Transitiоn-Metal Clusters and Cоmplexes: Ab-Initiо Studies оn Inducing Phоtоexcitatiоn and Phоtоcatalytic Activity

1. Electronic Structure Studies oF Clusters and Complexes Which Mimic Natural Dehydrogenase and Carbon-Monoxide Dehydrogenase Enzymes

The transition metal elements encompass two major assemblies: (i) elements with partly filled d-orbitals and (ii) elements with partly filled f-orbitals. The first transition row includes the elements with partly filled 3d shell. Many of their compounds have intense light absorption bands. The TiO₂ as electrode material was the first one studied for photochemical water splitting in a pioneering work of Fujishima in 1972 [1]. In the years to follow, a huge number of solids and molecular complexes of the 3d elements were studied in either water splitting or CO₂ reduction [2,3,4,5,6,7,8,9,10,11,12,13,14,15]. Theoretical modeling went parallel with experimental studies and in this way reaction mechanisms were discovered, active sites were elucidated, and the road to new active photo-catalytic compositions was traced [5,6,15,16,17,18,19,20,21]. Key parameters can be established by electronic structure methods: thermodynamic stability, proton and electron affinities, light-absorption spectra [16,17,18,19,20,21]. Starting with TiO₂, the studies continued with other oxide clusters and solids. In a solid crystalline material, the photochemical activity depends on the size of the optical band gap. In molecular systems the equivalent is the HOMO-LUMO gap, but in transition metal complexes there are often low-lying excitation states and their most intense excitation bands are determined by evaluation of the UV-VIS spectra. TiO₂, for example, and many other oxide materials have band gaps of the order or exceeding 3 eV, which requires UV light for photoactivation [1,5,7,17,22]. The oxides of the late 3d elements (Fe, Co, Ni, Cu) feature smaller band gaps but still above 2 eV [22,23,24]. The cupric oxide (Cu₂O) is among the promising materials for catalytic and photocatalytic water splitting [25,26], and nanoparticles are of particular interest in CO₂ reduction [20]. The best option, however, is to create materials which absorb largely visible light, and the natural enzymes (ferredoxins, CODH – carbon monoxide dehydrogenase) became a prototype in creating synthetic analogs [8,9,10,11,15,27,28,29,30]. This trend became fruitful, as it introduced transition metal sulfides in the form of clusters, solids, and molecular complexes with variable ligands [8,9,10,11,12,13,14,15,18,19,20,21]. The sulfide materials have smaller band gaps than the oxides, and the light absorption bands in sulfide clusters and complexes can be modified by coordination of ligands. The family of [Fe-Fe] iron hydrogenase enzymes and the complexes mimicking the enzyme active state were most extensively studied. The Fe₂S₂ rhombic core, part of redox enzyme structures, forms butterfly-like complexes with either nitrosyl or carbonyl ligands. The hexacarbonyl Fe₂S₂(CO)₆ has light absorption bands in the visible region, manifested by a strong band at 460 nm [31]. Theoretical studies proved that all of the late 3d transition metals, Fe-Cu, form rhombic disulfides and coordinate ligands [18,21,32].

The electronic structure of the clusters and complexes can be best examined by density functional theory using hybrid functionals such as B3LYP; the bare clusters were studied also by coupled-cluster methods. These calculations provide accurate description of the frontier orbitals, the charge distribution, the proton and electron affinities, and the UV-VIS spectra. Both the bare M₂S₂ clusters (M = Fe, Co, Ni, Cu) and their coordination compounds have a flexible geometry and the M₂S₂ core can be either planar, or deviate significantly from planarity, Figure 1 a,b. Up to six carbonyl ligands can be coordinated by M2S2, three on each metal center to form a hexacarbonyl, but the tetracarbonyl complexes are also stable, Figure 1c. With nitrosyl ligands, the tetra-nitrosyl was the highest coordination reported, Figure 1d.

The flexibility of the M₂S₂ core is a result of the highly delocalized bonding orbitals in these compounds, Figure 2. In the global minimum configuration of Fe₂S₂(CO)₆, the HOMO features Fe-Fe and S-S binding and is delocalized over the Fe₂S₂ core and the carbon atoms from the carbonyl group, while the LUMO is largely a non-bonding orbital located on sulfur centers. The flat-core isomer has its HOMO as non-bonding orbitals on sulfur atoms; the LUMO features higher delocalization. As the two isomers are separated by an energy gap of 0.923 eV, and both are diamagnetic, light absorption can easily induce isomerization. This plays a major role in the ability to efficiently absorb reactants.

The reactions of water splitting and of CO₂ reduction to either carbon monoxide (CO), formic acid (HCOOH), methanol (CH₃OH) or even deeper reduction to hydrocarbons, are all highly demanding in energy. Just the one-electron reduction of CO₂ to form the radical-anion CO₂^•− is endothermic and requires –206.5 kJ mol^-1. The situation changes when protons are included, in a mechanism following the route of CODH (carbon monoxide dehydrogenase) enzymes: CO₂ + 2H⁺ +2e⁻ ⇒ CO + H₂O already needs 74 kJ mol^-1, and the more desirable reaction to formic acid CO₂ + 2H⁺ +2e⁻ ⇒ HCOOH needs slightly higher energy of 82 kJ mol^-1 [9]. Numerous studies proved that the enzyme-mimicking transition-metal disulfides catalyse both water splitting and carbon dioxide reduction [18,19,20,21,29,33]. It has been argued, that sulfide-based catalysts may be unstable and undergo transformation to hydroxides, or oxo-hydroxides in course of the reaction of water splitting [14]. This, however, depends on the catalyst design. The thermodynamic stability of the disulfides as cores in coordination compounds was proven to be sufficiently high [8,9,11,15,18,19,21,29]. In water as solvent, iron sulphide clusters are stabilised by coordination of water molecules [34,35]. Ni-Fe sulphide photocatalysts, the direct analogue of CODH enzymes, also proved stable to fragmentation in thermodynamical studies [15]. An example of the possible fragmentation paths for Fe₂S₂(CO)₆ and Co₂S₂(CO)₆ is presented in Table 1. All decomposition steps are endothermic, the lowest decay reaction energy is for the loss of a carbonyl ligand from the cobalt complex, but it still requires 151 kJ mol^-1.

As proton transfers play a major role in CO₂ reduction, the proton affinity and the proton-electron affinity of the catalysts is a key marker in predicting their activity. Moreover, it has been experimentally determined, and proved in computational studies, that transition metal disulfide complexes with M₂S₂ core attach hydrogen as a ligand. While protonation always occurs at a sulfur center, hydrogen is attached either at sulfur, or at the metal cation center, usually midway coordinated to both cations, Figure 3. The highest energy gap of 79 kJ mol^-1 between S-H bonded hydrogen and M-H-M bonded hydrogen is found for the cobalt-sulfur hexacarbonyl complex Co₂S₂(CO)₆, Table 2. In the tetracarbonyl complex, however, this gap is reduced to 47 kJ mol-1 and interconversion between the two forms is much easier to achieve, as shown in Figure 3. These results illustrate the possibility of finely tuning the properties of these complexes by simply changing the number of ligands. For the Fe₂S₂(CO)₆ complex, the energy difference between the two isomeric hydrogen-bonded complexes is even smaller, 23 kJ mol^-1, which indicates that the two forms can coexist.

The proton affinity is always much higher than the proton-electron affinity, Table 2. As protons are always attached at sulfur, the proton-electron couples will also form on sulfur, and activation of the complexes to the active state of easily transferrable hydrogen can be reached by visible light irradiation. Similar trends in ligand binding, proton affinities and proton-electron affinities are observed with the next element of the chalcogenide family, selenium and the selenide complexes with [M2-Se2] core. Chalcogenide materials and more specifically, selenides, have been explored in electro-catalysis for HER (hydrogen evolution reaction), OER (oxygen evolution reaction) and also as energy storage materials [13,36,37]. As structural analogs of the rhombic transition metal disulfides they differ by the longer M-Se bonds, the preference to non-planar structure, and higher stability of tetracarbonyl complexes over hexacarbonyl complexes [21]. The proton affinities of selenide complexes are lower than those of sulfide complexes, but the proton-electron affinities are quite similar.

2. Light Absorption Properties oF Chalcogenide Complexes oF the 3D Elements and Photocatalytic Activity in Water Splitting

After the first elementary step of proton-electron couple [H⁺,e⁻] attachment, all of the complexes in their reduced form, absorb light in the visible and near-IR range, Table 3. Some complexes, namely [Fe₂S-SH](CO)₆ and [Co-H-CoSe₂](CO)₄ have very intense lines in the near-IR region. Thus, with respect to photoactivation in the HER reaction or in CO₂ reduction, all of the selenide and sulfide complexes with carbonyl ligands of the late 3d elements (Fe, Co, Ni, Cu), are eligible. It was proven experimentally, that Fe₂S₂(CO)₆ efficiently splits water upon visible light irradiation and, moreover, it was able to operate as a catalyst in dark conditions as well [11]. Theoretical modeling of the HER on selenide and sulfide complexes revealed that the iron sulfide and iron selenide complexes are most efficient [18,21], Figure 4.

The reaction bottleneck, however, is the oxygen evolution reaction, OER [5,6,18,21]. The activation barriers are much higher, than for HER. The reaction pathway is pH-dependent, because different intermediate species are formed in alkaline or acid media: peroxo- or hydroperoxo- species, (M₂-O-O) or (M₂-OOH). The route via an (M₂-OOH) intermediate is more favorable, and the selenide and sulfide complexes of cobalt and iron were more efficient than those of nickel and copper, Figure 4. For the OER photoactivation is of higher importance than it is for HER, because of the higher energy barriers, particularly in the lack of acidity. Fortunately, all of the chalcogenide complexes with oxygen intermediates, have their most intense light absorption bands in the visible region, Table 4.

3. Photocatalytic Activity oF M₂S₂(CO)₆ Complexes (M=Co, Fe) in Carbon Dioxide and Nitrogen oXides Reduction

The reaction mechanisms of energy demanding redox processes such as CO₂ and NO reduction, catalyzed by the hexacarbonyl complexes with Fe₂S₂ and Co₂S₂ rhombic cores were studied by transition state theory in combination with density functional methods. Nitrogen oxide reduction efficiently proceeds on Fe₂S₂(CO)₆ and Co₂S₂(CO)₆, the latter complex being more efficient, by accepting an electron-proton couple to the dimeric form (NO)₂. The N₂O₂H^• formed in the first reaction step is subsequently reduced to N₂O and H₂O [32]. The energy barriers to N₂O formation are within 33 – 68 kJ mol^-1 and while these reaction steps would benefit to photoactivation, the next stage, N₂O reduction is more challenging, as the barriers range from 77 – 133 kJ mol^-1. The first elementary step of all redox processes, in which M₂S₂(CO)_n (n=4,6) complexes are involved, include the attachment of proton-electron couple, represented as a hydrogen ligand (1). It is subsequently transferred to water molecules, (NO)₂ or CO₂ reactants [32,33].

[M₂S₂](CO)₆ + [H⁺,e⁻] ⇒ [M₂S₂H](CO)₆ + hν ⇒ [MHMS₂](CO)₆, where M=Co, Fe

(1)

Two different intermediates are formed in the first reduction step of CO₂: (i) hydrogen attached at the carbon atom to form OCHO^• intermediate and (ii) hydrogen attached at the oxygen end of CO₂, to form COOH^•. The OCHO^• intermediate reacts further to hydrogenation of the oxygen end and formation of formic acid HCOOH as a final product. The COOH^• intermediate also reacts further to hydrogenation of the other oxygen end-atom and thus leads to CO and H2O as a final product. A selective catalyst can predetermine the reaction products of CO2 two-proton-two-electron reduction. The Fe₂S₂(CO)₆ complex favors the formation of a COOH^• intermediate with an energy barrier of 87 kJ mol^-1 and thus CO and water end-products [33]. Though HCOOH can also be formed from the COOH^• intermediate by a second hydrogenation at the carbon center, this step is energetically less favorable. The Co₂S₂(CO)₆ complex favors the formation of an OCHO^• intermediate with an energy barrier of 53 kJ mol^-1, which yields selectively HCOOH as end-product. With respect to CO2 reduction, the Fe₂S₂(CO)₆ complex is less selective than Co₂S₂(CO)₆. Moreover, Fe₂S₂(CO)₆ as a structural analog of hydrogenase enzymes, has high activity in HER and it runs as a side reaction.

Among the 4d elements, (Y-Ag), notable redox and hydrogenation catalysts are present, among which the molybdenum disulfide, MoS₂ is an example of a catalyst with widespread application: cocatalyst in hydrodesulfurization, hydrogenation catalyst in organic chemistry; electronic and optical properties of the material in 2D layers. As either layered material, or as nanoclusters, MoS₂ proved efficient in photochemical water splitting [38]. When turning to the 4f elements, considered as inner-transition metals, the light-absorption and photochemical properties are mixed with light-emission properties, again tunable by the proper choice of ligands. Photochemical and photophysical properties go hand-in-hand in the lanthanides.

4. Sulfur-Containing oRganic Chromophores Enabling Charge Transfer Processes in Ln³⁺ Complexes.

The sensitization ability of the ligand-chromophore is specific for the luminescence of certain Ln(III) and the antenna effect could be achieved by UV excitation of new antenna-ligands or various combinations of known ligand chromophores [39,40,41,42]. Along with N,N- and O,O-donor ligands, the antenna potential of the S,S-donor ligand (dithiocarbamate (dtc) derivatives) in sensitizing lanthanide luminescence are of research interest from experimental and theoretical perspectives [43,44,45,46]. These compounds have been applied successfully in the production of lanthanide sulphide nanoparticles, for the purposes of photocatalysis sulfides [47,48] and as suitable materials for novel electronic and optical devices [43,49,50]. The S-donor chromophores are characterized by higher HOMO energy, compared to O-donor ligands, they are some of the strongest reducing agents and are considered to contribute to the low-lying ligand-to-metal charge transfer (LMCT) state of the Ln complexes [51,52,53,54,55,56,57]. It is known, that the N,N-donor chromophores such as phen (1,10-phenanthroline) and bipy (2,2′-bipyridine) enhance the luminescence of Eu³⁺ and Tb³⁺ in their complexes, but this does not occur for Sm^3+, Dy³⁺, and Pr³⁺ complexes [58]. The combination of phen and dtc ligands, however, results in sensitization of Sm³⁺ and Pr³⁺ luminescence and the corresponding complexes exhibit more intense emission than those of the Eu³⁺, Tb³⁺, and Dy³⁺ analogues [42,52,59]. Heteroleptic lanthanide complexes containing two chromophore ligands, one of which is dtc derivative, are suitable objects for estimation the type and role of charge transfer states in sensitizing lanthanide luminescence, opening an interesting perspective for tuning the optical properties of the lanthanide complexes (Figure 5)

A few hypotheses related with the emergence of charge transfer (CT) states in the excitation spectra of europium complexes with dtc^- have been proposed to understand the different impact of the dtc ligand on the optical properties of the Eu³⁺-phen and Sm³⁺-phen complexes, particularly inexplicably weak red emission of Eu³⁺: low-energy ligand-to-metal charge transfer (LMCT) state [51], which is the most efficient quenching channel for Eu³⁺ luminescence [52,53,56,60] or interligand charge transfer (ILCT) character of the lowest excited state [61]. Although the LMCT states can contribute to the sensitization mechanism of the lanthanide complexes, the corresponding molar absorption coefficients are rather modest compared to the intraligand transitions. Another excited state phenomenon, photoinduced electron transfer (PET), was also considered as typical quenching when excited Ln²⁺/L^• state is lower energy state than that of Ln³⁺ [62,63]. The R₂dtc⁻ is expected to be redox active ligand towards Eu³⁺ because of low oxidation potential of dtc anion (producing a radical (R₂NCS₂⁻/R₂NCS₂^•, ~0.24V)) [64] and the most reducible Eu³⁺ (reduction potential Eu³⁺/Eu²⁺ _aq = −0.34 V) [65]. The presence of dtc ligands in Eu³⁺-phen complexes unlike in the Sm³⁺-phen complex, leads to changes in UV-Vis absorption, especially in the visible range, and consequently to variations in the color of the complex depending on the environment and temperature. In solution (dichloromethane), the complex is colored in yellow, whereas in solid state it assumes unusual red-orange color. Upon cooling at 77 K, the red-orange complex in solid state turns yellow. The luminescence intensity of Eu³⁺ complex could be modulated by variation of substituents in dtc, for example it increases when pyrrolidine substituent in dtc ligand is replaced by diethyl one. Combined experimental and theoretically approaches, including ab initio, DFT and semi-empirical methods were developed and applied for thorough description of the antenna mechanism in the luminescent lanthanide complexes from ligand-centered absorption (S₀→S_n), geometry relaxation to the first singlet state, S_n→S₁ (Kasha’s rule), radiative relaxation S₁→S₀ (fluorescence) or nonradiative intersystem crossing process S₁→T₁, radiative (phosphorescence) and nonradiative T₁→S₀ relaxation or ligand-to-metal energy transfer (dtc→Eu³⁺) in excited state and metal-centered emission (luminescence from populated excited state acceptor levels (⁵D₀, ⁵D₁ for Eu³⁺). The Ln(S₂CR)₃(phen) complexes studied, (Ln ≡ Eu, Sm, phen ≡ 1,10-phenanthroline and R ≡ N(C₂H₅)₂, N(CH₂)₄) could be considered as benchmark models of the heteroligand luminescent lanthanide complexes with dtc [46]. The calculated energy level diagrams and rates of competitive radiation and nonradiation rates of isostructural the Eu³⁺ and Sm³⁺ complexes in gas phase and solution revealed: i) similar energy level diagrams; ii) ILCT (dtc-to-phen) character of the S₁ state, and small LMCT contribution to the S₁ state only for Eu(Et₂dtc)₃(phen) complexes with Et₂dtc without effect on the S₁ state energy, iii) comparable electronic relaxation paths and iv) a suitable energy gap between ligand-centered donor and metal-centered acceptor states for efficient L-to-Ln energy transfer (Figure 6) [46]. No distinct low-lying LMCT state was identified within the computational study. Therefore, a dominant LMCT-mediated quenching pathway appears unlikely, although a minor LMCT contribution cannot be completely excluded.

Two pathways were predicted for Eu³⁺ and Sm³⁺ complexes in solution: S₀ → S_n →S_1min → T₁ → Ln³⁺ or S₀ → S_n → S_1min → Ln^3+. (Figure 5). The predicted similar relaxation paths and the appropriate D*-A* energy gap favour the luminescent ability for both complexes. The evaluation of the antenna mechanism only, does not reproduce the difference in the luminescence quantum yield of Eu³⁺ and Sm³⁺. In addition to the sensitized luminescence mechanism (antenna effect), the degree of charge transfer and electron density distribution in ground state and formation of radical species, turns out to be a very important factor especially for S,S-ligands in the Ln(III) complexes. The calculated Ln−pdtc/phen bonding, charge decomposition analysis (CDA), Mayer bond order, and overlap populations predicted larger L(dtc)-to-M(Eu³⁺) charge transfer compared to that in the Sm³⁺ complexes. An important conclusion can be drawn that redox-activity of the pdtc anion ligand mainly towards Eu³⁺ occurs in the ground electronic state. These results were supported by the EPR measurements, which gave important information of appearance of signals for pdtc radical and Eu²⁺ in ground state (without UV excitation). Further, the EPR measurements at room- and low temperature, and without and with UV excitation give important information on the radical species formation and its dynamics upon variable conditions. The EPR measurements accompanied by photoexcitation could reveal the dynamics of dtc radicals. In case of increase of the amount of dtc radicals, the charge transfer character of excitation from ligand (dtc) to phen (ILCT) or to metal is expected. The experimentally predicted charge transfer upon excitation was supported by the calculated S₁ state character, namely dtc-to-phen, as well as a small dtc-to-Ln³⁺ charge transfer for Eu(Et₂dtc)₃(phen). In the Eu(dtc)₃(phen) complexes, there is a special case of observed two type of charge transfer, where i) LMCT occurs in the ground state, producing a partial Eu²⁺ and dtc anion radical and ii) photoinduced charge transfer proceeds in S₁ state, which is of interligand dtc-to-phen charge transfer character. Computations showed that the dtc ligand with partial radical character in the europium complex produces bands which contributed to the observed absorption of the europium complex in the 480-600 nm visible range and red-orange color in solid state. Other explanation of intense long-wavelength absorption (at 458 nm, red-brown color of solid state Eu³⁺ complex) was a possible overlap between the ILCT state and Eu²⁺ 4f⁷ ↔ 4f⁶ 5d¹ transition (Laporte allowed transition at ∼400÷480 nm) due to their close excitation energies (large extent). To a lesser extent, the partial interaction of ILCT state and Eu³⁺ electric dipole f−f states (460−470 nm, ⁷F₀ → ⁵D₂ and 470−480 nm, ⁷F₁ → ⁵D₂) is also expected. At a low temperature (77 K), this interaction breaks due to the blueshift of the ILCT band to ∼400−420 nm and as result the color of Eu(pdtc)₃(phen) complex changes from red-orange to yellow. The increase of vertical excitation energy (blueshifted band) with temperature decrease is explained by the large electron population of the lowest vibrational states of the ground electronic state. The combined approach of theoretical and experimental studies thoroughly reveals the factors (solvent, strength of intermolecular interactions in saturated solution or solid state, Ln³⁺ type and temperature) that affect the UV-Vis spectra (position and intensity of absorption bands). Furthermore, they could be used to tune the absorption ability of lanthanide complexes in relation to the mechanism of sensitized luminescence. The relatively weak emission of the europium complex was explained by partial reduction of Eu³⁺ in ground state.

5. Theoretical Description oF Photophysical Processes in Ln³⁺ Complexes with Antenna-Chromophores

The lanthanide luminescence is unique workable property, but its intensity is weak because of small molar absorption coefficients of 4f-4f transitions of lanthanide ions (Ln³⁺) (not higher than 10 M^-1 cm^-1). With the discovery of the possibility of enhancing the luminescence intensity of Ln³⁺ by introducing ligand-chromophores into the coordination sphere of lanthanides, the complexes have attracted great attention as clean energy conversion materials utilizing ligands for energy absorption and efficient photoexcitation leading to lanthanide emission [66]. It is characterized with well-defined narrow bands in various spectral ranges (including the visible region), relatively long lifetimes and high quantum yields. The luminescence quantum efficiency and yield can be modulated by varying the Ln(III) and the chromophores, called antenna ligand, which transfers the absorbed UV energy to the 4f excited state of lanthanide ion by a nonradiative mechanism. The optical characteristics of the luminescent lanthanide complexes have found numerous technological applications in organic light-emitting diodes (OLEDs), telecommunications, lasers, security marking, barcoding, luminescent probes in biomedical analysis, time-resolved microscopy, luminescent chemical sensors, coatings for photovoltaics, luminescent molecular thermometers, immunoassays or agriculture [67,68,69,70]. The basic steps of the energy transfer mechanism and the key factors responsible for efficient excited state energy transfer are well known. The energy gap between the triplet donor state and the acceptor level of the specific Ln³⁺ (ΔE_L-Ln) in range 0.35–0.6 eV, according to the Latva’s empirical rule is considered as important requirement for energy transfer [71]. Numerous synthesized homo- and heteroleptic lanthanide complexes aim to fulfill the established conditions and the most studied antenna-ligands, which form high thermal and chemical stability and good binding ability to the lanthanide ions (La³⁺, Pr³⁺, Sm³⁺, Eu³⁺, Gd³⁺, Tb³⁺, Dy³⁺) are N,N-donor heterocycles such as phen (1,10-phenanthroline) and bipy (2,2′-bipyridine) [72,73,74], O,O-diketones [75,76] and S,S-dtc [52,77]. A lanthanide-based photocatalytic platform producing CO₂ with the highest turnover to date has recently been reported [78].

The improved experimental spectroscopic techniques and theoretical approaches in the last decade have revealed a more complex mechanism of the energy transfers operating in the antenna effect encompassing all possible competing excited state processes and selection rules as well as the luminescence sensitization via ligand singlet state and charge transfer states [46,55,79,80,81,82,83]. This section highlights the progress of theoretical approaches to simulate and elucidate the sensitization mechanism in the luminescent lanthanide complexes by antenna effect. The general scheme and the main known steps of the sensitization mechanism were summarized firstly by Andrade et al. [84]: (1) intense absorption of UV radiation by the sensitizing ligand-chromophores; (2) rapid relaxation of the ligands to their lowest triplet state; (3) energy transfer from the triplet state to a quasi-resonant energy state of the lanthanide ion; (4) decay of the excited lanthanide metal ion to the ground state via photon emission in the visible region. Initially, the experimental UV–vis f-f excitations in the lanthanide ion were used to estimate the Judd-Ofelt intensity parameters Ω_λ, which were further processed to derive the Einstein coefficients A for each of the ⁵D₀ → ⁷F_J transitions. From these, branching ratios (relative intensities of each transition in the emission spectra), radiative decay rates and quantum yields were determined [85]. A number of structural, electronic and energetic parameters of lanthanide complexes as a function of the ligand type have been calculated and analyzed to predict a relation between the luminescence quantum yield and specific ground state properties [86,87]. In a series of Ln³⁺ complexes with coumarin-3-carboxylic acid, a correlation between specific Ln-O bonding characteristics in the ground state, especially larger orbital interaction term (covalent contributions) and shorter Ln-O bond lengths calculated with DFT/B3LYP method were found to correlate with larger luminescence quantum yield [88]. The efficiency of ligand-to-lanthanide energy transfer has been assessed in the pioneering work of Freidzon et al. at the CASSCF level based on the relative energies of the triplet excited states with respect to the emitting levels of the Ln³⁺ [89]. The luminescent properties of the lanthanide complexes were successfully modeled with a strategy that unites experimental techniques and theoretical approaches in close relation [90] and the execution of the procedure was included in the LUMPAC program code, developed for Eu³⁺ complexes [91]. For large lanthanide systems (as amorphous silicates, organic-inorganic hybrids and crystalline metal-organic frameworks containing lanthanide ions), semiempirical Sparkle model [87] has been progressed to determine ground state geometries with accuracy similar to that of ab initio/ECP calculations and significantly lower CPU times [92,93]. To assess the rates of the energy transfer between the ligand and the Ln³⁺_, Malta and collaborators [94] developed a formalism based on the Judd-Ofelt intensity parameters which takes into account the singlet and triplet excited states of the complex [95]. For the evaluation of the latter, semiempirical methods (INDO/S-CIS) can be employed, providing good estimates of excitation energies and oscillator strengths. The described theoretical approaches, however, do not give a detailed description of the discrete processes and responsible factors, which govern the photoluminescence.

The perspective design of new highly luminescent lanthanide complexes requires a thorough understanding of the relationship between the composition, geometry and photophysical properties and processes. An advanced entire computational strategy for reconstruction of the antenna-effect mechanism for lanthanide complexes was recently developed [46,81,83,96,97,98] that relies on ab initio, semiempirical and Judd-Ofelt’s and Malta’s methods. The computational approach is based on the long-established concept that the excited electronic states involved in the most intensive absorption transitions are localized on the antenna-ligands, while the main emissive states are metal-centered and weakly perturbed by the present ligand field. It is then possible to apply certain approximations, where a quantum-chemical modeling of the ligand-centered states can be performed, while the electronic structure of the lanthanide, including the 4f-subshells is rather treated in the form of effective potential. This assumption allows lanthanide complexes to be considered in a singlet ground state. The data for the relevant f-f absorption and emission transitions is normally extracted from experimental measurements or evaluated empirically, using data reported in available sources, such as the works of Carnall and coworkers [99,100]. The first step comprises DFT/TDDFT, MP2/RI-CC2 or CASSCF calculations to draw the energy level diagram (Jablonski diagram) of the luminescent lanthanide complexes, including geometry relaxation, energetics and nature of the ground (singlet) and excited state (singlet and triplet) as well as estimating the rates of all relevant radiative (fluorescence, phosphorescence) and non-radiative relaxation processes (ISC), which involve low-energy singlet and triplet excited states. In the next step, the resonance energy transfer rates from S₁ or T₁ states to the acceptor excited states of Ln³⁺, the Judd-Ofelt intensity parameters, as well as the quantum efficiency and quantum yield of the lanthanide emission are evaluated [91,101,102].

The DFT (functional and basis set) and semiempirical methods are validated in respect to geometrical parameters, vertical excitation energies, emission energy and luminescent quantum yield by comparison with the experimental structural (X-ray diffraction structure) and spectroscopic parameters (UV-vis, fluorescence and phosphorescence) in solution and solid state or with high ab initio methods as MP2(full), CC2, ab initio resolution-of-the-identity algebraic diagrammatic construction through second-order (RI-ADC(2)) and multireference configuration interaction method (DFT-MRCI) of Grimme and Waletzke [103].The theoretical methodology should also rely on the computational methods, which provide reliable energy and character of the Ln-ligand bonding and electron density distribution in the ground (S₀) and excited (S₁, T₁) states, long-range charge transfer excitations [104] and overcomes problems such as the “delocalization error” [105] and the triplet instability [106]. The main challenge in the quantum mechanical treatment of lanthanides is the relativistic effects, degeneracy and spin-orbit effect of the f orbitals and the DFT formalism allows inclusion of part of the relativistic effects through commonly used approaches: zero-order regular approximation (ZORA) [107], Pauli approaches [108] and relativistic effective core potentials (RECPs) [109]. For a structural description of the complexes with 4f elements, it is well acceptable to leave the 4f electrons in the core [110,111]. In line with the nonparticipation of the 4f electrons in the bonding, no effect of the spin-orbital (SO) coupling on the equilibrium geometries of Ln³⁺ complexes was found [112]. The TDDFT/4f-in core ECP approach is reasonable tool to examine the excitation, fluorescence and phosphorescence processes of lanthanide complexes, which comprise mainly the antenna-chromophore as its electronic density is insignificantly perturbed by the 4f electrons. One of the most widespread and proven theoretical approach for Ln³⁺/Ln²⁺ is the relativistic effective core potential (RECP) (small and large) optimized by the Stuttgart-Dresden group with the corresponding valence basis set [113].

On the basis of methodological studies, the long range-corrected hybrid DFT functional, ωB97XD, with dispersion correction appears to be suitable for describing the predominant electrostatic Ln³⁺-ligand bonding in the ground state, demonstrating excellent structural performance, especially for the Eu/Sm/Gd/Tb-N/O/S bond lengths, compared to those of previously applied DFT functionals [46,81,83,96,97,112,114,115] as well as correct location of the triplet states. The calculated bond lengths are slightly longer up to 0.4%. Some of the discrepancies found in the structural data may be due to the different environment in the model complexes (vacuum) and in the crystal structure (packing effect). The semiempirical, RM1 and PM7 methods most accurately provide the Eu/Gd–N(phen) bond lengths (dev. 0.01 ÷ 0.02 Å). It should be noted, that the estimated accuracy of the DFT methods is transferable, whereas the accuracy of the semiempirical methods varies for the different lanthanide complexes depending on the specific parameterization. The low orbital overlaps and triplet instability problems in TDDFT calculations are overcome by employment of the Tamm-Dancoff approximation for the triplet states [116].

The luminescent properties of lanthanide complexes are sensitive to the coordination polyhedron and shape around the Ln³⁺ and for the precise description, it is useful to apply continues shape measurements [46,117]. The luminescent Sm(pdtc)₃(phen) complex corresponds to the specific formula M(AA)₃(BB) and belongs to a geometric shape, biaugmented trigonal prism (BTPR-8). It holds out 32 stereoisomers − 30 (15 pairs of enantiomers) in the C₁ point group and 2 (1 pair of enantiomers) in the C_s point group, predicted using a program Complex Build [118,119]. According to the energy calculations (RM1 method), the possible stereoisomers for the lanthanide complex can be reduced to 12 (not counting the enantiomers) within which the relative enthalpy of formation (ΔH_f) is up to 1.7 kJ/mol and the structural parameters tolerate very small variations. The modifications (structural and energetic) of low-energy stereoisomers were estimated to have very little influence on the spectroscopic and photophysical properties studied in solution.

The influence of the surroundings on the sensitization mechanism in the luminescent lanthanide complexes in solution [83,96,97], solid phase [46] or incorporated in silicate matrix [98] is evaluated by simulating the environment for structure and properties close to the experimental one. The generally accepted approaches to reproduce the solvent effect on the geometry of the lanthanide complex in ground (S₀) and excited state (S₁, T₁) are i) micro-solvation (solvent cluster models that account for the specific solute-solvent interaction), ii) global solvation using the polarizable continuum model and its variations (integral equation formalism variant (IEFPCM), IPCM and SMD) [120] and the conductor-like screening model (COSMO), where the solute molecule forms a cavity within the dielectric continuum of permittivity ε that represents the solvent [121] and iii) micro- and global solvation [88]. The crystal structures of the lanthanide complexes are successfully modeled using the periodic projector-augmented wave (PAW)/DFT approach. Electron exchange−correlation interactions are treated using the generalized gradient approximation as parametrized by Perdew, Burke, and Ernzerhof (PBE functional) [122]. A number of luminescent Eu³⁺-based complexes have been incorporated into silica matrices for better mechanical-, thermal- and photo-stability and higher luminescence quantum yield than the respective pure complexes [123]. For reconstruction of SiO₂/Eu(phen)₂–based materials, sufficient siloxane model was used based on a hydrophilic (Si-O-Si-OH) and a hydrophobic (Si-O-Si(CH₃)₃) silicate surface and adsorbed Eu(Phen)₂X₃ (X = Cl^-,NO₃^–) complexes, Figure 7, [98]. Because of large studied system, a semiempirical Sparkle/AM1 approach yielded reasonable optimized geometries for all of the initially defined model systems. The influence of the environment on the structural and energy parameters of lanthanide complexes can be understood through a comparative analysis with the isolated complexes in the gas phase.

Vertical Excitation energy and energy of relaxation calculations. With elaboration of new exchange-correlation functionals and an inclusion of complex phenomena in the time dependent formalism, the TD-DFT method becomes suitable to reproduce the electronic excited state (EES) geometry and vertical excitation energies (VEEs), having local and charge–transfer character [124]. For both local n → π* and π →π* states, global hybrids such as B3LYP, PBE0 or M06 were found to provide accurate estimates whereas for charge-transfer or Rydberg EES, range-separated hybrids (CAM-B3LYP or ωB97xD) were recommended. The long-range corrected hybrid density functional, which includes D2- or D3 – dispersion correction of Grimme [125], ωB97xD [126] has proven itself for accurate description of energy and character of the electronic transition in different lanthanide complexes [46,81,83,97]. The gas phase VEEs calculations of the method was validated referred to DFT-MRCI and high-level RI-ADC(2) calculations [97]. The ωB97xD and the reference calculations predicted the same character of the S₁ state in gas phase, unlike PBE1PBE ones [81,97]. The TDDFT/ωB97xD calculations overestimated S₀→S₁ electron transition by ~0.3 eV (absorption band is blueshifted by ~25 nm) as compared to the experimental absorption bands. It was established that this deviation is due to the neglect of the vibronic contributions to the excitation energies. The vibrationally-resolved S₀→S₁ absorption, S₁→S₀ emission (fluorescence) and T₁→S₀ phosphorescence spectra of the Ln³⁺ complexes could be calculated in the frame of Time-Independent (TI) [127] and Time-Dependent (TD) formalisms and require optimization and harmonic frequencies calculations for ground and excited states. They are realized in the frame of Franck-Condon, Herzberg-Teller and Franck-Condon-Herzberg-Teller and Dushinsky matrix rotation and sum-over-states (SOS) approaches. Representation of the potential energy surface of the final state was achieved with adiabatic (the initial and final states are evaluated as harmonic potential energy surfaces) or vertical (the final state is evaluated around the equilibrium geometry of the initial state) Hessian models. The temperature effect at 298.15 K on one-photon absorption spectra and at 77 K on phosphorescence spectra is also included in the vibronic calculations [46,81,83,97].

For TDDFT calculation in solution, the solvation models like Polarizable Continuum Model (PCM) handle solvent-solute interactions via two formalisms - linear-response (LR) [128] or state-specific (SS) approach [129]. The LR, which responds to the transition density between the ground and excited state, is applicable to localized excitations, whereas SS accounts for the specific electron density of the excited state and is essential for accurate modeling of charge-transfer processes. In the frame of state-specific solvation, equilibrium and non-equilibrium solvation were used for excited state geometry optimization and vertical electronic excitations, respectively.

For better understanding the distinction in the diffuse reflectance spectra of powder lanthanide complexes as compared to that in solution, the linear optical properties of the crystal structures are estimated by means of the calculated frequency-dependent absorption coefficient using the periodic projector-augmented wave (PAW)/DFT approach [46]. Electron exchange−correlation interactions are treated using the generalized gradient approximation as parametrized by Perdew, Burke, and Ernzerhof (PBE functional) [122]. The optical parameters are assessed for incident photon energies in the range 1.8−6.2 eV (corresponding to the UV−vis spectra in the range 700−200 nm) and electric field polarization vectors along [100], [010], and [001] directions are assigned as xx, yy, zz ones. The different absorption spectra in the three directions indicate that the lanthanide complexes are three axes optical crystals in relation to UV excitations.

Rates and life estimation of the excited-state processes. The sensitization mechanism of luminescence lanthanide complexes proceeds through numerous excited-state processes, some of which are competitive. The calculated rate constants and lifetimes of the relaxation processes give essential information for the channel of energy relaxation through fluorescence, S₁-T₁ ISC, phosphorescence, luminescence and thermal deactivation processes. Upon UV irradiation, absorption occurs by the most absorptive and higher energy S₀→S_n>1 excitation and the absorbed UV energy rapidly releases to the lower vibrational level of S₁ state (for small energy difference (up to ~0.2–0.4 eV)) on the picosecond timescale (10⁻¹² s) by means of vibro-electronic mechanism (Kasha’s rule) state [130]. This statement was supported by calculated one step internal conversion S₂−S₁ lifetime of 1.2 × 10⁻¹² s. Whereas, the S₁ → S₀ internal conversion was estimated to be slower ~10⁻⁹ s due to the large S₁-S₀ energy gap of ~ 3.0 -4.0 eV (UV region). Therefore, although the absorption of Ln³⁺ complexes are promoted by higher excited state, the energy relaxation proceeds through the S₁ state in accordance with the experimental excitation spectra, showing that excitation at the S_n>1 and S₁ states leads to similar intense luminescence.

Further, a very fast geometry relaxation proceeds to the minimum of S₁ potential energy surface with energy stabilization. The process lifetime is of 10⁻¹⁰–10^–14 s, comparable to the period of a molecule vibration and faster than the energy-transfer processes [131]. The large energy stabilization and geometry reorganization of S₁ state leads to large Stokes shift. At this point, several scenarios are possible: the system could undergo an ISC transition to a triplet excited state T_n, returns into ground state (S₀), either by radiative or nonradiative mechanism or the energy could transfer from ligand S₁(D*) to Ln (A*). Accounting the ZPE correction to ΔE_ad(S₀(min)-S₁(min)) energy (TDDFT/ωB97xD method) shows a decrease in ΔE_ad by ~0.1 eV. The inclusion of entropy has less importance to the ΔE value [83].

The energy and character of the S_1(min) state (nπ* and ππ*) and oscillator strength of S₁→S₀ transition determine the radiative (k_r) and nonradiative (k_nr) rate constants and lifetime (τ) of energy relaxation (fluorescence). Previous studies have shown that the S₁ character depends on the theoretical method used [81,132]. The environment (vacuum, non-polar and polar solution) was found to alter the S₁ state energy, oscillator strength, character and fluorescence lifetime. In case of stabilization for S₁ state with nπ* character in non-polar solvent or solid state, the S₁-S₀ radiative process is suppressed and favors S₁-T_n ISC or excited-state energy transfer S₁(ligand)-Ln³⁺ [96,97]. The emission lifetimes (τ) of singlet states have been computed for spontaneous emission using Einstein transition probability according to the formula Eq. 2 [133]:

$$\tau ={\displaystyle \frac{1}{k_r}}={\displaystyle \frac{23.05}{f_{osc}{\left(\Delta E\right)}^2}}[\mathrm{n}\mathrm{s}]$$

(2)

where, f is the oscillator strength and ΔE is the energy change, characterizing the transition (TDDFT spectral data). Radiative and nonradiative rate constants and lifetimes of S₁→S₀ deactivation (fluorescence and internal conversion, respectively) were also calculated using the time dependent approach [134] as implemented within the DUSHIN code of the MOMAP2022A (Molecular Materials Property Prediction) package [135]. For further information about the methodology and application of this formalism, one could refer to Refs. [136,137].

The lack of an emission ligand band in the fluorescence spectrum of the lanthanide complexes suggests the occurrence of the preferred singlet-triplet Intersystem crossing (ISC) process. The direct connected ligand-chromophore to Ln³⁺ increases the interaction of spin and orbital angular momentum, which allows mixing of spin angular and orbital angular momentum of S₁ and T_n. The spin-orbit interaction increases in the order Sm³⁺, Eu³⁺ and Tb³⁺ according to the finding that it grows with fourth power of the metal atomic number Z. Apart from the strength of the spin-orbital coupling between two excited states, the electronic energy gap is also important factor for the rate of S-T nonradiative transition. Small energy gap less 0.3 eV supposed large forward and reverse rates of ISC S₁-T_n and not sufficient population of T₁ state. At small displacement of the S₁ and T₁ minima it is expected weak coupling where energy gap law is valid, whereas at large S₁ and T₁ minima displacement the coupling would be strong and the transition probabilities increases with increasing energy gap [138]. From the other side, the different orbital nature of T₁ and S₁ minima also enhances the spin-orbital coupling effect due to the nπ*→ ππ* transition that involves a rotation of a p-orbital at some heteroatomic center.

The quantitative expression of the nonradiative (ISC) rate constants was achieved within the frame of the Fermi Golden rule, Eq. 3 [139].

$$k_{\mathrm{I}\mathrm{S}\mathrm{C}}={\displaystyle \frac{2\pi }{\hslash }}{\left|V\right|}^2{\rho }_{\mathrm{F}\mathrm{C}},$$

(3)

where |V| is the strength of the perturbation or the interaction between the initial and the final state. For states with different parity the parameter coincides with the strength of the spin-orbit coupling (SOC) interaction between the respective singlet and triplet states. Last in eq. (3) stands the Franck-Condon-weighed density of states (FCWD), denoted ρ_FC. This parameter could be assessed on the basis of geometry optimization and harmonic frequency calculations for each of the states involved in the transition process. When the energy difference between the initial and the final state is rather small, it could be assumed that ρ_FC ~ 1/ΔE. This is the case, when a transition of the type S₁ → T_n (n = 1, 2) is examined.

For the lanthanide complexes, the spin-orbit coupling and the intersystem crossing rates of the transitions S₁→T₁, T₁→S₁ and T₁→S₀ were assessed on the basis of SA-CASSCF/NEVPT2 wave functions. The quasi-degenerate perturbation theory (QDPT) [140] was implemented to evaluate the spin-orbit coupling between low-lying singlet and triplet excited states as well as zero-filed splitting of the T₁ state of the molecules. More computational details are given in refs [46,81,83].

For phosphorescent emission, an average rate constant is calculated on the basis of ${\overline{\mathrm{k}}}_{\mathrm{p}\mathrm{h}}$ estimates for the particular T_nm-states (m = x, y, z), Eq. 4:

$${\overline{\mathrm{k}}}_{\mathrm{p}\mathrm{h}}={\displaystyle \frac{\sum _{m=x,y,z}{k}_p^{\left(m\right)}{e}^{-\frac{∆{\mathrm{E}}_m}{k_{\mathrm{B}}T}}}{\sum _{m=x,y,z}{e}^{-\frac{∆{\mathrm{E}}_m}{k_{\mathrm{B}}T}}}};k_{ph}^{\left(m\right)}={\displaystyle \frac{{\left(∆{\mathrm{E}}_m\right)}^2{f}^{\left(m\right)}}{23.05}}\left[{ns}^{-1}\right];f^{\left(m\right)}={\displaystyle \frac{2}{3}}{\left|⟨{\mathsf{\Psi }}_0^{SOC}|\widehat{\mu }|{\mathsf{\Psi }}_m^{SOC}⟩\right|}^2∆{\mathrm{E}}_m.$$

(4)

Here, k_ph^(m) is the rate of phosphorescence emission for an individual T_n - triplet level, k_B denotes the Boltzmann constant, f^(m) is the oscillator strength, $\widehat{\mu }$ – the dipole moment operator, while $\left|{\mathsf{\Psi }}_0^{SOC}\right.⟩$ and $\left|{\mathsf{\Psi }}_m^{SOC}\right.⟩$ are the wavefunctions of the mixed ground and m-th triplet excited state. It should be noted that the right-hand side of eq. (3) is correct, if the emission energy ΔE is evaluated in atomic units, while the transition dipole moment value is taken in length representation.

It should be mentioned that similar T₁ energy levels have been calculated for the ligand-chromophore and the corresponding homoleptic Ln³⁺ complexes. In this case, the T₁ state energy of Ln³⁺ complexes could be derived from the experimental phosphorescence spectra of the ligand and the corresponding Gd³⁺ complexes. The population of the triplet state and the theoretical phosphorescence quantum yield of the ligand and its Ln³⁺ complexes, however, should be different because of the contribution of the heavy metal to the strength of the spin-orbital coupling between T₁ and S₀.

Luminescence theoretical approach LUMPAC. The radiative and nonradiative energy transfer from S₁ or T₁ donor states of the ligand to the acceptor levels of Ln³⁺ would contribute to the luminescence emission. The radiative energy transfer process is considered to be less efficient because only a small fraction of the emitted light can be captured by the acceptor. The more probable nonradiative energy transfer is described by two mechanisms: electrostatic multipolar (Förster) and double-electron exchange (Dexter). The energy transfer from singlet state is operated by: i) Förster resonance mechanism, involving dipole-dipole interaction and long range interactions (up to ~30–100 Å and its distance dependence is 1/r⁶) or ii) Dexter mechanism known as exchange (short-range donor-acceptor distances and its distance dependence is e^-r) [141]. The triplet energy transfer is explained only according to the Dexter mechanism [142]. So, the total rate for energy transfer can be better described as k_ET (total)=k_ET (Förster)+k_ET (Dexter) by the theoretical model developed in the literature [143].

The resonance ligand-to-metal energy transfer rates were calculated, following the approach, developed by Malta and coworkers [144]. First, the intensity parameters Ω_λ (λ = 2, 4) and the radiative deactivation rate of ⁵D₀ (the key emissive state of Eu³⁺) were extracted from the experimental emission spectra of the Eu³⁺ complexes and lifetime of the lanthanide emission. Going further, theoretical estimates of the same values were obtained, following the method proposed by de Sá et al. [95] and extended by Freire and co-workers (QDC-scheme)[145]. Finally, the intensity parameter values and the rates of deactivation were used in the procedure for the theoretical estimation of the energy transfer rates L(S₁/T₁)→Ln and the luminescence quantum yield and efficiency.

The question arises of how to combine DFT/TDDFT and semi-empirical methods, as well as the Judd-Ofelt and Malta’s approach, to describe the complete photophysical process from absorption to luminescence for the lanthanide complexes. The latest approach for calculating the rate of ligand-to-lanthanide energy transfer, uses ligand-centered vertical excitations. The optimized geometry can be i) in ground state, obtained by the semiempirical RM1 method or DFT method, or ii) in excited states (S₁/T₁) obtained by TDDFT method. Further, vertical excitation energies could be calculated by INDO-S/CIS or TDDFT/TDA methods. The main requirement for a reliable estimation of the energy transfer rate to the potential donor states of Eu³⁺ (⁵D₀, ⁵D₁) is that the calculated energy level of the donor state T₁ (according to the methods mentioned above) corresponds to the energy level of the TDDFT triplet state minimum, and, consequently, to the experimental emission from T₁. The motivation for using the T₁ energy level, obtained at TDDFT level (T₁ geometry optimization and VEE) comes from the fact that in many cases, there are distinctive differences between the structural parameters in ground and excited state. At the same time, the energy transfer process is considered to be fast enough and no changes in molecular geometry are expected. The calculations also showed a similar geometry of the S₀ and T₁ states. Therefore, a more accurate evaluation of the energy transfer rates from a certain ligand-centered excited, singlet or triplet state, would involve intensity parameters and wavefunctions, calculated at the energy minimum of that state. While LUMPAC allows the use of both semi-empirical and TDDFT wavefunctions, it is our observation, that the rate estimates on the basis of first principles tend to be overestimated, leading to orders of 10¹¹ s^-1 and higher. As a consequence, it is considered that INDO-S/CIS wavefunctions [146] are more compatible for the computational scheme.

Theoretical prediction of excited state energy mechanism in luminescence lanthanide complexes with N- and O-donor ligands. The advanced computational strategy combining DFT/TDDFT and multireference ab-initio calculations with perturbative spin-orbit coupling corrections and Judd-Ofelt and Malta’s theories and QDC model in relation to experimental data allowed for a deep understanding of the role of the antenna-chromophore and the sensitization mechanism in luminescence lanthanide complexes: 1,10-phenanthroline (phen) (Eu(phen)₂(NO₃)₃ and Tb(phen)₂(NO₃)) [81,96,98], coumarin-3-carboxylic acid (HCCA) (Eu(CCA)Cl₂(H₂O)₂, Eu(CCA)₂Cl(H₂O)₂, Eu(CCA)₃(H₂O)₃, Tb(CCA)₂Cl(H₂O), Tb(CCA)₂(NO₃)(H₂O)) [97], phenacyldiphenylphosphine oxide (Phenac), Eu(Phenac)₂(NO₃)₃(H₂O), Eu(Phenac)₂(NO₃)₃, Eu(Phenac)₃(NO₃)₃ and Tb(Phenac)₂(NO₃)₃(H₂O)) [83]. The photophysical properties of the ligands, its binding ability to encapsulate and stabilize lanthanide complexes and L-M bonding strength and character are estimated in connection to the luminescence mechanism. The main steps of the sensitization mechanism: absorption, relaxations, intersystem crossing, energy transfer and quantum yield, are traced in the calculated energy level diagrams of the lanthanide complexes in gas phase and solution (polar and nonpolar).

In the homoleptic Ln(phen)₂(NO₃)₃ (Ln³⁺ = Eu, Tb, Gd), the phen is an antenna-chromophore and is N,N-bidentate bondеd to Ln³⁺, Figure 7. The counterion does not participate directly in absorption process, but can interfere the energy and character of the S₁ and T₁ states, involved in the excited state energy transfer. The small oscillator strength of S₁ state in gas phase (or non-polar solvent) and DMF produces a delay of the S₁-S₀ emission. For Tb(phen)₂(NO₃)₃, the calculated smaller radiative S₁→S₀ lifetime (~10^-9 s) than that of non-radiative S₁→T₂/S₁→T₁ ISC lifetime (~10^-12 s) indicated that the ISC process to triplet state is more probable than the fluorescence. The calculated similar energy gap between S₀ and T₁ state minima (S₀(opt)-T₁(opt)) for phen and Ln(phen)₂(NO₃)₃ (Ln=Eu, Tb, Gd) complexes in gas phase and DMF, showed that the solvent effect on the lowest triplet state is neglected and the T₁ state energy of phen chromophore determines this one in its Ln³⁺ complexes. Two relaxation channels of ISC, S₁→T₂→T₁ and S₁→T₁, are predicted for the population of T₁ state based on the calculated nonradiative rates of the forward and back-transfer S₁↔T₂, T₂↔T₁, S₁↔T₁ and T₁↔S₀ processes. Further, the energy transfer proceeds from a ligand-centered donor level T₁ to the main emissive metal-centered state, ⁵D₀ and ⁵D₄ of Eu(III) and Tb(III), respectively. The most effective electronic relaxation and energy transfer mechanism was predicted: S₀→S₁→T₁→⁵D₁/⁵D₀ for Eu³⁺ and S₀→S₁→T₂→T₁→⁵D₄ for Tb³⁺ complex. The other possible excitation channels S₁→⁵D₂/⁵L₆ (for Eu³⁺) and S₁→⁵D₃ and T₂→⁵D₃ (for Tb³⁺) were found to be less probable. It was established that phen acts as a better antenna for Eu³⁺ than for Tb³⁺. A possible explanation is that the transfer from the T₁ state could populate two possible Eu³⁺ centered acceptor states: ⁵D₀ and ⁵D₁. The energy differences ΔE(T₁→⁵D₀) ~ 0.64 eV, and ΔE(T₁→⁵D₁) ~ 0.42 eV are quite large compared to ΔE(T₁→⁵D₄) ~0.24 eV for the Tb³⁺ complex. At such large ΔE, reverse transfer rates ⁵D₀→T₁ and ⁵D₁→T₁ are low and the overall energy conversion efficiency is higher for the Eu³⁺ complex (QY of 35%, calc. 32%) than that of 13% for the Tb³⁺ complex. The process in Eu³⁺ complex is governed by the exchange mechanism and the main observed emission is known to be at a maximum at 615 nm due to ⁵D₀ → ⁷F₂ transition and red emission color. The spectrum of Tb³⁺ complex is dominated by the ⁵D₄→⁷F₅ transition, located at 541 nm, which is responsible for the green emission color.

Figure 8. Energy level diagram of Eu(phen)₂(NO₃)₃ predicted by DFT/TDDFT/PBE1PBE and INDO/S-CIS calculations and the most probable channels for the intramolecular energy transfer process From ref. [96].

Figure 8. Energy level diagram of Eu(phen)₂(NO₃)₃ predicted by DFT/TDDFT/PBE1PBE and INDO/S-CIS calculations and the most probable channels for the intramolecular energy transfer process From ref. [96].

The structural and photophysical properties (Jablonski diagrams and nonradiative rates) of two highly-emissive Eu³⁺-based luminescent complexes, Eu(Phen)₂Cl₃ and Eu(Phen)₂(NO₃)₃ adsorbed on hydrophilic Si-O-Si-OH and hydrophobic Si-O-Si(CH₃)₃ silicate surfaces were elucidated by combined semi-empirical (Sparkle/AM1; INDO/S-CIS) and simplified TDDFT calculations (Figure 7). The calculated data explained the highest luminescence quantum yield (experimental and theoretical) of Eu(Phen)₂(NO₃)₃ incorporated in hydrophilic matrix as compared to hydrophobic one. For Eu(Phen)₂Cl₃, the observation of large number of hydrogen bonding on hydrophobic silicate surface and strong adsorption interaction, appears an operative factor, which suppresses the nonradiative emission and assists the larger luminescence quantum yield of the complex incorporated in hydrophobic matrix than that in hydrophilic one. The calculated energy differences ΔE=((E(T₁)-E(⁵D₀)/E(⁵D₁)) are large enough to favour the T₁-⁵D₀ and T₁-⁵D₁ energy transfer relative to the back energy transfer. The theoretical research revealed other operative factors for the formation of the luminescence quantum efficiency and yield of Eu(Phen)₂X₃ when incorporated into SiO₂-based matrices.

The competitive excited state processes in the homoleptic Eu³⁺ and Tb³⁺ complexes of HCCA – fluorescence, intersystem crossing (ISC) and phosphorescence, were analyzed depending on the environment, number of the ligands (1-3), Ln³⁺ ion type (Eu and Tb) and counteranion (Cl⁻ and NO₃⁻) (Figure 9). The HCCA is bidentate bonded to Ln(III) through two oxygen atoms. It was found that the environment altered the S₁ state energy, oscillator strength, fluorescence lifetime as well as the S₁ character – polar solution stabilized the S₁(ππ*) state, whereas non-polar solution (gas phase, solid state) stabilized the S₁(nπ*) state. The S₁(nπ*) state characterizing with small oscillator strength was decisive factor for the resulting excited state energy transfer and Ln(III) luminescence. It suppressed the S₁ ligand emission in the complexes and favored S₁-T₁ ISC or direct transfer to the emitting levels of Ln³⁺. The HCCA triplet (T₁) state energy and ππ* character were preserved in the Eu/Tb-CCA complexes and are not changed by the environment. The energy gap between the higher energy T₁ donor state and the acceptor levels ⁵D₁ of Eu³⁺ (~0.5 eV) and ⁵D₄ of Tb³⁺ (~0.1 eV) provided optimal resonance conditions for effective energy transfer to Eu³⁺, but less probability for Tb³⁺. The excitation channel T₁→⁵D₀ through an exchange mechanism was predicted as the most probable one to populate the main emissive Eu-centered state in the complexes. The energy and the ππ* character of the populated T₁ state were not influenced by the H₂O-ligands, the Ln(III) type, the counterion type and environment. The larger quantum luminescence yield of Eu(CCA)₃(H₂O)₃ compared to Eu(CCA)₂Cl(H₂O)₂ (from experimental and theoretical point of view) was explained by the higher Eu³⁺ radiative emission rate, the lower nonradiative decay rate, the shorter distances (LR) between the singlet and triplet CCA donor states and Eu³⁺ acceptor states and the larger refractive index (solid state).

The sensitization capacity of Phenac ligand, combining C=O and P=O donor group was theoretically explored in the Eu³⁺ and Tb³⁺ complexes in vacuum, solution and solid phase. The influence of the coordination polyhedron (number of Phenac), ligand bonding type (mono- and bidentate) and strength, environment, character of the emissive singlet and triplet states stimulating the Eu³⁺/Tb³⁺ luminescence was elucidated to address the open question in Ref. [147] about the type of Eu³⁺ complexes formed in solution. The thermodynamic stability calculations for Eu(Phenac)₂(H₂O)(NO₃)₃, Eu(Phenac)₂(NO₃)₃ and Eu(Phenac)₃(NO₃)₃ in reactions predicted Eu(Phenac)₃(NO₃)₃ is mainly stabilized in solution and gas phase, while the other two complexes are in lower concentrations. Similar relaxation paths were found for all complexes, Figure 10. The absorbed energy by S_{n > 1}(π-π*) states relaxed to T_1min by major ISC mechanism S₁→T₂→T₁. The long radiative lifetime of S₁→S₀ due to n(p(C)(O))π-π*,p(P) character facilitated the forward ISC process. The environment, Phenac’s number and Ln³⁺ size affect the T₁ energy by ~0.1 eV. The calculated (Phenac→Eu³⁺ energy transfer rates suggested the most likely T₁ → ⁵D₁/⁵D₀ channels for a population of emitting metal-centered states. The larger luminescence quantum yield (experimental and theoretical) for the Tb³⁺-Phenac complexes was explained by the lower T₁ state of 0.1 eV and the more suitable ΔE(T_1min -⁵D₄) of ~0.5 eV for energy transfer as well as the possible enhancement of spin-orbital coupling effects by the heavier Tb³⁺ compared to Eu³⁺.