The highly perturbed A²Π _u– X²Π _g system of the chlorine cation has been investigated for nearly one century, however, knowledge about the structure of the electronic states, A²Π _u and X²Π _g, is still limited. This system has two components: A²Π _1/2– X²Π _1/2 and A²Π _3/2– X²Π _3/2. Initial spectroscopic data came from a series of dispersed emission studies.^{[1– 5]} It is difficult to assign the absolute vibrational quantum numbers to the observed bands because of the irregular vibrational internals and non-systematic chlorine isotopic effects resulting from the severely perturbed excited electronic state A²Π _u. In addition, since the spin– orbit splitting for the X²Π _g state is found to be very close to the vibrational energy spacing, the spin– orbit resolved vibrational bands have been analyzed ambiguously. In 1966, Huberman^[6] studied the emission spectrum and made the most comprehensive rotational assignment of the observed. In 1989, Tuckett and Peyerimhoff^[7] observed the emission spectrum in the supersonic expansion, which allowed them to make a fuller analysis due to the cold Doppler-free spectrum. One hundred vibronic bands were fitted into two Deslandres tables (one for A²Π _1/2– X²Π _1/2, and the other for A²Π _3/2– X²Π _3/2). Still, they could not assign vibrational quantum numbers to the Deslandres tables. Ab initio calculations show that A²Π _u is crossed near its origin by some states arising from , i.e. and ²Δ _u.

In 1989, the Λ -doublets was first observed in a jet-cooled A– X emission spectrum by Choi and Hardwick, ^[8] and four bands were unambiguously assigned as A²Π _1/2– X²Π _1/2. Later in 1991, they^[9] reported more bands of the Ω = 3/2 component. Among those, 10 bands were assigned to the A²Π _3/2– X²Π _3/2 system, which had been rotationally analyzed by Huberman earlier. These measurements improved the accuracy of the rotational constants for this system. Additionally, five bands were assigned to a new system of B²Δ _u3/2– X²Π _g3/2, and they, for the first time, rotationally analyzed the bands of the minor isotopologue ³⁵Cl³⁷Cl⁺ . They agreed with Huberman on vibrational numbering of the ground state, and their analyses were also in agreement with a new analysis by Gharaibeh et al.^[10] However, the vibrational numbering of the excited state seems to be modified. In 1990, Bramble and Hamilton^[11] reported the first laser-induced fluorescence spectra of and analyzed many bands of the (υ ′ , 0) progression of the Ω = 3/2 component together with some extra bands arising from the perturbation.

In 2012, Gharaibeh et al.^[10] studied the laser-induced fluorescence spectrum of jet-cooled with high sensitivity and rigorous vibrational and spin– orbit cooling. A full understanding of the A²Π _3/2– X²Π _3/2 band system was presented, including a complete treatment of the interaction between the A²Π _u3/2 and B²Δ _u3/2 states, based on the experimental spectrum and ab initio calculations. The (0, 0) band was identified for the first time. They studied two bands at high-resolution and obtained the accurate B values. For most of the low-resolution bands, they listed approximate upper state B values as well.

More information about the electronic states of has been derived from the photoelectron spectroscopy (PES).^{[7, 12, 13]} In 2007, Li et al.^[12] studied the isotopomer-resolved vibrational and spin– orbit energy structures of by one-photon zero kinetic energy (ZEKE) photoelectron spectroscopy. The precise values of the spin– orbit splitting for X²Π _g were reported. Most recently, Mollet and Merkt^[13] studied the X²Π _g, A²Π _u, B²Δ _u, and states of by partially rotationally resolved pulsed-field-ionization zero-kinetic-energy (PFI-ZEKE) photoelectron spectroscopy. A large number of Franck– Condon-forbidden transitions to vibrational levels of the X²Π _g (Ω = 3/2, 1/2) were observed. The potential energy function of was determined in a direct fit to the experimental data. Transitions to vibrational levels of the A²Π _u3/2, B²Δ _u3/2 states were identified by using the results of Ref. [10] and transitions to the upper spin– orbit component A²Π _u1/2 (υ = 2– 6) were detected. Additionally, the A²Π _u1/2 state was found to be perturbed by the ²Σ _u1/2 state. Unfortunately, their data were too scarce to model this interaction.

In the present work, we study the rotational structure of the Ω = 3/2 component of the X²Π _g, A²Π _u, and B²Δ _u3/2 states. Aiming to obtain the equilibrium rotational constants B_e and α _e of these states based on the experimental data, we reassign the vibrational quantum numbers for the previously observed bands and subsequently fit the rotational constants. Using the isotopic effect, the rotational spectroscopic constants for the minor isotopologues ³⁵Cl³⁷Cl⁺ and are predicted.

Two methods are used to calculate the values of origin T₀ for vibrational bands of the A²Π _3/2– X²Π _3/2 system of . The first method is to use the effective Hamiltonian matrix. The form of the matrix is shown in Fig. 8 of Ref. [10]. The diagonal energy terms are E_υ = T_e + G_υ , where

The off-diagonal terms are , where the first term is the electronic spin– orbit interaction matrix element and the second term is a vibrational overlap.^[14] The vibrational overlaps are calculated by using LeRoy’ s RKR1 2.0 and LEVEL 8.0 programs.^{[15, 16]} Additionally, for calculating the matrix elements, we adopt the reported vibrational constants and also the spin– orbit coupling term .^[10] The matrix is diagonalized and the eigenvalues are stored for calculating the values of origin T₀ (υ ′ , υ ″ ) = E_{υ ′} − E_{υ ″} of vibrational bands. Note that the adopted vibrational constants and the interaction constant are fitted by using the band heads rather than band origins in Ref. [10], because the band origins cannot be determined accurately without full rotational analysis. However, it was argued that this introduces a little error (only 0.3 cm^{− 1}– 0.5 cm^{− 1}), which is well within the overall standard error for the least square analysis. As to the vibrational assignment, it is accurate enough. Moreover, the band heads replace the band origins in the second calculation method.

The second method is much simpler: there is no need to calculate the vibrational overlaps nor to diagonalize the matrices. Using the experimentally observed positions of the band heads, such as T₀ (υ ′ , 0) of the (υ ′ , 0) bands, the values of band origin T₀ (υ ′ , υ ″ ) can be evaluated by using the following formulas:

and

The vibrational constants (645.6 cm^{− 1}) and (2.98 cm^{− 1}) of ^[13] are adopted. In Ref. [10], 70 band heads of the species are listed, which all originate from the υ ″ = 0 level of the X²Π _3/2 state. The relevant upper levels are A²Π _3/2(υ ′ = 0– 39) and B²Δ _u3/2(υ ′ = 0, 1, 3– 8, 10– 12, 14– 19, 21– 26, 28– 29, 31, 32). The extent of the upper levels and the accuracies of the band heads are sufficient for the purpose of this work.

Molecular chlorine has three isotopologues, ³⁵Cl₂, ³⁵Cl³⁷Cl, and ³⁷Cl₂ with relative abundances of 57.41%, 36.72%, and 5.87%, respectively.

The spectroscopic constants for two different isotopologues i and j have the following relationships:^[17]

and

with

where μ _i and μ _j are the reduced mass values for isotopologues i and j. The values of ρ for the pairs ³⁵Cl₂ and ³⁵Cl³⁷Cl, and ³⁵Cl₂ and ³⁷Cl₂ are 0.98640 and 0.97261 respectively. The investigations of the isotopic effect in electronic band spectra lead to the determination of the correlated constants^[18] and are helpful for discovering the new isotopologues of very small abundance.^[19]

The vibrational band origins of the A²Π _3/2– X²Π _3/2 and B²Δ _u3/2– X²Π _3/2 are calculated by using the above two methods respectively and subsequently the observed bands reported in Refs. [6], [7], and [9] are reassigned. Listed in Table 1 are the origins with the vibrational assignments in a Deslandres table for the A²Π _3/2u– X²Π _3/2g system of . The vibrational assignments for the emission spectra of and ³⁵Cl³⁷Cl⁺ observed by Choi and Hardwick^[9] are listed in Table 2. We agree on their ground state vibrational assignments but modify the vibrational numbering of the excited state A²Π _3/2. Moreover transitions to the B²Δ _u3/2 state have also been assigned, and they are also listed in Table 2.

Table 1.

Table 1.

Table 1. Deslandres table for . All data are in units of cm− 1. υ ″ υ ′ 0 1 2 3 4 5 5 21557a 20926.30b 20298.58b 19676.74b 19060.7b 6 21857a 21222.091c 20594.394c 19972.61b 19356a 7 22301.97b 21668.33b 21040.62b 20419.4b 8 22609.345c 21975.704c 21347.975c 20727a 20109a 9 22910.86b 22277.20b 21647a 10 23966a 23326.99b 22693.378c 22066a 11 24263a 23623.26b 22989.512c 22360a 21740.056c 21120a 12 24564.25b 23924.63b 23292a 13 24947.52b 24307.94b 23674.199c 14 25235.98b 24596.41b 23966a 22713.040c 15 16 25882.50b 25242.94b 17 26166.25b 25526.65b 24893a 18 19 26773.31b 20 27051.61b a From Ref. [7], b from Ref. [6], c from Ref. [9].

Table 1. Deslandres table for . All data are in units of cm− 1.

Table 2.

Table 2.

Table 2. Vibrational assignments for the emission spectra of and 35Cl37Cl+ observed by Choi and Hardwick[9]. Band origins/cm− 1 Upper state (υ ′ , υ ″ ) This work Ref. [9] 21222.091 A2Π 3/2u (6, 2) (5, 2) 20594.394 A2Π 3/2u (6, 3) (5, 3) 22609.345 A2Π 3/2u (8, 1) (7, 1) 21975.704 A2Π 3/2u (8, 2) (7, 2) 21347.975 A2Π 3/2u (8, 3) (7, 3) 22693.378 A2Π 3/2u (10, 2) (9, 2) 22989.512 A2Π 3/2u (11, 2) (10, 2) 21740.056 A2Π 3/2u (11, 4) (10, 4) 23674.199 A2Π 3/2u (13, 2) (12, 2) 22713.040 A2Π 3/2u (14, 4) (13, 4) 21513.817 B2Δ u3/2 (4, 2) (x, 2) 20886.082 B2Δ u3/2 (4, 3) (x, 3) 23035.716 B2Δ u3/2 (7, 1) (y, 1) 22402.080 B2Δ u3/2 (7, 2) (y, 2) 24036.785 B2Δ u3/2 (11, 1) (z, 1) 35Cl37Cl+ 21217.479 A2Π 3/2u (6, 2) (5, 2) 20598.076 A2Π 3/2u (6, 3) (5, 3) 22584.246 A2Π 3/2u (8, 1) (7, 1) 21959.075 A2Π 3/2u (8, 2) (7, 2) 21339.656 A2Π 3/2u (8, 3) (7, 3) 22671.110 A2Π 3/2u (10, 2) (9, 2) 22960.571 A2Π 3/2u (11, 2) (10, 2) 23887.800 A2Π 3/2u (12, 1) (11, 1) 22390.594 B2Δ u3/2 (6, 2) (y, 2)

Table 2. Vibrational assignments for the emission spectra of and 35Cl37Cl+ observed by Choi and Hardwick[9].

Based on the unambiguous vibrational assignments of the states, some vibrational quantum numbers υ with the rotational constants B_υ are modified consequently, such as in Ref. [9]. The rotational constants of the X²Π _3/2g, A²Π _3/2u, and B²Δ _3/2u states for and ³⁵Cl³⁷Cl⁺ in the previous work are listed in Table 3. The constants B_e and α _e are fitted according to B_υ = B_e − α _e(υ + 1/2). The derived values of B_e and α _e of the X²Π _3/2g and A²Π _3/2u states for and ³⁵Cl³⁷Cl⁺ and those of the B²Δ _3/2u state for are listed in Table 4. The values of equilibrium internuclear distance R_e for those three states of all the three isotopologues are determined from the values of B_e and listed in Table 4. The derived rotational constants of the X²Π _3/2g and A²Π _3/2u state are in good agreement with those from the ab initio PECS in Ref. [10]. For , the derived value of B_e is 0.1746 cm^{− 1} in comparison with the theoretical value of 0.1617407 cm^{− 1}.^[10]

Using Eq. (7), the rotational constants of the minor isotopologues ³⁵Cl³⁷Cl⁺ and are predicted and listed in Table 4 as well.

It should be mentioned that according to the calculated vibrational band origins, the bands at 16913, 17030, 17282, and 17323 cm^{− 1} would be assigned as (1, 6) (3, 7) and (2, 6) of and (2, 6) of ³⁵Cl³⁷Cl⁺ of the A²Π _3/2– X²Π _3/2 sub-system instead, which are tentatively assigned as (2, 7), (3, 7), and (4, 8) of and (3, 7) of ³⁵Cl³⁷Cl⁺ of the A²Π _1/2– X²Π _1/2.^{[20, 21]} When modifying the vibrational quantum numbers of the bands, the experimental rotational constants in Refs. [20] and [21] are in good agreement with those in Ref. [10].

Table 3.

Table 3.

Table 3. Experimentally determined rotational constants of the X2Π 3/2g, A2Π 3/2u, and B2Δ 3/2u states. All data are in units of cm− 1. X2Π 3/2g A2Π 3/2u B2Δ 3/2u υ ″ B′ υ ′ B′ υ ′ B′ υ ′ B′ 0 0.2686(8)a 0 16 0.1607a 0 0.179c 1 0.267309(39)b 1 0.19123c 17 0.1622a 1 0.170c 0.2670(4)a 2 0.18929c 18 0.157c 3 0.171c 2 0.265639(35)b 3 0.186c 19 0.1588a 4 0.166859(60)b 0.2654(0)a 4 0.179c 20 0.1568a 0.161c 3 0.263893(33)b 5 0.1788a 21 0.148c 5 0.161c 0.2637(6)a 6 0.173284(35)b 22 0.155c 6 0.160c 4 0.26234(22)b 0.1729a 23 0.150c 7 0.165263(48)b 0.2621(2)a 7 0.1749a 25 0.149c 0.161c 8 0.174426(33)b 26 0.149c 8 0.161c 0.1742a 27 0.145c 10 0.156c 9 0.1691a 28 0.143c 11 0.158755(88)b 10 0.168992(40)b 30 0.140c 0.157c 0.1694a 31 0.140c 12 0.157c 11 0.170458(42)b 32 0.135c 14 0.156c 0.1702a 33 0.124c 17 0.146c 12 0.1664a 34 0.121c 21 0.138c 0.1659c 35 0.124c 22 0.138c 13 0.163750(54)b 36 0.120c 23 0.145c 0.1641a 25 0.133c 14 0.16689(22)b 26 0.134c 0.1665a 28 0.129c 29 0.129c 32 0.123c X2Π 3/2g A2Π 3/2u B2Δ 3/2u υ ″ B′ υ ′ B′ υ ′ B′ 35Cl37Cl+ 0 0.26139(10)c 1 0.18611(9)c 6 0.159453(82)b 0.26145(14)c 2 0.18430(12)c 1 0.260135(50)b 6 0.169733(51)b 0.26a 8 0.169842(46)b 2 0.258471(52)b 10 0.16319(78)b 0.2585a 0.1644a 3 0.256789(53)b 11 0.166336(62)b 0.1663a 12 0.164038(54)b a From Ref. [6], b from Ref. [9], c from Ref. [10].

Table 3. Experimentally determined rotational constants of the X2Π 3/2g, A2Π 3/2u, and B2Δ 3/2u states. All data are in units of cm− 1.

Table 4.

Table 4.

Table 4. Fitted rotational constants, the predicted rotational constants (in cm− 1), and the equilibrium internuclear distances of the X2Π 3/2g, A2Π 3/2u, and B2Δ 3/2u states. 35Cl37Cl+ This work Previous work This work This work Previous work This work Previous work X2Π 3/2g Be 0.269791(85)a 0.26950b 0.262648(14) 0.262503e 0.255216e 0.2707(11)c α e 0.001665(27) 0.00164b 0.0016730(52) 0.001598e 0.001532e 0.0012(4)c σ 0.00006 0.0000073 Re 1.891 Å 1.890 Å 1.891 Å 1.891 Å A2Π 3/2u Be 0.1903(14) 0.1897495d 0.1878(24) 0.1852e 0.1846241d 0.1800e 0.1794985d α e 0.001770(63) 0.001375892d 0.00208(28) 0.0016988e 0.001320652d 0.001629e 0.001265973d σ 0.0038 0.003 Re 2.251 Å 2.25 Å d 2.235 Å 2.251 Å 2.254 Å f 2.251 Å 2.254 Å f B2Δ 3/2u Be 0.1746(12) 0.1617407d 0.1699e 0.1573721d 0.1652e 0.1530031d α e 0.001561(69) 0.001193303d 0.001498e 0.001146015d 0.001436e 0.001098608d σ 0.003 Re 2.350 Å 2.44 Å d 2.350 Å 2.442 Å f 2.350 Å 2.442 Å f a Numbers in parentheses are in the units of the last quoted digits. b From Ref. [6], c from Ref. [13], d from Ref. [10]. e Predicted using the isotopic effect. f Calculated based on the constants of the previous work.

Table 4. Fitted rotational constants, the predicted rotational constants (in cm− 1), and the equilibrium internuclear distances of the X2Π 3/2g, A2Π 3/2u, and B2Δ 3/2u states.

However there is less experimental information about the A state Ω = 1/2 levels than about the Ω = 3/2 levels. Theoretically, the A²Π _1/2u levels above ∼ υ = 4 will be perturbed by levels of the state.^[10] In the experiment, the jet-cooling only allows the Ω = 3/2– Ω = 3/2 transitions to be observed.^[10] If more bands of the other half transitions Ω = 1/2– Ω = 1/2 could be observed, combined with the data reported in previous work, the data could be fitted to understand the whole structure of and the spin– orbit coupling constant A for the A²Π _u state would precisely be obtained.

The vibrational and rotational structures of the A²Π _3/2, B²Δ _u3/2, and X²Π _3/2 states of the three isotopologues of (, ³⁵Cl³⁷Cl⁺ , ) are studied. Using the previous experimentally observed positions of the band origins and the vibrational constants, ^{[10, 13]} we calculate the vibrational band origins of the A²Π _3/2– X²Π _3/2 and B²Δ _u3/2– X²Π _3/2 systems and reassign the observed bands reported in Refs. [6], [7], and [9]. Some vibrational quantum numbers of the states are modified, and thus the experimental rotational constant B_υ are modified accordingly.

Based on the unambiguous vibrational assignments of the states, the values of B_υ reported in previous investigations are used to derive the constants B_e and α _e of the X²Π _3/2g and A²Π _3/2u states for and ³⁵Cl³⁷Cl⁺ and those of the B²Δ _3/2u state for . Then the rotational constants of the minor isotopologues ³⁵Cl³⁷Cl⁺ and are predicted based on the isotopic relations. Additionally, the values of equilibrium internuclear distance R_e for those three states of all the three isotopologues are calculated.