Charge-Transfer-Induced Predissociation in Rydberg States of Molecular Cations: MgAr.

Very little is known about the Rydberg states of molecular cations, i.e., Rydberg states having a doubly charged ion core. With the example of MgAr+, we present general features of the structure and dynamics of the Rydberg states of molecular cations, which we find are subject to the process of charge-transfer-induced predissociation. Our study focuses on the spectrum of low-n Rydberg states with potential-energy functions associated with the Mg+(3d and 4s) + Ar(1S0) dissociation asymptotes. In particular, we have recorded spectra of the 3dπΩ' (Ω' = 1/2, 3/2) Rydberg states, extending from the lowest (v' = 0) vibrational levels to their dissociation limits. This spectral range encompasses the region where the onset of predissociation by interaction with the mostly repulsive 2Σ and 2Π charge-transfer states associated with the Mg(3s2) + Ar+(2P1/2,3/2) dissociation asymptotes is observed. This interaction leads to very strong perturbations of the 3dπ Rydberg states of MgAr+, revealed by vibrational progressions exhibiting large and rapid variations of the vibrational intervals, line widths, and spin-orbit splittings. We attribute the anomalous sign and magnitude of the spin-orbit coupling constant of the 3dπ state to the interaction with a 2Π Rydberg state correlating to the Mg+(4p) + Ar(1S0) dissociation limit. To analyze our spectra and elucidate the underlying process of charge-transfer-induced predissociation, we implemented a model that allowed us to derive the potential-energy functions of the charge-transfer states and to quantitatively reproduce the experimental results. This analysis characterizes the main features of the dynamics of the Rydberg series converging to the ground state of MgAr2+. We expect that the results and analysis reported here are qualitatively valid for a broader range of singly charged molecular cations, which are inherently prone to charge-transfer interactions.

Introduction

High-resolution spectroscopic data on the electronically excited states of molecular cations are scarce, particularly on their Rydberg states. At the same time, Rydberg states of molecular cations are known to play an important role as transient resonances in ion-neutral collisions[1−4] and in plasmas.[5−9] Now that photoionization studies of molecular cations are becoming feasible at synchrotron and free-electron-laser facilities, the prospects of systematic studies of Rydberg states of molecular cations are excellent.[10−12] Rydberg electrons are primarily located outside the ionic core, so that, in first approximation, Rydberg states have potential-energy functions that closely resemble those of the ion-core electronic states to which the Rydberg series converge.[13−17] In the case of molecular cations M+ = F1F2+, the ionic core (M2+) is doubly charged and prone to dissociation into two charged atomic or molecular fragments F1+ + F2+ on a repulsive Coulomb potential (see Figure a). This situation does not favor the observation of discrete structures associated with Rydberg states (principal quantum number n) in the spectra of molecular cations, except those associated with shallow potential wells at long-range resulting from charge–induced-dipole interactions between F+ and F(n) (i, j = 1, 2). Such long-range wells are well-known for few-electron diatomic molecules such as H2+[1,18−20] and HeH+.[21−23]

Figure 1

Potential-energy functions of the Rydberg states of molecular cations, which form series converging on the low-lying electronic states of dications F1F22+ (F1 and F2 designate the two fragments). (a) Molecular cation with a first ionization threshold that leads to a repulsive state, for instance H2+. Whereas the ground state is bound by charge–induced-dipole and perhaps valence interactions, the Rydberg states are essentially repulsive because the Rydberg electron is mostly outside the repulsive F1F22+ core. (b) Molecular cation with a first ionization limit corresponding to a thermodynamically stable dication. The repulsive charge-transfer states (red) tend to cross Rydberg states converging to the first ionization threshold, which may lead to fast predissociation (red arrow).

The situation is qualitatively different when the doubly charged ion core F1F22+ is thermodynamically stable, or metastable with a potential well lying above the dissociation asymptote, as is the case, e.g., for He22+.[24−26] The case of interest in the present study is illustrated in Figure b and arises when the ionization energy of F1+ is smaller than that of F2. In this case, the ground electronic state can have a well depth of more than 1 eV, arising from the strong charge–induced-dipole interaction between F12+ and F2. Sharp spectral structures in the Rydberg manifold of F1F2+ are expected, provided that rapid charge-transfer-induced predissociation into F1 + F2+ does not render the Rydberg spectrum diffuse or prevent its observation altogether (see the red arrow in Figure b). One can indeed anticipate (see Section ) that the repulsive part of the potentials associated with charge-transfer states correlating at long-range with F1 + F2+ should cross the Rydberg-state potentials near their equilibrium positions. This was for example used to measure resonance-enhanced multiphoton dissociation spectra in CS2+.[27]

So far, only one thermodynamically stable diatomic dication, MgAr2+, has been characterized spectroscopically.[28] Its ground electronic state is strongly bound (D0 = 10690(3) cm–1,[28]Re = 2.318 Å[29]), and one thus expects the full Rydberg manifold of MgAr+ to be observable. Energetically bound Rydberg states of MgAr+ are expected to cluster below the successive dissociation limits Mg+(n) + Ar(1S0). The A+ 3pπ and B+ 3pσ states of MgAr+, correlating with the Mg+(3p) + Ar(1S0) dissociation limit, have been fully characterized.[30−36] Higher Rydberg states have, however, not yet been reported, except for a few long-lived vibrational levels of the 3dπ state located below the Mg+(3d) + Ar(1S0) dissociation limit, which were used as intermediate levels to record high-resolution photoelectron spectra of MgAr+.[28]

We report here on the full characterization of the 3dπ state and the observation of the 4sσ Rydberg state of MgAr+. These states both lie above the Mg(3s2) + Ar+(2P) dissociation limit (see Figure ) and are therefore ideally suited to quantify, for the first time, the effects of charge-transfer-induced predissociation on the structure and dynamics of the Rydberg states of a molecular cation by high-resolution spectroscopy.

Figure 2

Excitation scheme to study the 3dπ1/2 and 3dπ3/2 states of MgAr+. The ab initio potential-energy functions for the X+2Σ+, A+2ΠΩ″, B+2Σ+, 4sσ, 3dδΩ, 3dπΩ′, and 3dσ states are from ref (35). The potential-energy functions of the charge-transfer states correlating to the Mg(3s2) + Ar+(2P) dissociation limits, shown by the red curves, are from this work. The potential-energy function of the X2+1Σ+ ground state of MgAr2+ is from ref (28).

The relevant potential-energy curves of the 3s,p,d and 4s Rydberg states of MgAr+ from refs (28 and 35) are presented in Figure . They are crossed in their attractive region by the repulsive part of the potentials of the 2Σ and 2Π charge-transfer states associated with the Mg(3s2) + Ar+(2P) dissociation asymptotes and conform to the generic aspects of the Rydberg manifolds of molecular ions F1F2+ having a thermodynamically stable doubly charged ion core F1F22+ (compare with Figure b). The methods we employed to record the Rydberg spectra of MgAr+ and to analyze their structure are expected to be applicable to a broad range of molecular systems.

In the remainder of this article, we use the following nomenclature to designate the rovibronic states of MgAr+. The ground state (2Σ+ symmetry) is denoted as X+(v+, N+), where v+ and N+ are the vibrational and rotational quantum numbers, respectively. The first electronically excited states, which correlate to the Mg+(3p 2P) + Ar(1S0) dissociation limits, are denoted as (2ΠΩ″ symmetry) and B+(v″) (2Σ+ symmetry), with the vibrational and rotational quantum numbers v″ and J″. Higher-lying Rydberg states, which correlate to Mg+(n) + Ar(1S0) dissociation limits, are denoted as nλΩ′(v′, J′). For instance, the label 3dπ3/2(v′, J′) designates the 2Π3/2 state with dissociation limit Mg+(3d 2D3/2) + Ar(1S0) and vibrational and rotational quantum numbers v′ and J′, respectively. In cases where the rovibrational quantum numbers are not relevant for the discussion, they are omitted for legibility. If not otherwise specified, the results presented in the following are for 24MgAr+.

Experiment

The experimental setup has been described in detail in previous works.[37,38] We produced rotationally cold (∼5 K) MgAr molecules in their metastable a 3Π0(v = 0) state by laser ablation of Mg in a supersonic expansion of Ar gas. After passing through a 3-mm-diameter skimmer, the molecular beam entered a cylindrical electrode stack, where it was perpendicularly intersected by the beams of up to four Nd:YAG-pumped dye lasers (25 Hz repetition rate, ∼4 ns pulse duration) used to access and characterize the 3dπΩ′ state of MgAr+ in a resonant multiphoton excitation scheme. The electrode stack was used to apply pulsed electric fields in order to field ionize high Rydberg states and to extract photoions into a linear time-of-flight (TOF) mass spectrometer. All lasers were frequency doubled or tripled in β-barium-borate crystals to generate the desired radiation in the UV. Their fundamental wavenumbers were measured using a commercial wavemeter with a specified accuracy of 0.02 cm–1. The lasers are referred to as Lasers 1 to 4 hereafter.

A typical excitation scheme to the 3dπΩ′ state is shown as black arrows in Figure . Depending on the vibrational levels and the spin–orbit component of the 3dπΩ′ state, different intermediate states and detection schemes had to be used. Using Laser 1, we excited MgAr from the metastable a 3Π0 state to [X+(v+)]n Rydberg states, with v+ = 3 or 7 and n ∼ 130. Laser 2, delayed by ∼10 ns, then excited the MgAr+ ionic core to a rovibrational level of the A1/2+, A3/2+, or B+ state. Spectra of the transitions from the A1/2+, A3/2+, or B+ states to the 3dπΩ′(v′) levels were then recorded by tuning the frequency of Laser 3.

Transitions to the 3dπ1/2(v′ ≥ 5) and 3dπ3/2(v′ ≥ 3) core levels were detected by monitoring the charge-transfer-induced predissociation, which produced Mg atoms in their ground state and Ar atoms in a high Rydberg state (see horizontal red arrow in Figure ). The application of a ∼5-μs-long weak electric-field prepulse of −1.7 V/cm spatially separated prompt Ar+ ions generated by the laser pulses from the Ar Rydberg states in the electrode stack. A subsequent strong electric-field pulse of +172 V/cm was used to field ionize the Ar Rydberg states and extract all ions toward a microchannel-plate detector located at the end of the TOF tube. The spatial separation of the Ar atoms in Rydberg states from the prompt ions allowed us to distinguish them in the TOF spectrum. In this way, the predissociation products could be detected free of background as a function of the wavenumber ν̃3 of Laser 3. The process of dissociation of the ionic core within the orbit of a Rydberg electron and the subsequent detection of a Rydberg state by pulsed-field ionization are described in detail in ref (34).

Spectra of the transitions to the 3dπ1/2(v′<5) and 3dπ3/2(v′<3) levels, which are not strongly affected by charge-transfer-induced predissociation, were measured by resonance-enhanced multiphoton ionization (REMPI) to MgAr2+ using Laser 4 at a fixed wavenumber of ν̃4 = 46290 cm–1. In this way, we recorded the MgAr2+ signal as a function of the wavenumber ν̃3 of Laser 3, also under background-free conditions.

Experimental Results

We have measured transitions to vibrational levels (v′) of the 3dπ1/2 and 3dπ3/2 states of 24MgAr+ ranging from v′ = 0 up to the dissociation limit. The lowest levels, i.e., 3dπ1/2(v′≤5) and 3dπ3/2(v′≤4), are sufficiently long-lived so that their rotational structure could be partially resolved. Higher-lying levels are too short-lived for any rotational structure to be observable. To unambiguously assign the vibrational quantum number v′, we measured spectra of the transitions to the 3dπ3/2(v′=3–7, 9, 10) levels of 26MgAr+ and performed an isotopic-shift analysis. Just below the Mg+(4s) + Ar(1S0) dissociation asymptote, we observed transitions that do not belong to vibrational progressions of the 3dπ states. We attribute them to transitions to high-lying vibrational levels of the 4sσ state.

Rotationally Resolved Spectra of the 3dπΩ′ State at Low v′ Values

Figure a displays a rotationally resolved spectrum of the A3/2+(v″=1,J″) ← X+(v+=3,N+) transition, which was recorded by monitoring the Mg+ dissociation product as a function of the wavenumber ν̃2 of Laser 2 (see ref (36)). This spectrum illustrates how the rotational states of the intermediate A+ levels were selected. The sticks indicate the calculated positions and intensities of the rotational transitions for a rotational temperature of 4 K, determined using the standard expressions[13,39,40] for transitions between initial and final states that are well described by Hund’s angular-momentum-coupling cases (b) and (a), respectively. Lines drawn in blue, orange, green, and red correspond to the possible rotational branches with J″ – N+ = −1.5, – 0.5, + 0.5, and +1.5, respectively. The red arrow marks the position of Laser 2 used to select the intermediate J″ = 4.5, 5.5, and 7.5 rotational levels, when recording the spectrum of the 3dπ3/2(v′=2,J′) state depicted in Figure b. This spectrum shows the same characteristic features as all spectra we recorded for transitions to the 3dπ1/2(v′≤5) and 3dπ3/2(v′≤4) levels. It consists of three branches (P, Q, and R), corresponding to J′ – J″ = −1, 0, and +1, respectively. The transitions are labeled along the assignment bars, the colors of which correspond to the selected levels, as indicated in Figure a. To model the relative intensities of the transitions, we used the standard expressions for transitions between two Hund’s case (a) states,[13,39,40] and the results are depicted as sticks. Whereas the relative intensities of the P and R branches are in good agreement with the experimental results, the calculations underestimate the intensities of the Q-branch lines. We have no explanation for this observation. The lines in Figure b have a full width at half-maximum of about 0.5 cm–1, which reflects a slight power broadening.

Figure 3

(a) Spectrum of the A3/2+(v″=1) ← X+(v+=3) transition of MgAr+. The blue, orange, green, and red sticks correspond to individual lines of the rotational branches with J″ – N+ values of −3/2, −1/2, +1/2, and +3/2, respectively. (b) Spectrum of the 3dπ3/2(v′=2) ← A3/2+(v″=1) recorded after selecting the A3/2+ levels with J″ = 4.5, 5.5, and 7.5 at the position marked with a red arrow in panel a. See text for details.

We determined the rotational line positions by fitting Gaussian functions to the spectra. The band origins and rotational constants of the 3dπ states were determined from the rotational line positions in a least-squares fit based on the standard expression,[13,40]

In eq , is the rotational constant of the state reported in ref (36). In the case of the 3dπ1/2(v′=0) state, the resolution of the spectrum was not sufficient to determine the rotational constant from eq , and it was instead determined from linear extrapolation of the rotational constants of the higher vibrational levels. The band origin was subsequently deduced by matching the calculated rotational contour with the contour observed experimentally. The band origins and rotational constants are listed in Table .

Table 1

Measured Transitions to the 3dπΩ′(v′) States of 24MgAr+ a

v′	v″	Ω′	initial state	ν̃v′v″	Tv′	Bv′	Γv′
0	0	1/2	A1/2+	35352.9(3)	0.0(3)	0.192(3)	–
0	0	3/2	A3/2+	35313.94(20)	38.03(22)	0.1907(20)	–
1	0	1/2	A1/2+	35610.17(10)	257.27(14)	0.1883(17)	–
1	0	3/2	A3/2+	35570.58(10)	294.67(14)	0.1885(16)	–
2	1	1/2	A1/2+	35596.20(10)	509.28(14)	0.182(3)	–
2	1	3/2	A3/2+	35556.09(17)	545.54(20)	0.185(4)	–
3	1	1/2	A1/2+	35842.13(10)	755.21(14)	0.1813(13)	–
3	1	3/2	A3/2+	35801.31(10)	790.76(14)	0.1853(16)	–
4	2	1/2	A1/2+	35823.01(10)	995.64(14)	0.1778(22)	–
4	1	3/2	A3/2+	36040.49(10)	1029.94(14)	0.179(3)	–
5	2	1/2	A1/2+	36057.46(10)	1230.09(14)	0.1739(11)	–
5	5	3/2	A3/2+	35277.2(19)	1264.4(19)	–	9.4(24)
6	5	1/2	A1/2+	35543.90(20)	1456.60(22)	–	1.00(20)
6	5	3/2	A3/2+	35500(4)	1487(4)	–	18(4)
7	5	1/2	A1/2+	35763.7(10)	1676.4(14)	–	5.7(17)
7	8	1/2	A1/2+	35082(5)	1676.4(14)	–	5.7(17)
7	5	3/2	A3/2+	35721(7)	1708(7)	–	35(9)
8	8	1/2	A1/2+	35303(7)	1897(7)	–	36(9)
8	8	3/2	A3/2+	35290(30)	1950(30)	–	160(40)
9	8	1/2	A1/2+	35523(15)	2117(15)	–	77(19)
9	8	3/2	A3/2+	35499(10)	2166(10)	–	50(13)b
10	8	1/2	A1/2+	35738(10)	2329(4)	–	17(5)
10	12	1/2	A1/2+	34920(3)	2329(4)	–	17(5)
10	8	3/2	A3/2+	35690(3)	2357(3)	–	15(4)b
11	12	1/2	A1/2+	35111.3(24)	2520.4(24)	–	12(3)
11	12	3/2	A3/2+	35091(8)	2572(8)	–	38(10)
12	12	1/2	A1/2+	35320(4)	2729(4)	–	20(5)
12	12	3/2	A3/2+	35272(4)	2752(4)	–	21(5)
13	12	1/2	A1/2+	35499(3)	2908(3)	–	14(3)
13	12	3/2	A3/2+	35473(4)	2953(4)	–	22(5)b
14	12	1/2	A1/2+	35686(3)	3094.0(19)	–	9.0(23)
14	18	1/2	A1/2+	34668.0(15)	3094.0(19)	–	9.0(23)
14	12	3/2	A3/2+	35643(7)	3125(3)	–	13(4)
14	18	3/2	A3/2+	34630.6(22)	3125(3)	–	13(4)
15	12	1/2	A1/2+	35871.3(10)	3279.8(10)	–	4.9(12)
15	18	1/2	A1/2+	34853.6(10)	3279.8(10)	–	4.9(12)
15	12	3/2	A3/2+	35822.8(6)	3303.2(6)	–	3.0(7)
16	12	1/2	A1/2+	36038.1(9)	3447.4(11)	–	5.3(14)
16	18	1/2	A1/2+	35022.1(17)	3447.4(11)	–	5.3(14)
16	12	3/2	A3/2+	35998.4(19)	3478.3(22)	–	11(3)
16	18	3/2	A3/2+	34983(3)	3478.3(22)	–	11(3)
17	12	1/2	A1/2+	36197.0(10)	3606.2(13)	–	6.0(16)
17	18	1/2	A1/2+	35181(3)	3606.2(13)	–	6.0(16)
17	12	3/2	A3/2+	36154(5)	3639(3)	–	15(4)b
17	18	3/2	A3/2+	35145.6(24)	3639(3)	–	15(4)
18	18	1/2	A1/2+	35344.4(23)	3770.1(23)	–	11(3)
18	18	3/2	A3/2+	35299.6(23)	3794.2(23)	–	11(3)
19	18	1/2	A1/2+	35499.0(20)	3924.7(20)	–	10.0(20)
19	18	3/2	A3/2+	35456.1(8)	3950.7(8)	–	4.2(10)
20	18	1/2	A1/2+	35635.2(17)	4060.8(17)	–	8.5(21)
20	18	3/2	A3/2+	35603(3)	4098(3)	–	14(3)b
21	18	1/2	A1/2+	35786.2(12)	4211.9(12)	–	5.8(15)
21	18	3/2	A3/2+	35739(3)	4234(3)	–	15(4)b
22	4	1/2	B+	34274.6(21)	4347.1(21)	–	10(3)
22	18	3/2	A3/2+	35869.7(14)	4365.2(14)	–	6.7(17)
22	4	3/2	B+	34293.5(13)	4365.2(14)	–	6.7(17)
23	4	1/2	B+	34400.0(21)	4472.5(21)	–	11(3)
23	18	3/2	A3/2+	35998.7(6)	4493.8(6)	–	3.1(8)
23	4	3/2	B+	34421.8(6)	4493.8(6)	–	3.1(8)
24	4	1/2	B+	34521.7(12)	4594.2(13)	–	6.2(15)
24	18	3/2	A3/2+	36121.5(6)	4616.6(7)	–	3.4(8)
24	4	3/2	B+	34544.6(7)	4616.6(7)	–	3.4(8)
25	4	1/2	B+	34638.3(8)	4710.7(8)	–	4.0(10)
25	18	3/2	A3/2+	36236.4(13)	4731.7(14)	–	6.7(17)
25	4	3/2	B+	34660.1(14)	4731.7(14)	–	6.7(17)
26	4	1/2	B+	34748.7(10)	4821.2(10)	–	5.0(13)
26	4	3/2	B+	34766.5(20)	4838.9(20)	–	10(3)
27	4	1/2	B+	34851.2(18)	4923.7(18)	–	9.1(23)
27	4	3/2	B+	34865.9(15)	4938.4(15)	–	7.4(19)
28	4	1/2	B+	34947.3(16)	5019.8(16)	–	8.0(20)
28	4	3/2	B+	34961.3(13)	5033.8(13)	–	6.4(16)
30	4	1/2	B+	35120.0(19)	5192.4(19)	–	9.3(23)
30	4	3/2	B+	35135.0(5)	5207.4(6)	–	2.7(7)
31	4	1/2	B+	35199.2(7)	5271.7(7)	–	3.3(8)
31	4	3/2	B+	35211.7(3)	5284.2(4)	–	1.7(4)
32	4	1/2	B+	35272.8(9)	5345.3(9)	–	4.5(11)
32	4	3/2	B+	35286.0(9)	5358.4(9)	–	4.4(11)
33	4	1/2	B+	35343.2(11)	5415.7(11)	–	5.4(14)
33	4	3/2	B+	35353.3(7)	5425.7(7)	–	3.3(8)
34	4	1/2	B+	35403.5(3)	5476.0(4)	–	1.6(4)
35	4	1/2	B+	35461.1(8)	5533.5(8)	–	4.1(10)
35	4	3/2	B+	35471.1(13)	5543.5(13)	–	6.5(16)
36	4	1/2	B+	35513.4(7)	5585.9(7)	–	3.5(9)
36	4	3/2	B+	35521.4(11)	5593.9(11)	–	5.5(14)
37	4	1/2	B+	35562.4(6)	5634.8(6)	–	3.0(7)
37	4	3/2	B+	35568.7(10)	5641.2(10)	–	4.8(12)
38	4	1/2	B+	35603.6(11)	5676.0(11)	–	5.6(14)
38	4	3/2	B+	35610.7(15)	5683.1(15)	–	7.6(19)
39	4	1/2	B+	35642.9(20)	5715.4(21)	–	10(3)
39	4	3/2	B+	35648.6(9)	5721.1(10)	–	4.7(12)
40	4	1/2	B+	35678.0(8)	5750.4(8)	–	4.0(10)
40	4	3/2	B+	35682.7(7)	5755.2(7)	–	3.3(8)
41	4	1/2	B+	35709.1(7)	5781.5(8)	–	3.7(9)
41	4	3/2	B+	35712.8(6)	5785.2(6)	–	3.0(8)
42	4	1/2	B+	35737.7(6)	5810.1(6)	–	3.0(8)
42	4	3/2	B+	35740.3(6)	5812.8(6)	–	3.0(8)
43	4	1/2	B+	35760.9(5)	5833.4(6)	–	2.6(7)
43	4	3/2	B+	35764.0(5)	5836.4(6)	–	2.6(6)
44	4	1/2	B+	35782.3(3)	5854.7(4)	–	1.5(4)
44	4	3/2	B+	35784.6(3)	5857.1(4)	–	1.5(4)
45	4	1/2	B+	35800.3(3)	5872.8(4)	–	1.7(4)
45	4	3/2	B+	35802.3(3)	5874.8(4)	–	1.7(4)
46	4	1/2	B+	35815.5(4)	5887.9(5)	–	2.1(5)
46	4	3/2	B+	35817.7(4)	5890.1(4)	–	2.0(5)
47	4	1/2	B+	35828.6(5)	5901.0(5)	–	1.5(5)
47	4	3/2	B+	35829.5(5)	5901.9(5)	–	1.5(5)
48	4	1/2	B+	35838.6(5)	5911.1(5)	–	1.5(5)
48	4	3/2	B+	35839.5(5)	5911.9(5)	–	1.5(5)
49	4	1/2	B+	35847.0(5)	5919.4(5)	–	1.4(5)
49	4	3/2	B+	35847.8(5)	5920.3(5)	–	1.4(5)
50	4	1/2	B+	35853.8(5)	5926.3(5)	–	1.5(5)
50	4	3/2	B+	35854.7(5)	5927.1(5)	–	1.5(5)
51	4	1/2	B+	35859.2(5)	5931.7(5)	–	1.5(5)
51	4	3/2	B+	35860.1(5)	5932.5(5)	–	1.5(5)
52	4	1/2	B+	35863.2(5)	5935.6(5)	–	1.3(5)
52	4	3/2	B+	35864.1(5)	5936.5(5)	–	1.3(5)

The band origins and rotational constants for transitions to low-v′ levels were obtained from the analysis of the resolved rotational structure (see Section ). The band origins and widths for higher-lying vibrational levels were determined by fitting Lorentzian functions, unless specified otherwise. All measured quantities are in cm–1, and the numbers in parentheses represent 1 standard deviation in units of the last digit.

Line position and line width determined using a Fano line shape function.

We derived the vibrational constants ωe and ωexe, as well as the rotational constants Be and αe, from the observed positions of the rovibrational levels of the 3dπ1/2(v′≤5) and 3dπ3/2(v′≤4) states. Higher-lying levels of 3dπ states deviated strongly from a Morse-type behavior because of perturbations, as discussed below (see Figure , parts a and b), and were not included in this analysis. To determine the dissociation thresholds D0, we exploited the thermodynamic cyclesandfor the two spin–orbit components of the 3dπ state. In the equations, D0(A1/2+) = 5476.7(10) cm–1 and D0(A3/2+) = 5491.5(10) cm–1 are the dissociation wavenumbers of the A1/2+ and A3/2+ states,[36] respectively, and ν̃00(3dπ1/2) and ν̃00(3dπ3/2) are the band origins listed in Table . The quantities denoted E(Mg+...) stand for the atomic level energies of Mg+ (from ref (41)) at the relevant dissociation limits. The molecular constants determined for the 3dπ states are summarized in Table .

Figure 7

Vibrational intervals ΔG(v′) = T – T of the 3dπ1/2 (a) and 3dπ3/2 (b) states and spin–orbit splittings of the 3dπΩ′ state of MgAr+ (c). The open circles with error bars correspond to experimental data, the orange lines to exterior-complex-scaling calculations of the predissociation interactions between the 3dπ states and the repulsive states correlating to the Mg(3s2) + Ar+(2P) dissociation limits, and the blue lines to approximate perturbative calculations based on the same method.

Table 2

Molecular Constants Determined from the Analysis of the Spectra of the Low Vibrational Levels (v′ ≤ 5) of the 3dπΩ′ State of 24MgAr+ a

	3dπ1/2	3dπ3/2
D0	5944.7(10)	5907.7(10)
ωe	263.17(8)	262.48(6)
ωexe	2.857(15)	2.898(14)
Be	0.1933(10)	0.1920(13)
αe	0.0035(3)	0.0023(5)

All data are in cm–1 and the numbers in parentheses represent one standard deviation in units of the last digit.

Diffuse Spectra of the 3dπΩ′ State at High v′ Values

Most transitions to the 3dπ1/2 and 3dπ3/2 vibrationally excited levels were measured in overview scans from intermediate levels having large Franck–Condon overlap to the final states. In the following, we present selected spectra that illustrate the main characteristics of these transitions.

Figure depicts overview spectra of transitions to the 3dπ3/2(v′ = 4–16) levels from the A3/2+(v″) state with v″ = 5 (panel a), v″ = 8 (panel b), and v″ = 12 (panel c), which were recorded by monitoring the Ar+ signal, as explained in Section 2. Each panel in Figure covers a spectral range of 1000 cm–1, such that the spectra from the different panels can be compared directly. Peaks marked with an asterisk (*) correspond to spurious transitions that are observable through resonant multiphoton dissociation. In the spectrum depicted in Figure c, we could not assign the line labeled with a question mark. The spectra show a rapid broadening and the disappearance of rotational structures of the transitions to the 3dπ3/2(v′) levels above v′ = 4. The observed line widths of the 3dπ3/2(v′) levels reach a maximum at v′ = 8 (see Figure b), decrease again, and then oscillate as v′ increases further. The transition to v′ = 8 is so broad that it slightly overlaps with the transition to v′ = 9. In contrast, the transitions to v′ = 10 and 15 are particularly narrow. The transitions to the v′ = 9, 10, and 13 levels have asymmetric lineshapes, which can be described by Fano profiles, and the Ar+ signal never returns to zero between v′ = 11 and v′ = 13. These observations indicate the possibility of weak direct excitation to the dissociation continuum by Laser 3. The same qualitative behavior of the line widths was also observed for the 3dπ1/2(v′) levels (see Figure ). The origin of the rapid variation of the line widths with v′ is discussed in Section 4.

Figure 4

Overview spectra of the 3dπ3/2(v′) ← A3/2+(v″) transitions of MgAr+ with v′ in the range 4–16 recorded from A3/2+ levels with v″ = 5 (a), v″ = 8 (b), and v″ = 12 (c). The lines marked with asterisks are spurious lines corresponding to transitions. See text for details.

Figure 8

Predissociation widths of the vibrational levels of the 3dπΩ′ states with Ω′ = 1/2 (a) and Ω′ = 3/2 (b). The open circles with vertical error bars correspond to the experimental data. The full lines represent the results of exterior-complex-scaling calculations using the same color code as in Figure (see text for details).

The fact that only the 3dπ3/2 spin–orbit component is visible in Figure is a consequence of the ΔΩ = 0 selection rule for transitions between initial and final states that are both well described by Hund’s case (a). It results from the fact that the electron-spin projection quantum number Σ does not change in the transitions. At low vibrational excitation, the and the 3dπΩ′ states are indeed well described by Hund’s case (a), and they both have a well-defined orbital angular momentum projection quantum number of |Λ| = 1. Close to the dissociation limit, the states are better described in Hund’s case (c), see ref (36), and this selection rule breaks down.

The overview spectrum of the 3dπ1/2(v′) ← A1/2+(v″=18) transitions in the region v′ = 17–21 is depicted in Figure . Next to the transitions to the 3dπ1/2(v′) levels, additional lines, marked with asterisks, can be attributed to transitions. The remaining lines, which exhibit asymmetric lineshapes or are only visible as shoulders, are tentatively assigned to transitions to excited vibrational levels of the 4sσ state because the line positions match a vibrational progression that ends at the Mg+(4s) + Ar(1S0) dissociation limit (see also Figure ). Vibrational levels of the 3dδΩ and 3dσ states can be ruled out because the associated Franck–Condon factors calculated from the potential-energy functions in Figure are too small. The results obtained for the 3dδΩ state will be presented in a separate publication. The small intensity fluctuations with a period of about 10 cm–1 which are present in some parts of the spectrum, e.g., between 35740 and 35780 cm–1, are caused by fluctuations in the laser pulse energies. The transition to the 3dπ1/2(v′=18) level is saturated in this spectrum and was remeasured at lower laser pulse energies (not shown) for the analysis.

Figure 5

Overview spectrum of MgAr+ recorded from the A1/2+(v″=18) intermediate level and showing transitions to the 3dπ1/2(v′=17–21) levels, as well as transitions to vibrationally excited levels of the 4sσ state. The lines marked by asterisks are spurious lines of the band system.

Figure 6

Overview spectrum of the 3dπΩ′(v′) ← B+(v″=4) transitions of MgAr+ in the region from v′ = 18 to the Mg+(3d) + Ar(1S0) dissociation limit, marked by a dashed vertical line in the lower panel. In the low-wavenumber part below the Mg+(4s) + Ar(1S0) dissociation limit designated by a dashed vertical line in the upper panel, the spectrum also reveals transitions to the highest vibrational levels of the 4sσ state. Lines marked with asterisks correspond to spurious lines of the band system. The region designated by a diamond corresponds to a region dominated by the Mg 3s3p ← 3s2 transition where no reliable intensities could be measured.

Figure shows the overview spectrum of both 3dπΩ′ spin–orbit components from v′ = 20 up to their dissociation limits recorded via the intermediate B+(v″ = 4) state. Because the B+(v″=4) state is well described by Hund’s case (b), transitions to both 3dπ spin–orbit components are allowed. In the spectrum, asterisks indicate again spurious transitions. The region around 35051 cm–1 designated by a diamond (◊) corresponds to the atomic Mg 3s3p ← 3s2 transition. In this region, the Mg+ ion signal was so strong that it saturated the detection system and prevented the measurement of reliable intensities. Above v′ = 22, the spin–orbit splitting of the 3dπΩ′ state decreases and can no longer be resolved above v′ = 47. In the range between v′ = 23 and 30, the line widths of the 3dπ1/2 vibrational states first decrease and then increase again, whereas the 3dπ3/2 state exhibits the opposite behavior. The transition intensities to the 3dπ states are also subject to large variations. For instance, the strong intensities for both spin–orbit components at v′ = 33 are followed by very weak (3dπ1/2) or even vanishing (3dπ3/2) intensities at v′ = 34. These observations indicate strong perturbations, which are analyzed in detail in Section .

An additional vibrational progression is observed directly below the Mg+(4s) + Ar(1S0) dissociation limit at 34186.7 cm–1. We attribute it to transitions to high vibrational levels of the 4sσ state, for the reasons presented in the discussion of Figure .

To determine the line positions and widths (full width at half-maximum) of the measured 3dπ vibrational levels, we fitted Lorentzian functions to the corresponding lines. For a few transitions to vibrational levels of the 3dπ3/2 state with asymmetric line shapes, we also fitted Fano profiles. The vibrational term values T and line widths Γ of all observed levels are listed in Table . The specified uncertainties in the line positions and widths correspond to 20% and 25% of the determined values for the widths, respectively. These uncertainties also take into account the fact that the vibrational bands have an underlying unresolved rotational structure. For vibrational levels that were measured from more than one initial state, the term values and widths correspond to a weighted average of all measurements. Because the spin–orbit splittings could not be resolved for v′ ≥ 47, the positions of the two 3dπ spin–orbit components were estimated using the atomic spin–orbit splitting of −0.87 cm–1 of the Mg+(3d) states,[41] which we expect to be a good approximation for the molecular spin–orbit splittings just below the dissociation limit.

The vibrational assignments presented above are based on a standard isotopic-shift analysis.[14] For the 3dπ3/2 state we measured the positions of the vibrational levels v′ = 3–7, 9, and 10 of 26MgAr+ (not shown), in addition to those of 24MgAr+. The isotopic shifts range from 20(5) cm–1 for v′ = 3 to 49(5) cm–1 for v′ = 10, which is only compatible with the vibrational assignments presented in Table . The corresponding vibrational constants for 24MgAr+ are listed in Table .

All level positions, level widths, and molecular constants determined in this analysis of the 3dπΩ′ state of MgAr+ are presented in Tables and 2. The data obtained for both spin–orbit components (Ω′ = 1/2, 3/2) extend from the lowest (v′ = 0) level all the way to the dissociation threshold beyond v′ = 50.

To facilitate the discussion and interpretation of these observations, overviews of the observed vibrational intervals ΔG(v′) = T – T are presented in panels (a) and (b) of Figure . The evolution of the spin–orbit splittings and the line widths with v′ are presented graphically in Figure c and Figure , respectively. These data reveal pronounced perturbations above v′ = 6, discussed in detail in the next section. Moreover, the spin–orbit splitting of the 3dπΩ′ states reaches ∼40 cm–1 for the low-lying vibrational levels and the states with Ω′ = 1/2 are systematically lower in energy than those with Ω′ = 3/2. Asymptotically, the 3dπ1/2 state correlates to the Mg+(3d 2D5/2) + Ar(1S0) dissociation limit whereas the 3dπ3/2 state correlates to the Mg+(3d 2D3/2) + Ar(1S0) limit. In the Mg+ ion, the spin–orbit splitting of the 3d 2D3/2 and 2D5/2 states is −0.87 cm–1.[41] Using the model of Cohen and Schneider,[42] we would expect a spin–orbit splitting of the 3dπΩ′ states on the order of 2/5 × (−0.87) = −0.35 cm–1, the Ω′ = 3/2 levels being lower in energy than the Ω′ = 1/2 ones. Our experimental results show instead a splitting that is inverted and up to 2 orders of magnitude larger. The origin of this behavior is discussed in detail in Section .

Analysis and Discussion

Theoretical Model

To explain the experimental results and the strong perturbations for both spin–orbit components of the 3dπ state, in particular those affecting the line widths and level energies, we employed a model that describes the interaction of the 3dπ states with the energetically close-lying and essentially repulsive charge-transfer (CT) states associated with the Mg(3s2) + Ar+(3p5 2P1/2) and Mg(3s2) + Ar+(3p5 2P3/2) dissociation limits. For the description of the two spin–orbit components of the 3dπ states we used the ab initio potential-energy functions published in ref (35). The relevant molecular constants obtained from these potential-energy functions are D0 = 5873 cm–1, ωe = 263.5 cm–1, and ωexe = 2.82 cm–1 and D0 = 5836 cm–1, ωe = 262.9 cm–1, and ωexe = 2.83 cm–1 for the 3dπ1/2 and 3dπ3/2 states, respectively. These theoretical values are in good agreement with those determined experimentally (see Table ). No accurate ab initio calculations are available for the CT states. Consequently, we used the indirect information on these states that are encoded in the perturbations of the 3dπ states to construct their potential-energy functions based on model potentials.

In first approximation, the potential-energy functions of the CT states correspond to the eigenvalues of the matrix[13,42,43]where VΣ(R) and VΠ(R) are the diabatic potential-energy curves of the CT states of Σ and Π symmetry (R is the internuclear distance), and a = −954.389 cm–1 is the spin–orbit coupling constant, which is assumed to be independent of R and corresponds to the atomic spin–orbit splitting 3a/2 between the Ar+(2P1/2) and Ar+(2P3/2) states. We denote the eigenvalues of eq as V1/2+, V1/2–, and V3/2, where the subscript represents the total-angular-momentum projection quantum number Ω and the superscripts + and – denote the larger and smaller of the two eigenvalues with Ω = 1/2. The diabatic potential-energy functions are modeled with the functionswhich consist of a repulsive Born–Mayer term[44] and a charge–induced-dipole attractive long-range term. In eq , AΛ and bΛ are adjustable parameters that depend on the angular-momentum projection quantum number Λ (Λ = 0, 1), is the polarizability volume of ground-state Mg,[45] andis the Tang–Toennies damping function.[46] The values of AΛ and bΛ are given in Table , and their determination is discussed below. The potential functions for the CT states depicted in Figure correspond to these values.

Table 3

Model Parameters (AΛ, bΛ) Determined to Describe the Potential-Energy Functions of the Charge-Transfer States Correlating to the Mg(3s2) + Ar+(2P) Dissociation Limit and the Interactions (Hint) between These States and the 3dπ States According to Equations and 9

AΣ	25 Eh
bΣ	1.14 a0–1
AΠ	15 Eh
bΠ	1.09 a0–1
Hint,1/2+	75 cm–1
Hint,1/2–	200 cm–1
Hint,3/2	250 cm–1

To determine the effect of the interaction of the 3dπ states with the CT states, we first solved the nuclear Schrödinger equation,separately for each potential-energy function using a Legendre–Gauss–Lobatto finite-element discrete-variable-representation (FEM-DVR) technique in combination with exterior complex scaling.[47−49] The use of exterior complex scaling permits an accurate description of the continua above the dissociation thresholds, which can be treated on the same footing as the bound states. Using the eigenstates determined from eq , we constructed an effective Hamiltonian matrix describing the 3dπ and CT bound and continuum states, and their mutual interactions. We assume that the configuration interaction between the 3dπ and CT states preserves the projection of the total angular momentum onto the internuclear axis, thus only states with the same Ω value interact and the Ω = 3/2 and Ω = 1/2 states can be treated separately.

The matrix elements of the effective Hamiltonian matrix for Ω = 3/2 are given bywhere Hint,3/2 describes the CT interaction between the 3dπ3/2 state and the CT state with Ω = 3/2. The basis states in eq include bound and continuum states. The eigenvalues of the effective Hamiltonian matrix associated with predissociating resonances of predominant 3dπ3/2 character were selected by inspecting the eigenvectors. As usual in complex scaling, they possess a real part which is equal to the energy of the eigenstate and an imaginary part which is equal to – Γ/2, where Γ is the width of the eigenstate.

The interaction of the 3dπ1/2 state with the two Ω = 1/2 CT states is described by the effective Hamiltonian with matrix elements given bywhere the coupling elements Hint,1/2+ and Hint,1/2– are different. In principle, these coupling elements are R-dependent. However, in the absence of reliable data, we assumed R-independent couplings, which yields results in good agreement with the experimental data while restricting the number of adjustable parameters to a minimum (see also the discussion in the next section).

Line Positions and Line Widths

Panels a and b of Figure show Birge–Sponer plots[14,50] of the vibrational levels of the 3dπ1/2 and 3dπ3/2 states, respectively, i.e., the differences in the term values of adjacent levels, ΔG(v′) = T – T, versus v′ + 1/2. For the lowest vibrational levels (v′ ≲ 5), the data points form a straight line, indicating unperturbed states. Beyond v′ = 5 rapid oscillations of ΔG(v′) occur, which indicate perturbations. For the highest vibrational levels (v′ ≳ 35), the ΔG(v′) values evolve smoothly again, as do the spin–orbit splittings depicted in Figure c, which indicates that the perturbations become smaller in this range.

Figure depicts the experimentally determined line widths of all measured states. At low v′, the rotational structure could be resolved and the observed widths are limited by the instrumental resolution (∼0.1 cm–1). They are therefore set to zero in Figure . A rapid increase of the line widths is observed between v′ ∼ 5 and v′ ∼ 10, which is followed by an oscillating decrease toward the dissociation limit, as already stated in Section . The evolution of the widths with v′ closely reflects the perturbations in the level positions depicted in Figure , which suggests that both the widths and the perturbations in the energy-level structure have the same origin.

By carefully optimizing the parameters AΛ and bΛ of the model potentials as well as the interaction strengths Hint introduced in Section , we were able to satisfactorily reproduce the experimental results with our theoretical model, the results of which are shown as full orange lines in Figures and 8. All model parameters are listed in Table . The good agreement between the experimental results and those obtained with the theoretical model leads to the conclusion that the observed line widths and the perturbations of the vibrational level energies of the 3dπ states originate from the interaction with the CT states and their associated continua. Moreover, the Ar+ dissociation products detected experimentally at v′ values higher than 5 are direct evidence of CT-induced predissociation, and the line widths reflect the predissociation lifetimes of the 3dπ vibrational resonances.

Figure shows the potential functions of the 3dπ and CT states. Because the spin–orbit splitting of the 3dπ states is small on the scale of the figure, the two components lie almost on top of each other. In addition, the figure depicts all measured vibrational levels of the 3dπ3/2 state (the 3dπ1/2 levels are not shown for clarity). The 1/2– and V3/2 potential curves cross the 3dπ1/2 and 3dπ3/2 potential curves just below their v′ = 9 and v′ = 8 vibrational levels, respectively, where the observed predissociation widths are largest. This observation is an illustration of the well-known fact that predissociation is fastest in the vicinity of curve crossings.[13] In principle, predissociation is possible for all v′ levels, because they all lie above the Mg(3s2) + Ar+(2P3/2) dissociation limit. However, for the lowest 3dπ vibrational levels (v′ ≲ 5) predissociation is strongly inhibited because of the vanishing Franck–Condon overlap with the CT continuum states, which explains why, in this energy region, we did not detect Ar+ fragments on the time scale of our experiment.

Figure 9

Potential-energy functions used to calculate the positions and predissociation widths of the vibrational levels of the 3dπΩ′ state of MgAr+. The full and dashed lines correlating to the Mg+(3d) + Ar(1S0) dissociation limits correspond to the 3dπ1/2 and 3dπ3/2 states, respectively. The two full lines and the dashed line correlating to the Mg(3s2) + Ar+(2P) dissociation limits correspond to adiabatic potential-energy functions of the two Ω = 1/2 states and the Ω = 3/2 state. The gray potential-energy curves correspond to the first two diagonal elements of the matrix in eq . The horizontal lines correspond to the positions of the measured vibrational levels of the 3dπ1/2 state.

In general, the interaction strengths Hint depend on the internuclear distance R. However, our results show that in the present case the choice of constant Hint values is sufficient for a semiquantitative description of the perturbations because the coupling is only effective at R values near the crossings. The difference between the two interactions that describe the coupling to the Ω = 1/2 CT states (Hint,1/2+ = 75 cm–1 and Hint,1/2– = 200 cm–1) can be justified in terms of the Σ and Π characters of the V1/2+(R) and V1/2–(R) potentials of the CT states, which strongly depend on the internuclear separation. At large internuclear separations, i.e., for R ≳ 8 a0, the spin–orbit interaction mixes the two diabatic Hund’s case (a) Σ1/2 and Π1/2 states with Ω = 1/2 (see eq and the subsequent discussion) and the resulting V1/2±(R) adiabatic states correspond to Hund’s case (c) states. The V1/2+(R) adiabatic state has a predominant Π character (|cΠ+|2 = 0.67), whereas the V1/2–(R) state has predominant Σ character (|cΣ–|2 = 0.67) in this range. As R decreases and approaches 6.4 a0, the diagonal elements in eq corresponding to Σ1/2 and Π1/2 become degenerate (see gray curves in Figure ) and both Ω = 1/2 components become equal mixtures of Σ and Π character (|cΣ±|2 = |cΠ±|2 = 0.5). Below R ≈ 6 a0, the two Ω = 1/2 components can be well described by an almost pure Σ+ state in the case of the V1/2+ potential function and an almost pure Π1/2 state in the case of the V1/2– potential function. In the region of the curve crossings with the 3dπ1/2 state (R ≈ 5 – 6 a0), Λ is therefore a good quantum number for all states involved. Because homogeneous (ΔΛ = 0) perturbations are stronger than heterogeneous (|ΔΛ| = 1) perturbations, the CT state with the V1/2– potential function is coupled much more strongly to the 3dπ1/2 state than the state with V1/2+ potential, which explains why Hint,1/2– (200 cm–1) is larger than Hint,1/2+ (75 cm–1).

To gain a better understanding of the processes described in the previous paragraphs, we also calculated the level energies and widths of the 3dπ states in a perturbative treatment, in addition to the diagonalization of the complete Hamiltonians presented Section . Following Fano’s procedure,[13,51] the predissociation widths that arise from the interaction of the 3dπ3/2(v′) state (with unperturbed energy E) with the continuum of the V3/2 state are given bywhere |ε, V3/2⟩ is an energy-normalized continuum function with energy ε. The continuum-induced energy shifts are calculated aswhere denotes the Cauchy principal value. Equations and 11 are readily evaluated in the framework of exterior complex scaling, as explained in Appendix A. The widths and energy shifts of the 3dπ1/2 levels are given by incoherent sums of the interactions with the two Ω = 1/2 CT states,where the summands are calculated in the same way as in eqs and 11.

The results of the perturbative calculations are depicted as blue lines in Figures and 8. The agreement with the results obtained from the full calculation is so good that the blue lines are hardly distinguishable from the orange lines. This good agreement allows an interpretation of the processes based on eqs and 11. In these equations, the expression of the type is a Franck–Condon density, implying that the evolution of the widths and the energies with v′ is almost purely governed by the overlap of the 3dπ bound-state and CT continuum wave functions. This finding is also in accord with the observation that the predissociation widths are largest where the potential curves cross (see discussion above).

On the Energetic Order of the Spin–Orbit Components of the 3dπ States

As discussed at the end of Section , there are irregularities concerning the ordering of the spin–orbit components of the 3dπΩ′ state and the magnitude of the spin–orbit splitting. To resolve this issue, we complement our previous ab initio calculations[35] in the present work with relativistic Kramers-restricted ab initio complete-active-space configuration interaction (KR-CASCI) calculations[52,53] with variationally treated spin–orbit coupling at various Mg–Ar internuclear distances.

All KR-CASCI calculations were performed within the exact two-component Hamiltonian framework[54−56] including two-electron Coulomb and Gaunt contributions[57,58] in combination with very large, fully uncontracted aug-cc-pV5Z basis sets[59] (denoted as 5Z*) for Mg and Ar. The reference wave function at each Mg–Ar internuclear distance was obtained from an average-of-configurations self-consistent-field calculation[60] for the open-shell MgAr+ ion by considering all possible configurations of seven electrons in 32 Kramers-paired spinors (i.e., representing the Ar 3p and Mg 3s3p3d4s4p shells). The subsequent KR-CASCI correlation step encompassed the same correlation space, i.e., correlating the seven valence electrons of the MgAr+ molecular ion in 32 Kramers-paired spinors denoted as KR-CASCI(7,32)/5Z*. Consequently, the KR-CASCI(7,32)/5Z* calculations reported in this work provide useful qualitative insight, whereas for quantitative ab initio data we refer the reader to our previous work.[35]

All KR-CASCI(7,32)/5Z* calculations were carried out with the Dirac19 program package.[61,62] Molecular constants have been derived by a least-squares fit of the potential energy curves to a fifth-order polynomial by means of the TWOFIT utility program available in Dirac19.

Considering the qualitative nature of the KR-CASCI(7,32)/5Z* calculations, the resulting excited-state equilibrium internuclear distance of Re(3dπΩ′) = 4.67 a0 for both spin–orbit-split 3dπΩ′ states is in fair agreement with the experimentally determined value of Re(3dπΩ′) ≈ 4.56 a0. Moreover, we find a spin–orbit splitting of the Ω′ = 1/2, 3/2 components of approximately 33.20 cm–1 (v′ = 0) which compares well with the measured value of 38.03 cm–1 (v′ = 0, see Table ). In line with the experimental data and our previous ab initio results,[35] the Ω′ = 1/2 spin–orbit component of the 3dπΩ′ manifold is the lower state.

Figure shows the square of the CI coefficients (denoted as weight) of the leading configurations of the 3dπΩ′ states as a function of the internuclear Mg–Ar distance. The total weight for each electronic state is normalized to 1.0. As can be understood in view of Figure , starting from the asymptotic limit at large internuclear distances, the dominating configurations of the eigenvectors of the 3dπΩ′ states (with weights of ∼1.0) originate from the single occupation of a Mg 3dπΩ′ spinor in addition to the closed-shell [Ar 3p6] core. When approaching the equilibrium internuclear distance toward the strongly repulsive region of the potential energy curves for the 3dπΩ′ states (Mg–Ar internuclear distances smaller than ∼5.0 a0), the composition of the corresponding eigenvector markedly changes for each of the Ω′ = 1/2, 3/2 spin–orbit components as is evident from Figure . The additional configuration gaining simultaneously particular weight in both eigenvectors arises from the occupation of a spinor that exhibits predominantly a Mg-centered 4p character according to a Mulliken population analysis of the reference molecular spinor basis. Although such a contribution is dipole-forbidden in the asymptotic limit where the Mg+ ion is isolated, the presence of the Ar “atom” in the MgAr+ ion lifts this restriction at short internuclear distances. Moreover, since the energetic order of the spin–orbit components of the Mg 4p manifold is pπ1/2 < pπ3/2,[41] we attribute the observed large and reversed spin–orbit splitting of the 3dπΩ′ states to this somewhat unexpected configurational mixing that is most pronounced at Mg–Ar internuclear distances between 3.6 a0 and 4.7 a0.

Figure 10

Squares of the CI coefficient (denoted as weight) of the leading configurations of the 3dπΩ′ states as a function of the internuclear Mg–Ar distance as obtained from KR-CASCI(7,32)/5Z* calculations. Each leading configuration can be written as [Ar 3p6]x1 in a compact notation, where x denotes the nature of the additional occupied spinor(s). Blue-colored symbols refer to configurations contributing to the 3dπ1/2 and red-colored ones to the 3dπ3/2 state.

Finally, it is important to note that a coupling of the 3dπΩ′ states to the Ar CT states can be ruled out as a source of the observed inverse energetic ordering of the 3dπΩ′ spin–orbit components. As is illustrated in Figure , the largest contributions, although with weights <0.1, that can be attributed to an Ar CT configuration within the 3dπ3/2 eigenvector composition, can only be found at Mg–Ar internuclear distances at about 6.6–7.6 a0. As discussed in the previous section, it is at these internuclear distances where we expect the Ar CT states to cross the Mg-centered 3dπΩ′ states.

Conclusions and Outlook

In this article, we have presented a complete set of measurements of the 3dπ1/2 and 3dπ3/2 Rydberg states of MgAr+, which extends from the lowest vibrational levels (v′ = 0) up to the dissociation limits. The transitions to low vibrational levels (v′ ≲ 5) could be partially rotationally resolved, whereas higher-lying levels were observed as broad diffuse bands. From these measurements, we derived vibrational and rotational constants as well as the dissociation energies. The vibrational level positions and the line widths were interpreted using a model that describes the predissociation interaction of the 3dπ states with the CT states that correlate to the Mg(3s2) + Ar+(2P) dissociation limits. As basis states in the model, we used the solutions of the nuclear Schrödinger equation for the individual potential-energy functions using a FEM-DVR method in combination with exterior complex scaling. We also observed high-lying vibrational levels of the 4sσ state.

The fast predissociation of the 3dπ states is the result of the curve crossings with the CT states, as shown in Figure . The CT states have almost twice the bond length of the 3dπ states (Re = 4.56 a0 versus Re ≈ 7.5 a0). Consequently, their repulsive parts intersect the bound regions not only of the 3dπ states, but also of the entire manifold of MgAr+ Rydberg states that belong to series converging on the MgAr2+ X2+ state (see also Figure b). The large difference in the bond lengths can be understood in terms of the relevant atomic radii. The bond length of the CT states is determined by the atomic radii of Mg(3s2) and Ar+(2P). The 3dπ states, however, are part of a Rydberg series that converges on the MgAr2+ ground state and the relevant atomic radii are those from Mg2+(1S0) and Ar(1S0), as also revealed by the similar bond lengths of the 3dπ (Re = 4.56 a0) and X2+ (Re = 4.38 a0) states. The difference in the atomic radii of ground-state Mg and Mg2+ is much larger than for Ar(1S0) and Ar+(2P), which explains the different bond lengths for the 3dπ and CT states. Comparable bond-length differences are expected for other molecular cations F1F2+ forming thermodynamically stable doubly charged cations F1F22+ upon ionization; i.e., in general and , because F2 is typically much harder than F1. Consequently, the process of CT-induced predissociation observed in this article is likely to be a general property of the Rydberg states of such cations having the same electronic symmetry as the CT dissociative states.

The electronic coupling elements Hint that describe the CT interaction (see Section ) are essentially governed by the following two-electron integral,where r12 is the distance between the two electrons. For higher Rydberg states this integral scales as 1/n3/2, where n is the principal quantum number of the Rydberg electron, according to the well-known scaling of the wave function amplitude at the ionic core.[63,64] The observed line widths of Rydberg states therefore approximately scale with 1/n3 (see eq ). Starting from a line width of 10 cm–1 at n = 3, representative of our observations for the 3dπ states, we would expect line widths of ∼0.0001 cm–1 and ∼0.004 cm–1 (or lifetimes of ∼50 ns and ∼1.3 ns) at n = 130 and n = 40, respectively. These time scales are compatible with measurements we carried out in our recent study on the MgAr2+ ground state[28] by PFI-ZEKE photoelectron spectroscopy. In these measurements, the lowest Rydberg states with a MgAr2+ core we detected corresponded to n ≈ 130, which implies that these states have a lifetime of less than 1 μs, compatible with the estimated value obtained from the scaling law (∼50 ns). The onset of the pulsed-field-ionization signal in PFI-ZEKE photoelectron spectra is thus observed when Stark mixing by stray electric field sets in.[64,65] In the same study,[28] we observed autoionizing resonances corresponding to MgAr+ Rydberg states with principal quantum number n ∼ 40 and autoionization line widths of 0.3 cm–1 (corresponding to a lifetime of 180 ps), which is much broader than the predicted CT-induced predissociation width at n = 40 (see above). This confirms our simple order-of-magnitude estimate of predissociation lifetimes.

The comparably simple structure of MgAr+ and the thermodynamic stability of MgAr2+ make the Rydberg states of the MgAr+ ion an ideal model system for studying CT interactions in the Rydberg states of molecular cations. Because the ground state of MgAr2+ has a closed-shell electron configuration, the Rydberg states of MgAr+ are effective one-electron systems, if CT-induced predissociation is disregarded. In future work, one may therefore be able to calculate these states using the same approach as the one successfully employed in ref (66) to calculate the Rydberg states of ArH and KrH, extending it to CT processes.

We expect that the results presented here on the structure and dynamics of the Rydberg states of MgAr+ can be transferred to a broader class of singly charged cations. In particular, we expect Rydberg-state dynamics in these systems to be governed by CT-induced predissociation following the general mechanism presented in Figures and 2.