Abstract
Triplet excited states of the N2 molecule play an important role in electric discharges through air or liquid nitrogen accompanied by various afterglows. In the rarefied upper atmosphere, they produce aurora borealis and participate in other energy-transfer processes connected with atmospheric photochemistry and nightglow. In this work, we present spin–orbit coupling calculations of the intensity of various forbidden transitions, including the prediction of the electric dipole transition moment of the new band, which is strongly prohibited by the (+|−) selection rule, the new spin-induced magnetic transition, magnetic and electric quadrupole transitions for the B3Πg Wilkinson band, and the Lyman–Birge–Hopfield a1Πg ← X1Σg transition. Also, two other far-UV singlet–singlet quadrupole transitions are calculated for the first time, namely, the Dressler–Lutz a"1Σg+–X1Σg+ and the less studied z1Δg–X1Σg+ weak transitions.
Introduction
The great flux of solar energy through the upper atmosphere can be harvested by the rarefied gases of molecular and atomic components of the Earth’s mesosphere and lower thermosphere (MLT) regions (). The ground states of such abundant O2 (3Σg–), O (3P), and N (4S) species of MLT possess high multiplicity, and thus their lowest excited states are metastable, having a low electronic spin and strongly forbidden radiative relaxation (; ; ). Their long-lived emission to the ground state provides the possibility to harvest visible and near-UV solar radiation engaged in various energy transfer processes, which determine the climate, meteorology, and weather conditions (). In contrast, the ground state of the nitrogen molecule possesses zero spin and several high-energy triplet excited states with deep potential wells. The lowest of them,, can harvest a stock of 6.22 eV energy, being a strongly metastable triplet state with a relatively long radiative lifetime (τr) of 2 s (; ; ). Accounting for the short UV wavelength of the transition, this τr value is indeed unusually large.
N2 is a very stable and inert molecule in the ground state with high dissociation energy (De = 9.76 eV). At the same time, N2 possesses a variety of quite stable valence excitations of the πu–πg and 3σg–πg types; these excited states have large De values (around 4–6 eV) and are mostly metastable since their emission to the ground state is strictly forbidden by the electric dipole selection rules (; ; ; ; ; ). In gaseous electric discharges, when a molecule is irradiated by an electron flux, N2 dissociates into the ground state N (4S) atoms; they can recombine forming the lowest singlet (X1Σg+), triplet (A3Σu+), and quintet (A′5Σg–) basic states. The last two, shown in Figure 1, are involved in the so-called active nitrogen phenomenon detected by the characteristic “yellow afterglow” (; ). Its study together with aurora borealis involves a large number of metastable states and forbidden transitions in the N2 spectrum (Figure 2). The Lewis–Rayleigh afterglow () in the discharge consists of the first positive system of the nitrogen molecule, extending from IR to the blue edge, being the triplet–triplet B3Πg → A3Σu+ transition (1+ system) (). The visible part of the 1+ system was already investigated in 1902 by ); ab initio interpretation of its intensity was achieved by ) and a final form by ). It should be distinguished from the second positive system of the nitrogen molecule—the C3Πu → A3Σu+ transition (2+ system) and the infrared Hermann (HIR) band C″5Πu → A′5Σg+ (Figure 1). The main sources of emission of the first and second positive systems in N2 discharge are connected with the involvement of the N (2D) excited atom into a recombination reaction (Figure 1). The 2+ band system was observed as early as 1869 as it readily appears in ordinary air discharges (), but its rovibronic assignment came much later (). As opposed to the O2 molecule (), many visible and UV transitions between triplet excited states generated by electric discharge are possible in the nitrogen counterpart (; ; ). The quintet state A′5Σg+ and the HIR system of N2 have become clear only recently (; ). They are essentially important for the recombination of the N (4S) ground state atoms being the precursor of the Lewis–Rayleigh afterglow. The quintet A′5Σg+ can predissociate to the B3Πg state vibrational levels (v = 10–12, Figure 1), though the spin–orbit coupling (SOC) matrix element (ME) <A′5Σg+|Hso| B3Πg> is rather weak near the crossing in order to be efficient for generation of the spontaneous 1+ emission in the recombination of N (4S) atoms. At the same time, this SOC ME determines the high radiative probability (Einstein coefficient about 3∙104 s−1) of the newly predicted A′5Σg+ → A3Σu+ (0–6) transition, which borrows intensity from the 1+ system, as well as from the HIR band (). The latter source is attributed to a strong SOC between the A3Σu+ and C″5Πu states.
FIGURE 1
FIGURE 2
The excited metastable N (2D) and N (2P) atoms with energies of 2.4 eV and 3.6 eV above the N (4S) ground state, respectively (Figure 1), are present with low concentration in the discharge. Their recombination leads to a huge number of excited N2 states with varying degrees of stability and spontaneous emission probabilities (). Several other important states of nitrogen are shown in Figure 2.
Energy harvesting by triplet states of nitrogen
The triplet excited manifold of the N2 molecule is well studied in far-UV absorption and emission spectra (; ; ; ; ; ; ; ; ; ). In 1932, Vegard detected 120 weak bands in the red-degraded phosphorescence of solid nitrogen through the wide region of 670–170 nm (). Soon after, Kaplan observed similar bands in an N2 laboratory discharge (). The weak Vegard–Kaplan (VK) system was first detected by Wilkinson as absorption bands in a long-path spectrometer at 169 and 128 nm for highly excited vibronic levels (v’ = 6,7) (). Later on, the VK rovibronic intensity alternations were measured and analyzed very carefully (; ) including ab initio calculations for the VK transition probability and many other inter-combination systems (). SOC calculations within the quadratic response theory () explained why the Ogawa–Tanaka–Wilkinson system is much more intense (70 times) than the Vegard–Kaplan absorption and why the Tanaka transition is the most intense among all known triplet–singlet (T–S) absorption bands at that time (; ). The new T ← S transition in the far-UV region predicted by ) was later detected and analyzed by ). The upper state has been observed earlier in the pure nitrogen condensed discharge afterglow through the →B3Πg (0, v’‘) emission, which is now known as the fourth positive system (; ). The upper state was shown to be of Rydberg type () converging to the ground state N2+ ion. At longer N–N distances, it avoids crossing with the bound Rydberg state and the valence 33Σu+ state potential energy curve (PEC), demonstrating a repulsive character (). All theoretical predictions of the inter-combination D ← X transition () have mainly been supported by later experiments (). The predicted 0–0 transition is rather intense (f = 2×10−5), being the strongest inter-combination of a nitrogen molecule in agreement with measurements (; ). This far-UV region in N2 absorption is very dense, being covered by allowed transitions (b1Πu–X, for example) (), but the D–X (0, 0) band has a clear location in a fortuitous region of the b1Πu ← X allowed spectrum, just above its (4,0) band head, enabling the D–X (0, 0) observation (). All three sublevels of the triplet D state provide four rotational branches in agreement with ), according to rotational and parity selection rules of Hund’s case “b” (). The small negative zero-field splitting (λ = −0.036 cm−1 ()) of the D state is in agreement with SOC and spin–spin coupling calculations (λ = −0.041 cm−1) within the response approach (; ; ; ).
Thus, almost all important singlet–triplet transitions in the molecular nitrogen absorption spectra (up to the far-UV region) from the ground state to the triplet states of the “ungerade” symmetry—the A, W3Δu, C3Πu, and D states—have been calculated by the quadratic response theory within the multi-configuration approach (), giving results that are in good agreement with experimental intensity distributions (; ; ). The present work aims to calculate new forbidden transitions in the nitrogen spectra which have not been observed so far but can influence the triplet state harvesting and total kinetic balance of the upper atmosphere.
The B3Πg state produced by the second and fourth positive systems () can further generate 1+ bands, and the lowest triplet state by the cascade in the positive column of electric discharge. We have to note that the B3Πg → phosphorescence was not calculated in ), since even an account of SOC cannot overcome its parity prohibition in terms of electric dipole selection rules. The calculation of this transition intensity is an aim of the present work.
The VK transition satisfies the orbital electric dipole selection rule (EDSR) (), but being spin-forbidden it cannot be effectively induced by direct UV absorption. Thus, the N2 (A) state is primarily populated by collisions—in laboratory discharge and the upper atmosphere, this is accomplished through the electron impact and the cascade in the first positive system. The relatively long radiative lifetime enables N2 (A) to participate in collisions with the main background gases of the MLT region and to produce chemical reactions with N2, O2, N, and O species. In particular, the reactions are the most important ones (). A recent steady-state MLT model developed for the N2 (A) vibrational distribution in the terrestrial atmosphere is supported by comparison with the Vegard–Kaplan dayglow emission from atmospheric photochemistry and ionospheric spectroscopy measurements (). The steady-state N2 (A) vibrational distribution in the MLT region is found to be shifted to higher (v > 6) levels. This is in agreement with the VK absorption () and is important for our study. Direct excitation from the ground N2(X) state by the electron impact provides an essential contribution to populating the N2 (A, v > 6) sublevels, though their dominant excitation mechanism is the radiative cascade via the 1+ system (; ; ). The efficiency of this cascade depends on the B3Πg transition intensity, which in turn is determined by the EDSR-forbidden a1Πg magnetic-dipole-allowed band system. Intensity calculations of these strongly forbidden transitions are also the purpose of our work.
The N2 molecule, the most common and abundant component of the air, plays a crucial role in many high-energy photochemical processes caused by solar radiation in the upper atmosphere (; ). The discovery of new N2 transitions forbidden by the spin-selection rule and induced by SOC perturbation is an important part of optical nitrogen monitoring at different altitudes. The intensity origin of the known emission bands that are forbidden by the electric dipole selection rules is also an important task of N2 spectroscopy (; ; ; ; ; ; ; ; ; ; ; ; ). This work presents multi-reference configuration interaction (MRCI) calculations of the highly excited states of the nitrogen molecule and an explanation of the intensity origin of several forbidden optical transitions. With this aim and background, we have predicted the electric dipole transition moment (EDTM) of the unknown forbidden transition and calculated its dependence on the internuclear distance. This is a triplet–triplet (T–T) band, the intensity of which is entirely determined by spin–orbit coupling perturbations between various spin sublevels of the T states as was preliminarily shown in a recent work (). The upper state was earlier calculated by similar MRCI methods (; ), but no experimental manifestations of its existence have been evidenced so far, although the state is predicted with a deep minimum (De = 1.23 eV) and high energy above the ground state (Te = 12.15 eV) (). We believe that the state can be produced by N (2P) + N (4S) recombination (Figure 1), and that its low vibrational levels can avoid pre-dissociation at low pressure. The N2 state possesses a potential energy well located outside the Franck–Condon (FC) region, which is accessible from the metastable state as well as from the ground state. This explains the difficulties with the observation of the corresponding absorption bands. Under these conditions, the emissive transition from the lowest v’ = 0 sublevel will have the maximum FC factor for the v” = 7–8 vibronic levels of the A state. We provide evidence for the existence of this new band in the N2 molecule by calculating the transition probabilities through an account of SOC in the first order of the perturbation theory and comparing them with other known forbidden transitions to facilitate the validity of such a prediction. This would be a wide band of low intensity in the range of 209–450 nm with an approximate maximum at 328 nm; it is prohibited by the severe selection rule (+) → (–) but is allowed by spin-selection as a T–T transition (). Its spin-rovibronic structure would be analogous to the well-known Herzberg I band of molecular oxygen (; ; ).
Intensity borrowing mechanisms of the forbidden transition
For planning intensity calculations of the new band in nitrogen, we first take into account the corresponding well-known and intense transitions of the N2 molecule, relevant for our purpose. According to SOC selection rules, the new N2 band can be formed by spin–orbit coupling-induced mixing of the upper state with the state and by intensity borrowing from the first positive system , see Eq. 1. To include the SOC effect, we have to add the Ω quantization, where Ω = Lz + Sz is the z-projection of the total electronic angular momentum and Lz and Sz are orbital and spin angular momenta projections on the molecular axis (). The SOC operator can mix states with the same Ω; the EDTM selection allows transitions according to the rule ΔΩ = 0, ±1:
Figure 3 presents this mechanism as the type “I SOC” mixing. By a similar SOC mechanism, the studied forbidden band can borrow EDTM intensity from the newly predicted transition here, Eq. 2; this SOC mechanism of intensity borrowing refers to the “II SOC” type in Figure 3. Both these mechanisms have the perpendicular x, y polarization of EDTM; only the x component is shown in Eqs. 1–2 for one sublevel of the degenerate Π states (; ).
FIGURE 3
Figure 3 provides a good explanation of the relevant intensity sources of the studied transition, but it would be overloaded if all possible contributions are included. The type “II SOC” mechanism in Figure 3 includes also other states of the C′3Πu type (in total five 3Πu states are taken into account).
An additional source of intensity borrowing denoted as the type “III SOC” mechanism in Figure 3 includes parallel EDTM for the studied emission band (light polarization along the molecular z-axis). By symmetry arguments, the transition in N2 is similar to the Herzberg I band of the O2 molecule, and its probability can be calculated by a similar scheme of intensity borrowing (; ). In the oxygen molecule, the main contribution to the absorption intensity of the Herzberg I band origins from the SOC mixing between the state and the upper term of the Schumann–Runge system (), which is the most intense valence transition in molecular oxygen (). This provides a rather unusual (for the 3Σ−–3Σ+ band) type of Ω = 1–Ω = 1 parallel transition intensity, though the ΔΩ = 1 selection rule is more typical for such bands with prevailing perpendicular polarization ().
Let us consider the type “III SOC” mechanism in more detail. The SOC-induced mixing between the lowest state and the upper triplet of the Ogawa–Tanaka–Wilkinson system ( can be presented by the perturbation theory in the form:
We can also account for SOC perturbation for the counterpart as follows:
The EDTM between the perturbed states (3) and (4) is equal to
This means that the (+|−) forbidden transition can borrow intensity from ED-allowed and transitions. The latter contribution is a formal symmetry analog of the Schumann–Runge O2 transition. The SOC mixing mechanism shown in Eq. 3 is presented in Figure 3 by the intensity borrowing scheme “III-SOC”. The SOC-induced mechanism from Eq. 4 is not shown in Figure 3 to avoid overloading. The SOC matrix element (ME) in Eq. 3 is equal to zero in a semi-empirical approximation with the neglect of differential overlap:where is the SOC constant for the valence shell of the A atom and is a scalar product of the orbital and spin operators for the ith electron (; ). For the pure main configurations of the and states, the SOC ME is equal to ; this expression is zero with the neglect of differential overlap since (), but the account of overlap in normalization of the and molecular orbitals in the r-centroid approach (1.282 Å) leads to the different estimations = 85 cm−1and = 60 cm−1. Thus, the SOC ME in Eq. 3 reaches a non-zero value of 12.5 cm−1, which is rather close to the MRCI result. This scrutinized analysis shows the importance of the contribution expressed by Eq. 3 and the analogy with the Herzberg I Schumann–Runge transition coupling in the O2 molecule (). The denominator in Eq. 3 is rather small and homogeneously changes with r distance (Figure 3). Although the transition is relatively weak in the N2 molecule (EDTM = 0.026 ea0 at r = 1.4 Å) (), its contribution to the final EDTM of Eqs 1–5 is the largest. The EDTM of the transition (the Herman–Kaplan band system ()) has a smaller value (0.017 and 0.0105 ea0 at r = 1.28 and 1.4 Å, respectively) (), as well as the SOC ME in Eq. 4 at these distances (5.2 cm−1) ().
We have stressed before the EDTM component of the studied intensity borrowing from the band, which is a formal analog of the Schumann–Runge system of oxygen (). Thus, we can compare various contributions to the intensity of this so-far unknown transition with the well-known data for O2 and N2 spectra (; ; ; ). The intensity borrowing contribution from the first positive system in Eq. 1 can be compared with the Vegard–Kaplan S–T transition intensity presented in Eq. 7 (), which explains an extremely low spontaneous emission of the VK system.
As shown in Figure 2, the two denominators in Eq. 7 have opposite signs. The first denominator E(B)–E(X) decreases with r distance prolongation, whereas the second one, E(A)–E(b), increases by an absolute value with r. In the vicinity of the ground state equilibrium re distance (1.098 Å), both contributions tend to cancel each other, and the EDTM value crosses the zero point (). In the whole FC region, the EDTM is still close to zero, and the VK system has very low intensity both in absorption and emission. Although both the SOC ME values in the nominators of Eq. 7 are rather large (; ) as well as the transition moments of the 1+ and systems (), the cancellation of the two big terms in Eq. 7 is the only reason for the relatively large lifetime of the N2 state. To a large extent, this is also the reason for the efficient solar energy harvesting by the triplet states of nitrogen molecules and the aurora borealis phenomena.
For the studied transition , only the first “I SOC” mechanism provides an essential sign change with the internuclear distance (Figure 4). In the FC region, no big cancellations of different sign contributions are shown. The deteriorating “I SOC” mechanism is rather weak in the FC region 1.28–1.62 Å. For the most intense 0–7 vibronic band, the calculated EDTM is equal to 1.41×10−4 ea0, which corresponds to the radiative rate constant of 2.48 s−1. The total radiative lifetime of the zero vibrational sublevel of the state is estimated as 0.34 s.
FIGURE 4
The state can degrade much faster in the allowed T–T transitions (for example, through the emission). Thus, our estimation of the emissive transition is definitely negative. However, in absorption, the same transition can be observable since the calculated oscillator strength (f7–0 = 2.23*10−9) can be measured by modern techniques.
It is, at this point, relevant to estimate the other EDSR-forbidden inter-combination B3Πg → X1Σg+ transition of nitrogen (Wilkinson system) (), which so far has not been calculated by quantum chemical methods. This is a magnetic dipole transition that borrows intensity from the magnetic singlet–singlet counterpart a1Πg→X1Σg+ ().
Calculations of magnetic and electric quadrupole transition intensity
The Lyman–Birge–Hopfield (LBH) band system (a1Πg→X1Σg+) of the N2 molecule has been carefully studied in measurements of cascade-induced UV radiation to determine the intensity of this emission (). The LBH band has readily been seen in absorption as well as in emission though it is EDSR-forbidden by parity selection. Its magnetic and quadrupole transition moments are provided in Figure 5. They are calculated here at the level of the time-dependent density functional theory (TD DFT) using the B3LYP functional and 6-311G++(d, p) basis set with the Gaussian-09 package (). We have studied 40 singlet states and triplet excited states of N2 in the region 0.8–1.8 Å of the r distances. For the longer N–N bonds, the TD DFT approach produces untrustworthy PECs and cannot reproduce the proper dissociation limits. But for short r distances, all potential energy curves are quite reasonable and qualitatively reproduce MRCI results (; ). This DFT method provides equilibrium bond lengths of 1.205 and 1.598 Å for the triplet (B3Πg) and quintet (A′5Σg+) states of nitrogen, respectively. The latter is more realistic (), whereas the former re value deviates slightly from the experimental value of 1.213 Å ().
FIGURE 5
A similar approach has been successfully used for the permanent quadrupole moment calculations in N2 (). In addition to the LBH system, some other EDSR-forbidden bands are also calculated as quadrupole transitions, as shown in Figure 6. The Dressler–Lutz a"1Σg+–X1Σg+ quadrupole transition at 101 nm as well as the far-UV transition z1Δg–X1Σg+ (Figure 6) are calculated for the first time.
FIGURE 6
The growth of magnetic strength of the a1Πg→X1Σg+ transition (Figure 5A) and the decrease of its quadrupole moment are notable (Figure 5B). The X1Σg+ transition moment represents a complicated tensor with r-dependent anisotropy (Figure 6).
In the FC region (1.1–1.3 Å), our results in Figure 5 well coincide with the calculations of ) using the random phase approximation (RPA). The magnetic dipole transition moment (MDTM) of the LBH system (Figure 5A) increases with r, showing a trend of saturation at r = 1.3 Å, whereas the electric quadrupole transition moment (EQTM) decreases along the whole r range. Accounting for experimental FC factors and transition frequencies, we have obtained the radiative lifetime for the 0–0 vibronic transition of the LBH system equal to 65 μs in a reasonable agreement with experimental values in the interval 80–120 μs (; ). The calculated magnetic to quadrupole intensity ratio (m/eq) is equal to 92%, whereas experimental data are in the range of 67%–96% interval (). Emission from the higher vibrational levels has a lower probability of qualitative agreement with observations (; ). At the same time, we cannot accept the idea that the a1Πg state can decay solely into the X1Σg+ ground state (). From Figure 2, one can see that the infrared a1Πg→a’1Σu− emission is possible; its electric dipole transition moment is equal to 0.2 ea0 () using the r-centroid approach corresponding to the radiative lifetime for the 0–0 band of τr = 9 ms (FC factor is 0.219). We have also estimated a new quadrupole transition a1Πg→B3Πg,1. Accounting for SOC, in Eq. 8, this transition moment origins in the difference in the permanent quadrupole moments of these two states: Q (B3Πg) = 0.59 ea02 and Q (a1Πg) = 0.48 ea02. This difference is small as well as the quadrupole moment of transition a→B (4.9*10–4 ea02), but in principle, we could not disregard branching emission into other lower lying triplet states (B′, W, and A) in the calculation of the radiative lifetime of the LBH system. These S–T transitions are allowed in the EDSR approach with an account of spin–orbit coupling perturbation. Thus, we consider it more appropriate to present also the oscillator strength for the Lyman–Birge–Hopfield 0–0 band a1Πg ← X1Σg+ in absorption: f0–0 = 7.24×10−6.
The Dressler–Lutz a"1Σg+–X1Σg+ quadrupole transition in the far-UV absorption region (101 nm) is of the Rydberg type (); it is well reproduced by our TD DFT calculations. The triplet counterpart of the a"1Σg+ state is the known E3Σg+ Rydberg term, which was discussed previously when presenting our calculations of the Herman–Kaplan system transition). The Dressler–Lutz a"1Σg+←X1Σg+ band was observed in absorption at high pressure, and its intensity is mainly induced by collisions (). In this aspect, it is similar to the quadrupole Noxon band of O2, which is very sensitive to collision-induced intensity enhancement (). Both 1Σg+ states have similar re distance (about 1.1 Å) and FC factor close to unit. The calculated oscillator strength of the 0–0 band of the quadrupole a"1Σg+ ← X1Σg+ transition in nitrogen is equal to 1.5·10−7, and it is detectable even at low pressure.
Now, we can estimate the probability of the latter triplet–singlet B3Πg←X1Σg+ transition of the nitrogen molecule which, being strictly forbidden by ED selection, has not been included in previous calculations (). This Wilkinson band borrows intensity from the LBH band system (a1Πg←X1Σg+) of the N2 molecule because of the relatively strong spin–orbit couplingat the re distance and small energy gap between the B–a states. Only the Ω = 1 spin sublevel of the triplet B3Πg,1 state is active in the Wilkinson band absorption, and its rotational structure supports the magnetic transition nature (). The SOC of Eq. 8 and m1 magnetic moment (Figure 5A) provide the largest contribution (98.6%) to the B3Πg←X1Σg+ transition intensity. The other k1Πg state (1πu→3σu) shows a smaller magnetic moment for the k1Πg–X transition (m = 0.085 μB) and a much smaller SOC counterpart at the B state equilibrium. Although both parameters increase with r, their relative contributions remain rather small. The calculated magnetic transition moment for the 0–0 band of the Wilkinson absorption B3Πg←X1Σg+ is equal to 0.0073 μB. It corresponds to the oscillator strength f0–0 = 2.54∙10–10, and the magnetic intensity remains dominant for this transition. It is not strange that ) used an optical path as long as 20 m to detect this band.
Finally, we have estimated the spin-induced magnetic dipole moment for a new transition of the N2 molecule. According to Eq. 3, the perturbed A state has a small B’ state admixture for the Ms = ±1 sublevels: . Thus, the transition to the next Ms = 0+ spin sublevel of the state can borrow spin-current intensity from the microwave absorption band with the standard spin-magnetic transition moment that equals 2 μB. For the 0–0 absorption band , we have obtained oscillator strength f = 1.67 . 10–12, which is probably possible for detection.
Conclusion
The presence of nitrogen atoms in the discharge afterglow classifies “active nitrogen” as a free-radical phenomenon. This is relevant to the aurora borealis’ bright light and the yellow–orange Lewis–Rayleigh afterglow in the N2 gas discharge. The spectrum consists of several triplet–triplet emission bands of the 1+ and 2 + nitrogen systems (B3Πg–A3Σu+ and C3Πu–B3Πg transitions) and the →B3Πg infrared-visible afterglow system. The wide Wu–Benesh system B3Πg = W3Δu is another T–T transition of the afterglow (). One can see that many triplet states of the nitrogen molecule take part in discharge afterglow together with numerous T–S transitions and S–S cascades. The transitions allowed by the electric dipole selection rule are nowadays accurately calculated by sophisticated ab initio methods () including many T–S vibronic bands induced by SOC perturbation (). This is important for the kinetic balance of triplet harvesting in discharges and the Earth’s mesosphere and lower thermosphere regions. In the present work, we have calculated the probability of the magnetic and quadrupole Lyman–Berge–Hopfield transition a1ΠgX1Σg+, which is necessary for the intensity estimation of the Wilkinson B3ΠgX1Σg+ band (the only unknown intensity of a pure electronic T–S transition at zero pressure).
We have also calculated new transitions, and , that can be observed during absorption. The reason for finding such transitions is that the first excited triplet state of N2 possesses a relatively long radiative lifetime (about 2 s). Therefore, it is possible to excite the triplet–triplet transition from the state by two-photon experiments or other methods of flash photolysis in discharge. We know that the Herzberg I transition was discovered in the oxygen molecule as an excitation from the ground state , but in nitrogen, the situation is reversed since the symmetry corresponds to the upper state.
The state, non-observed so far, has an electronic wave function, which is mainly represented by the valence configuration (1πu)2 (1πg)2 in a form similar to a quintet state. The quintet–triplet – transition, also induced by SOC in the electric dipole approach, is the most intense among all studied intercombinations. The spin-induced transition in the visible region is interesting since it is rather unique in magnetic-origin borrowing intensity from the electron spin resonance in the state. The transition intensity could be sensitive to the external magnetic field in solid nitrogen. The band in N2 has common features with the visible A-band of molecular oxygen (; ).
Thus, we have noted many important comparable features in N2 and O2 spectra and also calculated for the first time the intensity of the predicted forbidden transitions including some magnetic dipole and quadruple S–S transitions in the nitrogen molecule. The main new predicted results are summarized in the following table.
| Transition | Absorption | Emission | Wavelength | |
| f0,0 = 5.2 10–12 | f0,7 = 2.2 10–9 | τ = 0.34 s | λ0,7 = 328 nm | |
| f0,0 = 4.9 10–8 | f0,6 = 2.0 10–4 | τ = 8.23 μs | λ0,6 = 598 nm | |
| f0,0 = 1.6 10–12 | Overlapped by 1 + band | τ = 3750 s | λ = 620 nm | |
Statements
Data availability statement
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.
Author contributions
OP: writing small fragments of the text, computer calculations of molecules, development of drawings, and correction of the text. BM: main author of the manuscript, writing most of the text, development of drawings, and selection and processing of literary sources. VM: writing text fragments and text correction. HÅ: writing text fragments and processing computer calculations.
Funding
This work was supported by the Ministry of Science and Education of Ukraine (project 0122U000760) and by the Swedish Wenner-Gren Foundations (project GFU 2022–0036).
Acknowledgments
The authors express gratitude to Ramon S. da Silva and Majdi Hochlaf for useful discussions. Boris Minaev acknowledges a grant from the Wennergren-Foundations through their program for support of international reserach, grant no. GFU2022-0036. The authors thank the Swedish National Infrastructure for Computing (SNIC 2021-3-22 and SNIC 2022-5-103) at the National Supercomputer Centre of Linköping University and High-Performance Computing Center North (Sweden) partially funded by the Swedish Research Council through grant agreement no. 2018-05973.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
triplet–singlet transitions, nitrogen molecule, Vegard–Kaplan band, Wilkinson band, Herzberg I band analog
Citation
Minaev BF, Panchenko OO, Minaeva VA and Ågren H (2022) Triplet state harvesting and search for forbidden transition intensity in the nitrogen molecule. Front. Chem. 10:1005684. doi: 10.3389/fchem.2022.1005684
Received
28 July 2022
Accepted
22 September 2022
Published
18 October 2022
Volume
10 - 2022
Edited by
Piotr Pander, Silesian University of Technology, Poland
Reviewed by
Filippo Tamassia, University of Bologna, Italy
Sergey V. Krasnoshchekov, Lomonosov Moscow State University, Russia
Updates
Copyright
© 2022 Minaev, Panchenko, Minaeva and Ågren.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: O. O Panchenko, panchenko9b@gmail.com
This article was submitted to Inorganic Chemistry, a section of the journal Frontiers in Chemistry
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