Dioxygen spectra and bioactivation

Abstract

Intensities of spin-forbidden transitions in electronic absorption and emission spectra of molecular oxygen are analyzed in order to understand the key mechanisms of spin-states mixing induced by spin-orbit coupling (SOC) and the ways to overcome spin prohibition for various photophysical and biochemical processes. Multireference configuration interaction calculations with SOC account are used to generalize spin-selection rules for the oxygen atmospheric and Herzberg bands in free O₂ molecule and in collision complexes. Intensity enhancement of the atmospheric a¹Δ_g → X³Σ, b¹Σ → X³Σ, and Noxon b¹Σ → a¹Δ_g transitions upon bimolecular collisions are compared with those for Herzberg III transitions A^′3Δ_u ← X³Σ. Electric quadrupole, dipole, and magnetic approximations are used for transition probability calculations. Intensity distribution in rotational lines is also considered. With this background, we propose some simple spin-selection rules for dioxygen activation in enzymatic reactions. © 2013 Wiley Periodicals, Inc.

Introduction

The spectra of the O₂ molecule (dioxygen) are a subject of permanent interest because of the great importance of molecular oxygen for the life at our planet.1–6 The excited states of dioxygen, besides of their obvious impact on chemistry of atmosphere, are involved in oxidative organic photochemistry and a number of photobiological oxygen effects including photosynthesis. Knowledge of dioxygen excited states is important even for dark enzymatic reactions.6–10 This is getting obvious if one considers the reaction rate in a framework of the full perturbation theory treatment with account of spin-orbit coupling (SOC).6, 11, 12 The SOC-induced mixing between states of different multiplicity is the key subject not only for dioxygen spectroscopy and photochemistry but also for oxidation reactions catalyzed by glucose oxidase (GO),7, 9 cytochrome P450,13, 14 and other enzymes.12

It is well established long time ago that the dioxygen molecule has a triplet ground state, X³Σ, and two singlet excited states, a¹Δ_g and b¹Σ, with low excitation energy, 0.98 and 1.63 eV, respectively, above the ground state, which are observed in the visible-near infrared (IR) regions15 (Fig. 1). To denote these states in the following, we shall use abbreviations X, a, and b, respectively. Transitions between the states are strictly forbidden by spin and spacial symmetry selection rules and can be observed in the optical absorption spectrum of the Sun light because of a vast amount of dioxygen and a long light pathway through the Earth atmosphere.

Figure 1. Energy diagram for atmospheric oxygen transitions in the nonrotating molecule.

The a – X and b – X transitions are induced by SOC perturbation; they occur in the near IR (1.27 μm) and in the red (762 nm) part of the spectrum, respectively.15 The b – X transition is much more intense; this red atmospheric band is among the bright features in the night-glow and among the Fraunhofer lines in the Sun light absorption spectrum.16 The reason for large intensity difference between the a – X and b – X transitions is very instructive for understanding of dioxygen chemical and biochemical reactivity.

Spectra and photochemistry of O₂ in the ultraviolet (UV) region below the first dissociation limit O(³P) + O(³P) at 242 nm are also of great interest.4, 17 These spectra are determined by π − π^* transitions and connected with the c¹Σ, A^′3Δ_u, and A³Σ states.16 These are weakly bound states lying close below the first dissociation limit at 242 nm; transitions to them from the ground state are forbidden in electric-dipole nonrelativistic approximation and occur as weak Herzberg bands. The Herzberg transitions are also observed in night-glow emission and provide important information on the O(³P) + O(³P) atomic recombination processes in the upper atmosphere.18 Thus, dioxygen spectra in gas phase demonstrate a number of forbidden transitions which are very weak at zero pressure19–23; in condensed phases, some of these transitions are strongly enhanced indicating that symmetry of electron distribution in the O₂ molecule is obviously perturbed by weak intermolecular interactions.24–27 Theoretical studies of such perturbations are also important for dioxygen chemistry analysis, especially for O₂ activation by enzymes.9, 11

Spin-prohibition: implications in dioxygen chemistry

Because of the paramagnetic nature of the triplet ground state of dioxygen, its chemical reactions with diamagnetic organic molecules are spin-forbidden and O₂ shows sluggish chemical reactivity at the ambient conditions (in the absence of radicals and paramagnetic metals). The majority of organic materials consists of the singlet ground-state molecules; their oxidation by O₂ in living organisms needs very active catalysts (enzymes) which can help to overcome the spin-prohibition of organic oxidation in order to gain energy for metabolism. The nature makes use of transition metal organic complexes, such as hemoglobin, chlorophyll, or cytochromes, to bind, transport, and activate O₂; at many stages of these bio-reactions, the various spin-change transformations are already established by electron paramagnetic resonance (EPR), ODMR, and Mössbauer spectroscopy.11, 14, 28, 29 Study of spin-selection rules in the dioxygen spectrum and the overcoming of spin-prohibition upon solvent effects is very instructive for the analysis of chemical reaction mechanisms of the O₂ molecule in oxidation processes including respiration and photobiology.9–11, 24

Almost all oxidation reactions are very exothermic, and it is quite strange from a thermodynamics point of view that our world has not been burnt so far. Occurrence of dioxygen in the Earth's atmosphere with the first photosynthesis from green algae about 1.4 billion years ago has provided the so-called dioxygen revolution. The advantages of aerobic life and oxidative metabolism were important factors during evolution: these are mainly connected with the high exothermicity of oxidation of organic molecules by dioxygen. At the same time, life strongly depends on the kinetic barriers to the molecular oxygen reactions. O₂ is used as an oxidant in respiration of mammals and oxidative metabolic processes, reducing food to water and carbon dioxide. Although these transformations are equivalent to combustion, the oxygenase enzymes control the specific reaction pathways that store and smoothly release energy by subtle spin selective processes, whereas combustion is a radical chain process. It requires a high temperature initiation step and develops like explosion without strong regulation of energy release. The kinetic constraints, which allow the controlled use of dioxygen by aerobic life in the presence of strong thermodynamic drive, are determined by spin selective interactions between dioxygen and oxygenase enzymes. The first step in understanding of these constraints should be connected with analysis of photophysical processes induced by O₂ interaction with solvents and gases, which provide interesting manifestations in spectroscopy.

In a practical sense, the molecular oxygen is almost transparent to the visible and near UV radiation (up to 200 nm). Nonetheless, dioxygen plays a key role in many important processes induced by solar radiation in the transparent region of the atmosphere, including aurora and night-glow, formation of the ozone layer, dioxygen assisted photobiology, photodynamic O₂ effects, and oxidative stress in living organisms exposed to sun light.11, 24, 27 This importance is connected with a great abundance of O₂ in the atmosphere and pressure dependence of the above-mentioned dioxygen forbidden transitions.

In this review, we will concentrate our attention on spectroscopy observations, which explain how the spin-prohibition can be overcome during O₂ interaction with light. This provides numerous instructions about internal magnetic perturbations involvement into small deformation of electronic wave functions of free dioxygen molecule, spin state mixing, and violation of strict spin selection rules in optical and EPR spectroscopy.

We then consider the influence of weak intermolecular interaction on additional relaxation of the symmetry selection rules and enhancement of spin-forbidden processes in dense gases, liquid, and solid solvents and in biomembranes. Taking into account these lessons from photophysics, photochemistry, and spectroscopy of dioxygen in condensed phase, we will use the new knowledge on internal magnetic perturbations for analysis of enzymatic reactions, where spin-selection rules are obviously and very often violated during dioxygen bioactivation.

Preparatory information on dioxygen electronic structure and spectroscopy

The triplet ground state, X³Σ, and two low-lying singlet excited states, a¹Δ_g and b¹Σ, are connected with the main electronic configuration of the O₂ molecule

• (1)

which is denoted shortly as “…π π” in the following. A small admixture of the doubly excited configuration … π π is also important even at semi-quantitative level for all states under discussion. Transitions between these states (X – a and X – b) provide very weak IR and red atmospheric bands at 1270 and 760 nm, respectively.15 These transitions are forbidden in the electric-dipole approximation and occur as very weak bands of magnetic15, 30 and quadrupole31, 32 nature induced by SOC.33, 34 The Noxon band b¹Σ − a¹Δ_g in the near IR region (1,9 μ)35 is completely allowed (to the highest possible degree) as quadrupole radiation,31 but still is not intense. Quadrupole contribution to the a – X transition was predicted to be very weak31 and has recently been detected by high-sensitivity cavity ring-down spectroscopy.32

Despite its biradical character, the O₂ molecule is pretty stable; it needs 5.21 eV excitation energy to reach the first dissociation limit O(³P) + O(³P).16 The cytochrome-c-oxidase can split the OO bond in mitochondrial membranes with low activation barrier by special spin-uncoupling through the Fe and Cu ions involvement in dioxygen activation.6 This spin-catalysis can be understood with account of energy expenses for all possible spin and orbital angular momenta transformations induced by exchange interactions and by internal magnetic perturbations. Analysis of the higher excited states of dioxygen is of great importance for the understanding of such spin-catalysis.8, 17, 36 In fact, a single electron excitation to the …π π configuration provides a series of states with various spin and orbital angular momenta projections: c¹Σ (4.1), A′ ³Δ_u (4.31), A³Σ (4.39), B³Σ (6.17), ¹Δ_u (8.4), and f¹Σ (10.1); in parentheses, the 0–0 transition energy (in eV) from the ground state is given.16 Thus, these transitions with single electron excitations cover a wide spectral region from near UV till vacuum UV wavelengths.

In 1903, Schumann first observed a series of discrete absorption bands below 200 nm and Runge in 1921 studied first the B → X emission in a high-voltage arc in dioxygen.16 Nowadays, there is a huge amount of research of this electric-dipole allowed transition including high-resolution studies in which the zero-field splitting of both triplet states is resolved; for a review, one can see Refs.37–40. In the upper atmosphere, a strong absorption of the UV sunlight in the Schumann-Runge (SR) band system (200–130 nm) B³Σ ← X³Σ leads to O₂ dissociation which prevents our Earth from such deadly radiation. The B³Σ state dissociates to the second dissociation limit O(³P) + O(¹D) providing a relatively large concentration of the metastable oxygen atoms in such rarefied air. The night-glow from O₂ is confined to the region near the altitude of 95 km, where the maximum oxygen O(¹D) atom density is observed.4 The energy transfer reaction between O(¹D) and O₂ generates the v = 0, 1 levels of the O₂(b¹Σ) state with high efficiency,41 which provides the night-glow b¹Σ → X³Σ transition.

In the lower atmosphere, the main fraction of the ground state oxygen atoms O(³P) arises from dissociation of dioxygen following absorption of the soft UV radiation in the transparent region 200–242 nm known as the Herzberg continuum.17, 27 Dissociation to the first limit O(³P) + O(³P) through an absorption to this continuum in the stratosphere is an important step in the ozone production. The most intense transition A³Σ ← X³Σ in this region was discovered by Herzberg in 1932,42 being the first of three forbidden absorption band systems of the O₂ molecule studied by him, which led to its designation as “Herzberg I” band.16 It is forbidden by the “+” – “−” selection rule and this is an example of the triplet–triplet transition induced by magnetic perturbation of the SOC nature.19, 43–45

The next weaker band system c¹Σ ← X³Σ has been discovered in 195346 and called “Herzberg II.” The set of the excited state vibrational levels that one would now label as v′ = 6–114 has been observed in this first absorption measurement by Herzberg, who used a high pressure and long absorption path up to 800 m.46 Later on, this transition has been detected in the night-glow of Venus by Soviet space-explorer “Venera-10” as a strong emission system including v′ = 0 to v″ = 5–13 sub-bands.19, 47, 48

The A′³Δ_u ← X³Σ transition called “Herzberg III” was also discovered in the famous absorption experiments of 1953.46 The A′³Δ_u state has both spin and orbital angular momenta and indicates a strong multiplet splitting by SOC. In the first observation of the “Herzberg III” band system,46 the term sequence for different Ω quantum numbers was proposed to be regular. Here Ω = Λ + Σ is the total electronic angular momentum projection on the interatomic axis (z); Λ and Σ are an orbital momentum and spin momentum projections on z-axis. Since both the π_u and π_g shells are more than half-occupied in the orbital configuration (π_u)³(π_g)³ of the A′³Δ_u state, the SOC-induced splitting should be inverted; this has been pointed out by Krupenie16 for the first time in his analysis of the oxygen spectra. The first emission analysis41 of this system supported the inverted structure of the A′³Δ_u multiplet. This means that the Ω = 3 state sub-level is the lowest one. This result has been confirmed by semiempirical43, 49 and ab initio2, 36, 50 calculations. Recently, a new O₂ emission band has been observed in the night-sky glow in the blue visible region (430 nm).4 This very weak band was identified as the forbidden c¹Σ → b¹Σ transition, induced by electronic-rotational coupling.4, 51

Nowadays, many rovibronic bands in the Herzberg systems are well studied in gas phase,1, 4, 52–54 but information about condensed phase spectra, especially on the “Herzberg III” band system17, 26, 27 is rather confusing.25, 55 Even the regular spin-orbit multiplet splitting for the A′³Δ_u state has been inferred from O₂ spectra in solid deuterium25, 55 in contrast with the accepted scheme in the gas phase,2, 36, 41 which seems to be rather strange. In the present paper, we will model the influence of intermolecular interaction on the splitting of the A′³Δ_u multiplet and enhancement of all Herzberg bands and oxygen atmospheric bands in water solvent. Photoinduced oxidation of hydrocarbons in supercritical CO₂ indicates that the formation of primary active oxygen species from O₂ proceeds at a wavelength longer than 242 nm.26 The explanation of this result is one aim of our study.

To complete a short review of O₂ spectra, one has to mention the far UV region. Besides the SR transition, the latter two states from the π π family, ¹Δ_u and f¹Σ, are accessible by allowed transitions from the first and second excited singlet states, respectively.39, 56–58 The vertical transitions, which reside in the vacuum UV region, lead to photodissociation. The only allowed transitions from the ground state, B³Σ ← X³Σ, provides the Schumann-Runge system: it starts at 200–175 nm as a discrete band system and at shorter wavelengths converges to the continuum. The allowed singlet-singlet transitions a¹Δ_g − ¹Δ_u and b¹Σ − f¹Σ are less intense.57, 59 Since they provide the largest contribution to the parallel polarizability component (α_zz) of the low-lying states, the α_zz constant (15.5 a) of the ground triplet X state is higher than the values α_zz of the low-lying singlet a and b states (14.1 and 13.6 a, respectively).6 This is important for the solvatochromic effect on these singlet-triplet a,b − X transitions60, 61; but in solvents, the polarizability of all these states is changed in a very peculiar manner,6 which will be discussed later on. We start with a preliminary analysis of the intensity of atmospheric bands in the free O₂ molecule and then consider their dependence on bimolecular collision in order to understand the general ideas behind the solvent effect on the background of molecular dynamics simulations.

Analysis of Intensity of the Dioxygen Weak Bands in Gas Phase

Despite the low transition probability, all weak bands of the O₂ spectra below the first dissociation limit (above 242 nm) can be observed in absorption owing to the vast amount of molecular oxygen in the terrestrial atmosphere and because of the long-path of the sunlight through this practically transparent optical window.20, 22, 23, 62 Some of these bands have been seen in night air-glow, day-glow, and Aurora spectra.4

Recent studies of the terrestrial night-glow have been carried out by an array of major new telescopes around the world.4, 20 They study a sky spectrum as a correction for astronomers who carry out analysis of stellar spectra. Such telescopes with large echelle spectrometers and CCD detection generate high-resolution data with broad spectral coverage and high sensitivity.4 Much of the terrestrial night-glow originates from O₂, and in most cases, the excited O₂ is pumped by three-body recombination of oxygen atoms.4 All these phenomena are now well studied experimentally with modern techniques,4, 40 but quantum theory treatment in their analysis is very often still based on rather old approaches.16, 42, 63, 64 Besides its great astronomical importance,20 the night-glow spectrum is now used for monitoring the photochemistry of dioxygen and the whole atmospheric energy budgets.40, 65 Weak spectral lines of O₂ are also present in day-glow emission from the stratosphere (b – X and a – X series) originating in the light absorption in the Hartley band of ozone followed by its photodissociation.40, 65 The day-glow spectrum is used for remote measurements of ozone concentration.4, 40 Dioxygen spectra are also used for geophysical control of pressure and temperature. Thus, the O₂ spectroscopy must not only be good enough for reduction of possible errors in remote sensing40; it must be clear in all details being based on consistent theoretical backgrounds. This is the reason for renovated interest to the theory of the dioxygen spectroscopy with high resolution and better detector sensitivity.4, 32, 40, 66–68

Analysis of the atmospheric bands in gas phase

Both singlet-triplet transitions, a – X and b – X, are triply forbidden by symmetry selection rules for electric-dipole radiation. They are doubly forbidden by inversion symmetry rule (gerade states) and by the spin selection. Behind this, the a¹Δ_g − X³Σ transition is additionally forbidden by the double change of an orbital angular-momentum projection (ΔΛ = 2) and the b¹Σ − X³Σ transition—by the Σ⁺ − Σ⁻ selection rule. All these selection rules are usually very rigorous and strictly forbid the electric-dipole radiation.69

Both these bands, a¹Δ_g − X³Σ and b¹Σ − X³Σ, are allowed in magnetic-dipole approximation with selection rule (ΔJ = 0, ±1) and in quadrupole (ΔJ = 0, ±1, ±2) approximation, where , and is a rotational moment and is a total spin.15, 30, 70, 71 With quantum-chemical calculations,31, 34 the quadrupole contributions have been shown to be practically negligible, and the main intensity of both bands has been explained as magnetic-dipole radiation31, 72, 73 in agreement with rotational analysis of the most intense observed spectral branches.30, 69–71 Nevertheless, extremely weak quadrupole contributions to particular rotational lines have been identified recently in both atmospheric bands32, 74, 75 indicating reliable predictions and success of the old theory.31, 34, 43, 76

For example, the 0–0 band of the singlet-triplet a¹Δ_g − X³Σ transition has been observed at 1270 nm as a very weak emission band in the night-glow.77, 78 O₂ is a dominant species in the night-glow emission from the upper atmosphere.40 This emission is known as the near IR atmospheric oxygen band.22 The night-glow originates by recombination of the O(³P) atoms created by daytime solar absorption and the O₂ cleavage; it covers the UV-visible part (from 240 nm) and extends to IR. Thus, all bound states studied in this paper are presented in night-glow emission.

The Einstein A-coefficient for the a → X spontaneous IR emission is extremely low: laboratory measurements extrapolated to zero pressure gave values from 2.58·10⁻⁴ s⁻¹ (Badger et al.22) to 1.47·10⁻⁴ s⁻¹ (Hsu et al.77). An old atmospheric absorption measurements (Ref.79) gave a value being close to the Hsu's result; the most recent cavity ringdown spectroscopy studies with a long-path absorption cell provide the Einstein coefficient close to Badger's result: 2.24·10⁻⁴ s⁻¹80 and 2.19·10⁻⁴ s⁻¹.81 Despite the difference of results, we can say that a → X transition is the most forbidden one in molecular spectroscopy.5 This transition rate constant is determined mostly by magnetic dipole radiation, which has been estimated for the first time in Refs.33, 82 as being induced by SOC mixing between a, X, and ^1,3Π_g states and by intensity borrowing from magnetic dipole transitions X − ³Π_g and b − ¹Π_g. Later, this calculation33 has been supported on ab initio level by Klotz et al.34 Intensity of vibronic a – X (0,1) band and (0,0)/(1,0) ratio is another important issue connected with a strong internuclear distance dependence of the magnetic transition moment.49, 62, 83

The quadrupole contribution to the IR atmospheric oxygen band radiative probability has been estimated for the first time by Sveshnikova and Minaev.31 One has to remind here about the IR Noxon band at 1.91 μ discovered 50 years ago,35 which is extremely weak and is determined by the b¹Σ → a¹Δ_g transition. The Einstein coefficient for spontaneous b → a emission (or radiative rate constant k_{b − a}) was estimated by Noxon to be equal to 0.0014 s⁻¹.35 The b¹Σ − a¹Δ_g transition is pure electric-quadrupole by nature; analysis of the a and b wavefunctions indicates that the Noxon band has a maximum possible quadrupole transition moment, which could exist in the O₂ system.31 Sveshnikova and Minaev31 have shown that the a – X band borrows quadrupole intensity from the Noxon band and found the quadrupole contribution to the IR atmospheric band radiative probability to be practically negligible (k_{a − X} = 2.3 ·10⁻⁶ s⁻¹).31, 43 In fact the recent measurements32 have detected this transition and supported an order of magnitude estimation.31, 43

The red atmospheric oxygen band (b¹Σ − X³Σ transition) is much more intense than the a – X system.4 The 0–0 band at 762 nm has the Einstein coefficient for spontaneous emission equal to 0.087 s⁻¹.23 This red atmospheric oxygen A band is among the brightest features in the night-glow and among Fraunhofer lines in the optical absorption spectrum of the Sun light. It was shown33 that the b¹Σ − X³Σ transition to the Ω = 1 component of the ground state is the most intense and borrows intensity from the “spin current” (spin-flip transition between spin sublevels Ω = 1 → Ω = 0 of the ground triplet state).33, 73 Just this peculiar mechanism is responsible for the observed radiative lifetime of the b¹Σ state (τ = 12 s) and intensity distribution in rotational structure of the A band.2, 30, 54 Radiative rate of vibronic satellites in the red atmospheric system has also found a perfect explanation on the ground of this mechanism.49, 84 Recently, both (IR and red) atmospheric oxygen band systems have been detected with high resolution from spectra of the night sky obtained from the echelle spectrographs at large Keck/HIRES telescopes.4, 20 Vibronic satellites of these bands are used to determine temperature, pressure, and density profiles in the atmosphere.4 These studies increase the need of theoretical analysis for better understanding of physical nature of atmospheric oxygen bands and their dependence on pressure and collisions. So far, a theoretical analysis of collision effects on Herzberg band was rather limited.26, 52, 85, 86 We shall start with zero pressure oxygen spectra and single O₂ molecule internal magnetic perturbations.

The singlet-triplet a¹Δ_g − X³Σ and b¹Σ − X³Σ transitions in a single nonrotating O₂ molecule

The main mechanisms of radiative activity in the atmospheric oxygen bands have been described elsewhere.33, 34, 62, 84, 87 Here we recapitulate shortly the essential details of these mechanisms, starting with the a – X band. With account of SOC, both transitions, a¹Δ_g,2 − X³Σ and b¹Σ − X³Σ (transitions 2 and 1 in Fig. 1, respectively), can borrow intensity from magnetic-dipole transitions of the type ³Π_g,0 − X³Σ. Here the low subscript corresponds to Ω (the total electronic angular momentum projection on the interatomic axis). There are also a¹Δ_g,2 − n¹Π_g,1 magnetic-dipole transitions which contribute additionally to the a − X₁ band intensity and b¹Σ − n¹Π_g,1 transitions which contribute destructively to the a − X₁ band intensity.33 All these contributions are induced by SOC and by orbital angular momentum projection perpendicular to the molecular axis, which agree with analysis of intensity distribution in rotational structure.30, 32, 33, 75, 81, 87 We want to remind that the a¹Δ_g,2 − X³Σ component of the a – X transition (2 in Fig. 1) is magnetic and the a¹Δ_g,2 − X³Σ component (transition 4 in Fig. 1) is pure quadrupole by nature.31, 33

We shall denote nonrelativistic wave functions (without SOC account) in terms of multiplicity (μ = 2S+1), spin projection Σ, orbital and total angular momentum projections Λ and Ω, respectively, by ^μΨ_n[Σ Λ Ω]. The main contribution to the triplet ground state wave function of the O₂ molecule can be presented by the scheme:

• (2)

where indicates the proper antisymmetrization product; π_{−, g} and π_{+, g} indicate two degenerate π_g molecular orbital (MO) with angular momentum projection λ = ±1 in units :

• (3)

Here r, θ, ϕ are coordinates of electron in the cylindrical system. An open shell of the main configuration, Eq. (2), can be presented by the following part of triplet ground state wave function:

• (4)

For the Ω=1 components, the spin parts are equal to α (1) α (2) and β (1) β (2) functions. All spin parts are symmetrical with respect to permutation and the spatial part is antisymmetrical as the Pauli principle requires. The latter property provides the “minus” sign of the Σ⁻ term.

The closed-shell part of all X³Σ, a¹Δ_g, and b¹Σ states is almost the same in a reasonable approximation. Two degenerate states a¹Δ_g have a simple open-shell part of the wave function:

• (5)

and

• (6)

The singlet state b has a similar configuration as a scheme given by Eq. (4). The main contribution to the open shell of the singlet b state wave function can be written in a form:

• (7)

The simple open-shell wave functions, Eqs. (4)–(7), illustrate an important symmetry property, which is essential for SOC analysis and spin selection rules. Reflection in any plane which contains internuclear axis corresponds to interchange (ϕ) → (−ϕ). For molecular orbitals in Eq. (3), this corresponds to transformation π_{+, g} → π_{−, g}. Thus, the spatial part of the triplet state wave function, Eq. (4), changes the sign, and the spatial part of the singlet state wave function, Eq. (7), does not change the sign upon reflection in molecular plane. These properties of the Σ and Σ spatial wave functions are very important for SOC analysis and explanation of optical spectra of oxygen molecule in the whole visible and near-IR regions.43, 88 Not only the b – X and a – X transitions intensity, but also the b¹Σ − B³Σ transition probability88, 89 and some other optical phenomena in diluted gases and solvents containing molecular oxygen43, 61, 90, 91 depend on the SOC-induced mixing between b¹Σ and X³Σ states. Because of the symmetry of the SOC operator,43 the spatial wave functions of the triplet and singlet ∑ and ∑ states are connected by the z-component of H_so

• (8)

where and the orbital part can be presented in a simple approximation43, 92

• (9)

Orbital part of the B_zS_z operator provides spatial rotation around z-axis; thus, it changes sign in respect to reflection in molecular plane and can mix the spatial parts of the wave functions of the Σ and Σ states. The spin operator S_z provides mixing of the spin functions in Eqs. (7) and (4). If one denotes the SOC-perturbed wave functions in the first order of perturbation theory by ϕ_n, where n is a spectroscopic term notation, then it follows:

• (10)

• (11)

• (12)

where, for example:

• (13)

In an effective one-electron approximation, Eqs. (8) and (9), the SOC matrix element in the numerator of Eq. (13) is equal to

• (14)

where ζ^O = 153 cm⁻¹, as it is derived from the multiplet splitting for the O(³P) atom.92 Ab initio calculations34, 50, 84 have given a larger value of 176 cm⁻¹. If we put this value into the nominator of Eq. (13) and submit the experimental energy gap (E_b − E_X = 13195 cm⁻¹16) into the denominator, the admixture coefficient will equal to C_b,X = 0.0134. This small coefficient is very important for dioxygen spectroscopy and photochemistry. It plays a crucial role in explanation of many optical phenomena, reactivity, magnetic, and radiative properties of dioxygen at zero pressure in the upper atmosphere,33 in real mixture of gases,43, 76 and even in solvents and solids.31, 49, 93–95 The singlet oxygen emission from the b¹Σ state and a great enhancement of the O₂(a¹Δ_g) oxygen emission in solvents are entirely determined by the coefficient of Eq. (13).33, 43, 61, 93

First, we explain the pure magnetic transitions from the b¹Σ and a¹Δ_g states to the ground X³Σ sublevels in a free (nonrotating) O₂ molecule.33

These spin sublevels of the ground triplet state (Ω = 1 and Ω = 0) are split in the absence of external magnetic field by the value D = 3.96 cm⁻¹.16 This zero-field splitting (ZFS) is determined by an expectation value of spin-spin coupling (D_ss = 1.44 cm⁻¹) and by the second-order contribution of perturbation theory with SOC account (D_so = 2.32 cm⁻¹)96; thus, the calculated value (D = 3.76 cm⁻¹) is in a reasonable agreement with the EPR and optical measurements.16, 96 In Figure 1, this ZFS is exaggerated (the doubly degenerate levels with Ω = 1 and Ω = 0 are presented in the split form for convenience). The five optical transitions, shown in Figure 1, have to be considered in a nonrotating oxygen molecule.43, 88 Transitions 1 − 4 are doubly degenerate; only transition 5 is a single one.

An additional EPR transition 6 between the ZFS spin sublevels, which occurs in the microwave region, is also shown in Figure 1. It is observed in the solid oxygen97 and is important for analysis of the optical absorption and emission of dioxygen.33, 88

The Noxon transition 3 is a pure quadrupole by nature and transition 6 (being an ordinary EPR transition) is a pure magnetic dipole in nature induced by the spin operator in Eq. (15) with perpendicular projection on the molecular axis. The magnetic dipole operator, μ, as given in Eq. (15),

• (15)

includes L and S—the orbital and spin angular momenta, respectively; here μ_B is the Bohr magneton and g_e = 2.0023 is a g-factor of free electron. Transitions 2 and 1 in Figure 1 are pure magnetic and have been considered as the following.33, 88

The magnetic dipole transition moment between b¹Σ and X³Σ states (transition 1, Fig. 1)

• (16)

includes both spin and orbital angular momenta contributions. Now we consider first the spin contribution, S, from Eq. (15). In this case, we can take into account that the ϕ_X,1 substates, Eq. (12), is almost pure, since the C_X,n admixtures are not important here. They are small and do not influence much the normalization constant; thus, we take for simplicity an approximation ϕ_X,1 ∼ Ψ(X³Σ). Perturbations of the Ω = 1 spin sublevels are important for the orbital, but not for the spin part of the transition moment Eq. (16) calculated with the total magnetic dipole operator (15). In a similar manner, we can neglect here the Π_g,0 substates admixtures in the total b state expansion, Eq. (10), since they do not contribute to the spin part of the magnetic transition moment, Eq. (16).

Thus, we can neglect for simplicity the C_b,n and C_X,n coefficients in Eqs. (10) and (12), respectively. Only the C_b,X coefficient in Eq. (10) produces contribution to the spin part of magnetic dipole transition moment76:

• (17)

Here S_∓ is the lower and increase operator. In fact we have found that the optical transition 1 (Fig. 1) of the atmospheric oxygen band in the red region of visible light can borrow intensity from the EPR transitions 6 (Fig. 1) in the terahertz microwave region. This is the first example in molecular spectroscopy, when the spin-flip magnetic dipole transition between spin sublevels of the ground triplet state M_s = 0 and M_s = 1 (spin current) “produces” the visible light. In the following, we can see that this is the largest contribution to the atmospheric red band intensity.73, 76

Of course, there is also an orbital angular momentum L contribution of the magnetic dipole operator (15) to the b¹Σ − X³Σ transition moment (transition 6 in Fig. 1)33, 88:

• (18)

The first semiempirical estimation33 based on INDO/S-type theory33, 73 has taken into account the lowest singlet and triplet 1^3,1Π_g states, produced by 3σ_g → π_g excitation. In order to see an essential feature of the intensity-borrowing mechanism is worse to consider the orbital contribution to magnetic transition moment Eq. (18) in terms of the old theory.33, 73 In fact the most important contributions to the sum of Eq. (18) are proved by the later ab initio34, 84, 88 to be produced with the lowest 1^3,1Π_g states. It is shown49, 50 that in the vicinity of the equilibrium internuclear distance, 1.15–1.35 Å, the 1^3,1Π_g states are really connected with the single electron excitations 3σ_g → π_g which provide the main contribution to the orbital magnetic transition moment Eq. (18). Thus, they can be approximated by a simple expression43:

• (19)

Here C_g,z = 0.653 is the LCAO-coefficient in the 3σ_g molecular orbital expansion for the 2p_z-AO in the INDO/S approximation. It is an essential feature of Eq. (19) that both singlet and triplet contributions come with opposite signs and indicate a trend to cancel each other. The total b − X,1 transition magnetic dipole moment μ_{b − X,1} = μ + μ is almost equal to the spin current, Eq. (17), which provides a radiative lifetime (τ = 12 s)76 in a good agreement with observations.23, 81 The magnetic b − X,1 transition 1 (Fig. 1) determines practically all intensity of the atmospheric oxygen red band at 762 nm. Since the first observation in a solar atmospheric spectrum, recorded with the air absorption path up to 150 km42 and its first analysis of rotational structure,30 the oxygen red band b¹Σ ← X³Σ is known to be magnetic by nature. The spin current contribution was only recognized a half century later.33

After the red atmospheric oxygen b − X band, let us consider the infra red a¹Δ_g − X³Σ band transition probability. Comparison between these two atmospheric oxygen band systems provides an instructive lesson for spin catalysis in chemistry and biophysics of O₂ activation. Especially important is comparison of magnetic (transitions 1 and 2 in Fig. 1) and quadrupole intensity (transitions 5 and 4, Fig. 1) in order to clarify the nature of big differences for all transitions in the red and IR atmospheric oxygen bands. The magnetic a − X,1 band intensity (transition 2, Fig. 1) in nonrotating oxygen molecule has only orbital angular momentum contributions.33 The structure of perturbation theory expression for the a¹Δ_g − X³Σ transition moment is generally similar to Eqs. (18) and (19)

• (20)

where . An obvious difference between Eqs. (18) and (20) is that in the former equation the decreasing (L₋) and increasing orbital angular momentum operator (L₊) are involved and in the later—only increasing operator contributes. In a similar simple approximation mentioned above in Eq. (21), one has an analogous expression for the infra-red atmospheric band33, 73, 43, 49:

• (21)

The magnetic dipole transition moments for the red and infra-red oxygen bands determined by the orbital magnetism are induced by the SOC mixing with the same Π_g states. In both transitions b − X and a − X, the orbital angular momentum expressions (19) and (21) are similar except the sign and value of the second term.

The multiplet mixing induced by SOC is rather small for all orbital magnetic contributions because of the large energy difference between the ^1,3Π_g and X,a,b states near the equilibrium.33, 34 The vertical excitation energy from the ground state to the 1³Π_g and 1¹Π_g states are estimated in these expressions to be equal to 8.06 and 9.33 eV, respectively.34, 43, 49

The semiempirically estimated value, Eq. (19), is in a good agreement with ab initio result84 of the quadratic response (QR) calculation with large active space at the multiconfiguration self-consistent field (MC SCF) level (4.6 × 10⁻⁴μ_B). The semiempirical value (21) is also in a reasonable accord with the vertical transition moment |μ_{a − X,1}| = 0.0025 μ_B, calculated by QR method.83, 98 The |μ| term is larger than the |μ| magnetic orbital contribution by 6.3 times. The square of transition moment correlates with the corresponding intensity; with account of the transitions energy, the ratio of the corresponding oscillator strength for the X → a and X → b absorptions, determined by orbital magnetism, is equal to 24.3.

But still the a − X,1 transition is very weak; the radiative rate constant for spontaneous emission (or Einstein A coefficient) calculated with the magnetic dipole transition moment, Eq. (21), provides k_{a → X} = 8.3 × 10⁻⁵ s⁻¹, being close to results of QR methods.83, 98 The ab initio calculations of Klotz et al.34 give |μ_{a − X,1}| = 0.00376 μ_B and k_{a − X} = 1.897 × 10⁻⁴ s⁻¹, which is in a reasonable agreement with the most recent experimental value of Newman et al. (2.19 × 10⁻⁴ s⁻¹).81

An internuclear distance (r) dependence of the magnetic dipole transition moments, Eqs. (16) and (20), is calculated in Refs.43, 83, 84, 99. The main b − X,1 spin transition moment, Eq. (17), increases with r, because the energy gap in the denominator of the C_bX expression, Eq. (13), decreases with the internuclear distance going to zero at the dissociation limit.43, 99 This behavior explains pretty well the vibronic lines intensity; for all known (1,0), (1,1), (2,0) and (3,0) bands, the deviation from the experimental measurements does not exceed 5%, though the radiative rate constants themselves differ by four orders of magnitude.84 Simple account of the Franck-Condon ratios16 cannot provide such accuracy.

Deviations from the Franck-Condon approximation are now known for the (0,0)/(1,0) band ratio of the atmospheric infra-red a − X,1 transition.100 The calculated r-dependence of the magnetic dipole transition moments, Eq. (20), indicates an increase in the range 0.8–1.6 Å, Table 1, (from 0.001 to 0.006 μ_B) and decrease at larger distances.83 The (0,0)/(1,0) intensity ratio still strongly depends on the complete active space (CAS) account in the MC SCF calculations.83 In Table 1, the internuclear distance dependence for the a − X,1 transition calculated at two different CAS is presented. Both CAS are reasonable; they have no frozen orbitals in the 1s core. But CAS-II includes much larger number of empty orbitals, thus it is better fitted for the Rydberg states. Account of the Rydberg potential curves increases the calculated a − X,1 transition magnetic moments only slightly near the equilibrium (r_e = 1.207 Å). Thus, the much stronger enhancement at larger distances (Table 1) can be ascribed to the better description of state mixing at the way to dissociation limit.36 Both calculated r-dependences start to diverge after r = 1.4 Å; CAS-II provides higher magnetic transition moments at larger interatomic distances because of better description of the excited ³Π_g states.

Table 1. Quadratic response calculations of the magnetic dipole a − X transition moment (μ_B), obtained at different intermolecular distances.

r (Å)	CAS-I	CAS-II
0.8	0.000790	0.000727
0.9	0.001166	0.001195
1.0	0.001474	0.001588
1.1	0.001789	0.002014
1.15	0.001952	0.002250
1.2	0.002117	0.002507
1.25	0.002282	0.002787
1.3	0.002444	0.003095
1.35	0.002597	0.003436
1.4	0.002734	0.003814
1.45	0.002845	0.004230
1.5	0.002916	0.004686
1.55	0.002929	0.005170
1.6	0.002862	0.005650
1.65	0.002685	0.006073
1.7	0.002373	0.006368
1.75	0.001901	0.006432
1.8	0.001260	0.006177

Rotational structure of vibronic IR bands has been considered in some details with account of the main magnetic intensity.83, 101 Recently, a weak quadrupole contribution became available from very sensitive measurements with high spectral resolution.32, 75 We shall start to consider this contribution with nonrotating molecule. This traditional application of quantum chemistry provides an integrated intensity source which depends entirely on electronic wave functions in the Born-Oppenheimer approximation.69 Next, we shall consider molecular rotation in order to clarify more carefully all reasons for differences in intensity distribution within rotational branches between the magnetic and quadrupole transitions which will be also useful in spin-catalysis of dioxygen reactivity.

The electric-quadrupole transition moments for the atmospheric oxygen bands

It is well known that the electric-quadrupole transitions are not so effective as dipole transitions in radiative processes69 and this explains the weakness of the b¹Σ − a¹Δ_g Noxon band at 1.91 mm (transition 3 in Fig. 1), which is pure quadrupole by nature. The b − a quadrupole transition moment is equal Q_{b − a} = 1.351 Debye × Å, which is approximately comparable with the permanent electric-quadrupole moment of the low-lying dioxygen states.31, 34 The corresponding radiative rate constant for the spin-allowed singlet-singlet b → a Noxon transition (k_{b − a} = 0.00139 s⁻¹)31, 34 is much smaller than that for the spin-forbidden singlet-triplet b → X transition (the k_{b − X} value is about 20 times larger).

The quadrupole component of the red band (transition 5, Fig. 1) is very weak: it has not been carefully identified until recent time, when Naus et al.74 and later Long et al.75 have made cavity ring-down spectroscopy measurements of high sensitivity. In the classical study by Herzberg,69 it was mentioned that rotational lines of the red b ← X system should in principle be accompanied by the thousands-time weaker lines of quadrupole nature. Some of these rotational lines (^pO and ^RS branches) coincide with much more intense magnetic transition branches and are overlapped and masked, but the ^NO and ^TS rotational branches are widely shifted.74 The ^TS lines are relatively easy to detect102 since they lie beyond the main R branch band head. Long et al.75 have determined the band strength of the quadrupole lines of the red b ← X system to be 8 × 10⁻⁶ weaker than the magnetic dipole intensity, determined by Eq. (17). From Eqs. (10) and (11), it follows that electric quadrupole tensor operator component along z-axis (Q_zz) provides the nonzero matrix element between the X³Σ and b¹Σ states in the form34, 43:

• (22)

In a nonrotating molecule, the only allowed quadrupole transition moment of the red system (5, Fig. 1) is determined by the difference of the permanent electric quadrupole moments of the X³Σ and b¹Σ states.34, 43

This difference is quite essential, because the wave functions calculated with account of configuration interaction are not exactly equal to the simple approximations, Eqs. (4) and (7). Instead of the simple …π π configurations, presented in these equations and denoted by square brackets like ³Ψ₀[X³Σ], there are essential admixtures of the doubly excited terms …π π. In the Ψ (b¹Σ) state, this admixture is larger than in the Ψ (a¹Δ_g) states and even much larger than in the ground triplet state. The corresponding contributions are equal to 6.4%, 4.3%, and 2.1%, respectively. The doubly excited configuration …π π results in a larger quadrupole component; Thus for the b state, the calculated electric quadrupole moment is equal to 1.746 Debye × Å, and for the ground state X, it is smaller, 1.37 Debye × Å. Similar results have been calculated by Klotz et al.34

The calculated quadrupole b → X₀ transition moment is equal to 0.0051 Debye × Å, which corresponds to the radiative rate constant k^r = 4.7 × 10⁻⁷, where degeneracy of both states is accounted32 (this result is approximately three times larger than the result of Klotz et al.34). The integrated electric quadrupole band intensity determined by Long et al.75 is 1.8 × 10⁻²⁷ cm/molecule, which corresponds to the radiative rate constant k^r = 7 × 10⁻⁷.32 Thus, the agreement between the theory, Eq. (22), and experiment75 for quadrupole intensity of the b → X₀ transition (5, Fig. 1) seems to be satisfactory, though few additional arguments of Refs.32, 103 provide some complications.

Now we need to consider quadrupole contributions to infra-red oxygen bands at 1.27 mm and 1.91 mm simultaneously. In order to calculate the quadrupole a¹Δ_g − X³Σ transition moment, one has to submit into the integral 〈ϕ_a|Q̂_zz|ϕ_X,0〉 = Q_{a − X,0}, the perturbed wave function of Eq. (11). Because of the nonzero admixture of the singlet ¹Σ state character into the ground triplet ϕ_X,0 spin-sublevel, Eq. (11), the a − X,0 transition (4 in Fig. 1) can borrow intensity from the b¹Σ − a¹Δ_g Noxon band.

• (23)

Since the b − a transition has an appreciable electric-quadrupole moment, this mechanism of intensity borrowing, Eq. (23), induces a nonvanishing quadrupole transition moment for the a¹Δ_g − X³Σ atmospheric oxygen band at 1.27 mm: Q_{a − X,0} = −CQ_{b − a} = 0.018 Debye × Å.31, 34 This corresponds to the a¹Δ_g − X³Σ transition radiative rate constant k_{a − X,0} = 5 × 10⁻⁷ s⁻¹.34 Semiempirical calculations by this scheme, Eq. (23), had provided a larger value of 2 × 10⁻⁶ s⁻¹.31

All the above conclusions concern the nonrotating O₂ molecule; they could be applied to the oxygen molecule frozen in a noble gas matrix, if the matrix does not perturb the internal O₂ structure. But this is not a case even for Ne matrix.104 The total band intensity estimations are not changed if the molecule rotates, but some additional details connected with intensity distribution among rotational branches occur. In that case, spin sublevels (Ω equals 1 and 0) are mixed by rotation and the prediction of different intensity of the a − X,0 and a − X,1 transitions cannot be verified directly. In order to check fine details of intensity distribution and make comparison with the observed rotational lines strength, we have to consider molecular rotation.

Molecular rotation

In 1934, Van Vleck30 has considered the nature of the atmospheric oxygen bands, just discovered that time, and presented short table for rotational line-strength formula for magnetic dipole radiation in the ¹Δ − ³Σ transition. He has also discussed different combinations of symmetries for both, red and IR atmospheric oxygen bands and concluded that the ¹Σ − ³Σ symmetry is a proper assignment for the red band, accounting that the same rotational structure would possess the ¹Σ − ³Σ transition. (Now it is known as the c − A band calculated in Ref.49 to have much higher magnetic transition moment 0.123 μ_B than the red band. Because of small c − A transition energy, 0.29 eV, its radiative rate constant is negligible, k_{A − c} =0.0054 s⁻¹ and cannot compete with other type of deactivation through Herzberg bands).

Van Vleck30 tried to explain the great difference in intensity of the red and IR bands by qualitative account of SOC-induced borrowing schemes, which correspond to those, presented in Eqs. (18) and (20) at a qualitative level. He has argued that the second term in Eq. (18) should be more important than the other possible contributions, which is absolutely correct, but he neglected the signs. Now we know that without account of spin-current contribution, Eq. (17), one cannot explain the red and infra-red bands intensity ratio.

The main achievement of Van Vleck's classical paper30 is determined by his rotational analysis and magnetic-dipole nature assignment of the atmospheric oxygen bands. The Van Vleck's formulas have been generalized by Bellary and Balasubramanian70 with account of spin splitting and centrifugal distortion in the ground state.70 Quantum chemical calculations in terms of this theory70 provide an agreement with fine details of rotational intensity distribution.83 In the following, we shall consider molecular rotation in order to clarify connections between magnetic and quadrupole intensity, the question which become quantitatively important in recent time.

The a¹Δ_g − X³Σ system of oxygen in the absence of electric-quadrupole branches with ΔJ = ±2 can be entirely ascribed to a magnetic dipole radiation with k_{a − X,1} = ∼ 2 × 10⁻⁴ s⁻¹.33, 34, 81 This is in a good agreement with predictions of the old theoretical models for quadrupole radiation: k_{a − X,0} = ∼(2–0.5) × 10⁻⁶ s⁻¹.31, 34 Thus, the electric-quadrupole contribution to the intensity of the a − X,0 transition in nonrotating O₂ molecule was considered before31, 83 as being practically negligible. But recently, a very weak electric quadrupole contribution to the a¹Δ_g − X³Σ transition rate constant has been identified and measured to be equal to (1.02 ± 0.10) × 10⁻⁶ s⁻¹.32 Such weak bands have been detected in the ground-based solar absorption spectra,32 thanks to a long absorption path through the atmosphere. Subsequently, the rotational analysis of the quadrupole-induced branches in the a¹Δ_g − X³Σ transition has been performed with the high-sensitivity cavity ring-down spectroscopy experiments.32 The peculiar rotational structure of these branches has been interpreted because of a good understanding of more intense magnetic dipole rovibronic transitions32, 34, 81, 84 and because of exact knowledge of the nature of quadrupole intensity borrowing scheme.31, 35, 43 Thus, a short discussion of the atmospheric oxygen bands in a rotating O₂ molecule is necessary here.

The rotational levels of the a and X states can be considered in terms of Hund's case (b) coupling approximation, though the lower levels of the ground state obey the intermediate Hund's coupling case.69–71, 103 The spin sublevels of the X³Σ state, Eqs. (11) and (12) in rotating molecule, are mixed and can be presented in a form70:

• (24)

• (25)

• (26)

where the F₁ component corresponds to J = N + S rotational sublevel, F₁ to J = N and F₂ to J = N − S, the e and f labels denote rotational-vibrational levels with rovibronic parity ±(−1)^J. The e parity rotational sublevels of the X³Σ and X³Σ states are mixed in spin-rotational terms F₁(J) and F₃(J), while the f parity rotational sublevel X³Σ corresponds to the F₂(J) spin-rotational term.16, 70 The wave functions in Eqs. (11) and (12) now depend on J because of additional account of Coriolis coupling, which includes BL_± terms (B is a rotational constant) besides the known SOC-induced corrections.70, 83 The J-dependent coefficients c_J and s_J for the X³Σ state of O₂ molecule are well known32, 70 for the intermediate and Hund's (b) coupling cases.

• (27)

In the Hund's (b) coupling limit, these coefficients are s_J = [(J + 1)/2J + 1)]^1/2 and c_J = [J/(2J + 1)]1/2.103 Account of rotation, Eqs. (24)–(26), SOC, Eqs. (11)–(13), and Coriolis perturbation terms, Eq. (29), following Refs.70, 103 and omitting some minor terms leads to the expression for the ground X³Σ state, which has only e parity:

• (28)

where the Coriolis coefficient accounts the triplet–triplet states mixing with the n³Π_g terms:

• (29)

The C_b,X coefficient given in Eq. (13) is still a crucial value of the whole rovibronic intensity theory. In a similar manner, the upper spin-sublevel X³Σ should also be accounted for quadrupole a − X transition in rotating molecule:

• (30)

The upper state a¹Δ_g,2f^e and all other Coriolis perturbation coefficients are given elsewhere70, 83 as well as the b¹Σ description.103 Intensity distribution in atmospheric oxygen bands with quadrupole approximation are studied in a series of works.32, 70, 71, 75, 83, 103 The line-strength formulas for the b − X band, for example, should contain the perturbed wave functions and the two quadrupole transition moments103 Q₂₀ = 〈Ψ (a¹Σ)|Q₂₀| Ψ (X³Σ)〉 and Q_{2 − 1} = 〈Ψ (a¹Σ)|Q_{2 − 1}| Ψ (X³Σ)〉, where Q_2k are spherical components of the molecular-fixed quadrupole E2 operator.103

The atmospheric oxygen red band (0–0) consists of four branches, two R branches forming a head and two P branches, separated from the former by a zero gap. There is an ^RR branch (J =+1, N =+1), a ^PP branch (J =−1, N =−1), an ^RQ branch (J = 0, N =+1), and a ^PQ branch (J = 0, N =−1).69 The ^PP and ^RR branches of the red atmospheric bands correspond to transitions from the F₂ component, the ^RQ and ^PQ branches to transitions from F₁ and F₃ components, respectively.

Long et al.75 have considered the b − X band quadrupole intensity with the Hund's (b) coupling limit. Balasubramanian et al.71, 103 have made the same, following the Hund's case (a) formulation route; they get a close agreement with results of Long et al.75 and found Q_{2 − 1}/Q₂₀ = −2.338. Accounting few reasonable approximations, Balasubramanian et al.71, 103 have shown connections between the fundamental Q₂₀ and Q_{2 − 1} parameters and the Q₁,Q₃ moments of the Long's et al.75 theory: Q_{2 − 1} = and Q₂₀ = Q₃ − Q₁. Using the values determined by Long et al.75 (Q₁ = 4.15 × 10⁻⁴² Cm², Q₁ = 2.61 × 10⁻⁴² Cm²), Balasubramanian et al.103 have interpreted these data with a complete success of their theory and explain the ratio Q_{2 − 1}/Q₂₀ found above.

Using similar approach and Eqs. (28)–(30), one can consider the quadrupole branches intensity in the a − X band system. Comparison with the observed lines intensity for the P,Q,R branches of the a¹Δ_g − X³Σ (F_i) transitions can be done using these eigenfunctions of rotational spin-sublevels and similar wave function for the upper ϕ_a(J) states83; the effective rotational transition moments in the Hund's (a) case are tabulated elsewhere.71, 101

Gordon et al.32 have used the measured weak quadrupole lines intensities for calculation of complete list of quadrupole a − X,0 transitions which obey the selection rule ΔJ = ±2, ±1, 0. These calculations have been performed for the intermediate Hund's coupling case, assuming that these transitions are possible only through the SOC-induced mixing of the X³Σ and b¹Σ states, as it is shown in Eq. (11) and through intensity borrowing from the Noxon band Eq. (23). The calculated complete list of lines agrees well with the experimental data and has been used to improve the residuals of the fitted solar atmospheric absorption spectrum.32 The most intense lines are those that obey the selection rule ΔJ = ± 2; namely these are lines in the ^TS and ^RS branches.32

Thus, we come to important conclusion that in free O₂ molecule the a − X emission band (1270 nm) consists practically only from magnetic radiation (transition a − X,1 to the Ω = 1 spin sublevel). This IR emission (transition 4, Fig. 1) is determined by the SOC-induced orbital magnetism, Eq. (20). The quadrupole a − X,0 transition is observable with very sensitive CRD spectrometer and its rotational structure supports the old mechanism, Eq. (23), proposed in Ref.31. The SOC-induced mixing between the b and X,0 states, Eq. (13), determines all peculiarities of the atmospheric oxygen bands.

Intensity of forbidden UV oxygen bands in gas phase

Rotational branch intensities in the “Herzberg” bands and in other UV spectra in the 240–400 nm region are more uncertain.4, 19, 20, 50, 53, 63 There are many attempts to fit the mechanism of their intensity borrowing to the observed intensity distribution in the rotational lines.4, 19, 20, 50, 54, 63, 105 All “Herzberg” bands are electric-dipole allowed if SOC is taken into account, but without SOC perturbation they are spin-forbidden (even the triplet–triplet transitions, X − A^′ and X − A).43, 45 Thus, their photoabsorption cross-sections are 4–6 orders of magnitude smaller than that in the SR continuum. Since the “Herzberg” bands absorption is very weak, it affords the near UV sunlight to penetrate deep into the Earth atmosphere, where dioxygen concentration is quite large. The absorption in the “Herzberg” band systems is a key step in the stratospheric ozone formation; the most intense and important is the Herzberg I band. Its analysis has a long history.

In 1935, Present63 considered rotational branch strengths for the A³Σ − X³Σ Herzberg I transition through possibility of mixing of either A or X states with a ³Π states by SOC or rotational Coriolis coupling. His simulations of strong Q-branch and weaker O− and S-form branches neither were satisfactory.63, 64 Huestis and Slanger, in their paper on the A^′3Δ_u − X³Σ Herzberg III band44 mentioned on simulation of the A³Σ − X³Σ transition that included a new additional source of intensity borrowing, that is the SOC-induced mixing of the A³Σ triplet with the Schumann-Runge state B³Σ. This phenomenological fitting provided a significant improvement over the previous attempts, but the nature of SOC matrix element between the A³Σ and B³Σ states has not been considered.106 This mechanism of the Herzberg I band intensity borrowing has been studied on quantum chemical level independently,43, 45 where SOC calculation has been performed with account of two-center interaction for the first time in a long series of molecular SOC effects studies.43 Thus, intensity borrowing from the Schumann-Runge system determines the main source of radiation rate in the “Herzberg-I” A³Σ − X³Σ transition45 and this mechanism agrees with intensity distribution in rotational branches since the electric-dipole transition moment is along molecular axis.20

The scheme of the “Herzberg” transitions in nonrotating O₂ molecule is presented in Figure 2. The most intense “parallel” transition in the Herzberg I band system is transition 5 (Fig. 2), which includes in fact four degenerate transitions between sublevels with Ω = 1 in both states, each one has an electric dipole moment D(X,1 − A,1) = 0.00041 a.u. as follows from our direct multireference configuration interaction (MRCI)+SOC calculations. The perpendicular polarized transition 3 (Fig. 2) has only 0.00013 a.u. transition moment as well as the transition 6 (Fig. 2), having very close to this value electric dipole moment.

Figure 2. Energy diagram for the Herzberg's transitions in the nonrotating O₂ molecule.

The perpendicular polarization occurs because of intensity borrowing from the A³Σ − ³Π_g,1,0 transitions by SOC-induced mixing between A³Σ and ³Π_g,1,0 states. Intensity calculations for these and other Herzberg's bands, c − X and A^′ − X with perpendicular polarization have been presented by Klotz and Peyrimhoff.50 Similar results (but without vibronic band calculations) have been performed on semi-empirical MINDO/S CI level.43, 49 The mechanisms of intensity borrowing for the perpendicular transitions of the Herzberg's bands are presented elsewhere.50, 53, 54, 105, 107 Buijsse et al.105 have detected atomic fragments O(³P_J) with fine structure resolved by REMPI (2+1) techniques. The one-photon Herzberg's transitions to the continuum give rise to an angular distribution of the photofragments which depend on the laser polarization and the fragment recoil directions.105 Buijsse et al. have also developed a semiempirical theory based on extrapolation of the latest experimental data on the discrete Herzberg's bands spectroscopy and on the theory.50 The amount of the parallel character of the Herzberg's transitions to the continuum has been measured from this ion-imaging REMPI experiment and has been compared with theoretical predictions. In fact it was comparison with the original quantum-chemical theory from Ref.45, though only results of Klotz and Peyrimhoff50 have been mentioned by Buijsse et al.105 We have to stress that the parallel-perpendicular polarization ratio predicted by the older theory43, 45 is in a reasonable agreement with the REMPI experiment by Buijsse et al.105 More recent theories53, 54, 107 and experiments4, 20, 40, 52 provide the state-of-art level of understanding of the Herzberg's transitions in the isolated O₂ molecule.

Slanger et al.19 have mentioned that they showed a new source of the A − X transition probability through SOC between A³Σ and B³Σ states and refer to the earlier paper from 198344; unfortunately, one cannot find any discussion of the A − X transition mechanism in Ref.44. One can stress that this mechanism has been independently proposed and calculated in 1983 by the authors of Refs.43, 45, who had no possibility to visit 38th Symposium in Ohio, which also has been mentioned in Ref.19 as a source of the new mechanism presentation. The authors of Refs.43, 45 had performed the SOC matrix element calculation

• (31)

using configuration interaction (CI) wavefunctions and estimated the parallel component of the A³Σ − X³Σ electric dipole transition moment. It was stressed that the SOC matrix element Eq. (31) is the only important exclusion in the theory of diatomic spectra where zero-differential overlap approximation cannot be applied for semi-quantitative explanation of SOC effects.43

Let us consider now the Schumann-Runge bands and absorption near the second dissociation limits O(³P) +O(¹D). Our direct CI+SOC calculations indicate for the first time that in the intense SR system there are spin-forbidden crossing transitions of the type B³Σ − X³Σ. This weak perpendicular electric-dipole transition moment (0.0008 a.u.) borrows intensity from the transitions to the ³Π_u states and is about five orders of magnitude smaller than the allowed vertical SR transition moment (0.774 a.u.). Since the upper and lower states of the SR system have rather similar ZFS parameters (3.96 and 3.26 cm⁻¹, respectively),49, 96 each intense line in the SR bands could have very weak satellites which all together look like a triplet. Because of predissociation in the upper B³Σ state, the SR lines are wide and weak satellites are overlapped and have not been nonobservable so far.

Transitions from metastable states

Absorption from the metastable b state to the upper state of the SR system is one of the most interesting finding in dioxygen spectroscopy in recent time.108 The B³Σ ← b¹Σ absorption has been detected in two-step photo-dissociation experiment with the fragment O(¹D) atom recoil measurements.108 Electric dipole transition moment for this absorption can be explained by the same coefficient Eq. (13), which provides the ground triplet state admixture to the b state, Eq. (10).88 Thus, the B − b transition borrows intensity from the intense SR band, which provides the largest singlet-triplet B³Σ ← b¹Σ transition moment in dioxygen (D(B,1 − b) =0.0089 a.u.17, 88, 89 In the present review, we give our new direct CI+SOC calculations which include aug-ccPVTZ basis, small CAS-I, and 27 spin states. This calculation provides the parallel component of the electric dipole transition moment D(B,1 − b) =0.00973 a.u. being also in a good agreement with experiment.89, 108 The calculated oscillator strengths for the B(v^′) − b(v^″ = 0) bands are in the range f(B − b) ∼ 10⁻⁸ for the most intense (v^′ = 10) − (v^′ = 16) vibronic bands, thus explaining the observed absorption cross-section and recoil direction of the aligned O(¹D) fragment.89, 108 The perpendicular component of the B³Σ ← b¹Σ transition moment is determined by the SOC-induced mixing with the Π states (D(B,1 − b) =0.0009 a.u.) and also agree well with the previous results.88, 89

We have to mention that a new possible transition from the metastable b state to the split A^′ triplet is predicted in our MRCI+SOC calculations to be relatively active. It touches only the Ω =1 spin-component of the A^′3Δ_u state (the calculated vertical transitions moment D = 0.00019 a.u. provides expectation that this transition could be observable; for the most intense (v^′ = 9) − (v^″= 0) vibronic band, the calculated oscillator strength is equal to f(A^′,1 ← b)27 × 10⁻¹⁰).

In Ref.17, a set of spin-allowed transitions from the triplet A,A^′ states to the quasistable 1³Π_g state have been calculated and predicted to be observable. The 1³Π_g state is irregular with the SOC splitting equal to 88 cm⁻¹. Thus, the Ω =2 spin-component is the lowest one.

The 1³Π_g triplet state has a very shallow (if any) minimum on the potential curve produced by avoiding crossing17, 105 and the above-mentioned 1 ³Π_g ← A,A^′ transitions can provide the narrow continuum bands in the visible (∼470 nm) region. With our new direct CI+SOC calculations, we support this prediction; for example, we get D(1³Π_g,2 ← A′³Δ_u,3)=0.173 a.u. at 1.21 Å. In the short-wavelength edge of same region, we obtain a relatively intense singlet-triplet transition 1 ³Π_g,1 ← c¹Σ with the SOC-induced perpendicular transition dipole moment D(1³Π_g − c) =0.013 ea₀. Intensity ratio for the nearest bands (0.0057) provides a weak but still detectable singlet-triplet transition (f ∼ 10⁻⁷). These visible absorption can be detected in electric discharge in the air when the c¹Σ and A^′3Δ_u states are formed by the O(³P) atoms recombination.

Transitions from the A^′3Δ_u states

Now we shall concentrate attention on the Herzberg III band (transitions 2,4 in Fig. 2) and on the Chamberlaine A^′3Δ_u − a¹Δ_g band, transitions 7 (Fig. 2),20 which show a specific spin-selectivity.43, 49

The Herzberg III band, A^′3Δ_u − X³Σ, being rather weak, is the most important from the instructive point of view in connection with dioxygen chemical reactivity and spin-catalysis. The spin-sublevels of the A^′3Δ_u state are highly split by the first-order SOC effect.69 This inverted splitting (Fig. 2) is equal to ζ^O in a simple theory,43 which is now supported by all experimental dioxygen spectra in gas phase. The reassignment of the regular splitting of the A^′3Δ_u state, originally used in Ref.69, has been first proposed by Krupenie16 and supported on the ground of the Chamberlaine A^′3Δ_u − a¹Δ_g band analysis.44 That means that the Ω =3 spin-component is the lowest one (Fig. 2) and is the most intense in the Chamberlaine band emission. In fact our calculations provide for the Chamberlaine band system the largest electric dipole moment for the transition 7; each of four degenerate components is equal to 0.00019 a.u. (All other transitions moments are calculated to be less than 10⁻⁵ a.u.) Transition 8 in Fig. 2 is also predicted to show a relatively large intensity (D(A − a) = 0.00013 a.u.)

In condensed phase, all spin-prohibition selection rules are changed dramatically because intermolecular interactions reduce the high symmetry of the electron cloud inside the O₂ species.43, 93 Even nonreactive collisions at low pressure and temperature provide important deformations in the electronic shell of dioxygen. Of course, these deformations become more important when the collision energy will be comparable with activation barrier of chemical reaction.

It is well-known that all forbidden transitions (especially the a − X, b − a, A − X, c − a and A^′ − X bands) are strongly enhanced when O₂ is perturbed by nearby molecules in the gas or condensed phase.4, 6, 26, 61, 104, 109–111 Especially, the radiative decay of singlet molecular oxygen, a¹Δ_g → X³Σ, is very sensitive to collisions in gas phase109, 112, 113 and to weak intermolecular interactions in solid104, 110, 114 and liquids.61, 115–117 The basic understanding of this sensitiveness is essential because of importance of the O₂(a¹Δ_g) in photophysics (I₂ laser pumping), photochemistry,61 biochemistry, and medicine, including the mechanisms of certain human diseases, cell aging,118 and laser cancer treatments.61

In this review, we are going to consider intermolecular perturbations in dioxygen spectra by direct CI+SOC calculations of some complexes between O₂ and nitrogen and water containing environment.

Collision-Induced Transition Intensity

A theory developed 30 years ago93, 94 was grounded on a prediction of a great enhancement of the Noxon b¹Σ → a¹Δ_g transition probability (3, Fig. 1) by intermolecular interaction. In order to recapitulate it shortly, let us consider the O₂ + H₂ collision with the C_2v symmetry and analyze the singlet excited states of the oxygen moiety in such a system. The electronic structure of both molecules at a typical intermolecular distance (R≃2.6 Å) is almost not perturbed; there are no big changes in their wave functions and the molecular orbitals are almost completely localized on each molecule. Thus, we can use MO notations following the intrinsic symmetry of each species. Instead of complex MOs, Eq. (3), we have to use real wave functions for the two quasidegenerate π_x,g and π_y,g MO in the collision complex, since there is no molecular axis and thus the orbital angular momentum projection is quenched. If the planar O₂ + H₂ collision complex lies in the yz plane, where z is the OO axis, the strongest deformation occurs in the π_y,g MO; it acquires some admixture of the σ_u MO of the H₂ moiety.93 In a minimal basis set, the π_x,g MO has no admixture, since there is no π-MO on hydrogen. The Noxon band transition moment in the O₂ moiety now acquires a form93

• (32)

being oriented along the intermolecular y-axis. Here D_y is an expectation value of electric dipole moment for the π_y,g MO. Even small hydrogen admixture to this MO provide large D_y value because of the long intermolecular distance; at the same time D_x = 0. Thus, Eq. (32) explains a nonzero electric dipole transition moment for the Noxon band induced by any collision. The intramolecular SOC perturbation does not change much upon intermolecular interaction (the coefficient C_b,X, Eq. (13), is only slightly reduced upon collision); thus, Eq. (23) leads to electric dipole transition moment for the singlet oxygen emission a¹Δ_g → X³Σ (if we substitute the quadrupole operator Q̂ by the electric dipole operator ). This prediction is supported by all calculations93 including our relativistic approach. Thus, the a − X,0 band enhancement is explained by intensity borrowing from the collision-induced b − a transition. In the following calculations of the complex between the O₂ and N₂O molecules, we shall concentrate on some important sequences of the previous qualitative consideration.

The collision-induced transitions in the complexes O₂ + N₂, O₂ + N₂O and in water

We have optimized geometry of two collision complexes O₂ + N₂ and O₂ + N₂O in the ground triplet state by density functional theory (DFT) with the B3LYP functional and also by multiconfiguration self-consisted field (MCSCF) method in a CAS for the singlet and triplet states. A well-known 2p-CAS88 has been used for O₂ and N₂ molecules; similar CAS has been implemented for NNO species. The b¹Σ → a¹Δ_g transition intensity of oxygen moiety in collision complexes has been calculated by linear response (LR) method119; the a¹Δ_g → X³Σ transition probability has been calculated by QR function at the excitation frequency of the first S-T transition.120 A total scheme of this approach is described elsewhere36, 120 and the DALTON code has been used.121

The optimized geometry of the O₂–N₂O complex is presented in Figure 3. The O₂–N₂ complex is close to similar structure with the optimized N(4)-N(3)-O(2) angle equal to 112^° and larger N(3)-O(2)-O(1) angle (132^°). The nearest intermolecular distances are 3.14 and 3.23 Å for the complexes with N₂ and N₂O molecules, respectively.

Figure 3. Optimized structure of the collision complex NNO + O₂. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

In Figure 4, the result of water + O₂ aggregate simulation is given. Hyper-fine coupling (HFC) constants at the ¹⁷O and ¹⁴N nuclei and Mulliken atomic population analysis are presented in Tables 2 and 3. Calculated IR and Raman spectra in the ground triplet state of the O₂ − N₂ complex are given in Table 4. Similar IR spectrum is obtained for the O₂ − N₂O system.

Figure 4. A snapshot of oxygen molecule and solvent molecules in the first solvation shell. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Table 2. Anisotropic (B_ii) and isotropic (a) hyper-fine coupling constants (spin-dipole and Fermi contact couplings, MHz) in the the ground triplet state of the O₂-N₂ complex at the ¹⁷O and ¹⁴N nuclei, Mulliken atomic spin densities (ρ), and Mulliken charges (Q).

Atom	Baa	Bbb	Bcc	a	ρ	Q
O(1)	167.579	−83.743	−83.836	−42.308	1.00149	0.00196
O(2)	167.067	−83.294	−83.772	−42.357	0.99799	−0.00175
N(3)	−0.246	−0.229	0.474	0.017	−0.00002	0.00104
N(4)	−0.225	−0.210	0.435	0.005	0.00054	−0.00125

Table 3. Anisotropic (B_ii) and isotropic (a) hyper-fine coupling constants (spin-dipole and Fermi contact couplings, MHz) in the ground triplet state of the O₂-N₂O complex at the ¹⁷O and ¹⁴N nuclei.

Atom	Baa	Bbb	Bcc	a
O(2)	182.609	−91.205	−91.404	−25.826
O(3)	180.093	−89.703	−90.390	−26.264
N(1)	−0.258	−0.252	0.510	0.010
N(4)	−0.254	−0.250	0.504	0.131
O(5)	0.363	0.293	−0.656	0.005
N(4)	−0.225	−0.210	0.435	0.005
O(1)	167.579	−83.743	−83.836

Table 4. Calculated IR and Raman spectra in the the ground triplet state of the O₂-N₂ complex.

Cs	ν	f	IR intensity	Raman activity	Depolar (P)
Frequencies (ν, cm^−1), force constants (f, mDyne/Å), IR intensities (KM/Mole), Raman scattering activities (A^4/amu), depolarization ratios for plane polarized incident light.
a′	7.41	0.001	0.002	2.479	0.744
a″	22.11	0.001	0.000	0.976	0.745
a′	27.46	0.001	0.001	2.479	0.750
a′	52.52	0.024	0.003	1.387	0.724
a′	1658.33	25.916	0.015	12.822	0.295
A′	2457.70	49.834	0.011	15.669	0.254

As it follows from Table 2, the spin density of paramagnetic oxygen only slightly penetrates to diamagnetic N₂ molecule during collision. Isotropic HFC constant at the nearest nitrogen atom N(3) is higher than at the terminal one, but the total atomic spin density has an opposite trend. This is connected with spin polarization and negative spin density at atom N(3). Anisotropy of spin dipole couplings is very similar for both nuclei. In the IR spectrum of the O₂N₂ complex, the oxygen bond stretching vibration gets higher intensity than the N-N vibrational mode (Table 3). These intramolecular vibrations are active in Raman spectrum but IR-forbidden in isolated molecules. Collsions induce some small electric polarization and corresponding IR activity, including nonzero dipole moment derivatives for some intermolecular modes. Intermolecular vibrations provide rather weak far-IR bands except the ν₄ = 52.5 cm⁻¹, which could be observable.

The orbitals of the complexes are mostly localized on each partner of collision, thus all excited states can be classified in terms of local excitations and charge-transfer excitations. The calculated Einstein coefficient for the b¹Σ → a¹Δ_g transition as a function of the O₂ − N₂O distance (R) indicates a strong increase at the distances slightly shorter than the equilibrium value (R_e = 3.14 Å). At R = 3.2 Å, the calculated Einstein coefficient is equal to 23.5 s⁻¹ and at R = 2.9 Å it exceeds 100 s⁻¹. (For collision with hydrogen, this value has been obtained at shorter distance, 2 Å91). This strong intermolecular distance dependence for the b¹Σ ← a¹Δ_g transition moment is obvious and natural. But we have found quite unexpected result that the collision-induced b − a transition dipole moment depends on the intramolecular OO bond distance. Accounting an expansion of the collision-induced b − a transition dipole moment D_{b − a} as a function of the oxygen bond length (r) in a series in a vicinity of the equilibrium bond distance (r_e)D_{b − a}(r) = D_{b − a}(r_e) + (dD_{b − a}/dr)_e(r − r_e) +, one can predict a strong enhancement of the (0,1) vibronic band in the Noxon transition. This band is in a typical IR region (about 3700 cm⁻¹), which is difficult for detection in kinetic transient spectroscopy, since both states are strongly metastable. This specific intramolecular OO bond distance dependence of the D_{b − a} transition dipole moment in the collision complexes immediately transfers into similar dependence for the D_{a − X,0} transition, because of their connections through Eq. (23) and its dipole moment analog. The Eq. (23) leads to the proportional electric dipole transition moment for the a − X singlet oxygen emission. Thus, we come to conclusion that not only the 0–0 a − X transition is strongly enhanced, but also 0–1 band has a particular enhancement which deviates from the Franck-Condon ratio.

We need to pay attention to the fact that in a free nonrotating O₂ molecule the final state of magnetic a → X spontaneous emission is the Ω =1 sublevel (transition 2 in Fig. 1), while the collision-induced emission is connected with other spin sublevel (Ω =0, transition 4 in Fig. 1). We have simulated O₂ in water solvent in isothermal-isobaric ensemble using molecular dynamics simulation. From the simulation trajectory, we have chosen a single snapshot (see Fig.4) of oxygen and its first solvation shell of water (containing 24 water molecules) for analysis. The results of molecular dynamic simulation in the time scale of 5 ps qualitatively reproduce scenario of transition enhancement for a simple collision complex in gas phase. The collision-induced b → a and a → X emission at low gas pressure (2.5 Torr) in electric discharge through O₂ and addition of foreign gases has been studied in Refs.109, 113. Fink et al.113 have produced the excited O₂(b¹Σ) state by microwave discharge and studied the emission by Fourier transform IR spectrometer. The detected collision-induced transition appears as a continuum emission under the discrete lines of pure quadruple b → a and pure magnetic a → X,1 rotational branches.109, 113 Upon addition of a few Torr of PCl₃ vapor, the integrated intensity of the continuum becomes much higher than that of the discrete lines, which explain strong increase in intensity of the unresolved band in the spectra of low resolution. From the pressure dependence of observed emission, Fink et al.109, 113 have determined the second-order rate constants k and k for collision-induced transitions. They have stressed that the values of k strongly depend on the nature of collider. These values increase by a factor of about 400 on going from noble gas Ne to the polar PCl₃ molecules, but the ratio k/k remains almost constant for all studied gases and quite close to the result predicted by Eq. (33),49, 93, 94, 109, 122 which is derived from the quantum theory described above. Exactly from Eq. (23), it follows that the ratio of the collision-induced b → a and a → X radiative rate constants does not depend on the buffer gas collider because both transition moments are connected through the crucial C_b,X coefficient determined by Eq. (13), which is an entirely dioxygen property:

• (33)

where K = d_b/d_a depends on the degeneracy factors (d), which need some additional comments. Formally K=1/2, but the choice of degeneracy ratio in the dioxygen transitions is complicated by spin-multiplicity change6 and by nuclear spin statistics effect on rotational energy levels.81 It was shown78 that the degeneracy ratio for the A and B Einstein coefficients is weakly temperature dependent for the a → X atmospheric band; it tends rapidly to a high-temperature limit valid for T > 100K.

Thus, we use simple ratio K=1/2. In this case, Eq. (33) is equal to 3.05 × 10⁻⁴ in a good agreement with the measurements in gas phase, liquids and solids.104, 109, 113, 114

This ratio has been measured in matrix-isolated oxygen by Schurath and coworkers104, 114 In Ne and Ar, they obtained 3.1 × 10⁻⁴ and 3.7 × 10⁻⁴ which are in agreement with our general prediction (3.05 × 10⁻⁴) from Eq. (33). The ratio increases to 4.1 × 10⁻⁴ in Kr matrix and this supports the general trend, since all these small deviations from theoretical prediction, Eq. (33), can be explained by the external heavy atom effect of rare gases73 which enhances the singlet-triplet a → X transition, but does not influences the spin-allowed b → a transition. The SOC-induced splitting in the ³P_J excited state of Kr atom is quite strong; in general, it is proportional to the fourth power of nuclear charge.92 Because of weak intermolecular interaction with the Kr matrix some charge-transfer admixtures produce deformation of the dioxygen wave functions and increase SOC inside the O₂ moiety enhancing the singlet-triplet a → X transition.73

The vibronic transition intensity of the a¹Δ_g[v^″= 0] → X³Σ [v^′ = 1] emission band (0,1) can be analyzed through the derivative (dD_{b − a}/dr)_e estimation and relation given by Eqs. (23).

The (0,1) rate constant is estimated to be equal 0.0072 s⁻¹. The calculated ratio of the vibronic collision-induced a − X emission rate constants is equal to k_(0,0)/k_(0,1) = 35. It differs considerably from the ratio determined by the Franck-Condon factors q_0,0 ν/q_0,1 ν = 147.2.16 The collision-induced (0,1) band of the a¹Δ_g → X³Σ transition has been observed in organic solvents as a weak band at 1558 nm123, 124 and as a continuous emission in gas mixtures.109 In the late experiment,109 the intensity of the continuum in the (0,0) band is only slightly larger than that in the (0,1) band (just by few times); their ratio highly deviate from the Franck-Condon factors estimation. This is determined by a strong dependence of the collision-induced a − X transition dipole moment on the intramolecular OO bond distance inside dioxygen. Such unexpected dependence has been obtained for all calculated models of the collision complexes6, 83 and probably has a general character.

Intermolecular interactions in collision complexes of dioxygen reveal interesting manifestations of internal magnetic perturbations and solvatochromic effects in spectroscopy and photochemistry of singlet and triplet O₂ species.6, 61, 83, 100, 103, 125 It was shown for the first time in Refs.7–9 that some similarity in magnetic perturbations can be anticipated for enzymatic reactions at the starting stages of dioxygen activation.

Of course, strong chemical interaction at the final stage of oxidation process will fully destroy the orbital and spin arrangements of reactants, including the triplet O₂ species. Because of very complicated character of dioxygen reactions with oxidases and other similar enzymes, it is of principal importance to show a possibility to use the spectroscopic experience in biochemistry of dioxygen.

In the following chapter, we shall move to the bioactivation of dioxygen by some enzymes and try to explain the origin of spin multiplicity changes being based on the similar spin and spin-orbital interaction ideas, which have been described above in the spectroscopic chapters.

We are not going to say that the same pure molecular oxygen wave functions which have been considered before in Eqs. (2)–(7) can be used for description of O₂ reactions with enzymes. Instead, some new atomic and molecular orbitals and their overlaps come into play. Nevertheless, it was shown that at particular stages of enzymatic dioxygen activation, where redox processes occur, the same SOC integrals, which operate in the O₂ spectroscopy, are responsible for the triplet–singlet transition in the active center of enzyme.8, 7, 10–12, 28 Interaction of such small molecules like O₂ with metalloproteins is of coordination nature which is well described in coordination chemistry.126 Though the structure of O₂ and of the central iron metal in cytochrome or in myoglobin can change and acquire various redox forms during dioxygen activation process, one can still use some common “finger-print rules” implemented in molecular spectroscopy of dioxygen.28, 83 In the following chapter, we shall show how these common rules can be used in a number of enzymes.

Enzymatic Reactions

Spin-forbidden organic oxidation reactions of dioxygen are effectively activated by enzymes. GO catalyzes the triplet ground state dioxygen reduction into H₂O₂ by effective overcoming of spin prohibition.7

GO is a homodimeric protein mostly found in fungi; in the graphical abstract taken from the structure of the enzyme127 from Aspergillus niger,128 we present only one protein subunit. In the GO active center, there is noncovalently bound flavin adenine dinucleotide (FAD) shown inside and outside the GO protein chains. GO catalyzes the oxidation of D-glucose into D-glucono-1,5-lactone in reductive half-reaction (i) in Figure 5, which then hydrolyzes to gluconic acid (Fig. 5, iii). At the stage (i), FAD is reduced to FADH₂. Thus, GO mediates hydrogen transfer from glucose to FAD in the reductive half-reaction (i) and then GO catalyzes the peroxide H₂O₂ production by concerted electrons and protons transfer in the systems of FADH₂ + O₂ giving FAD + H₂O₂, which is illustrated by oxidative half-reaction (ii) in Figure 5. This half-reaction is spin-forbidden9 since GO, H₂O₂, and GOH₂ are in singlet ground states; it overcomes spin prohibition because of specific SOC inside dioxygen.7

Figure 5. General reaction mechanism of glucose oxidase.

The oxidation mechanism proposed7–9 started when O₂ occupies a cavity between hystidine residue (His516 shown by green color in the graphical abstract) and the flavin moiety of FAD. The measured limiting rate constants at high and low pH128 indicate that only one prototropic form of GO with protonated hystidine (at low pH) is working. At the first step of oxidation, a fast electron transfer from the reduced cofactor FADH₂ to dioxygen occurs in the small cavity. This is exothermic process as follows from DFT calculation including dielectric effect; the reason for such easy electron transfer is that electron affinity (EA) of dioxygen is extremely high (5.1 eV) in the presence of the protonated hystidine at low pH. The EA for free dioxygen is only 0.45 eV16, 69 and strong attraction of the “jumping electron” to the protonated hystidine His516 is of fundamental importance for spin-catalysis by GO. The electron jump is initiated just by occurrence of O₂ at the active site of GO and by intermolecular and OO vibration (see Table 4 for typical values of the modes frequency). The zero vibrational level of O₂ is almost isoenergetic with the v=3 level of the O and the corresponding Franck-Condon factor is relatively large.9

The generated ion-radical pair FADH – O is obviously in the triplet state. In order to trigger the subsequent nonradical-chain chemistry, the triplet–singlet transition has to be induced in the ion-radical pair. If the system occurs on the singlet state potential energy surface (PES), a simultaneous 2H⁺ + e⁻ transfer to superoxide O anion can generate the final product—hydrogen peroxide. DFT calculation of the transition state for such reaction on the singlet state PES in the radical pair FADH – O provides the free-energy barrier equal to 6.6 kcal/mol. This corresponds to the rate constant of 10⁸ s⁻¹.9 In the electron-coupled protons transfer, 2H⁺ + e⁻ from the GO active center, FADH + His516H⁺, both protons reach the superoxide O anion almost simultaneously. The new nascent radical O₂H⁻ attracts proton very fast and this process can occur only on the singlet state PES. The final step of reaction cycle FADH⁺ + His516 → FAD + His516H⁺ includes back proton transfer to histidine through water and protein environment, which is calculated to be essentially thermoneutral,9 after which the product O₂H₂ leaves the cavity and the catalytic GO cycle is completed.

The DFT/B3LYP calculation of the Hessian at the transition state on the singlet state PES provides the imaginary frequency of 567 cm⁻¹. From this Hessian, a clear deuterium isotope effect is predicted (and no ¹⁸O isotope effect),9 which is exactly opposite to experimental study.128 Because the predicted isotope effects do not match the experimental observations, the calculated singlet state reaction with the discussed transition state cannot be the rate-limiting. Such conclusion is supported by the calculated barrier and the rate constant of 10⁸ s⁻¹9 which is much higher than the experimental rate of only 10⁶ s⁻¹.128 Since the first electron transfer from cofactor to dioxygen is predicted to be exothermic process without activation barrier, we come to a conclusion that the triplet–singlet transition has to be the rate-limiting step.7, 9

In order to compete with other chemical processes (back electron transfer, radical-chain reactions), the triplet–singlet transition could be induced by relatively large SOC matrix element. In such cases, a typical intersystem crossing rate constant can reach a high limit of 10⁶ s⁻¹7, 92 which is in agreement with experimental kinetics of GO.128 The mechanism of strong SOC in this system was prompted by analysis of the famous integral in Eq. (13), the orbital part of which is similar to the < π_x|B_z|π_y > matrix element in Eq. (14). In other words, the large SOC integral, Eq. (13), which determines states mixing, Eqs. (11) and (10), and intensities of various spin-forbidden transitions in dioxygen spectrum, at the same time determines dioxygen activation in GO and in other flavanoid enzymes.6, 7

In order to explain the rate-limiting step in GO reaction cycle, we need to consider the mechanism of singlet O₂(¹Δ_g) quenching by amines.6, 8 It has been shown that the quenching rate constant (K_q) increases upon going from primary to secondary and to tertiary amines, i.e., with a decrease in the ionization potential (IA) of the amine. The dependence of K_q on temperature and pressure indicates a purely “collisional” character for the quenching. The observed linear dependence of logK_q on IA led to conclude that the charge-transfer states (CT) should affect the quenching mechanism. With account of two CT states, one is a singlet state with transfer of an electron from the lone pair n-MO of the amine to the incompletely filled π_g,x-MO of the dioxygen (¹CT_x) while the second charge-transfer state (³CT_y) is a triplet with transfer of an electron from n-MO of the amine to the π_g,y-MO, it was shown that small admixtures of these CT states to the singlet and triplet dioxygen states, respectively, lead to increase of the quenching K_q rate constant.6 This is because the SOC matrix element <³CT_y|H_so|¹CT_x > = (1/2)< π_x|B_z| π_y > is determined by the same orbital integral, Eq. (14), which enters the important spectroscopic expression Eq. (13). It entirely depends only on oxygen, since the amine lone pair n-MO is the same in both CT states and does not contribute to the S-T matrix elements of the single-electron SOC operator, Eq. (14).

This is in agreement with the observed ¹⁸O isotope effect and an absence of the deuterium isotope effect in the GO catalytic cycle.128 In order to provide T-S crossing between the charge-transfer states (³CT_y and ¹CT_x) inside the ion-radical pair FADH – O (which actually corresponds to degeneracy of the ²Π_g state in free O), vibration and rotation of dioxygen moiety in the GO active site should occur. This is an additional prerequisite for the T-S transition besides the SOC, which explains the observed ¹⁸O isotope effect.

Other significant effects of spin change on biochemical functions are connected with dioxygen binding to heme proteins and to hemoglobin (Hb) in particular. They have the similar charge-transfer nature of SOC perturbation. The heme unit commonly includes an iron-protoporphyrin-IX complex linked to globular protein by amino acid residue (an axial ligand) to the heme iron.13, 14, 29 One coordination position at the iron heme active site is empty and can be used for dioxygen binding, like in myoglobin (Mb).

It is often assumed that one can overcome the spin prohibition to oxidation of organic substrates with atmospheric oxygen by successive addition of a single electron and proton in the successive reduction of the O₂ molecule. It is assumed that the removal of spin prohibition in such reactions proceeds as in the case of radical chain oxidation, where the spin prohibition can be removed upon formation of primary radicals. One has though to stress a fundamental difference between the enzymatic reactions and the radical reactions in chain oxidation processes. In the latter case, radicals go to the bulk and do not save the spin memory about their precursors. In contrast, all participants of biochemical oxidation reactions, i.e., dioxygen and electron transfer agents, are confined within the same active site of the enzyme. The electron transfer to the oxygen molecule from a diamagnetic enzyme produces a triplet ion-radical pair. In such triplet precursor, all spins remain correlated, the spin memory is saved, and the spin prohibition to subsequent reactions of the radical ion pair is still valid and cannot lead to a singlet product. For example, the O₂ reaction with GO9 involves flavine adenine dinucleotide (FAD) and includes two stages; namely, glucose oxidation to glucosolactone with reduction of FAD to FADH₂ and the reverse cycle FADH₂ → FAD, with reduction of O₂ to hydrogen peroxide. It is interesting to consider only the second stage. After formation of a triplet radical pair, FADH O, the T→S transition needs to occur in order to provide the final products FAD + H₂O₂. The last phase of the catalytic cycle accompanied by the formation of hydrogen peroxide can occur only in the singlet state. The T-S transition has been explained by a relatively large SOC between the S and T states of the radical pairs, which have different orbital structures inside the superoxide ion.

It has been shown that a similar mechanism of SOC enhancement by charge transfer can be applied for spin dependent reaction of dioxygen binding to heme. At the intermediate distances 2.5 Å, the starting ³A^″(2) state from the entrance channel transfers to the triplet Fe³⁺…O radical pair. In this region, there are few crossing points between S and T states, including the ³A^″(2) — ¹A^′ (1) states crossing, where spin change could occur.28 The both radical pairs could be in T and S states; all four states are almost degenerate at the intermediate distances.

The starting triplet radical pair corresponds to a charge transfer (CT) state described by ³A^″ wave function and the final singlet radical pair also represent CT state but with other space symmetry ¹A^′. Both are similar to charge-transfer states Fe³⁺ … O of the ³CT_y and ¹CT_x type described above.

The SOC matrix element between these states is equal to half of the integral shown in Eq. (13) as it was shown in the case of the singlet oxygen quenching by amines and in the GO catalytic cycle. It entirely depends only on oxygen, since the Fe orbital is the same in both CT states and does not contribute to the S-T matrix elements of the single-electron SOC operator, Eq. (14). Account of more sophisticated CAS CI can lead to nonzero SOC contribution from Fe ion as well. It is important to stress that contributions from dioxygen and from the iron cannot quench each other. In principal, these SOC contributions interference could be destructive but it is not so in dioxygen binding to Hb. The reason is that the orbital angular momentum quantization axes for Fe ion and for O₂ do not coincide in hemoglobin.28 The porphyrin macrocycle in heme is characterized by significant electron and spin delocalization including charge transfer to the Fe ion. This mobility of unpaired spin density is important for various heme functions. A high probability of spin transitions between close lying states of different multiplicity11, 29 in heme proteins determines their reactivity in respect to O₂ and spin-catalysis of various enzymatic oxidation process.6, 12 Such spin-capacity in Hb and Mb provides fast spin transformations during dioxygen binding to heme Fe(II) site.11, 29 The enzymatically active heme species in horse reddish peroxidase and in cytochrome P450 family include so-called compound I as a key intermediate.14 On the way to “compound I,” the dioxygen insertion into substrate bound ferric and ferrous species leads to the oxy-ferrous form, which is similar to oxy-myoglobin. Numerous spin changes in these reactions are now well-established by kinetic spectroscopy14, 29 and quantum-chemistry methods.

The reaction of the ferrous heme quintet ground state with the triplet dioxygen provides a number of spin multiplets in the entrance channel, but none of them can lead to the oxy-heme product, which has an open-shell singlet ground state structure.14, 129 Analysis of wave function of this product at the oxygen moiety indicates that it is similar to the upper state of the Herzberg III A^′3Δ_u ← X³Σ transition in O₂. This is not a simple O superoxide anion with the π open shell; there is also a hole in the π orbitals, which interact with the 3d-AOs of the iron. From such analogy, we can apply various spin-selection rules, developed during analysis of the Herzberg III bands intensity distribution. The Chamberlein bands A^′3Δ_{u, Ω} → a¹Δ_g intensity ratio can also be instructive. On this background, we can predict semiqualitatively the relative rate constants for different spin transitions during the O₂ − heme reaction and in the oxy-heme product.12, 28 We come to a conclusion that the SOC in dioxygen moiety is the most important in these reactions. Of course, the SOC in the iron 3d-shell can also contribute to the spin-selective reaction rate constants, but both SOC contribution can be considered separately, since the dioxygen and iron quantization axes do not coincide in the oxy-heme product (the angle between FeO and OO axes is about 134^°). Thus, one cannot anticipate that the Fe and O₂ SOC contributions are able to interfere; in the case of the parallel axes, the destructive quenching of both contributions is possible.

The A^′3Δ_{u, Ω} spin substrates are doubly degenerate in free O₂ molecule and show the inverted character (Ω =3 is the lowest state and the Ω =2 is the middle one, Fig. 2). In collision with N₂ and Fe-ion, the A^′3Δ_{u, Ω = 2} spin substrate is not doubly degenerate, but strongly splitted by about 200 cm⁻¹, while the general multiplet SOC splitting (142 cm⁻¹) is only slightly reduced. The Herzberg III A^′3Δ_u ← X³Σ transition in collision complex is strongly increased only for the A^′3Δ_{u, Ω = 2} spin substrate (D=0.004 a.u.). At the same time, the Chamberlein band for the same spin substrate is predicted to be completely allowed (D=0.259 a.u.) in our MRCI + SOC calculation of the O₂ − N₂ collision complex (In spite the fact that this is a singlet-triplet transition). From this finding, we can conclude that spin-forbiddenness can be effectively overcome in dioxygen activation by enzymes, if the open shell π³π³ configuration is admixed during exchange perturbation with the catalyst.

Conclusions

The a¹Δ_g state is doubly degenerate in free O₂ molecule, but this degeneracy is slightly shifted in O₂ collisions with N₂, NNO, and water. The splitting is very small; it is about 3 cm⁻¹ for the optimized geometry of the O₂ + N₂O complex. This splitting has nothing in common with the band width of singlet oxygen emission in solvents, which depends on collision-induced polarizability changes in the a and X states.6 The enhancement of the Noxon band, b¹Σ − a¹Δ_g, is about 10⁴ times, which induces the corresponding a¹Δ_g → X³Σ (M_s = 0) transition enhancement. Vibronic intensity response to intermolecular interactions is very peculiar. Additional enhancement of the (0,1) vibronic band in the a → X emission is explained by a rather strange collision-induced transition-moment dependence on the OO bond distance in dioxygen. Such dependence is completely unexpected from the general principles of intermolecular interactions. This is in qualitative agreement with the collision-induced (0,1) band intensity of the a → X phosphorescence in dense gases.109 The ratio of the collision-induced radiative rate constants for the a → X and b → a emission does not depend on the solvent or on the nature of the colliding gas particle.

Intensity distribution in rovibronic lines of the red atmospheric oxygen band systems has been calculated and analyzed by ab initio methods for a free O₂ molecule and its complexes with N₂, NNO, and water. We have to mention some additional enhancement of the (0,1) vibronic band of the b¹Σ − X³Σ system. It depends on collision-induced electric dipole moments of the b and X states. Calculations show that their difference strongly depends on the OO bond length vibration during the O₂ + N₂O collision.

From our MRCI +SOC calculations, we have found that the Schumann-Runge band system can include besides the very intense bands, like X³Σ → B³Σ, also a very weak transition with different Ω values in the both states. Thus, our calculation with the aug-ccPVTZ basis set (small CAS) provides a perpendicular component of the electric dipole transition moment X³Σ → B³Σ equal to 0.00008 a.u. The same result is obtained for the Δ Ω = −1 spin component.

Since the ZFS parameters in both states are rather close (3.96 and 3.26 cm⁻¹ in the ground and upper states, respectively), the intersystem transitions could not be overlapped by the intense Δ Ω = 0 transitions. For the lower v values, the Schumann-Runge linewidths are not very wide and the Δ Ω = ±1 intersystem transitions could be observable, despite their low intensity.

All these spectroscopic details are important for O₂ reactivity and for biochemical activation of dioxygen by enzymes. The same SOC integrals which are responsible for spin-forbidden transitions in dioxygen spectra do occur in the key steps of the O₂ reactions catalyzed by oxidases and cytochromes. FAD is a common component in biological oxidation-reduction reactions. The redox reactions involve a gain or loss of electrons. In the GO-catalyzed redox reaction, FAD works as the initial electron acceptor and is reduced to FADH₂. Then FADH₂ is oxidized by the final electron acceptor, dioxygen, which can do so because it has a higher reduction potential and a special arrangement of protonated protein residue in GO. O₂ is then reduced to hydrogen peroxide (H₂O₂) transforming triplet ground state to the singlet. In GO, the triplet–singlet transition occurs at the ion-radical stage and is determined by SOC matrix element in the superoxide ion, which is known from dioxygen spectroscopy. Thus, the key step of the oxidative half-reaction in GO (Fig. 5, ii) is an electron transfer and the triplet–singlet transition inside the O anion, being bound with the FADH cation. This is in agreement with the observed ¹⁸O isotope effect in GO oxidative half-reaction and a strange absence of deuterium isotope effect. The rate constants estimations also support the predicted spin-flip mechanism of the rate-limiting step in GO catalytic cycle. Similar SOC parameters are applicable for the most important biochemical reaction—dioxygen binding to hemoglobin. The π_u − π_g excitations in the Herzberg bands simulate in some way an electron promotion during dioxygen activation by cytochrome P450. Thus, the analysis of spin-forbidden transitions in dioxygen spectroscopy provides some useful information for the enzymatic spin-catalysis.

Acknowledgements

The authors are grateful to Gumilev National University in Astana for a great hospitality during the Fock conference.

Biographical Information

Boris Minaev was born in Yekateriburg (Russia) in 1943. He has graduated from Tomsk State University (in Siberia) and received there his PhD in physics devoted to EPR and phosphorescence lifetime calculations. In 1974, Minaev received an assistant professor position in Karaganda State University (Kazakhstan) and the chief of physical chemistry chair (1976). He has defended habilitation work in Moscow Institute of chemical physics (1983) and created a quantum chemistry chair in Karaganda SU. Minaev moved to Ukraine just before the collapse of the Soviet Union and received a chemistry chair in Cherkassy Institute of Technology. Since 2007, he is a chief of Organic Chemistry Department and director of Scientific Institute of Physics and Chemistry of Functional Materials at Bogdan Khmelnitsky Natioanl University in Cherkasy. As a guest professor, during last decade, Minaev delivered lectures at the Royal Institute of Technology and collaborates with a group of Hans Ågren. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Biographical Information

N.A. Murugan was born in Paramakudi, India, in 1976. He received his master degree in general chemistry in 1998 from GRI Deemed University, Dindigul, India. He was awarded with a PhD degree from Indian Institute of Science in 2005 for his contributions in understanding temperature- and pressure-induced structural transitions in organic molecular crystals using variable shape Monte Carlo simulations. Until 2010, he had postdoctoral visits to ULB (Brussels), UPC (Barcelona), KTH (Stockholm). From 2011, he is working as a researcher at the Department of Theoretical Chemistry and Biology at the Royal Institute of Technology, Sweden. His current research is mostly devoted to understanding the structure, dynamics, and properties of molecular probes for medical diagnosis application. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]

Biographical Information

Hans Ågren was born 1950 in the town of Skelleftea in northern Sweden. In 1979, he received his PhD in experimental atomic and molecular physics at the University of Uppsala under the supervision of Kai Siegbahn. After a few PostDoc years in USA, he became assistant professor in Quantum Chemistry in Uppsala 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. The Department of Theoretical Chemistry and Biology at KTH, which he currently heads, houses ca 20 scientists and 40 PhD students, with research activities in theoretical modeling primarily in the areas of molecular/nano/bio photonics and electronics, in catalysis, and in X-ray science. The research is a mix of method development and problem-oriented applications in collaboration with experimentalists. Hans Ågren participates in several national and international networks in his research areas. He spends his spare moments in the summer cottage close to his home town up north, with some fishing and mountain hiking. [Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.]