Black-body radiation-induced photodissociation and population redistribution of weakly bound states in H₂⁺

Figures & data

Figure 1. Energy-level diagram of the weakly bound states of H${}_2^{+}$. The ground electronic state, X${}^{+}{}^2{\mathrm{\Sigma }}_g^{+}$, is depicted on the left-hand side, displaying the dissociation continuum on the top and the two highest vibrational levels. Levels with v = 13−17 are not shown for clarity but are taken into account in the calculation On the right-hand side, the first excited electronic state, A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$, is shown with a single vibrational level, v = 0, and the dissociation continuum. For both electronic states, the rotational levels denoted with a dashed line are the ortho-levels with I = 1. The arrows represent a simplified version of the transitions present for para-H${}_2^{+}$  with the whole population starting in the X${}^{+}$($v=19,N=0,G=1/2$) level; wavy red lines indicate spontaneous emission, the double-headed solid blue lines show stimulated emission and absorption, and the single-headed dashed blue line stands for absorption into the continuum (all states experience absorption into the continuum, but only a few are explicitly shown). The forbidden transitions, allowed by g/u-mixing, are also shown. Neglecting spin-rotation interactions, the levels F are degenerate and not shown.

Figure 2. (a) Bound-bound A${}^{+}$ – X${}^{+}$  stick spectrum of H${}_2^{+}$, with the labels indicating the vibrational bands. The forbidden transitions (allowed by g/u-mixing) are indicated as dashed lines. (b) Photodissociation cross-section for X${}^{+}(v=13-19,N=1)$ (solid line) and A${}^{+}(v=0,N=1)$ (dash-dotted line). For comparison, X${}^{+}(19,0)$ is also shown. (c) Black-body photon density at room (solid line) and at liquid nitrogen (dashed line) temperature. See text for details.

Figure 3. Time evolution of the rovibronic state populations at 293 K with the ions selectively prepared in: (a) Para-X${}^{+}(19,0)$ with no g/u-mixing. (b) Para-X${}^{+}(19,0)$ including g/u-mixing. Labels are added to the now-accessible ortho-states and the repeated labels from panel (a) are omitted. (c) Ortho-X${}^{+}(19,1)$ without g/u-mixing. (d) Ortho-X${}^{+}(19,1)$ with g/u-mixing.

Figure 4. Time evolution of the rovibronic state populations at 77 K with the ions selectively prepared in: (a) Para-X${}^{+}(19,0)$ with no g/u-mixing. (b) Para-X${}^{+}(19,0)$ including g/u-mixing. Labels are added to the now-accessible ortho-states and the repeated labels from panel (a) are omitted. (c) Ortho-X${}^{+}(19,1)$ without g/u-mixing. (d) Ortho-X${}^{+}(19,1)$ with g/u-mixing.

Figure 5. Effective lifetimes of the weakly bound states of H${}_2^{+}$. A line is drawn connecting states with N = 0; dashed blue line for 77 K, solid orange line for 293 K and dash-dotted red line for 400 K.

Table 1. Effective lifetimes of the weakly bound states of H${}_2^{+}$  at 293 K. Level v = 0 corresponds to A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$ , the other levels to X${}^{+}{}^2{\mathrm{\Sigma }}_g^{+}$.

${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(293K\right)$ (s)
v	N = 0	1	2	3
0	1.0620[−01]	1.2558[−01]	1.8600[−01]
19	3.1955[−01]	5.2335[−01]
18	1.7810[−02]	1.9270[−02]	2.2940[−02]	3.2010[−02]
17	4.4200[−03]	4.4800[−03]	4.6400[−03]	4.9300[−03]
16	6.5200[−03]	6.3200[−03]	5.9600[−03]	5.4900[−03]
15	5.1070[−02]	4.7850[−02]	4.2070[−02]	3.4890[−02]

Note: Table entries $a\left[b\right]$ stand for $a\times {10}^b$.

Table 2. Effective lifetimes of the weakly bound states of H${}_2^{+}$  at 77 K. Level v = 0 corresponds to A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$ , the other levels to X${}^{+}{}^2{\mathrm{\Sigma }}_g^{+}$.

${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(77K\right)$ (s)
v	N = 0	1	2	3
0	1.9700[−01]	2.3700[−01]	3.7400[−01]
19	5.1190[+00]	8.3910[+00]
18	3.6300[−01]	4.0900[−01]	5.2000[−01]	8.0700[−01]
17	3.2600[−01]	3.0300[−01]	2.6100[−01]	2.1600[-01]
16	5.8191[+01]	4.8358[+01]	3.3474[+01]	1.9685[+01]
15	8.9116[+04]	1.9420[+05]	3.3290[+05]	4.3669[+05]

Note: Table entries $a\left[b\right]$ stand for $a\times {10}^b$.

Table 3. Effective lifetimes of the weakly bound states of H${}_2^{+}$  at 400 K. Level v = 0 corresponds to A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$ , the other levels to X${}^{+}{}^2{\mathrm{\Sigma }}_g^{+}$.

${\tau }_{\mathrm{e}\mathrm{f}\mathrm{f}}\left(400\mathrm{K}\right)$ (s)
v	N = 0	1	2	3
0	8.4630[−02]	9.9690[−02]	1.4399[−01]
19	1.5871[−01]	2.6052[−01]
18	8.7600[−03]	9.4600[−03]	1.1230[−02]	1.5650[−02]
17	2.1500[−03]	2.2000[−03]	2.2900[−03]	2.4600[−03]
16	2.0500[−03]	2.0200[−03]	1.9600[−03]	1.8900[−03]
15	7.1100[−03]	6.8100[−03]	6.2700[−03]	5.5500[−03]

Note: Table entries $a\left[b\right]$ stand for $a\times {10}^b$.

Table B1. Effective coupling constants of the Fermi interaction for the weakly bound states of H${}_2^{+}$  in X${}^{+}{}^2{\mathrm{\Sigma }}_g^{+}$.

$b_F$ (MHz)
v	N = 1	3
13	728.41	726.70
14	722.46	721.02
15	717.73	716.58
16	714.25	713.41
17	712.07	711.58
18	711.27	711.17
19	711.38

Table B2. Effective coupling constants of the Fermi interaction for the weakly bound states of H${}_2^{+}$  in A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$.

$b_F$ (MHz)
v	N = 0	2
0	711.32	711.32

Table C1. Bound-bound line strengths and Einstein rate coefficients of the weakly bound states of H${}_2^{+}$  for transitions involving A${}^{+}(0,1)$.

A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$ $v=0,N=1$
v	N	ν (THz)	${\mu }_e\left(e{a}_0\right)$	$S\left(e^2{a}_0^2\right)$	A (s${}^{-1}$)	$\overline{B}$ (s${}^{-1}$)
*19	0	5.2890[−02]	7.8405[+00]	1.2295[+02]	6.8403[−04]	7.8613[−02]	2.0409[−02]
18	2	3.5197[−01]	4.7023[+00]	8.8448[+01]	4.8334[−02]	8.1445[−01]	1.9704[−01]
18	0	6.4562[−01]	3.7955[+00]	2.8812[+01]	9.7175[−02]	8.7118[−01]	1.9615[−01]
17	2	3.8904[+00]	6.4155[−01]	1.6463[+00]	1.2150[+00]	1.3632[+00]	1.1796[−01]
17	0	4.5903[+00]	5.4325[−01]	5.9025[−01]	7.1549[−01]	6.3828[−01]	4.3417[−02]
16	2	1.1809[+01]	9.7880[−02]	3.8320[−02]	7.9082[−01]	1.3361[−01]	5.0338[−04]
16	0	1.2872[+01]	8.5060[−02]	1.4470[−02]	3.8679[−01]	5.3463[−02]	1.2688[−04]
15	2	2.3980[+01]	1.6470[−02]	1.0900[−03]	1.8762[−01]	3.7676[−03]	6.0557[−08]
15	0	2.5368[+01]	1.4500[−02]	4.2000[−04]	8.6064[−02]	1.3712[−03]	1.1694[−08]

Note: Rate coefficients for absorption can be obtained using the tabulated values and Equation (21(21) $B_{ji}=\left(g_j/g_i\right)B_{ij}.$(21) ). The first and second column of $\overline{B}$ present the values at 293 and 77 K, respectively. * Level X${}^{+}\left(19\right)$ lies above A${}^{+}\left(0\right)$ and thus the rate coefficients, A and B, reported in this row take the former as the upper state. $a\left[b\right]$ stands for $a\times {10}^b$.

Table C2. Bound-bound line strengths and Einstein rate coefficients of the weakly bound states of H${}_2^{+}$  for transitions involving A${}^{+}(0,2)$.

A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$ $v=0,N=2$
v	N	ν (THz)	${\mu }_e\left(e{a}_0\right)$	$S\left(e^2{a}_0^2\right)$	A (s${}^{-1}$)	$\overline{B}$ (s${}^{-1}$)
*19	1	1.7610[−02]	9.1888[+00]	3.3774[+02]	2.3115[-02]	8.0022[+00]	2.0945[+00]
18	3	1.5664[−01]	5.5070[+00]	1.8196[+02]	5.2584[+00]	2.0234[+02]	5.1276[+01]
18	1	5.9369[−01]	3.5779[+00]	5.1205[+01]	8.0575[+01]	7.8894[+02]	1.7994[+02]
17	3	3.2774[+00]	6.2199[−01]	2.3212[+00]	6.1447[+02]	8.6476[+02]	9.1554[+01]
17	1	4.4040[+00]	4.6558[−01]	8.6704[−01]	5.5692[+02]	5.2677[+02]	3.8241[+01]
16	3	1.0827[+01]	8.9435[−02]	4.7992[−02]	4.5809[+02]	9.3651[+01]	5.3778[−01]
16	1	1.2565[+01]	7.0481[−02]	1.9870[−02]	2.9641[+02]	4.3391[+01]	1.1773[−01]
15	3	2.2672[+01]	1.4634[−02]	1.2849[−03]	1.1260[+02]	2.8149[+00]	8.2142[−05]
15	1	2.4953[+01]	1.1808[−02]	5.5775[−04]	6.5168[+01]	1.1125[+00]	1.1468[−05]

Note: See caption of Table A3 for details. Line strengths are given for states with $G=1/2$; the corresponding value for states with $G=3/2$ can be obtained according to Equation (11(11) $S_{{\eta }^{\prime }{v}^{\prime }{N}^{\prime }G,\eta vNG}=(2G+1){\mathcal{S}}_{N^{\prime },N}|{\mu }_{{\eta }^{\prime }{v}^{\prime }{N}^{\prime },\eta vN}{|}^2.$(11) ) and are twice as large. $a\left[b\right]$ stands for $a\times {10}^b$.

Table C3. Bound-bound line strengths and Einstein rate coefficients of the weakly bound states of H${}_2^{+}$  for transitions involving A${}^{+}(0,0)$.

A${}^{+}{}^2{\mathrm{\Sigma }}_u^{+}$ $v=0,N=0$
v	N	ν (THz)	${\mu }_e\left(e{a}_0\right)$	$S\left(e^2{a}_0^2\right)$	A (s${}^{-1}$)	$\overline{B}$ (s${}^{-1}$)
*19	1	9.6431[−02]	6.2200[+00]	7.7376[+01]	8.6959[−01]	5.4620[+01]	1.4038[+01]
18	1	5.1487[−01]	4.2445[+00]	3.6032[+01]	1.8491[+02]	2.1014[+03]	4.8869[+02]
17	1	4.3252[+00]	6.2061[−01]	7.7032[−01]	2.3435[+03]	2.2734[+03]	1.6960[+02]
16	1	1.2486[+01]	9.7541[−02]	1.9029[−02]	1.3927[+03]	2.0692[+02]	5.8105[−01]
15	1	2.4875[+01]	1.6667[−02]	5.5558[−04]	3.2150[+02]	5.5611[+00]	5.9428[−05]

Note: See caption of Table A3 for details. Line strengths are given for states with $G=1/2$; the corresponding value for states with $G=3/2$ can be obtained according to Equation (11(11) $S_{{\eta }^{\prime }{v}^{\prime }{N}^{\prime }G,\eta vNG}=(2G+1){\mathcal{S}}_{N^{\prime },N}|{\mu }_{{\eta }^{\prime }{v}^{\prime }{N}^{\prime },\eta vN}{|}^2.$(11) ) and are twice as large. $a\left[b\right]$ stands for $a\times {10}^b$.

Table C4. Continuum absorption rates of the weakly bound states of H${}_2^{+}$  at 293 K.

${\overline{B}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}\left(293\mathrm{K}\right)$ (s${}^{-1}$)
v	N = 0	1	2	3
0	4.1756[−01]	5.6180[−01]	6.6154[−01]
19	3.0503[+00]	1.8482[+00]
18	5.3561[+01]	4.9919[+01]	4.3118[+01]	3.1106[+01]
17	2.2450[+02]	2.2149[+02]	2.1496[+02]	2.0232[+02]
16	1.5328[+02]	1.5815[+02]	1.6787[+02]	1.8211[+02]
15	1.9577[+01]	2.0902[+01]	2.3772[+01]	2.8667[+01]

Note: $a\left[b\right]$ stands for $a\times {10}^b$.

Table C5. Continuum absorption rates of the weakly bound states of H${}_2^{+}$  at 77 K.

${\overline{B}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}\left(77\mathrm{K}\right)$ (s${}^{-1}$)
v	N = 0	1	2	3
0	9.7873[−02]	1.3250[−01]	1.2351[-01]
19	1.7430[−01]	1.0228[−01]
18	2.1915[+00]	2.0166[+00]	1.8094[+00]	1.2036[+00]
17	2.9510[+00]	3.2089[+00]	3.7731[+00]	4.5669[+00]
16	1.6840[−02]	2.0366[−02]	2.9630[−02]	5.0481[−02]
15	2.3250[−06]	3.0429[−06]	5.1857[−06]	1.1280[−05]

Note: $a\left[b\right]$ stands for $a\times {10}^b$.