Precision measurement of the rotational energy-level structure of the   three-electron molecule He$_2^+$

Luca Semeria; Paul Jansen; Fr\'ed\'eric Merkt

arXiv:1705.00869·physics.chem-ph·May 3, 2017

Precision measurement of the rotational energy-level structure of the three-electron molecule He$_2^+$

Luca Semeria, Paul Jansen, Fr\'ed\'eric Merkt

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TL;DR

This study precisely measured the rotational energy levels of the He₂⁺ molecule's ground state, revealing discrepancies with recent ab initio calculations that grow with higher rotational states.

Contribution

The paper provides highly accurate experimental rotational energy levels of He₂⁺ and compares them with theoretical calculations, highlighting significant deviations at higher rotational quantum numbers.

Findings

01

Measured rotational levels with 8 x 10⁻⁴ cm⁻¹ accuracy.

02

Discrepancies between experimental data and ab initio calculations increase with N⁺.

03

Differences reach up to 0.07 cm⁻¹ at N⁺=19.

Abstract

The term values of all rotational levels of the $^{4}$ He $_{2}^{+} X^{+}^{2} Σ_{u}^{+} (ν^{+} = 0)$ ground vibronic state with rotational quantum number $N^{+} \leq 19$ have been determined with an accuracy of 8 x 10 $^{- 4}$ cm $^{- 1}$ ( $\sim 25$ MHz) by MQDT-assisted Rydberg spectroscopy of metastable He $_{2}^{*}$ . Comparison of these term values with term values recently calculated ab initio by Tung et al. [J. Chem. Phys. 136, 104309 (2012)] reveal discrepancies that rapidly increase with increasing rotational quantum number and reach values of 0.07 cm $^{- 1}$ ( $\sim 2.1$ GHz) at $N^{+} = 19$ .

Tables4

Table 1. Table 1: Rotational term values of the a 3 Σ u + ( ν ′′ = 0 ) superscript 𝑎 3 superscript subscript Σ 𝑢 superscript 𝜈 ′′ 0 a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0) state of He 2 (3rd column) and the X + Σ u + 2 ( ν + = 0 ) superscript 𝑋 superscript superscript subscript Σ 𝑢 2 superscript 𝜈 0 X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0) ground state of He + 2 superscript subscript absent 2 {{}_{2}}^{+} (6th column). For comparison, term values calculated with the molecular constants for the a 3 Σ u + ( ν ′′ = 0 ) superscript 𝑎 3 superscript subscript Σ 𝑢 superscript 𝜈 ′′ 0 a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0) state of He 2 reported by Focsa et al. Focsa et al. ( 1998 ) and the term values of the X + Σ u + 2 ( ν + = 0 ) superscript 𝑋 superscript superscript subscript Σ 𝑢 2 superscript 𝜈 0 X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0) state of 4 He + 2 superscript subscript absent 2 {{}_{2}}^{+} calculated ab initio by Tung et al. Tung et al. ( 2012 ) , are shown in the 2nd and 5th columns respectively. All values are given in cm -1 .

$N^{''}$	Ref. Focsa et al. (1998)¹¹1Calculated using the rotational constants reported by Focsa et al. Focsa et al. (1998).	This work	$N^{+}$	Ref. Tung et al. (2012)	This work²²2The observed rotational term values can be described by $B_{0}^{+} = 7.101 143 (25)$ cm^-1, $D_{0}^{+} = 5.2156 (36) × 10^{- 4}$ cm^-1, $H_{0}^{+} = 3.29 (16) × 10^{- 8}$ cm^-1, and $L_{0}^{+} = - 7.3 (22) × 10^{- 12}$ cm^-1
	(FTIR)			(ab initio)	(MQDT fit)
1	$0$	$0$	1	$0$	$0$
3	$75.8129$ p m 0.0003	$75.8136$ p m 0.0008	3	$70.936$	$70.9379$ p m 0.0008
5	$211.9937$ p m 0.0008	$211.9950$ p m 0.0008	5	$198.359$	$198.3647$ p m 0.0008
7	$408.0605$ p m 0.0016	$408.0614$ p m 0.0008	7	$381.822$	$381.8346$ p m 0.0008
9	$663.322$ p m 0.003	$663.3231$ p m 0.0008	9	$620.683$	$620.7021$ p m 0.0009
11	$976.879$ p m 0.004	$976.8809$ p m 0.0008	11	$914.112$	$914.1367$ p m 0.0008
13	$1347.638$ p m 0.007	$1347.6396$ p m 0.0008	13	$1261.089$	$1261.1242$ p m 0.0008
15	$1774.307$ p m 0.011	$1774.3072$ p m 0.0008	15	$1660.420$	$1660.4627$ p m 0.0009
17	$2255.413$ p m 0.017	$2255.4133$ p m 0.0008	17	$2110.736$	$2110.7932$ p m 0.0009
19	$2789.31$ p m 0.03	$2789.3056$ p m 0.0008	19	$2610.505$	$2610.5744$ p m 0.0009

Table 2. Table 2: Observed transitions from the a 3 Σ u + superscript 𝑎 3 superscript subscript Σ 𝑢 a\,^{3}\Sigma_{u}^{+} ( ν ′′ = 0 , N ′′ ) superscript 𝜈 ′′ 0 superscript 𝑁 ′′ (\nu^{\prime\prime}=0,N^{\prime\prime}) state of 4 He 2 to the n 𝑛 n p N N + subscript superscript 𝑁 𝑁 N^{+}_{N} Rydberg states belonging to series converging to the X + Σ u + 2 superscript 𝑋 superscript superscript subscript Σ 𝑢 2 X^{+}\,{}^{2}\Sigma_{u}^{+} ( ν + = 0 , N + ) superscript 𝜈 0 superscript 𝑁 (\nu^{+}=0,N^{+}) states of 4 He + 2 superscript subscript absent 2 {{}_{2}}^{+} and comparison with the results of MQDT calculations. The symbol Δ Δ \Delta represents the difference between observed and calculated line positions ( Δ = obs. − calc. Δ obs. calc. \Delta=\text{obs.}-\text{calc.} ). All values are given in cm -1 . Missing entries correspond to transitions that were not observed or to strongly blended transitions.

	$1 \to n p 1_{1}$		$1 \to n p 1_{2}$		$3 \to n p 3_{3}$		$3 \to n p 3_{4}$
$n$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$
$97$	$34 289.526 24$	$- 0.000 04$	$34 289.562 34$	$0.000 33$	$34 284.650 92$	$- 0.000 25$	$34 284.580 58$	$- 0.000 19$
$98$	$34 289.763 34$	$- 0.000 25$	$34 289.794 96$	$0.000 12$	$34 284.888 50$	$0.000 02$	$34 284.799 30$	$- 0.000 11$
$99$	$34 289.993 61$	$- 0.000 13$	$34 290.021 53$	$0.000 34$	$34 285.118 83$	$0.000 20$	$34 285.012 32$	$- 0.000 22$
$100$	$34 290.217 09$	$0.000 08$	$34 290.241 41$	$0.000 32$	$34 285.341 80$	$- 0.000 10$
$101$	$34 290.433 68$	$- 0.000 00$	$34 290.455 30$	$0.000 63$	$34 285.558 61$	$0.000 04$
$102$	$34 290.644 09$	$0.000 08$	$34 290.662 49$	$0.000 41$	$34 285.769 19$	$0.000 29$
$103$	$34 290.848 40$	$0.000 17$	$34 290.863 74$	$0.000 29$	$34 285.973 26$	$0.000 14$	$34 286.031 32$	$- 0.000 45$
$104$	$34 291.046 69$	$0.000 10$	$34 291.059 38$	$0.000 25$	$34 286.171 32$	$- 0.000 16$	$34 286.224 84$	$0.000 63$
$105$	$34 291.239 61$	$0.000 31$	$34 291.249 28$	$0.000 93$	$34 286.364 07$	$- 0.000 12$	$34 286.412 41$	$0.000 32$
$106$	$34 291.426 71$	$0.000 13$	$34 291.433 64$	$0.001 36$	$34 286.551 46$	$- 0.000 01$	$34 286.595 61$	$0.000 27$
$107$					$34 286.733 20$	$- 0.000 33$	$34 286.774 26$	$0.000 29$
$108$					$34 286.909 93$	$- 0.000 61$	$34 286.948 13$	$0.000 10$
$109$	$34 291.958 07$	$- 0.000 01$	$34 291.949 02$	$0.000 23$	$34 287.082 15$	$- 0.000 55$	$34 287.117 89$	$0.000 27$
$110$	$34 292.124 79$	$- 0.000 55$	$34 292.107 16$	$- 0.000 53$	$34 287.249 73$	$- 0.000 46$	$34 287.283 64$	$0.000 80$
$111$	$34 292.287 99$	$- 0.000 31$	$34 292.257 90$	$- 0.000 48$	$34 287.412 91$	$- 0.000 25$	$34 287.444 33$	$0.000 54$
$112$	$34 292.446 64$	$- 0.000 28$	$34 292.399 93$	$- 0.000 38$

	$5 \to n p 5_{5}$		$5 \to n p 5_{6}$		$7 \to n p 7_{7}$		$7 \to n p 7_{8}$
$n$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$
$95$	$34 275.399 57$	$0.000 04$	$34 275.427 33$	$0.000 11$	$34 262.802 52$	$0.000 09$	$34 262.806 98$	$- 0.000 10$
$96$	$34 275.651 56$	$- 0.000 52$	$34 275.678 34$	$0.000 45$
$97$	$34 275.896 92$	$0.000 07$	$34 275.921 23$	$0.000 36$	$34 263.299 79$	$0.000 05$
$98$	$34 276.134 04$	$- 0.000 12$	$34 276.156 71$	$0.000 22$
$99$	$34 276.363 93$	$- 0.000 37$	$34 276.385 36$	$0.000 33$
$100$	$34 276.587 38$	$- 0.000 20$	$34 276.607 01$	$0.000 25$
$101$	$34 276.803 83$	$- 0.000 42$	$34 276.822 30$	$0.000 34$
$102$	$34 277.014 20$	$- 0.000 37$	$34 277.031 16$	$0.000 27$	$34 264.417 82$	$0.000 39$	$34 264.407 66$	$0.000 47$
$103$	$34 277.217 90$	$- 0.000 90$	$34 277.233 94$	$0.000 14$	$34 264.621 58$	$- 0.000 04$	$34 264.609 38$	$0.000 16$
$104$			$34 277.431 31$	$0.000 09$	$34 264.819 30$	$- 0.000 30$	$34 264.805 33$	$- 0.000 02$
$105$			$34 277.622 36$	$0.000 24$
$106$			$34 277.808 48$	$0.000 29$	$34 265.200 41$	$0.000 32$	$34 265.180 98$	$0.000 19$
$107$	$34 277.978 14$	$- 0.001 06$	$34 277.989 26$	$0.000 21$	$34 265.381 88$	$- 0.000 24$	$34 265.360 24$	$- 0.000 25$
$108$	$34 278.155 11$	$- 0.001 10$	$34 278.165 09$	$0.000 20$	$34 265.558 76$	$- 0.000 37$	$34 265.534 86$	$- 0.000 25$
$109$			$34 278.336 71$	$0.000 80$	$34 265.731 58$	$0.000 29$	$34 265.704 51$	$- 0.000 34$
$110$			$34 278.503 42$	$0.001 14$			$34 265.869 85$	$- 0.000 03$

	$9 \to n p 9_{9}$		$9 \to n p 9_{10}$		$11 \to n p 11_{11}$		$11 \to n p 11_{12}$
$n$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$
$95$					$34 226.284 84$	$- 0.000 40$	$34 226.327 83$	$0.000 28$
$96$					$34 226.537 70$	$- 0.000 07$	$34 226.578 89$	$0.000 52$
$97$					$34 226.782 51$	$- 0.000 03$	$34 226.821 74$	$0.000 23$
$98$					$34 227.019 52$	$- 0.000 32$	$34 227.057 85$	$0.000 57$
$99$					$34 227.250 19$	$0.000 21$	$34 227.285 85$	$- 0.000 12$
$100$
$101$	$34 247.811 32$	$- 0.000 84$	$34 247.794 65$	$- 0.000 56$	$34 227.690 02$	$0.000 10$	$34 227.723 74$	$0.000 52$
$102$	$34 248.023 25$	$- 0.000 11$	$34 248.003 94$	$- 0.000 50$	$34 227.900 13$	$- 0.000 11$	$34 227.932 72$	$0.000 42$
$103$	$34 248.227 07$	$- 0.000 44$	$34 248.207 04$	$0.000 86$	$34 228.103 35$	$- 0.001 11$	$34 228.135 95$	$0.000 61$
$104$			$34 248.402 10$	$- 0.000 01$	$34 228.301 68$	$- 0.001 13$	$34 228.333 51$	$0.000 94$
$105$			$34 248.587 08$	$- 0.003 17$	$34 228.495 11$	$- 0.000 41$	$34 228.524 68$	$0.000 47$
$106$	$34 248.806 35$	$0.000 53$	$34 248.780 80$	$- 0.000 30$	$34 228.682 71$	$- 0.000 09$
$107$	$34 248.987 51$	$- 0.000 35$	$34 248.960 20$	$- 0.000 14$	$34 228.864 15$	$- 0.000 69$	$34 228.891 15$	$- 0.000 41$
$108$			$34 249.136 06$	$0.001 18$	$34 229.041 69$	$- 0.000 16$
$109$			$34 249.304 31$	$- 0.000 73$
$110$	$34 249.505 50$	$0.000 99$	$34 249.470 92$	$0.000 30$			$34 229.405 79$	$0.000 23$
continued on next page

Table 3. Table 2: – Continued

	$13 \to n p 13_{13}$		$13 \to n p 13_{14}$		$15 \to n p 15_{15}$		$15 \to n p 15_{16}$
$n$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$
$95$					$34 175.184 30$	$- 0.000 85$	$34 175.232 64$	$- 0.000 70$
$96$	$34 202.766 38$	$- 0.000 54$	$34 202.819 75$	$- 0.000 30$	$34 175.437 46$	$- 0.000 22$
$97$	$34 203.011 09$	$- 0.000 58$	$34 203.062 27$	$- 0.000 39$	$34 175.681 58$	$- 0.000 85$	$34 175.728 09$	$0.001 02$
$98$	$34 203.248 13$	$- 0.000 83$	$34 203.297 39$	$- 0.000 55$	$34 175.919 62$	$- 0.000 11$	$34 175.963 13$	$0.000 41$
$99$	$34 203.478 55$	$- 0.000 54$	$34 203.526 25$	$0.000 08$	$34 176.149 54$	$- 0.000 33$	$34 176.192 44$	$0.001 14$
$100$	$34 203.701 53$	$- 0.000 82$	$34 203.747 80$	$0.000 17$			$34 176.412 76$	$- 0.000 32$
$101$	$34 203.918 40$	$- 0.000 61$	$34 203.962 97$	$0.000 38$	$34 176.590 00$	$0.000 21$	$34 176.628 50$	$0.000 17$
$102$	$34 204.129 53$	$0.000 21$	$34 204.172 03$	$0.000 74$	$34 176.799 39$	$- 0.000 71$
$103$	$34 204.333 55$	$0.000 02$
$104$	$34 204.533 44$	$0.001 56$	$34 204.571 22$	$0.000 34$	$34 177.202 82$	$0.000 16$
$105$	$34 204.724 62$	$0.000 04$	$34 204.764 64$	$0.002 43$	$34 177.394 19$	$- 0.001 18$	$34 177.429 73$	$0.000 82$
$106$	$34 204.911 72$	$- 0.000 13$	$34 204.948 54$	$0.000 37$	$34 177.581 25$	$- 0.001 39$	$34 177.614 78$	$- 0.000 30$
$107$	$34 205.093 57$	$- 0.000 33$	$34 205.129 12$	$0.000 15$	$34 177.763 08$	$- 0.001 60$	$34 177.796 00$	$- 0.000 06$
$108$	$34 205.270 96$	$0.000 06$
$109$	$34 205.442 72$	$- 0.000 34$	$34 205.476 35$	$0.000 53$
$110$	$34 205.610 31$	$- 0.000 22$			$34 178.280 90$	$- 0.000 42$

	$17 \to n p 17_{17}$		$17 \to n p 17_{18}$		$19 \to n p 19_{19}$		$19 \to n p 19_{20}$
$n$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$	${\tilde{ν}}_{obs}$	$Δ$
$95$	$34 144.410 06$	$0.000 41$	$34 144.442 31$	$0.000 18$			$34 110.275 32$	$0.001 58$
$96$	$34 144.661 17$	$- 0.001 01$	$34 144.694 19$	$0.000 75$	$34 110.551 35$	$- 0.000 08$	$34 110.524 19$	$- 0.001 12$
$97$	$34 144.906 04$	$- 0.000 89$	$34 144.937 09$	$0.000 07$	$34 110.796 83$	$0.000 68$
$98$	$34 145.143 92$	$- 0.000 31$	$34 145.173 17$	$- 0.000 03$	$34 111.033 32$	$- 0.000 11$	$34 111.006 06$	$0.000 39$
$99$	$34 145.372 91$	$- 0.001 45$	$34 145.402 04$	$- 0.000 24$	$34 111.264 53$	$0.000 98$	$34 111.234 89$	$- 0.000 17$
$100$	$34 145.597 43$	$- 0.000 19$	$34 145.624 02$	$- 0.000 51$	$34 111.488 17$	$0.001 37$	$34 111.457 59$	$- 0.000 03$
$101$	$34 145.814 64$	$0.000 36$			$34 111.702 24$	$- 0.001 21$
$102$	$34 146.024 88$	$0.000 29$	$34 146.049 52$	$- 0.000 11$
$103$	$34 146.228 90$	$0.000 10$	$34 146.254 15$	$0.001 18$			$34 112.086 48$	$- 0.000 52$
$104$					$34 112.317 75$	$0.001 45$
$105$					$34 112.508 60$	$- 0.000 40$
$106$	$34 146.807 23$	$0.000 12$			$34 112.696 34$	$0.000 08$	$34 112.663 06$	$- 0.000 88$
$107$	$34 146.988 07$	$- 0.001 08$			$34 112.879 55$	$0.001 25$	$34 112.845 05$	$- 0.000 55$
$108$	$34 147.166 22$	$0.000 06$	$34 147.186 99$	$0.000 48$	$34 113.054 43$	$- 0.000 87$	$34 113.022 26$	$- 0.000 01$
$109$	$34 147.337 84$	$- 0.000 47$	$34 147.359 28$	$0.001 28$	$34 113.227 23$	$- 0.000 22$	$34 113.193 94$	$- 0.000 18$
$110$	$34 147.504 69$	$- 0.001 09$	$34 147.526 05$	$0.001 21$	$34 113.394 80$	$- 0.000 12$

Table 4. Table 3: Estimated statistical ( 1 σ 1 𝜎 1\sigma ) and systematic contributions to the uncertainty in the determinations of the ionization threshold of the a 3 Σ u + superscript 𝑎 3 superscript subscript Σ 𝑢 a\,^{3}\Sigma_{u}^{+} state of He 2 (second column) and rotational intervals of the He X + 2 + Σ u + 2 , ν + = 0 subscript superscript superscript 𝑋 2 superscript superscript subscript Σ 𝑢 2 superscript 𝜈 0 {{}_{2}}^{+}\,X^{+}\,{}^{2}\Sigma_{u}^{+},\,\nu^{+}=0 ground state (third column). All values are given in MHz.

Source		$E_{I}$	${\tilde{ν}}_{N^{+} + 2, N^{+}}$
Statistical
	Wavenumber calibration	0	25
	Determination of line centers	2	3
	MQDT fit	3	5-8
	Total statistical\bigstrut	4	26
Systematic
	Wavenumber calibration	40	0
	AC-Stark shift	2	0
	DC-Stark shift	10	5
	Zeeman shift	1	0
	Doppler shift	5	0
	Pressure shift	1	0
	Total systematic\bigstrut	42	5

Equations2

\tilde{ν}_{n ℓ} = \tilde{ν}_{N^{''}, N^{+}} - \frac{R _{He_{2}}}{( n - δ _{ℓ} ) ^{2}}, with h c \tilde{ν}_{N^{''}, N^{+}} = E_{I} + E_{rv}^{+} - E_{rv}^{''},

\tilde{ν}_{n ℓ} = \tilde{ν}_{N^{''}, N^{+}} - \frac{R _{He_{2}}}{( n - δ _{ℓ} ) ^{2}}, with h c \tilde{ν}_{N^{''}, N^{+}} = E_{I} + E_{rv}^{+} - E_{rv}^{''},

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Full text

Precision measurement of the rotational energy-level structure of the three-electron molecule He ${{}_{2}}^{+}$

Luca Semeria

Paul Jansen

Laboratory of Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland

Frédéric Merkt

[email protected]

Laboratory of Physical Chemistry, ETH Zurich, CH-8093 Zurich, Switzerland

(March 2, 2024)

Abstract

The term values of all rotational levels of the 4He ${{}_{2}}^{+}\,X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0)$ ground vibronic state with rotational quantum number $N^{+}\leq 19$ have been determined with an accuracy of $8\text{\times}{10}^{-4}$ cm*-1* ( $\sim{25}$ MHz) by MQDT-assisted Rydberg spectroscopy of metastable He ${}_{2}^{*}$ . Comparison of these term values with term values recently calculated ab initio by Tung et al. [J. Chem. Phys. 136, 104309 (2012)] reveal discrepancies that rapidly increase with increasing rotational quantum number and reach values of $0.07$ cm*-1* ( $\sim{2.1}$ GHz) at $N^{+}=19$ .

pacs:

34.50.Gb, 33.80.Rv, 34.80.Gs, 37.10.Mn

I Introduction

Predicting the energy level structure of molecules from first principles is one of the major goals and tasks of theoretical molecular physics and chemistry. Few-electron molecules such as H ${{}_{2}}^{+}$ , H2 and He ${{}_{2}}^{+}$ are particularly attractive systems for comparison with experimental results because numerically “exact” predictions of molecular properties can in principle be obtained, i.e., predictions that are only limited in accuracy by the uncertainties of fundamental constants. For H ${{}_{2}}^{+}$ and H2, remarkable progress has been achieved by theoretical calculations in the past ten years Korobov (2006, 2008); Korobov et al. (2014); Piszczatowski et al. (2009); Komasa et al. (2011); Puchalski et al. (2016). The most precise theoretical calculations in these molecules are based on an effective nonrelativistic form of QED that treats relativistic and radiative effects perturbatively using the nonrelativistic wave functions. Remarkable progress has also been made in experimental determinations of energy intervals in these molecules Osterwalder et al. (2004); Koelemeij et al. (2007); Liu et al. (2009); Salumbides et al. (2011); Dickenson et al. (2013); Haase et al. (2015); Biesheuvel et al. (2016) so that the prospect of exploiting the comparison of theoretical and experimental results in these systems to either improve the accuracy of fundamental constants Wing et al. (1976) or uncover new physical phenomena Schiller and Korobov (2005); Ubachs et al. (2016) becomes increasingly attractive. For instance, Salumbides et al. argue that precision measurements in molecules may reveal (or at least set upper bounds to potential effects of) so far unknown forces acting on nuclei at distances typical of molecular bonds Salumbides et al. (2013); Ubachs et al. (2016). In general, precision measurements in simple systems are ideal to test theoretical predictions at the most fundamental level. The three-electron molecule He ${{}_{2}}^{+}$ , with nuclei four times as heavy as those in H ${{}_{2}}^{(+)}$ , is also an attractive molecular system in this context.

Ab initio calculations in He ${{}_{2}}^{+}$ were carried out almost immediately after Heitler and London showed how to efficiently treat electron correlation in H2 Heitler and London (1927). Majorana Majorana (1931) and Pauling Pauling (1933) extended the Heitler-London method to predict the dissociation energy, the equilibrium distance, and the harmonic vibrational frequency of He ${{}_{2}}^{+}$ . Since then, numerous calculations have been reported on He ${{}_{2}}^{+}$ (see, e.g., Refs. Weinbaum (1935); Moiseiwitsch (1956); Csavinszky (1959); Reagan et al. (1963); Liu (1971); Maas et al. (1976); Khan and Jordan (1986); Bauschlicher et al. (1989); Cencek and Rychlewski (1995, 2000); Xie et al. (2005); Tung et al. (2012)). The most recent ab initio calculations of the energy-level structure of He ${{}_{2}}^{+}$ by Tung et al. Tung et al. (2012) provided almost exact nonrelativistic energies and an accuracy comparable to the accuracy of the best experimental results available on He ${{}_{2}}^{+}$ in the literature, i.e., the 9 infrared rovibrational transition frequencies recorded between low-lying rovibrational levels in 3He4He+ by Yu and Wing Yu and Wing (1987) and the 7 rovibronic transition frequencies measured between high-lying vibrational levels of the $X^{+}\,{}^{2}\Sigma_{u}^{+}$ electronic ground state of 4He ${{}_{2}}^{+}$ and the weakly bound levels of the $A^{+}\,{}^{2}\Sigma_{g}^{+}$ state near the He+ + He dissociation limit by Carrington and coworkers Carrington et al. (1995). Tung et al. Tung et al. (2012) reported the accuracy of their calculations to be limited by the neglect of relativistic and QED corrections, which they estimated to be on the order of 0.004 cm*-1* (120 MHz) for the low-lying rovibrational levels of He ${{}_{2}}^{+}$ by comparing such corrections in H2 with the relativistic correction calculated for the $\nu=0-1$ interval of 3He4He+ Stanke et al. (2009).

Experimental measurements of the low-lying levels of 4He ${{}_{2}}^{+}$ are complicated by the fact that 4He ${{}_{2}}^{+}$ is apolar and has no electric-dipole-allowed vibrational and rotational transitions. Recently we have started systematic measurements of the energy-level structure of H ${{}_{2}}^{+}$ and He ${{}_{2}}^{+}$ by multichannel-quantum-defect-theory(MQDT)-assisted Rydberg spectroscopy of H2 Osterwalder et al. (2004); Liu et al. (2009); Sprecher et al. (2011); Haase et al. (2015) and He2 Sprecher et al. (2014); Jansen et al. (2015). In the case of He ${{}_{2}}^{+}$ , the procedure consists of measuring high-resolution spectra of triplet ( $S=1$ ), $n$ p Rydberg states of very high principal quantum number $n$ from the $a\,^{3}\Sigma_{u}^{+}$ metastable state of He2 and extrapolating the Rydberg series to their limits using MQDT Jungen (2011). We report here the determination of the term values of all rotational levels of the $X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0)$ state of He ${{}_{2}}^{+}$ up to the rotational level with rotational quantum number $N^{+}=19$ , which is located more than 2600 cm*-1* above the ground state of He ${{}_{2}}^{+}$ . Surprisingly, we find that the deviations between measured and calculated term values rapidly increase with increasing $N^{+}$ value.

II Experimental procedure

The experimental setup is presented schematically in Fig. 1 (see also Refs. Jansen et al. (2015) and Jansen et al. (2016)). The measurements were performed with a three-stage pulse-amplified laser system Hollenstein et al. (2000) that was seeded with the output of a continuous-wave (cw) ring dye laser and pumped by the second harmonic of a Q-switched neodymium-doped yttrium-aluminium-garnet (Nd:YAG) laser. The pulse-amplified output of the ring laser was frequency doubled in a beta-barium-borate crystal, resulting in near-Fourier-transform-limited pulses of UV radiation with a bandwidth of 180 MHz. The frequency was adjusted by scanning the seed-laser frequency.

Calibration of the laser fundamental frequency to an accuracy of 20 MHz (1 $\sigma$ ) was achieved by coupling a fraction of the pulse-amplified output of the ring laser into a wavemeter (HighFinesse WS7). To quantify the effects of the frequency chirp arising in the pulse-amplification process, part of the cw output of the ring laser was used to record the laser-induced-fluorescence spectrum of I2 so as to determine the difference between the frequencies of the cw and pulse-amplified outputs of the ring dye laser.

A supersonic beam of metastable helium molecules was produced in a source chamber by striking an electric discharge through an expansion of pure helium gas at the exit of a pulsed valve Raunhardt et al. (2008). The body of the valve was cooled to a temperature of 77 K, resulting in a supersonic beam with a velocity of approximately 1000 m/s Motsch et al. (2014). The molecular beam was collimated with a 2-mm-diameter skimmer before entering a second, differentially-pumped vacuum chamber containing a 55-stage Zeeman decelerator Vanhaecke et al. (2007); Motsch et al. (2014); Wiederkehr et al. (2011). Although the Zeeman decelerator was not used to obtain the results presented below, it played an important role in the initial assignment of the spectra and in the assessment of the systematic uncertainty resulting from the Doppler effect, as explained in Refs. Jansen et al. (2015) and Jansen et al. (2016).

After about 1 m of flight, the molecules entered a third vacuum chamber that was used for photoexcitation and detection. The photoexcitation region was surrounded by a cylindrically symmetric stack of electrodes for the application of ionization and extraction electric fields. The molecular beam crossed the UV laser beam at right angles on the symmetry axis of the electrode stack, which was oriented perpendicularly to the plane defined by the laser and molecular beams. High- $n$ Rydberg states were ionized by a pulsed electric field, which was also used to extract the ions toward a microchannel-plate (MCP) detector. The spectra were obtained by monitoring the pulse-field-ionization yield as a function of the UV laser wavenumber. The spectra were greatly simplified by applying a small electric field shortly after photoexcitation but before the field-ionization pulse. This first “discrimination” pulse did not only field ionize Rydberg states with $n\gtrsim 200$ , but also shifted all ions produced by direct ionization or rapid autoionization (prompt ions) out of the photoexcitation volume. Consequently, the ions generated by the second, field-ionization pulse had different flight times to the MCP detector than the prompt ions and could be monitored separately. Stray electric fields were determined by recording spectra in the presence of different applied potentials across the stack and fitting the observed Stark shifts to a quadratic polynomial and were compensated to a level of better than 1 mV/cm, as explained in Ref. Osterwalder and Merkt (1999). To suppress stray magnetic fields, two concentric mu-metal tubes were used to shield the photoexcitation region.

III Rydberg series of $\text{He}_{2}$ and Multichannel Quantum-defect Theory calculations

With the exception of the single bound level of the electronic ground state, all bound states of He2 are Rydberg states Herzberg (1987). The rotational levels of the $a\,^{3}\Sigma_{u}^{+}$ metastable state and of high- $n$ Rydberg states are schematically depicted on the left-hand side of Fig. 2. The bosonic nature of the 4He nuclei excludes the existence of even rotational states in the $a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0)$ state of the neutral molecule and in the $X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0)$ state of the ion, so that half of the energy levels is missing. In the following, we use double-primed symbols to indicate the quantum numbers of the metastable He2 molecule and symbols with a superscript “ $+$ ” to indicate the quantum numbers of the molecular ion. Single-photon excitation from the metastable $a\,^{3}\Sigma_{u}^{+}$ state, which is a triplet ( $S^{\prime\prime}=1$ ) Rydberg state of s character Ginter (1965), primarily results in the excitation of p Rydberg series converging on the different rotational levels of the $X^{+}\,{}^{2}\Sigma_{u}^{+}$ ground state of He ${{}_{2}}^{+}$ . The spin-rotation splittings of the rotational levels of the $X^{+}\,{}^{2}\Sigma_{u}^{+}$ ground state of He ${{}_{2}}^{+}$ are not resolved in our spectra, so that the ionic rotational levels are adequately labeled by the rotational quantum number $N^{+}$ . The total angular momentum (excluding spins) $\vec{N}$ of the high Rydberg states results from the vector addition of $\vec{N}^{+}$ with the Rydberg-electron orbital angular momentum $\vec{\ell}$ so that $N=N^{+}+\ell,N^{+}+\ell-1,\ldots,|N^{+}-\ell|$ . For the p Rydberg states of interest here, this results in $N=N^{+},N^{+}\pm 1$ (see Ref. Raunhardt et al. (2008) for details). Consequently, the Rydberg states can be labeled as $n\text{p}N^{+}_{N}$ . Each rotational level of the $a\,^{3}\Sigma_{u}^{+}$ metastable state is split into a triplet by the spin-spin and spin-rotation interactions Lichten et al. (1974). The rotational levels are labeled by the rotational quantum number $N^{\prime\prime}$ and the total angular momentum quantum number $J^{\prime\prime}$ ( $J^{\prime\prime}=N^{\prime\prime},N^{\prime\prime}\pm 1$ ). Each rotational level consists of a triplet with characteristic splittings, which are known extremely precisely from the high-resolution microwave studies of Lichten et al. Lichten et al. (1974); Lichten and Wik (1978) and Bjerre et al. Kristensen and Bjerre (1990); Hazell et al. (1995). Consequently, all transitions have a triplet structure with characteristic splittings and intensity patterns that can be used to assign the initial level in a straightforward manner. Examples of these patterns for transitions originating from the $N^{\prime\prime}=1-11$ levels are given on the right-hand side of Fig. 2. Three Gaussian functions of full width at half maximum of 180 MHz, separated by the known fine-structure intervals Lichten et al. (1974); Lichten and Wik (1978); Kristensen and Bjerre (1990); Hazell et al. (1995); Focsa et al. (1998) and with relative intensities weighted by the $2J^{\prime\prime}+1$ degeneracy factors, were used to convert each observed triplet into its “fine-structure-free” center of gravity. All transition wavenumbers reported in this article correspond to such centers of gravity.

Rydberg series converging on a particular ( $\nu^{+},N^{+}$ ) rovibrational level of the $X^{+}\,{}^{2}\Sigma_{u}^{+}$ electronic ground state of He ${{}_{2}}^{+}$ are approximately described by Rydberg’s formula

[TABLE]

where $\tilde{\nu}_{n\ell}$ represents the spectral position of the Rydberg states of principal and orbital angular-momentum quantum numbers $n$ and $\ell$ , respectively, and quantum defect $\delta_{\ell}$ . The quantities $\mathcal{R}_{\text{He}_{2}}$ and $E_{\text{I}}$ represent the mass-corrected Rydberg constant for He2 and the adiabatic ionization energy of metastable He2, corresponding to the energy difference between the $X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0,N^{+}=1)$ level of He ${{}_{2}}^{+}$ and the $a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0,N^{\prime\prime}=1)$ metastable level, respectively. $E_{\text{rv}}^{+}$ and $E_{\text{rv}}^{\prime\prime}$ are the rovibrational energies of the He ${{}_{2}}^{+}$ ion core and of metastable He2 with respect to the ( $\nu^{+}=0,N^{+}=1$ ) and ( $\nu^{\prime\prime}=0,N^{\prime\prime}=1$ ) levels, respectively. Although Rydberg’s formula accurately describes the essentially unperturbed $n\text{p}N^{+}_{N=N^{+}}$ series, it does not account for the interactions between series of the same $N$ value that converge on different rotational states of He ${{}_{2}}^{+}$ . For the $n\text{p}[N^{+}=N+1]_{N}$ series, these interactions give rise to spectral perturbations below, and to rotational autoionization above, the $N^{+}=N-1$ thresholds (see Fig. 2) and can be accounted for in the framework of multichannel quantum-defect theory (Jungen, 2011, and references therein).

In this work, we determined the ionization thresholds corresponding to the different rotational levels of He ${{}_{2}}^{+}$ with respect to the $N^{\prime\prime}=1$ rotational state of He ${}_{2}^{*}$ by extrapolating Rydberg series of metastable helium molecules using the MQDT program developed by Jungen Jungen (2011). The data used for the extrapolation consist of $n$ p Rydberg series with $n$ ranging from 95 to 110, excited from the $\nu^{\prime\prime}=0$ and $N^{\prime\prime}\leq 19$ levels of metastable He2.

The quantum-defect parameters for the gerade triplet $n$ p and $n$ f Rydberg states of He2, derived by Sprecher et al. Sprecher et al. (2014), was adopted without change for the extrapolation of the Rydberg series. In Ref. Sprecher et al. (2014), internuclear-distance( $R$ )-dependent and energy( $\epsilon$ )-dependent $\eta$ eigenquantum-defect matrices were extracted from data available in the literature on $n$ p and $n$ f Rydberg levels with $n$ in the range $2-39$ , $v^{+}\leq 2$ , and $N^{+}\leq 19$ in a least-squares-fit procedure. High- $n$ Rydberg states were not used in the fit, because the level energies of high- $n$ Rydberg states are insensitive to adjustments of the quantum defects, a fact that is also exploited in the present analysis (see Ref. Sprecher et al. (2014) for details).

IV Data analysis and results

The dominant $n\text{p}N^{+}\leftarrow a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0,N^{\prime\prime})$ Rydberg series in the spectrum of He2 are those with $N^{+}=N^{\prime\prime}$ . There are three such series with $N=N^{+},N^{+}\pm 1$ . However, all $n\text{p}N^{+}_{N^{+}-1}$ Rydberg states with $N^{+}>1$ located above the $N^{+}-2$ ionization thresholds are subject to rapid autoionization, because of the strong coupling, indicated by the curved arrows in Fig. 2, between channels of the same value of $N$ , but differing in $N^{+}$ by 2 Raunhardt et al. (2008); Sprecher et al. (2014). The corresponding transitions are strongly broadened and thus not detectable in our high-resolution spectra. Below the $N^{+}-2$ ionization thresholds, the autoionization of the $n\text{p}N^{+}_{N^{+}-1}$ Rydberg states is suppressed but the coupling between the $n\text{p}N^{+}_{N^{+}-1}$ and $n\text{p}[N^{+}-2]_{N^{+}-1}$ channels leads to strong perturbations and mixing of the Rydberg series. The $n\text{p}N^{+}_{N^{+}-1}$ series is always observable from both the $N^{\prime\prime}=N^{+}$ (full arrows in Fig. 2) and $N^{\prime\prime}=N^{+}-2$ (dashed arrows in Fig. 2) rotational levels of the metastable state, which enables one to determine the relative position of the $N^{\prime\prime}-2$ and $N^{\prime\prime}$ levels by building combination differences.

To obtain a full set of rotational intervals in the $a\,^{3}\Sigma_{u}^{+}$ state of He2 and the $X^{+}\,{}^{2}\Sigma_{u}^{+}$ state of He ${{}_{2}}^{+}$ , we proceeded in two steps. First, initial-state combination differences $\tilde{\nu}_{N^{\prime\prime}-2,N^{\prime\prime}}$ , i.e., the energy intervals between the $N^{\prime\prime}-2$ and $N^{\prime\prime}$ rotational levels, were determined as illustrated in Fig. 3 with the example of the $N^{\prime\prime}=15$ and 17 levels, where the upper and lower panels show the $n\text{p}15_{16}$ series around $n=57$ recorded from the $N^{\prime\prime}=15$ and $N^{\prime\prime}=17$ levels, respectively. Each member of this series observed from both rotational levels represents an independent measurement of $\tilde{\nu}_{15,17}$ . The differences between the centers of gravity of the triplets (marked by vertical dashed lines) give a value of $481.1061(7)$ cm*-1* for the interval between the $a\,^{3}\Sigma_{u}^{+},\,\nu^{\prime\prime}=0,N^{\prime\prime}=15$ and 17 levels. All transitions used to determine the rotational structure of the $a\,^{3}\Sigma_{u}^{+}$ state are listed in Table S1 of the Supplementary Material. These transitions form a redundant network, from which the relative positions of all states connected by the transitions can be determined in a global fit, as described in Ref. Schäfer et al. (2010). The rotational term values of the $a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0)$ metastable state of He2 presented in the third column of Table 1 could be determined in this way up to $N^{\prime\prime}=19$ . These term values agree, within the respective uncertainties, with the rotational term values we calculated with the rotational constants reported by Focsa et al. Focsa et al. (1998) and presented in the second column of Table 1.

In the second step, the positions of the rotational levels of the He ${{}_{2}}^{+}$ ion were determined by extrapolating $n\text{p}[N^{+}=N^{\prime\prime}]_{N^{+},N^{+}=1}\leftarrow N^{\prime\prime}$ series. Figure 4 presents spectra of the $n\text{p}[N^{+}=N^{\prime\prime}]_{N=N^{\prime\prime},N^{\prime\prime}+1}\leftarrow N^{\prime\prime}$ Rydberg series recorded for this purpose, and Table 2 lists the corresponding transition wavenumbers. The choice of the range of principal quantum number between 95 and 110 used for the extrapolation represented the best compromise between maximal accuracy of the Rydberg-series extrapolation, which is reached at the highest possible values of $n$ , and minimal systematic uncertainties in the wavenumber measurements. In our experiments (see discussion below), the dominant source of systematic uncertainty comes from Stark-shifts induced by residual electric fields, which could be compensated to 1 mV/cm. This uncertainty amounts to 10 MHz at $n=110$ and scales with $n^{7}$ , which made extrapolations based on transitions including Rydberg states with $n>110$ insufficiently accurate.

The spectra depicted in Fig. 4 all reveal at least one of the two expected series. In some spectra, such as the spectrum of the $n\text{p}1_{1,2}\leftarrow 1$ transitions, perturbations lead to regions where the two series are clearly distinguishable (e.g., in the region $n=100-105$ and above $n=109$ in the $n\text{p}1_{1,2}$ series). In other spectra, the two series are difficult to distinguish because of the partially overlapping fine structure and reliable “fine-structure-free” positions can only be derived from the analysis of lineshapes.

In an initial fit, the $n\text{p}19_{20}\leftarrow 19$ series was excluded, because of its interaction with the $n\text{p}21_{20}$ series which could not be measured at sufficiently high $n$ values for the $N^{+}=21$ threshold to be accurately determined. The rotational energy levels with $N^{+}\leq 19$ obtained in this way were used to derive a set of effective rotational constants that was subsequently used to predict the term value of the $N^{+}=21$ level with respect to the $N^{+}=1$ level (3158.115 cm*-1*). The ionization thresholds of all series, including the $n\text{p}19_{20}$ series, were then determined in a second fit for which the $N^{+}=21$ threshold was fixed at the predicted value. The relative positions of the ionic rotational states obtained from the transition wavenumbers listed in Table 2 and from the known positions of the rotational levels of the metastable state (see third column of Table 1) are listed in the last column of Table 1, where they are compared with the results of the ab initio calculations of Tung et al. Tung et al. (2012). The adiabatic ionization energy of He2 ( $a\,^{3}\Sigma_{u}^{+}$ ), defined as the interval between the $a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0,N^{\prime\prime}=1)$ state of He2 and the $X^{+}\,{}^{2}\Sigma_{u}^{+}\,(\nu^{+}=0,N^{+}=1)$ state of He ${{}_{2}}^{+}$ , is found to be $$34,301.205,65(12) $\pm 0.0014_{\text{sys}}$ cm*-1*, as already reported in Ref. Jansen et al. (2015).

Table 3 summarizes the main sources of systematic and statistical uncertainty in the determination of ionization thresholds and He ${{}_{2}}^{+}$ rotational intervals. The main source of statistical uncertainty comes from the lineshape analysis and the wavenumber calibration (25 MHz), resulting from the accuracy of the wavemeter. The systematic uncertainty of the ionization thresholds is larger (40 MHz) than that of the rotational intervals because the absolute calibration error cancels out when building combination differences. Although the absolute accuracy of the wave-meter measurements is specified as 20 MHz, the intrinsic Allan deviation over periods shorter than 1 min lies typically around 300 kHz Sanguinetti et al. (2009); Deiglmayr et al. (2016). Long-term fluctuations of the wavemeter were monitored by recording a small part of the spectrum prior to each measurement and performing a statistical analysis of the fitted line positions. The resulting histogram is shown in Fig. 5 and resembles a Gaussian distribution with a standard deviation of 25(5) MHz in the UV. This uncertainty of 25 MHz approximately corresponds to the specified relative accuracy of the wavemeter of 10 MHz in the fundamental and contributes a statistical uncertainty of 25 MHz to the rotational intervals. The uncertainty of the Rydberg-series extrapolation by MQDT is dominated by the uncertainty in the quantum-defect parameters reported in Ref. Sprecher et al. (2014) and is only 3 MHz for the ionization energy and 6 MHz for the rotational intervals. This uncertainty does not cancel out when determining the relative positions of the ionic levels because the series of different $N$ values were extrapolated separately.

The remaining systematic uncertainties are dominated by the contribution of the Stark shift resulting from the residual electric field. The other sources of systematic uncertainties only affect the value of the ionization energy because they cancel out when determining the relative positions of the ionic rotational levels.

As explained in Sec. II, the stray electric field was determined and subsequently compensated by recording Rydberg spectra in the presence of different dc electric-field strengths and fitting the observed Stark shifts to a quadratic polynomial. Stray magnetic fields were suppressed by surrounding the photoexcitation region by two concentric mu-metal shields. The contribution of Stark and Zeeman shifts induced by residual electric and magnetic fields are 10 and 1 MHz, respectively. No shift in the line positions was observed when the laser power and the nozzle-stagnation pressure were varied, and the contributions of ac-Stark and pressure shifts to the systematic uncertainties is estimated to be at most 2 and 1 MHz, respectively. Uncertainties from Doppler shifts were determined to be around 5 MHz.

V Discussion and conclusions

The most precise ab initio calculations of the rovibrational energy levels of He ${{}_{2}}^{+}$ reported so far have been carried out by Tung et al. Tung et al. (2012). The calculations relied on the use of 1200 explicitly correlated Gaussian functions with shifted centers in the range of internuclear distances between $R=0.9$ and $100\,a_{0}$ to obtain the Born-Oppenheimer potential energy function. To derive accurate rovibrational energies, Tung et al. Tung et al. (2012) also calculated the adiabatic corrections and included nonadiabatic corrections by using $R$ -independent effective reduced masses in the vibrational and rotational kinetic operators in the Hamiltonian. Relativistic and radiative corrections were neglected, but using an argument based on scaling these corrections from the values they take in H2 Komasa et al. (2011) to the relativistic correction for the first vibrational interval of 3He4He+ Stanke et al. (2009) and assuming that these corrections are mass independent, they estimated their predictions of the lowest vibrational interval in He ${{}_{2}}^{+}$ to have an accuracy of about 0.004 cm*-1*, or 120 MHz.

Comparison of our experimental results with the predictions of Tung et al. Tung et al. (2012), we find that the experimental term values always lie above the calculated ones, and that the discrepancies rapidly increase with increasing rotational quantum number (see Fig. 6), reaching a value of almost 0.07 cm*-1* (2.1 GHz) at $N^{+}=19$ , which is more than ten times larger than the estimated uncertainty of the calculations. In reconstructing the argument of Tung et al. Tung et al. (2012), we found that the relativistic (-0.0026 cm*-1*) and radiative (0.0010 cm*-1*) corrections of the first rotational interval of H2 Komasa et al. (2011), rather than the first vibrational interval, were used to scale the relativistic correction of the first vibrational interval in 3He4He+ (-0.006 cm*-1*) Stanke et al. (2009). When we scaled the relativistic (-0.0234 cm*-1*) and radiative (0.0213 cm*-1*) corrections of the $(\nu,J):(0,0)\rightarrow(1,0)$ interval in H2 Komasa et al. (2011), we found a correction of 0.0005 cm*-1*, or 15 MHz, for this interval in 3He4He+. Following the argument of Tung et al., we scaled the relativistic and radiative corrections of the rotational states of H2 Komasa et al. (2011) by the factor 3.9. The results are displayed as the dashed line in Fig. 6. The scaled corrections appear to account for the discrepancies between experimental and calculated rotational term values reasonably well up to $N^{+}=11$ , but rapidly become inadequate beyond $N^{+}=11$ . Further work is required to understand the origin of this discrepancy. It is conceivable that the approximate treatment of the nonadiabatic correction through $R$ -independent reduced masses in Ref. Tung et al. (2012) becomes increasingly inaccurate as $N^{+}$ increases.

The results presented in this article suggest that even the best ab initio calculations available in the literature on the three-electron molecule He ${{}_{2}}^{+}$ are still far from reaching the same level of agreement with experimental data as in the two-electron molecule H2 Liu et al. (2009); Piszczatowski et al. (2009); Komasa et al. (2011); Salumbides et al. (2011); Dickenson et al. (2013).

Supplementary Material

See Supplementary Material at [address to be inserted by Editor] for the wavenumbers of all transitions used to derive the rotational level energies of the $a\,^{3}\Sigma_{u}^{+}\,(\nu^{\prime\prime}=0)$ state of He2.

Acknowledgements.

We thank Christian Jungen (Laboratoire Aimé Cotton du CNRS, Orsay) for allowing us to use his MQDT program and Hansjürg Schmutz and Josef Agner for their expert technical assistance. This work is supported financially by the Swiss National Science Foundation under Project Nos. 200020-159848 and 200020-149216 and the NCCR QSIT. P.J. gratefully acknowledges ETH Zurich for support through an ETH fellowship.

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TL;DR

Contribution

Findings

Abstract

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Precision measurement of the rotational energy-level structure of the three-electron molecule He2+{{}_{2}}^{+}2​+

Abstract

pacs:

I Introduction

II Experimental procedure

III Rydberg series of He2\text{He}_{2}He2​ and Multichannel Quantum-defect Theory calculations

IV Data analysis and results

V Discussion and conclusions

Supplementary Material

Acknowledgements.

Precision measurement of the rotational energy-level structure of the three-electron molecule He ${{}_{2}}^{+}$

III Rydberg series of $\text{He}_{2}$ and Multichannel Quantum-defect Theory calculations