Theory and Calculation of the Phosphorescence ... - ACS Publications

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Theory and Calculation of the Phosphorescence Phenomenon Gleb Baryshnikov,†,‡ Boris Minaev,†,‡ and Hans Ågren*,†,§ †

Division of Theoretical Chemistry and Biology, Royal Institute of Technology, SE-106 91 Stockholm, Sweden Bohdan Khmelnytsky National University, 18031 Cherkasy, Ukraine § Institute of Nanotechnology, Spectroscopy and Quantum Chemistry, Siberian Federal University, Svobodny pr. 79, 660041 Krasnoyarsk, Russia ‡

ABSTRACT: Phosphorescence is a phenomenon of delayed luminescence that corresponds to the radiative decay of the molecular triplet state. As a general property of molecules, phosphorescence represents a cornerstone problem of chemical physics due to the spin prohibition of the underlying triplet-singlet emission and because its analysis embraces a deep knowledge of electronic molecular structure. Phosphorescence is the simplest physical process which provides an example of spin-forbidden transformation with a characteristic spin selectivity and magnetic field dependence, being the model also for more complicated chemical reactions and for spin catalysis applications. The bridging of the spin prohibition in phosphorescence is commonly analyzed by perturbation theory, which considers the intensity borrowing from spin-allowed electronic transitions. In this review, we highlight the basic theoretical principles and computational aspects for the estimation of various phosphorescence parameters, like intensity, radiative rate constant, lifetime, polarization, zero-field splitting, and spin sublevel population. Qualitative aspects of the phosphorescence phenomenon are discussed in terms of concepts like structure−activity relationships, donor−acceptor interactions, vibronic activity, and the role of spin−orbit coupling under charge-transfer perturbations. We illustrate the theory and principles of computational phosphorescence by highlighting studies of classical examples like molecular nitrogen and oxygen, benzene, naphthalene and their azaderivatives, porphyrins, as well as by reviewing current research on systems like electrophosphorescent transition metal complexes, nucleobases, and amino acids. We furthermore discuss modern studies of phosphorescence that cover topics of applied relevance, like the design of novel photofunctional materials for organic light-emitting diodes (OLEDs), photovoltaic cells, chemical sensors, and bioimaging.

CONTENTS 1. Introduction 2. General Principles of Phosphorescence 2.1. Phosphorescence in a System of Molecular Photophysical Processes 2.2. Short Historical Background 3. Principles of Computational Phosphorescence 3.1. Spin−orbit Coupling and S-T Mixing 3.2. Perturbation Theory for S-T Transitions 3.3. Phosphorescence Intensity and Lifetime 3.4. Vibronic Perturbations in Phosphorescence 3.5. Quadratic Response Theory and Phosphorescence 3.6. Density Functional Response Theory for Phosphorescence 3.7. Nonradiative S-T Transitions 3.8. Simple Qualitative Rules for Intersystem Crossing and Phosphorescence analysis 4. Computational Phosphorescence of Molecular Systems 4.1. Phosphorescence of Molecular Nitrogen 4.2. Magnetic Phosphorescence of Dioxygen 4.3. Some Examples of Phosphorescence from Triatomic Species

© 2017 American Chemical Society

4.3.1. Ozone 4.3.2. Water, Hydrogen Sulfide, and Hydrogen Cyanide 4.3.3. Hypohalogenous Acids 4.4. Phosphorescence of Unsaturated and Aromatic Hydrocarbons 4.5. Phosphorescence of Azabenzenes 4.6. Phosphorescence of Porphyrins 4.7. Internal and External Heavy-Atom Effects 4.7.1. Unsaturated Hydrocarbons 4.7.2. Halogen-Substituted Benzenes and Pyridines 4.8. Phosphorescence of Heavy Metal Complexes for Light-Emitting Diode Applications 5. Some Recent Applications of Singlet-Triplet Transitions 5.1. Reverse Intersystem Crossing and Delayed Fluorescence 5.2. Phosphorescent Emission from Exciplexes 5.3. Photobiology Applications of the Triplet State 6. Conclusions

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Received: January 23, 2017 Published: April 7, 2017

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DOI: 10.1021/acs.chemrev.7b00060 Chem. Rev. 2017, 117, 6500−6537

Chemical Reviews Author Information Corresponding Author ORCID Notes Biographies Acknowledgments Acronyms References

Review

ions and lanthanides. They are expensive, toxic, and not friendly for the environment; thus, it is desirable to substitute them by pure organic phosphors. Metal-free pure organic dyes with persistent phosphorescence at ambient conditions are much more preferable because organic materials are typically cheap, biocompatible, easy processable, and characterized by a general variability of substituents with high relative stability. However, the design of new organic dyes with persistent phosphorescence at room temperature in open air is a very difficult task as the internal magnetic perturbations in organic molecules are too weak to effectively overcome spin-prohibition for tripletsinglet (T-S) phosphorescent transitions and provide a high quantum yield for emission. The process can in general not compete with nonradiative electronic energy transfer to vibrational and translational modes. Thus, the phosphorescence quenching by vibrational relaxation upon molecular collisions is typically a main reason for the low quantum yield of such emission for organic dyes at ambient conditions. The quenching is suppressed in solid media at cryogenic conditions, and molecular phosphorescence is typically detected at 77 K or lower temperatures from organic chromophores prepared in the form of a glass or doped crystals under degassed conditions.18 Thus, applications of pure organic phosphorescence for OLEDs and modern biotechnologies are strongly desirable but limited in practical realization at ambient conditions. Organic phosphors with persistent emission at room temperature are also desirable for medical applications, especially for imaging, but the general principles and structure− activity relationships for synthesis of such compounds are not well understood yet. Theoretical studies of phosphorescence requires an analysis of the excited state electronic structure with account of internal magnetic interactions and vibronic perturbations, the spontaneous emission, and nonradiative quenching channels. The common nonrelativistic Born−Oppenheimer approximation and perturbation theory,1 and in particular the quadratic response (QR) formalism,22−30 can be used as a starting approach where the phosphorescence acquires dipole activity via spin−orbit coupling (SOC). On that basis, it is readily possible to calculate radiative rate constants and phosphorescence radiative lifetimes22−30 as well as to estimate the role of nonradiative processes under the degradation of the first excited triplet state.31 Contemporary theoretical approaches make it possible to understand the origin of phosphorescence as one of the fundamental phenomena of molecular spectroscopy. For a long time, the general definition of phosphorescence seemed to be the following: “A delayed luminescence that persists after removal of the excitation source”.19 But according to the modern IUPAC definition, a huge number of delayed luminescence processes originally considered as genuine phosphorescence should be excluded from the scope of this review. First of all, this concerns a large part of semiconductor physics encompassing luminescence and photoconductivity of metal salts.20 In line with this notion, phosphorescence was also referred to emission of light by bioluminescent plankton17 and some other forms of chemiluminescence.4−7 In these cases, the excited state is created as the product of a chemical reaction. Thus, the light emission delay follows the accumulation of the reaction product and tracks the kinetics of the chemical transformation. We will here provide a more particular mechanistic specification of this definition. In this review, we present the computational principles of molecular phosphorescence using its typical definition as a spin-

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1. INTRODUCTION According to the IUPAC definition, phosphorescence is a radiative transition between states of different electronic spin multiplicities. Most often it refers to triplet → singlet transitions in stable molecules [i.e., between the first excited triplet (T1) and the ground singlet (S0) state]. As well, it can correspond to the S1 → T0 emission (like in the oxygen molecule with the triplet ground state) and to quartet-doublet transitions (like in a quartz laser). Phosphorescence can be induced by light absorption, electric current (electrophosphorescence),1−3 and by a chemical reaction (chemiluminescence).4−7 Though chemiluminescence can often be released in the form of spin-allowed emission, it is in some cases also triggered by the T → S transition in the reaction products.6 The rich chemistry of singlet oxygen provides not only very characteristic S → T phosphorescence but also many examples of chemiluminescent processes.4−6 Photo- and electrophosporescence provide many important applications such as lightemitting technologies,1−3,8−10 chemosensors,12−14 and bioimaging.15−17 Light-induced phosphorescence constitutes a large domain of molecular spectroscopy being of great importance for the photophysics and photochemistry of molecules18 and for fundamental electronic structure theory in organic chemistry.19 Historically, the term “phosphorescence” has been ascribed to more general phenomena which comprise various types of afterglow in order to distinguish the long-lived emission from the fast fluorescence when light is emitted during short time (less than 10−8 s) after cessation of irradiation. The phenomenological definition of the term “phosphorescence” as a distinction of the time duration of afterglow compared with fluorescence was used until recent time19−21 and led to some ambiguity; the term was equally applied to luminescence of organic and inorganic substances being of completely different mechanistic origin.7,18 This confusion is widely spread over the literature on luminescence of metal salts (photoconduction accompanied phosphorescence) and other “glow-in-the dark” materials. The etymology of the word “phosphorescence” and some historical aspects of delayed luminescence studies also represent a number of inconsistent and contradictory features. Thus, we need to pay tribute to these aspects of the phosphorescence notion in order to deal with the discrepancies one can meet in the literature.17−21 Phosphorescence is very sensitive to molecular aggregation, temperature, and access of oxygen. Thus, most studies of molecular phosphorescence have been performed in cryogenic air-proof matrices. These obstacles limit the processability of molecular phosphors and prevent technological applications of the phosphorescence phenomenon in medicine, photobiology, and optoelectronics. Until recent time, the most efficient materials which exhibit phosphorescence at room temperature were metal sulfides, oxides, and other inorganic compounds, as well as organometallic luminophores with Ir(III) and Pt(II) 6501


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(−1, 0, and +1). In a many-electron molecule, the total wave function satisfies the following equations for the total spin S⃗ operator

forbidden T-S emission. The implementation of these principles is very important for the future development of new phosphorescent materials suitable for low-cost and highefficient organic electronic devices, high-sensitive and selective sensors for bioimaging and medicine, new analytical methods and cheap luminophore coatings, to mention a few of many examples.

S⃗ Ψ = ℏ2S(S + 1)Ψ

(1)

Sz Ψ = M sℏΨ

(2)

where S is the total spin quantum number; S = 0 for singlet state and S = 1 for triplet, where Ms = 0, ±1. In the case of heavy elements (HE) and HE-containing molecules, a strong spin contamination occurs due to spin− orbit coupling (SOC) and S is not a good quantum number. Thus, the IUPAC definition of phosphorescence is only qualitatively correct in the framework of small deviations from the spin selection and quantization rules, eqs 1 and 2. For example, in cyclometalated Ir(III) complexes, a number of lowlying S and T states are quasi-degenerate and SOC-induced mixing is so strong that some spin-sublevels responsible for “phosphorescence” can not be predominantly classified as triplet or singlet states. It is necessary to stress the obvious fact that the overwhelming majority of chemically stable molecules possess an even number of electrons (2N); the lowest in energy N molecular orbitals (MO’s) are doubly occupied by electrons in pairs with opposite spins. Thus, they have a ground singlet (S0) state with zero spin quantum number S = 0 in eq 1 and are diamagnetic since the orbital angular momentum is also equal to zero. A simple generalization of Hund’s rule predicts that the first excited state (typically of the highest occupied MO → lowest unoccupied MO nature) should be the triplet T1 state lying lower than the corresponding first excited singlet (S1) by the double exchange integral calculated with the HOMO (ψi) and LUMO (ψu) wave functions:

2. GENERAL PRINCIPLES OF PHOSPHORESCENCE 2.1. Phosphorescence in a System of Molecular Photophysical Processes

The interconnection between different types of luminescence can be presented schematically in the form of a simplified energy diagram, given in Figure 1, which is often called the

Figure 1. Modified Jablonski diagram for the main photophysical processes in molecular systems.

“Jablonski diagram”32 in honor of the Polish scientist Jablonski who considered the metastable (triplet) level as the origin of delayed emission. However, Jablonski did not guess that this level has a triplet state nature. A brief history of the “triplet state” will be described later, but first we discuss Figure 1 keeping in mind that the T1 level possesses the parallel orientation of two spins similar to a biradical structure. As can bee seen from Figure 1, the lowest-lying triplet state (T1) can be populated through the following sequence of photoprocesses: the S0 → Sm absorption induced by the incident UV light and followed by a fast internal conversion (IC) with the kIC rate constant usually higher than 1012 s−1; this serves as a background for the well-known Kasha rule: (1) fluorescence occurs from the lowest S1 state; (2) the nonradiative direct S1∼T1 intersystem crossing (ICS) with the kISC rate produces a T1 state population; and (3) the ISC between the higher-lying Sm and Tn states with the subsequent Tn-T1 internal conversion (IC) with a kIC rate constant that in some cases also can lead to the phosphorescent state. The fluorescence channel (kf) is a main competitive process with respect to the T1 state optical pumping. However, after the T1 state population, the nonradiative T1∼S0 relaxation (knr) can compete effectively with the T1-S0 phosphorescence radiation (kr). The T1 → S0 emission is spin forbidden as an electric dipole radiation which renders the phosphorescence process to be usually much less intense compared with the spin-allowed fluorescence emission. The spin prohibition is not so severe for nonradiative T1∼S0 relaxation, but it still obeys the general quantum mechanics law of spin conservation.33 An important feature of the T1 state, as well of other Tn manifolds, is that it consists of three spin sublevels (Tx, Ty, and Tz, Figure 1) being analogous of eigenstates with different Ms quantum numbers

⎛

E(S1) − E(T) 1 = 2

⎞ ⎟⎟ψi(2)ψu(1) dv1 dv2 1,2 ⎠

∫ ∫ ψi*(1)ψu*(2)⎜⎜⎝ re

2

(3)

Though this approach is a definite oversimplification, it corresponds to a qualitatively correct general scheme of the low lying excited states presented in Figure 1. In accordance with radiation theory, the probability of spontaneous emission is measured by the reciprocal radiative lifetime (τr), while the emission intensity is determined by the quantum yield (QY), a ratio of the absorbed and emitted photons. Thus, the expressions “high transition probability” and “high intensity” are not synonymous (e.g., the low-probability spin forbidden T1 → S0 transition with a long lifetime can in some cases produce a high quantum yield and intense phosphorescence). As the triplet-singlet transitions are improbable in certain materials, the absorbed radiation may be re-emitted during several seconds or even minutes after the original excitation by light. At the same time, the nonradiative intersystem crossing (S1∼T1) is usually observed in the nanosecond scale due to the small S1-T1 energy gap.18 So, we can see that the kinetics of the spin-forbidden relaxation depends on the T-S band gap (ΔETS). However, the question, “How the spin prohibition vanishes for ISC and phosphorescence?” remains open and will be discussed in section 3 below. 2.2. Short Historical Background

Luminescence can be treated as emission of light without burning or without appreciable heat as occurs in slow oxidation of phosphorus.20,21 At the household level, the glow-in-the-dark 6502


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phenomenon of radioactivity.18 Following immediately the discovery of X-rays, Becquerel had the mistaken idea that minerals made phosphorescent by visible light might emit Xrays. Becquerel had a wide collection of minerals, many of which exhibit phosphorescence. His experiment was to wrap a photographic plate in a black paper, to place a phosphorescent uranium salt mineral on it, and to expose it to bright sunlight. After the first exposure, the sun did not shine in Paris for several days and Becquerel stopped his experiments. He placed the uranium and wrapped the photoplate together in a drawer. After several days he developed the plate and found the image to be as intense as in the original sunlight experiment. Thus, he decided that the exposure came from uranium itself, and this was a great discovery. Use of zinc sulfide for alarm glow-signals became popular in the 1930’s. Nowadays, ZnS is substituted by more efficient strontium aluminate, the luminance of which is ten times greater. As mentioned above, the phosphorescence of these metal salts is determined by electron−hole recombination in the p−n junctions of the semiconductor materials and has nothing in common with molecular phosphorescence. Unfortunately, the strict separation between organic and inorganic long-lasting luminescence in terms of the spin selection rule can not be exact for heavy-element containing materials due to violation of the Russell-Sounders scheme. In some cases, the time delay in metal oxide luminescence can be connected not only with the excited electron trapping by impurities but also with the spin flip inside the activator. Nevertheless, for organic molecules the realm of delayed emission is well-determined as being connected with singlet−triplet transitions. Phosphorescence of organic dyes which exactly corresponds to the IUPAC definition was observed from viscous or solid organic solvents for the first time in 1880.20 In spite of a great interest in experimental studies of molecular luminescence since that time and a powerful breakthrough in atomic and diatomic spectroscopy along with the discovery of the electron spin, a complete understanding of the role of the molecular triplet state in phosphorescence was not realized until a few decades after the introduction of quantum mechanics.18 The spin selection rule, which states that only the same multiplicity states can combine and produce intense atomic lines, occurred simultaneously with the idea of spin quantization. Spin−orbit coupling (SOC) as a reason for multiplet splitting and for appreciable intensity of intercombination series in heavy atoms and diatomics was though realized quite early.35 The main difficulty in the generalization of these ideas to polyatomic luminophores was connected with understanding the role of nonradiative transitions (which are almost absent in diatomic molecules and isolated atoms) and the expansion of the molecular orbital concept to large molecules. Determining of the electronic origin of molecular phosphorescence is inextricably connected with the names of A. N. Terenin, G. Lewis, and M. Kasha who explained the nature of the “secret” metastable energy level proposed by Jabl̷oński in 1933.32 The latter concluded the existence of such a level based on Kautsky’s experiment on phosphorescence of “energetically isolated” fluoresceine adsorbed by convenient adsorbents.35−37 Jablonski was absolutely right that the phosphorescence process involves metastable (M) low-lying state. He assigned not only phosphorescence as a direct M-S0 radiation transition but also an indirect M → S1 → S0 delayed fluorescence emission (without mentioning of the “singlet” and “triplet” terms). It is interesting that the efficient delayed T1 → S1 → S0 emission

toys and other examples of glowing paint and clock dials that glow for some time after being illuminated are usually considered as phosphorescence.19 Such light emission then slowly fades away in a dark room, typically within a few minutes or even hours. This is though not a real phosphorescence in the sense of the IUPAC definition determined by the quantum nature of spin prohibition for the origin of light. In a historical perspective, the term “phosphorescence” has undergone a few transformations since it became known in ancient time. A lack of concord in the common use of this term still exists in contemporary literature of physics of semiconductors and organic chemistry. Historically, phosphorescence was first attributed to inorganic materials, such as metal salts and phosphorus. The earliest revelation of a solid state phosphorescence phenomenon and its practical use came to us from ancient China with the description of paintings which look different in day lighting and in the dark.20,21 The ink used in the painting was, possibly, the first man-made long-lasting phosphor material. The science in Europe was alerted much later; the first phosphorescent material was discovered when alchemists tried to heat minerals in the attempt to create gold. Thus, Vincenzo Cascariolo in Bologna heated natural barite (barium sulfate) under reducing conditions and obtained the famous luminous material, the so-called Bolognian stone. We do not know which dopant occurred to be responsible for the observed phosphorescence, but we can suppose that the host material was BaS.14 Thus, barium sulfide was the first metal salt ever synthesized as a long-lasting phosphor material. Nowadays, it is known that the long-living afterglow of some minerals and metal salts is determined by migration and trapping of excited electrons in the crystal; they can be capped by the traps or by emission centers. The electron “is sitting” in the trap until it gets sufficient activation energy from phonons and can be released into the conduction zone. Finally the electron reaches the luminescence center emitting the light. The luminescence center (or activator) represents a defect introduced into the crystal lattice as a natural chemical admixture or as an artificial dopant. For example, the wellknown phosphor ZnS contains in its crystal lattice a small amount of copper. The word “phosphorescence” was applied to these inorganic materials until recent time, but such practice is not recommended anymore according to the IUPAC convention. The word “phosphor” (or “light bearer” in Greek) was used for a long time, though the element “phosphorus” was obtained first by Hennig Brand in 1669.34 When phosphorus is slowly oxidized in the open moist air it becomes luminescent, and this long-lasting emission provides the name “phosphorescence”. Thus, this very general phenomenon was erroneously connected with the glowing white phosphorus. Now we know that this is a result of an oxidation reaction, and the term is not appropriate in the mechanistic sense applied in this review. The 19th century evidenced much advance in synthesis and measurement of phosphorescent materials, but the time was still too early for proper interpretation. The synthesized CaS and SrS sulfides should be mentioned as interesting phosphor materials. The culmination point was achieved in 1866 when Theodor Sidot obtained hexagonal crystals of ZnS (similar to mineral wurtzite), with a controlled amount of dopant, and studied their phosphorescence.34 The results of Sidot were used by Becquerel who specialized in luminescence of inorganic salts, and these studies led him in 1986 to discover the 6503


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spin-allowed and are also usually allowed by the orbital symmetry selection rules (for many molecules the symmetry elements are absent and thus no symmetry restrictions exist). In great contrast, the absorption or emission involving singlet and triplet states (Tn) are strongly spin-prohibited since they proceed as second-order, or nonlinear, processes involving both spin−orbit and dipole couplings. A large body of phosphorescence measurements has been accumulated over the years, covering simple diatomic molecules7 and more complicated compounds like polyacenes,47−49 porphyrins,8,50,51 circulenes,47,52 heavy metal complexes,1−3,8 and others.53−55 Despite this fact, investigations on phosphorescence at the level of modern quantum mechanics are quite limited and a detailed description of phosphorescence mechanisms has therefore remained elusive.11

found a great application recently when Adachi et al. in 2012 constructed T1 → S1 → S0 based pure organic light-emitting diodes (OLEDs).38,39 The delayed T1 → S1 → S0 emission was activated by heating, and this phenomenon was therefore coined as thermally activated delayed fluorescence (TADF). The TADF phenomenon was not utilized for a long time because the S1 level is often located much higher in energy (by 1−2 eV) with respect to the T1 state for typical organic chromophore molecules. In this case, the thermal activation of the reverse spin-forbidden T1 → S1 transition requires special temperature heating. Adachi et al. developed a proper design of organic chromophores to achieve a very small S1-T1 energy gap (less than 0.1 eV), which finally leads to a competitive rate of the T1 → S1 reverse intersystem crossing (RISC) at room temperature.38,39 An idea that the Jablonski’s metastable energy level being responsible for phosphorescence is a triplet state was advanced by A. N. Terenin in 1943.40 The same conclusion was made independently by G. Lewis and M. Kasha in 1944.41 It is quite clear that they did not know about each other’s research due to the second world war. Initially, the triplet origin of phosphorescence met strong resistance from Jabl̷oński himself and from the school of J. Frank, E. Teller, and R. Livingston.42 The strong obstruction was continued until 1958 when the first EPR signal of the triplet state was observed for naphthalene.18 In contrast to Jabl̷oński’s views, Terenin, Lewis, and Kasha used the “spin” term to assign the metastable state as the triplet spin state. Particularly, Terenin wrote: “Evidence pointing to the existence a long-lived metastable electronic state in the near derivatives of benzene in the condensed phase at low temperatures as well in the gaseous state is being summarized and the view advanced that this state corresponds to a triplet term of the aromatic ring”.40 This was in complete agreement with the later words of Lewis and Kasha: “Having concluded that the existence of the phosphorescent state is not contingent upon a rigid solvent, there seems no reasonable alternative to the assumption that the phosphorescent state is the triplet state of the molecules”.41 During the 70 years that have passed since 1944, the theory and practice of phosphorescence has been developed in all details due to the circumstance that “persistent luminescence” has a real impact in human life. New capabilities for scientists and ramifications of the phosphorescence phenomenon have turned out to be really great. One can mention here biomolecules bearing long-lasting phosphorescent labels (phosphor-reporters) that provide important medical diagnostics. For example, Chermont et al.43 proposed in 2007 a novel in vivo imaging biotechnique by using rare-earth doped nanoparticles with red to near-infrared long-lasting emission (phosphorescence), which greatly improve the signal-to-noise ratio. These new optical labels permit a deep penetration into tissue and constitute the next-generation in modern bioimaging technology.15−17,43−46

3.1. Spin−orbit Coupling and S-T Mixing

Neglecting spin−orbit coupling and other relativistic effects, any singlet state (1Ψ) can be represented as the product of a symmetrical (with respect to permutation) spatial wave function (Φ+) and an antisymmetrical spin wave function (Ω−): 1

Ψ = Φ+Ω−

(4)

For two electrons, the spatial and spin parts for the singlet state can be written as follows: Φ+ =

1 [φ (1)φb(2) + φa(2)φb(1)] 2 a

(5)

Ω− =

1 (αβ − βα) 2

(6)

where φa(1) and φb(2) are the molecular orbitals which depend on the spatial coordinates x1, y1, and z1 and x2, y2, and z2 for the first and second electrons, respectively; α and β are the eigenfunctions of the one-electron Sz operator, with quantum numbers MS = ± 1/2. The triplet state wave function (3Ψ) is also asymmetrical and can be represented as a product between an asymmetrical spatial wave function (Φ−) and three different symmetrical spin 0 wave functions (Ω1+, Ω−1 + , and Ω+): 3

Ψ = Φ−Ω+

(7)

1 [φ (1)φb(2) − φa(2)φb(1)] 2 a

Φ− =

Ω1+ = αα , Ω−+1 = ββ , Ω+0 =

1 (αβ + βα) 2

(8)

(9)

If Ψ and Ψ correspond to the same electronic configuration [φa(1) and φb(2) are the same orbitals for the S and T states], the “−” sign in the spatial part of 3Ψ (eq 8) leads to the fact that the T state energy is lower than the S state energy (the first Hund’s rule) since the exchange integral between the orthogonal MOs φa and φb is always positive. Therefore, the lowest excited state is always triplet for singlet ground state molecules (most known molecules exhibit a singlet ground state, in other words, all electrons are paired that determine their diamagnetism). Actually, the 1Ψ and 3Ψ functions are never “pure” singlets or triplets (even for helium and noble gases) because the spin− orbit coupling always mixes the 1Ψ and 3Ψ eigenfunctions.24,56−59 In a physics sense, SOC is the force which aims to “flip” the spin angular momentum (S) of an electron due to 1

3. PRINCIPLES OF COMPUTATIONAL PHOSPHORESCENCE The great importance of triplet states in chemistry owes to the exchange correlation effect. The triplet excited states are always positioned lower in energy than the corresponding singlet excited states, which are formulated as a molecular analogue of Hund’s rule in atomic physics (eq 3). In the framework of the electric-dipole approximation, the electronic transitions between the ground singlet (S0) and excited singlet (Sn) states are 6504

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the following relation of S with the spin magnetic momentum (μ):

μ=−

|e| S mc

−1 ⟨1Ψ|HSO|3Ψ⟩ = ⟨Φ+Ω−|HSO|Φ−Ω1,0, ⟩ + Z = α 2 ∑ ∑ A3 ⟨Φ+|Li|Φ−⟩ i A r

(10)

αα ⎛ ⎞ ⎜ ⎟ ββ 1 ⎟ (αβ − βα)|S|⎜ ⎜ 1 ⎟ 2 (αβ + βα)⎟ ⎜ ⎝ 2 ⎠

where e, m, and c are universal constants. The interaction energy between spin magnetic momentum (μ) and the magnetic field caused by electron orbital motion can be represented as H = −μ[Ev], where v is the speed of electron motion in the nuclear electric field E. In a relativistic approximation and accounting of eq 10 for the single electron, the main part of the spin−orbit Hamiltonian (HSO) equals:18 HSO =

Ze 2 LS 2m2c 2r 3

(14)

However, this simplification is not entirely correct. In the considered case of two electrons ∑i Li = l1 + l2 and S = s1 + s2 and with account of functions (4)−(9), both spatial and spin parts of eq 14 are equal to zero because of the difference in permutation symmetry of the components in eqs 4 and 7. Thus, only the antisymmetric combinations of l1 − l2 and s1 − s2 operators with respect to permutations (eq 15) provide the proper permutation symmetry in order to mix the singlet and triplet state:

(11)

where r is the orbital radius and L is the orbital angular momentum. Eq 11 can be rewritten for many-electron molecular systems in a simplified form accounting for the summation of i electrons and A nuclei:24 HSO = α 2 ∑ ∑ A

i

ZA ⃗ l s = 3 Ai i⃗ rAi

∑ ςA ∑ lAi⃗ si⃗ A

i

∑ LiSi = l1s1 + l2s2 i

(12)

=

where α is the fine structure constant (α ≈ 1/137 if l, s and r are in atomic units). Eq 12, known also as the one-electron microscopic Breit−Pauli Hamiltonian,60 thus couples the orbital and spin angular moments through the interaction with the effective nuclear charge for each atom ZA. In eq 12, we have neglected the repulsive interaction of li⃗ and si⃗ with the electron j and other types of two-electron interactions of si⃗ and lj⃗ . These terms decrease relatively with an increase of Z and r (i.e., with the increase of element number), and thus, the spin− orbit Hamiltonian is well-represented by eq 12 for most cases.56 For two-center integrals in light diatomics, these two-electron terms exhibit a trend to be canceled by the single-electron contribution. Thus, in eq 12, a semiempirical approximation (the right part) can be effective, where the one-center integrals are estimated by the corresponding atomic SOC constant ςA, determined from spectral multiplet splittings, while all twocenter terms are neglected.7 The expectation value of the r−3 operator with the Slater atomic orbitals includes Z3, thus the SOC splitting depends on nuclear charge power four. The SOC constant thus strongly increases with atomic number and is very high in heavy elements and can even be comparable with electron repulsion integrals of type (3) inside the atom. Deviations from the Russel-Saunders scheme in this case gets essential, and a perturbational SOC treatment cannot be applied. The complete form of the Breit−Pauli many-electron SOC operator reads:24 ⎡ Z HSO = α 2⎢∑ ∑ 3A lAi⃗ si⃗ − ⎢⎣ r Ai A i

∑∑ i

j≠i

⎤ 1 ⃗ ⎥ + l ( s 2 s ) j⃗ 3 ij i⃗ ⎥⎦ rAi

1 1 (l1 + l 2)(s1 + s2) + (l1 − l 2)(s1 − s2) 2 2 (15)

The symmetric combinations in the right part of eq 15 can mix two triplet states of different spatial symmetry and of different spin projections. The antisymmetric permutation parts of the SOC operator, eq 15, can mix the S and T states. Only the s1 − s2 operator provides mixing of the spin parts in eq 14. This is a very important peculiarity of the SOC operator which is crucial for the theory of phosphorescence. The S-T mixing is responsible for phosphorescence intensity, lifetime, and quantum yield. The T-T perturbations by SOC provide anisotropic deviation of the g-factor in EPR spectra and of the Zeeman energy in an external magnetic field; they also provide an important contribution to the zero-field splitting (ZFS) parameters being in strong balance with the corresponding S-T contributions (in the second order of perturbation theory). These peculiarities of the SOC operator are important in the complete treatment of the phosphorescence phenomenon by the optical detection of the magnetic resonance (ODMR) technique in low temperatures for molecular solids.1,24−30 In order to calculate the SOC energy value, the spatial parts of the SOC matrix elements in eq 13 need to be summed. This task is complicated for polyatomic molecules because many contributions to SOC should be accounted for (at least singleand multicenter, one- and two-electron integrals). For the large number of diatomic species and some other small systems, the spin−orbit coupling energy can be accurately calculated by ab initio methods with results that often are found to be in good agreement with experimental spectroscopic data.61−64 For large systems (aliphatic hydrocarbons, polyacenes, etc.), semiempirical approximations, like the all-valence-electron CNDO/S or MINDO−CI models, were used earlier.7,65,66 Generally, these methods provide a good interpretation of experimental data with spectroscopic parametrization that restricts the choice of studied objects. In this context, the search for balance between the reasonable computational scaling and adoptable errors for the account of SOC remains

(13)

where the second term represents the two-electron contribution (i and j electrons) into the SOC value in addition to the one-electron part (eq 12). In order to separate the spin−orbit coupling integral into spatial and spin parts, the following combination of eqs 4−9 and eq 12 are often used:56 6505


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open. In computational chemistry, there is growing competition between the nonrelativistic and fully relativistic four-component methodologies,67,68 the former being based on the Schrödinger equation and the latter on the Dirac equation. Unfortunately, an application of relativistic four-component theory for largescale systems is very limited due to the high computational cost, which is an argument that has spurred the continued development of nonrelativistic approaches capable of relativistic “corrections” in modern computational phosphorescence. Among them, the scaled-charge spin−orbit (SCSO)67 approximation is one of the best balanced methods for computations of SOC comparing with the more complicated Breit-Pauli and atomic mean field spin−orbit approximations. The SCSO approximation implies the usage of an effective-core oneelectron HSO operator (ECSO) which is similar to the BreitPauli operator (eq 12) but that consists of the effective charges 67 Zeff A instead of ZA. The ECSO approximation usually provides a correct distribution of valence electron density for heavy atoms elements, while other nonrelativistic methods tend to fail in that respect. That underpins the applicability of the linear scaling ESCO technology to describe the phosphorescence of heavy atom complexes. The practical applications of the ECSO operator method is additionally discussed in section 4.8. Another treatment of the HSO operator is given by the atomic mean field (AMF) approximation,69−71 which describes the screening effect in a computationally efficient way. The AMF approach includes SO corrections for the valence electrons through the exact one-electron term (eq 12) using the mean field two-electron screening procedure. Considering only valence−valence excited state determinants, the Fock-type generalization of the HSO operator gives71

T1̃ = T1 +

∑

(17)

n

where δn is an admixture coefficient: δn =

⟨Sn|HSO|T1⟩ E(T1) − E(Sn)

(18)

Typically the admixture coefficient values vary in the range of 0.001−0.1 depending on the system. As follows from eq 18, the δn value is larger the larger the ⟨Sn|HSO|T1⟩ integral in the numerator and the smaller the ΔEST gap in the denominator. For radiative T1 decay (phosphorescence), the intensity of the S0-T1 interaction is proportional to the square of the ⟨S0|M|T1⟩ transition moment (M is the electric dipole moment operator). Accounting for eqs 17 and 18 and a similar mixing of the Tn and S0 states one can get: ⟨T1|HSO|S0⟩ E(T1) − E(S0) ⟨Sn|HSO|T1⟩ + ∑ ⟨Sn|M|S0⟩ E(T1) − E(Sn) n ⟨Tn|HSO|S0⟩ + ∑ ⟨T1|M|Tn⟩ E S0) − E(Tn) ( n

⟨S0̃ |M|T1̃ ⟩ = [⟨T1|M|T1⟩ − ⟨S0|M|S0⟩]

(19)

where the first term is a permanent dipole difference contribution, while the second and third terms correspond to the borrowing of intensity from spin allowed Sn−S0 and Tn−T1 transitions. It should be noted that the |T1⟩ state consists of three individual components (sublevels) that are characterized by distinct decay rates and different polarization directions along the x, y, and z axes. Here we need to clarify that this is a correct statement for high-symmetry molecules (C2v, D2h, and higher point groups) for which the emission from individual spin-sublevels is polarized along one of the x, y, and z axes of symmetry. In this case, the irreducible representation of the rotational vector coincides with the individual irreducible representations of the symmetry point group, which is also correct for the translation vector. In such highly symmetrical molecules, the quantization axes of the zero-field spin-sublevels Tx, Ty, and Tz strictly coincide with the symmetry axes and each triplet state sublevel provides strict x-, y-, and z-polarized emission. For low symmetry molecules (CS, C2, etc.), the quantization axes of the zero-field spin-sublevels may not be identical with the symmetry axes and therefore the emission may be of mixed polarization. It is caused by the symmetry of rotational and translational vectors that define the polarization direction and SOC anisotropy. In general, the multipole expansion for interaction between a molecular electronic shell and a light electromagnetic wave can be truncated with account of the electric dipole, magnetic dipole, and electric quadrupole terms. Thus, the transition moment for the S-T radiation process has the following general definition:

MF ‐ orbs

HijSO − MF = ⟨i|H(1) SO − MF|j⟩ +

∑ δnSn

occ(M ) ×

M

⎡⟨iM |H(1,2)|jM ⟩ + ⟨iM |H(1,2)|jM ⟩ − ⟨iM |H(1,2)|M j⟩− ⎤ α β SO β α SO α ⎥ ⎢ α SO ⎢ ⎥ (1,2) − ⟨iMβ|H(1,2) ⎢ ⎥ SO |Mβ j⟩ − ⟨Mαi|H SO |jMα⟩ ⎢ ⎥ − ⟨Mβ i|H(1,2) ⎣ ⎦ SO |jMβ ⟩

(16)

where occ(M) is the occupation number for orbital M, i, and j denote spin−orbitals, and Mα and Mβ are partially occupied orbitals. The HijSO−MF operator in eq 16 remains computationally expensive because it requires an evaluation of all twoelectron SO integrals. Due to this reason, eq 16 can be simplified by reducing all multicenter atomic orbital parts for each atom separately: it is indeed possible to calculate SO integrals for each separate atom. Accounting for the spherical atomic symmetry, it is possible to separate integrals into pure radial and angular parts with the subsequent generation of a one-center, one-electron, SO integral matrix. The AMF method provides excellent results for most cases, particularly when heavy-atom effects on S-T transitions are considered71 (see section 4.7). 3.2. Perturbation Theory for S-T Transitions

Due to the SOC effect, the singlet states of polyatomic molecules contain some admixture of triplet state character. At the same time, the T1 state is contaminated by singlet state wave functions that can be described by first-order perturbation theory.1,31,56 The total wave function of the perturbed T̃ 1 state can be presented as the sum of the zero-order unperturbed T1 wave function and some admixture of the nth singlet-state wave functions Sn: 6506


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⎛ 1 ⎞ ∑ e ri⎟⎟ |MS − T|2 = ⟨S0|∑ e ri|T1⟩2 +⟨S0|⎜⎜ ⎝ 2mc i ⎠ i ⎛ℏ ⎞ ∂ × ⎜⎜ ∑ ⎟⎟|T1⟩ ⎝ i i ∂ri ⎠ 3π 3(ΔES − T )3 + ⟨S0|(m ∑ e ri) × (∑ ri)|T1⟩2 10c 2h3 i i

state wave function contaminated by the zero-order Tn admixtures in the first-order perturbation theory analogous to eq 17:

S0̃ = S0 +

The radiative lifetime (τα) and corresponding rate constant (kα) for the selected spin sublevel Tα1 can be estimated as1 kα =

where ri is the ith electron radius-vector, m is the electron mass, e is the electron charge, and ΔES‑T is the energy difference between S0 and T1 states. The first component in eq 20 is the electric dipole moment contribution, while the second and third terms correspond to the magnetic dipole moment and electric quadrupole moment contributions, respectively. Both magnetic dipole and quadrupole terms are negligibly small compared with the first component, but in the case when the electronic transition is forbidden in the electric dipole approximation due to some selection rules, these small magnetic dipole and quadrupole contributions become very important for molecular phosphorescence (like in the case of molecular oxygen). The first “dipole difference” term in eq 19 is important for considering phosphorescence processes accompanied by charge transfer (CT) upon an S-T transition. It is especially important for the phosphorescence of heavy-metal complexes for which the T1 state usually has a metal-to-ligand CT nature. Considering configuration interaction (CI) for the localized and charge-transfer states of the same symmetry, we can present the wave function for the T1 state similarly to eq 17 involving the CT admixture (with a coefficient) as a perturbation to the pure local triplet state wave function: Ψ1 = |T1local⟩ + a| 3CT ⟩

τav =

(25)

γ

(26)

1 + e−(ΔExy / kBT ) + e−(ΔExz / kBT ) kx + k ye−(ΔExy / kBT ) + kze−(ΔExz / kBT )

(27)

where ΔExy and ΔExz correspond to the energy difference between the xy and xz spin-sublevels, respectively. Since the radiative phosphorescence emission competes with the fast nonradiative quenching processes, we need to distinguish separately the observable phosphorescence lifetime (τobs) and the radiative lifetime (τav). The τobs for rigid media can often be estimated from experimental data:56

(21)

1 = k r + k nr τobs

(28)

where kr is a rate constant for the radiative T1-S0 transition and knr is a rate constant for the intramolecular nonradiative T1-S0 transition. We should note that the τav parameter is a peculiar property of the isolated molecule when all quenching factors are absent by definition. As a general rule, τobs < τav because the nonradiative processes quicken the T1 state deactivation, and for the correct calculations of τobs we therefore need to take into account the knr value. Nonradiative T1 decay can be estimated within time-dependent perturbation theory by the Fermi’s Golden Rule31,76 through the ⟨S0|HSO + Tnuc|T1⟩ integral. Here Tnuc is the nuclear kinetic energy operator and ⟨S0|, |T1⟩ are eigenfunctions of the full Hamiltonian (including Tnuc and HSO).76,77

∑ |⟨S0̃ |M γ |T1̃ ⟩|2 (22)

3.4. Vibronic Perturbations in Phosphorescence

and is equal to 2 ΔES ‐ TMS2‐ T 3

∑ |M γ (T1α)|2

Distinctive spin orientation of the triplet state sublevels can be detected at extremely low temperatures, about 1−4 K (when the SLR processes are almost suppressed), by the optical detection of magnetic resonance method (ODMR).72−75 In some cases, particularly in the case of heavy metal complexes, eq 26 is no longer strictly valid in the hightemperature limit due to the large ZFS. It can be corrected by Boltzmann-weighted averaging over the kx, ky, and kz decay rates for the individual spin-sublevels of the T1 state:

The oscillator strength of the S0 → T1 absorption is proportional to the square of the transition moment value MS‑T:

fS ‐ T =

x ,y,z

3 1 1 1 = + + τav τx τy τz

3.3. Phosphorescence Intensity and Lifetime

γ

64π 4(ΔETS)3 1 = τα 3h4c 3

At room temperature and even at 77 K, the triplet spin sublevels are populated evenly due to the fast spin−lattice relaxation (SLR) processes. Therefore, the average radiative lifetime (τav) can be estimated as

Thus, the locally excited T1 state of organic molecules can be perturbed by charge transfer from the lone pair of a heavy atom (e.g., 4px of Br− ion in KBr solvent). Eq 21 can be rewritten in the same manner for the singlet ground state wave function. Since the T1 and S0 states usually have different spatial symmetry, the CT admixtures can include excitations from different lone pairs of the heavy atom. As a result, the SOC matrix elements between the T1 and S0 states ⟨Tlocal + a3 CT| 1 1 HSO|Slocal + β CT⟩ will contain αβ⟨4p |l |4p ⟩ contributions 0 x z y from the external heavy atom due to the CT contamination of an initially localized wave function through the intermolecular CI coupling. Accounting for the large first term in eq 19, it is easy to explain the enhancement of phosphorescence transition moments by the external heavy atom effect.

MS2‐ T =

(24)

n

(20)

3

∑ δnTn

In the above discussion, we have used the “pure” electronic Hamiltonian perturbation (∑nδnSn term in eq 17), whereas the nuclear coordinates are considered to be fixed. In order to take into account the nuclear motion, we need to use the Maclaurin series expansion for the HSO operator in terms of the normal coordinates of the emissive T1 state:56

(23)

where M and ΔES‑T values are in atomic units (a.u.), γ sets the polarization of the transition momentum M along the x, y, or z axes. In eq 22, the S0̃ term corresponds to the ground singlet 6507


Chemical Reviews HSO =

0 HSO

Review

⎛ ∂H 0 ⎞ + ∑ ⎜⎜ SO ⎟⎟Q a ∂Q a ⎠ a ⎝

∫ ∫ ∫

(29)

In the right part of eq 29, both terms can mix T1 and S0: the first term corresponds to the electronic Hamiltonian for the motionless nuclei (superscript “0”), while the second term provides mixing of the T and S states being coupled by the firstorder vibronic perturbation due to the nuclear displacement along the 3N−6 normal modes Qk. Actually, the matrix element of the operator determined by eq 29 can be rewritten in the extended form: 0 w = ⟨S0|HSO |T1⟩ +

∑∑ k

0 ⟨Sn |HSO |T1⟩

E(T) 1 − E(Sn )

+

S0

n

∑∑

0 ⟨Tn|HSO |S 0 ⟩ Cn′, k E(S0) − E(T) 1

k

n

∫ ∫ ∫

where |0⟩ is the initial reference state and ⟨⟨A;Vω1⟩⟩, ω1 ω2 ω3 ⟨⟨A;Vω1,Vω2⟩⟩, ∫ ∞ −∞⟨⟨A;V , V , V ⟩⟩ correspond to the linear, quadratic, and cubic response functions, describing how a property referring to operator A changes under perturbations of second, third, and fourth order in the light−matter interaction. If one considers HSO as the first-order perturbation (Vω1 → HSO), the molecular property of interest in this review is the lth component of the dipole operator xl:

∂ Sn Cn , k ∂Q k T1

∞

̃ |A|0(t) ̃ ⟩ = ⟨0|A|0⟩ + ⟨0(t) ⟨⟨A; V ω1⟩⟩e−iω1dω1 −∞ ∞ ∞ 1 + ⟨⟨A; V ω1, V ω2⟩⟩e−i(ω1+ ω2) 2 −∞ −∞ dω1 dω2 ∞ ∞ ∞ 1 + ⟨⟨A; V ω1, V ω2 , V ω3⟩⟩ 6 −∞ −∞ −∞ e−i(ω1+ ω2 + ω3) dω1 dω2 dω3... (31)

∂ Tn ∂Q k

lim (ω − ωf )⟨⟨x l ; HSO, V ω2⟩⟩0, ω2

ω → ωf

(32)

where f is the final state of the system (the triplet state in our case), Vω2 being an arbitrary second-order perturbation. The response function (eq 32) actually corresponds to the S-T transition moment of eq 19 and can be represented by the Einstein summation convention as

(30)

where the second and third terms can mix the T1 and S0 states by vibronic perturbations for the S0-Sn and T1-Tn couplings by further accounting of vibrational integrals with the Qk displacement length operator, included in the Cn,k coefficients.77,78 Note again that |T1⟩ has three components, each of which has its own decay rate. The w2 integral multiplied by a Franck−Condon factor can be related to the ISC process and to vibronic activity of phosphorescence.78 Accounting of vibronic perturbations is very important for the cases when the T-S mixing with eq 19 is symmetry forbidden by the SOC and dipole selection rules. The best known example of this phenomenon is the benzene molecule where the phosphorescence is purely vibronically induced.79−81 The time-dependent method in the Condon approximation82 and the multidimensional harmonic oscillator model83 can also be used for an account of vibronic interactions. For large and nonrigid molecules, we can neglect in principle the nonadiabatic coupling between the S0 and T1 states and describe the vibronic structure of the phosphorescence spectra by the displaced harmonic oscillator model (DHOM).84,85 In this model, the distortion effect of the potential energy surfaces can be ignored, so the so-called Duschinsky rotation effect,86 which characterizes the difference between the potential energy surfaces (PES) of ground S0 and excited T1, is small and thus the normal frequencies and eigenvectors are assumed to be the same for singlet and triplet states despite the relative shift of the equilibrium positions for these states. The DHOM formalism is simple but usually describes the shape of phosphorescence spectra profiles quite well for many molecular systems, like organometallic Ir(III) phosphors, for example.87,88

k lim (ω − ω1)⟨⟨x l ; HSO ; V ω2⟩⟩0, ω

ω → ω1

[2] [2] = −N jr(ωf )HSOjl Xlf − N jSO(r[2] jl + r lj )Xlf [3] [3] SO + N jr(ωf )(E[3] jml + E jlm − ωf S jlm)Nm Xlf r

where the linear response vectors N (ωf) and N from the linear response equations:

(33) SO

originate

N r(ωf ) = [(E[2] − ωf S[2])−1r [1] *]*, [1] N SO = (E[2])−1HSO

(34)

and the triplet excitation vectors (Xf) and frequencies (ωf) are calculated by solving the triplet excitation eigenvalue equation: (E[2] − ωf S[2])X f = 0

(35)

Equations 33 and 34 include the E[2] and S[2] response matrices that correspond to the Hamiltonian and overlap matrices in the T operator basis, respectively: * E[2] jk = ⟨0|[T j , [Ho , Tk ]]|0⟩ , * S[2] jk = ⟨0|[T j , [Ho , Tk ]]|0⟩

(36)

The matrices with superscript [1] (over the perturbing operators) denote the gradient vectors: * r[1] j = ⟨0|[T j , r ]|0⟩ ,

3.5. Quadratic Response Theory and Phosphorescence

* r[1] SO , j = ⟨0|[T j , HSO]|0⟩

22−27,89−91

Response theory can be considered as a special representation of time-dependent (TD) perturbation theory31,76 where the selected molecular property (at an external perturbation) is characterized by response functions. In the presence of a TD-perturbing field Vt, a molecular property A can be Fourier-transformed yielding coefficients that define the response functions of different orders (indices 1, 2, 3,···):

(37)

for the dipole and SOC operators, respectively, while the matrices with superscript [2] can be similarly represented through the double-commutator expressions.23,24 The QR theory approach has been used successfully for “ordinary” electric-dipole phosphorescence in a series of articles.22,81,24 In magnetic phosphorescence,92 the electric 6508


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variation perturbation theory60,104 and spin−orbit coupling configuration interaction (SOC−CI) methods105−107 can be especially noted.

dipole operator is replaced by the magnetic dipole operator expression: μ ⃗ = βB ∑ ( li ⃗ + 2 si⃗) i

3.6. Density Functional Response Theory for Phosphorescence

(38)

where βB is the Bohr magneton. The second and fourth terms in eq 31 correspond to the linear and cubic response functions, respectively.22 The first of them (linear response) describes second-order characteristics such as polarizability, and the residue (at some excitation energy) provides the corresponding transition moment.93 The cubic response function provides the fourth-order parameters like second-order hyper-polarizabilities and is usually applied for the computations of exotic nonlinear optical features like the frequency-dependent hyperpolarizability to estimate intensities in hyper-Raman spectra.94 A main feature of the response theory is that the usual algorithms for the excited state summation is replaced by the solving of linear equation sets that can be accomplished without prior calculations of the excited states.22 It makes it possible to apply the QR theory to phosphorescence problems of largescale molecular systems like heavy-metal complexes,95−98 organic oligomers,99,100 and other nanoscaled objects.101−103 In the framework of conventional approaches (accounting for SOC), the first-order corrections to the S0 and T1 states provide a nonzero transition moment which can be presented as an energy weighted sum over intermediate states coupled to S0 and T1 states through the SOC and dipole matrix elements. In a finite basis set calculation, the number of intermediate states is finite and calculations of the transition moments have often been realized by evaluating and summing the individual contributions to this interaction. The number of terms are though mostly prohibitively large for an explicit summation to be possible, and one is therefore resorted to truncation, socalled few-state or truncated sum-over-state approaches. However, the summation can be very slowly convergent.22−27 This is often the case already for linear properties and even more so for nonlinear properties like the transition moments for the singlet−triplet transitions, where partial contributions of intermediate states can vary in size and have an arbitrary sign. There is thus a clear advantage to compute phosphorescence by response function theory,22−27 where the phosphorescence parameters are determined from the residues of QR functions containing both the electric dipole and SOC operators, and where the complete sum-overstate value is implicitly obtained through iterative solutions of linear sets of equations and/or eigenvalue equations. With response theory, one so obtains an analytic transferability between the wave function and its properties, in that once the parameters of the former has been determined, all its properties are obtained analytically without further approximations or interference from the user. The property calculation, like phosphorescence matrix elements, is not much more costly than the optimization of the reference state and its total energy. Thus, the evaluation of phosphorescence parameters in the QR methodology is described not in terms of individual contributions to a sum over states but as an implicit summation which is carried out by solving sets of linear equations. This represents a well-defined and much more efficient alternative to property calculations than sum-over-state procedures used in conventional approaches for calculation of molecular properties.23 At the same time, there are some other efficient methods, alternative to QR, that can be used for computing phosphorescence parameters, among them the

The general theory of spin-dependent response functions implemented for ab initio wave function electronic structure theories (like SCF and MCSCF) have been successfully applied to describe the phosphorescence of small molecules.22,24,67 The response methodology implemented for wave function theory is rigorous and accurate26 but has limitations in terms of size of the studied system which have prohibited progress of QR methodology relative to phosphorescence of larger classes of molecules. One of the limitations is that the usage of the twoelectron Breit-Pauli SOC operator introduces a strong increase of computational time due to the calculation of a large number of two-electron integrals. The second problem relates to the fact that the accounting of SOC with the consideration of triplet perturbations induces a latent “triplet instability” problem for the random phase approximation (RPA) response applied to HF reference wave functions. Moderately correlated MCSCF wave functions with small active spaces can solve, in principle, this triplet instability problem, but it leads to a limitation to small species because of the poor computational scaling of active space wave functions. To avoid the first limitation (regarding two-electron integrals), the two-electron part of the SOC operator should be omitted (one-electron microscopic Breit-Pauli SO operator, for example) or accounted for through the one-electron schemes which include two-electron terms implicitly via the atomic mean-field approximation, nuclear screening, or by the Ziegler scheme which neglects spin-other-orbit terms, discussed in section 3.1.26 The combination of such one-electron approximations for the SO integrals together with single determinant theory like DFT for response calculations makes it possible to overcome the limitations reviewed above (although a latent triplet instability problem remains for TDDFT). As a result, we get an effective QR DFT methodology, which is generally accurate and applicable to prediction of phosphorescence parameters as well as of other triplet state features.26 The starting point of the density functional response formalism26 for general response properties including phosphorescence, and that makes it compatible with corresponding response theory for ab initio wave functions, is an exponential parametrization of the time evolution |t⟩ operator which acts on the Kohn−Sham (KS) determinant |0⟩ |t ⟩ = e−κ(̂ t )|0⟩

(39)

where κ̂ is an anti-Hermitian operator κ̂ =

∑ κpqσap†σ aqσ pqσ

(40)

The matrix elements of the κ̂ operator provide the nonredundant rotations between occupied and virtual orbitals. The κ̂ operator implicates a summing over the orbitals p, q, and spin projections σ; a†pσ (aqσ) is the creation (annihilation) operator of an electron in orbital φp (φq) with spin σ. This parametrization allows for the evaluation of the density response by the DFT functional used to optimize the Kohn− Sham orbitals and the density itself. The adiabatic approximation is still actual because the time-dependent functional 6509


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of spin-unrestricted DFT, thus excluding phosphorescence. A future viable alternative, “in between” spin restricted and spin unrestricted DFT, is probably the implementation of QR to spin-flip response theory.109,110

depends on the time-dependent density by the same principle like the time-independent functional versus time-independent density dependence; thus the “time evolution” is strictly allocated to the Kohn−Sham determinant and density and not to the functional. The density of electrons with spin σ can be represented through the expectation of the σ-density operator as follows: ρσ (r, t ) = ⟨0| e κ ρ̂ σ̂ (r) e−κ |̂ 0⟩

3.7. Nonradiative S-T Transitions

In the general sense, nonradiative S-T transitions, so-called intersystem crossings or ISC, can not be considered as phosphorescence as they are of a different physical origin. However, on the other hand, both ISC and phosphorescence processes have a common spin-forbidden origin. Therefore, a proper analysis of the ISC phenomenon can not be avoided in our review. Even after the advent of quantum mechanics, there were difficulties in understanding nonradiative transitions in polyatomic molecules. That is the reason why the triplet nature assignment of the metastable state in the Jablonski diagram was not accepted until much later. The seminal paper of Terenin40 started with the normal coordinate analysis on the potential energy surface (PES) of polyatomic molecules and with the Duschinsky effect for electronic transition between two PES’s.

(41)

that can be rewritten in the Kohn-Shem formalism as ρσ̂ (r) =

∑ φp(r)*φq(r)ap†σ aqσ (42)

pq

where ρα and ρβ for α and β spins are treated as independent variables.26 The equivalence between α and β densities is finally imposed, and we can present the Hamiltonian operator through the variation in the spin density energy functional δE[ρα,ρβ] H=

∑ ∫ dτρσ̂ (r) σ

δE δρσ (r)

(43)

Q T = JQ S + D

Applying the Ehrenfest theorem to an arbitrary operator q in the presence of a time-dependent perturbation V(t) gives26 ⎡ ⎛ d⎞ ⎤ ⟨0|⎢q, e κ ̂⎜H + V − i ⎟e−κ ̂⎥|0⟩ = 0 ⎝ ⎣ dt ⎠ ⎦

where QT and QS are the mass-weighted normal coordinates of the triplet and singlet states, J is the Duschinsky matrix which determines rotation of the two coordinate systems, and D is the displacement vector showing the geometry change between the two PES’s. Thus, the general probability of nonradiative transitions in organic chromophores through vibrational relaxation was first analyzed in ref 40. Modern computational technology83 affords a complete simulation of excited states PESs and their vibrational assignments. Not only internal conversion but also intersystem crossing (ISC) have been analyzed in detail in terms of modern DFT methods.57,83,111,112 Two types of ISC transitions, the pumping S1 → T1 and deactivating T1 → S0 processes, can be considered in terms of the same methodology. For the further qualitative analysis of the ISC probability, the “Fermi Golden Rule” approximation is usually applicable31,76 when the SOC integral between the initial |i⟩ and final |f⟩ states is much smaller than the adiabatic energy gap Ei − Ef. Under these conditions, the implementation of the “Fermi Golden Rule” leads to the ISC rate constant k:57

(44)

where q is a column vector with nonredundant orbital rotation operators. To evaluate the QR function within V ω 1 and V ω 2 perturbations, we must evaluate the first-order parameters that correspond to these perturbations. As a result, the secondorder Hamiltonian consists of contributions from the first- and second-order densities: H ω1, ω2 =

∑ ∫ ∫ dτ dτ′ρσ̂ (r) σσ ′

+P12

∑

∫∫∫

δ 2E ρ ω1, ω2 (r′) δρσ (r)δρσ ′(r′) σ ′

dτ dτ′dτ″

σσ ′ σ ″

ρσ̂ (r)

δ 3E ρ ω1 (r′)ρσω′ 2 (r″) δρσ (r)δρσ ′(r′)δρσ ″(r″) σ

(47)

(45)

kISC = 2π

and the second-order densities consist of the first- and secondorder terms

1 ℏ

∑ |H1if |2 δ(Ei − Ef )

(48)

f

where is the matrix element of the first-order Hamiltonian and the δ function provides the energy conservation for the nonradiative transition. For the rate constant of the Si → Tf nonradiative ISC process, one can write within the “Fermi Golden Rule” approximation and neglecting the spin-vibronic (second-order) perturbation terms H1if

ρσω1, ω2 (r) = ⟨0|[κ ̂ω1, ω2 , ρσ̂ (r)] + P12[κ ̂ω1 , [κ ̂ω2 , ρσ̂ (r)]]|0⟩ (46)

where P12 is the symmetrizing operator. Eq 45 for the secondorder Hamiltonian looks quite complex, and it becomes quite complicated to implement for certain functionals, like for GGAtype functionals.26 Despite the algebraic character of the QR DFT formalism, this methodology demonstrates an advantageous cost/accuracy relationship not only for benchmarking small compounds but also for large size systems. The quadratic response theory of phosphorescence is readily applicable for MCSCF wave functions also of molecules with ground state open shells and, thus, to phosphorescence from states different from triplets. Molecular oxygen serves here as a good example.92 In the realm of density functional theory, the QR formalism extends to high-spin coupled open shell ground states,108 while there is so far no implementation for low-spin coupled cases. Also, to the best of our knowledge, quadratic response theory has yet not implemented for the popular form

kISC = 2π

1 ℏ

∑ ∑ |⟨Si , θik|HSO|Tγf , θf l⟩|2 δ(Eik − Efl) γ

l

(49)

where γ designates the ZFS spin-sublevel, i and f numerate the electronic states of the singlet and triplet multiplicity, and k and l label the vibrational states wave functions θ for each PES. In the second order of perturbation theory, the nonadiabatic coupling integrals are necessary to be calculated with eq 29, and this formulation of the nonradiative ISC theory remains rather complicated.57,83 Vibrational contributions in eq 49 can be separated from the electronic terms in the framework of the Condon approach83 6510


Chemical Reviews k kISC = 2π

= 2π

Review

1 ℏ

∑ |⟨Si|HSO|Tγf ⟩|q2

1 ℏ

∑ |⟨Si|HSO|Tγf ⟩|q2 F k(Eik)

γ

0

×

l

0

γ

the Huang−Rhys factor explained above in this section. In order to use the Herzberg−Teller theory for vibronic phosphorescence bands with the DHOM approach, we shall denote the Born−Oppenheimer (BO) wave functions as

∑ |⟨θik|θf l⟩|2 δ(Eik − Efl) (50)

λ

where Fk (Eik) is the final vibrational state density at the point of the initial state energy level weighted by Franck−Condon (FC) factors ∑1|⟨θik|θfl⟩|2. The ISC rate is always lower than the internal vibrational relaxation; thus, the total nonradiative S1-T1 transition rate can be determined by a summation of the Boltzman-weighted kkISC rate constants (eq 50) using the population distribution multiplier exp[(Ek − E0)/kBT]. The FC-weighted density of states (Fk) can be estimated by the Marcus semiclassical approach in the room-temperature regime:111 k

−1/2

F = (4πλkBT )

Ψ n , μ(r , Q ) = λ ψ n(r , Q )Θn , μ(Q )

where total vibrational wave function Θn,μ (Q) is a product of harmonic oscillator polynomials θn,μ in the electronic quantum state n, μ is a particular vibrational quantum number, Θn,μ (Q) = ∏k θn,μ (Qk), and Qk is a normal mode. In the framework of the DHOM approach, the normal modes are the same in the S and T states. The vibronic radiative T1 → S0 transition moment can be expanded into series around the T1 state equilibrium:

∫ θT ,0θS ,μ dQ + ∑ ηTS , k ∫ θT ,0Q kθS , μ dQ

MTS = MTS(Q 0)

∑ exp(−Sk) n

⎡ (ΔE + nℏω + λ)2 ⎤ k ⎥ exp⎢ − ⎥⎦ n! ⎣ 4λkBT

k

Skn

Sk =

(54)

where (51)

⎡ ∂ ⎤ ηTS , k = ⎢ MTS(Q )⎥ ⎢⎣ ∂Q k ⎥⎦ 0

Here ΔE is the energy gap between the S1 and T1 states at their equilibrium geometries, ωk is the frequency of an effective normal mode involved in the ISC process, ωk ΔQ k2/2ℏ

(53)

(55)

The electronic transition moment MTS (Q) = ⟨3ψ1(r,Q)| e∑ i r i | 1 ψ 0 (r,Q)⟩ can be calculated at different nuclear coordinates along the Qk normal mode by the quadratic response method; the overlap integral in eq 54 is evaluated through the gradients, shift in equilibria, and the Huang−Rhys factor (eq 52).103,112 The second way to calculate vibronic activity of phosphorescence is to implement the expression of the type (eq 30) with account of the appropriate electric dipole Cn,k = ⟨θSn,0|eQk|θS0,1⟩⟨ΘS′n|ΘS′0⟩ coefficients, where the primed Θ excludes the k-vibrational mode. This approach was used for analysis of benzene phosphorescence.79

(52)

is the dimensionless Huang−Rhys factor with the equilibrium position offset ΔQk. The Marcus reorganization energy λ accounts for low-frequency vibrational modes and solventinduced relaxation.57,111 In any case, the overall rate constant is usually very sensitive to the energy gap and the geometry shift ΔQk of the states PES’s involved in the ISC process. Thus, it is very significant to employ the most accurate methods for geometry optimization and frequency calculations while the SOC integrals are usually less sensitive to the accuracy of the electron correlation method used. During the ISC process the change in the PES structure of the S and T states is accompanied by a change in vibrational frequencies and expansions of normal modes. Duschinsky (as assistant of prof. Terenin)40,86 analyzed this effect and indicated that the normal modes of both states are not orthogonal to each other any more; thus the FC factors cannot be separable into one-dimensional overlap integrals for individual harmonic oscillator polynomials. In the general case of a complex polyatomic molecules it is rather difficult to find a relation between S and T normal modes involved in the ISC process. By Duschinsky’s proposal these two sets of vibrational modes are related by the linear matrix transformation (eq 47). This relationship is not exact in the general case for common normal modes of the S and T states which span the same irreducible representation as rotation around one of the axes. This is especially important for the S-T state SOC matrix elements since they transform as rotation symmetry. But so far the Duschinsky approximation has been proven to be a reasonable approach.83,111,112 If the molecule consists of common symmetry elements in the triplet and singlet state, the J matrix in eq 47 has a block-diagonal form and the displacement D vector possesses nonvanishing elements only for the totally symmetric vibrations. It is relevant to connect the analysis of ISC to vibronic phosphorescence in terms of gradients, shift in equilibria, and

3.8. Simple Qualitative Rules for Intersystem Crossing and Phosphorescence analysis

First order of perturbation theory (see section 3.2 and eqs 17 and 18) shows that the singlet−triplet mixing is increased with the increase of the SOC matrix elements (condition 1), as well as with the decrease of the S-T energy gap (condition 2). Both factors are important for radiative (eq 19) and especially for nonradiative (eqs 50 and 51) S-T transitions. Condition 1 can be interpreted in the framework of the conceptual El-Sayed rules,56,57,59 which says that the ISC rate is larger when the nonradiative relaxation proceeds between S and T states of different orbital symmetry with respect to the molecular plane reflection, thus involving a change of orbital type. Particularly, this rule predicts that the rates of radiationless ISC transitions of the 1(π,π*) ⇝ 3(n,π*) type are larger than ISC for the 1(π,π*) ⇝ 3(π,π*) transitions and vise versa [i.e., the 1(n,π*) ⇝ 3(π,π*) transitions are usually faster than the 1(n,π*) ⇝ 3(n,π*) transitions]. This rule is grounded on the single-electron nature of the main part of the SOC operator (eq 12) and the rotation symmetry of the orbital angular momentum in real space. Let us consider the zcomponent of the scalar product l ⃗s ⃗ and a carbonyl molecule R2CO which lies in the yz plane (Scheme 1). Inside an atom, we have lzpx = −iℏpy, and the spin part in eq 15 mixes the S and T (Ms = 0) states. Thus, the operator lz rotates the px orbitals on the oxygen atom (a part of the out-of-plane π MO) and 6511


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4.1. Phosphorescence of Molecular Nitrogen

Scheme 1. Orbital Rotation for One Electron at the Oxygen Atom in the Excited Carbonyl Moleculea

a

Molecular nitrogen is a main component of Earth’s atmosphere and plays an important role in many processes, including the green aurora phenomenon. Despite that N2 is a very simple diatomic molecule, its electronic spectrum is quite complicated113 and includes a number of symmetry- and spinforbidden transitions. The spin-allowed transitions of N2 were early well studied theoretically,114,115 while the S-T transitions of N2 remained poorly studied until the 1990’s,25,116 when ab initio computations accounting for relativistic effects started to flourish. The most important S-T transitions of the N2 molecule are A3∑+u → X1∑+g , B′3∑−u → X1∑+g and C3∏u ← X1∑+g transitions which correspond to the Vegard-Kaplan (AX), Ogawa-Tanaka-Wilkinson (B′-X), and Tanaka (C-X) band systems, respectively.113,114 The A-X (ν0−0 = 49754.8 cm−1) and B′-X (ν0−0 = 65585.4 cm−1) bands are well-observed both in absorption and emission (phosphorescence) spectra of molecular nitrogen, while the more high-lying C-X band (ν0−0 = 88977.9 cm−1) is observable only in absorption.114 We should note that the intensity of the Tanaka C-X band is comparable with that for the spin-allowed S-S absorption in the same region (above 100 nm). The experimentally estimated oscillator strength for the C-X band117 (0−0 transition) is about 2.2 × 10−6, which is comparable with the intensity of the singlet− singlet Lyman-Birge-Hopfield transition (a1∏g ← X1∑g+) allowed by magnetic and quadrupole interactions.117 The next observable 0−1 and 0−2 transitions of the C-X band are two- and four-times less intensive than the 0−0 line (1.1 × 10−6 and 5.6 × 10−6, respectively), which gives the upper intensity limit of 3.86 × 10−6 for the Tanaka band. The total oscillator strength for the first three bands calculated by multiconfiguration quadratic response (MCQR) is 1.05 × 10−6, which is of the same order as the experimental values.25 The B′-X band113 is less intense than the C-X band but more intense than the Vegard-Kaplan A-X transition. The overall oscillator strength calculated by the MCQR method for the B′X system was found to be equal 1.65 × 10−7 (notably, that all three spin sublevels of the B′3∑−u state are active).25 The most known and most studied singlet−triplet transition in the electronic spectrum of molecular nitrogen is the VegardKaplan band system A3∑+u → X1∑+g .25,50,116,118,119 In contrast to the B′-X band, the A-X transition possesses a spin sublevel resolved phosphorescence. The experimentally measured lifetimes are 2.5 s (Ms = ± 1) and 1.27 s (Ms = 0) for the lowest vibrational state of X1∑+g .119 In the electric dipole and quadrupole approximations, the emission from the Ms = 0 spin substate is forbidden, but it is not prohibited in a magnetic dipole approximation. At the same time the radiative decay from the Ms= ± 1 spin sublevel is an allowed electric dipole transition. Therefore, the Ms = 0 decay component should be significantly less intense than the Ms = ± 1 components.116 This is in some contradiction with the Shemansky’s experimental119 data presented above in the text that refers to the Hund’s rule (b) for the rotational line intensity.120 While the MCQR calculations consider the molecules as nonrotating systems, the two equal lifetimes for the Ms = ± 1 and infinite lifetime for the Ms = 0 sublevel have been predicted.116 The long lifetime of the A3∑+u state (couple of seconds) means that all the spin sublevels should be in a thermal equilibrium under ambient conditions (usually used for the studies of VegardKaplan bands system), and therefore, only an averaged radiative lifetime could be correctly measured in laboratory conditions. By this statement, the overall radiative lifetime of the A3∑+u (Ms

Pure AO rotation produces T-T excitation.

transfers it into the py in-plane AO, which almost coincides with the lone pair n-MO. Thus, the 3(n,π*) state can be coupled with the 1(π,π*) state or the singlet ground state. Pure orbital rotation of one (red) electron in Scheme 1 represents a magnetic Tnπ*→ Tππ* transition induced by orbital angular momentum. This will determine a very weak optical absorption. If the process includes the other (blue) electron orbital rotation with simultaneous spin flip, it corresponds to ISC into the ground state. A relatively large SOC contribution from the oxygen atom in this case will contribute to ISC, and only the Tz (Ms = 0) spin sublevel will be active in the deactivation process. The symmetric part of eq 15 is responsible for SOC mixing between Tnπ* and Tππ* states having different spin projections (Ms = ± 1). Such SOC integrals can influence the zero-field splitting of the triplet state and its effective g-factor in EPR spectra but do not contribute directly to ISC and phosphorescence. Condition 2 implies two limiting cases:57 a weak coupling limit (small ΔEST gap) and a strong coupling limit (high ΔEST gap). In the first case, the ΔQk displacement for each normal kmode should be rather small; therefore, the Franck−Condonweighted density of states (eq 51) and the transition rate constant depend exponentially on the energy ΔEST gap (eq 50) (i.e., the smaller the energy gap the larger the ISC rate). The inverse relation between the ISC rate constant and the adiabatic ΔEST energy gap is often realized in the case of the high-energy limit: a higher ΔEST gap corresponds to a lower ISC probability. These qualitative El-Sayed rules are commonly summarized as the energy gap law.57

4. COMPUTATIONAL PHOSPHORESCENCE OF MOLECULAR SYSTEMS In this section, we focus on the theoretical explanation of phosphorescence for concrete molecular examples, including simple symmetrical systems like di- and triatomics, polyenes, benzene, and related acenes, such as the azabenzenes, halogensubstituted benzenes, and pyridines, as well as more complicated polyatomic systems like porphyrins and Ir(III) phosphors. 6512


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= ± 1) substate was predicted by MCQR theory to be equal to 2.58 s and the corresponding averaged radiative lifetime to 3.87 s (accounting eq 26 and supposing that one of the time components is equal to ∞)116 that can be compared well with the experimental τav = 1.9 s.119 Concluding briefly, the VegardKaplan emission band is a clear example of zero-point averaging of spin-sublevel-resolved phosphorescence due to the SLR phenomenon for the long-lived triplet excited states under laboratory conditions and moderate temperatures. The photophysics and spectroscopy of molecular nitrogen deserves to be published in a separate review because of the atmospheric importance not only of N2 neutral molecules but also N2+ species and their collision complexes with oxygen and other gas-phase particles.121,122 Here we have focused only on the three most demonstrative spin-forbidden A-X, B′-X, and C-X bands and, particularly, on the applicability of the QR formalism to the case of N2 as a representative of diatomics.

the theoretical point of view, the a-X and b-X transitions are allowed in both the quadrupole and magnetic-dipole approximations.122−124,128,129 Neglecting the small quadrupole contributions, Minaev explained these emission bands as magnetic-dipole allowed radiation130 in agreement with the rotational spectra analysis.128,129,131,132 Magnetic-dipole transitions are induced by the orbital angular momentum L and are usually observable in optical emission spectra of diatomic species. In both b-X and a-X transitions of molecular oxygen, the L term is affected by spin− orbit coupling mixing with the triplet states of the same Πg symmetry (Figure 2).122−124 Hereinafter, we introduce the low subscript numbers “0” and “1” to indicate the different spinsublevels in the triplet state and, corresponding to Ω, the total electronic angular momentum projection on the interatomic axis. In accounting for zero-field splitting, the ground state of molecular dioxygen consists of three spin sublevels: doubly degenerate levels with Ω = 1 (quantum number MS = +1 and −1) and a single one with Ω = 0 (M = 0). These spin sublevels of the T0 state (Ω = 1 and Ω = 0) are split in zero magnetic field by 3.96 cm−1 (Figure 2).122 It is important to note that the a-X1 component of the a-X transition is magnetic and that the a-X0 is pure quadrupole by nature.124 Remembering the fact that the quadrupole contribution is very small,130 we consider the a-X infrared band as the a-X1 transition. The a-X1 band intensity (Figure 2) in the nonrotating oxygen molecule consists of only an orbital angular momentum contribution to the magnetic dipole moment due to the SOC-induced mixing of the a, X, and 1,3Πg states.123 Owing to the large energy differences between these states, E(1Πg) − EX = 78000 cm−1 and E(3Πg) − Ea = 48000 cm−1, the mixing of the singlets and triplets is negligible and the spin prohibition to the a-X transition remains strict. In the simple INDO approximation,130,133 the magnetic dipole moment induced by orbital angular momentum L equals |MLa‑X1| = 0.0023iμβ (μβ is the Bohr magneton) in good agreement with subsequent ab initio calculations,134 including the QR level of theory.22,92 It is also interesting to discuss the intensity of the red atmospheric b-X1 band for which the magnetic dipole moment has a dominant contribution induced by spin angular momentum, S. In the general case,122−124

4.2. Magnetic Phosphorescence of Dioxygen

Another important diatomic component of the Earth’s atmosphere is dioxygen. It is a truly amazing molecule supporting biological life on Earth and possessing a triplet “biradicaloid” ground state. In addition to the triplet ground state of X3∑−g symmetry dioxygen also possesses two low-lying singlet excited states a1Δg and b1∑+g (shortly X, a, and b, see Figure 2). The phosphorescence of dioxygen is also specific and

Figure 2. (a) The mechanism of intensity borrowing of spin magnetic moment for the b-X1 transition (red lines). The blue lines denote the mechanism of intensity borrowing of the orbital contribution to the magnetic transition moment for the b-X1 (---) and a-X1 (− · −) transitions (SOC-2 ≈ SOC-3 ≈ SOC-4). (b) Scheme for the quadrupole transitions: the Qb transition is determined by the difference of the static quadrupole moments in the b and X0 states accounting for SOC between them (SOC1). The Qa transition borrows intensity from the quadrupole allowed Qab Noxon transition via SOC1.

Mb ‐ X1 = MbL‐ X1 + MbS‐ X1 = μβ (L + 2S)

(56)

In accordance with the calculations, the orbital contribution to the transition moment of the b-X1 band is negligible (about | MLb‑X1| = 4 × 10−4iμβ).92,123 The major contribution to the b-X1 transition intensity gains from the spin term in eq 56 with the magnetic moment polarization vector directed perpendicular to the molecular axis. Taking into account that123

corresponds to the S1,2 → T0 emission being reversed to the usual phosphorescence associated with the T1 → S0 emission. Electronic transitions between the X3∑−g ground state and the doubly degenerated a1Δg singlet states are forbidden both by the gerade−gerade parity and spin selection rules. The a1Δg → X3∑−g transition is also prohibited by the change of orbital angular-momentum projection (ΔΛ = 2), whereas b1∑+g → X 3 ∑ g− is additionally forbidden by the ∑ + → ∑ − selection.92,122−124 It is therefore somewhat surprising that both a-X and b-X bands have been well-observed experimentally in the red (762 nm, b-X transition)125 and in the near-infrared (1270 nm, a-X transition) regions.126,127 From

Ψb = |b1Σ+g ⟩ + c|X3Σ−g,0⟩

c=

(57)

⟨X3Σ−g,0|HSO|b1Σ+g ⟩ E(b1Σ+g ) − E(X3Σ−g,0)

(58)

one has, 6513


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mixed with the 3B1 state.143−151 This pure theoretical conclusion is in good agreement with the latest experimental measurements and high-level ab initio calculations for the electronic structure: “the Wulf spectrum is a superposition of two almost independent spectra corresponding to the two diabatic electronic states, 3A2 and 3B1”.148 4.3.2. Water, Hydrogen Sulfide, and Hydrogen Cyanide. Singlet−triplet absorption transitions for the important triatomic molecules H2O and H2S are illustrating examples. The lowest electronic transitions in their spectra take place in the UV region in the range of 186−145 nm and 270− 190 nm for the H2O and H2S species, respectively,135 that corresponds to the H + O(S)H photodissociation process. The S−T transitions in H2O and H2S are very weak:136 the transition moment to the lowest (the most active) spin-sublevel of the lowest triplet state 3B1 for the H2O molecule equals 2.6 × 10−3a.u.; for the H2S molecule, the same parameter is about twice as high and equals 5.7 × 10−3 for the lowest triplet state 3 A2. Accounting for eq 23, Minaev et al. estimated the average oscillator strength value to be 2.9 × 10−6 and 6.5 × 10−6 for the singlet−triplet absorption of H2O and H2S molecules in the lowest 3B1 and 3A2 states, respectively (QR DFT/B3LYP level of theory).136 In contrast to the H2O molecule, where the first 3 B1 ← X1A1 transition overlaps with the stronger S−S absorption, the 3A2 ← X1A1 transition in the H2S molecule provides a clear visible tail at about 240 nm because it is separated from the more intense and high-lying S−S absorption.136,137 Despite the weakness of the S-T transitions for the H2O and H2S species, the kinetics of their photodissociation cannot be explained correctly without accounting for the T1-S0 transition contribution to the bond breaking.136 It is also interesting to discuss the S-T absorption of another triatomic species, hydrogen cyanide (HCN), because of its importance in the CN laser system and for dense interstellar cloud spectroscopy.152,153 Fluorescence of HCN has been reported,154 but phosphorescence remains unobservable despite the fact that the T1 state of HCN has been predicted155 to be physically stable. It is known136 that the HCN molecule has a linear structure in the ground singlet electronic state (X1∑+) while hydrogen cyanide becomes nonlinear (∠HCN = 124°) upon excitation into the a3A′ triplet state. Therefore, the a3A′ ← X1∑+ absorption band consists of a vibrational progression that is affected by a linear-to-bent molecular transformation upon the electronic transition. It was shown in ref 136 that the vertical S0-T1 excitation of the HCN molecule is characterized by a relatively large oscillator strength (f = ∼10−8) compared with ozone, for example. The second closelying S0-T2 b3A″ ← X1∑+ transition has comparable oscillator strength, thus both of them could be well-resolved in the absorption spectrum. An interesting feature of the a3A′ and b3A″ states of the HCN molecule is a principal difference in their g-tensors (particularly in the gxx components).136 Comparing with the free electron the gxx value for the a3A′ exhibits a positive deviation (+0.00154) while for the next b3A″ state the deviation is negative (−0.00813). This means a principal difference in the spin-rotation coupling constants for the two states (Mxx = −2AΔgxx, where A = 25.195 and 21.202 cm−1 for the a3A′ and b3A″, respectively).136 This is an important computational prediction that could help to identify experimentally the a3A′ and b3A″ states by rotational analysis for the corresponding electronic transitions. 4.3.3. Hypohalogenous Acids. Other important triatomic species are the hypohalogenous acids, HOHal, that play a

MbS− X1 = μβ ⟨Ψb|2S|ΨX1⟩ = μβ 2 c⟨X3Σ−g,0|Sx ± iSy|X3Σ−g,1⟩ = −2μβ c = −0.0268iμβ

MLb‑X1

(59)

MSb‑X1

Summing up the and terms in eq 56, we receive a total magnetic dipole moment equal to −0.0264iμβ123 that is almost equal to the value obtained from experimental intensity measurements.135 As can be seen from eq 59, the b-X1 band “borrows” its intensity from the microwave transition X3∑−g,0 → X3∑−g,1 between the spin sublevels with Ω = 1 and Ω = 0 (Figure 2). This is a unique example in molecular spectroscopy.122−124 This model provides an excellent explanation for the experimentally proved higher probability of the red b-X1 atmospheric band compared to that of the infrared a-X1 singlet oxygen emission. 4.3. Some Examples of Phosphorescence from Triatomic Species

The measuring of phosphorescence parameters for triatomic species is usually impossible, since for most cases the low-lying triplet state has a dissociative nature.127 A number of important compounds like water, ozone, hydrogen sulfide, and hypohalogenous acids possess a “saturated” structure for which the S-T transition leads to bond cleavage and to subsequent destruction of the compound. Therefore, one can often observe S-T absorption136−141 but not phosphorescence. At the same time, some species like SO2,142,143 NO2−,144 and GeHal2136,145 exhibit observable phosphorescence with a mechanism that can be well-explained theoretically combining with rovibronic spectroscopy analysis. Certainly, we can here concentrate our attention only on the most interesting examples of triatomics such as O3, H2O, H2S, HOCl, and HCN. 4.3.1. Ozone. Ozone plays an important role in the atmosphere, protecting life on Earth from fatal cosmic radiation due to strong absorption in the Hartley system X1A1 → 11B2 (300−214 nm) and in the Huggins X1A1 → 21A1 (354−300 nm) band.146 At the same time, a very weak absorption in the near IR region 1048−700 nm (so-called Wulf band) remained overlooked for a long time and many controversial assignments of this band were proposed because of the manifold of ozone excited states that exist in the low-energy region (1−2 eV).146 It was not until 1992147 that the Wulf band was correctly assigned as a singlet−triplet 3A2 ← X1A1 absorption, and in 1994 this assignment was proved by ab initio QR calculations.146 In 2007, the predominant 3A2 ← X1A1 nature of the Wulf band was finally proven by the study of Grebenshchikov et al.148 on the basis of calculations of rotational states dynamics for the O3 molecule. As follows from the ab initio calculations,146,149 absorption to the first low-lying 3B2 state can be neglected due to the very small intensity ( f = 8 × 10−10) that corresponds to a long radiative lifetime of the 3B2 state of about 83 s.146 The transition dipole moment for the 3B2 state is about ten times smaller than for the close-lying 3A2 and 3B1 states. Thus, the excitation of the 3B2 state of ozone provides only a very small contribution to the Wulf band. The oscillator strengths estimated by the MCQR approximation for the next 3A2 ← X1A1 and 3B1 ← X1A1 electronic transitions are of the same order of magnitude, being 5.7 × 10−7and 1.3 × 10−7, respectively.146 Both of these transitions therefore contribute to the Wulf band and can be finally assigned as the 3A2 state 6514


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Table 1. Complete Map of Selection Rules for the Vibronically Induced Phosphorescence in Benzenea N

mode

pathway (Ms = 0, HSO = A2g)

M

N

pathway (Ms = ± 1, HSO = E1g)

M

1

e2g b2g e2g b2g e2g b2g e2g b2g e2g b2g e2g b2g

X1A1g ― 1E1u ··· 1B2u --- 3B1u

xy

X1A1g ··· 1E2g --- 3E2g ― 3B1u

xy

X1A1g ··· 1E2g ― 1B2u --- 3B1u

xy

X1A1g --- 3A2g ··· 3E2g ― 3B1u

xy

X1A1g ― 1E1u --- 3E1u ··· 3B1u

xy

X1A1g --- 3A2g ― 3E1u ··· 3B1u

xy

7 8 9 10 11 12 13 14 15 16 17 18

X1A1g ― 1A2u ··· 1E2u --- 3B1u X1A1g ― 1E1u ··· 1E2u --- 3B1u X1A1g ··· 1E2g --- 3B2g ― 3B1u X1A1g ··· 1B2g --- 3E2g ― 3B1u X1A1g ··· 1E2g ― 1E2u --- 3B1u X1A1g ··· 1B2g ― 1E2u --- 3B1u X1A1g --- 3E1g ··· 3B2g ― 3B1u X1A1g --- 3E1g ··· 3E2g ― 3B1u X1A1g ― 1A2u --- 3E1u ··· 3B1u X1A1g ― 1E1u --- 3A2u ··· 3B1u X1A1g --- 3E1g ― 3E1u ··· 3B1u X1A1g --- 3E1g ― 3A2u ··· 3B1u

z xy z xy z xy z xy z xy z xy

2 3 4 5 6

The sign “―” denotes dipole coupling, “---” spin-orbit coupling, and “···” vibronic coupling. The benzene molecule is oriented in the xy plane, the z axis is perpendicular to the molecular plane, and M is the dipole moment operator.

a

4.4. Phosphorescence of Unsaturated and Aromatic Hydrocarbons

destructive role in the atmospheric ozone depletion. In contrast to most of the triatomics, the S0-T1 absorption for HOHal species has the same order of intensity as the S 0 -S 1 transition.139,156 Thus, the S-T transitions could definitely contribute to the photodissociation processes of HOHal molecules in the region above 350 nm, since a number of low excited states of the hypohalogenous acids are dissociative. Really, the first S0-T1 (13A″ ← X1A′) transition provides a very weak band in the simulated absorption spectrum of the HOCl molecule (in the range of 385−420 nm depending on the choice of basis set, while the experimental value is 380 nm). An estimated intensity for this transition was about f = (5−6) × 10−6, which is only slightly less intense than that for the spinallowed singlet−singlet absorption at 304 nm.138 SOC analysis indicates that the T1-S0 electronic transition is strongly polarized along the O−Cl bond and borrows intensity from the well-allowed n1A′ ← X1A′ UV transitions of the nπ* type.156 Therefore, the direct S-T absorption can be considered as a competitive pathway of HOCl photodissociation together with the spin-allowed singlet−singlet absorption, both being in the atmospheric transparency window. It is remarkable that the 13A″ triplet state of HOCl plays a crucial role in gas-phase chlorination of methane. Litvinenko and Rudakov recently showed157 that the reaction CH4 + HOCl + H2O → CH3Cl + 2H2O entails the singlet−triplet preactivation of HOCl. They also showed that the involving of the 13A″ dissociative state into the chlorination of methane by HOCl−H2O system makes it possible to reduce the TS energy barrier by 21.4 kcal/mol compared with the singlet transition state reaction pathway. This reaction is a magnificent example of how the triplet states can beneficially affect the chemical kinetics. Thus, one can never underestimate the role of spin-forbidden S ↔ T transitions in chemistry, even if these transitions have low probability. The triplet state of HOBr was first predicted by Francisco et al.139 without intensity estimation. Ingham et al.140 observed a weak band of HOBr (maximum at 460 nm) with an intensity being in good agreement to that which was predicted by independent theoretical SOC calculations.138 Barnes et al.141 observed production of HO radicals upon HOBr photolysis in the range of 400−600 nm that additionally confirms the S-T nature of the long-wavelength absorption. In the case of HOI, the relatively strong absorption has definitely been ascribed to the two lowest S-T transitions.138

The low-lying excited states of conjugated unsaturated and aromatic hydrocarbons are usually associated with delocalized π-electrons.18,24 The role of radiative S ↔ T transitions in electron spectroscopy of conjugated hydrocarbons has been substantially less studied comparing with singlet states. Notably, the triplet states of polyenes play an important role in the spectroscopy of natural carotenes158−160 and synthetic electroactive polymers55 like polyfluorene98,161 and polyphenylenevinylene.162 Particularly, Guha et al.161 have shown that the triplet yields for ISC starting from low-lying singlet excited states for different polyenes are significantly increased (about three orders) with the increase of the polymer chain. These triplets provide observable electrophosphorescent emission under inert atmosphere (i.e., in the absence of quenchers). Moreover, in the presence of heavy metal salts or complexes of transition metals such as Ir, Pd, and Re, there is considerable enhancement of the T-S emission intensity observed due to the spin−orbit coupling effects.161 Transition metals are known to help in spin crossover processes,163−166 and here, we refer works on spin-catalysis theory for more details and applications.167−171 Returning to the discussion on short polyenes it should be noted that the estimated oscillator strengths of the S0 → T1 transition for pure species like ethylene, butadiene, and hexatriene are on the order of 10−101,99,168 that corresponds to a very large radiative lifetime of the T1 state (from seconds to few minutes) and being devoid of any phosphorescence patterns, because of efficient quenching. In contrast to polyenes, the aromatic molecules exhibit not only S0 → T1 absorption but also long-lived T1 → S0 phosphorescence in rigid solvents and crystals.24 These phenomena are rather complicated to analyze from the theoretical point of view because the spin prohibition of the S↔T transitions is complemented by orbital symmetry selection rules. Thus, we must apply not only electronic structure theory, like spin−orbit quadratic response theory but also take into account vibronic perturbations of the phosphorescence.24,79,172−175 Starting with benzene which is a milestone species of aromatic compounds and which exhibits a T1−S0 transition that is a great challenge to theory because of the dual spin and orbital symmetry prohibition. However, at the same time the low-temperature phosphorescence spectrum of benzene has been successfully detected and characterized by 6515


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Figure 3. Scheme of the most efficient pathways for the vibronically induced phosphorescence of benzene (blue lines, channels (15) + (17); pink lines, channels (16) + (5); z,x ̅ y̅ ̅ means the emission polarization along the z axis and xy plane, respectively). This figure correlates with Table 1 but additionally illustrates the energy disposition of the benzene excited states and the vibronically active e2g and b2g vibrations. The zero-field splitting is shown schematically.

--- 3E1u is really weak since both states are of ππ* nature and has only two-center integral contributions. A detailed computational study of the spin sublevel selectivity for the benzene phosphorescence indicates that the x and y degenerate sublevels (out-of-plane polarization) are characterized by the shortest radiative lifetimes in the hightemperature limit approximation. The phosphorescence from the T1x,y sublevels (in-plane polarization) induced by the vibrations of b2g symmetry [channels (8) and (16) are the most active among the other b2g promoted channels, Table 1]24 leads to an order of magnitude longer radiative lifetime compared with the e2g-induced bands (ν8 and ν9 vibrations). At the same time, in-plane polarization from the Tz1 spin sublevel is negligibly weak, which corresponds to the huge radiative lifetime of more than 1 h.24 Due to this fact, the average phosphorescence radiative lifetime (τav) equals to about 1.5 τx,y. All these features are in good agreement with MIDP experiments.172 The average phosphorescence radiative lifetime estimated at the QR level of theory with different basis sets varies in the range of 23−96 s that is in agreement with the socalled “best experimental” value of 30 s.177 However, the estimations for the nonradiative quenching processes and the quantum yields for the benzene phosphorescence still present an open problem178,179 and now provide a more accurate radiative lifetime estimation of 100 s. It is natural to consider next the phosphorescence of linear polyacenes, including the simplest naphthalene example. In the framework of the D2h symmetry point group for all these molecules (naphthalene, anthracene, tetracene, pentacene, etc.), the lowest energy level has 3B2u symmetry (molecule is in the xy plane, x is the long axis of C−C skeleton, and z axis is perpendicular to the molecular plane). Since the spatial restriction is absent for the 3B2u → X1Ag transition in the linear polyacenes, only the spin prohibition still remains strict; the HzSO and HxSO components couple the 13B2u state with the 1 B3u and 1B1u states, respectively.66 The electric dipole allowed transitions can possess only x- and z-polarization, introducing

the well-resolved rich vibronic structure.176 It is commonly established now that the first excited triplet state of benzene is of 3B1u symmetry (in the framework of D6h symmetry point group).24,79,172−176 By this reason, the main problem for the benzene phosphorescence is that the 3B1u → X1A1g transition is strongly forbidden both by spin and symmetry selection rules (actually the 0−0 transition is missing). Even accounting for “pure electronic” SOC effects the 3B1u → X1A1g phosphorescence remains disallowed and is only allowed through the coupling of nuclear and electronic motions including SOC perturbation (so-called vibronically induced phosphorescence). This is quite difficult to calculate and analyze, only the quadratic response level of theory seems to be adequate for this case.24,79 The theoretical analysis of vibronic-resolved phosphorescence spectra of benzene indicates that the emissive activity can be promoted only by b2g and e2g vibrations.24,80 Accounting for these vibronic perturbations and also for the fact that the lowest 3 B1u state consists of three spin sublevels with Ms = 0 and Ms= ± 1 magnetic quantum numbers, we can summarize the full possible scheme for the benzene phosphorescence as presented in Table 1. As follows from the multiconfiguration quadratic response (MCQR) calculations, the major contribution to the benzene phosphorescence intensity originates from the channels (15) + (17) (Table 1 and Figure 3),24,75,174 which are promoted by the e2g vibrational mode (namely ν8 within the standard Wilson classification) and are determined by the intensity borrowing from the out-of-plane polarized dipole-allowed πσ excitations of X1A1g ― 1A2u and 3E1g ― 3E1u types. The microwaveinduced delayed phosphorescence (MIDP) measurements (at 4.2 K) for the C6H6/C6D6-doped crystals indicates that the main part of the e2g-promoted phosphorescence should be outof-plane polarized175 that correctly corresponds to the channels (15) + (17) (Table 1 and Figure 3). Channel (5) provides inplane polarization, but it does not include one-center SOC integrals and is shown to be less effective. The SOC mixing 1E1u 6516


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finally confirmed these results.24,174,181 Only one τx-spin sublevel is responsible for the radiative decay of the lowest 3 B2u state providing the out-of-plane phosphorescence polarization common also for all studied Tl(ππ*)-S0 transitions of polyenes and polyacenes and related aromatic compounds. DFT QR calculations with the B3LYP/6-31G** approximation provide similar results for naphthalene with the radiative lifetime 84 s and the band maximum at 454 nm. The corresponding results for anthracene are 797 nm and τ = 129 s (with 89% of perpendicular polarization of phosphorescence) do not contradict any known experimental data.18

the 3B1g and 3B3g triplet states, respectively, for the T1−Tn dipole strength borrowing in the triplet manifold. Therefore, four possible mechanisms can be proposed to explain the total intensity borrowing for the 3B2u → X1Ag phosphorescence (see Table 2). Table 2. Complete Map of Selection Rules for the Phosphorescence in Naphthalenea N

pathway (Ms = 0, HSO = B1g)

M

N

pathway (Ms = 1, HSO = B3g)

M

1 2

X1Ag ― 1B3u --- 3B2u X1Ag --- 3B1g ― 3B2u

x x

3 4

X1Ag ― 1B1u --- 3B2u X1Ag --- 3B3g ― 3B2u

Z Z

4.5. Phosphorescence of Azabenzenes

Sign “―” denotes dipole coupling; “---” spin-orbit coupling; and M is a dipole moment operator.

a

Azabenzenes are isoelectronic analogues of benzene. The formal substitution of CH groups by nitrogen atoms introduces the lone-pair of nitrogen into nπ* excitations and also a molecular symmetry reduction. Both these reasons provide interesting features of azabenzene phosphorescence.24,182,183 It is a well-known rule56,57 that ISC coupling is strongest between states of different orbital configurations in terms of a strong SOC between them. In the case of azabenzenes, it relates to the strong coupling between the 1(nπ*) ↔ 3(ππ*) and 1(ππ*) ↔ 3 (nπ*) states.24 The strength of SOC for the 3(nπ*) states originates from the local character of the nonbonding n-orbital providing a 2 order of magnitude higher SOC for the 3(nπ*) states than for the 3(ππ*) states. This has been confirmed by QR theory calculations of azabenzenes: the 3(nπ*) states emit during ∼10 ms comparing with the more than 1 s for the 3 (ππ*) states.24,182 The strong SOC for the 3(nπ*) states is clearly manifested for the pyrazine and pyrimidine molecules for which the 3(nπ*) state is the phosphorescence responsible T1 state. Moreover, these species have been studied by phosphorescence microwave double resonance MIDP184,185 techniques resolving spin sublevel rates. In 1960, Cohen and Goodman 186 reported strong phosphorescence for the 3(nπ*) state of pyrazine and pyrimidine in the high temperature limit. The phosphorescence quantum yields were found to be equal 0.3 for pyrazine and 0.14 for pyrimidine that corresponds to the radiative phosphorescence lifetime of 0.02 and 0.01−0.02 s, respectively.

The pathways (1) + (2) provide the decay of the zcomponent (Ms = 0) of the 3B2u triplet state, while the schemes (3) + (4) represent the decay of the x-component (Ms = 1). This corresponds to polarization of phosphorescence along the x- and z-axis for τz and τx sublevels, respectively. The third ycomponent sublevel of the 3B2u state can be induced only by vibronic perturbations. The lowest 1B1u (path 3) and 3B3g (path 4) are the πσ* and σπ* states, respectively. The SOC matrix elements consist of the primary contributions from the onecenter and one-electron terms. The 1B3u state (path 1) is predominantly of ππ* type for which there are only one-center two-electron and multicenter contributions to SOC that are nonzero. The higher-lying 3B1g state can be of ππ* or σσ* symmetry. In the case of ππ*, only the multicenter terms contribute, while for the case of σσ* state symmetry, the 3 B1g-3B2u transition is dipole forbidden. The quantitative analysis of phosphorescence polarization for the linear polyacenes in the framework of the all-valence-electron CNDO/S CI approximation66 indicates that the main role is played by the out-of-plane component within pathways 3 and 4 in good agreement with experiment: more than 75% of the total naphthalene phosphorescence is out-of-plane polarized, while the in-plane long-axis component is only 18%.180 MCQR calculations with the usage of the full Breit-Pauli HSO operator

Figure 4. Scheme of spin-sublevel resolved phosphorescence of pyrazine (left) and pyrimidine (right). x,y ̅ ̅ denotes the emission polarization along the x and y axes, respectively; blue solid and wavy lines denote the dipole and spin−orbit coupling, respectively. 6517


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Figure 5. Structure and choice of axes for the free-base porphyrin molecule (H2P) together with the illustration of the internal metal ion (IMI) and external heavy atom (EHA) models.

Table 3. QR DFT Calculations of Phosphorescence Intensity at the Ground State Geometry of H2P (D2h), MgP, and ZnP (both in D4h)103 Molecules Optimized by the B3LYP/6-31G(d,p) Methoda E(S−T) (eV) 3

H2P ( B2u) MgP (3Eu) ZnP (3Eu) a

191

1.46 (1.56) 1.72 (1.70)191 1.81 (1.82)206,207

⟨Tx1|Mz|S0⟩ (a.u.)

⟨Tz1|Mx|S0⟩ (a.u.)

−5

−7

2.4 × 10 6.6 × 10−5 7.1 × 10−5

6 × 10 1.2 × 10−6 1.8 × 10−4

kx (s−1) −3

kz (s−1) −3 205

1.8 × 10 (2.0 × 10 ) 2.4 × 10−2 (2.1 × 10−2)181 3.1 × 10−2 (1.6 × 10−1)205

−7

6.0 × 10 1.8 × 10−5 2.0 × 10−1 (10 × 10−1)205

τav, s (4 K)

τav, s (77 K)

540 (500)205 126 4.9

1619 377 (47)191 12.8 ( 9. In the most long-wave absorption tail around 240 nm, a weak S-T absorption was predicted: the calculation with the QR method by the DALTON code302 predicts a relatively high intensity for the S0-T1 absorption ( f = 0.51 × 10−6). The T1 state of the glycine anion has A″ spatial symmetry (nπ* nature), while the second triplet state T2 is a totally symmetrical 3A′ state; the energy of the S0-T2 transition (about 5.6 eV) overlaps with an intensive band of spin-allowed absorption for the glycine anion. It should be noted that the intensity of this S0-T2 transition ( f = 0.44 × 10−4) is quite large as for the spin-forbidden process. Really, the intensity of singlet−triplet absorption for the aromatic hydrocarbons and related systems is 5 or even 6 orders of magnitude weaker.24 Similar values have been obtained for some other amino acids. Particularly, for the sulfur-containing methionine molecule the

has been successfully applied in OLEDs. Particularly, Adachi et al.278,280 and Wang et al.279 fabricated pure organic exciplexbased OLEDs with emission that is completely of the RISCinduced delayed fluorescence type (Figure 15b). The principal possibility of exciplex emission in OLEDs through TADF was additionally proved in ref 279 on the basis of magnetic experiments with m-MTDATA:Bphen exciplexbased OLEDs. These devices clearly demonstrate a positive magnetoelectroluminescence and magnetoconductivity (at 100 mT external magnetic field) that confirms the TADF nature of the exciplex electroluminescence via the RISC stage rather than the triplet−triplet annihilation process. 5.3. Photobiology Applications of the Triplet State

Participation of triplet excited states is at focus in many modern studies of biochemical processes like photodynamic therapy, vision, respiration, photochemistry of DNA components, peptides and retinals. The low-temperature phosphorescence spectra of DNA are shown to consist of two basic bands originating predominantly from the thymine and adenine bases.291,292 The prevalence of these two molecules and the absence of triplet guanine signals have not been explained until recent time.293 The involvement of triplet states in the fast energy deactivation and vibrational relaxation processes for the pyrimidine-type DNA nucleobase components have been elucidated quite precisely by Marian294,295 and Marquetand296 both theoretically and experimentally, but further future studies are certainly needed (in particular for the thymine nucleobase291). Terenin40 was the first to point out that the triplet states of biomolecular systems could serve as intermediates in various photochemical reactions because of their relatively long lifetime and typical biradical character. This notion fully applies to DNA nucleobases. For example, triplet excited states of DNA/RNA nucleobases take part in such important photochemical reactions as the synthesis of phototherapeutic nucleobasepharmacon adducts and photodimerization of pyrimidine-type bases. The latter process is the main reason for the genetic damage (mutations) caused by UV-light irradiation.293 The thymine and uracil nitrous bases of DNA exhibit formation of triplet states in light-induced dimerization of neighboring pyrimidine pairs being stacked in the DNA double chain.291 This dimerization is obtained with triplet sensitized population pumping. The triplet produced without sensitization (just after excitation of the isolated nucleobases in gas phase) was postulated to be inferred from the presence of a long-lived dark state.291 However, this assignment was not supported in view of the small changes produced by oxygen coexpansions. As is well-known, because of the triplet ground state of the O2 molecule, it usually selectively quenches excited triplet molecules.291 The appearance of triplet molecules in solution is generally deduced from transient absorption experiments in time scales of a few picoseconds. Several theoretical works support the importance of the triplet state in DNA components and their formation in stepwise photoprocesseses.297,298 Particularly, González-Luque et al.297 have presented a simple scheme for the explanation of deactivation channels for the S1 and T1 excited states of the nitrous bases from DNA and RNA: adenine (A) and guanine (G) of the purine type and uracil (U), thymine (T), and cytosine (C) of the pyrimidine type. They studied the S-T crossing and SOC effects in order to characterize the mechanisms for triplet state population of these species. At 6527


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levels and where the spin−orbit operator can be accounted for exactly or by various effective approximations. The quadraticresponse theory rests on an implicit sum-overstate principle, which allows one to describe correctly and largely clarify the phosphorescence mechanism for a number of difficult cases such as dioxygen (magnetic phosphorescence), benzene (vibronically induced phosphorescence), porphyrins (external and internal heavy atom effects), and cyclometalated iridium complexes (spin-sublevel selectivity and tuning of transition dipole moment orientation). The greatest challenge of computational phosphorescence of today is the purposeful design of novel materials for modern devices of organic electronics, especially for organic light emitting devices (OLEDs). The triplet states in OLED materials play a crucial role because upon electron−hole recombination 3/4 of the formed excitons reside in the triplet state. The involvement of triplet excitons for enhancing of photoluminescence is impossible without overcoming the spin prohibition for singlet−triplet transitions. Therefore, the search of novel effective schemes for triplet-to-singlet spin crossover is a very important topic for modern phosphorescence spectroscopy as well as for theoretical modeling. For today, the most hopeful pathway to enable the emission from triplets in OLEDs is the thermally activated delayed fluorescence (TADF) process acting through the reverse intersystem crossing (RISC) process. In this review, we have presented the recent achievements in this field and the theoretical background for some new schemes of TADF in binary systems like intermolecular triplet-singlet energy transfer and RISC between singlet- and triplet-state exciplexes. Finally, we also reviewed some photobiology applications of the triplet state. The field remains wide open, and there is much room for new theory and modeling with strong ramifications for the rational design of novel electronic and photonic devices. We hope that this review can contribute to the further progress in computational phosphorescence and raise the awareness of researchers on the principles of modern phosphorescence spectroscopy.

S-T transition intensity is predicted seven times higher than for glycine (f = 3.7 × 10−6). The QR DFT calculations provide an average radiative lifetime for the T1 state of methionine equal to 0.32 ms (at the high temperature limit) comparable with some phosphorescent azabenzenes (see section 4.5). There is no strict sublevel selectivity for the T-S emission of methionine due to the absence of symmetry restrictions for SOC; however, the partial transition rates relate as 1:8.4:21, which indicates an anisotropic nature for methionine phosphorescence. The origin of quite high S-T transition intensity of methionine can be attributed to the strong EHA from the sulfur moiety; however, the EHA effect enhances simultaneously the nonradiative T1∼S0 probability that should complicate the optical detection of T-S emission of amino acids. The measuring of S-T absorption should still be possible, in principle, at high spectral resolution at the long-wavelength absorption shoulder in order to separate the S-T absorption from ordinary singlet−singlet absorption. Concluding, one can say that the account of S-T transitions is very important for the explanation of the weak long-wavelength light absorption (usually in the near-UV) by numerous aliphatic amino acids. Certainly, similar effects could take place in the absorption spectra of small peptides and real proteins, but this task is still open for future consideration. Accounting for the fact that triplet excited states of aliphatic amino acids (like glycine) usually are dissociative, it can be expected that the S-T transitions provide an additional channel for the radiation damage of amino acids together with the generally accepted spin-allowed UV light absorption.7 Unfortunately, the singlet− triplet absorption of aliphatic amino acids has not been clearly observed at this time, but we believe that future investigations of the Zeeman effect in molecular crystals could help to discover their S-T transitions.

6. CONCLUSIONS In the present work, we have made an historical account of the achievements in the theory and computations of phosphorescence and related spin-forbidden phenomena. Our notion is that although a lot of experimental data have been collected over the years, theoretical investigations shedding light on the principles and mechanisms of phosphorescence remained quite limited until the last couple of decades. The triplet origin of phosphorescence remained unproven until 1943, and the debates around the triplet-singlet nature of phosphorescence did not cease until 1958 when Hutchinson and Magnum discovered EPR absorption of the triplet state in crystals and finally could prove the paramagnetic nature of this triplet state.303 However, the main contribution to the establishment of triplet state spectroscopy belongs, independently, to Terenin, Lewis, and Kasha in the fifties. Accounting for perturbation theory, phosphorescence gains intensity from spin-allowed S-S and T-T electronic transitions via the spin−orbit coupling mixing of the singlet excited states manifold (Sn) with the first triplet (T1) excited state and of the triplet excited states manifold (Tn) with the ground singlet (S0) electronic state. A proper account of spin−orbit coupling makes it possible to estimate the main parameters of phosphorescence, like the transition dipole moment, radiative lifetime, and rate constant for each of the spin sublevels of the triplet state. The best equipment for the theoretical study of phosphorescence of today is, in our opinion, the quadratic-response theory formalism implemented in the DALTON program package,302 where this method is accessible at the HF, MCSCF, or DFT

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Hans Ågren: 0000-0002-1763-9383 Notes

The authors declare no competing financial interest. Biographies Gleb Baryshnikov was born in 1989 in the town of Kramatorsk in eastern Ukraine. He graduated in the chemical department of Bohdan Khmelnytsky National University in 2012 and defended his Ph.D. in physical chemistry three years later at V. N. Karazin Kharkiv National University (Ukraine) under the supervision of prof. B. F. Minaev. He is now a postdoc in the group of prof. Hans Ågren at KTH−the Royal Institute of Technology (Stockholm). Gleb is an author of 80 publications in international journals and two monographs in organic electronics. His research interests are focused on theoretical studies of materials for dye-synthesized solar cells and organic light-emitting diodes. Boris Minaev was born in 1943 in the main city of the Urals mountains region−Sverdlovsk (Russia). He graduated in physical department of Tomsk State University in 1967 and defended a Ph.D. five years later 6528


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HE heavy element HF Hartree−Fock approximation H2P free-base porphyrin HOMO highest occupied molecular orbital IC internal conversion IHA internal heavy-atom ILCT intraligand charge transfer IMI internal metal ion IQE internal quantum efficiency IR infrared ISC intersystem crossing IUPAC International Union of Pure and Applied Chemistry KS Kohn−Sham formalism LLCT ligand-to-ligand charge transfer LUMO lowest unoccupied molecular orbital MCQR multiconfiguration quadratic response method MCSCF multiconfiguration self-consistent field approximation MgP magnesium-porphyrin MIDP microwave-induced delayed phosphorescence MLCT metal-to-ligand charge transfer m-MTDATA 4,4,4-tris(N-3-methylphenyl-N-phenylamino)triphenylamine MNDOC−CI modified neglect of diatomic overlap parametrized for configuration interaction mphmq 2-(3,5-dimethylphenyl)-4-methylquinoline anion mphq 2-(3,5-dimethylphenyl)quinoline anion ODMR optically detected magnetic resonance OLED organic light-emitting diode PES potential energy surface PhOLED phosphorescent organic light-emitting diode pic picolinate anion piq phenylisoquinoline anion PL photoluminescence PPT 2,8-bis(diphenylphosphoryl)dibenzo-[b,d]thiophene ppy 2-phenylpyridinate anion PQY phosphorescence quantum yield QR quadratic response methodology QY quantum yield RISC reverse intersystem crossing RPA random phase approximation RT room temperature conditions S singlet state SCF self-consistent field SCSO scaled-charge spin−orbit approach SLR spin−lattice relaxation SOC spin−orbit coupling SOC−CI spin−orbit coupling configuration interaction method T triplet state TA thermally activated TADF thermally activated delayed fluorescence t-Bu-PBD 2-(biphenyl-4-yl)-5-(4-tert-butylphenyl)-1,3,4-oxadiazole TCTA 4,4,4-tris(carbazol-9-yl)-triphenylamine TD DFT time-dependent density functional theory tmd 2,2,6,6-tetramethylheptane-3,5-dione anion TPBi 1,3,5-tris(N-phenyl-benzimidazol-2-yl) benzene TSET triplet-singlet energy transfer WOLED white organic light-emitting diode ZFS zero field splitting ZnP zinc-porphyrin

in optics and spectroscopy under supervision of prof. N.A. Prilezhaeva (former assistant of N.A. Terenin). He subsequently received a position at Karaganda State University and there created the first USSR quantum chemistry chair. In 1984, he defended his Dr.Sc. work (habilitation) at Moscow Institute of Chemical Physics. In 1989, he moved to Cherkassy (Ukraine). Now he is the head of the chemistry and nanomaterial science department at Bohdan Khmelnitsky National University. He has developed the concept of spin catalysis and explored photo- and bioactivation of dioxygen and the mechanisms of magnetic-field effects in luminescence quenching. Hans Ågren was born 1950 in the town of Skellefteå in northern Sweden. He graduated in 1979 with a PhD in experimental atomic and molecular physics at the University of Uppsala under the supervision of Kai Siegbahn. After a couple of postdoc years in the USA he became assistant professor in Quantum Chemistry at Lund University in 1981 and associate professor in the same subject at Uppsala University in 1983. He became the first holder of the chairs in Computational Physics at Linköping University in 1991 and in Theoretical Chemistry at the Royal Institute of Technology, Stockholm, in 1998. He is active at the Department of Theoretical Chemistry and Biology at KTH and, since 2017, at the Siberian Federal University, Krasnoyarsk, Russia, doing theoretical modelling research in molecular/nano/biophotonics and electronics and X-ray science.

ACKNOWLEDGMENTS This research was supported by the Ministry of Education and Science of Ukraine (project number 0115U000637). B.F.M. acknowledges the grant of the Chinese Academy of Science in the framework of the President’s International Fellowship Initiative for Visiting Scientists in 2015. Hans Ågren acknowledges the support from the Swedish Science Research Council through contract 201211191404-1. ACRONYMS 3TPYMB tris-[3-(3-pyridyl)mesityl]borane Acac acetylacetonate AMF atomic mean field approximation AO atomic orbital B3LYP Three-parameter Becke−Lee−Yang−Parr exchange−correlation functional BIPC3 3,6-bis(5-methoxyindol-1-yl)-9-(4-methoxyphenyl)carbazole Bo benzoxazole anion Bt benzothiazole anion CASSCF complete active space self-consistent field method CIS conical intersection CNDO/S complete neglect of differential overlap for the spectroscopy semiempirical approach CT charge transfer DFT density functional theory DHOM displaced harmonic oscillator model DMAC27 2,7-di[di(4-methylphenyl)amino]-9-ethylcarbazole DMAC36 3,6-di[di(4-methylphenyl)amino]-9-ethylcarbazole ECP effective core potential ECSO effective-core spin−orbit operator EHA external heavy atom EL electroluminescence EPR electronic paramagnetic resonance EQEs external quantum efficiency GGA generalized gradient approximation 6529


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