Hyperfine coupling of the iodine ${\boldsymbol{D}}{0}_{{\boldsymbol{u}}}^{+}$ and β1_g ion-pair states

Analysis we have carried out before measurements has shown that there are pairs of the β, v_β, J_β and D, v_D, J_D rovibronic levels which can be mixed by the HFI according to selection rules, ∣ΔΩ∣ ≤ 1, ∣ΔJ∣ ≤ 1 for the MD and ∣ΔΩ∣ ≤ 2, ∣ΔJ∣ ≤ 2 for the EQ interactions [6–9], (here Ω is the projection of electron total angular momentum onto the molecular axis, J is a molecular total angular momentum), since accidental resonances between them occur, and FCFs are large enough. Spectroscopic parameters of the β and D states from [25, 26] were used in calculations. Rovibronic levels that could be used in the experiments are the following (see figures SD1–3 in supplementary data is available online at stacks.iop.org/JPB/51/095101/mmedia):

$$\begin{eqnarray*}&&\begin{array}{l}\beta ,13,{J}_{\beta }\approx 17,\,32,\,63\sim D,12,{J}_{D},{J}_{D}={J}_{\beta }+2,\\ \,+\,\mathrm{1,}\,0,{\rm{F}}{\rm{C}}{\rm{F}}=5\times {10}^{-2};\end{array}\\ &&\begin{array}{l}\beta ,28,{J}_{\beta }\approx \mathrm{22,}\,\mathrm{37,}\,\,62\sim D,28,{J}_{D},{J}_{D}={J}_{\beta }+2,\\ \,+\,\mathrm{1,}\,0,{\rm{F}}{\rm{C}}{\rm{F}}=2\times {10}^{-2};\end{array}\\ &&\begin{array}{l}\beta ,47,{J}_{\beta }\approx \mathrm{5,}\,\mathrm{10,}\,27\sim D,48,{J}_{D},{J}_{D}={J}_{\beta }+2,\\ \,+\,1,\,0,{\rm{F}}{\rm{C}}{\rm{F}}=7\times {10}^{-3}.\end{array}\end{eqnarray*}$$

The β state rovibronic levels can be populated in perpendicular transitions utilizing the standard two-step two-color, hν₁ + hν₂, OODR excitation scheme [27]

$$\begin{eqnarray}&&{I}_{2}(\beta {1}_{g},{v}_{\beta },{J}_{\beta }\mathop{\leftarrow }\limits^{h{\nu }_{2}}B{0}_{u}^{+},{v}_{B},{J}_{B}\mathop{\leftarrow }\limits^{h{\nu }_{1}}X{0}_{g}^{+},{v}_{X}=0,{J}_{X}).\end{eqnarray} \tag{ 1 }$$

Experiments have shown that we can find a distinct manifestation of the HFI under study at a low vibronic state, v_β = 13, if the scheme (1) is utilized. For high vibronic states, it was found to be more convenient to populate the D state rovibronic levels optically using the hν₁ + 2hν₂ OODR excitation scheme (see [28] and references):

$$\begin{eqnarray}&&{I}_{2}(D{0}_{u}^{+},{v}_{D},{J}_{D}\mathop{\leftarrow }\limits^{2h{\nu }_{2}}B{0}_{u}^{+},{v}_{B},{J}_{B}\mathop{\leftarrow }\limits^{h{\nu }_{1}}X{0}_{g}^{+},{v}_{X}=0,{J}_{X}).\end{eqnarray} \tag{ 2 }$$

In the present work, we have studied the β, 13, J_β ∼ D, 12, J_D and β, 47, J_β ∼ D, 48, J_D pairs, only, to understand, if HFI depends on a vibrational quantum number of the states. The B, v_B = 19 and 20 vibronic levels were used as intermediate for an optical population of the β, 13 and D, 48 vibronic states, respectively.

A detailed description of experimental apparatus has been given elsewhere (see [4, 27–29] and references). Briefly, the Quantel system consisting of two tunable dye TDL90 lasers pumped with a pulsed YG981C Nd:YAG laser has been utilized.

The first TDL laser operated on the B–X transition (hν₁), while hν₂ laser radiation was produced by mixing of the second TDL laser fundamental output (ν₂') and fundamental harmonic of the YG981C laser (${{\rm{\nu }}}_{f}^{i}$) in a KDP crystal, ${\nu }_{2}={\nu }_{2}^{\prime} +{{\rm{\nu }}}_{f}^{i}$ [27]. The fundamental YG981C laser harmonic consists of four spectral components [29]. Therefore, the hν₂ laser radiation also consists of four spectral components after mixing. In our experiments, two most intensive components corresponding to the mixing ${{\rm{\nu }}}_{f}^{1}$ = 9395.12 ± 0.02 cm⁻¹ ('main' component) and ${{\rm{\nu }}}_{f}^{2}$ = 9393.53 ± 0.07 cm⁻¹ ('satellite' component) were observed. The spectral width of each ν₂ generation mode was found to be FWHM ≈ 0.13 cm⁻¹ (see figure SD4).

The ν₁ and ν₂ values were determined using the WS6 wavelength meter (Ångstrom). The counter-propagating, temporally overlapped laser beams were directed through the chamber filled with iodine vapor at a fixed pressure, usually ∼100 mTorr. One can expect to observe no collision-induced non-adiabatic transitions (CINATs) at such pressure [27].

If β, v_β, J_β and D, v_D, J_D rovibronic levels are coupled by HFI, mixed states (see section 2.2) can be populated using scheme (1) or (2). Therefore, luminescence from D as well as from β state should be observed simultaneously. To find such coupled rovibronic levels we measured excitation spectra (scanning λ₂ at fixed λ₂ and λ_lum) of the D → X, λ_lum ≈ 2830 Å, and β → A λ_lum ≈ 3400 Å, luminescence. However, one should bear in mind that E → A, C transitions luminescence also occurs near λ_lum ≈ 3400 Å (see [30] and references). Cross-sections of the β ← B transitions are approximately 30 times less than those of the E ← B in I₂ (see [25] and references), and the D, v_D, J_D rovibronic states can be populated in the E, v_E, J_E $\mathop{\to }\limits^{{I}_{2}(X)}$ D, v_D, J_D CINATs (see [27] for details). To be sure that the D rovibronic levels under study were not populated in the CINATs, temporal behaviors of the D → X, E → B and β → A luminescence intensities were measured (figure SD5). It was found, that all the temporal behaviors corresponded to an optical population of the D and β state rovibronic levels; no features of the E $\mathop{\to }\limits^{{I}_{2}(X)}$ D, β CINATs were found (see details in [29]).

The total parity + ↔ — selection rule is valid for the β1_g—$B{0}_{u}^{+}$ optical transition and only one of the Ω doubling component can be populated. So, even J_β populated from odd J_B (−) is not coupled with J_D = J_β and, similarly, there is no HFI between even J_D = J_β + 1 and odd J_β if an optical transition from even J_B (+) rovibronic state occurs (figure 1). Therefore, one has to populate J_β rovibronic states from even and odd J_B states to observe HFI of even and odd rovibronic states under study.

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In general, to describe HFI between two rovibronic states correctly, one should take into account their hyperfine structures. In our experiments, the resolution of measured spectra is limited by the laser bandwidth, so we utilized a two-state perturbation model discussed in [10] (see [23], also) to estimate matrix elements averaged over hyperfine components.

In the case of two interacting rovibronic states Φ and Ψ, the mixed ∣1〉 and ∣2〉 states are the following:

$$\begin{eqnarray}&&| \,1\rangle =\,\cos \,\theta | {\rm{\Phi }}\rangle \,+\,\sin \,\theta | \,{\rm{\Psi }}\,\rangle \,;\\ &&| 2\rangle =\,-\,\sin \,\theta | {\rm{\Phi }}\,\rangle \,+\,\cos \,\theta \,| \,\,{\rm{\Psi }}\,\rangle \,,\end{eqnarray} \tag{ 3 }$$

where the mixing angle θ is given as

$$\begin{eqnarray}&&\tan \,2\theta =\displaystyle \frac{2{H}_{hf}}{{\rm{\Delta }}E},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{H}_{hf}=\sqrt{{\rm{F}}{\rm{C}}{\rm{F}}}\langle {\rm{\Phi }}| {\hat{H}}_{hf}^{el}| \,{\rm{\Psi }}\rangle .\end{eqnarray} \tag{ 5 }$$

${\hat{H}}_{hf}^{el}$ is the operator of the HFI between the Φ and Ψ states,

ΔE is an energy gap between unperturbed rovibronic levels of the Φ and Ψ states under study (see figure 2).

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The ∣1〉 and ∣2〉 wave functions have small admixtures of the interacting states. In our case, for the most pairs of D and β state rovibronic levels given in tables 1, 2 (see section 3), transitions to both mixed states occur within the laser bandwidth (∼0.13 cm⁻¹), and the ratio of the ∣1〉 and ∣2〉 state excitation rates is equal to cos² θ/sin² θ [23]. No appearance of saturation effects in absorption in a wide range of the hν₂ radiation intensity has been observed in experiments, discussed in the paper.

The Φ → i emission spectrum, observed in the experiment, corresponds to Φ-components of both mixed states. The Φ → i luminescence intensity can be expressed as follows:

$$\begin{eqnarray}&&I({\rm{\Phi }}\to i)\sim ({\nu }_{| 1\rangle i}^{3}{\cos }^{4}\,\theta \,+{\nu }_{| 2\rangle i}^{3}{\sin }^{4}\theta ){\langle {\rm{\Phi }}| {\hat{M}}_{{\rm{\Phi }}\to i}| i\rangle }^{2},\end{eqnarray} \tag{ 6 }$$

The Ψ → j luminescence intensity is equal to

$$\begin{eqnarray}&&I({\rm{\Psi }}\to j)\sim ({\nu }_{| 1\rangle j}^{3}{\cos }^{2}\theta \,{\sin }^{2}\theta +{\nu }_{| 2\rangle j}^{3}{\cos }^{2}\theta \,{\sin }^{2}\theta )\\ &&\,\times {\langle {\rm{\Psi }}| {\hat{M}}_{{\rm{\Psi }}\to j}| j\rangle }^{2},\end{eqnarray} \tag{ 7 }$$

$\hat{{\boldsymbol{M}}}$ are dipole moment operators of the transitions under study.

The ratio of the total emission intensities for the transitions in the case of an optical population of the Φ state is:

$$\begin{eqnarray}\displaystyle \frac{I({\rm{\Psi }}\to j)}{I({\rm{\Phi }}\to i)} & = & \displaystyle \frac{2{{\rm{\cos }}}^{2}\theta {{\rm{\sin }}}^{2}\theta }{{{\rm{\cos }}}^{4}\theta +{{\rm{\sin }}}^{4}\theta }\displaystyle \frac{\displaystyle \int {v}^{3}{M}_{{\rm{\Psi }}\to j}^{2}{\rm{d}}v}{\displaystyle \int {v}^{3}{M}_{{\rm{\Phi }}\to i}^{2}{\rm{d}}v}\\ & = & \displaystyle \frac{2{{\rm{\tan }}}^{2}\theta }{1+{{\rm{\tan }}}^{4}\theta }\displaystyle \frac{{A}_{{\rm{\Psi }}\to j}}{{A}_{{\rm{\Phi }}\to i}}.\end{eqnarray} \tag{ 8 }$$

Hence, an equation for the intensity ratio is the following:

$$\begin{eqnarray}&&{R}_{{\rm{\Psi }}\to j/{\rm{\Phi }}\to i}=\displaystyle \frac{1}{1+\displaystyle \frac{{\rm{\Delta }}{E}^{2}}{2\,{H}_{hf}^{2}}}\cdot \displaystyle \frac{{A}_{{\rm{\Psi }}\to j}}{{A}_{{\rm{\Phi }}\to i}}.\end{eqnarray} \tag{ 9 }$$

Therefore, one can estimate matrix element H_hf from measurements of relative intensities of luminescence from both coupled states.

In fact, for a molecule composed of identical atoms, an additional factor K should be added to equation (9). The origin of this factor is the following: due to Pauli Exclusion Principle for two identical nuclei, the symmetry of a molecular term is related to the parity of the total nuclear spin I [20, 31]. For the u negative and g positive terms, the I is even. For the u positive and g negative terms, the I is odd. The sign of the rotational components of the $D{0}_{u}^{+}$ state is determined by the number (−1)^J. The iodine nuclear spin is I = 2.5. Hence, the rotational levels with even J_D have odd I = 1, 3, 5 and with odd J_D have even I = 0, 2, 4. A total number of the spin states with even I is 15 and with odd I is 21. The hyperfine Hamiltonian is of even parity and therefore conserves sign. Thus, any u ∼ g hyperfine coupling implies an odd nuclear spin change [8]. Since ΔI = 0 coupling is forbidden, it always corresponds to a change in nuclear spin orientation.

In the excitation scheme (1), the β1_g state levels are populated. In the case of odd J_D, the ratio of the degrees g_D, g_β of the nuclear spin degeneracy of the $D{0}_{u}^{+}$ and β1_g states is g_D/g_β = 15/21. From 21 β1_g spin states, one can construct 15 'bright' states which interact with 15 $D{0}_{u}^{+}$ spin states and six 'dark' states which do not interact with the $D{0}_{u}^{+}$ state. The population of the dark states does not lead to the D → X transition resulting in K = g_D/g_β = 15/21. In the case of even J_D, a number of the β1_g spin states is less than a number of the $D{0}_{u}^{+}$ spin states, each β1_g spin state interacts with those of $D{0}_{u}^{+},$ and a population of every β1_g spin state leads to both β → A and D → X transitions which results in K = 1. The same arguments can be applied to excitation scheme (2).

So, we obtain the following equation for the intensity ratio after population in the scheme (1):

$$\begin{eqnarray}&&{R}_{D-X/\beta -A}=\displaystyle \frac{1}{1+\displaystyle \frac{{\rm{\Delta }}{E}^{2}}{2\,{H}_{hf}^{2}}}\cdot \displaystyle \frac{{A}_{D\to X}}{{A}_{\beta \to A}}\cdot K({J}_{D}),\end{eqnarray} \tag{ 10a }$$

where K = 15/21 for odd J_D and K = 1 for even J_D. A similar equation can be derived for the R_β-A/D-X ratio in the scheme (2):

$$\begin{eqnarray}&&{R}_{\beta -A/D-X}=\displaystyle \frac{1}{1+\displaystyle \frac{{\rm{\Delta }}{E}^{2}}{2\,{H}_{hf}^{2}}}\cdot \displaystyle \frac{{A}_{\beta \to A}}{{A}_{D\to X}}\cdot K({J}_{D}),\end{eqnarray} \tag{ 10b }$$

where K = 15/21 for even J_D and K = 1 for odd J_D.

Einstein coefficients of the transitions calculated using temporal histories (see details in [32]) and branching ratios for the D → X, $a^{\prime} {0}_{g}^{+},$ ${0}_{g}^{+}$(ab) and β → A, C, 1 _u(ab) transitions obtained in [32, 33] were found to be A_D→X = 6.9 × 10⁷ s⁻¹, A_β→A = 8.1 × 10⁷ s⁻¹ for the β1_g, 13 ∼ D, 12 states and A_D→X = 7.4 × 10⁷ s⁻¹, A_β→A = 7.8 × 10⁷ s⁻¹ for D, 48 ∼ β, 47 pairs.

To determine ratios of integrated intensities of the D → X and β → A luminescence (see section 3) we measured luminescence spectra in the λ_lum ≈ 2400–4500 Å range at the excitation lines corresponding to a simultaneous population of the D, v_D, J_D and β, v_β, J_β rovibronic states. Uncertainties of the total D → X, β → A luminescence intensities depend mainly on those of spectra simulation (see [4, 5, 27–29] for details).

The spectroscopic constants and/or potential curves from the following sources: X state [34, 35], A state [36], B state [34, 37], β state [25], D state [26] were used for the experimental data analysis.

Excitation scheme (1) was used for study of the β, v_β = 13, J_β ∼ D, v_D = 12, J_D coupling in vicinities of the J_β ≈ 17, 32 and 63 rovibronic levels. To determine, at which rovibronic levels hyperfine coupling occurs, we measured excitation spectra of the D → X and β → A luminescence at each intermediate B, v_B, J_B rovibrational levels used. Besides, excitation spectra of the E → B, λ_lum ≈ 4300 Å, luminescence were measured to exclude possible admixture of E → A, C luminescence to the β → A one [30]. One of the spectra is given in figure 3.

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It has been found no features of the coupling in a vicinity of the J_β ≈ 17 levels, where levels with J_D = J_β + 2 are expected to be coupled, i.e. no EQ hyperfine coupling between the I₂(β, v_β = 13 and D, v_D = 12) vibronic states has been observed. The MD coupling has been observed in vicinities of the J_β ≈ 32 and 63 levels, namely, at the J_β = 32–36, J_D = J_β + 1 and J_β = 61–66, J_D = J_β levels (see figure 3 as an example).

The R_D-X/β-A luminescence intensity ratios at each coupled β, 13, J_β ∼ D, 12, J_D rovibronic state have been determined using luminescence spectra (see figure 4, as an example). The R_D-X/β-A values and since coupling are the largest at minimal energy gaps between coupling states. We found the largest coupling at the J_β = 34 and 64 (table 1), ∼ (2–3) rotational levels higher than it was predicted by calculations (see figure SD1), that means that calculated energies are altered, maybe due to uncertainties of spectroscopic constants of the β and D states used.

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To determine correct energy gaps between the coupled rovibronic states, we measured excitation spectrum of the D → X and β → A luminescence at the β, 13, 61 ∼ D, 12, 61 coupled rovibronic state corresponding to a maximal energy gap and a minimal R_D-X/β-A value, which we could determine (figure 5). Integral intensity of the D → X luminescence is ∼ 130 times less than that of the β → A for each line of the J_β = J_D = 61 pair (figure SD6). Therefore, scattered light of the β → A luminescence presents at the λ_lum ≈ 2830 Å where the D → X luminescence excitation spectrum has been measured (see 'quasi D → X luminescence' at the β, 13, 60 rovibronic level, at which we have not observed even a trace of the D → X luminescence). Scattered light has been taken into account (see figure 5).

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One sees the β, 13, 61 ∼ D, 12, 61 ← B, 19, 60 'quasi-doublet' in excitation spectrum of the D, 12, J_D → X luminescence. The ν₂ = 24 270.92 and 24 271.08 cm⁻¹ lines correspond to the D, 12, 61 rovibronic states with a minor admixture of the β, 13, 61 one (the ∣2〉 state) and β, 13, 61 rovibronic states with a minor admixture of the D, 12, 61 one (the ∣1〉 state). The D → X luminescence intensity is the same at both lines. As to intensity of the β → A luminescence, its ratio for population of the ∣1〉 and ∣2〉 state is $\tfrac{I{(\beta \to A)}_{| 1\rangle }}{I{(\beta \to A)}_{| 2\rangle }}\,$≫ 1, and similar quasi-doublet cannot be observed in the excitation spectra of the β, 13, J_β → A luminescence (equations (10a) and (10b)). The energy gap between two components of the 'quasi-doublet' is equal to 0.16 cm⁻¹ whereas calculated one is 0.012 cm⁻¹. Therefore, one should correct calculated energy gaps given in table 1.

The R_D-X/β-A ratios as a function of correct energy gap for D, 12 ∼ β, 13 coupling are presented in figure 6.

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A position of quasi-Lorentzian maximum corresponds to ΔE = 0, and the R_D-X/β-A value at the maxima is equal to the ratio of Einstein coefficients. Therefore, one has to vary its full width of half maximum (FWHM), only, which was found to be equal to FWHM = 0.031. The FCF of the vibronic states coupled is 0.05, so one can obtain ${H}_{hf}^{el}$ = 0.049(3) cm⁻¹.

Excitation scheme (2) was used to study the β, 47, J_β ∼ D, 48, J_D coupling. One of excitation spectra of the D → X and β → A luminescence measured is given in figure SD4.

As shown in section 2.1 and figure SD3, one can expect to find the hyperfine coupling in vicinities of the J_D ≈ 7, 10, 33 and 75 rovibronic levels for J_β = J_D−2, −1, 0 and +1, respectively. It has been found no features of the coupling in a vicinity of the J_D ≈ 7 level, i.e. no EQ hyperfine coupling between the β, 47 ∼ D, 48 vibronic states has been observed. The MD coupling has been observed in vicinities of the J_D ≈ 14, 33 and 78 levels, namely, at the J_D = J_β + 1 = 13–15, J_D = J_β = 30–35 and J_D = J_β−1 = 77–79 levels. As for the case of D, 12 ∼ β,13 coupling, the R_β-A/D-X luminescence intensity ratios at the coupled D, 48, J_D ∼ β, 47, J_β rovibronic states have been determined at each rotational lines of the D, 48, J_D ← B, 20, J_B transitions where the β → A luminescence had been observed.

The R_β-A/D-X values and, since, coupling are the largest at the J_D = 14 (not included in the table 2), 33 and 78 (table 2), ∼(3–5) rotational levels higher than those obtained in calculations (see figure SD3) due to uncertainties of spectroscopic constants of the β and D states used.

To determine correct energy gaps between the coupled β, 47, J_β ∼ D,48, J_D rovibronic states, we measured excitation spectrum of the D → X and β → A luminescence in transitions to the D, 48, 33 and β, 47, 33 rovibronic states, respectively (figure SD4). The energy gap between experimental Q lines of the transitions is 0.011 cm⁻¹, whereas calculated one is −0.179 cm⁻¹. Therefore, calculated energy gaps in table 2 were amended by +0.19 cm⁻¹.

The R_β-A/D-X values obtained plotted as a function of energy gaps for the β, 47 ∼ D, 48 coupling are presented in figure 7. The procedure of electronic matrix element determination is identical to that described in section 3.1. FWHM of quasi-Lorentzian in figure 7 is 0.015 cm⁻¹, and the FCF of the vibronic states coupled is 0.007. Therefore, it follows that ${H}_{hf}^{el}$ = 0.063(7) cm⁻¹.

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The determined electronic matrix elements for β, 13 ∼ D, 12 and β, 47 ∼ D, 48 couplings slightly differ from each other. It can be assumed that there is dependence on vibrational excitation.

Coupling of the D and β states at large energy gaps are well described in the framework of the two-state perturbation model. The correct description of the quasi-resonant rovibrational levels requires taking into account hyperfine structure. Beyond the two-state perturbation model, the total emission intensity ratio is given by

$$\begin{eqnarray}{R}_{D-X/\beta -A} & = & \displaystyle \frac{\displaystyle {\sum }_{{i},{j}=1}^{3}\displaystyle {\sum }_{{n}=1}^{2}\displaystyle \sum {| \langle {\rm{X}}| {{d}}_{{i}}| {n}\rangle \langle {n}| {{d}}_{{j}}| {B}\rangle | }^{2}}{\displaystyle {\sum }_{{i},{j}=1}^{3}\displaystyle {\sum }_{{n}=1}^{2}{\displaystyle \sum | \langle {\rm{A}}| {{d}}_{{i}}| \rangle {n}\langle {{nd}}_{{j}}\rangle {B}| }^{2}}\\ & & \cdot \displaystyle \frac{{{\rm{\nu }}}_{D\to X}^{3}}{{{\rm{\nu }}}_{\beta \to A}^{3}},\end{eqnarray} \tag{ 11 }$$

and similarly for the R_β-A/D-X. Here ∑ means summation over all hyperfine components of initial, intermediate (n = ∣1〉,∣2〉) and final states, i, j are spatial coordinates, and d is a transition dipole moment operator.

As it was mentioned above, we did not find any evidence of EQ interaction between D and β states, so we can take the molecular Hamiltonian including electronic-vibrational-rotational and magnetic HFI.

The MD interaction between $D{0}_{u}^{+}$ and β1_g states is determined by $\sqrt{{\rm{F}}{\rm{C}}{\rm{F}}}{A}_{\perp }$ product where A_⊥ parameter is hyperfine structure constant

$$\begin{eqnarray}&&{A}_{\perp }=\displaystyle \frac{{\mu }_{I}}{i}\left\langle \beta {1}_{{\rm{g}}}\,\left|\sum _{j}{\left(\displaystyle \frac{{{\boldsymbol{\alpha }}}_{j}\times {{\boldsymbol{r}}}_{j}}{{r}_{j}^{3}}\right)}_{+}\right.\phantom{\rule{}{4.0ex}}| D{0}_{u}^{+}\right\rangle ,\end{eqnarray} \tag{ 12 }$$

μ_I = 2.81 μ_N and I = 2.5 are the magnetic moment and spin of ¹²⁷I nucleus, μ_N is nuclear magneton, α_j are the Dirac matrices for the jth electron, r_j is its radius vector in the coordinate system centered on an iodine atom. It was assumed that hyperfine splitting of near-resonant levels is determined only by the MD interaction between them.

Following the computational scheme of [38, 39] the mixed states ∣1〉 and ∣2〉 states can be obtained by numerical diagonalization of the molecular Hamiltonian on the basis set of the spin-electronic-vibrational-rotational wave functions

$$\begin{eqnarray}&&| \beta {1}_{{g}},{\rm{\Omega }}=1\rangle {\chi }_{\nu \beta }({R}){{\rm{\Theta }}}_{{{M}}_{\beta },{\rm{\Omega }}=1}^{{{J}}_{\beta }}{{U}}_{{{M}}_{{I}\beta }}^{{{I}}_{\beta }}+{(-1)}^{{{J}}_{\beta }}{P}\\ &&\,\times \,| \beta {1}_{{g}},{\rm{\Omega }}=-1\rangle {\chi }_{\nu \beta }({R}){{\rm{\Theta }}}_{{{M}}_{\beta },{\rm{\Omega }}=-1}^{{{J}}_{\beta }}{{U}}_{{{M}}_{{I}\beta }}^{{{I}}_{\beta }},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&| {D}{0}_{u}^{+}\rangle {{\rm{\chi }}}_{{\rm{\nu }}{D}}({R}){{\rm{\Theta }}}_{{{M}}_{{D}},{\rm{\Omega }}=0}^{{{J}}_{{D}}}{{U}}_{{{M}}_{{ID}}}^{{{I}}_{{D}}},\end{eqnarray} \tag{ 14 }$$

where:

• –  
  $| \psi \rangle ,{\chi }_{{\rm{\nu }}}({R}),{{\rm{\Theta }}}_{{M},{\rm{\Omega }}}^{{J}},{{U}}_{{{M}}_{{I}}}^{{I}}$ are electronic, vibrational, rotational, and nuclear spin wave functions, respectively,
• –  
  M and Ω are the electron-rotational angular momentum projection onto the laboratory and internuclear axis, respectively,
• –  
  M_I is the projection of the nuclear angular momentum onto the laboratory axis,
• –  
  P = ±1 is the sign (parity) of the β1_g state.

The electronic-vibrational-rotational part of the Hamiltonian is diagonal on the basis set (13) and (14). Its diagonal matrix elements correspond to energy levels obtained using spectroscopic constants of the β and D states.

With a given parameter A_⊥, matrix elements on the basis set (13) and (14) can be calculated using the angular-momentum algebra [40]. Fitting the experimentally observed R_D-X/β→A and R_β-A/D-X ratios, we obtained A_⊥ = 980 and 1150 MHz for the D, 12 ∼ β, 13 and D, 48 ∼ β, 47 couplings, respectively. Observed and calculated R_D-X/β-A and R_β-A/D-X ratios are presented in figures 6, 7. One sees that they agree well.

Previously the experimental, ${{{A}}_{| | }}^{\exp }$ = 477 MHz, and theoretical, ${{A}_{| | }}^{{\rm{calc}}}$ = 967 MHz, values for the hyperfine structure constant were obtained for the I₂(β1_g, 16) state [24]. Two approximations were made in the calculation. First, the calculation was made in the framework of the asymptotic model, where it is assumed that wave functions of the IP states (for all internuclear separations) are given by the I⁻(¹S) + I⁺(³P₂) dissociation limit [41]:

$$\begin{eqnarray}&&| D{0}_{{u}}^{+}\rangle \mathop{\to }\limits^{R\to \infty }\displaystyle \frac{1}{\sqrt{2}}[{| {}^{3}{\rm{P}}_{2},{\rm{\Omega }}=0\rangle }_{{a}}{| {}^{1}{\rm{S}}_{0}\rangle }_{{b}}\\ &&\quad +\,{| {}^{1}{\rm{S}}_{0}\rangle }_{{a}}{| {}^{3}{\rm{P}}_{2},{\rm{\Omega }}=0\rangle }_{{b}}],\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&| \beta {1}_{{g}}\rangle \mathop{\to }\limits^{R\to \infty }\displaystyle \frac{1}{\sqrt{2}}[{| {}^{3}{\rm{P}}_{2},{\rm{\Omega }}=1\rangle }_{{a}}{| {}^{1}{\rm{S}}_{0}\rangle }_{{b}}-{| {}^{1}{\rm{S}}_{0}\rangle }_{{a}}| {}^{3}{\rm{P}}_{2},{\rm{\Omega }}={1}_{{b}}\rangle ].\end{eqnarray} \tag{ 16 }$$

Second, the hyperfine structure constant for the ion I⁺(³P₂) was calculated using the known hyperfine structure for the neutral I(³P_1/2,3/2) atom and assuming that I⁺(³P₂) and I(²P) are composed of the same one-electron orbitals in the single-configuration approximation. The calculation scheme is rather rough. Therefore the difference of about two times between ${{{A}}_{| | }}^{\exp }$ and ${{{A}}_{| | }}^{{\rm{calc}}}$ is not surprising.

The hyperfine structure constant for the I₂(β1_g, v_β = 16) state is given by

$$\begin{eqnarray}&&{A}_{| | }=\displaystyle \frac{{\mu }_{{I}}}{{i}}\left\langle \beta {1}_{{g}}\,\left|\sum _{{j}}{\left(\displaystyle \frac{{{\boldsymbol{\alpha }}}_{{j}}\times {{\bf{r}}}_{{j}}}{{{r}}_{{j}}^{3}}\right)}_{+}\phantom{\rule{}{4.0ex}}| \beta {1}_{{g}}\right.\right\rangle .\end{eqnarray} \tag{ 17 }$$

It is interesting to note, that within the asymptotic model, the hyperfine structure constant A_∣∣ for β1_g state is equal to A_⊥ given by equation (12) since β1_g and $D{0}_{{u}}^{+}$ states correlate with the same I⁻(¹S) + I⁺(³P₂) dissociation limit. However, experimentally determined ${{A}_{| | }}^{\exp }\,$ = 477 MHz for the β1_g state in [24] is approximately two times lower than the constant A_⊥ obtained in this work. Accidentally the calculated value, ${{A}_{| | }}^{{\rm{calc}}}$ = 967 MHz, is very close to our results.

Within the asymptotic model and assuming the single-configuration approximation for the I⁺(³P₂) ion, the constant for quadrupole HFI can be estimated as [8]

$$\begin{eqnarray*}&&\displaystyle \frac{{{e}}^{2}{Q}}{2}\left\langle \beta {1}_{{g}}\left|\displaystyle \sum _{{j}}{\left(\displaystyle \frac{{C}_{21}}{{r}^{3}}\right)}_{{j}}\right|D{0}_{{u}}^{+}\right\rangle =\displaystyle \frac{{{e}}^{2}{Q}}{4}\\ &&\,\,\times \left\langle {}^{3}{\rm{P}}_{2},{\rm{\Omega }}=1\,\left|\displaystyle \sum _{{j}}{\left(\displaystyle \frac{{C}_{21}}{{r}^{3}}\right)}_{{j}}{}^{3}{\rm{P}}_{2},{\rm{\Omega }}=0\right|\right\rangle \\ &&\,=\,\displaystyle \frac{{{e}}^{2}{Q}}{8\sqrt{3}}\left\langle 5{{p}}_{0}\left|\displaystyle \frac{{C}_{21}}{{{\rm{r}}}^{3}}\right|5{{p}}_{-1}\right\rangle \approx 80\,\text{MHz},\end{eqnarray*}$$

where Q = 0.689 b is the quadrupole moment for the ¹²⁷I nucleus [42], ${C}_{2q}=\sqrt{\tfrac{4{\rm{\pi }}}{5}}{{\rm{Y}}}_{2q},$ Y_kq is the spherical function, e is the charge of the electron, 5p is the valence orbital for the I⁺(³P₂) ion. The constant for quadrupole HFI is ∼10 times less than A_⊥ = 980 and 1150 MHz for the D, 12 ∼ β, 13 and D, 48 ∼ β, 47 couplings, respectively (see above). Therefore, the R_β-A/D-X ratio for this interaction is ∼100 times less than those observed in the experiments. This is beyond the experimental sensitivity.