Some of these aspects involving IET are: (i) evaluation of matrix elements for the dipole-multipole and exchange mechanisms, (ii) selection rules and pathways, (iii) definition of emission quantum yields and emission efficiencies, (iv) rate equations, (v) decay lifetimes, (vi) appropriate use of the intensity parameters, emphasizing that only the contribution from the forced electric dipole mechanism (Judd-Ofelt theory) should be taken into account, and (vii) discussion and rationalization of the reasons why IET rates involving triplet (T1), singlet (S1), and ligandto-metal charge transfer (LMCT) states, might have considerably different values (in some cases, by several orders of magnitude) as found in the literature. These are the focus of the present chapter, keeping in mind the limitations of the theoretical and computational approaches, aiming at facilitating its proper use and application and, of course, to encourage further developments of the treatments of IET in coordination compounds of lanthanide ions. Even though it is a complex subject from the theoretical point of view, it is possible to provide expressions and computational procedures that might be not so complicated. The chapter has, therefore, a clear theoretical bias, using experimental data and application of the theory, whenever appropriate for the sake of illustration, and does not aim at providing an exhaustive overview on IET and luminescence processes in lanthanide chelates. It should be mentioned that a substantial material on this exciting and ever growing subject is available in the literature [17,36–38].

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FIG. 1 Schematic general organogram of the highlights and main goal to be presented and discussed in this chapter.

The organogram in Fig. 1 intends to very briefly summarize and highlight the main steps and general goal that will be discussed in this chapter. We emphasize that the CGS unit system is adopted here. Energy unit may also be used in wavenumbers (cm
1).

2 Theoretical background The present section is focused on the theoretical background of the mechanisms of non-radiative intramolecular energy transfer in lanthanide chelates. The following subsections will present and call attention to important aspects that might have been overlooked in the literature when donor-acceptor transfer rates are evaluated. These aspects (the use of Judd-Ofelt intensity parameters, the inclusion of shielding factors, and selection rules) might lead to orders of magnitude differences in calculated IET rates. The independent systems model, in which the total donor-acceptor wave functions are assumed to be a simple product of the individual donor and acceptor wave functions, and the bipolar expansion for a two-center interacting electronic clouds, as defined for example in Ref. [39], are used here as in the previous works [35].

2.1 4f-4f Intensity parameters The Ωλ (λ ¼ 2, 4, and 6) parameters are the well-known intensity parameters within the Judd-Ofelt theory. From them, information can be extracted about the chemical environment around the Lnz+; in absence of a center of inversion at the Lnz+ position, these parameters contain two main contributions: the forced electric dipole (FED), from the original Judd-Ofelt theory [40,41], and the dynamic coupling (DC) mechanisms [42].

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Jørgensen and Judd (1964) introduced the DC mechanism 2 years after the Judd-Ofelt theory. At that moment, these authors related the behavior of the so-called hypersensitive transitions to the pseudoquadrupole mechanism [42], an alternative description to explain some transitions that do not comply with the Judd-Ofelt theory (FED). One decade later, the discussion about those hypersensitive transitions was reopened by Mason, Peacock, and Stewart [43]. They applied what was called Dynamic Coupling terms of H€ohn and Weigang [44], introducing them as a mechanism in the treatment of 4f-4f intensities. In 1979, Judd demonstrated that the equations obtained by Mason et al. are equivalent to his 1964 publication with Jørgensen [45]. The FED mechanism occurs due to a small mixture of electronic configurations with opposite parities (f-d and f-g mixing), relaxing the Laporte’s rule as originally dealt with in the Judd-Ofelt theory [40,41]. This theory has been very important in the interpretation of the 4f-4f transition intensities. Since then, besides the classical point-charge model by Bethe [46], several models became available to describe the ligand field: angular overlap model (1963) [47,48], exchange charge model (1970) [49,50], superposition model (1971) [51], simple overlap model (1982) [52,53], point-multipolar electrostatic model (1982) [54], covalo-electrostatic model (1985) [55], and density functional (DFT) based ligand field theory (LF-DFT) [56–58]. The parameters Ωλ, Eq. (1), only appear from the odd part of the ligand field, expressed by the odd rank ligand field parameters γ tp and the odd rank components Γtp of the DC Hamiltonian, while the Stark energy levels splitting are defined by the even part of the ligand field expressed by the so-called ligand field parameters Bkq [59,60].   X Bλtp 2 DC Ωλ ¼ ð2λ + 1Þ , Bλtp ¼ BFED (1) λtp + Bλtp 2t + 1 t, p where the main contributions (FED and DC) can be calculated by Eqs. (2) and (3): 2  t + 1 r Θðt, λÞγ tp ΔE   D   E ðλ + 1Þð2λ + 3Þ 1  λ  2 r ð1
σ λ Þ f CðλÞ f Γt δt, λ + 1 BDC ¼
λtp p ð2λ + 1Þ BFED λtp ¼

(2)

(3)

where the indexes t and p are the rank and component, respectively, that define the complex conjugate of the spherical harmonics (Yt∗ p ) inside the quantities γ tp and Γtp. Θ(t, λ) is a numerical factor that depends only the Lnz+ ion, hrt+1i and hrλi are radial integrals. In the DC mechanism, δt, λ+1 is the Kronecker’s delta function and (1
σ λ) are the shielding (or screening) factors. The ΔE is the energy difference 4f n– 4f n
15d of the Lnz+ ion, as given by analogy with the average energy denominator method of Bebb and Gold [61,62].

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FIG. 2 Illustration on how covalency effects are taken into account in the BOM for the DC mechanism. Adapted with permission from R.T. Moura Jr, A.N. Carneiro Neto, R.L. Longo, O.L. Malta, On the calculation and interpretation of covalency in the intensity parameters of 4f–4f transitions in Eu3+ complexes based on the chemical bond overlap polarizability, J. Lumin. 170 (2016) 420–430, © 2016 Elsevier Science B.V.

Recently, Moura et al. [63] introduced covalency effects also in the Γtp term of the DC mechanism via the bond overlap model (BOM), as shown in Fig. 2. This was done with the partition of the polarizability (α) into two contributions: the overlap polarizability (αOP) and a ligand effective polarizability (α0 ). With the inclusion of covalency in the DC mechanism, the screening factors (1
σ λ) must be removed from Eq. (3), because shielding is naturally taken into account in αOP through the overlap integral ρ (Eq. 5). The BDC λtp contribution is then given by:  1  E ðλ + 1Þð2λ + 3Þ 2  λ D  r f CðλÞ  f ð2λ + 1Þ h i  1 X 2β t + 1 α 0

+ α OP, j 2 j j 4π t Yp,∗ j δt, λ + 1  t + 1 2t + 1 Rj j

BDC λtp ¼


(4)

where hf kC(λ)k fi are monoelectronic reduced matrix elements for f orbitals, that is, f ¼ 3. The αOP can be calculated by [64]: αOP ¼

e 2 ρ 2 R2 2Δε

(5)

where ρ, Δε, and R are the overlap integral between the valence shells, the excitation energy corresponding to a diatomic-like (pair Ln–Ligating atom),

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and the distance of the Lnz+ ion to the ligating atom (ion) in the first coordination sphere, respectively, and e is the elementary charge. The 4f-4f intensities are treated through the odd part of the ligand field (t ¼ 1, 3, 5 and 7). Therefore, the radial integrals hrt+1i of interest to the FED mechanism (Eq. 2) are hr2i, hr4i, hr6i and hr8i. The values of these integrals (with exception of hr8i) can be found in Ref. [65]. In this chapter, we present the values of hr8i obtained by extrapolation (Appendix A). As mentioned before, Θ(t, λ) are numerical factors that depend on the Lnz+ ion and they are obtained by:   D   ED  ðtÞ  E f t g ð1Þ     Θðt, λÞ ¼ f C g g C f 1 f λ   D  E D    E f t d + ð1
2δt Þ f Cð1Þ d d CðtÞ f (6) 1 f λ where the first term in Eq. (6) contains the  f-g mixingcontributions, and j1 j2 j3 is a 6-j symbol, the last term contains the f-d mixing ones, j4 j5 j6 and δt represents the relative contribution of the core (3d and 4d) orbitals to the Ωλ [66], X 1 (7) δt ¼ h4f j rj n0 d ihn0 dj r t j 4f i t + 1 h4f j r j 4f i n0 ¼3, 4 Electronic structure calculations for the Eu3+ ion at the Hartree-Fock level of theory yielded δ1 ¼ 0.539, δ3 ¼ 0.223, δ5 ¼ 0.082 and δ7 ¼ 0.000 [67]. The δt involve radial integrals, whose values can be used for other Lnz+ ions without significant inconsistencies. It is important to take into account that the core contributions (4f-3d and 4f-4d) have higher energies as compared to the 4f-ng excitations. According to the Simple Overlap Model (SOM, [52,53]), the γ tp in Eq. (2) is given by:

 

t + 1 Ypt∗ θj , φj 4π 1 X 2 t 2 e gj ρj 2βj (8) γp ¼ 2t + 1 Rtj + 1 j where g is a charge factor located at R/(2β), where β ¼ 1/(1 ρ), and the plus sign (+) is used when the ionic radius of Lnz+ is greater than the ionic radius of the ligating atom or ion, whereas the minus sign (
) holds for the inverse situation [53]. The SOM considers covalency effects through the product gρ(2β)t+1. The Ωλ parameters can thus be theoretically calculated from structural and nature of the chemical environment data in the Lnz+ chelates (see Section 4). It is important to emphasize again (as will be discussed later) that only the forced electric dipole mechanism (FED) should be considered in the evaluation of IET rates.

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The intensity parameters (including both FED and DC mechanisms) define the dipole strengths SED, and SMD, which can be calculated exactly when they are allowed by parity: D X   E2 1 Ω λ ψ 0 J 0  U ð λÞ  ψ J (9) SED ¼ 2J + 1 λ¼2, 4, 6 ħ2 1 2 (10) hψ0 J 0 kL + 2Skψ J i 4m2 c2 2J + 1 where m is the electron mass, J is the total angular momentum quantum number of the rare earth ion, and ψ J stands for the 4f states in the intermediate coupling scheme. The quantity hψ0 J0 kU(λ)kψ Ji is a reduced matrix element of the unit irreducible tensor operator U(λ), and the angular momentum operators L and S are in units of ℏ. The radiative spontaneous emission rate is given by: " # 2 4e2 ω3 nðn2 + 2Þ 3 SED + n SMD (11) AJ!J0 ¼ 3ħc3 9 SMD ¼

where n is the index of refraction of the medium and ω is the angular frequency of the J ! J0 transition. The oscillator strength of a J0 ! J transition is expressed as: PJ0 !J ¼

2J + 1 mc2 AJ!J0 2J 0 + 1 2ω2 e2 n2

(12)

We must subtract the magnetic dipole contribution (SMD) in order to obtain experimental Ωλ, when it is necessary.

2.2 Intramolecular energy transfer A Jablonski-type energy level diagram depicted in Fig. 3 will serve as the basis for the discussions on the IET processes developed along this chapter together with the definitions of the relevant channels, energy levels and rates involved. For some specific cases in the following sections, this diagram will be appropriately redrawn.

2.2.1 Energy transfer rates through the Coulomb direct and exchange interactions In the treatment of ligand-Lnz+ energy transfer processes, we consider a model of independent systems, as shown in Fig. 4A and used previously in similar approaches [19,68,69]. It is important to emphasize that the calculation of the matrix elements involving two centers can be performed by two different approaches. The first one requires transformation of both donor and acceptor wave functions to the same center [19,68,69]. The second one, adopted in our works, in a more simplified way, involves the use of the

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FIG. 3 Schematic energy levels diagram involving the ligand and Lnz+ ion states. j S0i and j S1i are the ligand ground and lowest singlet excited states, respectively. j T1i is the lowest triplet ligand level. Abs. is the initial absorption, IC the internal conversion, Fluor. the ligand fluorescence, ISC the intersystem crossing, Phosp. the ligand phosphorescence, Wnrad and Anrad are non-radiative decays between the Lnz+ levels, Em. is the emission, and Arad the spontaneous emission coefficient. WET and WbET are the forward and backward energy transfer rates, respectively.

FIG. 4 (A) Simplified energy level diagram and (B) coordinate systems used to describe the interaction between the electrons involved in the ligand-Lnz+ energy transfer processes.

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bipolar expansion in the independent systems model. In Fig. 4B, the index i refers to electrons of the Lnz+ ion in a coordinate system centered at its nucleus, whereas the index j refers to electrons of the ligand in a coordinate system located at the electronic barycenter of the intraligand states. The lanthanide ion states j ψJMi are described in the intermediate coupling scheme. The two coordinate systems are separated by a distance RL, see Eq. (87) in Section 5.2. Within an independent systems model and neglecting vibrational couplings (weak coupling limit), the initial jii and final j f i states of the ligand-Lnz+ pair, as indicated in Fig. 4A, are written as: jii ¼ j ΨN
1 Π∗ ij ψJMi

(13)

j f i ¼ jΨN
1 Πij ψ 0 J 0 M0 i

(14)

and where j ΨN
1Πi represents a determinant describing the ground state of the N electrons in the ligand and j ΨN
1Π∗i that of an intraligand excited state that corresponds to a single electronic excitation from the molecular spin-orbital Π to Π∗. This representation imposes a single-determinantal description of both ground and excited states of the ligand employing the same spin-orbitals in the j ΨN
1i determinant. The states j ψJMi and j ψ 0 J0 M0 i are 4f states of Lnz+ for which the total angular momentum quantum number J is considered approximately a good quantum number, as well as its component M, and these states are described in the intermediate coupling scheme, as mentioned previously and will be recalled later, due to the importance of this coupling scheme. The interaction Hamiltonian between the two groups of electrons is given by H ¼ HC + Hex ¼

X e2 i, j

rij




X e2 i, j

rij

Pij

(15)

where HC and Hex are the direct Coulomb and exchange interactions, respectively, and Pij is the exchange operator [70]. Starting from the bipolar expansion, in terms of irreducible tensor operators [39], the direct Coulomb interaction can be expressed as: ðk1 + k2 Þ  1  X ð2k1 + 2k2 + 1Þ! 2 k1 k2 k1 + k2 C
Q ðLÞ ð
1Þk1 H C ¼ e2 q q0
Q ð2k1 Þ!ð2k2 Þ! RLk1 + k2 + 1 i , j , k 1 , k2 q, q0 , Q k2 ð k2 Þ  rj Cq0 ð jÞrik1 Cðqk1 Þ ðiÞð1
σ k1 Þ (16) where the polar angles in the Racah tensors [71] Cðk1 + k2 Þ (L) and C(k1)(i) refer to the coordinate system centered at the Lnz+ ion and the polar angles in

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C(k2)( j) refer to the coordinate system centered at the barycenter of the intraligand states, σ k1 is a factor that describes the shielding effects produced by the 5s2 and 5p6 subshells on the 4f electrons. In Eq. (16),  it is assumed that j1 j2 j3 the distance RL is greater than ri and rj, and the quantity is a m1 m2 m3 3-j symbol [71]. The exchange operator Pij is given by the Dirac identity [70]: 1 Pij ¼ + 2^ si  s^j 2

(17)

where b s is the spin operator. This identity is accurate only in an orthogonal spin-orbital basis. In the case of functions belonging to different centers, because of the neglect of higher-order permutations and the requirement that the functions be orthogonal, it becomes approximated [19,68,69]. Notice that despite the operators acting only on the spin coordinates, it can be shown [19,68–70] that this identity is equivalent to the permutation of electrons i and j within the (orthogonal) spin-orbital basis. The factor 12 in this identity, Eq. (17), can be incorporated into HC, thus leading to the following representation of the exchange interaction: X s^i  s^j 0 ¼
2e2 (18) Hex rij i, j By using the bipolar expansion [39] at the same coordinate system as before, the exchange interaction, Eq. (18), becomes:    X k1 + m ð2k1 + 2k2 + 1Þ! 1 k1 k2 k1 + k2 0 2 2 ð
1Þ Hex ¼
2e q q0
Q ð2k1 Þ!ð2k2 Þ! i, j, k1 , k2 q, q0 , Q ðk1 + k2 Þ C
Q ðLÞ ðk Þ (19)  k1 + k2 + 1 rjk2 s
m ð jÞCq0 2 ð jÞrik1 sm ðiÞCðqk1 Þ ðiÞð1
σ k1 Þ RL where sm is a spherical component of the spin operator. According to Fermi’s golden rule, the energy transfer rate between the initial j ii and final j fi states, Eqs. (13) and (14), will be, in the weak coupling limit, given by: WET ¼

2π jh f j Hj iij2 F ħ

(20)

where the factor F, which is associated with the spectral overlap between the functions that describe the spectral profiles of the emission and absorption bands of the donor and acceptor states, respectively, determines the dependence of the energy transfer rate with the resonance energy conservation

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condition, as will be discussed in Section 2.2.2. Substituting Eq. (15) into Eq. (20) the energy transfer rate becomes:   2  2π 2π  HC 0 2 WET ¼ jh f j HC + Hex j iij F ¼  f j HC
+ Hex j i  F 2 ħ ħ ¼

 2   π 2π  2π 0 0 j i  F + h f j HC j ii f j Hex ji F jh f j HC j iij2 F +  f j Hex 2ħ ħ ħ

(21)

Notice that the bipolar expansions yielding Eqs. (16) and (19) for HC and 0 , respectively, often converge rapidly with the increase of k1 and k2, Hex because of their dependence with (RkL1+k2+1)
1 beyond the first coordination sphere. Based on this fact, we consider only the cases for k2 ¼ 0 and k2 ¼ 1 from the intraligand states. At first, the same could have been done for the expansion involving the Lnz+ states, i.e., considering only the cases for k1 ¼ 0 and 1. However, the well-known effect of mixing excited configurations of opposite parities with the fundamental configuration [Xe]4f n, promoted by the odd components of the ligand field [40,41], may be large enough to make contributions from k1 ¼ 2, 4 and 6 comparable to the one for k1 ¼ 1. Thus, the values of k1 ¼ 0, 1, 2, 4, and 6 will be considered. For k2 ¼ 0, the direct and exchange contributions, Eqs. (16) and (19), respectively, reduce to: HC ¼ e2

X

ð
1Þq

i, j, k1 , q

ðk Þ

ðk Þ

C
q1 ðLÞrik1 Cq 1 ðiÞð1
σ k1 Þ RLk1 + 1

(22)

and 0 Hex ¼ 2e2

X

ð
1Þm
q

i, j, k1 , q, m

ðk Þ

ðk Þ

C
q1 ðLÞCq 1 ðiÞrik1 sm ðiÞs
m ð jÞð1
σ k1 Þ RLk1 + 1

(23)

In this case, the matrix elements of the direct (Coulomb) interaction in Eq. (22) with respect to the initial j ii and final jf i states become h f j HC j ii ¼ e2 N hΠj Π∗ i

X k1 , q

ð
1Þq

E C
q1 ðLÞ D 0 0 0 ðk1 Þ ψ J M j D j ψJM ð1
σ k 1 Þ q RLk1 + 1 ðk Þ

(24) where N is the number of electrons in the ligand, which arises from the 1) is normalization of the j ΨN
1i determinant, and the tensor operator D(k q defined as X k Dðqk1 Þ ¼ ri 1 Cðqk1 Þ ðiÞ (25) i

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whereas the exchange operator in Eq. (23) has the following matrix elements 

ðk Þ X  C
q1 ðLÞ 0 f j Hex j i ¼ 2e2 ð
1Þm
q k1 + 1 hΨN
1 Πj s
m ð jÞj ΨN
1 Π∗ i RL Di, j, k1 , q, m E 0 0 0 k1 ð k1 Þ  ψ J M j rj Cq ðiÞsm ðiÞj ψJM ð1
σ k1 Þ

(26)

Taking into account the orthogonality of the intraligand states results in the matrix element of Eq. (24) being identically zero. On the other hand, the matrix elements of the spin operators involving the intraligand states in pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (26) are proportional to SΠ ðSΠ + 1Þð2SΠ + 1ÞδSΠ∗ ,SΠ , where SΠ and SΠ∗ are the values of the total spin for the ligand ground state and for its excited state, respectively. For closed-shell ligands, SΠ ¼ 0, and therefore, the contribution via the exchange mechanism for k2 ¼ 0 will be zero. This is valid for most cases of practical interest because radical ligands (open-shell species) are not commonly used due to their high reactivity and kinetics lability. Thus, only k2 ¼ 1 needs to be considered in Eq. (16), so that the matrix element of the direct Coulomb interaction becomes ðk1 + 1Þ  1  X 2 k C
Q ðLÞ k1 ð2k1 + 3Þ! 1 1 k1 + 1 2 ð
1Þ h f j HC j ii ¼ e 0
Q q q ð Þ!2 2k RLk1 + 2 1 k1 , q , q 0 , Q D ED E ð1Þ  ψ 0 J 0 M0 j Dðqk1 Þ j ψJM ΨN
1 Πj Dq0 j ΨN
1 Π∗ ð1
σ k1 Þ (27) Both ground and excited states of the ligand can conveniently be described in a representation that allows the application of the Wigner-Eckart theorem. This can be achieved by representing the states | ΨN
1Πi and | ΨN
1Π∗i as | βΠΓΠγ Πi and | βΠ∗ ΓΠ∗ γ Π∗i, where the set of states specified by γ Π and γ Π∗ form bases for the irreducible representations ΓΠ and ΓΠ∗, respectively, of the symmetry point group of the system. The symbols βΠ and βΠ∗ represent additional quantum numbers needed to specify different states with equal quantum numbers Γ and γ. The tensor operators D(1) q0 can be described as a linear combination [8], defined by a unitary transformation of the components of a tensor T(Γ) that forms the basis for an irreducible representation Γ of the point group of the system, such that ΓΠ∗  ΓΠ ⨂ Γ, namely, X ð1Þ cq0 γ TγðΓÞ (28) Dq0 ¼ γ

Then, from the Wigner-Eckart theorem, we can write D E X


1 ð1Þ ΨN
1 Πj Dq0 j ΨN
1 Π∗ ¼ cq0 γ dΓΠ∗ 2 hβΠ ΓΠ γ Π j βΠ∗ ΓΠ∗ γ Π∗ i γ, β D E    β Π Γ Π j  T ð ΓÞ j β Π ∗ Γ Π ∗

(29)

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where the index β is required if ΓΠ∗ with dimension dΓΠ∗ appears more than once in the direct product ΓΠ ⨂ Γ. The quantities hβΠΓΠγ Π j βΠ∗ ΓΠ∗ γ Π∗i are symmetrical coupling coefficients analogous to the Clebsch-Gordan coefficients in spherical symmetry (or to the 3-j symbols) and thus, form a unitary orthogonal matrix. Substituting Eq. (29) into Eq. (27) and taking its square, summing over the components M, M0 , over γ Π and γ Π∗, and dividing by the degeneracies G and (2J + 1) of the states j Ψ N
1Π∗i and j ψJMi, respectively, the following expression for the square of the matrix element of the interaction HC between the donor and acceptor states is obtained: ðk + 1Þ ð k + 1Þ X C
Q1 ðLÞC
Q1 ðLÞ∗ ð1
σ k1 Þ2 e4 ð k 1 + 1Þ h f j H C j ii ¼

k1 + 2 2 Gð2J + 1Þ k , Q RL 1 E2 D   E2 D    ψ 0 J 0 Dðk1 Þ ψJ βΠ ΓΠ T ðΓÞ ; βΠ∗ ΓΠ∗ 2

(30)

The quantity e2hβΠΓΠ jj T(Γ)jj βΠ∗ ΓΠ∗i2 can be identified as the dipole strength SL of the electronic transition j ΨN
1Π∗i ! j ΨN
1Πi, which can be measured spectroscopically or calculated with quantum chemical methods. P ð k + 1Þ ð k + 1Þ Using the identity Q C
Q1 ðLÞC
Q1 ∗ ðLÞ ¼ 1, Eq. (30) is rewritten as:  0 0  ðk Þ  2 X ψ J D 1 ψJ ð1
σ k1 Þ2 e2 SL ð k 1 + 1Þ h f j HC j ii ¼

k1 + 2  2 Gð2J + 1Þ k ¼2, 4, 6 RL 1 2

(31)

Because the reduced matrix element hψ 0 J0 jj D(k1)jj ψJi are proportional to hf jj C(k1)jj f i, the values of k1 are limited to 2, 4, and 6 in Eq. (31). Therefore, the selection rules in the quantum numbers J established for the dipolemultipole mechanism are j J
J0 j  k1  J + J0 (except for J ¼ J0 ¼ 0). In the treatment of the contribution of the terms corresponding to k2 ¼ 1, !

two approximations have been used. The first one was to take the vector R L as being collinear to the z axis of the coordinate system centered at the Lnz+ ion (Fig. 4B). Thus, only the cylindrical component (Q ¼ 0) of the tensor operapproximation takes the values of the ator C(2)(L) is required. The  second 1 1 2 1 squares of the 3-j symbols as an average value equal to 15 . With q
q 0 these approximations applied to Eqs. (25) and (27), we obtain for k1 ¼ 1, after the usual configuration interaction procedure like in the Judd-Ofelt theory [8]: h f j H C j i i2 ¼

 E2 e2 SL ð1
σ 1 Þ2 X FED D 0 0  U ðλÞ ψJ Ω ψ J λ Gð2J + 1ÞR6L λ¼2, 4, 6

(32)

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where hψ 0 J0 jjU(λ)jj ψJi2 are the squares of the reduced matrix elements of the unit tensor operators U(λ). The selection rules, in this case, are also are those from j J
J0 j  λ  J + J0 (except for J ¼ J0 ¼ 0). The parameters ΩFED λ the Judd-Ofelt theory, considering only the contribution from the FED mechanism, without the DC contribution, as mentioned in Section 2.1, and here emphasized to avoid the misconception of using experimental Ωλ, since both mechanisms are present in the experimental Ωλ values and cannot be distinguished [12,72]. For the exchange interaction corresponding to k2 ¼ 1, in Eq. (19), only the terms corresponding to k1 ¼ 0 were taken into account, which reduces the expression for this interaction to: 0 ¼
2 Hex

0 e 2 ð1
σ 0 Þ X ð1Þ ð1Þ ð
1Þ1
q + m C
q0 ðLÞrj Cq0 ð jÞs
m ð jÞSm R2L j, q0 , m

(33)

where the total spin operator Sm with spherical component m corresponds to the sum over the 4f electrons of the spin operators sm. As in the previous case, ! RL

is considered collinear to the z axis of the coordinate system centered at the Lnz+ ion. Using the Wigner-Eckart theorem it can be shown that 

 0  2e2 ð1
σ 0 Þhψ 0 J 0 kSkψJ i X J 1 J m + J 0
M0 0 f j Hex ji ¼ ð
1 Þ
M0 m M R2L m   * + X    ð1Þ  ΨN
1 Π  rj C0 ð jÞs
m ð jÞ ΨN
1 Π∗  j 

(34)

Taking the square of the matrix elements in Eq. (34), summing over M and M0 , and dividing by the degeneracies (2J + 1) and G, yields: 

2 4e4 ð1
σ 0 Þ2 hψ 0 J 0 kSkψJ i2 0 f j Hex ji ¼ 3Gð2J + 1ÞR4L   * +2 X  X   ð1Þ ∗  ΨN
1 Π  r C ð jÞs
m ð jÞ ΨN
1 Π  j j 0  m

(35)

where it was considered that a ligand-to-Lnz+ energy transfer step departs from a one-level component of the donor state and that the donor level components are approximately equally populated, thus generating the degeneracy G. The reduced matrix element of the spin operator S for the lanthanide ion establishes the following selection rules: J
J0 ¼ 0,  1 (except for J ¼ J 0 ¼ 0). In summary, it was shown that for k2 ¼ 0, the matrix elements of the direct and exchange Hamiltonians are zero, whereas for k2 ¼ 1 and k1 ¼ 0 the selections rules are complementary, that is, when h f j HC j ii is nonzero then

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0 hfj Hex j ii ¼ 0 and vice-versa. Thus, for the dominant contributions from k2, the 0 cross term h f j HC j iih f j Hex j ii in Eq. (21), becomes zero. The energy transfer rates via direct and exchange mechanisms, within the validity of the aforementioned approximations, are, respectively, simplified to      2    0 0 0 1   2π  C     ∗ WET ¼ WET ¼  ΨN
1 Π  ψ J M  HC ψJM  ΨN
1 Π  F (36) ħ 2

if j J
J0 j  k1  J + J0 (except for J ¼ J 0 ¼ 0; k1 ¼ 2, 4, 6), or ex ¼ WET ¼ WET

   2 2π  0 j ψJM j ΨN
1 Π∗  F ΨN
1 Πj ψ 0 J 0 M0 j Hex ħ

(37)

if J
J0 ¼ 0,  1 (except for J ¼ J 0 ¼ 0). By combining Eqs. (31), (32) and (36), the direct Coulomb interaction rate becomes D   E2 2π e2 SL F X C ¼ Λλ ψ 0 J 0 U ðλÞ ψJ (38) WET ħ Gð2J + 1Þ λ¼2, 4, 6 where  ð1
σ 1 Þ2 Λλ ¼ 2ΩFED λ

!  λ 2 D  ðλÞ  E2 1 ð λ + 1 Þ 2 + r f C f ð1
σ λ Þ 2 R6L ðRλL + 2 Þ

(39)

corresponding to the energy transfer between the intraligand and the Lnz+ ion excited states, due to the direct Coulomb interaction. It is important to remind, as mentioned before, that the dipole strength SL contains the square of the elementary charge (e2). It should be called attention to the fact that in Eq. (39) the last term in the right-hand side is by far the dominant one (orders of magnitude) for λ ¼ 2 [12]. Also important is to notice the difference of this special case in comparison with the multipolar expansion of values of B(k) q ligand field parameters and Ωk intensity parameters. Ligand field interactions are ˚ ). extremely close to the Lnz+ ion (distances of the order of 2.5 A Inserting Eq. (35) into Eq. (37) yields: ex ¼ WET

8π e4 ð1
σ 0 Þ2 F 0 0 2 hψ J kSkψJ i 3ħ R4L Gð2J + 1Þ   * +2 X  X   ð1Þ ∗  ΨN
1 Π  r C ð jÞs
m ð jÞ ΨN
1 Π  j j 0  m

(40)

which corresponds to the energy transfer rate between the intraligand excited state and the Lnz+ ion, due to the exchange interaction. The matrix elements from the ligand, appearing in Eqs. (34), (35) and (40), involve the coupled electric dipole and the spin operators, which can be evaluated by quantum chemical methods analogous to the ones used to evaluate spin-orbit matrix

Modeling intramolecular energy transfer in lanthanide chelates Chapter

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elements in organic molecules [73]. It is important to emphasize that these coupled operators break down the usual selection rules on the multiplicities of the initial and final states of the ligand. A simple demonstration of how this matrix elements may be expressed in terms of monoelectronic molecular spinorbitals is provided in Appendix B and typical values will be discussed later. In the approach proposed in 1997 [10], the shielding effects were not considered in the exchange interaction. As a result, the calculated values of the transfer rates due to the exchange interaction were significantly overestimated. In subsequent work, the same authors included the shielding effect in the bipolar expansion [11]. In a revisited article [12], it was proposed to replace the screening factor (1
σ 0)2 in Eq. (40) by h4f j Li4, where h4f j Li represents the overlap integral between a 4f orbital and a molecular orbital describing the donor state (orbital Π∗ in Fig. 4A). This ad hoc approximation was based on the well-known Mulliken approximation [74,75]; however, at the current level of the theoretical development, we consider that the formulation presented in Eq. (40) is the most consistent one, when appropriate shielding parameters are employed. To maintain consistency with some ligand-field models [9,52,53], the following expression has been proposed to estimate these parameters: ð1
σ k1 Þ ¼ ρð2βÞk1 + 1

(41)

which, as mentioned in the reference [9] is appropriate for k1  2, but not for k1 > 2 as it was suggested inadvertently in reference [12]. Here we propose that the (1
σ 0) factor be treated as an adjustable parameter with σ 0 > 0.9. This is based on the fact that the observational relationship given in Eq. (41) works quite well for k1 ¼ 2 (as in the SOM model for the ligand field [52,53]). However, for k1 < 2, there are no detailed calculations in the literature and so, for k1 ¼ 0 (a monopole), the value of (1
σ 0) may differ considerably from that provided by Eq. (41). Notice that there are alternative expressions for the exchange contribution to the energy transfer rate between lanthanide-lanthanide ions [19,68,69] and ligand-lanthanide ions [19,68,69] available in the literature. These approaches are based upon the bipolar expansion around a single center (e.g., at the lanthanide ion) [19,68,69], which leads to extremely complicated expressions for the transfer rates that are difficult to simplify and to calculate. In addition, apparently, the selection rules that might eventually be extracted from these expressions are not evident. An alternative, which is consistent with the formulations we have been using, is to employ Eq. (41) for k1 ¼ 1 and 2, and the following relationship for the specific case k1 ¼ 0:  Rmin 7=2 ð1
σ 0 Þ ¼ ρ (42) RL

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where ρ is the overlap integral (the same appearing in the BOM and SOM, Eqs. (4), (5) and (8), in Section 2.1, with values ca. 0.05) between the valence shells of the Lnz+ ion and the ligating atom (ion), Rmin (in previous works denoted as R0 [35]) is the smallest distance in the first coordination sphere and RL is the distance from the Lnz+ nucleus to the electronic barycenter of the ligand donor (or acceptor) state. The right-hand side of Eq. (42) contains implicitly the screening effects due to the filled 5 s and 5p sub-shells of the Lnz+ ion and gives the exchange Hamiltonian, in Eq. (33), an appropriate dependence on RL, as it should be expected for a donor-acceptor exchange interaction.

2.2.2 The energy mismatch spectral overlap factor F This factor that appears in the expressions for the transfer rates given in Section 2.2.1 (given in units of erg
1; we are using the CGS unit system, in which ħ ¼ 1.05  10
27 erg s is Planck’s constant divided by 2π) has been used in previous works in the following simplified form: rffiffiffiffiffiffiffiffiffi Δ 2 1 ln 2
ħγL ln 2 e (43) F¼ ħγ L π where it is assumed that ligand bandwidth at half-height, γ L (in s
1), is much larger than the widths of the 4f-4f transitions, γ Ln (in s
1), of Lnz+ ions, i.e., γ L ≫ γ Ln, and Δ (in erg) is the energy difference between donor and acceptor energy levels [8,12]. A more complete formulation of the factor F as a function of temperature is based on the schematic energy level diagrams in Fig. 5 [14,76].

FIG. 5 Schematic configurational coordinate diagrams for direct (left) and reverse (right) thermally activated ligand-to-metal energy transfer processes.

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Energy conservation implies in phonon creation when j ψ 0 J 0 M0 i is below the donor state, and in phonon annihilation in the opposite case. By assuming a Boltzmann distribution for the vibrational components of the electronic states, one may show that [76]: F



X

1
e

m , m 0 , n, n0



 1
e Z 







ħωJ 0 kb T

ħωΠ∗ kb T

!

! e

e









n0 ħωJ 0 kb T

mħωΠ∗ kb T



 Π∗ Π 2 χ m j χ m0

 J J 0 2 ln j ln0

0

∗ 0 0 0 gJn0 ðE0 ÞgΠ m ðEÞδðΔE + mħωΠ
m ħωΠ∗
n ħωJ 0 + nħωJ ÞdEdE

(44) J0 0

Π∗

J where χ m , χ Π m0 , ln, and ln are vibrational components, with frequencies ωΠ, ωΠ∗, ωJ, and ħωJ0 , associated with the electronic states j Πi, j Π∗i, j ψJMi, and j ψ 0 J 0 M0 i, respectively, ΔE is the zero-point energy difference between 0 ∗ the donor and acceptor states, gJn0 (E0 ) and gΠ m (E) are the band profiles of the vibronic components of the 4f and intraligand states, respectively, T is the 0 temperature, and kb is the Boltzmann constant. Assuming that hlJn j lJn0 i ¼ δn, n0 , which is an approximation often used in lanthanide spectroscopy, and that both parabolas shown in Fig. 5 have approximately the same curvature, the F factor simplifies to:

!

ħω ∗ mħωΠ∗ X  Π∗ Π 2
k ΠT
kb T b e 1
e χ m j χ m0 F m, m0 ! 0 # Z "X ħω 0 n ħω 0
k TJ
k TJ 0 ∗ 0 b b e gJn0 ðE0 Þ gΠ  1
e m ðEÞdEdE

n0

 δðΔE + ðm
m0 ÞħωΠ Þ

(45)

where the term in brackets corresponds to the band-shape function for the electronic transition j ψ 0 J 0 M0 i ! j ψJMi, which might be approximated by a Gaussian function. This type of band-shape is also assumed for each vibronic component of the state j Π∗i. Thus, it may be shown that

!

ħω ∗ vħω ∗ D E2 X
k ΠT
k TΠ ∗ Π b b FF 1
e e χΠ (46) v
np j χ v v¼ max ð0, np Þ ΔE where np  ħω is the number of phonons with energy ħωΠ∗ created when Π∗ ΔE  0 or annihilated when ΔE > 0, in the energy transfer process, and ωΠ∗

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(the promoting mode) for the j Π∗i state. F is an overlap integral between two Gaussian functions, and is given by [8,76]: ln 2 1 F ¼ pffiffiffi 2 π ħ γ Ln γ vib

(" 

1 ħγ vib

0 2Δ

)
1 2  # 2 1 2 + ln 2 ħγ Ln !2

1

B C ln 2  B1 C ðħγ Ln Þ2 Δ 2 B " C #  exp B  ln 2
C 2  2 B4 C ħγ Ln 1 1 @ A + ln 2 ħγ vib ħγ Ln

(47)

where γ vib is the bandwidth at half-height (in s
1) of the vibronic component of the intraligand transition, and Δ is the dissonance between ΔE and npħωΠ. The sum in Eq. (46) can be described as a function of the so-called HuangRhys parameters, S, leading to [76]:

v

v
np X e
SΠ h2m + 1i SΠ hmi SΠ h1 + mi

 (48) FF v! v
np ! v¼ max ð0, np Þ where SΠ is the Huang-Rhys parameter corresponding to the intraligand electronic transition and hmi is the average vibrational quantum number given by:

mħω X
k TΠ b me m

hmi ¼ (49) X
mħωΠ e kb T m

2.2.3 Energy transfer rates involving LMCT states In lanthanide compounds, especially those ones with Eu3+, LMCT states can be important channels for the deactivation of both the ligand and lanthanide ion excited states, via energy transfer, thus promoting quenching of the luminescence. In order to describe the effects of the LMCT states on the process of luminescence, models for the description of the energy transfer processes ion Ln3+ ! LMCT (case 1) and ligand ! LMCT (case 2) have been developed [76]. For the treatment of case 1, the following initial and final states are considered: jii ¼ j ΨN
1 ϕ0 ij ψ 0 J 0 M0 i

(50)

j f i ¼ j ΨN
1 4f ij ψJMi

(51)

and

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where j ΨN
1ϕ0i refers to the ground state of the N electrons in the ligand and j ΨN
14fi represents the LMCT state which corresponds to excitation of an electron of the molecular orbital ϕ0 in the ligand to a 4f orbital in the lanthanide ion. The states j ψJMi and j ψ 0 J 0 M0 i are 4f states for which the quantum number of total angular momentum J can be considered approximately a good quantum number, as previously mentioned. By considering that the interaction Hamiltonian between the electrons coming from the ligands and those of the lanthanide ions is similar to Eq. (15). Following the same steps given by Eq. (15) to Eq. (37), it is possible to obtain: D X   E2 2π F C 0 0  ð λÞ  ¼ ωCT ψ J U (52) ψJ WET ħ Gð2J + 1Þ λ¼2, 4, 6 λ where CT means charge transfer, ωCT λ

! 2 4  λ 2 D  ðλÞ  E2 ðλ + 1Þe2 SCT 2 ð2λ + 1Þe h4f j ϕ0 i   ¼ r f C + f ð1
σ λ Þ 2 2 ðRλL + 1 Þ ðRλL + 2 Þ ! 2 4 e2 SCT 2 e h4f j ϕ0 i FED + (53) + 2Ωλ ð1
σ 1 Þ R4L R6L

where SCT is the dipole strength of the LMCT transition. Eq. (52) corresponds to the energy transfer rate between the Lnz+ ion and the LMCT state, due to Coulomb interaction, and: ex ¼ WET

8π e4 ð1
σ 0 Þ2 2 Fhψ 0 J 0 kSkψJ i 3ħ R4L ð2J + 1Þ   * +2  X X   ð1Þ  ΨN
1 4f  r C s ðk Þh Ψ ϕ  k k 0
m  N
1 0 m

(54)

is the energy transfer rate between the Ln3+ ion and the LMCT state due to the exchange interaction. The energy mismatch spectral overlap factor F that appear in Eqs. (52) and (54) is similar to the one presented in either Eq. (43) or Eq. (46), replacing the intraligand transition parameters by those corresponding to the LMCT state [76,77]. A model for ligand ! LMCT energy transfer process was developed by modeling both ligand and LMCT states as combinations of Slater determinants [77]. In this case, the following initial and final states are considered:     1  π ∗ ð1Þ ϕ0 ð2Þ   π ∗ ð1Þ ϕ0 ð2Þ  j ii ¼ (55) 2  π ∗ ð 1Þ ϕ 0 ð 2Þ   π ∗ ð 1Þ ϕ 0 ð 2Þ  and j fi¼

    1  π ð1Þ 4f ð2Þ   π ð1Þ 4f ð2Þ  2  π ð1Þ 4f ð2Þ   π ð1Þ 4f ð2Þ 

(56)

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where ϕ0, π and π ∗ are molecular orbitals of the ligand and 4f is an atomic orbital of the Lnz+ ion, all with α spin, whereas ϕ0 , π, π ∗ , and 4f are orbitals with β spin. The positive signal is used for a singlet state while the negative is used for a triplet state of the component Sz ¼ 0. By the present approach, the exchange interaction is conveniently taken into account without the need to use exchange operators (Eq. 15). Except for the shielding factors, the interaction Hamiltonian is the same as Eq. (16), with the sum over the electrons restricted to the interaction of two electrons. The most relevant difference between the present case and case 1 is that the isotropic term, corresponding to k1 ¼ 0 and k2 ¼ 0, is non-zero and becomes the dominant contribution. Thus, because the states π and π ∗ are orthogonal, the matrix elements of HC can be reduced to: h f j HC j ii ¼

e2 h4f j π ∗ ihϕ0 j π i RL

(57)

Substituting Eq. (57) into Eq. (20), the energy transfer rate between an excited state of the ligand and an LMCT state can finally be given as: L ¼ WCT ¼ WET

2πe4 ħðRL Þ2

h4f j π ∗ i2 hϕ0 j π i2 F

(58)

Taking into account that the bandwidths of both LMCT and intraligand transitions are of the same order, Eq. (43) cannot be used as an approximation to the energy mismatch spectral overlap factor F in Eq. (58). In this case, one should use either Eq. (47), after replacing γ Ln and γ vib by γ L and γ CT, that correspond to the intraligand and LMCT bandwidths, respectively, as well as Δ by the energy difference between the intraligand excited state and the LMCT [76], or: 9 8

i

i
np ðkÞ > > k = <
S h 2m + 1 i
S CT Xe πS X e SCT hmi SCT h1 + mi π

 FF > k! > i! i
np ðkÞ ! ; :i¼ max ð0, np ðkÞÞ k (59) where np(k)  (ΔE + kħωπ)/ħωπ is the number of phonons created when ΔE  kħωπ or annihilated when ΔE kħωπ in the energy transfer process, and SCT is the Huang-Rhys parameter corresponding to the LMCT transition. The factor F in Eq. (59) corresponds to Eq. (47), after replacing γ 4f by γ vib that corresponds to the bandwidth at half height for a vibronic component of the transition π∗ ! π corresponding to an effective promoting normal mode.

2.2.4 A summary of selection rules From the expressions for the direct and exchange IET rates, we can extract the following selection rules as summarized in Table 1. These selection rules are

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TABLE 1 Selection rules from the WET rates. In all these cases J 5 J 0 5 0 is excluded. Mechanism

Lnz+

Dipole-dipole

jJ
J 0 j  λ  J + J 0

Dipole-multipole

jJ
J 0 j  λ  J + J 0

Exchange

ΔJ ¼ J
J 0 ¼ 0,  1

Ligand

SL ! [10
34
10
36]

Transfer through Singlet

SL ! [10
34
10
36]

Transfer through Triplet

SL ! [10
40
10
42]

These selection rules apply as far as J mixing effects are negligible, as usual. SL are in units of esu2 cm2.

very useful in determining relevant IET pathways. Concerning selection rules involving the LMCT, the isotropic contribution is used (Eq. 58), no restrictions are imposed.

2.3 Assigning the ligand state as donor or acceptor Over the last three decades, the determination of the ligand triplet energy position for the IET process in lanthanide complexes has been reported in different ways. Namely, the so-called T1 ! S0 zero-phonon transition energy (at the higher side energy of the phosphorescence band), the triplet (T1) barycenter energy (Fig. 6), or the phosphorescence band maximum. This issue is, of course, dependent on the molecular structure and vibrational modes of the ligands, as well as on the energy mismatch condition Δ. It is not difficult to perceive that most vibronic transitions, as illustrated in Fig. 6, might contribute to the energy transfer process depending on their mismatch with the lanthanide energy difference (E2
E1) and their corresponding FranckCondon factors. Once the density of vibrational states inside a potential curve of a ligand electronic state is very high, it is reasonable to take the energy barycenter or the maximum of the phosphorescence band (they not always coincide) in the evaluation of IET rates. Besides, it is important to emphasize that this argument also applies to the cases of IET from a ligand singlet state and energy back-transfer. However, formally this argument does not exclude the choice of zero-phonon energy, depending on the experimental situation. In fact, all these three quantities have been used in calculations of the IET rates. The experimental estimate of the energy of the donor state will be presented in Section 3.2.

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FIG. 6 Schematic ligand-Lnz+ partial energy level diagram (in the harmonic oscillator representation) relating the triplet (T1) energy barycenter (Eb) and the zero-phonon energy used to evaluate the energy mismatch condition Δ in the spectral overlap factor F (Eq. 43).

2.4 Rate equations The kinetics of an IET process in a given Lnz+ chelate involve a balance between rates of absorption, IET (both forward and back energy transfer) rates, radiative and non-radiative decay rates in both the ligand and in the Lnz+ ion. This kinetics is described by an appropriate system of rate equations based on rates for a microscopic ligand–Lnz+ unit; the ensemble of these units enables the description of macroscopic photophysical properties such as luminescence, intrinsic emission quantum yield (QLn Ln), emission quantum yields (QLLn), lifetimes, risetimes, etc., as defined in the next sections. In the present treatment, each of these units is considered as independent, except in the cases where ligand–ligand interactions between two neighboring units are sufficiently strong to compete with the fast energy distribution within a unit.

2.4.1 Approximate analytical solution Fig. 7 depicts a simplified schematic energy level diagram indicating the main rates that will be considered in the luminescence process. Only energy levels relevant to the IET kinetics are considered. Also, no crossing IET terms (S1 ! j 4i and j 3i ! T1) are considered, as well as no LMCT states for the moment. We now consider that Ni is the normalized population of level j ii, i.e., 0  Ni  1. We further assume that the ground states are very little depleted,

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FIG. 7 Illustrative schematic energy level diagram used to describe the kinetics of IET process. S1 and T1 are the lowest singlet and triplet states, respectively.

so that N0  N5  1. Under these conditions, in the steady-state regime, the appropriate system of rate equations corresponding to the diagram in Fig. 7 is:  dN4 1 T T N4 + W43 N3 + WET ¼
+ WbET N1 ¼ 0 (60) dt τ4  dN1 1 1 T T ¼
N1 + N2 + WbET + WET N4 ¼ 0 (61) dt τT τS  dN2 1 S S N2 + ϕ + WbET ¼
+ WET N3 ¼ 0 (62) dt τS

 dN3 S S ¼
W43 + WbET N3 + WET N2 ¼ 0 dt

(63)

Solving this system of equations we find:

 S S S WET WbET WET 1 S  N2 !

 N2
+ WET N2 + ϕ ¼ 0 N3 ¼

S S τS W43 + WbET W43 + WbET N2 ¼ "



ϕ

1 WS WS S 
ET bET + WET S τS W43 + WbET

#

WT N + bET 4 1 1 T T + WET τS + WET τT τT

N1 ¼ 

N2

From Eq. (60),  T 1 WET N2 WT WT T +  ET bET N4 ¼ 0
+ WbET N4 + W43 N3 +  1 1 τ4 T T τS + WET + WET τT τT

(64) (65)

(66)

(67)

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2 3  T T 6 1 7 W W T N2 ¼ 6
 ET bET 7 W43 N3 +  + WbET (68) 4 5 N4 1 1 τ4 T T τS + WET + WET τT τT 2 2 3 3 ,  S S T T T 6 WET WbET 6 1 7 WET WET WbET 7 T 6 7 7 N4 ¼ 6 4 W + W S  +  1 5N2 4 τ4 + WbET
 1 5 43 T T bET + WET τS + WET τT τT (69) T WET

The emission quantum yield can then be calculated as: QLLn ¼

I54 A54 N4 A54 N4 ¼ ffi I20 ϕN0 ϕ

(70)

In Eq. (70), II5420 is the ratio of the number of emitted photons by the Lnz+ ion, in the j4i ! j5i transition, to the number of absorbed photons by the ligand. The approximation in Eq. (70) is due to the assumption that N0 ffi 1 and because the population of the emitting state N4 is proportional to N2 and thus to ϕ (see Eq. 65), the quantum yield becomes independent of ϕ, and A54 is the spontaneous emission rate of the j 4i ! j5i transition. The lifetime of the emitting level j 4i, in the absence of IET, commonly has a radiative (Arad) and a non-radiative (Anrad) component, as will be discussed in Section 3. However, in the presence of IET (both forward and back transfer) the definition of the lifetime of the emitting level requires special attention, as will also be discussed in Section 3. Another important aspect is the triplet T1 lifetime, which at room temperature might be dominated by the ISC rate T1 ! S0. This rate is not easy to evaluate from molecular spectroscopy. It could be done either by using a vibronic interaction Hamiltonian (ca. 1
5 cm
1) or by a direct spin-orbit matrix element, hS0 j HSO j T1i (ca. 1 cm
1), in the Fermi golden rule, where HSO is the spin-orbit coupling in the molecular ligand generally including the heavy atom effect. The density of states entering the Fermi golden rule formulation may be evaluated in different ways and depends strongly on the energy difference ES
ET (zero-phonon energy) and the temperature [78,79]. In our case, devoted to Lnz+ chelates, several rough estimates of T1 ! S0 ISC rates have shown that they are ≲ 106 s
1 even at room temperature. On the other hand, the ISC rates S1 ! T1 are usually much higher, mainly due to a much smaller energy gap ES1
ET1. They are estimated to be ≳108 s
1 (also at room temperature). We have systematically observed that our descriptions and evaluations of WET and QLLn are very consistent when introducing these orders of magnitude in the rate equations.

Modeling intramolecular energy transfer in lanthanide chelates Chapter

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2.4.2 Numerical solution The kinetics of the luminescence process are given by a set of differential equation relating the populations of each relevant state, X dNi X ¼ kji Nj
kij Ni dt j¼1 j¼1

(71)

where the summations run over all levels of the system, Ni is the population of j ii, and kij is the transition rate from level jii to level j ji. To avoid inconsistencies, the transition rates from and to the same state are zero, that is, kii ¼ 0, or the summations in Eq. (71) are restricted to j 6¼ i. The rate equations, Eq. (71), are simple to interpret because the temporal dependence of a given population is the difference between the summation of all processes contributing to increase and to decrease the level population. As shown previously, these equations have been frequently solved analytically by invoking the steady-state approximation. Alternatively, they can be solved numerically by propagating the populations in time from an initial condition. One of the most robust numerical method for solving coupled differential equations is the fourth-order Runge-Kutta method with an adaptive integration step [80]. This method is simple to implement and even the non-optimized code [80] is very fast, where the rate equations for 12-levels can be solved in a few minutes in a core-i7 notebook; it takes only a few seconds for the more common 7-levels system. In fact, with this method, ca. 40,000 simulations of the rate equations were performed to provide data for chemometric analyses of the dependence of the luminescence quantum yield upon the transition rates for systems with 5- and 8-levels with and without LMCT states [81]. This method has also been used to model hundreds of different conditions for determining the emission lifetime of chelates and other materials containing Eu3+ [82] as well as the quantum yields and emission intensities of luminescent compounds [18,83] and the dynamics of excited states in upconversion systems [84]. In order to employ this method, all that is needed is the initial population of each level and the transition rates between the levels. For most cases, the initial populations are set equal to zero, except the population of the ground state that is set to one, namely, N1(t ¼ 0) ¼ 1 and Ni6¼1(t ¼ 0) ¼ 0. The initial step size can be taken as the inverse of the largest transition rate and the propagation is performed until the steady-state of the populations is achieved, typically a final simulation time of 0.01 s. This steady-state regime provides the populations of the ground and the emitting levels needed to calculate the emission quantum yield. In addition, the temporal dependence of the populations can provide the risetime of any level of interest. The steady-state populations can also be used as the initial populations for determining the emission lifetime, simulating the situation where the excitation source is turnoff.

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FIG. 8 Schematic energy level diagram describing the IET processes leading to the photophysical properties (emission quantum yield, lifetime and risetime) of typical lanthanide chelates and other materials. Attention should be paid to the fact that the system could be the same as in Fig. 7 with the notation being adapted to the numerical solution of the rate equations and the widths of the ligand levels are not emphasized.

As an illustrative schematic example, consider a 6-levels system as used in Fig. 7, for which the levels have been relabeled to be consistent with the summations in the rate equations as shown in Fig. 8. For the sake of illustration, to solve numerically the system described in Fig. 8, the following transition rates were used (in s
1): k13 ¼ 10, k21 ¼ 1, k25 ¼ 107, k31 ¼ 1, k32 ¼ 104, k34 ¼ 103, k35 ¼ 102, k45 ¼ 106, k52 ¼ 103, k56 ¼ 1.2  103, and k61 ¼ 109. The remaining transition rates have zero values. As mentioned before, the initial populations were zero, except N1(t ¼ 0) ¼ 1, the initial time step was 10
10 s and the propagation time was 0.01 s. Notice that because there are distinct ground states for the ligand (level j1i) and the lanthanide ion (level j 6i), a very large transition rate k61 is necessary to maintain the ligand ground state populated. The numerical solution provided the temporal dependence of all levels and Fig. 9 displays the populations of levels j2i, j 3i, and j 5i. It can be observed from Fig. 9 that the populations of all levels reach ss
6 steady-state regime and these populations are Nss 1 ¼ 0.991, N2 ¼ 1.718  10 , ss ss ss ss
4
7
3 N3 ¼ 8.926  10 , N4 ¼ 8.926  10 , N5 ¼ 8.256  10 , and N6 ¼ 1.140  10
7. Thus, the quantum yield can be calculated as, QLLn ¼

I5!6 A5!6 N5 A5!6  8:256  10
3 ¼ 8:331  10
4 A5!6 ¼ ¼ I1!3 k1!3 N1 10  0:991

(72)

If the radiative rate is equal to the total rate, i.e., A5!6 ¼ k56 ¼ 1.2  103, then QLLn ¼ 1.00 (100%), despite the rates k21 and k31 being non-zero. However, the IET rates are much larger than the other transition rates so that the

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FIG. 9 (A) Temporal dependence of the populations of levels j 2i, j3i, and j5i (Fig. 8) during continuous excitation. (B) Temporal dependence of the population of level j 5i after the excitation source is turnoff.

ligand-to-lanthanide ion pathway dominates and the quantum yield can become very close to one if the nonradiative decays are zero: this is the case where QLLn ffi QLn Ln. These steady-state populations can be used as the initial populations in a, simulation with the same transition rates, but with the excitation source turned off, i.e., k13 ¼ 0. All populations decrease to zero, except that of the ligand ground state (j 1i) that reaches 1. The decay of the emitting state is depicted in Fig. 9, with its fit to an exponential function: y ¼ A1e
x/t1 + y0. Fitting the whole decay curve yields t1 ¼ 8.658  10
4  4.723  10
7 s with χ 2 ¼ 1.9481  10
9. Notice, however, that because the populations of levels j 2i and j 3i are still significant at the steady-state, they continue to populate level j 5i, so it takes longer to decay and the beginning of the curve should not be taken into account because it is not an exponential decay. When the fit is performed after the population of level j5i reached 0.006 the value obtained is t1 ¼ 8.343  10
4  1.2675  10
8 s with χ 2 ¼ 6.6079  10
13, which is more consistent with the inverse of the decaying rate 1/k56 ¼ 8.333  10
4 s. In fact, this agreement occurs because the IET rate k25 is several orders of magnitude larger than the back-IET rate k52, so the presence of a non-radiative level coupled to the emitting state does not affect the lifetime [82]. Thus, the photophysical properties of any luminescent chelate or materials can be easily investigated by changing the relevant states describing the system and/or by varying the transitions rates. For the example in Fig. 8, increasing the decaying rates at the ligand levels to k21 ¼ 103 and k31 ¼ 103, while keeping the other rates the same as before, gives the following steady-state populations for
3 ss the ground and emitting states: Nss 1 ¼ 0.9916 and N5 ¼ 7.579  10 , respecL
4 tively. The quantum yield is then QLn ¼ 7.579  10 A5!6, with a maximum value of 0.917 (91.7%) for A5!6 ¼ k56 ¼ 1.2  103. So, despite the energy transfer processes being dominant, the maximum quantum yield will be ca. 92% due to the losses at the ligand.

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3 Relevant photophysical properties This section discusses some relevant (in the context of the present chapter) photophysical properties of lanthanide compounds. Specifically, the properties described in Sections 3.1, 3.2, and 3.3 are essential for treating IET processes, allowing the modeling of quantum yields, emission lifetimes, and other properties.

3.1 An important remark concerning intensity parameters The theoretical background of the 4f-4f intensity theory was discussed in Section 2.1. In the present section, we focus on the experimental determination of the intensity parameters (Ωλ) for trivalent lanthanide compounds. The intensities of intraconfigurational 4f transitions in Eu3+ complexes are usually expressed in terms of the areas under the curves in their emission spectra. The experimental intensity parameters (Ω2 and Ω4) can be calculated according to Eq. (73). Ωλ ¼

3ħc3 A0!λ 7 2 4e2 ω3 χ F λ kUðλÞ k5 D 0

(73)

where χ ¼ n(n2 + 2)2/9 is the Lorentz local field correction due to the medium with refraction index n, ω is the angular frequency of the transition, and h7FλkU(λ)k5D0i2 are the squared reduced matrix elements with values 0.0032 and 0.0023 for λ¼ 2 and 4, respectively [85]. A0!J are the spontaneous emission coefficients of 5D0 ! 7FJ transitions of the Eu3+ ion. The 5D0 ! 7F1 transition is governed practically by the magnetic dipole mechanism; consequently its spontaneous emission coefficient is, in principle, insensitive to the ligand field, namely, A0!1 ffi 0.31  10
11n3ν30!1, where ν0!1 is the transition energy in wavenumbers (cm
1) [35]. Therefore, depending on the way the experimental spectra are measured, e.g., emitted power or number of photons per second versus wavelength, which is the most common nowadays, the A0!J values for Eu3+ compounds can be calculated by Eqs. (74) and (75).   S0!J ν0!1 (74) Emitted power : A0!J ¼ A0!1 S0!1 ν0!J  S0!J Number of photons : A0!J ¼ A0!1 (75) S0!1 where S0!1 and S0!J are the areas under the emission curves of the 5D0 ! 7F1 and 5D0 ! 7F0
6 transitions, respectively. ν0!1 and ν0!J correspond to the barycenter of their transitions energies. The 5D0 ! 7F6 transition is usually not observed experimentally. In addition to Eu3+ complexes, several other lanthanide compounds have their intensity parameters been determined using absorption spectroscopy.

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FIG. 10 Structure of [Sm(X2PPh2)3(THF)2] complexes (X ¼ S or Se) generated from the crystallographic data in Ref. [86].

The following chelates were selected because they have known structures and Ω2, Ω4, and Ω6 were determined experimentally, so they are illustrative candidates for applying theoretical and computational calculations. For instance, the [Sm(S2PPh2)3(THF)2] and [Sm(Se2PPh2)3(THF)2] complexes, illustrated in Fig. 10, are well-characterized [87]. Selected values of the intensity parameters (experimental and calculated FED mechanism contribution) for some lanthanide chelates are shown in Table 2. The FED values were calculated, in the present work, using the JOYSpectra program (to be published). Overall, the theoretical calculations show values are much smaller than the contributions from the DC that the ΩFED λ mechanism, particularly in the case of Ω2. It is relevant to mention that the DC mechanism is the one invoked to explain the so-called hypersensitivity of certain transitions to changes in the chemical environment [42,43,45,96]. These transitions obey, mainly, the selection rules ΔJ ¼  2 (considering the intermediate coupling scheme), although it has been shown (both experimentally and theoretically) [97,98] that there are deviations from these rules. From these works, the conclusion is that a combination of the coordination geometry and the polarizabilities of the ligating atoms (ions) is determinant [63,88,98,99].

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TABLE 2 Experimental intensity parameters Ωλ (λ 5 2, 4, and 6), in 10220cm2, of several lanthanide chelates and the forced electric dipole (FED) contribution, in parentheses, calculated in this work, which are the contributions entering in the WET rates, to which the dynamic coupling (DC) mechanism must not be taken into account. Entry

Ln3+–chelate

Ω2(ΩFED 2 )

Ω4(ΩFED 4 )

Ω6(ΩFED 6 )

References

1

[Pr(bpyO2)4](ClO4)3

14.10

16.75

12.50

[88,89]

(0.55)

(4.61)

(8.11)

2

[Nd(bpyO2)4](ClO4)3

2.18

1.10

0.41

(0.024)

(0.17)

(0.29)

3

[Eu(bpyO2)4](ClO4)3

19.70

2.50

-

(0.029)

(0.090)

(0.18)

4

Cs3[Tb(dpa)3]

44.0

14.1

2.68

(0.21)

(0.40)

(0.56)

5

[Eu(dpa)3]3


10.5

5.31

8.32

(0.29)

(0.34)

(0.51)

6

[Sm(S2PPh2)3(THF)2]

25.49

6.88

6.11

(0.12)

(0.16)

(0.42)

7

[Sm(Se2PPh2)3(THF)2]

13.11

6.50

6.26

(0.24)

(0.14)

(0.36)

8

[Er(tta)3(tppo)]

34.31

1.45

1.85

(0.029)

(0.072)

(0.113)

[88,90]

[89,91]

[92,93]

[94]

[87]

[87]

[95]

The very high sensibility of Ω2 to angular variations of the coordination geometry can be understood from the theoretical expressions, Eqs. (4) and (8), whereas Ω4 and Ω6 are more sensitive to the distances and, therefore, to covalency effects as explored in Refs. [63,99]. In a general way, the FED mechanism contribution in the same compound < ΩFED < ΩFED (values in Table 2 in parentheses), with the has the trend ΩFED 2 4 6 exception of [Sm(Se2PPh2)3(THF)2] (entry 7). This behavior can be explained in terms of the competition between the FED and the DC mechanisms (including interference effects between both), where the DC contribution increases almost exclusively for the Ω2 in the case of [Sm(S2PPh2)3(THF)2] (entry 6), causing a decrease of the FED values by a factor of 12. The explanation of this result can be traced to the differences between the Sm-S and Sm-Se overlap integrals (ρ) in Eqs. (4) and (8) [52,53,63], a discussion which is outside the scope of this chapter. However, it is certainly a curious issue that will be

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discussed in a forthcoming work, on the significance and importance of the Ωλ intensity parameters taking into account covalency effects. For the present purposes, the relevant aspect to be extracted from Table 2 is that should be considered in the evaluation of the WCET IET the values of ΩFED λ rates (Eqs. 38 and 39), because they require opposite parity configuration mixing as in the Judd-Ofelt theory.

3.2 Experimental determination of the ligand donor states It must be pointed out that when we are dealing with integrated intensities, spectral shifts or bandwidths, the emission spectra must be represented as a function of energy and not wavelength, to circumvent misleading and incorrect conclusions. For this (and besides the normal correction for the instrumental response of the equipment), the photon flux per constant wavelength interval function, ϕ(λ), must be converted to photon flux per energy interval [100,101], ϕ(E), accordingly to Eq. (76):  dλ d hc hc ¼
ϕðλÞ 2 (76) ϕ ð EÞ ¼ ϕ ð λ Þ ¼ ϕ ð λ Þ dE dE E E where h is the Planck constant, c is the speed of light, and the minus sign - that can be ignored - simply indicates the distinct directions of integration in λ and E. This correction (known as a Jacobian transformation) has seldom been performed in luminescence spectroscopy, but can be critical, especially when the spectrum displays multiple peaks over a wide energy range and with different bandwidths [101]. Despite the little effect that the Jacobian transformation has on 4f transitions, especially when they are closely-spaced, its use is strongly encouraged for an adequate description of the electronic structure and a proper evaluation of the performances of the material. Gd3+ complexes are most commonly used to determine the energy of the ligand triplet state in lanthanide coordination compounds. The Gd3+ ion forms isomorphic (often referred as isostructural) compounds to those containing Eu3+ or Tb3+, while also having the excited 6P7/2 level with energy sufficiently high for not being populated by energy transfer from the ligand states. In addition, the presence of Gd3+ increases the probability of the S1 ! T1 intersystem crossing through not only the external heavy atom, but also via the paramagnetic effect. However, in several cases where the efficiency of IET is high, large decrease of ligand phosphorescence in Gd3+ complexes is observed due to presence of small amounts of Eu3+ [102]. In such situations, Y3+, which has similar ionic radius to Eu3+ and Tb3+, La3+ or Lu3+ complexes are also used to determine the ligand triplet and singlet states. It should be noted that, due to its smaller ionic radius, Lu3+ may not create coordination compounds isomorphic to their Eu3+ or Tb3+ analogues. Furthermore, Y3+, La3+, and Lu3+

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FIG. 11 Phosphorescence spectrum of Na[Gd(sk)4] and illustration of the determination of the Eb of the ligand triplet donor state by using a Gaussian curve fitting procedure (single peak fit) and of the zero-phonon line (ZPL) energy. sk is di(4-methylphenyl)-phenylsulfonyl-amidophosphate. Adapted with permission from P. Gawryszewska, O.V. Moroz, V.A. Trush, V.M. Amirkhanov, T. Lis, M. Sobczyk, M. Siczek, Spectroscopy and structure of LnIII complexes with sulfonylamidophosphatetype ligands as sensitizers of visible and near-infrared luminescence, ChemPlusChem 77 (2012) 482–496. © 2012 John Wiley and Sons.

complexes, due to the absence of paramagnetic screening, may have higher intensity of phosphorescence than in the case of Gd3+ compounds. The barycenter energy (Eb) determination procedure presented in Fig. 11 works well when the phosphorescence or fluorescence bands are symmetrical and the curve obtained from the Gaussian curve fitting procedure (single peak fit) represents properly the experimental emission band, despite the vibrational progression sometimes poorly observed. This procedure was used to determine the barycenter of the ligand triplet state in Ref. [9]. If the ligand emission band is asymmetrical, then a multiple Gaussian peak fitting procedure should be performed. Two methods for determining the ZPL (zero-phonon line) of the T1 ! S0 transition are described in literature. In numerous publications concerning the IET analysis, the estimation of ZPL energy value is performed in the same manner as shown in Fig. 11. For instance, the ZPL energies of the T1 ! S0 transition in the case of lanthanide coordination compounds with di(4-methylphenyl)phenylsulfonylamidophosphate were estimated in this way [103]. The value reported was 25,350 cm
1 [103], while the value shown in Fig. 11 is 25,627 cm
1. The difference between these results arises from the lack of unit conversion from nm to cm
1 when using the Jacobian transformation in Ref. [103]. Significantly greater differences in the estimations of ZPL energy value are encountered when using the second method, relying on carrying out the multiple Gaussian curve fitting procedure, where the ZPL value is

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acquired as the maximum intensity of the band with the highest energy among those generated in the fit. This procedure for obtaining the ZPL presented in Fig. 11 yields a value (21,420 cm
1) that differs significantly from those obtained in previous estimations (24,582  75 cm
1). This method of ZPL determination was carried out, inter alia, in Refs. [102,104]. In fact, most of the energy values of Eb and ZPL available in literature were obtained not only using different methods, but also derived from bands expressed in nm as well as in cm
1, resulting in poor comparability between the energy values of Eb and ZPL estimated for different lanthanide complexes. As a result, comparisons of these values from different sources must be done with care. Attention should be paid to methodological details affecting the final result. Accurate comparison of Eb and ZPL values is possible only when these parameters are determined using the same method.

3.3 Ligand-to-metal charge transfer (LMCT) states In terms of the molecular orbital approach, electronic transitions from an initial state represented by a molecular orbital located on the ligand to a final state predominantly at the metal ion are called ligand-to-metal charge transfer (LMCT) transitions. On the other hand, if a transition occurs from a metal orbital to a molecular orbital localized on the ligand, it is denoted metal-to-ligand charge transfer (MLCT) [105–107]. This latter plays an important role in the spectroscopic properties of d-transition metal compounds, but for lanthanides, it is only observed in the electronic spectra of few Ce3+ and Tb3+ compounds [108]. Since Jørgensen [109] assigned, for the first time, the strongest broad absorption band in the absorption spectra of the trivalent lanthanide complexes as either LMCT or 4f ! 5d transitions, the role played by the LMCT state on the luminescence sensitization mechanism has been under intense investigation. In contrast to the 4f ! 5d transitions that require quite higher energies to participate in the luminescence sensitization of lanthanide chelates, LMCT states may be located close to the first excited state of the ligand and to the main emitting levels of some Ln3+ ions. Usually, low energy LMCT states are observed for lanthanide complexes presenting ligands with low oxidation potentials and easily reducible ions such as Sm3+, Eu3+, Tm3+, and Yb3+. Furthermore, the energies of Ln3+ emitting states, which are in the spectral range of 18,000 to 25,000 cm
1 in some complexes [110], have also been correlated with the small covalent contribution to the ligand-lanthanide ion interaction in the first coordination sphere, mainly when ligands induce LMCT states with low-lying energies, as for example those containing N-donors. Although the LMCT states can participate in the sensitization mechanism in the lanthanide complexes, these states generally present modest values of molar absorption coefficients as compared with intraligand S0 ! S1 transitions [111]. Moreover, LMCT states of low energies have been considered one of

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the most efficient luminescence quenching channels for Eu3+ complexes, mainly when their energies are close to either the T1 ligand state or the 5D0 and 5D1 excited levels centered on the europium ion [76,77,112–115]. In 1975, Napier et al. [116] associated the low emission quantum yields for some Eu-acetylacetonate (acac) complexes to an efficient luminescence quenching via LMCT state. Among Eu-β-diketonates, those containing dipivaloylmethanate (dpm) in both monomeric and dimeric forms have been extensively investigated in view of their very low emission intensity, which has been attributed to efficient luminescence quenching via the LMCT state [117–119]. These studies have been performed by experimental and theoretical techniques [77,118,120,121]. It is noteworthy that both Tb-acac and Tb-dpm complexes display intense emission in the green spectral region. The effect of LMCT on the luminescence of europium complexes with dibenzoylmethanate (dbm), hexafluoroacetylacetonate (hfa), trifluoroacetylacetonate (tta) and dibenzoylacetonate (bzac) ligands has been reported [122–124]. Interestingly, the luminescence intensity of these Eu-β-diketonates is significantly increased when ancillary neutral ligands such as sulfoxide, phosphine oxide and heteroaromatic are coordinated to the europium ion. This behavior is not only due to a decrease in the multiphonon relaxation but it has been also assigned to changes in the energies of the LMCT excited states [125–127]. The spectroscopic properties of europium complexes with benzo-15crown-5 in the presence of different nitrates, thiocyanates, and perchlorates counterions have also been investigated. The weak luminescence observed for the nitrate and thiocyanate complexes have been associated with the presence of charge transfer states [128]. A particular type of systems presenting LMCT states with very low energies is Eu3+ dithiocarbamate complexes [122,129,130]. The emission efficiencies for [Eu(Et2NCS2)3phen], [Eu(Et2NCS2)3bpy] and [Eu(Ph2NCS2)3phen], where Et2NCS2: diethyldithiocarbamate, Ph2NCS2: diphenyldithiocarbamate, phen: 1,10-phenanthroline and bpy: 2,20 -bipyridine, have been investigated via a theoretical methodology taking into account the energy transfer rates from the ligands and 4f levels to the LMCT states [120]. LMCT states have also shown to play an important role in the luminescence properties of carboxylate complexes, mainly for those based on salicylate and their derivatives ligands [131,132]. The ability of the salicylate in quenching the luminescence from Eu3+ ion has been investigated in terms of decoupling in π–π or p–π conjugation in the ligand. The results showed that this decoupling may lead to a significant lowering of the LMCT state energies in these europium complexes, just down to ca. 27,800 cm
1 [133]. The role of substituents on the salicylate ligand has been also investigated and it was shown that the presence of electron-withdrawing groups increases the energy of LMCT states, increasing the luminescence intensity in the complexes. The absorption band assigned to LMCT transitions lie usually at high energy typically at 50000 cm
1 or it usually overlaps with bands from transitions

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localized at the ligands (e.g., π ! π* and n ! π*). Furthermore, many trivalent lanthanide ions present Laporte allowed interconfigurational transitions (4f n ! 4f n
1 5d) in the same spectral region [134]. However, the presence of LMCT states in Eu3+ complexes has been invoked based on different experimental evidences. The difference between absorption or diffuse reflectance spectral profiles of the analogous Eu3+ and Gd3+ coordination compounds may be used to assign the LMCT transitions. If the absorption band assigned to the ligand-to-europium charge transfer transitions are located in the visible spectral region, Eu3+ complexes generally appear more colorful than Gd3+ ones [115,116,128]. Another strong evidence of LMCT state of low energy has been invoked from positron annihilation lifetime spectroscopic data [135]. Experimental results have shown that Eu3+ compounds generally exhibit low probability of positronium formation as compared with analogous Gd3+ and Tb3+ systems. A plausible mechanism to explain the low probability of positronium formation in Eu3+ compounds was proposed [136,137] by taking into account the competition between the ligand-to-europium charge transfer induced by the positron-molecular electron interaction and positron formation. Interestingly, this mechanism is consistent with experimental luminescence properties, considering that the complexes with the lowest positronium formation probabilities are also those ones with the lowest intrinsic emission yields QLLn. The use of reflectance spectra for the La3+ or Gd3+–complexes are worthy for comparison with their analogous complexes containing other Ln3+ ions to determine the LMCT states [104,138]. Fig. 12 shows two examples by using lanthanum and gadolinium complexes as references for identifying the LMCT band in the europium analogous complexes [104,138]. Qualitative analysis of the LMCT state of low energy in lanthanide coordination compounds may be also investigated based on excitation data. The band attributed to this charge transfer state may appear as weak broadband in the excitation spectra of the Eu3+ complexes. Furthermore, the absorption bands involving ligand-localized and intraconfigurational-4f 6 centered transitions with energies above the LMCT state generally are very weak and exhibit strong temperature dependence of their intensities [119,120]. This experimental evidence together with the unexpected low luminescence quantum yield for complexes in which ligand localized S1 and T1 states present appropriate resonance conditions to an efficient intramolecular energy transfer are among the main reasons to take into account the presence of LMCT states in the intramolecular energy transfer mechanisms of Eu3+ complexes. A direct luminescence quenching of the Eu3+-dipivaloylmethanate complex through the LMCT state has been extensively investigated in the literature [119,120,124]. Berry et al. have determined experimentally the activation barrier for LMCT state quenching of the 5D0 and 5D1 excited levels based on temperature-dependence of the luminescence decay rate constant [115,118,121].

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A [Ln2(sub)3(H2O)4] Solid state 300 K Eu3+ 7

F0

La3+

5

G2,5,6

Normalized intensity (a.u.)

LMCT

H3

250

300

5

D4

L6

5

5

5

350

5

D2

D3

400

450

5

D1

500

550

Wavelength (nm) B [Ln(keto)3(H2O)] (i)

450

400

Wavelength (nm) 350 300

LMCT

Reflectance

Reflectance

(ii)

250

22000

300

350

400

27000 32000 Energy (cm-1)

450

37000

500

Wavelength (nm) FIG. 12 (A) Diffuse reflectance spectra using La3+ complex as reference (sub ¼ suberate). (B) Diffuse reflectance spectra of (i) [Eu(keto)3(H2O)] and (ii) [Gd(keto)3(H2O)] (keto ¼ ketoprofen); the inset shows the arithmetic difference between the two spectra. Panel (A): Adapted with permission from I.P. Assunc¸a˜o, A.N.C. Neto, R.T.M. Jr., C.C.S. Pedroso, I.G.N. Silva, M.C.F.C. Felinto, E.E.S. Teotonio, O.L. Malta, H.F. Brito, Odd-even effect on luminescence properties of europium aliphatic dicarboxylate complexes, ChemPhysChem. (2019), © 2019 John Wiley and Sons.Panel (B): Adapted with permission from M.G. Lahoud, R.C.G. Frem, D.A. Ga´lico, G. Bannach, M.M. Nolasco, R.A.S. Ferreira, L.D. Carlos, Intriguing light-emission features of ketoprofen-based Eu(III) adduct due to a strong electron–phonon coupling, J. Lumin. 170 (2016) 357–363, © 2016 Elsevier Science B.V.

The excitation spectra of the [Eu(bpyO2)4]3+ and [Tb(bpyO2)4]3+ complexes (bpyO2: 2,20 -bipyridine-1,10 -dioxide) in the UV region exhibit the intraligand electronic transitions involving the bpyO2 as well as the presence of ligand-tometal charge transfer transition (LMCT) centered at 370 nm (Fig. 13) [90].

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10

C-T

Absorbance

8

Eu(bpyO2)43+

6

4 Tb(bpyO2)43+ 2

0 250

300

350

400

450

500

550

Wavelength (nm) FIG. 13 Excitation spectra of [Tb(bpyO2)4](ClO4)3 (λem ¼ 546.4 nm) and [Eu(bpyO2)4] (ClO4)3 (λem ¼ 612 nm) complexes at 77 K. C-T means LMCT. Adapted with permission from E. Huskowska, I. Turowska-Tyrk, J. Legendziewicz, J.P. Riehl, The structure and spectroscopy of lanthanide(III) complexes with 2,20 -bipyridine-1,10 -dioxide in solution and in the solid state: effects of ionic size and solvent on photophysics, ligand structure and coordination, New J. Chem. 26 (2002) 1461–1467, © 1987 Royal Society of Chemistry.

Excitation spectra of the Eu3+ complexes with dithiocarbamate ligands [Eu(Et2NCS2)3bpy] and [Eu(Et2NCS2)3phen] recorded at 77 and 300 K (Fig. 14) present low lying LMCT states that may act as an efficient luminescence quenching channel [120]. Furthermore, such a study may serve as a starting point for other research involving photoluminescence in the uncommon lanthanide sulfur-coordinated complexes, thus contributing to a better knowledge of the Ln-S chemical bond and, consequently, of the chemical properties of these complexes. In this context, the relative value of the emission quantum efficiency for the Eu3+-dithiocarbamate complexes as well as its dependence with the temperature was satisfactorily modeled by a theoretical approach [14].

3.4 Emission quantum yield The emission quantum yield is an important experimentally evaluated quantity characterizing the optical performance of luminescent materials. When excitation is performed in the surroundings of the emitting activator which then is excited through energy transfer, the corresponding quantum yield is called overall or external quantum yield. Generally, for lanthanide-based organic and organic-inorganic hybrid materials, determination of the emission

LMCT

350

450

500

F0 --> 5D0

7

F0 --> 5D0

LMCT

7

F0 --> 5D2 7

400

F1 --> 5D2

7

7

300

[Eu(Et2NCS2)3bipy]

7

F0 --> 5L6

7

7

F0 --> 5D0

F0 --> 5D2

[Eu(Et2NCS2)3phen]

7

F0 --> 5L6

F0 --> 5D4 7

7

F0 --> 5G6

S0 --> S1 F0 --> 5D4

Intensity (a.u.)

S0 --> S1

F1 --> 5D1

F0 --> 5D1

LMCT

7

7

[Eu(Ph2NCS2)3phen]

7

7

7

S0 --> S1

F0 --> 5D2

F0 --> 5L6

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F0 --> 5D4

94

550

600

λ(nm) FIG. 14 Excitation spectra of the Eu3+ complexes with dithiocarbamate ligands at room temperature (dash line) and at 77 K (solid line). The 5D0 ! 7F2 emission near 612 nm was monitored. Reproduced with permission from W.M. Faustino, O.L. Malta, E.E.S. Teotonio, H.F. Brito, A.M. Simas, G.F. de Sa´, Photoluminescence of europium(III) dithiocarbamate complexes: electronic structure, charge transfer and energy transfer, J. Phys. Chem. A. 110 (2006) 2510–2516, © 2006 American Chemical Society.

quantum yield is accompanied by measurements of the absorption efficiency (or absorption cross-section), the intra-4f non-radiative decay paths efficiency, and the quantum efficiency of the 4f emitting level, also called intrinsic or internal quantum yield. The ligand-to-metal (or host-to-ligand-to-metal) sensitization efficiency, is then equal to the ratio of the overall (external) to intrinsic (internal) quantum yields. [108,139,140]. The absolute emission quantum yield of lanthanide complexes have usually been recorded at room temperature ( 300 K) upon excitation in the ligand by using, for example, an integrating sphere as a sample chamber for collecting photons, coupled with a photon counting system. Usually, this method presents experimental errors of ca. 10%. The emission quantum yield is defined as the ratio of the number of emitted photons by the Lnz+ ion to the number of photons absorbed by the ligand (Eq. 77), which is thus denoted as QLLn, QLLn ¼

number of emitted photons by the lanthanide ion ðLnz + Þ number of absorbed photons by the ligand ðLÞ

(77)

It is important to emphasize in Eq. (77) that the number of emitted photons refers exclusively to the photons emitted by the lanthanide ion, while the number of absorbed photons refers exclusively to photons absorbed by the ligand. This is of paramount significance in interpreting experimental data,

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because (i) spurious emissions (e.g., from the ligands, even weak, or from impurities) can affect the numerator of Eq. (77), and (ii) resonant 4f-4f (or 4f-5d) absorptions by the lanthanide ion, despite their oscillator strengths (ca. 10
6) being orders of magnitude smaller than the ligand absorption (ca. 10
2 to 10
1), can affect the denominator of Eq. (77), may be present in the experimental measurements. When necessary, time-resolved experiments can be performed to separate these effects. Notice that this notation for the quantum yield, QLLn, is the same employed previously [17,36–38], however, it now includes these restrictions regarding the absorption and emission measurements. Similarly, if the excitation occurs exclusively at the emitting level of the lanthanide ion, then the ratio of the number of emitted photons by the lanthanide ion to the number of absorbed photons by the lanthanide ion gives the intrinsic quantum yield, QLn Ln.

3.4.1 Experimental determination of emission quantum yield (powdered samples) For powdered samples, and if an integrating sphere is not available (which is a common situation in many laboratories), the less precise and robust methods described by Wrighton et al. [141] and Brill et al. [142] can be alternatively used to estimate the emission quantum yield. In the former case, the emission quantum yield (Eq. 78) is obtained as [141]: qLLn ¼

As Rsd
Rs

(78)

where As is the area under the emission spectrum (excited in the ligand) of the sample, Rsd and Rs are the diffuse reflectance of a reflecting standard (designated as white standard) and of the sample, respectively. The white standard should have reflectance values close to the unit (e.g., KBr, MgO, and BaSO4). The experimental error is estimated as 25% [141]. Notice that qLLn becomes QLLn when the emission of the sample is restricted to only the lanthanide emission. Brill et al. [142] proposed a method for measuring emission quantum yield values (experimental error within 10%) in inorganic luminescent materials and for a given sample it can be determined by comparison with a standard phosphor, whose emission quantum yield has been previously determined by absolute measurements. This method can provide relative quantum yield data, avoiding absolute measurements, and indeed it has been used frequently. The quantum yield values qLLn (Eq. 79) using this method are thus determined by the following expression:   1
rST ΔϕS QST (79) qLLn ¼ 1
rS ΔϕST

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where rST and rS are the amount of exciting radiation reflected by the standard and by the sample, respectively, and QST is the emission quantum yield of the standard phosphor (that must be experimentally determined under ligand excitation to be consistent with the adopted notation), ΔϕS and ΔϕST are the integrated photon flux (photons s
1) for the lanthanide-based material and the standard phosphors, respectively. The ΔϕS and ΔϕST values must be determined by integrating the emission intensity over the total spectral range in the emission spectra, plotted as quanta per wavelength interval (photons s
1 nm
1) versus wavelength (nm). The emission spectra must be previously corrected for the spectral dependencies of the photomultiplier response and the grating reflectivity (instrumental function). The Jacobian transformation referred above must also be applied. Notice that qLLn becomes QLLn when the emission of the sample is restricted to only the lanthanide emission. The reflection coefficients r are established by scanning the emission monochromator through the excitation wavelength region, and integrating the intensities of the spectra thus obtained. In order to have absolute r values, MgO is used as a reflectance standard (r ¼ 0.91). There are several compounds that can be used as standard phosphors, depending on the excitation wavelength, color and full width at half maximum of the emission. The most widely used standard phosphors for the red, green and blue spectral regions are Y2O3:Eu, Gd2O2S:Tb, and ZnS:Ag, respectively [143]. All these phosphors have an absolute emission quantum yield around QST ¼ 0.95 for an excitation wavelength around 262 nm. For full-color emitters, sodium salicylate (C7H5NaO3), which presents a large broad emission band peaking around 425 nm, with QST ¼ 0.60 for excitation wavelengths between 220 and 380 nm, is an adequate standard phosphor for ultraviolet absorbing complexes [144]. The overall emission intensity may also be quantified through the so-called radiance, which quantifies the number of emitted photons that falls within a given solid angle in a specified direction. The radiance can be estimated using an integrating sphere complemented with excitation and emission spectrometers or a calibrated charge coupled device. Although often mentioned in the literature, the comparison between the emission of different samples based on the detected intensity is an incorrect procedure leading, in general, to erroneous conclusions, even if all the experimental conditions—similar density of emitting centers, excitation wavelength, light source intensity, excitation/emission slits—are kept constant during the measurements. For instance, different intensities can be only ascribed to distinct absorption cross-sections (and consequently distinct effective optical paths) and not to distinct emission quantum yields. Therefore, emission quantum yield or radiance measurements are required to support a quantitatively and reliable comparison between the emission performances of different luminescent materials.

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3.4.2 Intrinsic emission quantum yield The intrinsic quantum yield (quantum efficiency or internal quantum efficiency) of a 4f emitting level, QLn Ln, defined previously, depends on the radiative (Arad) and non-radiative (Anrad) rates due to deactivation processes of the emitting level as follows: QLn Ln ¼

Arad τobs ¼ Arad + Anrad τrad

(80)

where τrad and τobs are the radiative and experimental lifetimes of the 1 excited state, respectively. The total decay rate corresponds to Atotal ¼ τobs ¼ P 0 Arad + Anrad , where Arad ¼ J AJ0 !J (J is the emitting level) and the Anrad rate depends on the phonon coupling between the Ln3+ ion and its chemical environment. The QLn Ln is characteristic of one emitting electronic level. In fact, if several excited states of an Ln3+ ion emit light, then each of them will have a characteristic intrinsic quantum yield, as already stressed by B€unzli and Eliseeva [140] noting a lot of misleading statements (due to notation) about this that are commonly found in the literature. Moreover, and much less noted, in the presence of near-resonant non-radiative energy transfer processes, the experimental lifetime of a given emitting state, thus, its intrinsic quantum yield, depends on the excitation energy. In a general way, the lifetime of an emitting level is defined as the time interval required for its population to decrease by a fraction 1/e from its initial value. As this decay depends strongly on the absence or presence of energy transfer channels (e.g., longand short-range energy migrations, ion-to-ion energy transfer, and intraand intermolecular energy transfer) [82], proper use of lifetime definition in the presence of energy transfer is required. The natural (or intrinsic) lifetime is defined as the lifetime in the absence of any energy transfer channels, and it has been demonstrated that the presence of an intermediate energy level (e.g., ligand-to-metal charge transfer states, 5d states or defects), in near-resonance with an emitting level mediating energy transfer mechanisms strongly increases the lifetime of an emitting state relative to its intrinsic or natural lifetime [82]. In other words, an increase in the experimental lifetime and, therefore, in the intrinsic quantum yield, is observed when the excitation wavelength opens energy transfer pathways involving states in near-resonance with the emitting level. Determination of the intrinsic quantum yield by Eq. (80) requires evaluation of the radiative lifetime which is related to Einstein’s rates of spontaneous emission. This procedure was reviewed numerous times in the literature and, so, we will not consider it here. The reader is referred to references [35], [143] and [145], for detailed examples. It should be mentioned that the Anrad rate depends mainly on 4f-4f multiphonon decay rates. However, it may also depend on other factors such as

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traps, impurities, back energy transfer, near-resonant excited states like LMCT states. Considering that the Atotal value is experimentally obtained from the emitting level lifetime of the lanthanide, determination of the QLn Ln value from Eq. (80) necessitate the determination of A and A . The spontaneous emisnrad rad P sion coefficient (Arad ¼ J0 AJ!J0 ) of a transition between two manifolds J and J0 is given by Eq. (11). We should stress that accordingly to Eqs. (77) and (80), QLn Ln is always an upper limit of QLLn. Obviously, emission from the ligands or from impurities (even weak) must not be taken into account in the experimental determination of QLLn. If QLLn > QLn Ln is observed, this reflects an inadequate procedure in the determination of the quantum yields. Examples of two of the most common errors are (i) not checking for ligand or impurities emissions (even looking at spectral ranges different from that of the Lnz+ ion) and (ii) using different excitations to measure the emission spectrum and the lifetime of the excited level for which QLn Ln is determined. Moreover one has to bear in mind that quantum yield determination are usually affected by an uncertainty of 10%, which sometimes makes QLLn > QLn Ln.

3.4.3 An interesting theoretical example: Joint effect of ligand (singlet and triplet) and LMCT states on the emission quantum yield The presence of a LMCT state may affect considerably QLLn, not only through a direct action on the 4f excited levels but also through the ligand singlet and triplet states. This situation has been theoretically analyzed in Ref. [77]. Here we present the results of the dependence of QLLn on the energy of the LMCT state, depicted in Fig. 15. Energy transfer between the ligand states and the LMCT state can be quite efficient, resulting in an overall effect that might lead to either limiting energy transfer from the ligand, or to luminescence quenching of the f-emission, depending on resonance conditions and the balance provided by an appropriate system of rate equations. In this analysis [77], the intrinsic quantum yield QLn Ln was assumed to be 50%.

4 Modeling and calculations The IET rates are one of the key elements to model the luminescent properties of lanthanide complexes and materials. In fact, the modeling of their luminescent properties is a multistep sequential procedure, which can be pictured as an in silico experiment. In fact, this computer experiment follows the usual chemical/physical experimental procedure, namely, preparation of the system, control of variables, measurements, and analyses. The equivalent to the preparation of the system is the construction of a structural model representing the luminescent center and the determination of its geometry. This model should represent as close as possible the real system and should describe properly the

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FIG. 15 Emission quantum yield (QLLn) as a function of the relative energy position of the LMCT state with respect to the ligand and the Eu3+ states. hπ j ϕ0i are estimated values in Eq. (57). Adapted with permission from W.M. Faustino, O.L. Malta, G.F. de Sa´, Intramolecular energy transfer through charge transfer state in lanthanide compounds: A theoretical approach, J. Chem. Phys. 122 (2005) 054109, © 2005 AIP Publishing.

measurement conditions such as the sample (crystal, powder, solution), the measurements (power source, pulsed or continuous, photobleaching, and temporal resolution), the conditions (temperature, pressure, solvent), which becomes equivalent to the control of variables. Then, the electronic states and their properties of the ligands and lanthanide ion in the complex, such as, transition energies and their oscillator strengths and location of the corresponding electron densities, vibrational frequencies, are determined and used to calculate the intensity parameters, the IET rates and other relevant transition rates. This step could be equivalent to the measurements on the system after preparation. Afterward, these rates are used to build the rate equations, which are solved to provide the luminescent properties, e.g., quantum yield and quantum efficiency as well as lifetime, risetime and the temporal dependence of any relevant state population, which is equivalent to the analyses of the measurements. This computer experiment as with any chemical experiment must be reproducible, which means that each step should be clearly stated and defined. In addition, the modeled photophysical properties will depend upon the molecular structure (geometric and electronic) employed, equivalently to the preparation of the system in the actual experiment as well as the accuracy of the analyses will be limited by the accuracy of the measurements, the reliability of the calculated properties will be dictated by the physical descriptions of the parameters and variables contained in the model.

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4.1 Molecular structures Modeling properties of compounds and materials (e.g., luminescence) requires as a first step the determination of the geometry (e.g., Cartesian coordinates of each atom) of the system of interest, because it will be employed in the quantum calculations of the electronic states and properties as well as the intensity parameters and IET rates. The most important feature of this geometry is that it should reliably represent the relevant structural properties of the luminescent center. Thus, when modeling lanthanide complexes, it is helpful to distinguish two aspects: (i) the chemical or structural model (step 1.1. in Section 4.5) and (ii) the structural calculation method (step 1.2. in Section 4.5). The structural model is related to the chemical representation of the luminescent center, minimally, the lanthanide ion and its ligands for a moleculartype structure. This model needs to be consistent with all the experimental data available, at least, chemical formula, ligand structures, counterions, etc. For extended systems, the construction of a representative structural model requires special attention and considerations, such as, where to truncate the extended structure, which interactions are relevant for the structure, IET rates, and luminescence properties. Such systems are beyond the scope of this Chapter and will not be discussed further: we thus limit the presentation to modeling molecular-type compounds and materials. To obtain an adequate chemical representation, the first consideration is the physical state of the sample, namely, crystalline or solution (or occasionally in the gas phase). Modeling molecular complexes in the solid state may require the inclusion of neighboring species, such as counterions, crystallization molecules or even other complex moieties, to provide the intermolecular interactions that will lead to the proper structure of the complex in the crystal. Indeed, it has been known that the counterion can affect the luminescent properties of lanthanide compounds [133,146–149], most likely by structural changes induced by their interactions. For instance, the emission lifetime of the 1G4 level of C[Tm(acac)4], where acac is acetylacetonate, varies from 344 to 360 to 400 ns when the counterion C+ changes from Li+ to Na+ to K+ [148]. In ionic liquid crystals, these effects are even more remarkable [150,151], for example, when Ln3+ are coordinated with neutral ligands (e.g., salicylaldimine Schiff bases LH) to form lanthanide-containing metallomesogens [Ln(LH)3]X3, with X being the counterion, the emission spectra of the Eu3+ containing materials are completely different for X ¼ NO
3,
CH3(CH2)5CH2SO
, or CHF (CF ) CH SO [152]. 4 2 2 5 2 4 Interactions with neighboring complex moieties in the crystal can impose some restrictions on the structure of the luminescent center. For instance, the terminal aromatic rings in the [(H2O)3L2EuL2LnL2(H2O)3] complex, with L ¼ pentafluorobenzoate, C6F5COO
, [153] interact with the image rings at neighboring units in the crystal as displayed in Fig. 16. These interactions

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FIG. 16 (A) Crystalline structure of the [(H2O)3L2EuL2LnL2(H2O)3] complex (L ¼ pentafluorobenzoate, C6F5COO
) viewed along the a-axis, with b- and c-axes in green and blue, respectively [153]. (B) B3LYP calculated structure of the dimeric unit, without the neighboring units [154]. The illustrations were generated from the data in Refs. [153,154].

cause the terminal aromatic rings to assume a nearly parallel orientation with the c-axis. On the other hand, the geometry optimization of only a single unit of this complex using the B3LYP functional provides an accurate structure; however, the terminal aromatic rings have a perpendicular orientation with respect to the c-axis as can be observed in Fig. 16 [154]. Notice that the

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modeled structure also did not take into account one crystallization water molecule per unit of complex, which might affect the optimized geometry. However, investigations regarding these and other structural effects onto the luminescent properties, especially onto the IET rates, are still lacking. Crystallization molecules and/or neighboring species in the crystal may be important to maintain a given structure of the luminescent center, especially when these molecules generate hydrogen bonds or ion-dipole or π-stacking interactions with the complex. In solution, these effects would be even more relevant, so the solvent effects can have an impact on the structures of luminescent centers. These effects are mainly due to the dielectric medium generated by the solvent as well as the specific interactions between solute and solvent, such as, ion-dipole interactions and hydrogen bonds. The dielectric effects do not require any special attention regarding the structural model of the luminescent complex and can be taken into account with a proper structural calculation method. Notice that the theoretical treatment of the dielectric solvent effects onto the intensities of intra-4f transitions has been worked out for quite a while [155] and experimental data are also available regarding the dependence of spectroscopic properties of lanthanide complexes with respect to the solvent [156,157]. However, if the solute-solvent specific interactions are relevant for the structure of the luminescent center, then some solvent molecules need to be explicitly included in the structural model. In this case, some care must be exercised because these explicit solvent molecules can interact strongly with the complex, if there are no additional solvent molecules to compete with. Thus, the number and placement of these solvent molecules around the complex have to be adequately chosen to mimic as closely as possible the real system. Indeed, if the solvent molecules can form hydrogen bonds, they will compete with intramolecular hydrogen bonds, if present, which affects the structure of the luminescent center. There are a few systematic investigations of the effects of explicit solvent molecules onto the structure of lanthanide complexes, except for aqua complexes [158–161] and for magnetic resonance imaging contrast agents (MIR-CAs) [162–166]. For instance, the geometry optimizations of some selected lanthanide [Ln(H2O)8]3+(H2O)16 and [Ln(H2O)9]3+(H2O)18 complexes with a second hydration shell led to the migration of four water molecules to the third hydration shell yielding structures such as [Ln(H2O)8]3+ (H2O)12 and [Ln(H2O)9]3+(H2O)14 [160]. The addition of the second hydration shell causes a significant increase of the asymmetry between the Ln
OH2 distances, that is, the distribution of these distances in the coordination sphere increases from tenths to hundredths of angstrom when the second hydration shell is taken into consideration [160]. In the case of complexes for MIR-CAs, explicit inclusion of two secondsphere water molecules is required to obtain accurate Gd3+
OH2 distances and 17O hyperfine coupling constants [163]. For instance, the Gd3+
OH2 dis˚ for tance in the [Gd(dota)(H2O)]
xH2O chelate is 2.606, 2.523, and 2.494 A

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 103

x ¼ 0, 1, and 2, respectively, showing the need of explicit water molecules to ˚ [163]. Despite the lack of better mimic the experimental distance of 2.456 A systematic studies regarding the effects of explicit second solvation shell on the luminescent properties of lanthanide complexes, it can be inferred from the above-described structural changes that several parameters of the model could be affected, such as RL, Ωλ, and SL, and thus the IET rates. Another relevant aspect of introducing explicit solvent molecules in the model is related to the lability of most lanthanide complexes. In view of the very large concentration of solvent molecules in solutions, the possibility of ligand-solvent exchange resulting in solvent coordination needs to be considered when proposing a realistic and reliable structural model. In fact, water lability is one of the most relevant features of MIR-CAs, which have been extensively investigated by experimental and computational methods, thus showing that this behavior is quite general for lanthanide complexes. Regarding their photophysical properties, the Eu3+ complex with edta4
(ethylenediamine tetraacetate) in aqueous solution shows some intriguing spectroscopic behavior, for instance, two lines for the 5D0 7F0 transition were observed with an energy difference of 14 cm
1 [167]. The intensities of these two lines varied with the temperature and pressure [167,168], which has been attributed to the change in the number of coordinated water molecules, namely, [Eu(edta)(H2O)x]
(aq) > [Eu(edta)(H2O)x
1]
(aq) + H2O(l), with x ¼ 3 [169]. In addition to the lability of lanthanide complexes, the flexibility of their structures is relevant to the structural model. Indeed, many lanthanide complexes are fluxional in the sense that the ligands interchange their positions in a given timescale. For instance, crystallographic structures of tris-β-diketonate (β-dik) complexes [Ln(β-dik)3L], with L being an ancillary neutral ligand such as phenanthroline or bipyridine, show that one of the anionic ligand is distinct from the other two. Approximately, one β-diketonate is opposite (trans) to the L ligand, whereas the other two β-diketonate ligands are perpendicular to L. However, room temperature 1H NMR spectra show only one set of signals for the β-diketonate ligands, indicating that these ligands interchange positions faster than the NMR timescale, so they become magnetically equivalent. There are several other examples of fluxionality and structurally flexible lanthanide compounds, such as the higher average molecular symmetry in solution than in the crystal of a Eu3+ complex with a tris-tridentate ligand [170] or of homoleptic tris-chelates [171] as well as the fac-mer isomerization and Λ-Δ racemization of six- [172] and nine-coordinate [173] homoleptic trischelates. Lanthanide complexes with macrocyclic ligands are also prone to fluxionality due to the conformation of the ring(s) [162–166,174] and, in some cases simultaneous Λ-Δ racemization, which can have different rate constants [174]. Despite being quite common, the effects of the lability and fluxionality on the luminescence of lanthanide chelates are still unexplored by modeling techniques.

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Once the structural model has been proposed, the next step consists of selecting a method to provide the geometry or, more specifically, the (Cartesian) coordinates of each atom constituting the structural model. These methods can be classified into experimental or computational ones. X-ray crystallography of single crystals is the most employed method to geometry of luminescent complexes. In fact, the CCDC (The Cambridge Crystallographic Data Centre) database [175] (https://www.ccdc.cam.ac.uk accessed June 10, 2019) has recently celebrated the mark of over 1 million structures from which the number of structures containing lanthanide elements is present in Table 3 [176]: The most luminescent ions (Eu3+, Tb3+, Sm3+, Nd3+, and Yb3+) and Gd3+ (used as blank or reference) are the most investigated ones. Despite X-ray crystallography being quite accurate and reliable, there are some warnings when using the corresponding molecular structures in the modeling of luminescent centers. Assuming that the structural resolution is below a reasonable threshold, there are possible structural disorders [177,178] and distances involving hydrogen atoms that need some consideration. In particular, positional disorder (rotamer, conformer, or isomer) can lead to quite unrealistic distances. For instance, disorder around a single bond Ar-X-CH3 (with Ar ¼ aromatic ring and X ¼ CH2, O, S, etc.) can lead to very short average X-C distances. Because this geometry is then used in subsequent steps of the modeling, these very short bond distances could unrealistically distort the electronic density and localize, for instance, the triplet state in this region of the complex instead of the appropriate location, possible onto an aromatic ring or conjugate moiety. Another concern is related to the short distances involving hydrogen atoms usually reported, which could also distort the electronic density of the ligands in the complex leading to inadequate estimates of the ground and excited states properties (e.g., transitions energies, oscillator strengths, and state location). In these cases, it is wise to employ a computational method to optimize the disordered bonds and/or the bonds involving hydrogen atoms. However in such a case, it is recommended to employ a crystallographic structure as the starting geometry. In addition of saving substantial computational resources it limits considerably the number of initial geometries that would be required to optimize the geometry of the structural model. In fact, when modeling new compounds for which there are no crystallographic structures available, it is a good practice to investigate if similar structures exist which could provide an educated initial guess for the structural model. However, this approach is be reliable for lanthanide complexes, for which small changes in a ligand formula or structure or variations of the synthetic conditions can lead to different structures for the chelate. For instance, substituting the hydrogen atom at the 6-position of the phenol ring in the oxazolylphenol ligand by a methyl group causes a change from a dimeric [Y(iPr-Ph)3]2 to a monomeric [Yb(iPr-MePh)3] structure [172]. In another example, variation

TABLE 3 Number of crystallographic structures of lanthanide compounds in CCDC database (checked in June 10, 2019). La

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Ho

Er

Tm

Yb

Lu

3483

2564

2029

3624

4

3841

4692

3967

3551

1387

2528

627

3468

1353

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of the synthetic conditions leads to dimeric or to monomeric structures for the reaction of several lanthanide salts with pentafluorobenzoic acid [153]. In addition, care must be exercised when invoking the isomorphic behavior of lanthanide complexes when creating the structural model, because it is common for the coordination number and/or the structure to change along the lanthanide series. For complexes in solution, NMR techniques are useful and the most employed to obtain the structures of chelates [164,170,172,179–183] as well as combined with quantum chemical computational calculations [165,174,184,185] or with CD (circular dichroism) spectroscopy for describing chiral complexes [186]. This approach has been particularly important for investigating the structures and dynamics of MIR-CAs in solution. It is noteworthy that these are average structures, meaning that if the lanthanide complex is fluxional, then the NMR analyses yield distances between the magnetic nuclei and the lanthanide ion that represent an average of those accessible structures at the NMR timescale. And, as mentioned before [170–172], the symmetry of this average structure is higher than that in the crystal, because the inequivalent ligands become equivalent due to the fluxional process. Thus, the geometry obtained by NMR analyses, when employed directly, need to be used with caution to avoid distances that are too short or too long, which would cause distortions of the electronic density and unreliable properties of the ground and excited states. Computational methods based on quantum chemical approaches [187–191] are by far the most ones employed in obtaining the geometry of lanthanide complexes to be used in the modeling of their luminescent properties. These methods are usually classified into WFT (wave functional theory) [192–195] and DFT (density functional theory) [196–199] based approaches. WFT-based methods are denoted ab initio because they do not employ any experimental data, except for some physical constants and atomic numbers, and they have a hierarchical improvement, e.g., HF (Hartree-Fock), MPn (nth-order MøllerPlesset), CCSD (coupled-cluster singles and doubles), CCSDT (CCSD with triples), CCSDTQ (CCSDT with quadruples), etc., which can provide, at least in principle, the exact solution of the electronic Schr€odinger equation. Whereas this hierarchical improvement straightforward for DFT-based methods, most implementations employ some empirical parameters or functions adjusted to experimental and/or computational data. For determining the geometry of lanthanide complexes the semi-empirical (SE) Sparkle/SE methodology [200], with SE ¼ AM1 [201–203], RM1 [204], PM3 [205–207], PM6 [208], and PM7 [209], has been used and improved over the last two decades [200,210]. These approaches are based on simplifications of the Fock operators and the Hartree-Fock equations and describe the trivalent lanthanide ion as +3 point charges with a repulsive potential and analytical expressions for the core-core integrals. These functions depend on parameters that are adjusted to minimize the unsigned mean errors (UME)

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of the coordination polyhedron between crystallographic and calculated structures. Because of the simplifications in the treatment of the lanthanide ion and the ligands by semi-empirical methods, the computational demand of the Sparkle/SE approach is the smallest for quantum chemical methods and allows performing many calculations for a given complex or calculations of very large systems (few hundreds to thousands of atoms). However, the Sparkle/SE approach has some caveats, namely, difficulties in treating small (ionic) ligands, complexes with carbon, sulfur, selenium, and other ligating atoms, because none or just a few compounds were employed in the parametrization procedure, restricted to some specific ligands such as carboxylates as will be discussed below. Thus, it is wise to validate the geometries obtained by the Sparkle/SE approach as well as by WFT (e.g., Hartree-Fock, HF, second-order Moller-Plesset, MP2, etc.) or DFT (e.g., GGA, generalized gradient approximation, meta-GGA, etc.), because the latter methods employ approximations and require some choices that need to be validated. One of these choices, for both WFT- and DFT-based methods, is the description of one-particle functions, for which in the case of non-extended systems (i.e., molecules), Gaussian basis sets are commonly employed [211,212], although Slater-type orbitals (STOs) have been successfully used in the Amsterdam Modeling Suite [213,214]. Because the relevance of the relativistic effects for describing lanthanide ions [187,188,215], this choice of the basis sets impacts also on choosing the appropriate level of calculation for the manyelectron part. For instance, choosing an all-electron basis sets [215–217] implies that all electrons will be treated explicitly and the level of calculation should include the relevant relativistic effects (solving approximated relativistic quantum equations) as well as open-shell (spin multiplets) and static correlation (multiconfigurational description) [218], so the basis sets need to be specialized to this type of calculation [215–217,219–226]. Despite the continuing development of computer hardware and software as well as theoretical approaches, calculations employing all-electron basis sets are still not practical for obtaining geometries of large lanthanide complexes. Indeed, for such a task, basis sets employing relativistic pseudopotentials (PPs) [227–230] are quite suitable and frequently employed because of the considerable decrease of the computational demand without a significant loss in accuracy. This is achieved by decreasing of the number of electrons treated explicitly as well as by including relativistic effects implicitly into the PP. For lanthanides the most used PPs are (quasi)relativistic ECPs (effective core potentials), which can be energy-consistent or shape-consistent PPs [227–230] with different choices of the core, e.g., MWB28 [231], SBKJC [232], and CRENBL [233] with 28, 46, and 54 electrons in the core, respectively, whereas the 4f electrons are outside the core and thus have to be treated explicitly. Thus, in the description of the cerium atom there will be 30 (4s2 4p6 4d10 4f1 5s2 5p6 5d1 6s2), 12 (4f1 5s2 5p6 5d1 6s2), or 4 (4f1 5d1 6s2) valence electrons when using the MWB28, SBKJC, or CRENBL ECPs, respectively.

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For the two latter ECPs, the basis sets describing these valence electrons were designed and presented with the development of the respective ECP [232,233], whereas for the MWB28 ECP several basis sets have been developed, including generalized contracted (14s13p10d8f6g)/[6s6p5d4f3g] [234] and segmented contracted (14s13p10d8f6g)/[10s8p5d4f3g] [235] ANO (atomic natural orbital) basis sets as well as segmented contracted basis sets of double-ζ to quadruple-ζ valence quality, def2-XZVP and def2-XZVPP (X ¼ D, T, and Q) [236]. Alternatively, because the core-like behavior of the 4f electrons and their negligible effects upon the geometry, as can be ascertained by the Sparkle/SE approach, 4f-in-core ECPs like MWB46 + x, where x is the number of electrons in the 4f subshell (e.g., Eu3+ 4f6 MWB52), were developed for di- [237,238], tri- [237,238], and some tetravalent (Ce-Nd, Tb, and Dy) [239] lanthanide ions. Segmented contracted (7s6p5d)/[5s4p3d] basis sets were developed for the 4f-in-core ECPs of di- and trivalent lanthanide ions [237,238], which were improved by core-polarization potentials [240], in addition to basis sets designed for crystal orbital (periodic) calculations [241]. These 4f-in-core ECPs and basis sets usually yield excellent results for the molecular structure of lanthanide complexes, and because of their significant savings of computational resources they became very popular. It is noteworthy that with these MWB46 + x ECPs, the lanthanide ions and, therefore, most complexes become closed-shell, which facilitates the SCF (self-consistent field) convergence and decreases the CPU time of the calculations. Another aspect favoring the use of these 4f-in-core ECPs is because, depending on the choice of the density functional, the 4f subshell can become overfill, especially for Eu3+ and to a lesser extend for Yb3+, when performing an all-electron or a MWB28 ECP calculation [242,243]. This spurious population of the 4f subshell can affect the coordination polyhedron of Eu3+ and Yb3+ complexes. In addition to the choice of description of the one-electron functions, the approximations employed in the wave function [194,244,245], e.g., HF, MP2, CCSD, etc., or in the density functional [199,244,245], e.g., GGA, hybrid-GGA, meta-GGA, hybrid meta-GGA, etc., can have significant impact in the calculated geometry of lanthanide complexes. Because of the effects of electron correlation in 4f-compounds, the HF method is usually not recommended and correlated WFT based-methods are employed [244–246]. Although high-level correlated WFT methods have been occasionally applied to Ln3+ complexes [242,244–247], the scaling of these methods with the size of the system currently prevents their application to systems of practical relevance. Thus, the DFT-based approaches are more commonly employed, e.g., B3LYP, PBE0, M06-2X, etc. [248], to investigate lanthanide complexes [154,164–166,174,184,185,248–250]. Notice, however, that the experiences with the applications of DFT functionals to transition metal complexes cannot be directly extended for lanthanide complexes because the ligand-lanthanide

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ion interactions are mainly electrostatic. The structures of the lanthanide complexes are then dictated mainly by the ligand-ligand interactions balanced by the ligand-lanthanide attractions. Thus, the choice of the DFT functional should be based on the proper description of these interactions, however, despite of the known flaws of B3LYP in describing intermolecular interactions, it is quite often employed in structural calculations of lanthanide complexes. In fact, the use of B3LYP and other functionals, such as PBE0, for determining the structures of these complexes has been validated by the RMSD (root mean square deviations) between the calculated and the crystallographic geometries. Dispersion corrections of some functionals, including B3LYP and PBE0, by empirical functions like the popular DFT-D3 [251] as well as the use of dispersion corrected functionals have to be regarded with caution and need to be validated. For instance, the asymmetric feature (see Fig. 16A) of the bridging ligands observed in the [(H2O)3L2LnL2LnL2(H2O)3] complexes (L ¼ C6F5COO
) [153] and many other dimeric complexes with bridging carboxylates was reproduced only by the B3LYP functional [154]. In fact, the structures calculated by Sparkle/SE approaches (SE ¼ AM1, PM3, or PM7) as well as dispersion corrected DFT functionals (APFD and WB97XD) or with empirical D3 corrections (B3LYP-D3 and B97-D3) presented symmetric coordination of the bridging ligands, even though the starting geometry was the crystallographic one [154]. So, it is always wise to validate the calculated structures by the RMSD with respect to the crystallographic ones or, in cases where these structures are lacking, employ different Sparkle/SE approaches and distinct DFT functionals and basis sets. The properties of the excited states of the ligands in the complex are required in the calculations of the IET rates, specifically RL, SL, Δ, and ΓL. Some of these quantities cannot be obtained from experiments; however, photophysical data such as absorption spectrum (transition energies and intensities), phosphorescence of the complex (e.g., with La3+, Gd3+, Y3+, and Lu3+), including emission lifetimes and time-resolved spectra, are important to calibrate and validate calculations.

4.2 Excited states Once the geometry of the lanthanide complex has been determined, several computational approaches can be employed to determine its excited state properties [252], such as semi-empirical INDO/S-CIS method [253,254], DFT based time-dependent approaches (time-dependent DFT, TD-DFT) [248,255–259], and ab initio multireference wave function for static correlation [260–262] with a quasi-degenerate dynamic correlation approach [263,264] and spin-orbit effects [265]. This latter technique is recent, difficult to perform, specialized and quite expensive, so it has not been widely applied and a multireference SA-CASSCF/XMCQPDT2/SO-CASSCF (state-averaged complete active space

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SCF with quasi-degenerate second order perturbation theory and spin-orbit CASSCF) has been applied to lanthanide complexes: [Eu(tta)3phen], [Eu(tta)3bipy], and [Tb(acac)3bipy], where tta is thenoyltrifluoroacetonate, acac is acetylacetonate, phen is 1,10-phenanthroline, and bipy is 2,20 bipyridine [266,267]. In the first model implemented for the calculations of the IET rates, the properties of the excited states were determined with the INDO/S-CIS approach, where the lanthanide ion was replaced by a + 3e point charge [268]. This approach has been quite successful and widely applied in the modeling of luminescent properties of lanthanide complexes [18,83,113,269–283]. However, this approach has some caveats related to the choice of the configuration space (occupied and unoccupied molecular orbitals employed in the configuration interaction singles—CIS) and the strong distortions of the electronic density caused by the +3e point charge. Indeed, the INDO/S-CIS method was parametrized using a limited configuration space, so its application should also employ a small number of selected occupied and unoccupied MOs. The arbitrariness of selecting the configuration space remains one of the main difficulties of this approach. The point charge model has been improved by making the charge dependent on the distance to the lanthanide ion [284], however, this correction has not been implemented in recent versions of the programs employed in the INDO/S-CIS calculations. In summary, the INDO/S-CIS with +3e point charge is very fast, requires minimum computational resources, and provides satisfactory results, except for some cases such as tetrakis β-diketonate and dimeric carboxylate complexes. Thus, the TD-DFT approach has gained attention lately and is becoming a useful computational tool to determine the excited states of the ligands in the complex [285–299], especially when combined with 4f-in-core ECPs. In fact, this approach (TD-DFT/4f-in-core ECP) is consistent with the independent model used to calculate the IET rates, because the 4f electrons do not affect the ligand electronic density except by shielding the nuclear charge of the lanthanide. The TD-DFT approach has been quite successful in predicting the energies of the low-lying triplet states and remarkable correlations were obtained with the luminescent properties of the lanthanide complexes [285–287,297,299]. It has also been validated by comparing with the experimental values of the energies of the lowest triplet states. It should, however, be emphasized that most of these comparisons were performed with the calculated singlet ground state structure, whereas the phosphorescence measurements are related to the geometry of the complex at the excited triplet state. In fact, four [Eu(β-diketonate)3phen] complexes were investigated with the PBE1PBE and CAM-B3LYP functionals by calculating the energy of the lowest triplet excited state via ΔSCF approach [300]. In this approach, the geometries of these complexes were optimized for the singlet ground state S0 and the lowest triplet excited state T1 using the 4f-in-core ECP/basis set for the lanthanide ion and 6-31G* basis set for the remaining atoms. Then the energy difference between the triplet and the ground states

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can be adiabatic: ΔEadia ¼ E(T1)
E(S0), where E(S0) and E(T1) are the ground and triplet states SCF energies calculated with the geometries optimized at the S0 and T1 states, respectively, and vertical: ΔEvert ¼ E(T1)
E(S0; T1), where E(S0; T1) is the ground state SCF energy calculated with the geometry optimized at the T1 state. For the adiabatic transition, the zero-point energy (ZPE) corrections were included for both states. In addition, the calculated vibrationally resolved phosphorescence spectra and UV-vis absorptions were simulated within the Franck-Condon approximation using the adiabatic Hessian (AH) model and Gaussian functions with half-widths at half-maxima of 500 cm
1. The CAM-B3LYP functional provided the best calculated triplet state energy values with deviations from the zero-phonon experimental values smaller than 1200 cm
1 and the Mulliken atomic spin densities were used to ascertain the location of the lowest triplet state [300]. Similar TD-DFT and ΔSCF approaches were successfully employed for a series of six complexes of Gd3+ coordinated to modified cryptates of tris-bipyridines with errors smaller than 2000 cm
1 compared to zero-phonon experimental data [292,296,298].

4.3 Charge factors and effective polarizabilities Several interesting aspects can be discussed about the FED and the DC mechanisms. An analysis of typical values of the quantities that appear in the two mechanisms shows that, in general, they contribute with opposite signs and, therefore, interference effects may be relevant. Although the ΩFED λ is the portion of the total Ωλ that contributes to the WCET direct mechanism in and IET rate constants (step 6 in Fig. 19), a good description of both ΩFED λ is important to properly model the emission properties. In this sense, ΩDC λ it is relevant to point out some issues about the estimate of the effective ligand polarizability (α0 , Eq. 4) and the charge factor (g, Eq. 8), that are used within BOM and SOM to estimate Ωλ. The charge factor can be calculated using mainly two different approaches: (i) a fitting procedure to adjust the theoretical Ωλ to the experimental values or (ii) calculations with a theoretical/computational model. The fitting procedure can be performed using different methodologies, including manual intuitive approaches or sophisticated fitting procedures based on optimization algorithms (see Section 4.4). For the calculation of the charge factor, within the frame of the SOM model [52,53], the charge factor is related to the ionic specific valence definition and is given by [64]: rffiffiffiffiffiffiffiffi k g¼R 2Δε

(81)

where Δε is the same excitation energy that appears in Eq. (5) for αOP, and k is the force constant that must be assigned to each individual Ln–X bond.

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FIG. 17 (A) Pseudo-diatomic model for calculating the force constant in Lnz+ complexes. (B) Partition scheme for the force constant calculation involving bidentate ligands. GC and g.c. means geometric center. Adapted with permission from R.T. Moura Jr, A.N. Carneiro Neto, R.L. Longo, O.L. Malta, On the calculation and interpretation of covalency in the intensity parameters of 4f–4f transitions in Eu3+ complexes based on the chemical bond overlap polarizability, J. Lumin. 170 (2016) 420–430, © 2016 Elsevier Science B.V.

The value of k, in this sense, can be calculated using a pseudo-diatomic model, as depicted in Fig. 17. To avoid the contributions of chemical groups inside the ligand molecules (e.g., O]C in a diketone) to the Ln3+–X stretching, the ligand geometries are kept rigid and the second derivative of the energy is calculated as a function of the Ln3+–X distance. For monodentate ligands, the second derivative associated with the ligand
Lnz+ stretching is directly defined as the force constant k. Whereas, for bidentate ligands, there are two Ln3+–X bonds for only one calculated force constant. This force constant is defined as belonging to the geometric center of two bonded atoms at the ligand, as presented in Fig. 17B. Therefore, it is necessary to partition the force constant associated with the geometric center into contributions from each bonded atom, as depicted in the partitioning scheme in Fig. 17B. !

!

!

The vectors F GC , F L1 and F L2 are the forces (potential energy derivatives) associated with the geometric center (GC), and atoms L1 and L2 connected to the central ion, respectively. They can be written as ! F GC

b GC ¼ kGC δxGC D

(82)

! F L1

b L1 ¼ kL1 δxL1 D

(83)

! F L2

b L2 ¼ kL2 δxL2 D

(84)

where kGC, kL1 and kL2 are the force constants associated with the geometric center and to the atoms L1 and L2, respectively, δxGC, δxL1 and δxL2 are their

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b GC , D b L1 and D b L2 are unit vectors defining the directions displacements and D of the respective forces. Centering the coordinate system at the central ion and rotating it so that the three atoms are laying in the xy-plane, then ! ! ! b L1 Eqs. (82)–(84) form a set of equations such that F GC ¼ F L1 + F L2 , with D b L2 being bidimensional unitary vectors (in plane xy), with solutions. and D  δxGC y2 (85) kL1 ¼
kGC δxL1 y1 x2
x1 y2  δxGC y1 (86) kL2 ¼ kGC δxL2 y1 x2
x1 y2 where δxGC is the displacement used for calculating kGC, δxL1 and δxL2 are the displacements for L1 and L2, respectively, and can be calculated from the geometrical parameters. The (x1, y1) and (x2, y2) values are the L1 and L2 coordinates for the ion-centered and rotated system, where z1 ¼ z2 ¼ 0. Currently, the computationally accessible methodology to perform the force constant calculations using the pseudo-diatomic model are the same as those depicted in Section 4.1. Moreover, it is necessary to use an equilibrium geometry obtained with a computational methodology. This means that it is not possible to use directly the crystallographic structure to obtain the charge factors with the pseudo-diatomic model. It is necessary to include the appropriate environment descriptions (periodic boundary conditions, solvent, etc.) in the structural model to obtain an equilibrium geometry. In addition, it is necessary to use the latter structural model for calculating the force constant. The charge factor in Eq. (81) is general and can be applied for any system, as long as the force constant is calculated at the equilibrium and within the same level of theory. Some recent examples can be found in the literature [63,98,99]. Although general, caution is required in obtaining k with semiempirical methods. Sparkle/SE models include a repulsive potential for the lanthanide ion [200]. The balance of this term with the core repulsions and the Coulomb attractions makes the potential energy surface too steep around the equilibrium geometry, exhibiting high values of force constants (second derivatives of the energy with respect to the position of the ligand). Thus, when using the pseudo-diatomic approach to calculate charge factors to study a set of compounds, the recommendation is to use a subset of selected cases and run comparative tests using DFT or WFT as reference to establish a correction factor to the SE force constant calculations. The polarizabilities of the chemical environment around the lanthanide ion are factors that strongly influence the intensities of the 4f-4f transitions and therefore they are important for describing the total intensity parameters Ωλ. A satisfying calculation of the intensity parameters is an important indication that the structure and chemical environment are well described.

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Polarizabilities for monoatomic ligands are much easier to infer [301] compared to polyatomic ligands. In the latter case, obtaining an effective polarizability for the coordination polyhedron requires considerable attention. The total dipolar mean polarizability of the complex structure without the Ln3+ ˚ 3 to >1000 A ˚ 3 [63], depending on the ligand composiion can vary from 200 A tion and volume. These values are obviously not appropriate to be included in the DC mechanism. Different works highlight the importance of the polarizability of the ligands (in the dynamic coupling mechanism) for 4f-4f transitions in Ln3+ compounds [43,302–305]. The contributions of anisotropic ligand polarizability were evaluated for Lnz+ compounds with atoms (in an inorganic crystal) or organic molecules as ligands [302–304]. In these works, the polarizabilities of the ligands were estimated using geometrical assumptions and considering the polarization of electric dipoles associated with group of atoms at the ligand molecules. There is a general consensus that the charge distribution localized at the chemical bond and on the substituents in the ligand environment makes significant contributions to the crystal-field potential of Eu3+ compounds [304]. It has been proposed to use localized molecular orbitals (LMOs) and to determine their contributions to the molecular polarizability in order to provide the effective polarizabilities of the ligating atoms or groups [63]. The localization of the canonical molecular orbitals by the Pipek-Mezey [306] procedure has been quite successful in providing LMOs for the ligands in different coordination compounds. The decomposition of the molecular dipolar polarizability into LMOs components is available in the GAMESS program [307] and provides the effective ligand polarizability in the context of the BOM. More specifically, the LMO polarizabilities are calculated from the differences between the LMO coefficients determined in the presence of small finite electric fields [308]. A spatial region is defined for each ligand according to its distance from the metal ion. Fig. 18 illustrates the superposition of the LMOs associated with chemical groups near the lanthanide ion. The effective ligand polarizability is defined as the sum over the polarizabilities of these LMOs.

4.4 Computational programs A few computational programs are available for the treatment of lanthanidebased-systems and for modeling their spectroscopic properties (e.g., LUMPAC, JOES, and JOYSpectra). The LUMPAC program, developed by Dutra et al. [309], has been designed to be an user friendly program for Windows® operating systems. It is divided in four modules ranging from geometry optimization to calculation of the emission quantum yield. Geometry optimization is based on the Sparkle/SE approaches mentioned before within the MOPAC program [310]. The theoretical intensity parameters Ωλ are based on the expressions from the SOM for the

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FIG. 18 Superposition of localized molecular orbitals (LMOs) in chemical groups near the lanthanide ion for tta. An isosurface of 0.1 e/a30 was used to generate these LMOs.

ligand field [52,53] (Eq. 8) and Mason et al. [43] traditional DC mechanism. The latter mechanism does not take into consideration covalency effect as considered explicitly in the BOM [63], Eq. (4). The values of charge factors g (Eq. 8) and the ligand polarizabilities α, which are different from the α0 in Eq. (4), are obtained with optimization procedures based on a generalized simulated annealing (GSA) algorithm [311] to fit to the experimental values of Ωλ. Nevertheless, it should be emphasized to users that the present version of the program LUMPAC needs to be adapted to become consistent with the formulation of IET rates given by Eqs. (38)–(42) in Section 2.
c et al. [312], was recently launched The JOES program, developed by Ciri
with the purpose to calculate the experimental intensity parameters Ωλ, by using Eqs. (11), (73) and (74), from emission spectra of Eu3+-based compounds. It is easy to download the program (no license is required) and a large database of refractive indexes (n) is available. This point is crucial, as discussed in Section 3.1, to obtain the spontaneous emission coefficient for the 5 D0 ! 7F1 transition (A0!1) that depends on the third power of n. This coefficient is used as an internal reference for the calculations of the experimental intensity parameters Ωλ. Therefore, any variation in n may lead to different values for the experimental parameters Ωλ in Eu3+ compounds. The Judd-Ofelt Spectroscopy (JOYSpectra) program was built to use the SOM and BOM models, taking covalency effects of the Ln–L bonds into consideration. In addition, charge factor calculations using the pseudo-diatomic approach (step 3 in Section 4.5) is available in JOYSpectra. The program intends to be useful to theoreticians as well as to experimentalists, it is free of

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charge and has relevant features that facilitate a better understanding of the chemical-physics information behind the Ωλ parameters At the moment, the program is restricted to key aspects of 4f-4f intensity calculations, including covalency effects. Modules to calculate the non-radiative energy transfer rates between lanthanide ions and the IET rates in lanthanide compounds will be available soon. JOYSpectra has different functionalities to manipulate geometry, to produce specific symmetry changes (natural distortions and vibrational or thermal effects), and to follow the intensity parameters variations with coordination geometry changes. The program also provides the symmetry point group of the coordination polyhedron according to the input geometry and in each step of a specific geometry variation. The αOP quantities are automatically calculated using the input geometry and internal parameters (see Eq. 5). The g and α0 quantities can be adjusted within physically reasonably acceptable values, to reproduce, as close as possible, the experimental Ωλ values. In addition, JOYSpectra is able to produce the necessary data of g and α0 (see Section 4.3), using external programs such as Gaussian [313], MOPAC [310], ORCA [314] and GAMESS [307], to enable ab initio calculation of the intensity parameters without any experimental data. The JOYSpectra package includes a web-based platform (JOYMaker program) for creating input files and downloading the program, user manual, examples, tutorials, etc. It can be accessed at http://www.cca.ufpb.br/gpqtc/joyspectra (last accessed, July 27, 2019).

4.5 Modeling procedures The calculation of the IET rates (WET) involves a multistep procedure, where different choices of models and methodologies are made. The flowchart in Fig. 19 summarizes the most relevant steps of a general procedure to obtain the luminescent properties of lanthanide complexes and materials.

Step 1. Compound structure The initial step requires the geometry (e.g., Cartesian coordinates of each atom) of the system of interest. This is a key step and is discussed in details in Section 4.1.

Step 2. Excited states Once the structural aspects are modeled and the compound geometry determined, different computational approaches can be employed to determine their excited state properties, as described and discussed in Section 4.2. This step should provide quantities such as SL (Table 1), Δ (Eq. 43), ΓL and RL, which can be estimated from molecular orbitals calculations as:

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FIG. 19 Flowchart for a multistep sequential procedure to model the luminescent properties of lanthanide complexes and materials.

X

c2i RLi RL ¼ iX 2 ci

(87)

i

where ci is the molecular orbital coefficient of atom-i at the ligand donor (or acceptor) state and RLi is the distance of atom-i to the Lnz+ ion [9].

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Step 3. Calculations of α0 and g The charge factors (Eq. 81) and ligand effective polarizabilities can be calculated using the compound geometry, as described in Section 4.3.

Step 4. Intensity parameters calculations Once the structure of the compound is optimized (step 1) and the Ln–L parameters defined (step 3), Eqs. (1)–(8) can be used for calculating the theoretical intensity parameters.

Step 5. Choice of relevant states From the excited states modeled in step 2, and described in Section 4.2, it is possible to choose the most relevant states for which the energy transfer rates should be calculated. Further details can be found in Section 2.4.

Step 6. Direct mechanism WCET The Coulomb direct mechanism, described in Section 2.2.1, can be calculated using Eq. (38). It is important to highlight that only the forced electric dipole mechanism should be considered in the evaluation of IET rates.

Step 7. Exchange mechanism Wex ET The exchange mechanism, described in Section 2.2.1, can be calculated using Eq. (40). It is important to emphasize that the screening factor (Eq. 41) that appears in Wex ET is also discussed in Section 2.2.1.

Step 8. Rate equations How to set the rate equations is addressed in Section 2.4, where an appropriate system of rate equations, based on rates for a microscopic ligand– Lnz+ unit, can be solved using analytical (Section 2.4.1) or numerical (Section 2.4.2) approaches.

Step 9. Photophysical properties The definition of the quantum yields (QLLn and QLn Ln, Eq. (77) and Eq. (80), respectively) can be found in Section 3.4. After solving the rate equations, it is possible to calculate QLLn or QLn Ln using analytical (Section 2.4.1; Eqs. 60–70) or numerical (Section 2.4.2; Eqs. 71 and 72) solutions. The latter method can also provide the temporal dependence of the level populations and photophysical properties such as rise times and lifetimes.

5 Selected cases This section is devoted to the discussion of three aspects. The first one is the case of IET via ligand singlet states, which has caused some controversies in the literature. Our conclusion is that IET via ligand singlet states is a ubiquitous

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channel that in certain situations may become operative or even the dominant one in Lnz+ chelates. Nevertheless, this fact does not contradict the majority of the cases, in which the IET process is dominated by IET via ligand triplet states. In this context, the LMCT state (either as a sensitizer or a suppressor) may play a crucial role in Eu3+ chelates. The overall balance between transition rates in a given chelate, extracted from the solutions of an appropriate system of rate equations, may indicate the relevant mechanisms and pathways. The second aspect concerns a discussion on experimental evaluation and values of IET rates from the literature. Finally, the third aspect concerns a concrete example of application of the theory described in this chapter. This application is carried out in the simplest form in order to illustrate some relevant points in the prediction of the emission quantum yield (QLLn).

5.1 Energy transfer via singlet The commonly observed sensitization by IET processes in luminescent Lnz+ coordination compounds involves a triplet pathway, in which the nonradiative energy transfer from the first excited triplet state (T1) of ligands takes place. This pathway has been unambiguously proven experimentally, and corroborated by theory, in Ref. [14]. However, the energy transfer from the ligand singlet state cannot be ruled out. Kleinerman [315] has argued that if the intersystem crossing (S1 ! T1) rate is smaller than about 1011 s
1, a direct ligand singlet energy transfer to the lanthanide ion may play an important role in luminescence sensitization. Though this rate value (1011 s
1) is certainly not reasonable, either for non-radiative energy transfer or intersystem crossing (a value of ca. 108 s
1 would be more reasonable, as known from molecular spectroscopy) [73,316,317], the issue of IET through the singlet state remains a challenge and should be clarified. Indeed, experimental results for some complexes indicate that the sensitization via the singlet state is feasible [16,318]. As it was discussed in Section 2.2, the IET pathway via either triplet or singlet states is a matter of appropriate resonance conditions, distance of the donor barycenter state to the Lnz+ ion, and selection rules. Thus, designing Lnz+ complexes with a singlet pathway for sensitized emission could be an interesting strategy that possesses several advantages [321–323]. This sensitization process is characterized by the following photoluminescent properties: it is insensitive to quenching by oxygen, known to be a triplet quencher; it allows to increase the absorption range towards the visible spectrum and contributes, for example, to the possibility of obtaining compounds for white-light-emitting materials in which the red Eu3+ luminescence is combined with the blue luminescence from the chromophore. However, dominant singlet energy transfer has seldom been observed, as the intersystem crossing is usually very fast due to the external heavy atom effect of lanthanide ions, and therefore an increase in the weak spin-orbit interaction in the ligand. As a result, only few works have demonstrated the singlet pathway as being a dominant process in the sensitization of the lanthanide ion

120 Handbook on the Physics and Chemistry of Rare Earths

emission [16,20,90,315,318–339]. Furthermore, it is difficult to prove the participation of the S1 state in feeding the excited levels of the Lnz+ ion and, moreover, the contribution of a singlet pathway to the overall transfer is often only partial. Nevertheless, singlet energy transfer is proposed as the main pathway in the luminescence sensitization of Ln3+ ions in the near-infrared [322,323,325,330, 333,334] and visible [16,20,90,315,318–321,324,326–329,331,332,335–339] spectral range. The NIR sensitization emission has been investigated based on intensity, quantum yield and lifetime analysis from the ligand-related fluorescence of the free ligand, as well as the Gd3+ complex, to mimic the investigated luminescent lanthanide complexes. The similarity of the ligand fluorescence intensity (or fluorescence lifetimes) for the Gd3+ complex recorded at 300 K in comparison with the free ligand, combined with the absence of the ligand phosphorescence broad band at 77 K of the Gd3+ complex are the basis for finding that the S1 ! T1 intersystem crossing induced by the external heavy atom effect is not very operative. Decrease of the ligand fluorescence intensity (or shortening of the fluorescence lifetime) in lanthanide complexes in comparison with the Gd3+ complex, as well as the lack of differences in 4f-4f luminescence intensities for aerated and deoxygenated solutions of the Ln3+ complexes, indicates the energy transfer from the ligand singlet state to the metal ion. In these cases, the S1 state is partially quenched by the Ln3+ ions and the non-radiative depopulation of the T1 states by dioxygen may not be reflected in the decrease of the 4f-4f emission intensity. Singlet sensitization of Nd3+ and Er3+ luminescence by dansyl and lissamine derivatives and by a dendrimer with dansyl units have been successfully reported [322,323,325]. Using the strategy described above, in combination with donor-acceptor spectral overlap conditions and energy transfer selection rules, a ligand-to-metal singlet energy transfer mechanism has been proposed. The sensitization of Er3+ and Yb3+ complexes with boron-dipyrromethene (bodipy) ligands with methoxyphenyl moieties was also proposed from the lowest S1 state, which is resonant to the 4F9/2 emitting level of the Er3+ ion, but is out of good resonance conditions with the Yb3+ excited level. A similar strategy was used for Ln3+ complexes with dendritic 9,10-diphenylanthracene ligands [330] and the ones with a tetraaryl porphyrin bearing four hydroxyquinolinyl chelating units and its Pd2+ complex [333]. The photophysical data of the free ligand fluorescence were also reported in the case of the dendritic complexes with Nd3+, Er3+, and Yb3+, while the Gd3+ complex was only used to confirm the absence of phosphorescence at 77 K [330]. In the substituted porphyrin case, the energy transfer process in Nd3+ complexes has been assigned to the singlet excited state of the porphyrin and the phosphorescence of the compounds containing both Pd2+ and Nd3+ or only Pd2+ were used to support this mechanism [333]. Singlet energy transfer as the main pathway of Eu3+ and Tb3+ visible luminescence sensitization has also been proposed [16,20,90,315,318–321,324,328, 329,331,332,335–339]. For Tb3+ chelates, the S1 singlet process may be of

Modeling intramolecular energy transfer in lanthanide chelates Chapter

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relevance particularly when the T1 state lies below or very close to the emitting 5 D4 state, a situation in which luminescence quenching is normally efficient [90,324,329,338,339]. In addition, singlet sensitization has been described for Eu3+ and Tb3+ complexes even for a T1 state with energy above the 5D0 and 5 D4 emitting levels, respectively [16,20,315,318–321,328,331,332,335–337]. For instance, nonradiative energy transfer rates for a Tb3+ complex with 1-phenyl-3-methyl-4-(trimethylacetyl)pyrazol-4-one ligand derived from time-gated and time-correlated single photon counting measurements and subsequent comparison with the Gd3+ complex were also used to support this mechanism [328]. Several spectroscopic techniques such as fast kinetics measurements, time-resolved luminescence and triplet-triplet transient absorption spectroscopy (TAS) have been shown very useful for investigating the sensitization pathway via singlet state. These techniques were used to prove the sensitization of Eu3+ complexes via S1 state composed of 2,6-pirydine dicarboxylic acid with a polyoxyethylene arm, fitted with a coumarin ligand at its extremity [319]. A direct correlation between the risetime of the 5D1 level and the decay time of the S1 level, and also between the risetime of the 5D0 level and the decay time of the 5D1 level, the absence of Eu3+ luminescence during the long-lived triplet emission and the identical time evolution of T-T TAS in Eu3+ and Gd3+ complexes were the basis of the proposed mechanism. The same techniques, in combination with the determination of energy transfer rates, were used by Ward and co-authors [320], to investigate the photophysical properties of the Eu3+ complex with dota-appended 1,8-napthalimide ligand in solution. The sensitization of lanthanide emission by the naphthalimide chromophore has both fast (WET  109 s
1) and slow (WET  104 s
1) components. The authors suggested that the energy transfer from the ligand S1 and T1 excited states take place, with the balance between the two pathways due to S1 ! Eu3+ energy transfer and intersystem crossing (S1 ! T1) occurring on similar timescales [320]. Eu3+ luminescence sensitization from both triplet and singlet states was also proposed for the diphenylphosphanoethanate complex in solution [331]. Based on the TAS measurements, energy transfer via S1 state as well as LMCT state was suggested for the Eu complex with β-triketonate ligands in solid state [336]. Systems containing two types of ligands, excited states of which can commonly participate in energy transfer processes, are particularly difficult to analyze. For example, the Eu3+ complex containing three thenoyltrifluoroacetonate (tta) and an ancillary dipyrazolyltriazine ligand (L) were analyzed twice by the same authors [321,332]. Using the fast-kinetic measurements in the range of nano- and millisecond timescales, as well as time-resolved luminescence, the singlet pathway of sensitization was proposed as the governing mechanism [321]. Carlos et al. [20] proposed for the same system the following sensitization path: 1ILCT(S1(L)) ! T1(L), in which the T1(L) state transfers energy to both T1(tta) and the Eu3+ ion with rather low rates. In this scheme, even though the transfer rate from the

122 Handbook on the Physics and Chemistry of Rare Earths

T1(tta) state to the Eu3+ ion is considerably high (108 s
1), its lifetime is governed by the lifetime of the T1(L) state. Besides, it was noticed that the observation of 5D0 emission after 1 s is a clear indication that the 5D0 level is still being populated in the long-time scale via triplet states. Carlos et al. [20] emphasized that the proposed sensitization pathway does not exclude the participation of the S1(L) state in the direct population of the 5D1 level of Eu3+, which might be considerably high. The Eu(tta)3L complex was once again investigated by Fu et al. [332] by means of ultra-fast spectroscopic technics. The participation of 1LMCT state in ligand-to-metal energy transfer was proposed, in which the 1LMCT state effectively depopulates the S1(L) state and feeds the 5D1 excited level of Eu3+. Dominant contribution of IET from the ligand S1 state to the Eu3+ and Tb3+ ions by sulfonylamidophosphates (L) was suggested in references [16,318]. Chemical formulae of the ligands are shown in Fig. 20. The IET processes occurring in this new family of compounds, Na[Ln(L)4], were analyzed on the basis of experimental data and detailed theoretical results. The energy transfer rates WET from S1 and T1 ligand states were calculated using the theoretical model described in Section 2. Moreover, the higher lying excited levels of Eu3+ (5DJ, 5LJ, 5GJ) [16,318] and Tb3+ ions (5DJ, 5GJ, 5LJ, 5HJ, 5FJ, 5IJ) [16] were included in the calculations for the first time and their crucial role in the IET process was shown. Additionally, the important role of the 7F5 level of Tb3+ in the energy transfer process was also emphasized [16]. Namely, new energy transfer paths becoming possible from S1 to higher 4f excited levels as well as an energy transfer from T1 to the 5D4 level are facilitated due to the abnormally long lifetime of 7F5 level as described by Souza et al. [340]. For the acquisition of experimental data, spectroscopic techniques such as fast kinetic measurements in the range of nano- and millisecond timescale, as well as absorption and emission measurements, were used for these complexes [16,318].

FIG. 20 The structural formula of the sulfonylamidophosphates ligands.

Modeling intramolecular energy transfer in lanthanide chelates Chapter

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The theoretical approach for energy transfer rates was successfully applied to the rationalization of experimental data. The transfer rates from the S1 state to the 5D3, 5L6, 5L7, 5G2, 5G3, 5G5 and 5G6 excited levels of Eu3+ were theoretically calculated. Taking into account the selection rules (Section 2.2.4) and the 7F0,1 thermal populations at 300 K, the IET from the S1 to 5D3, 5L6, 5L7, 5 G3, 5G5 levels becomes allowed through the multipolar (dipole-2λ pole and dipole-dipole) mechanisms and the energy transfer to the 5D0, 5D2, 5G2 becomes allowed through the exchange mechanism. All calculated values of the energy transfer rates from the singlet and triplet states (dipole-multipole and exchange interactions as well as back energy transfer), and other quantities are listed in the Supporting Information of Refs. [16,318]. The main channels of energy transfer in Na[Eu(L1)4], Na[Eu(L2)4], Na[Eu(L3)4], Na[Tb(L1)4] and Na[Tb(L2)4], as well as the energy transfer rates from S1 and T1 states are presented in details in Refs. [16,318]. Based on these data, the authors concluded that the energy transfer in these Eu3+ and Tb3+ complexes occurs from both ligand singlet and triplet states. However, the strong experimental and theoretical evidence indicated a dominant role played by the ligand excited singlet state in this process. The following comments on the ligand-to-metal energy transfer via singlet state can be made. Sensitization of the lanthanide emission via ligand S1 state requires that the S1 ! T1 be slow enough for the S1 ! Ln3+ energy-transfer to become competitive. It is known that ISC depends, among other factors, on the energy difference between the S1 and T1 states. When the energy difference is large, the efficiency of this process becomes limited. An additional factor decreasing the ISC can be the presence of a tertiary amine, which has an abnormal quadrupole moment responsible for the highly forbidden character of the S1 ! T1 transition [325]. The amine moiety is found in many of the complexes that were suggested to exhibit singlet pathway sensitization [16,318,320,322,323,325,328]. Moreover, additional ISC contribution from the external heavy-atom effect should be not operative in the presence of S1 ! Ln3+ nonradiative energy transfer. A large distance between the organic chromophore and the lanthanide ion favors this fact. Indeed, a lot of Ln3+ coordination compounds which reveal sensitization via singlet are characterized by a large donor-acceptor distance. In some of these complexes, lanthanide ions are separated from the chromophore unit by a spacer [319,320,325,330,331,334]. For multipolar mechanisms, only the 4f-4f transitions with significant squared matrix elements (U(λ)) may be considered. For the exchange mechanism, only the levels which have a reduced matrix hψ 0 J0 kSkψJi 6¼ 0 can be taken into account. However, if an IET process is allowed through the exchange mechanism, which is disfavored by the very small orbital overlap between the donor and the acceptor wave functions, despite a good energy mismatch, they tend to give a small contribution to the intramolecular energy transfer process. This is the case of large donor-acceptor distances.

124 Handbook on the Physics and Chemistry of Rare Earths

The investigation of the role of the singlet state in the IET process in lanthanide complexes continues to be a challenge. Suitable tools are needed for this purpose, such as sophisticated equipment for measuring fast kinetics. Energy transfer processes in lanthanide coordination compounds are a complex phenomenon due to the presence of numerous deactivation pathways competing with each other. For this reason, any conclusions regarding these phenomena should always be made with utmost caution and the crucial roles of experience exchange and scientific discussion make them indispensable to further advances in this issue. Detailed calculations of IET rates, for several pathways in the Eu3+ and 3+ Tb complexes with sulfonylamidophosphates (Fig. 20) have been performed [16]. Dipole-quadrupole (Wd
q), dipole-dipole (Wd
d) and exchange (Wex) mechanisms were taken into account, as well as the total energy transfer rates (WET) from the singlet and triplet states. The large donor–acceptor distance, RL, and an expected significant decrease of the ISC (S1 ! T1) rate have been evoked to explain, in these particular cases, the large predominance of a IET via the singlet state. For the sake of illustration, some representative data extracted from Ref. [16] are displayed for the Na[Eu(L1)4] and Na[Tb(L1)4] chelates in Tables 4 and 5, respectively. An interesting aspect in the case of Tb3+ chelates is the participation of the 7F5 level in the kinetics of IET processes due to its abnormally high lifetime [340].

TABLE 4 Some representative values of the energy transfer rates via different pathways from the 7F0 ground level in the Na[Eu(L1)4] chelate [16]. Transfer

Mechanism

Δ

Wd2q

Wd2d

Wex

W∗

S1 ! 5D4

Multipole

7356

2.39  104

4.27  104

0

4.26  104

S1 ! 5G6

Multipole

8235

172.64

2.09  105

0

1.34  105

S1 ! 5L6

Multipole

9751

450.98

5.46  105

0

3.49  105

S1 ! 5D1

Exchange

16,048

0

0

5.81

T1 ! 5D1 T1 ! D4 5

T1 ! L6 5

3.72 5.26  105

WET from S1 Exchange

4216

Multipole


4476

a

0

Multipole


2081

a

0.15 8.20  10


3

0

347.91

222.66

0.26

0

0.26

9.94

0

6.36

WET from T1

229.29

Negative values of Δ correspond to ligand donor state lying below the Eu acceptor state. W ∗ is the sum over the dipole-quadrupole (Wd
q), dipole-dipole (Wd
d) and exchange (Wex) mechanisms multiplied by the thermal population factor of the 7F0 (at room temperature ca. 64%) in the same pathway. Δ (in cm
1) is the energy mismatch that appears in the factor F, Eq. (43). All the energy transfer rates are in s
1. The values of the total energy transfer rates WET (via S1 or T1) are highlighted in bold characters. For the 7F0 ! 5D1 transition, the matrix elements of the spin operator (Eq. 34) is h5D1kSk7F0i ffi
0.165. a

3+

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 125

TABLE 5 Some representative values of the energy transfer rates of different pathways from the 7F6 level for the Na[Tb(L1)4] chelate [16]. Transfer

Mechanism

Δ

Wd-q

Wd-d

Wex

W*

S 1 ! 5L 7

Multipole and Exchange

5556

8.56  105

6.26  104

0.03

9.19  105

S1 ! 5G5

Multipole and Exchange

7132

1.22  106

5.22  104

20.27

1.27  106

S1 ! 5D3

Multipole

8752

150

3.58  103

0

3.74  103

14,474

6.34  10

476

0

6.39  104

S1 ! D4 5

Multipole

4

2.26  106

WET from S1 T1 ! 5D3

Multipole


3080a

1.8  10
3

0.042

0

0.044

T1 ! 5D4

Multipole

2642

7.42

0.056

0

7.47

T1 ! 5G6

Multipole and Exchange


3412a

11.70

0.376

635

647

WET from T1

654

a Negative values of Δ correspond to ligand donor state lying below the Tb3+ acceptor state. W∗ is the sum over the dipole-quadrupole (Wd
q), dipole-dipole (Wd
d) and exchange (Wex) mechanisms in the same pathway. Δ (in cm
1) is the energy mismatch that appears in the definition of factor F, Eq. (43). All the energy transfer rates are in s
1. The values of the total energy transfer rates WET (via S1 or T1) are highlighted in bold characters. The matrix elements of the spin operator (Eq. 34) are h5L7kSk7F6i ffi 6.38  10
3, h5G5kSk7F6i ffi
0.195 and h5G6kSk7F6i ffi
0.745.

5.2 Experimental energy transfer rates The experimental evaluation of intramolecular energy transfer rates in trivalent lanthanide chelates has been usually performed by three procedures depending on the availability of experimental photophysical data [17,315,321,338,341,342]: (i) Fluorescence or phosphorescence lifetime, either from the isolated ligand or from the ligand coordinated to a non-luminescent lanthanide ion (La3+, Lu3+, Y3+, or Gd3+) complex, is measured. Subsequently, the same ligand lifetime measurement is carried out on the complex with a luminescent Ln3+ ion (for example, Eu3+ or Tb3+), giving a shorter ligand state lifetime if an operative ligand-to-Ln3+ IET is present. Then the IET rate (WET) is assumed to be the difference of the inverse of these lifetimes. Obviously, this method can only be used if the ligand luminescence can still be detected in the presence of the IET process. at low (ii) The radiative lifetime of the ligand luminescence (τr) is measured τexp temperature and in the absence of IET. Then the ratio is taken τr ligand

126 Handbook on the Physics and Chemistry of Rare Earths

as being proportional to the IET rate, where τexp is the ligand luminescence lifetime in the presence of IET, also at low temperature. Again, as in the previous case, this method can only be applied provided the ligand luminescence can be detected. (iii) The risetime in the transient curve of the lanthanide emitting level is measured, and its inverse is taken as the IET rate. This method can be used provided the IET rate is much higher than any rate from other potential channels populating the lanthanide emitting level. In this case, this method leads to a direct and unambiguous experimental evaluation of the IET rate. The inherent experimental error in each of these methods will depend on the characteristics of the experimental setup. However, a crucial aspect is the harvesting and interpretation of experimental data, particularly when the IET kinetics is not described by an appropriate system of rate equations (see Section 2.4), as this is usually overlooked in methods i) and ii). Thus, for example, the intersystem crossings (ISC) S1 ! T1 and T1 ! S0, play a role of paramount importance in the IET overall kinetic balance in the rate equations. Using an appropriate system of rate equations has shown that over or underestimates of these ISC rates may change the final lanthanide emission quantum yield (QLLn) very significantly, depending on the predominant pathway channel (singlet or triplet). The heavy atom effect is certainly important with respect to this issue. Order of magnitude estimates from molecular spectroscopy [343] makes it possible to assume typical values for the non-radiative rates of ISC (S1 ! T1 108 s
1) and (T1 ! S0 106 s
1), assuming that the energy differences are S1
T1 ≳ 5000 cm
1 and T1
S0 ≳ 20000 cm
1. These ISC non-radiative rate values are in clear contrast with values assumed in the literature (see Ref. [315] and references therein). Theory indicates [9,11,12,18] that, for example, in the cases where the ISC non-radiative rate T1 ! S0 is higher than 106 s
1, the emission quantum yield (QLLn), when the triplet pathway is the dominant one, decreases very fast. To the best of our knowledge, the only application of method iii) together with a detailed theoretical analysis has been performed for the typical [Eu(tta)3(H2O)2] complex (entry 10 in Table 6) in Ref. [14]. It is seen in Table 6 that the reported experimental values of IET rates may vary by several orders of magnitude. The transition metal (d) to Lnz+ (4f ) IET rates, in some cases considered to be mediated by bridging ligands, are considerably high even at large d-4f distances [345]. These values could be explained by the fact that d elements do not suffer shielding effects compared to 4f ions and they are much more covalently bonded to the ligand. Moreover, d donor dipole strengths (SL) are high, in comparison with the values for 4f-4f transition. The large widths of d bands and energy mismatch conditions (the F factor) favor large IET rates. In the cases of organic ligand-to-Lnz+ IET, the situation is more complex. There is in the literature much discussion on the IET channel (pathway).

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TABLE 6 Some experimental values of WET collected from the literature. Entry

Complex

WET (s21)

References

1

Ir(pdt)2(μ-bpmc)Nd(tta)3

3.6  109

[344]

Os–Ph-Er(tta)3

6.6  10

[345]

Ir(ppy)2(Hdcbpy)-Yb(NO3)3

5.0  10

[346]

[(Znq2)2](m-CH3COO){Tb(hfac)2}

3.6  10

[347]

Eu(hfaa)4ButNH3

1.1  10

[348]

10

[315]

>10

[349]

3 4 5 6 7

Various Ln 3+

3+

5

chelates

3+

with tta, hfa, dpo and

5 7 6

10 7

8

Eu and Tb acac

9

Ln3+ (Sm3+, Eu3+, Tb3+, Dy3+) with dbm and acac

[107–1011]

[350]

10

[Eu(tta)3(H2O)2]

1.4  107

[14]

On this issue, our conclusion is that both IET channels through ligand triplet and singlet states may be operative. However, theoretical support indicates that the dominant pathway depends on several parameters. As mentioned before, the first one is the donor-acceptor distance RL, which is a quantity that must be evaluated. In our modeling scheme this quantity has been determined either from previous knowledge on where, in the ligand, the donor state is approximately localized (the center of an aromatic ring, for example) or by using the Eq. (87) from electronic structure calculations [9]. Last but not least, are the selection rules (Table 1) from both sides donoracceptor (SL for the ligand and ΔJ for the Lnz+). Together with the energy mismatch factor F, these points define the values of WET for each channel. Of course, in treating the overall kinetics of IET processes, back-IET must be taken into account. These facts might be one of the reasons for possible misinterpretations of experimental data values of WET. We conclude that theoretical support is crucial in interpreting these experimental data. Thus, for example, we have reasons to disagree with the conclusions in Refs. [315,349]. It might be useful to consider the arguments evoked in Ref. [20] contradicting the specific case considered by Yang et al. [321]. Our arguments are based on the fact that an appropriate setup and solution of rate equations would be necessary as it has been done in several cases [9,18,20,35,351]. Sato and Wada [352] studied IET processes in Lnz+ (Eu3+ and Tb3+) chelates, using Gd3+ analogous complexes for the determination of triplet ligand state by using the zero-phonon energy. They described the IET kinetics using the appropriate rate equation in the steady-state regime to evaluate emission quantum yields (QLLn).

128 Handbook on the Physics and Chemistry of Rare Earths

In the work of Latva et al. [353], it is proposed that efficient IET would occur provided the energy difference Δ between donor (ligand) and acceptor (Lnz+ ¼ Eu3+ and Tb3+), should not be higher than 1850 cm
1, though no detailed discussion was made on the determination of the value of Δ, if the ligand zero-phonon or the barycenter energy was used and, moreover, the reported results were obtained in solution and do not take into consideration quenching processes. This point can make a considerable difference in the IET mechanisms and rates. Hayes and Drickamer [354] have performed an interesting experiment by applying high pressure on Eu3+ chelates, inducing significant changes in the energies of the singlet and triplet states. The emission intensity from the Eu3+ ion was studied as a function of these energy changes. They claimed that the IET process is dominated by the ligand triplet state. However, a possible role of the singlet state was not discussed. A theoretical detailed treatment of their experimental data, based on the triplet state, was successfully performed in Ref. [18]. In addition to the experimental aspects, from the theoretical point of view, we are able to fundamentally describe the IET rates and rationalize why their values can differ, in some cases, by orders of magnitude. A quantity that affects enormously the IET rates, besides the selection rules, is the donor-acceptor distance RL, mainly for the exchange mechanism, which ˚ [12]. These two becomes practically negligible for RL larger than 4–5 A aspects together with the energy mismatch factor F determine the relevant IET channels.

5.3 Application of the theory: An example Our purpose now is to present a concrete example of an application of the theoretical scheme described in the previous sections. For this purpose, we have selected the Ln3+ chelates [103]: Na[Ln(sk)4] (Ln ¼ Eu and Tb, sk ¼ di (4-methylphenyl)-phenylsulfonyl-amidophosphate). These two cases were chosen due to: (i) their crystallographic structures have been well resolved (the ligands are the same) as well as (very approximately) both chelates structures (Fig. 21); (ii) the energies of ligand singlet and triplet, and the LMCT, have been assigned; (iii) among several other spectroscopic data, emission spectra (Fig. 22) and quantum yields (QLLn) have been measured; (iv) there has been found an enormous difference between the QLLn values for the Eu3+ and Tb3+ chelates. Formally, the flowchart in Fig. 19 should be followed in order to get a more accurate description of IET and modeling processes, including preferably the numerical solutions of the rate equations. Despite this premise, for the sake of illustration and demonstration on how to get relevant information on IET processes and pathways, analysis of the present example is carried out following a simpler way. The weak absorption by the LMCT state [103] does not imply that it may not act as an efficient 4f-4f luminescence sensitizer or quencher.

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 129

FIG. 21 Crystallographic determined structure of Na[Ln(sk)4] (Ln ¼ Eu and Tb) chelates. Structure obtained in the Cambridge Crystallographic Data Centre (CCDC) [175] via www.ccdc.cam. ac.uk, CCDC code: 860890.

FIG. 22 Emission spectra of (A) Na[Eu(sk)4] and (B) Na[Tb(sk)4]. Adapted with permission from P. Gawryszewska, O. V. Moroz, V.A. Trush, V.M. Amirkhanov, T. Lis, M. Sobczyk, M. Siczek, Spectroscopy and structure of LnIII complexes with sulfonylamidophosphate-type ligands as sensitizers of visible and near-infrared luminescence, ChemPlusChem 77 (2012) 482–496, © 2012 John Wiley and Sons.

130 Handbook on the Physics and Chemistry of Rare Earths

In the present case it does act as a very efficient suppressor of the absorption of ligand S1 state, when it is very closely below it [103]. Moreover, as we have calculated from Eq. (58) the IET (LMCT) rate is extremely high, depopulating strongly the S1 state The LMCT state is known to decay extremely fast, perturbing considerably the populating channel by energy transfer LMCT ! T1, which in this case has been evaluated to be small, on the basis of the factor F and the overall balance in the rate equations. Let us consider the energy level diagram in Fig. 7. According to Gawryszewska et al. [103], the LMCT barycenter practically coincides with the onset of the singlet S1 band. In our example, we assume that the LMCT lifetime is extremely short, making energy back-transfer from the LMCT to S1very little operative in the overall kinetic process. A simple and efficient way to analyze the effect of the S1 ! LMCT transfer channel is to introduce in Eq. (62) (the S1 population) a transfer rate WCT as shown in Eq. (88),

↘

 dN2 1 S S ¼
+ WET + W CT N2 + ϕ + WbET N3 ¼ 0 dt τS

(88)

inclusion of the LMCT state in the population of the S1 state: From the Na[Eu(sk)4] luminescence spectrum (Fig. 22A), at room temperature, we have determined the experimental Ω2 and Ω4 intensity parameters using the areas (S0!J) under the spectral emission curves (Eqs. 73 and 75). By using the fitting procedure in the JOYSpectra program, we have obtained charge factors ( g) and effective polarizabilities (α0 ) values leading to theoretical Ωλ. These g values were then used to calculate the FED contributions to ); these contributions will be the intensity parameters in both chelates (ΩFED λ used in the evaluation of the multipolar IET rates. From Table 7 it may be noticed that in these two cases the DC mechanism is by far the dominant

TABLE 7 Underlying properties to calculate the IET rates and the QLLn. Na[Eu(sk)4]

Na[Tb(sk)4]a

Ω2(ΩFED 2 )

8.54(1.210
3)

6.44(10
3)

Ω4(ΩFED 4 )

1.19(2.1410
2)

3.11(1.1210
2)

Ω6 (ΩFED 6 )

– (3.7610
2)

0.34(1.9110
2)

Arad

281.3

335.2b

a

All the properties were theoretically estimated. h D4 j jL + 2S jj 7F5i2 ffi 2.2. Intensity parameters Ωλ and the FED contribution (ΩλFED) in units of 10
20 cm2. The spontaneous emission radiative coefficients Arad (s
1). b 5

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 131

one. Indeed, from a collection of our theoretical calculations for lanthanide chelates this seems to be a general aspect, in contrast to the cases of inorganic materials doped with Lnz+ ions. Based on available structural data we have estimated the distance ˚ as an average of geometric centers between the aromatic rings RL ¼ 4.81 A in the same ligand, which may be considered satisfactory. The aromatic rings and groups close to the first coordination sphere are rather far apart. The values of the intraligand overlap integral hπ j ϕ0i have been estimated in Ref. [77]. This overlap integral involves ligand molecular orbitals coupled with the LMCT state (Section 2), and a quantum chemical computational method can estimate them, as well as the distance RL. In the present example, we have proceeded, as emphasized before, in the simplest, but meaningful way to evaluate IET rates and QLLN. The intraligand overlap integral hπ j ϕ0i is assumed to be ca. 0.01, while the overlap integral h4f j πi value was calculated using DFT/BP86 functional [355,356], TZ2P basis set [357], and the inclusion of ZORA scalar relativistic effects [358–360] in the ADF program [214]. The other necessary input data to the appropriate rate equations are shown in Table 8. Once again, in this example, the purpose is to make estimates and to illustrate relevant points that may be extracted from theory, so that we use Eqs. (60)–(70) (Section 2.4.1), substituting Eq. (62) by Eq. (88) in the case of the Eu3+ chelate. For the Na[Tb(sk)4] chelate, we have estimated the spontaneous emission coefficients AJ!J0 (Eq. 11) for each 5D4 ! 7FJ (J ¼ 6, 5, 4, 3) transition using the g and α0 values from the Na[Eu(sk)4] compound to calculate the theoretical Ωλ. This can be assumed because both lanthanide ions are surrounded by very similar chemical environments. In addition, in the case of Na[Tb(sk)4], the large magnetic dipole contribution of the 5D4 ! 7F5 transition (AMD ffi 140 s
1) must be taken into account in the calculation of the total Arad. The final results for the emission quantum yields are considered to be very satisfactory. They are indeed remarkable in the sense that the very small QLLn value for the Eu3+ chelate

TABLE 8 IET rates (s21) for Na[Eu(sk)4] and Na[Tb(sk)4] chelates. Channel

IET rate

Na[Eu(sk)4]

Na[Tb(sk)4]

2.08  10

–

S1 ! LMCT

WCT

S1 ! j 3i

WSET

ca. 10

ca. 108

T1 ! j 4i

WTET

3.14  108

60.5

QLLn

Experimental

0.42

20.0

Theoretical

0.40

28.5

(%)

a

10

8

a The values of h4f j π∗i ¼ 0.067 and hϕ0 j πi ¼ 0.01 were used in Eq. (58). The donor states energies used were 35,210 cm
1 (S1) and 21,420 cm
1 (T1).The donor-acceptor distance is RL ¼ 4.81 A˚ for both cases.

132 Handbook on the Physics and Chemistry of Rare Earths

can be rationalized. If WCT in Eq. (58) is made equal to zero, this emission quantum yield value changes drastically to over 40%, and if WCT is two orders of magnitude smaller, the QLLn jumps to ca. 30%. Another remarkable aspect is that these results for QLLn are predicted in excellent agreement with the experimental values, despite the large difference in the IET rates from the triplet T1 to the lanthanide ion due to the large contribution from the exchange mechanism to the T1 ! 5D1 channel, as may be observed in Table 8. The extremely high value of WCT is due to the very favorable energy mismatch condition in the factor F and favorable overlap integrals involving the LMCT state. We would like to stress that for the Eu3+ chelate in Table 8, the value of the theoretical QLLn compared to the experimental value, should be regarded rather as a coincidence. The relevant and important fact to be stressed here is the predicted enormous decrease in QLLn produced by the LMCT effects on the singlet state.

6 Challenges and perspectives As mentioned in the Introduction, this chapter is grounded on a theoretical development we have been using during the last two decades to rationalize IET rates and emission quantum yields (QLLn) in lanthanide chelates, in particular, trivalent europium and terbium ions, albeit the theory has a general character. The theoretical expressions for the IET rates described in Section 2, particularly their contributions from both the Coulomb direct and exchange interactions, allowed us to identify the relevant factors for each specific case (Lnz+ chelate). This, together with appropriate systems of rate equations enabled to decipher the dominant IET mechanisms and pathways and to understand the role of singlet, triplet, and LMCT states, which had stirred interesting discussions in the literature. We emphasize that the knowledge of structural data and ligand energy levels are of paramount importance in this procedure. As stressed before, the relevant quantities are: the distance RL (formally defined in Eq. 87), matrix elements and selection rules (from both sides, ligand (SL) and Lnz+, Table 1), forced electric dipole (FED) Judd-Ofelt intensity parameters, the energy mismatch factor F and, highly important, the shielding factors σk (k  2). The conjunction of these quantities determines the values of IET rates, which may vary considerably from case to case, and together with an appropriate system of rate equations may lead to correctly predicted values of QLLn, lifetimes and other photophysical properties. The influence of the donor ligands and the structure of the chelates on these parameters can be understood by this modeling. Unlike the formulation of the direct Coulomb interaction that has not varied during the last two decades, the exchange interaction contribution to the IET rates has been formulated in different ways depending on the form of the exchange interaction Hamiltonian assumed. In the context of the independent systems model and the two centers bipolar expansion, we suggest (see Section 2) that, the exchange interaction Hamiltonian should be

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 133

expressed in terms of the shielding factor σ0, which is certainly very close to 1, and that (1
σ 0) should be conveniently given by Eq. (42), which is an overlap integral and confers to (1
σ 0) a high sensitivity on the distance RL. As for the shielding factors σ 1 and σ 2, (1
σ 1) entering in the expression of the dipoledipole mechanism while (1
σ 2) appearing in the dipole-quadrupole one, the use of Eq. (41) is suggested, as in previous works [9,12,35]. It is important to emphasize that whenever overlap integrals involving the 4f sub-shell are present, there is no need of including shielding factors, once shielding is naturally included in the overlap values. The shielding factor σ 0 can be related to the overlap integral h4fj Li according to Eq. (42), which is equivalent to the formulation described in Ref. [12]. Estimated values of these overlap integrals may be consistently obtained, at least in terms of order of magnitude, from structural data and molecular electronic calculations as described in Section 4. Each case must be specifically treated, though some general points are quite clear concerning IET processes. In fact, to quantify the model for calculating the IET rates, chemometric and sensitivity analyses are needed [361]. This is a relevant perspective because the equations for the IET rates involve quite a number of variables (RL, Δ, ΓL, σ k, SL, Ωλ, etc.), which requires a multivariate treatment to establish the principal factors and their interactions. A challenge for the experimentalists consists of measuring accurately and reproducibly the emission radiance of lanthanide chelates. This photophysical property is relevant, even more than the quantum yield, for applications related to luminescence. Its calculation with the IET rates and the corresponding rate equations is quite straightforward from the population of the emitting state and the radiative decay rate, as long as the absorption rate is known. In fact, such measurements and comparisons would represent a stringent test for validating the model and the equations for the IET rates. Remarkable results can be obtained with the model, as demonstrated in the selected cases considered (Section 5), for which application of the theory in its simplest and illustrative form, elucidated the role of singlet states in the IET and produced accurate results compared to experimentally measured IET rates. IET processes in Lnz+ chelates are still under intense investigation, motivating several fundamental and experimental questions on this subject. This is not only a matter of technological applicability, which, of course, has been of paramount importance [21,34,139,362–365]. One example is the (nano)thermometry based on luminescence for which the accurate calculation of ligand–ion and ion–ion energy transfer (and back-transfer) rates is crucial for a complete understanding of the thermal sensitivity of the materials. Nevertheless, understanding and, consequently, designing, controlling, and modeling these marvelous luminescent molecular systems is important and always asking for deeper comprehension.1

1

If necessary, under request, any specific data or information may be obtained from the authors.

134 Handbook on the Physics and Chemistry of Rare Earths

Acknowledgments The authors (A.N.C.N., E.E.S.T., G.F.S., H.F.B., M.C.F.C.F., R.T.M.Jr., R.L.L., W.M.F., O.L.M.) would like to thank the Brazilian agencies CNPq, CAPES, FACEPE, FAPESP, and FINEP for providing partial financial support under grants PRONEX APQ-0675-1.06/14, APQ-1007-1.06/15, CNPq-PQ fellowships, as well as the Institutions Universidade Federal de Pernambuco (dQF-UFPE), Universidade Federal da Paraı´ba (DQ-UFPB and DQF-UFPB), Universidade de Sa˜o Paulo (IQ-USP), Instituto de Pesquisas em Energia Nuclear (IPEN). This work was partially developed within the scope of the project CICECO-Aveiro Institute of Materials, FCT (Portuguese agency) Ref. UID/CTM/50011/2019, financed by national funds through the FCT/MCTES. A.N.C.N. also thanks SusPhotoSolutions—Soluc¸o˜es Fotovoltaicas Sustenta´veis, CENTRO-01-0145-FEDER-000005. The authors are grateful for Dr. Ewa Kasprzycka for providing data from her doctoral dissertation.

Appendix A Radial integrals values The values of hr8i were obtained from the values of hr2i, hr4i and hr6i by using extrapolation of the type:  k 2 (A1) r ¼ eða + bk + ck Þ where a, b and c are fitted parameters (Table A1) obtained to adjust the values of hr2i, hr4i and hr6i from Ref. [65]. Table A2 shows the values of the radial integrals.

TABLE A1 Values of a, b and c used in the calculations to obtain the values of hrki. Ln3+

a

b

c

Ce


0.17684

0.05627

0.08295

Nd


0.17225


0.03118

0.08521

Sm


0.16976


0.10382

0.08742

Eu


0.16771


0.13734

0.08866

Gd


0.16788


0.16764

0.08973

Tb


0.16932


0.19562

0.09072

Dy


0.16832


0.22387

0.09193

Ho


0.16796


0.25052

0.09311

Er


0.16961


0.27475

0.09412

Tm


0.17029


0.29873

0.09525

Yb


0.27715


0.25171

0.08761

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 135

TABLE A2 Values of hrki obtained from the data in Table A1 introduced into Eq. (A1). The hrki are in atomic units, ak0 ffi (0.529 A˚)k. Ln3+

hr2i

hr4i

hr6i

hr8i

Ce

1.31

3.96

23.27

265.63

Pr

1.21

2.94

15.29

201.92

Nd

1.11

2.90

15.00

153.20

Pm

1.04

2.27

10.60

123.91

Sm

0.97

2.26

10.53

98.94

Eu

0.92

2.02

9.03

82.09

Gd

0.87

1.82

7.82

68.97

Tb

0.82

1.65

6.84

58.66

Dy

0.78

1.50

6.04

50.61

Ho

0.74

1.38

5.37

44.12

Er

0.71

1.27

4.81

38.71

Tm

0.68

1.17

4.33

34.32

Yb

0.65

1.12

3.92

31.56

Lu

0.64

1.03

3.66

29.49

Appendix B The ligand matrix element for the exchange mechanism As emphasized in Section 2.2.1 this matrix element is crucial in the evaluation of transfer rates by the exchange mechanism. It breaks down the usual selection rules on the multiplicities of the ligand initial and final states. Let us then consider: X¼

X m

* ΨN
1

Π∗

  +2 X    ð1Þ r C ð jÞs
m ð jÞ ΨN
1 Π   j j 0 

where the monoelectronic molecular spin-orbitals are:     1 0 1 0 ∗ jΠi ¼ φ, ms and jΠ i ¼ φ , ms 2 2

(B1)

(B2)

136 Handbook on the Physics and Chemistry of Rare Earths

The following relations hold for μz( j) ¼ rjC(1) 0 ( j):   * +  X   ΨN
1 Π∗  μz ð jÞs
m ð jÞ ΨN
1 Π   j   * ! + N
1  X    ¼ ΨN
1 Π∗  μ ð jÞs
m ð jÞ + μz ðj0 Þs
m ðj0 Þ ΨN
1 Π  j z    * + X  N
1   ¼ ΨN
1  μ ð jÞs
m ð jÞ ΨN
1 hΠ∗ j Πi + hΨN
1 j ΨN
1 ihΠ∗ j μz s
m j Πi  j z  D E 1 1 (B3) ¼ hφ0 j μz j φi m0s j s
m j ms 2

2

∗

because hΠ j Πi ¼ 0 and hΨN
1 j ΨN
1i ¼ 1. Using the Wigner-Eckart theorem for the spin matrix element, we obtain for the above matrix element: ! 1 1 D E 1 0 1 1 1
m 2 2 (B4) hφ0 j μz j φið
1Þ2 s ksk 2 2
m0s
m ms   pffiffiffiffiffiffiffiffi where the reduced matrix element 12 ksk12 ¼ 3=2. Taking the square of the above expression, using the orthonormality relation for the 3-j symbols and subsequently summing up over ms and ms0 and dividing by the spin degeneracy (2s + 1 ¼ 2) we obtain: 3 2 X ¼ hφ0 j μz j φi 4

(B5)

If we wish to relate X with the dipole strength (SL) of the ligand transition (a measurable quantity), the following relationship is used: 1 2 2 hφ0 j μz j φi ¼ hφ0 j μtotal j φi 3

(B6)

and in this case we have: 1 2 X ¼ hφ0 j μtotal j φi 4

(B7)

The dipole strength SL is related to the ligand donor state radiative lifetime by: 1 32 π 2 σ 3 SL ¼ τRL 3 ħ

(B8)

where σ is the energy barycenter in cm
1 (or zero-phonon energy, depending on the treatment choice).

Modeling intramolecular energy transfer in lanthanide chelates Chapter

310 137

Abbreviations and symbols acac BOM bpmc bpy bpyO2 bzac CASSCF dbm DC DFT dota dpa dpm dpo Eb ECP edta Et2NCS2 Et2NCS2 F FED Hdcbpy hfa, hfaa, hfac IET INDO INDO/S ISC keto LMCT LMO MLCT pdt Ph2NCS2 phen ppy L QLn Ln QLn S1 S2PPh2 sal Se2PPh2

acetylacetonate bond overlap model 5-bromopyrimi- dine-2-carboxylic acid 2,20 -bipyridine 2,20 -bipyridine-1,10 -dioxide dibenzoylacetonate complete active space self-consistent field dibenzoylmethanate dynamic coupling mechanism Density functional theory 1,4,7,10-tetraazacyclododecane-1,4,7,10-tetraacetate dipicolinate dipivaloylmethanate diphenylene oxide barycenter energy effective core potential ethylenediamine tetraacetate diethyldithiocarbamate diethyldithiocarbamate energy mismatch spectral overlap factor forced electric dipole mechanism 2,20-bipyridine-4,40-dicarboxylic acid hexafluoroacetylacetonate intramolecular energy transfer intermediate neglect of differential orbitals intermediate neglect of differential orbitals screened approximation intersystem crossing ketoprofen ligand-to-metal charger transfer localized molecular orbital Metal-to-ligand charge transfer 1,3-dimethyl-5-phenyl-1H-[1,2,4]triazole diphenyldithiocarbamate 1,10-phenanthroline 2-phenylpyridinate emission quantum yield intrinsic quantum yield or quantum efficiency first excited singlet state dithiophosphinate salicylate diselenophosphinate

138 Handbook on the Physics and Chemistry of Rare Earths

sk SOM sub T1 TAS TD-DFT THF tppo tta C WET ex WET WFT ZPL

di(4-methylphenyl)-phenylsulfonyl-amidophosphate simple overlap model suberate first excited triplet state transient absorption spectroscopy time-dependent density functional theory tetrahydrofuran triphenylphosphine oxide 4,4,4-trifluoro-1-(thiophene-2-yl)butane-1,3-dionate (thenoyltrifluoroacetonate) direct Coulomb energy transfer rate exchange energy transfer rate wave functional theory zero-phonon line

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