Analysis of the atmospheric bands in gas phase
Both these bands, a1Δg − 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 a1Δg − 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 1,3Πg states and by intensity borrowing from magnetic dipole transitions X − 3Πg and b − 1Πg. 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 → a1Δg transition. The Einstein coefficient for spontaneous b → a emission (or radiative rate constant kb − a) was estimated by Noxon to be equal to 0.0014 s−1.35 The b1Σ − a1Δg 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 (ka − 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 a1Δg − 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, a1Δg,2 − X3Σ and b1Σ − X3Σ (transitions 2 and 1 in Fig. 1, respectively), can borrow intensity from magnetic-dipole transitions of the type 3Πg,0 − X3Σ. Here the low subscript corresponds to Ω (the total electronic angular momentum projection on the interatomic axis). There are also a1Δg,2 − n1Πg,1 magnetic-dipole transitions which contribute additionally to the a − X1 band intensity and b1Σ − n1Πg,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 a1Δg,2 − X3Σ component of the a – X transition (2 in Fig. 1) is magnetic and the a1Δg,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 μΨn[Σ Λ Ω]. The main contribution to the triplet ground state wave function of the O2 molecule can be presented by the scheme:
where indicates the proper antisymmetrization product; π−, g and π+, g indicate two degenerate πg molecular orbital (MO) with angular momentum projection λ = ±1 in units :
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:
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, a1Δg, and b1Σ states is almost the same in a reasonable approximation. Two degenerate states a1Δg have a simple open-shell part of the wave function:
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:
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 Hso
where and the orbital part can be presented in a simple approximation43, 92
Orbital part of the BzSz 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 Sz 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 ϕn, where n is a spectroscopic term notation, then it follows:
where, for example:
In an effective one-electron approximation, Eqs. (8) and (9), the SOC matrix element in the numerator of Eq. (13) is equal to
where ζO = 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 (Eb − EX = 13195 cm−116) into the denominator, the admixture coefficient will equal to Cb,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(a1Δg) 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 a1Δg 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 (Dss = 1.44 cm−1) and by the second-order contribution of perturbation theory with SOC account (Dso = 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),
includes L and S—the orbital and spin angular momenta, respectively; here μB is the Bohr magneton and ge = 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)
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 ϕX,1 substates, Eq. (12), is almost pure, since the CX,n admixtures are not important here. They are small and do not influence much the normalization constant; thus, we take for simplicity an approximation ϕX,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 Πg,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 Cb,n and CX,n coefficients in Eqs. (10) and (12), respectively. Only the Cb,X coefficient in Eq. (10) produces contribution to the spin part of magnetic dipole transition moment76:
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 Ms = 0 and Ms = 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:
The first semiempirical estimation33 based on INDO/S-type theory33, 73 has taken into account the lowest singlet and triplet 13,1Πg states, produced by 3σg → πg 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 13,1Πg states. It is shown49, 50 that in the vicinity of the equilibrium internuclear distance, 1.15–1.35 Å, the 13,1Πg states are really connected with the single electron excitations 3σg → πg which provide the main contribution to the orbital magnetic transition moment Eq. (18). Thus, they can be approximated by a simple expression43:
Here Cg,z = 0.653 is the LCAO-coefficient in the 3σg molecular orbital expansion for the 2pz-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 μb − 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
The multiplet mixing induced by SOC is rather small for all orbital magnetic contributions because of the large energy difference between the 1,3Πg and X,a,b states near the equilibrium.33, 34 The vertical excitation energy from the ground state to the 13Πg and 11Πg 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 × 10−4μB). The semiempirical value (21) is also in a reasonable accord with the vertical transition moment |μa − X,1| = 0.0025 μB, 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 ka → 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 |μa − X,1| = 0.00376 μB and ka − 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 CbX 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 μB) 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 (re = 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 3Πg states.
Table 1. Quadratic response calculations of the magnetic dipole a − X transition moment (μB), obtained at different intermolecular distances.
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 − a1Δg 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 Qb − 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 (kb − a = 0.00139 s−1)31, 34 is much smaller than that for the spin-forbidden singlet-triplet b → X transition (the kb − 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 (Qzz) provides the nonzero matrix element between the X3Σ and b1Σ states in the form34, 43:
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 (a1Δg) 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 kr = 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 kr = 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 a1Δg − X3Σ transition moment, one has to submit into the integral 〈ϕa|Q̂zz|ϕX,0〉 = Qa − X,0, the perturbed wave function of Eq. (11). Because of the nonzero admixture of the singlet 1Σ state character into the ground triplet ϕX,0 spin-sublevel, Eq. (11), the a − X,0 transition (4 in Fig. 1) can borrow intensity from the b1Σ − a1Δg Noxon band.
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 a1Δg − X3Σ atmospheric oxygen band at 1.27 mm: Qa − X,0 = −CQb − a = 0.018 Debye × Å.31, 34 This corresponds to the a1Δg − X3Σ transition radiative rate constant ka − 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.
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 a1Δg − X3Σ system of oxygen in the absence of electric-quadrupole branches with ΔJ = ±2 can be entirely ascribed to a magnetic dipole radiation with ka − 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: ka − 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 a1Δg − 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 a1Δg − 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:
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 cJ and sJ for the X3Σ state of O2 molecule are well known32, 70 for the intermediate and Hund's (b) coupling cases.
In the Hund's (b) coupling limit, these coefficients are sJ = [(J + 1)/2J + 1)]1/2 and cJ = [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:
where the Coriolis coefficient accounts the triplet–triplet states mixing with the n3Πg terms:
The Cb,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:
The upper state a1Δg,2fe 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 Q2 − 1 = 〈Ψ (a1Σ)|Q2 − 1| Ψ (X3Σ)〉, where Q2k 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 Q2 − 1/Q20 = −2.338. Accounting few reasonable approximations, Balasubramanian et al.71, 103 have shown connections between the fundamental Q20 and Q2 − 1 parameters and the Q1,Q3 moments of the Long's et al.75 theory: Q2 − 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 Q2 − 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 a1Δg − X3Σ (Fi) transitions can be done using these eigenfunctions of rotational spin-sublevels and similar wave function for the upper ϕa(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.
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.
The perpendicular polarization occurs because of intensity borrowing from the A3Σ − 3Πg,1,0 transitions by SOC-induced mixing between A3Σ and 3Πg,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(3PJ) 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.